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Full text of "AIRCRAFT NAVIGATION"

NASA TECHNICAL 

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AIRCRAFT NAVIGATION 



by S. S, Fedchin 

"Transport" Press 
Moscotv, 1966 



LOAN COPY: RETURN TO 

AFWL (WLIL-2) 

KWTLAND AFB, N MEX 



NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • FEBRUARY 1969 



TECH LrBRARY KAFB, NM 



001=8=152 



AIRCRAFT NAVIGATION 
By S. S. Fedchin 



Translation of: "Samoletovozhdeniye." 
"Transport" Press, Moscow, 1966 



NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 



For sale by the Clearinghouse for Federal Scientific and Technical Information 
Springfield, Virginia 22151 - CFSTI price $3.00 



TABLE OF CONTENTS 

ABSTRACT xl 

INTRODUCTION xlii 

CHAPTER ONE. COORDINATE SYSTEMS AND ELEMENTS OF 

AIRCRAFT NAVIGATION 1 

1. Elements of Aircraft Movement in Space 1 

2. Concepts of Stable and Unstable Flight 

Conditions 4 

3. Form and Dimensions of the Earth 7 

4. Elements Which Connect the Earth's Surface 

with Three-Dimensional Space 9 

5. Charts, Maps, and Cartographic Projections 12 

Distortions of Cartographic Projections 14 

Elt-ipse of Dis tort-ions 14 

Distortion of Lengths 15 

Distortion of Directions 16 

Distortion of Areas 17 

Classification of Cartographic Projections 18 

Division of Projections by the Nature 

of the Distortions 18 

1. Isogonal or conformal projections 18 

2. Equally spaced or equidistant projections 19 

3. Equally large or equivalent projections 19 

4. Arbitrary projections 20 

Division of Projections According to the 

Method of Construction (According to the 

Appearance of the Normat Grid) 20 

Cylindrical Projections 20 

Normal (equivalent) cylindrical projection 20 

Simple equally spaced cylindrical projection 22 

Isogonal cylindrical projection 23 

Isogonal oblique cylindrical projections 24 

Isogonal transverse and cylindrical Gaussian 

projection 25 

Conic Proj actions 27 

Simple normal conic projection 28 

Isogonal conic projection 29 

Convergence angle of the meridians 30 

Polyconic projections 31 

International projection 32 

Azimuthal (Perspective) Projections 34 

Central polar (gnomonic projection) 36 

Equally spaced azimuthal (central) projection 38 

Stereographi c polar projection 38 

Nomenclature of Maps 41 



111 



Maps Used for Aircraft Navigation 42 

6. Measuring Directions and Distances on the Earth's 
Surface 45 

rthodrome on the Earth's Surface 45 

Orthodrome on Topographical Maps of Different 

Projections 55 

Loxodrome on the Earth's Surface 60 

General Recommendations for Measuring Directions 

and Distances 65 

7. Special Coordinate Systems on the Earth's Surface.... 66 

Orthodromic Coordinate System 67 

Arbitrary (Oblique and Transverse) Spherical and 

Polar Coordinate Systems 71 

Position Lines of an Aircraft on the Earth's 

Surface 73 

Bipolar Azimuthal Coordinate System 74 

Goniometric Range- F i nd i ng Coordinate System 77 

Bipolar Range- F i nd i ng (Circular) Coordinate 

System 78 

Lines of Equal Azimuths 80 

Difference-Range-Finding (Hyperbol ic) 

Coordinate System 81 

Ove ra 1 1 -Range-F i nd i ng (Elliptical) Coordinate 

System 85 

8. Elements of Aircraft Navigation 88 

Elements which determine Flight Direction 88 

1. Assymetry of the Engine Thrust or Aircraft 

Drag (Fig. 1.59) 94 

2. Allowable Lateral Banking of an Aircraft in 
Horizontal Flight 94 

3. Coriolis Force 95 

h. Two-dimensional Fluctuations in the Aircraft 

Course 95 

5. Gliding During Changes in the Lateral Wind 

Speed Component at Flight Altitude 95 

Elements Which Characterize the Flight Speed of 

an Aircraft 96 

Navigational Speed Triangle 98 

Elements Which Determine Flight Altitude 101 

Calculating Flight Altitude in Determining 

Distances on the Earth's Surface 103 

Elements of Aircraft Roll 107 

1. Combination of Roll with a Straight Line 110 

2. Combination of two rolls 110 

3. Linear prediction of roll (LPR) Ill 

CHAPTER TWO. AIRCRAFT NAVIGATION USING MISCELLANEOUS 

DEVICES 113 

1. Geotechnical Means of Aircraft Navigation 113 

2. Course Instruments and Systems 114 



IV 



Methods of Using the Magnetic Field of the 

Earth to Determine Direction 114 

Variations and Oscillations in the Earth's 

Magnetic Field 119 

Magnetic Compasses 121 

Deviation of Magnetic Compasses and its 

Compensation 123 

Equalizing the Magnetic Field of the Aircvaft.... 126 

Deviation Formulas 128 

Calculation of Approximate Deviation 

Coefficients 131 

Change in Deviation of Magnetic Compasses as a 
Function of the Magnetic Latitude of the Locus 

of the Aircraft 133 

Elimination of Deviation in the Magnetic 

Compasses 134 

Gyroscopic Course Devices 141 

Principle of Operation of Gyroscopic 

Instruments 142 

Degree of Freedom of the Gyroscope 144 

Direction of Precession of the Gyroscope Axis ... . 146 
Apparent Rotation of Gyroscope Axis on the 

Earth ' s Surface 146 

Gyroscopic Semicompass 149 

Distance Gyromagnetic Compass 152 

Gyroi nducti on Compass 158 

Details of Deviation Operations on Distance 

Gyromagnetic and Gy ro i nduct i on Compasses 162 

Methods of Using Course Devices for Purposes 

of Aircraft Navigation 165 

Methods of Using Course Devices Under Conditions 

Included in the First Group 166 

Methods of Using Course Devices Under Conditions 

of the Second Group 168 

Methods of Using Course Devices Under the 

Conditions of the Third Group 172 

3. Barometric Altimeters 175 

Description of a Barometric Altimeter 180 

Errors in Measuring Altitude with a Barometric 

Altimeter 183 

4. Airspeed Indicators 186 

Errors in Measuring Airspeed 193 

Relationship Between Errors in Speed Indicators 

and Flight Altitude 196 

5. Measurement of the Temperature of the Outside Air.... 199 

6. Aviation Clocks 201 

Special Requirements for Aviation Clocks 202 

7. Navigational Sights 204 

8. Automatic Navigation Instruments 210 

9. Practical Methods of Aircraft Navigation Using 
Geotechnical Devices , 214 



V 



Takeoff of the Aircraft at the Starting Point 

of the Route 215 

Selecting the Course to be Followed for the 

Flight Route 218 

Change in Navigational Elements During Flight 221 

Measuring the Wind at Flight Altitude and 
Calculating Navigational Elements at Successive 

Stages 224 

Calculation of the Path of the Aircraft and 
Monitoring Aircraft Navigation in Terms of 

Distances and Direction 227 

Use of Automatic Navigational Devices for 
Calculating the Aircraft Path and Measuring 

the Wind Parameters 230 

Details of Aircraft Navigation Using Geotechnical 

Methods in Various Flight Conditions 233 

10. Calculating and Measuring Pilotage Instruments 234 

Purpose of Calculating and Measuring Pilotage 

Instruments 234 

Navigational Slide Rule NL-10M 235 

CHAPTER THREE. AIRCRAFT NAVIGATION USING RADIO-ENGINEERING 

DEVICES 2 50 

1. Principles of the Theory of Radionavi gational 
Instruments 250 

Wave Polarization 251 

Propagation of Electromagnetic Oscillations in 

Homogeneous Media 253 

Principles of Superposition and Interference 

of Radio Waves 257 

Principle Characteristics of Rad i onav i ga t i ona 1 

Instruments 257 

Operating Principles of Rad i onav i gat i ona 1 

Instruments 258 

2. Goniometric and Goniometri c-Rangef i nding Systems.... 259 

Aircraft Navigation Using Groun-Based Radio 

Direction-Finders 263 

Select-ion of the Course to be Followed and 

Control of Ftight Direot'ion 265 

Path Control in Terms of Distance and Deter- 
mination of the Aircraft ' s Location 269 

Determination of the Ground Speed, Drift Angle, 

and Wind 270 

Automatic Aircraft Radio D i s tance- F i nders 

(Radiocompasses) 273 

RadioGompass Deviation 279 

Aircraft Navigation Using Radiocompasses on 

Board the Aircraft 283 

Special Features of Using Radiocompasses on 
Board Aircraft at High Altitudes and Flight 
Speeds 292 



VI 



Details of Using Radiooompasses in Making 
Maneuvers in the Vicinity of the Airport 

at Which a Landing is to be Made 295 

Ultra-Shortwave Goniometric and Goniometric- 

Range Finding Systems 296 

Details of Using Goniometric-Range Finding 

Systems at Different Flight Altitude^ 304 

Fan-Shaped Goniometric Radio Beacons 306 

3. Difference-Rangef indi ng (Hyperbolic) Navigational 
Systems 310 

Operating Principles of Differential Range- 
finding Systems 312 

Navigational Applications of Differential- 

Rangefinding Systems 317 

Methods of Improving Differential RAngefinding 
Navigational Systems 318 

4. Autonomous Radio-Navigational Instruments 320 

Aircraft Navigational Radar 320 

Indicators of Aircraft Navigational Radars 325 

Nature of the Visibility of Landmarks on the 

Screen of an Aircraft Radar 327 

Use of Aircraft Radar for Purposes of Air- 
craft Navigation and Avoidance of Dangerous 

Meteorological Phenomena 328 

Autonomous Doppler Meters for Drift Angle and 

Ground Speed ; 339 

Schematic Diagram of the Operation of a 

Meter with Continuous Radiation Regime 347 

Use of Doppler Meters for Purposes of 

Aircraft Navigation 350 

Preparation for Flight and Correction of 

Errors in Aircraft Navigation by Using 

Doppler Meters 357 

5. Principles of Combining Navigational Instruments 366 

CHAPTER FOUR. DEVICES AND METHODS FOR MAKING AN 

INSTRUMENT LANDING 370 

SYSTEMS FOR MAKING AN INSTRUMENT LANDING 370 

Simplified System for Making an Instrument 

Landing 374 

Marker Devices 375 

Low-Altitude Radio Altimeters 376 

Gyrohorizon 378 

Variometer 380 

Angle of Slope for Aircraft Glide 380 

Typical Maneuvers in Landing an Aircraft 381 

Calculation of Landing Approach Parameters 

for a Simplified System 386 

Calculation of Corrections for the Time for 

Beginning the Third Turn 387 



Vll 



Cat cut at -Ion of the Covveotvon for the Time 

of Starting the Fourth Turn 388 

Calculation of the Moment for Beginning 

Descent Along the Landing Course 389 

Calculation of the Vertical Rate of Descent 

Along the Glide Path 390 

Determination of the Lead Angle for the 

Landing Path 391 

Landing the Aircraft on tiie Runway and Flight 
along a Given Trajectory with a Simplified 

Landing System 391 

Course-Glide Landing Systems 394 

Ground Control of Course-Glide Systems 396 

Aircraft-Mounted Equipment for the Course- 
Glide Landing System 400 

Location and Parameters for Regulating the 
Equipment for the Course-Glide Landing 

System 401 

Landing an Aircraft with the Course-Glide 

System 403 

Directional Properties of the Landing 

System Apparatus 406 

Directional Devices for Landing Aircraft 408 

Radar Landing Systems 410 

Bringing an Aircraft In for a Landing 

with Landing Radar 415 

CHAPTER FIVE. AVIATION ASTRONOMY 418 

1. The Celestial Sphere 418 

Special Points, Planes, and Circles in the 

Celestial Sphere 418 

Systems of Coordinates 421 

Apparent System of Coordinates 421 

Equatorial System of Coordinates 422 

Graphic Representation of the Celestial Sphere.... 424 

2. Diurnal Motion of the Stars.... 426 

Motion of the Stars at Different Latitudes 427 

Rising and Setting, Never-Rising and 

Never-Setting Stars 428 

Motion of Stars at the Terrestrial Poles 431 

Motion of Stars at Middle Latitudes 432 

Motion of Stars at the Equator 433 

Culmination of Stars 433 

Problems and Exercises 435 

3. The Motion of the Sun 436 

The Annual Motion of the Sun 436 

Motion of the Sun Along the Ecliptic 437 

Diurnal Motion of the Sun 439 

The Motion of the Sun at the North Pole 439 

Motion of the Sun between the North Pole and 

the Arctic Circle 439 



Vlll 



Motion of the Sun above the Arctic Civcle 441 

Motion of the Sun at Middle Latitudes 441 

Motion of the Sun at the Tevrestviat 

Equator 442 

4. Motion of the Moon 442 

Intrinsic Motion of the Moon 442 

Biveotion and Rate of the Moon ' s Motion 443 

Phases of the Moon 443 

Nature of the Motion of the Moon around 

the Earth 445 

Location of the Moon Above the Horizon 445 

5. Measurement of Time 446 

Essence of Calculating Time 446 

Sidereal Time 446 

True Solar Time 447 

Mean Solar Time 448 

Local Civil Time 449 

Greenwich Time 449 

Zone Time 451 

Standard Time 453 

i^elation Between Greenwich, Local and Zone 

(Standard) Time 454 

Measuring Angles in Time Units 455 

Time Signals 457 

Organization of Time Signals in Aviation 458 

A Brief History of Time Reckoning 459 

6. Use of Astronomical Devices 461 

Astronomical Compasses 467 

Astronomical Sextants 469 

CHAPTER SIX. ACCURACY IN AIRCRAFT NAVIGATION 470 

1. Accuracy in Measuring Navigational Elements and 

in Aircraft Navigation as a Whole 470 

2. Methods of Evaluating the Accuracy of Aircraft 
Navigation 474 

3. Linear and Two-Dimens i onal Problems of 

Probability Theory 478 

4. Combination of Methods of Mathematical Analysis 
and Mathematical Statistics in Evaluating the 

Accuracy of Navigational Measurements 490 

5. Influence of the Geometry of a Navigational 
System on the Accuracy of Determining Aircraft 
Coordinates 493 

6. Evaluation of the Accuracy of Measuring a 
Navigational Parameter 497 

7. Calculation of the l^lind with an Evaluation of the 
Accuracy of Aircraft Navigation 499 

8. Consideration of the Polar Flattening of the Earth 
in the Determination of Directions and Distances 

on the Earth's surface 501 



XX 



ll 



CHAPTER SEVEN. FLIGHT PREPARATION 507 

1. Goals and Problems of Flight Preparation 507 

2. Preparing Flight Charts and Marking the Route 508 

3. Studying the Route and Calculating a Safe 

Flight Altitude 514 

4. Special Preparation of Charts and Aids for 

Using Various Navigational Devices in Flight 517 

5. Calculating the Distance and Duration of Flight 518 

Calculating the Fuel Supply for Flight on 

Aircraft with Low-Altitude Piston Engines 518 

Calculating the Fuel Supply for Flight in Air- 
craft with High-Altitude Piston Engines 521 

Calculating the Fuel Supply for Flight on 

Aircraft with Gas Turbine Engines 521 

Calculating the Greatest Distance of the 

Aircraft's Point of Closest Approach to a 

Reserve Airport 530 

6. Pre-flight Preparation and Flight Calculation 532 

CHAPTER EIGHT. GENERAL PROCEDURE FOR AIRCRAFT NAVIGATION 536 

1. General Methods of Aircraft Navigation along 

Air Routes 536 

2. Stages in Executing the Flight 538 

Tal<e-Off and Climb 539 

Executing a Flight Along a Route 540 

Descent and Entrance to the Region of the 

Landing Airport by an Aircraft 542 

Maneuvering in the Vicinity of the Airport 

and the Landing Approach 543 

Supplement 1. Composite Chart of Topographical Maps 545 

Supplement 2. Spherical Trigonometry Formulas 547 

Supplement 3. Map of the Heavens 549 

Supplement 4. Map of Time Zones 550 

Supplement 5. Table of Greenwich Hour Angles of the 
Sun and Chart of Their Corrections for 

the Flight Date 551 

Supplement 6. Table of Values of the Function $ (a; - a). 552 
Supplement 7. Units often Encountered in Aircraft 

Navigation and Their Values 554 



ABSTRACT : The theory and practice of aircraft 12_ 

navigation at the modern level of aviation tech- 
nology are summarized in this hook; the most im- 
portant practical problems of the utilization of 
general 3 radio-engineering, and astronomical means 
of aircraft navigation are set forth; the proce- 
dure of the pilot's preparation for flight, the 
means of calculating the distance and duration 
of a flight, and the carrying out of pve-landing 
maneuvering and landing of the aircraft under 
complex meteorological conditions during the day 
or at night are elucidated. 

The basic material of the book, sufficient 
for the practical mastery of the means and methods 
of aircraft navigation, is presented with the ap- 
plication of mathematics within the limits of a 
secondary school course. The problems which are 
necessary for a deeper study of the material are 
discussed in terms of principles of higher math- 
ematics . 

The book is intended for pilots and naviga- 
tors. It can be used as a textbook for students 
of civil aviation educational institutions . 



XI 



INTRODUCTION 



/3 



Aircraft navigation or aerial navigation is a science which 
studies the theory and practical methods of the safe navigation of 
airplanes as well as other aircraft (helicopters, dirigibles, etc.) 
in the airspace above the Earth's surface. 

By the process of aircraft navigation, we mean the complex of 
activities of the aircraft crew and the ground traffic control, 
which are directed toward a constant knowledge of the aircraft's 
location and which ensure safe and accurate flight along a set course 
as well as arrival at the point of destination at a set altitude and 
at an established time. 



During the initial period of the development of aviation, air- 
craft did not have equipment for piloting when the natural horizon 
was not visible and for orientation when the ground was not visible, 
so that visual orientation was the basic method of aircraft naviga- 
tion. The position of the aircraft was determined by comparing vis- 
ible landmarks In the area over which the aircraft was flying, with 
their representation on a map. 

However, at this time the necessity for instrumental methods 
of aircraft navigation was already felt. The most simple devices 
for measuring airspeed, flight altitude, the aircraft's course, and 
several other flight parameters were Installed on aircraft. This 
period saw the appearance of the first navigator's calculating in- 
struments (wind-speed indicators and navigational slide rules). 

At the beginning of the 1920's, the first hydroscopic devices 
appeared on aircraft; they were turn and glide indicators which (in 
combination with indicators of airspeed and vertical velocity) in- 
dicators (variometers) made it possible to judge in a rather primi- 
tive way the position of the aircraft in space when the natural 
horizon was not visible. By means of these devices, the aircraft 
crews (after special training) were already able to carry out 
flights in the clouds and above the clouds. 

s and the beginning of the 30's, more re- 
developed: gyrohorizons and gyrosemi- 
ime reliably ensured pilotage of aircraft 

on 
llotage 




xne Deginning or xne ou's, moi 
loped: gyrohorizons and gyros 
liably ensured pilotage of all 
'ound was not visible. Later, 
ces for automatic aircraft plJ 



/^ 



Xlll 



Achievements in the area of piloting aircraft when the Earth 
was not visible, as well as the growth by that time of the speed, 
altitude, and distance of aircraft flights, required the creation 
of means to ensure aircraft navigation independent of the visibility 
of terrestrial landmarks . 

During these years, zone radio beacons which allowed the air- 
craft's flight direction to be maintained along a narrowly directed 
radial line which coincides with the direction of the straight part 
of an aerial route began to appear. Ground radiogoniometers also 
appeared, by means of which direction was determined in an aircraft, 
as well as the position of the aircraft along two intersecting di- 
rections . 

Another aspect of the development of aircraft navigation at 
this time was astronomical orientation. To determine the location 
of an aircraft, various sextants were constructed and special com- 
putation tables and graphs of the movement of heavenly bodies were 
compiled for use with the sextants. In the mid-30's, devices ap- 
peared for determining the course of an aircraft according to the 
heavenly bodies . 

At the same time, optical sighting devices were used, by means 
of which (during visibility of the terrestrial landmarks) the ground- 
speed, flight direction and drift angle of the aircraft were meas- 
ured, all of which were later used for some time as constants for 
calculating the path of an aircraft according to flight time and 
direction . 

A very important stage in the development of means of aircraft 
navigation of the mid-30's was the appearance of aircraft radio- 
goniometers ( radiosemicompasses ) , a further modification of which 
were the automatic aircraft radiocompasses . Radiosemicompasses and 
radiocompasses were, for a period of more than 20 years, the basic 
means of aircraft navigation in aircraft with piston engines. 

During World War II and especially in the postwar years, radio- 
engineering systems of long and short-distance navigation of a dif- 
ferent kind as well as radio-navigation landing systems became wide- 
spread. Essentially, these were not autonomous means of aerial 
navigation but systems which included both ground-based facilities 
for the security of aircraft navigation and aircraft equipment. 

Radical changes in the area of means and methods of aircraft 
navigation occurred (and are occurring at the present time) in con- 
nection with the development of jet aviation technology. 

The sharply growing speed, altitude, and distance of flights 
have required automation of the most laborious processes of aircraft 
navigation. Magnetic course devices and non-automatic radio navi- /5 
gational systems were of little use for ensuring the automation of 
aircraft navigation and the piloting of high-speed aircraft. There 

xiv 



J 



arose a necessity for developing highly stable gyroscopic compasses, 
autonomous speed and flight direction meters , and stricter consider- 
ation of the aircraft's flight ' dynamics to ensure the rapid and ac- 
curate solution of navigational problems by computers. 

The science of "aircraft navigation" grew and developed along 
with the development of aviation and navigation technology. The 
works of the outstanding Russian scientists and inventors, M. V. 
Lomonosov, N. Ye. Zhukovskiy, K. E. Tsiolkovskiy , and A. S. Popov 
were the basis of aircraft navigation theory. 

A large contribution to the science of aircraft navigation was 
made by the following Soviet navigators and scientists: B. V. 
Sterligov, S. A. Danilin, I. T. Spirin, G. S. Frenkel', A. V. Bely- 
akov, L. P. Sergeyev, R, V. Kunitskiy, G. 0. Fridlender, G. F. 
Molokanov, B, G. Rats, V. Yu . Polyak, et al . 

The successes achieved in the development of aircraft naviga- 
tion as a science made it possible, even in 1925-1929, to accomplish 
long flights by Soviet aircraft along the routes: Moscow-Peking 
(M. M. Gromov), Moscow-Tokyo and Moscow-New York (S. A. Shestakov). 

Further nonstop flights by Soviet aviators, organized from 
1936-1939 (V, P. Chkalov, M. M. Gromov, and V. K. Kokkinaki) both 
over the territory of the Soviet Union and especially over the North 
Pole to the USA, were like a great school, in which the examinations 
were the successes achieved by Soviet scientists in the area of 
aircraft navigation. 

World War II was a verification of all the achievements in the 
theory and practice of aircraft navigation, especially in the field 
of long-distance aviation, with the carrying out of long-distance 
night flights. During this period, a rich store of experience was 
accumulated and further improvements in aircraft navigation methods 
were carried out. 

In the postwar period, the science of aircraft navigation under- 
went an especially vigorous development in connection with the ap- 
earance of high-speed jet aircraft, and also in connection with 
he great achievements of the radio and electronics industry. 

Long-distance flights of high-speed aircraft along aerial 
utes which include international and intercontinental flights, as 
11 as flights to the Arctic and Antarctic, are becoming routine 
r civil aviation crews. 

At the present time, aircraft navigation science has been dis- 
guished as an Independent and orderly science in which the 
ievements of a number of the general and special branches of 
wledge are employed: physics, mathematics, geodesy, astronomy, 
Physics, aerodynamics, radio engineering, radio electronics, etc. 



XV 



Navigation technology is developing at a rapid pace; aircraft /_6_ 
and ground facilities for aircraft navigation are' continually being 
perfected and the professional training and navigational prepara- 
tion of flight and ground personnel has improved. All this has 
radically raised the reliability of aircraft navigation, its accu- 
racy, and its chief criterion, safety. 

Modern technical means of aircraft navigation are divided into 
four basic groups according to the principle of operation. 




2. Radio-engineering means of aircraft navigation^ which are 
based on the operating principle of radio-electronic technology. 
These include goniometer radio-engineering systems (radio compasses 
with ground transmitting radio stations , ground radiogoniometers 
with aircraft receiving-transmitting radio stations, and radio bea- 
cons with aircraft receiving radio equipment), rangefinding systems, 
goniometer-rangef inding systems, ground and aircraft radar, Doppler 
meters and systems, radio altimeters, course-landing beam systems 
with their ground and aircraft equipment, etc. 

3. Astronomical (radio astronomical) means of aircraft navi- 
gation^ which are based on the principle of measuring the motion 
parameters of heavenly bodies. These include aviation sextants, 
astrocompasses , astronomical orientators , etc. 

4. Light engineering means of aircraft navigation^ which are 
based on the principle of using light energy radiation. These in- 
clude ground light beams, light and pulse-light equipment for take- 
off and landing strips as well as aircraft, enclosures for the light- 
ing equipment of the routes and airports (housings for ground in- 
stallations) , various pyrotechnic devices, etc. 

At the heart of a safe and accurate flight according to a set 
route, in the vicinity of the airport, or during take-off and land- 
ing, lies the principle of the overall usage of all the available 
technical means of aircraft navigation, both ground facilities and 
those aboard the aircraft. 



xvi 



■■■■■■■I ■l^im IIUHHI 



NASA TT F-524 



CHAPTER ONE 
COORDINATE SYSTEMS AND ELEMENTS OF AIRCRAFT NAVIGATION 



n* 



1. Elements of Aircraft Movement in Space 

The fundamental problem of aircraft navigation in all stages 
of flight is maintaining a given trajectory of aircraft movement in 
altitude, direction and time by means of a complex utilization of 
navigational means and methods. A successful solution to these 
problems depends on constant and accurate information concerning 
the position of the craft relative to a given flight trajectory, 
the nature of the aircraft movement, and the actions of the crew. 

As a result of the curvature of the Earth's surface, any given 
flight trajectory of an aircraft is curvilinear. However, by taking 
into account the large radius of curvature of the Earth's surface, 
a small area can always be .delineated on it whose surface can be 
assumed to be plane (Fig. 1.1). 

Let us erect a perpendicular 0^1 from 
the center of the small area which we have 
chosen and continue it until it intersects 
the center of the Earth. Obviously, this 
will be a perpendicular line, which we can 
call the vevticat of the Zoaus. 

In the plane of the small area which 
we have chosen, let us draw a straight 
line through the point Oi and take it as 
the X axis ; then let us draw another 
straight line through the point O^ in the 
plane of the area, perpendicular to the 
first, and call it the Z axis. 

Thus, at point 0^ on the Earth's sur- 
face, we will obtain a rectangular system 
of space coordinates X, Y, Z. 




Fig. 1.1. Rectangular 
Coordinate System on 
the Earth's Surface. 



Numbers in the margin indicate pagination in the foreign text 



^ 



The travel of an aircraft over the Earth's surface will in- 
volve both a shift in the point Oi (origin of the coordinates) and 
the rotation of the axes of the coordinate system around the center _/_8 
of the Earth (point 0) . 

However, the system of coordinates which we have obtained can 
be used for determining the directions of the aircraft axes and the 
component flight speed vectors. Since the origin of this system is 
being continuously shifted, let us designate it as a gliding rea- 
tangulav system of eoord-inates . 

In this coordinate system, the 
following elements can be distin- 
guished : 

(a) Position of the longitu- 
dinal axis of the aircraft in the 
horizontal plane {aivovaft course) , 

(b) Position of the longitu- 
dinal axis of the aircraft in the 
vertical plane (angle of pitch of 
the aircraft ) , 




Fig. 1.2. Dip Angle of the 
Trajectory of Altitude Gain, 



(c) Position of the lateral axis of the aircraft in the ver- 
tical plane (lateral banking) , 

(d) Distance along the vertical from the Earth's surface (the 
area which we have chosen) to the aircraft (flight altitude) , 

(e) Vertical speed (altitude gain and loss), 

(f) Component flight speed along the X and Z axes or the vec- 
tor of groundspeed and its direction (groundspeed and flight angle), 

(g) Angular velocity of aircraft roll, 

(h) Component wind speed along the X and Z axes of the system, 
or the wind vector and its direction (wind speed and direction) . 

Usually the position of the craft on the Earth's surface is 
treated in surface-coordinate systems, the most widely used of which 
are the geographic system and the reference system whose major axis 
coincides with a given flight trajectory on the Earth's surface. 

The position of the aircraft in surface-coordinate systems is 
assumed to be the position of the origin of the gliding system. To 
analyze the elements of aircraft navigation, let us combine the X 
axis of the gliding-coordinate system with a given flight trajectory 
of the aircraft. 

In order to keep the aircraft in the rectilinear horizontal 



segment of this trajectory, the crew must maintain a flight condi- 
tion in which the aircraft will not be shifted along the vertical 
(altitude gain and loss), there will be no lateral deviation (to 
the right or left), i.e., the vertical velocity Vy and the lateral 
component of the velocity V^, will be equal to zero, and the longi- 
tudinal flight velocity V^ (along the X axis) will be as given. 

If the flight trajectory is inclined (segments of altitude gain 
and loss), the crew must hold this trajectory by maintaining the 
vertical and longitudinal flight velocities ( 7y and V^) t i.e., /J_ 
maintain a given dip angle of the trajectory 9 (Fig. 1.2). 

Obviously, at a constant dip angle of the flight trajectory, 
the latter will have a curvature in the vertical plane just as in 
horizontal flight. Therefore, if we neglect the curvature of the 
horizontal flight trajectory, we may assume 



X^-Xx ' (1.1) 

where Q is the dip angle of the flight trajectory; Xi, X2 are the 
coordinates of the initial and final points of the sloping segment 
of the trajectory; Hi, H2 represent a given altitude at the initial 
and final points. 

When the aircraft travels from the initial point Xi to the mov- 
ing point J, the flight altitude is changed by the value 

iiff=(X-Xi)tgg, (1.2) 

and the value of the moving flight altitude is 

H = ff^ + ^H^Hl + ^X—Xl)^gfl (1.3) 

or if we take Formula (1.1) into account, 

H^ff, + (X-X^) f-^' . (1.4) 

■n.2 — ^1 

Since the altitude during a sloping trajectory is a variable 
value, a given flight trajectory is maintained at a constant value 
of the vertical velocity 

Vy^V^tgQd r Vy^V,-^^^. (1.5) 

Checking of the position of the aircraft at given values of 
the varying flight altitude is carried out only at specific points 
on the sloping trajectory. 



Translator's note: tg = tan. 



2. Concepts of Stable and Unstable Flight Conditions 

A navigational flight condition is determined by the motion 
parameters of an aircraft along a trajectory or by navigational 
elements of flight: course, speed, and altitude. 

The motion parameters of an aircraft are usually measured rel- 
ative to airspace. However, considering that the airspace also 
shifts , they are selected in such a way as to ensure retaining the 
given flight trajectory relative to the Earth's surface. 

Based on the nature of the trajectory and the conditions of 
aircraft navigation, four main flight conditions are distinguished: /.lO 
horizontal rectilinear flight, altitude gain, altitude loss, and 
roll. 

Horizontal rectilinear flight is characterized by two constant 
parameters: height and flight direction. 

Altitude gain and loss conditions each have two constant param- 
eters: flight direction, and vertical velocity or dip angle of the 
traj ectory . 

The condition of roll is always combined with one of the first 
three flight conditions , so that the flight direction becomes vari- 
able and can be replaced by a parameter which characterizes the 
curvature of the roll trajectory through the radius of roll or the 
angular velocity. 

A flight condition is stable if its parameters acquire constant 
values, and unstable if its parameters are variable. 

Flight practice shows that flight conditions , strictly speak- 
ing , are never fixed for any prolonged time, since there are always 
factors changing the aircraft's motion parameters. 

The main sign of a stable flight condition is the equality to 

zero of the first derivative of the given parameter with time 

d'^S 
or of the second derivative path with time — . 

For example, for the velocity parameter V = const, if 

= or r-=0. 



(dl) 



df dfl 



Analogously, for the flight direction parameter (.^) and the 
altitude parameter (ff) : 

dii dH 

.l;=const, If — T- = 0, //=const, if = 0. 

dt dt 



If forces arise during flight which change the aircraft's mo- 
tion parameters, the extreme values of the motion parameters (i.e., 
the points of the maxima and minima on the curve which characterizes 
the change of the given parameter with time) indicate equilibrium 
of these forces. 



A stable flight con 
on a given parameter exi 
the extreme points, sine 
derivative parameters ba 
at these points are equa 
while the disturbing for 



The disturbing fore 
maximum value at points 
i.e., when the second de 
rameters based on time a 
zero (Fig. 1.3). On a c 
structed for the velocit 
the points of a stable c 

designated by one line, while points of maximum distur 

are designated by two lines. 



Fig. 1.3. Graph of the 
Changes of a Navigational 
Parameter and Points with 
a Stable Flight Condition 



dition based 
sts only at 
e the first 
sed on time 
1 to zero 
ces are absent 

es acquire a 
of inflection, 
rivative pa- 
re equal to 
urve con- 
y parameter, 
ondition are 
bing forces 



From aerodynamics, we know that in horizontal flight at a ve- 
locity significantly less than the speed of sound, the drag of an 
aircraft in a counterflow is 



/ll 



Qx = cj,s 



pV2 



where a^ is the coefficient of drag of the aircraft, S is the cross- 
sectional area of the midship section, and p is the air density at 
flight altitude. 

It is obvious that the airspeed will be stable if the thrust 
of the engines (P) is equal to the drag of the aircraft P - Qx- 

With a disturbance of this equilibrium, there arises a disturb- 
ing force which changes the flight velocity. For example, with an 
increase in the thrust of the engines the disturbing force will be 
equal to : 



&P^P-—CjcS 



pV2 



which causes an initial acceleration of the aircraft 

dt ~ m ' 



where m is the mass of the aircraft in kg. 



Later, with an increase in velocity, the drag of the aircraft 
will also increase. The value of this drag will approach the value 
of the thrust of the engines, i.e., the velocity very slowly ap- 
proaches a stable value logarithmically. 

Changes in airspeed which are analogous in nature arise during 
changes in the velocity of the headwind or the incident airflow at 
flight altitude. For example, with an increase in the velocity of 
the incident airflow, the airspeed diminishes. This provides a 
surplus of engine thrust. Subsequently, an increase in airspeed 
occurs logarithmically. 



If the lateral component of the wind speed changes 
pressure on the surface of the aircraft arises: 



a lateral 



Qz~CzSt 



pvl 



where Cg is the coefficient of lateral drag of the aircraft ; 
the cross-sectional area of the aircraft in the XI plane; V^ 
lateral velocity component equal to Ug . 

The initial lateral acceleration of the aircraft is: 

rfi^z Qz 



dt 



at 



is the 



Subsequently, the lateral velocity of the aircraft will log- 
arithmically approach the lateral component of the wind velocity, 
i.e., the flight condition will approach a condition which is stable 
in direction. 



Usually, during navigational calculations for each parameter, 
its mean value for a definite length of time is called a stable 
flight condition: mean velocity, mean vertical velocity, mean di- 
rection, etc. 

From the point of view of maintaining flight direction, air- 
craft roll is an unstable condition. If a given trajectory is curv- 
ilinear, the roll condition is also examined as stable or unstable. 
The entrance or exit of an aircraft from roll , as well as roll with 
variable banking, can serve as examples of unstable roll conditions. 

The rolling of an aircraft is considered to be coordinated if 
the longitudinal axis of the aircraft constantly coincides with the 
tangent to the trajectory of its movement, i.e., external or inter- 
nal aircraft glide is absent. This is achieved by tilting the rud- 
der of the aircraft for banking in a roll. 

During banking of an aircraft, its lift (J) is directed not 
along the vertical plane but along the axis of the aircraft, which 
is deflected from it (Fig. 1.4). 



/12 



Rolling of an aircraft without descent or with stable vertical 
velocity is possible only when the vertical component of the lift 
(.Yl) is equal to the weight of the aircraft G. 

In this case, the horizontal (centrip- 
etal) component of the lift is: 

K«=GtgP. 




Fig. 1.4. Resolution 
of Forces During 
Rolling of an Air- 
craft . 



where 
craft 



is the banking angle of the air- 



Since we are examining a coordinate 
roll (without gliding of the aircraft), 
the centrifugal force in the roll 

mV2 



R 



will be equal to the centripetal force , 
i.e., 

= Otgp, 



/? 



where m is the mass of the aircraft; and R is the radius of the co- 
ordinated roll. 



Transforming this equation, taking into account that m = 



g 



we will obtain formulas for determining both the radius and path of 
the aircraft with coordinated roll: 



R 



1/2 



fi-tgP 



; ^'^ 2t:R. 



(1.6) 



Formulas (1.6) relate the radius of stable coordinated roll of 
the aircraft with the airspeed and also with banking in rolling, 
and they are used in calculations of the radius and path of the air- 
craft along a curvilinear flight trajectory. 

3. Form and Dimensions of the Earth 



/13 



In the practice of aircraft navigation, it is necessary first 

of all to deal with distances and directions on the Earth's surface 

which are the result of the mutual distribution of objects through 
which the flight path passes. 

The Earth's surface, its relief and mutual distribution of ob- 
jects can be most accurately expressed on a model of the Earth (a 
globe). However, a globe with a representation of the Earth's sur- 
face that satisfies the demands of aircraft navigation would be so 
large that its use in flight would be impossible. Therefore, dif- 
ferent means of representing the surface of the Earth, which is 
curved in all directions, on a plane (sheets of paper) are used. 



The Earth has a complex form called a geoid (without consider- 
ing the local relief, if we imagine that its entire surface is cov- 
ered with water at sea level). The surface of a geoid at any point 
is perpendicular to the direction of the action of gravity. A de- 
scription of a geoid by mathematical expressions is very complex, 
and if we consider the folds in the relief of the Earth's surface, 
then it is practically impossible to express its form mathematically, 
Therefore, in calculations the form of the Earth is taken as an 
e'Lt't'pso'id of revotuiion, the form closest to a geoid. 




Fig. 1.5. Great and Small Circles on the Earth's Surface. a) Semi- 
axis of the Earth and Great Circle; b) Small Circle. 



According to measurements made by Soviet scientists under the 
supervision of F. N. Krasovskiy, the major semiaxis of this ellip- 
soid (a), which coincides with the radius of the equator, is equal 
to 6,378,245 km. The minor semiaxis of the ellipsoid {b) , which 
coincides with the axis of the Earth's rotation, is equal to 
6,356,863 km (Fig. 1.5, a). 



/14 



The flattening of the Earth at the poles is 

— '^~* 1 

~ a ^ 298^3 ' 



These dimensions show that the Earth's ellipsoid of revolution 
is practically close to a sphere; to simplify the solution of the 
majority of problems in aircraft navigation, it is taken as a true 
sphere, equivalent in volume to the Earth's ellipsoid. The radius 
of such a sphere is equal to 6371 km. 

The maximum distortion of distances caused by the replacement 



of the Earth's ellipsoid by a sphere does not exceed 0.5%, and the 
distortion of directions is not more than 12 minutes of angle. 

In geodesy and cartography, the plotting of maps, as well as 
in other branches of science where more accurate calculations of 
distances and directions are necessary, the Earth's surface is taken 
as an etZipso'id of revolution. 

4. Elements Which Connect the Earth's Surface 
with Three-Dimensional Space 

Taking the Earth as a true sphere , we will locate a perpendicu- 
lar (a resting pendulum) at any point above the Earth's surface. 
Then, disregarding the possible insignificant deviations caused by 
the varying relief, the irregularity of distribution of the densest 
masses in the Earth's crust, and the tangential accelerations con- 
nected with the Earth's rotation, it is possible to consider that 
the line of the perpendicular runs in the direction of the center 
of the Earth. 

The perpendicular line (see Fig. 1.5, a) joining the center of 
the Earth with the point of the observer's position, and continued 
in the direction of the celestial sphere (Y) , is called the geo- 
oentr-io veTtiaal of the locus. 

The plane on the Earth's surface, tangent to the sphere at the 
point of the observer and perpendicular to the true vertical of the 
locus, is called the plane of the true horizon. 

The direction and velocity of aircraft movement at every point 
on the Earth's surface are examined in the plane of the true hori- 
zon, while the altitude change is examined in the direction of the 
true vertical. 

If we cut the plane of this true horizon in any direction by 
another plane along the true vertical (through the center of the 
Earth), the line formed by the intersection of this plane with the 
Earth's surface forms a closed great circle, the mean radius of 
which will be equal to the radius of the Earth. 

The shortest distance between two points AB on the Earth's sur- 
face or part of the arc of a great circle is called the orthodrome 
(see Fig . 1.5, a ) . 

The mean radius of a great circle is assumed to be equal to 
6371 km. The length of the circumference of such a radius is equal 
to 40,000 km. One degree of arc of a great circle is equal to 
111.1 km, while one minute of arc is equal to 1,852 km. The length 
of a segment of the arc of a great circle at one minute of angle is 
called a nautical mile. 



/15 



With an intersection of the Earth's sphere by a plane which 



does not pass through the center of the Earth, the line of inter- 
section of this plane with the Earth's surface forms a closed smatt 
a-VTote y the radius of which will always be less than the mean radius 
of the Earth. The small circles parallel to the plane of the equa- 
tor are called iparattets (see Fig. 1.5, b). 




M(X,!iZ} 



Fig. 
a Sp 
nate 



R fr 
gles 
dius 
and 



1.6. 
heric 
s and 



om th 
: an 
-vect 
the d 



For the purposes of aircraft 
navigation, a coordinate system 
which unequivocally determines 
the position of an aircraft and 
objects on the Earth's surface 
is necessary. Obviously, a 
spherical coordinate system will 
be the most convenient (Fig. 1.6), 



A spherical coordinate sys- 
tem is distinguished from a rec- 
tangular system (Cartesian) by 
the fact that instead of deter- 
mining three distances to a 
point in the directions of the 
X, J, and Z axes, we determine 
the length of the radius-vector 
e center of the coordinate system to a point, and two an- 
gle A between the XY plane and the projection of the -ra- 
or (i?) to the plane XZ , and angle (j) between the XZ plane 
irection of the radius-vector (i?) . 



Relationship Between 
al System of Coordi- 
a Rectangular System. 



There is an obvious relation between spherical and rectangular- 
coordinate systems : 

X= Rcostf cos X; \ 
K=/?sin(f; I 

2 = ;? cos if sin X. J 



(1.7) 



With a constant length of the radius-vector R, if angles X and 
^ assume all possible values, the geometric location of the points 
of the end of the vector radius will be a sphere. 

To determine coordinates on the Earth's surface, there is no 
need to indicate the radius of the Earth (i?) each time. This coor- 
dinate is considered, once and for all, constant. 



/16 



Thus, the spherical coordinate system is transformed into a 
two-dimensional surface system which is called a geographic system 
of coordinates. 

The plane of the equator and the plane of the prime (Greenwich) 
meridian are taken as the initial reference planes in a geographic 
coordinate system. The point coordinates on the Earth's surface 
bear the name "longitude of the locus" and "latitude of the locus" 
(Fig. 1.7). 



10 



The dihedral angle between the plane of the prime meridian and 
the plane of the meridian of a given point is called the longitude 
of the point (X). Determination of the longitude can be given in 

arc values : the length of the 
arc of the equator (or the paral- 
lel), expressed in degrees, be- 
tween the prime meridian and the 
meridian of a given point is 
called the longitude of the point. 




Fig. 1.7. Spherical Coordinate 
System on the Earth's Surface. 



Reading 


of the longitude is 


carried 


out 


from to 180° east 


of the 


rpime 


meridian {east long- 


itude ) 


and f 


rom to 180° west 


of the 


rpime 


meridian {west long- 


itude ) . 


In 


navigational calcu- 


lations 


, east longitude is taken 


as posi 


tive 


and is designated by 


a plus 


sign , 


while west longitude 


is nega 


tive 


and is designated by 


a minus 


sign 


However, in carry- 


ing out 


navi 


gational calculations , 


it is more convenient to carry- 


out a readin 


g of longitude in the 


easterl 


y direction from zero to 


360° . 







The angle between the plane of the equator and the true verti- 
cal of a given point (or the length of the meridian arc, expressed 
in degrees, from the plane of the equator to the parallel of a given 
point) is called the latitude of the -point ( (j) ) . Since a set of true 
verticals at a constant latitude forms a cone with the vertex in the 
center of the Earth and an angle at the vertex equal to 90°-((), then 
in contrast to the dihedral angle between the planes of the m.eridlans , 
we shall call a similar angle in other spherical systems, the conia 
angle . 

Reading of the latitude is carried out from the plane of the 
equator to the north and south from to 90° {north and south lati- 
tude) . In navigational calculations, north latitude is considered 
positive and south, negative. 

A geographic coordinate system is a surface curvilinear system, /17 
i.e. , the meridians of the coordinate grid on the Earth are not 
parallel. However, if we examine the meridians and parallels on 
any unit area of the Earth's surface, they turn out to be orthogonal 
(perpendicular in one plane). Two special points on the Earth's 
surface (the geographic poles) are an exception. 

A geographic coordinate system is used not only to determine 
the location of a point (object) on the Earth, but to determine 
direction from one point to another. 



11 



The angle included between the northern direction of the meri- 
dian which passes through a given point and the orthodrome direction 
to a point setting a course is called the heaving or azimuth. Read- 
ing of the angles of bearing or azimuth is done clockwise from to 
360° . 

Since the meridians on the Earth's surface are generally not 
parallel, the value of the azimuth changes with a change in the mov- 
ing longitude along the line which joins the two points; the greater 
the latitude , the more it changes . Therefore , for the orthodrome 
direction together with an indication of the azimuth, it is neces- 
sary to mention from which meridian thia direction is measured. 

The change in azimuth with a change in the moving longitude 
does not make it possible to use magnetic compasses for moving along 
the orthodrome without introducing corresponding corrections, espe- 
cially when the two points are far apart. 

If the magnetic declination does not change, following a con- 
stant magnetic course will cause the meridians to intersect at iden- 
tical angles. The line which intersects the meridians at a constant 
angle is called the loxodrome . 

In order to proceed to a more detailed examination of the ele- 
ments of aircraft navigation and their measurement, it is necessary 
to become acquainted with the making of maps, their scales, and 
some features of cartographic projections. 

5. Charts, Maps, and Cartographic Projections 

The representation of a small part of the Earth's surface on a 
plane is called a chart. Distortion as a result of the curvature 
of the Earth's surface is practically absent on a chart. 

The conventional representation of the Earth's surface in a 
plane is called a map. 

A map is a continuous representation of the surface of the 
Earth or a part of it without discontinuities and folds, made with 
a variable scale according to a definite rule. The sphericity of 
the Earth's surface does not allow it to be represented with com- 
plete accuracy on a plane surface. Therefore, there are many ways /18 
of projecting the Earth's surface onto a plane which make it possi- 
ble to represent most accurately on the map only those parameters 
(elements) which are most necessary under the given conditions of 
application . 

Methods or laws of representing the Earth's surface on a plane 
are called cartographic projections. 

A common geometricat projection is the point of intersection 
of the line of sight (which passes through the eye of the observer 

12 



and the projected point) with the plane onto which the given point 
is projected. It is a special case of cartographic projection. 

A cartographic projection is set analytically as a function of 
geographical coordinates on the Earth (sphere) between the coordi- 
nates of a point on a plane. 

If we call one of the main directions on a map the X axis and 
the perpendicular to it the Z axis , then 

^=^i('P; I) and 2 = ^2(9; ^); 



P = ^3(?; ^) an d 8 = Ft (<f,; X), 

where p and 6 are the main directions on maps of conic and azimuthal 
projections, and (J) and X are the geographical coordinates of a point 
on the Earth (sphere). 

The properties of the projections will depend on the properties 
of these functions (Fi, F2 , F3, and F14 ) , which must be continuous 
and well-defined, since the map is made without discontinuities so 
that a single point on the map corresponds to every point in the 
locat ion . 

Map Scales 

The map-making process is divided into two stages. 

a) The Earth is decreased to the definite dimensions of a 
globe . 

b) The globe is unrolled to form a plane. 

The extent of the overall decrease in the Earth's dimensions 
to the fixed dimensions of a globe is called a principal scale. 

A principal scale is always indicated on the edge of a map and 
makes it possible to judge the decrease of the length of a segment 
in transferring it from the Earth's surface to the globe. 

A principal scale is numerically equal to the ratio of the 
distance on the globe to the actual distance at a location: 



M: 



A. S, 



AS 



e. s . 



where M is the principal scale, ASg is a segment on the globe, and 
^"^e . s . is a segment on the Earth's surface which corresponds to the 
segment on the globe. 

On maps, the principal scale is usually shown as a fraction 
(numerical scale) and by means of a special scale (linear scale). 



13 



The numeTicat soate is a fraction, the numerator of which is 
one, while the denominator shows how many such units of measurement 
fit into the location. 

For example, 1:1,000,000 means that if we take 1 cm on a map, 
then 1,000,000 cm at a location (i.e., 10 km) wil^ correspond to it. 

A t'lneav soate is a scale on a map in which a definite number 
of kilometers at a location correspond to special segments of the 
scale . 

However, a principal scale (numerical and linear) is insuffi- 
cient for accurately measuring distances on the entire field of a 
map. It is necessary to know the laws of distortion of distances 
and directions. The laws of change in the principal scale along 
the map field are determined by a special scale. 

A special scale is the ratio of an infinitely small segment in 
a given place on the map in a given direction, to an analogous seg- 
ment in a location (globe). At each point on the map, the special 
scale is different. It is either somewhat larger or somewhat small- 
er than the principal scale. 

Distortions of Cartographic Projections 

Ettiipse of Distortions 

Let us draw on a sphere (globe), an infinitely small circle 
with radius t\ let us also designate a rectangular coordinate system" 
on the sphere by x and z (Fig. 1.8, a). Then 



/■2 = ;e2 + z2. 



(1.8) 





Fig. 



Distortion of Scales on a Plane: (a) Scale on a Globe; 
(b) Scale on a Plane. 



14 



In the transfer of the coordinate system from the sphere 
(globe) to the plane, the direction of the coordinate • axes is dis- 
torted (Fig. 1.8, b). 

Having designated the special scales on a plane (map) by m in 
the direction X and n in the direction s, we obtain: 

*i = mxi 



m 'iwhile*= '=~ 
m n 



/20 



Substituting the latter in (1.8), 



and then dividing both sides of the equation by p^ , we obtain 

\mr ) [ nr )~ 



(1.9) 



From mathematics , it is known that this is the formula of an 
ellipse with conjugate diameters; therefore: 

a) Any infinitely small circle on the surface of the Earth's 
sphere in any projection is represented by an infinitely small el- 
lipse . 

b) On the surface of the Earth's sphere (globe), it is pos- 
sible to choose two mutually perpendicular directions which will be 
transferred to a map without any distortions. 

These directions are called principal directions. 

Knowing the special scales (m and n) in the principal direc- 
tions, it is always possible to construct an ellipse of distortions 
which will make it possible to judge the nature of the distortions 
of the projection as a whole. In the majority of projections, the 
directions along the meridians and parallels are taken as the prin- 
cipal directions. 

Distortion of Lengths 

If an infinitely small circle on the Earth is represented by 
an ellipse (Fig. 1.9, b) with its transfer to a plane, the distor- 
tion of the special scale in any direction {hS^) can be expressed 
as follows: ,y 



/21 



AS.= 



OM 



(1.10) 



15 



but from the circle in Figure 1.9, a: 

jf=slnor, wh i 1 ei^ = cosar, 

AS, = ym2sin2a + n2cos2o , 



then 



(1.11) 





Fig. 1.9. Distortion in a Plane: (a) Length on a Globe; (b) Length 

on a Plane . 




Z Z, 




Fig. 1.10. Distortion of Directions on a Map. (a) Direction on a 

Globe; (b) Direction on a Map 

i.e., knowing the special scales for the principal directions, we 
can always judge the value of the distortion of the special scale 
in any direction (and therefore, the distortion of the length of the 
segment as a whole). 

Distovt-ion of D-ireat'ions 

Let us take the radius r = 1 (Fig. 1.10) of an infinitely small 
circle on the Earth; then 

tg«=,-^, while tgp=-^. (1.12) 



16 



Dividing Equations (1.12) into one another, we obtain: 



tgP= tga. 

m 



(1.13) 



Obviously, knowing the special scales for the principal direc- 
tions , it is always possible to find an angle g on a map for an an- 
gle a in a location, and vice versa. 

Distortion of Areas 

The distortion of areas AP can be determined by a comparison 
or division of the area of the ellipse (.Sq±) by the area of a circle 
(5(2i); see Figure 1.11: 



/22 



AP=- 



' e 1 nab ab 



'Cl 



7cr2 



r2 ' 



(1.14) 



but if we take the radius of the circle on the Earth as equal to 1, 
then 

^P = ab 

or 5 if we express a and b by special scales for the principal direc- 
tions, we obtain: 

'^P="tn,, (1.15) 





Fig. 1.11. Distortion of Areas on a Map. (a) Area on a Globe; 

(b) Area on a Map. 

The distortion of areas is equal to the product of the special 
scales for the principal directions . 

Hence , we see that if we know the special scales for the prin- 
cipal directions , we can give the complete characteristics of any 
map projection.. 



17 



Classification of Cartographic Projections 

There are many cartographic projections. They can be divided 
according to two basic characteristics: 

(a) according to the nature of the distortions, and 

(b) according to the means of construction 'or the appearance 
of the normal grid. 

By normal grid we mean the coordinate system on a globe which 
is most simply represented on a map. Obviously, this is a system 
of meridians and parallels . 

Division of Projections by the Nature of the Distortions 

The choice of cartographic projections depends on the problems 
for whose solution they are intended. According to the nature of 
the elements which have the least distortion on a map, cartographic 
projections are divided into the following groups: 

1. Isogonal or conformal projections 

These projections must satisfy the requirement of equality of 
angles and similarity of figures ( conf ormability ) within the limits 
of unit areas of the Earth's surface, i.e., so that in projecting a 
surface of a globe onto a plane (map)., the angles and similar fig- 
ures do not change. 



X 



I 



b) 



11 




Fig. 1.12. Conf ormability of Figures on Maps. (a) Preserving the 
Conformability of a Unit Area; (b) Destroying the Conf ormability of 

a Long Strip . 



According to the stipulation, the angle on a map must be equal 
to the angle at the location: L & = La, but from (1.13) it is ob- 
vious that in this case m = n . 

Therefore, the equation of special scales for principal direc- 
tions is a condition for isogonality. 



On large parts of the surface, within the limits of which it is 
impossible to disregard the change in scale, the conf ormability 
(and therefore the isogonality) are not preserved. Figure 1.12 
gives an example of preserving the conf ormability of a unit area 
and destroying the conf ormability of a long strip. 



The unit area (Fig. 1.12, a) is transferred to the map on a 
definite scale without distortions. The long strip (Fig. 1.12, b) 
can be divided into a number of unit areas , each of which will be 
transferred to the map on a somewhat changed scale. Since the scales 
mx and ns are increased proportionally in the direction of the strip, 
each of the small areas is represented on the map with the conforma- 
bility being preserved, only on a different scale. By equating the 
lateral limits of the small areas , we do not obtain a conf ormal 
figure, i.e., the similarity of small figures in isogonal projec- 
tions is preserved, while the similarity of large figures (large 
lakes, seas, etc.) is destroyed. 

2. Equally spaced or equidistant projections 



/2H- 



The equivalence to unity of the special scales for a principal 
direction (m = 1 or n = 1) is a necessary condition of this group 
of projections. 



.) 



b) 



o 

L] 



CJ 



-E^ 



.•^t 



Fig. 1.13. 
Conf ormabi 
Spaced Pro 
(a) Appear 
ure in a L 
Appearance 
on a Map . 



Distortion o 
lity in Equal 
j ect ions : 
ance of a Fig 
ocation ; ( b ) 

of the Figur 



f 



This means that the map scale will 
be preserved in one of the principal 
directions. Therefore, when using such 
a map we can measure the distance in 
one of the directions by means of a 
scale. The nature of the distortion 
of conf ormability in these projections 
is shown in Fig. 1.13. Here m = const, 
while n is a function of Z. 

3. Equally large or equivalent 
projections 



This group of projections must 
satisfy the condition of equivalence 
of areas, i.e., the product of the 
special scales for the principal direc- 
tions must equal unity (mn = 1); there- 
fore, the relation between the special 
scales for the principal directions will be inversely proportional: 



1 



m> = 



These projections do not have an equivalence of angles and a 
similarity of figures. 



19 



ti 



4. Arbitrary projections 

Projections of this group do not satisfy any of the conditions 
mentioned above. However, they are also used when comparatively 
small portions of the Earth's surface are projected onto a plane 
where the distortions of the angles and the scales for the principal 
directions and along the entire map field are insignificant and the 
similarity of figures and areas which satisfy the needs of their 
practical application is preserved. This group of projections in- 
cludes a basic flight map on a scale of 1:1,000,000, which is con- 
structed according to a special law and which has been accepted by 
international agreement. 

For the purposes of aircraft navigation, the most necessary 
conditions are (obviously) isogonality and equal scale of the maps. 
Equally large and equally spaced projections of maps are used in 
aircraft navigation only as survey maps for special applications. 
They include maps of hour zones, magnetic declinations, composite /2 5 
diagrams of topographical map sheets, climatological and meteorolo- 
gical maps, etc. 

B-ivis-ion of ProQections Aoaording to the Method of Construation 
(Aaaording to the Appearance of the Normat Grid) 

Depending on the method of construction, cartographic projec- 
tions are divided into several groups, the bases of which are the 
following : 

(a) group of cylindrical projections; 

(b) group of conic projections and their variants, polyconic 
pro j ections ; 

(c) group of azimuthal projections; 

(d) group of special projections. 

Each of these projections is divided in turn into the following 
categories: normal ^ if the Earth's axis concides with the axis of 
the figure onto which the Earth's surface is projected; transverse ^ 
if the Earth's axis forms an angle of 90° with the axis of the fig- 
ure, and oblique J if the axis of the Earth does not coincide with 
the axis of the figure and intersects it at an angle which is not 
equal to 90° . 

Cylindrical Projections 

Normal (equivalent) cylindrical projection 

All cylindrical projections are formed by means of the imagin- 
ary transfer of the Earth's surface (globe) to a tangential or in- 
tersecting cylinder, with subsequent unrolling. 

In Figure 1 . 14- , a simple normal cylindrical projection is 
given, i.e., a projection of the Earth on a tangential cylinder, 

20 



the axis of which coincides with the axis 'of the Earth (globe), 
while the height of the cylinder is proportional to the length of 
the axis • 




© 

e 



Fig. 1.14. Normal (Equivalent) Cylindrical Projection 



^ 


j3zr~~~^ 
























— r* '" 










































- o- 






















^' 




































































___— --'^ 























Fig. 1.15. Simple Equally Spaced Cylindrical Projection 



In this projection, the meridians are compressed while the 
parallels are extended to a degree which increases with latitude. 
The projection includes a category of equally large and equivalent 
projections, since it satisfies the condition of an equivalence of 
areas . 



Its equation can be written in the following form: 



(1.16) 



where X represents the coordinates of a point along the meridian; 
Z represents the coordinates of a point along the equator; and R 
is the Earth's radius. 

Let us determine what the special scales for the directions 
are equal to in this projection: 



/26 



21 



m = 



dSmap rdf 



d5,globe R'if 
dS^map RdX 



R cos yrfy 
Rdf 

Rd\ 



■■ COS f ; 



''•^■qlobe '"''^ Rcosdl cosf 



= secf , 



(1.17) 
(1.18) 



where m is a partial scale along a meridian; n is a partial scale 
along a parallel; dS^-^^ is an increase in distance on the map; 
"-^•^elobe ^^ ^" increase in distance on the globe. 



The product of the special scales is 

' =1 or m = -^, whi le n = — 



mn ==cos ^1 



COS<f 



Therefore, the given projection is equal. Since m ^ n; m ^ 1 and 
n 7^ 1 in the principal directions (meridians and parallels) it is 
not isogonal and not equally spaced. Only in the equatorial band, 
in the limits from to ±5° along its latitude, is it practically 
possible to consider it isogonal and equally spaced. 

Simple equally spaced cylindrical projection 

If we take the height of a cylinder to be proportional not to 
the length of the Earth's axis, but to the length of a meridian, 
and instead of simply projecting we unfold the meridians to the 
cylinder walls, as shown in Fig. 1.15, then a simple, equally spaced 
cylindrical projection is obtained. It is regarded as normal since 
the axis of the globe coincides with the axis of the cylinder. 

In this projection, the meridians will be transformed to their 
full size during their transfer from the globe's surface to a map 
(i.e., m = 1), and the equator also will be transformed to full 
size (at the equator, n = 1), while the parallels will be extended 
just as in a normal (equivalent) projection. The magnitude of the 
effect increases with latitude. 

The coordinate grid of the map of this projection has the ap- 
pearance of a uniform rectangular ruling. Its equations have the 
form : 

x==Rr, Z^R\. 

The special scales are equal to: 



/27 



along the meridian 



along the parallel 



dS 



map 



dS 



globe 
1 

COSip 



Rdtf 
Rdf 



= 1; 



= sec y. 



(1.19) 



(1.20) 



22 



Since m = 1, the projection is equally spaced along the meri- 
dians and also along the equator. Since m ^ n and mn ^ 1, the pro- 
jection is not isogonal and not equally large, except for the equa- 
torial band in the limits from to ±5° along the latitude, where 
it is practically possible to consider it Isogonal and equally large. 

Maps in normal (equivalent) and simple, equally distant cylin- 
drical projections are used in aviation only as references: maps 
of hour zones, maps of natural light, etc. 

Isogonal cylindrical projection 

An isogonal cylindrical projection (Mercator projection) is 
the most valuable of all the cylindrical projections for navigation. 
It is obtained from a simple, equally spaced cylindrical projection 
by artificially extending the scale along the latitude (lengthening 
the meridians), proportional to the change in scale along the longi- 
tude. The coordinate grid of the map of this projection is shown 
in Figure 1.16. 

The reason for its use is the fact that the angles measured on /28 
the map are equal to the corresponding angles at the location, i.e. , 
m - n = sec(i). 



c„' W" 20''J0' 4 0' SO' SO' 70' SO' SO' Wd' IW' 120' 
"" I I I 1 1 1 — -T — 1 1 1 A~, .gg- 



50' 

40' 
30° 
20° 
10° 

W 

20' 
30' 
MO' 

SO' 



60' 



SO' 

fO' 

30' 
70' 

10° 



W 

20' 

30' 

40' 

50' 



Let us write an equa- 
tion of this map projection 
along a meridian (J-coor- 
dinate) for which we can 
find m: 

dS map ax 



''^ globe ^'^f 



where dS is an increase of 
distance along the meridian 
on the map; and Rd(^ is an 
increase in distance along 
the meridian at the loca- 
tion. 

We must have 
m = sec<p, 

We shall then equate the 
right-hand sides of these 
equations : 



10' 20' 30° «r SO' eo° jo' so' so' loo'm' bo^"' 



Fig. 1.16, Coordinate Grid of an Iso- 
gonal Cylindrical Projection. 



-^^='sec*>hencedA-=-^^, (l.21) 
Rdf ^' cos<p 



23 



I 



After integrating (1.21), we will obtain the J-coordinate 
along the meridian: 



/29 



A" = /? In tg 



(«-.f). 



(1.22) 



while the Z-coordinate along the parallel is determined by the sim- 
ple equation : 

^'=^^- (1.22a) 

Since m= n, the projection is isogonal but not equally spaced 
(m 7^ 1 and n / 1) and not equally large imn ^ 1). 

The basic advantage of maps in an isogonal cylindrical projec- 
tion is the simplicity of their use with magnetic compasses for 
moving from one point on the Earth to another, since the loxodrome 
in this projection has the appearance of a straight line. Therefore, 
the isogonal cylindrical projection has been used widely, primarily 
in marine navigation during the compilation of naval maps. 

The change in scale with latitude is a disadvantage of normal 
cylindrical projections. Here, in normal (equivalent) and simple, 
equally spaced cylindrical projections, the map scale is not identi- 
cal in the principal directions (north-south and east-west), so 
that the distance between two points in directions not parallel to 
the lines of the grating can be determined only by calculation. 

In an isogonal cylindrical projection, the map scale along the 
latitude is also variable, but at any point on the map it is identi- 
cal in the principal directions. This makes it possible to measure 
distances by means of compasses, for which a scale (varying with 
the latitude) is drawn on the western and eastern edges of the map. 
Means for measuring distances on maps with such a projection are 
indicated in manuals for marine navigation. 

Isogonal oblique cylindrical projections 

The basis for creating maps in an isogonal cylindrical projec- 
tion is a property of the Mercator projection: its isogonality. 
Such projections are used in the preparation of special flight maps 
on scales of 1:1,000,000, 1:2,000,000, and l:^, 000, 000 which are 
used in civil aviation. 

The tangential (Fig. 1.17) or intersecting (Fig. 1.18) cylinder 
is situated at such an angle to the axis of the globe that the tan- 
gent of the cylinder's surface to the globe or the intersection runs 
along the flight path. Usually the strip along the tangent does not 
extend more than 500-600 km to either side of the route (or the mid- 
dle line of the route, if it has discontinuities), while on the in- 
tersecting cone it does not extend more than 1000-1^+00 km to either 
side of the given middle line of the routes. 



24 



In practice, such flight maps are isogonal, equally spaced, 
and equally large; however, since the cylinder is in contact with 
the globe along the arc of a great circle or cuts the globe compar- 
atively close to the arc of a great, circle, the orthodrome on these 
maps will in practice be represented by a straight line. 

The distortions of lengths on flight maps of oblique tangential 
projections do not exceed 0.5%; for intersecting projections they 
do not exceed 0.8%-1.2%. 



/30 





Fig. 1.17. Isogonal Oblique 
(Tangential) Cylindrical 
Pro j ect ion . 



Fig. 1.18. Isogonal Oblique 
(Intersecting) Cylindrical 
Pro j ection . 



Isogonal transverse and cylindrical Gaussian projection 

The axis of the cylinder in Gaussian projections is perpendicu- 
lar to the axis of rotation of the Earth (globe). The construction 
of maps with this projection is similar to the construction of maps 
with oblique cylindrical projections. For example, a flight map 
on a scale of 1:1,000,000 for Leningrad-Kiev has been compiled on 
such a projection. However, on the whole, isogonal transverse cyl- 
indrical Gaussian projection is used for compiling maps on a large 
scale, where the special principles of construction are used. 

A spheroid (Earth's ellipsoid) is taken as the figure from 
which the Earth's surface is projected, while the tangential cylin- 
der on which the Earth's surface is projected has an elliptical base 
according to the form of the Earth's ellipsoid. 

The entire Earth's surface is divided by meridians into zones, 
each of which has a latitude of 6° and is projected onto its own 
cylinder which is tangential to the Earth's surface along the mid- 
dle meridian of the given zone. 

Thus, in order to project the whole surface of the Earth, it 
is necessary to turn the elliptical cylinder mentally around the 



25 



axis of the Earth's ellipsoid through 6° at a time, In Figure 1.19, 
a, the projection of only one zone for 6° longitude is shown, while 
in Figure 1.19, b, the unrolling of a semicyllnder after its rota- 
tion around the Earth's axis in order to project several zones is 
shown. With such a projection, all maps are constructed on the 
scales: 1:500,000, 1:200,000, 1:100,000, 1:50,000, and 1:25,000. 
The latter are essentially charts. 



/31 



V- 0' 3' 3' 15' 21' 27° 33' 39' 




Fig. 1.19. Isogonal Transverse-Cylindrical Gaussian Projection. 

Each zone on maps with a scale of 1:200,000 and larger has its 
own special X and J(Z) rectangular coordinate system, which is 
called the Gaussian kilometer system. Meridians and parallels on 
maps of this projection are curved lines and do not coincide with 
the Gaussian system. The vertical lines of the rectangular Gaus- 
sian system are parallel to the central meridian of the zone and do 
not coincide with other meridians of the zone. 

The angle between the vertical line X of the Gaussian system 
and the line to the object (point) is called the d'ivect'tonal angle. 
In order to obtain the true or magnetic direction (angle), the 
angles of the convergence of the system with the true and magnetic 
meridians are indicated on the edge of the map. In addition, the 
vertical section of a map (frame) always runs in the direction of 
the true meridian. 

By means of the Gaussian system and figures in the frames of 
the maps, it is possible to determine the distance from the equator 
and from the central meridian of the zone to the object (point). 
Distortions of lengths on these maps are insignificant and do not 
exceed 0.14-% along the edges of the zone in the latitude which is 
equal to zero (l^lO m at 100 km). 

Maps on an isogonal transverse-cylindrical Gaussian projection 
are used both in aviation for a detailed orientation and location 



26 



of targets , and in many branches of the national economy for linking 
projects, equipment, and radio engineering facilities in a location, /32 
for determining geodesic reference points, and for accurate geodesic 
calculations of distances and directions, etc. 

Conic Projections 

Conic projections are constructed by projecting the surface of 
the Earth's spheroid (globe) on a tangent or intersecting cone, with 
its subsequent unrolling to form a plane surface (Fig. 1.20, a). 



°' /\ 


N 

N J 


/- 


K. 


/^ 


~~>v 


/,trr: 


4~~] 





/ ^ 


\_ 


_y 



b) 



Pn 




Fig. 1.20. Construction of Conic Projections: (a) Tangent (inter- 
secting) cone; (b) Unrolling of the Cone to Form a Plane. 

According to the positions of the axes of the globe and cone, 
conic projections can be normal, transverse, and oblique. However, 
in our publications normal projections are generally used when the 
axis of the cone coincides with the axis of the globe. 

In a normal conic projection, meridians are represented by 
straight lines, while parallels are represented by arcs of concen- 
tric circles (Fig. 1.20, b). 

From Figure 1.20, a, it is easy to see that the radius of a 
parallel of tangency (Pq) can be expressed by the Earth's radius: 

po = /? ctg <po. 

where R is the radius of the Earth (globe) and tj) q is the latitude 
of the parallel of tangency. 



form ; 



The equation of this projection is written in the following 



P = Po + ^ ("Po + t); 



Translator's note: ctg = cot. 



(1.23) 



27 



It 



where 6 and p are the principal directions in the polar coordinate 
system along the parallel and meridian, respectively, and a is the 
coefficient for the angle of convergence of the meridians. 

Simple normal conic projection 

A simple normal conic projection is constructed with the con- 
sideration that the meridians on the whole map and the parallel of 
tangency be transferred from a globe without distortions to their 
natural value (i.e., m = 1), while for the parallels of tangency 
((fiQ ) w = n = 1. 

Such a projection forms the basis of the improved intersecting 
conic Kavrayskiy projection (Fig. 1.21, a). It is equally spaced, 
since m - 1, while on the intersecting parallels it is isogonal and 
equally large (Fig. 1.21, b). 



/33 



^^ Inte rsection p aral lels 




Fig. 1.21. 



Simple Normal Conic Projection, 
(b) Unfolding of the Cone to 



(a) Intersecting Cone; 
a Plane . 



Many aircraft maps with scales of 1:2,500,000, 1:2,000,000, 
and even 1:1,500,000, which are used in aircraft navigation for 
general orientation and the approximate determination of the posi- 
tion of an aircraft by means of radio engineering facilities (air- 
craft radio compasses, ground radiogoniometers, etc.), have been 
published . 

Their positive feature if the insignificant distortion of 
lengths in the strip ±5° from the intersecting parallels, which 
does not exceed 0.34% (340 m for 100 km). Their disadvantage is 
the distortion of directions , which increases with distance from 
the intersecting parallels. 



28 



Isogonal conic projection 

By analogy with the construction of an isogonal cylindrical 
Mercator projection, destroying the equal spacing, a simple normal 
conic projection is transformed into an isogonal projection by re- 
ducing (equating) the scale along the meridians to the scale along 
the parallels (m = n) . This is more valuable for use in aviation. 

Aircraft maps with a scale of 1:2,000,000 and survey maps on 
scales of 1:3,000,000, 1:4,000,000, and 1:5,000,000 are published 
with a normal isogonal conic projection for aviation. 

Maps with a scale of 1:2,000,000 in this projection, besides 
having the basic advantage of isogonality, also have distortions of 
length which are permissible in the practice of aircraft navigation. 
On an intersecting cone in a strip from 40° to 70° in latitude, the 
maximum length distortions do not exceed ±1.8 km for 100 km. 



/34 




Fig. 1.22. Angle of Convergence of the Meridians of a Tangent Conic 
Projection: (a) Arc of a Parallel on a Globe; (b) Arc of a Parallel 

on a Map. 

The orthodrome on maps of an isogonal conic projection for dis- 
tances up to 1200 km appears as a practically straight line. This 
valuable quality is used during flights on civil aviation airlines 
of average length by using gyroscopic and astronomical compasses 
for following the orthodrome. At great distances, the orthodrome 
(as a result of a change in scale) is bent by a bulge tending to- 
ward a larger scale. 

The loxodrome is represented by an arc of a logarithmic spiral. 



29 



This creates dif f iculties • in aircraft navigation by means of magnetic 
compasses. In these instances, for distances up to 500-800 km in 
directions which intersect the meridians on a map, a straight line 
is constructed, while measurement of the flight angle is carried 
out along the central meridian of the route which is maintained in 
flight by means of a magnetic compass. 

It is also possible to construct (continue) the loxodrome along 
an angle measured in the middle of the straight line joining the /35 
control (rotating) landmarks of the route. 

The disadvantage of all maps with conic projections is the pre- 
sence of an angle of convergence of the meridians from the parallels 
of tangency (parallels of intersection) to the pole. It is neces- 
sary to consider this angle when determining directions (flight 
angles) or the location of the aircraft by means of aircraft radio 
compasses. In addition, depending on the parallels of intersection 
or tangency, the angle of convergence of the meridians will be dif- 
ferent . 

Convergence angle of the meridians 

The principal scale of conic projections is taken along the 
meridians and parallels of tangency or intersection ( (j) q ) • There- 
fore, the arc MU is equal to the arc Mi^i (Fig. 1.22). It is known 
that on a globe (spheroid) (Fig. 1.22, a), the arc MN - rAA , where 
T is the radius of the parallel. On a map of a conic projection 
(Fig. 1.22, b) the arc M^^^ = PqAS; then 



rAX = poA8. 



(1.24) 



But T = R cos (j) and p q = -R cot cjiQ, and from the equation of a conic 
projection (1.23), A6 = aAX. 

Substituting the values of r, pq, and A6 in (1.24) and carrying 
out the necessary reductions, we obtain: 



0= iinipo. 



(1.25) 



Obviously, on the equator the coefficient of convergence of 
the meridians a = 0, since sin 0° = 0; at the poles a = 1, since 
sin 90° = 1, and in the general case for central latitudes, 0<a<l. 

Knowing the coefficient a, it is not difficult to determine 
any angle of convergence of the meridians 6 along a parallel of 
tangency or intersection : 

* = ^^*' (1.26) 

where AX is the difference in longitude between the given meridians. 
At any other latitude, the coefficient a will be different from 



30 



the coefficient a at a latitude of tangency (intersection). There- 
fore, for approximate calculations in the practice of aircraft navi- 
gation during the determination of flight angles or location of the 
aircraft, the mean latitude of the route, part of the route, or the 
distance between the aircraft and the radio station, is taken as 



6 = (A2-Xi) sincfi^^^ or 6 = (Xr.-Xa) 



sine 



mid 



where X2 ^''^'^ ^l s^e the longitudes of the final and initial points, 
Xp and Xg are the longitudes of the radio station and aircraft , re- 
spectively, and (|>niid is the middle latitude between the indicated 
points (places). 

In some cases , for approximate determinations of the location 
of an aircraft or the flight angles, the coefficient a is assumed 
constant for a given map of a conic projection. Thus, for example, 
for a map with a scale of 1:2,000,000 and a normal isogonal conic 
projection, it is possible to let a (w 0.8, which corresponds to the 
sine of the latitude of the middle parallel between the intersection 
parallels, where the map scale will be minimum. 

Polycom ic projections 

Polyconic (multiconic) projections are the greatest perfection 
of conic projections for the purpose of decreasing distortions of 
lengths and angles in projecting the Earth's surface onto a plane. 

The principle of construction of such projections is shown in 
Figure 1.23, a. The central meridian of the projections is a 



/36 





Fig. 1.23. Polyconic Projection: (a) Intersecting Cones on the 
Globe; (b) Unrolling of Cones on a Plane. 



31 



straight line, while meridians in the form of curved lines are sit- 
uated to the west and east of it. The parallels are concentric 
circles with different centers, lying on the central meridian (Fig. 
1.23, b). As a result of the increase in scale in proportion to 
the distance from the central meridian to the west and east, such 
projections are used only to represent the Earth's surface in coun- 
tires extended along a meridian. 

International projection 

In terms of the method of construction, an international pro- 
jection is related to a modified polyconic projection; in terms of 
the nature of the distortions , it is related to an arbitrary pro- 
j ect ion , 



E^ 
h° of 
to 64°; 
for alj 
the pr; 
as a Ti 
parall« 
central 
1 (Fig. 
sheet c 
cipal £ 
of the 



map wi 
6° of 1 

icted ac 
of a gi 

i is giv 
interse 
= 1) an 

' the sh 
In the 
of long 

56 sheet 

igh 4°) 



range 
itude ; 
s is ; 
and a'. 



1,000,000, which encompasses 
a range of latitudes from 
:s own law, which is general 
lal strip. On each sheet, 
outer parallels of the sheet 
globe by a cone along these 
leridlans , separated from the 
the west and east, where m = 
Etudes from 64° to 80° , each 
-om 80 to 88°, 24°. The prin- 
^en along the outer parallels 
; meridians which are distant 



/37 



—7—2 


=/ 
















L^ 


■/ 






^6' -"' ■^^^zmsiiilr^rwmB'^''^ 



Fig. 1.24. International Projection: (a) Construction of the Sheet; 
(b) Breaks in the Splicing of Sheets. 



32 



from the central meridian of the sheet by 1+ and 8°, respectively. 
The regions of the poles are projected onto separate sheets in a 
central (polar) projection. 

The meridians in this projection are represented by straight 
lines which have an angle of convergence to the poles, similar to 
the conic projections, while the parallels are curved lines which 
are constructed according to a special mathematical law. The centers 
of the circle-parallels are situated on the central meridian of a 
given sample of sheets, while their radii are proportional to the 
cotangents of the intersection latitudes: 

Ri = ctg 9i: R2 = ctg 92 e t c •., 

According to studies by Limnitskiy, distortions of lengths on 
maps with a scale of 1:1,000,000 with such a projection, in the mid- 
dle latitudes does not exceed 0.076% (76 m in 100 km), while distor- 
tion of directions is 5'. The greatest distortions arise in the 
region of the equator: distortion of lengths up to 0.1'+%, angles 
up to 7' . 



Insignificant distortions make it possible to consider the map 
as a practically isogonal, equally spaced, and with equally large 
pro j ectlon . 

f maps with scales of 
d. In a range of latl- 
th a scale of 1:2,000,000 
ude (nine sheets of a 
e of 1:4,000,000 occupies 
cale of a 1:2,000,000 map 
sheet and the meridians 
of the sheet by 6° to the 
ap with a scale of 
t by 8°50' to the north 
Istortions , and the meri- 
e central parallel and 





According to this principle, s 


heets 


1 : 


2 ,000 ,000 


and 1:4,000,000 are constructe 


tudes from 


to 64° , the 


sheet of a 


map wi 


occupies 12° 


of latitude 


and 18° of 


longit 


mi 


llionth , 3 


X 3) , while 


a map with 


a seal 


24 


and 36° , 


respectively . 


The principal s 


is 


given along the outer 


parallels 


of the 


wh 


ich are di 


stant from th 


e central 


median 


we 


St and eas 


t (Fig. 1.24, 


b ) , while 


on a m 


1 : 


4 ,000 ,000 


the parallels 


which are 


distan 


an 


d 8°10' to 


the south are given wi 


thout d 


dians are 12 


° to the west 


and east 


from th 


th 


e central 


meridian , res 


pectively . 





/38 



The distortion of lengths in the middle latitudes on maps with 
a scale of 1:2,000,000 reaches 0.5%, and the distortion of the an- 
gles is 30'; on 1:4,000,000 maps, distortion of lengths reaches 1.5%, 
that of angles, 1°30'. 

A disadvantage of maps in the international projection on all 
scales is the presence of discontinuities in the splicing of several 
sheets, as a result of the features of its construction. Sheets of 
maps of only one strip or one column are spliced without breaks. 
During splicing of nine sheets of maps on a scale of 1:1,000,000 
(3 X 3), the discontinuities which arise are partially evened out 
by deformation of the paper, and the use of such a map does not re- 
sult in perceptible distortions of lengths and angles. Splicing of 
a large number of sheets is not recommended. 



33 



It is even impossible to splice a map with a scale of 1:2,000, 
000 from four sheets (2 x-2) without a break. At a latitude of 60°, 
the discontinuity of the spliced sheets reaches 1.8 cm, i.e., 36 km 
(see Fig. 1.24, b). Therefore, it is possible to splice only one 
strip or one column of these maps. 

The orthodrome with a length up to 1200 km on, maps with a 
scale of 1:1,000,000 and 1:2,000,000 (within the limits of one 
sheet) appears in practice as a straight line, while the loxodrome 
is the arc of a logarithmic spiral. Usually, in directions which 
intersect the meridians, the loxodrome sections with a length up to 
600 km are likewise constructed in the form of a straight line, while 
the flight angle is measured in the middle of a part of a route in 
order to lessen by a factor of 2 the error of the measured angle 
during flight with the use of a magnetic compass. 

During the determination of the position of an aircraft by 
means of radio compasses , a correction is allowed for the convergence 
of the meridians just as in maps of conic projections, with an ap- 
proximate formula 

^ f a "mid 

where X-^ is the longitude of the radio station; Xg^ is the longitude 
of the aircraft; "^mid is the mean latitude between the radio station 
and aircraft, or the mean latitude of the sheet (sheets) if the ap- 
proximate position of the aircraft is unknown. 

In civil aviation, maps with a scale of 1:1,000,000 and 1:2, /39 
000,000 on an international projection are used as flight maps, pri- ' 
marily on piston-engine aircraft and helicopters , and secondly on 
aircraft with gas-turbine engines. Maps with a scale of 1:4,000, 
000 are used as aircraft maps for general orientation and approxi- 
mate determination of the location of an aircraft by means of radio- 
engineering facilities. 

Azimuthal (Perspective) Projections 

Azimuthal (perspective) projections are constructed according 
to the laws of a simple geometric perspective; therefore, they are 
often called perspective projections. 

According to the position of the plane of the figure, azimuthal 
projections are divided into normal or polar (Fig. 1.25, a), trans- 
verse or equatorial (Fig. 1.25, b), and oblique or horizontal (Fig. 
1.25, c); depending on the position of the center of the projection 
relative to the plane of the figure, they can be of the following 
types (Fig. 1.26): 

a) Centval or gnomonioj when the center of the projection co- 
incides with the center of the Earth (globe): point A; 



31+ 



b) Steviogvaphio y when the center of the projection is sepa- 
rated from the point of contact with the plane of the figure by a 
distance equal to the diameter of the Earth (globe): point B; 

c) Orthographic J when the center of the projection is infi- 
nitely separated from the plane of the figure: point Cj 

d) External J when the center of the projection is located 
above the plane of the figure: point D. 



Plane of the'figure 




Fig. 1.25. Azimuthal Projections: (a) Normal; (b) Transverse; 

( c ) Oblique . 



PI ane 
the f 




Fig. 1.26. Position of the 
Centers of Projection in 
Azimuthal Projections. 



As is evident from Fig. 1.26, 
on such projections points M and N 
on the Earth's surface will be pro- 
jected at a different distance from 
the point of tangency of the plane 
of the figure with the Earth's sur- 
face . 

Meridians in azimuthal (polar) 
projections are represented by 
straight lines which converge to a 
pole at an angle equal to the dif- 
ference in longitude: 6 = AX. 

Parallels are represented by 
concentric circles, the radii of 
which depend on the center of the 
projection and the latitude of the 
position . 

In aviation, central polar and 
stereographic polar projections are 
generally used. 



/i+0 



35 



Central polar (gnomonic projection) 

The center of projection in this projection coincides with the 
center of the Earth (globe) at the point (Fig. 1.27, a). 

From Figure 1.27, it is possible to write the equation of this 
pro j action : 

p = /?ctg<i>. 



In order to have a complete idea of the projection, let us find 
the special scales (m, n) for the principal directions (meridians 
and parallels): 

dSmap 



_ -dp _ -d(Rctgf) 



where dp is the increase in the radius of the unrolling, i.e., a 
positive increase in latitude ( (j) ) corresponds to a negative increase /M-1 
in the radius (p). Integrating the latter, we obtain: 

__±Rdf 1 



Rd<f sec2 <f 



[.sec2,y 



or 



ft = 



Here , r = i? cos 



dS 



mcosec2<p; 
map priB /? ctg <fd\ ctgy 1 



"^^globe '■'^^ RcosfdX cosiy sinip ' 

is the radius of the parallel, i.e., 
n = cosec <p.. 



(1.27) 



(1.28) 



a) Plane of 
the F i gu re 




Fig. 1.27. Central Polar (Gnomonic) Projection: (a) Position of 
the Plane of the Figure; (b) Appearance of the Projection. 

Translator's note: cosec = esc. 

36 



Therefore, the projection is not isogonal {m ^ n) , not equally- 
spaced (m 7^ 1 and n 7^ 1 ) , and not equally large {mn ^ 1). 

Although the projection is not isogonal, the orthodrome on it 
is represented by a straight line. This remarkable property is ex- 
plained by the fact that the plane of the circumference of a great 
circle (plane of the orthodrome) always passes through the center 
of the Earth, which in this case appears -^s the center of the pro- 
jection, while the intersection of the plane of a great circle with 
the plane of the figure is a straight line. 

Since the projection is not isogonal, the moving azimuth of 
the orthodrome on it, if it is not equal to 0, 180, 90, or 270°, 
does not correspond to the azimuth on the Earth's surface. 

Distortion of directions on the map will be equal: /42 

. „ n cosec 9 c t o o ^ 

tgP = tga = ^ (g a != sin <p tg a, (.1.29) 

m cosec2()) t & > 

while it is possible to calculate the actual direction of the ortho- 
drome at the location analogously with the aid of the measured angle 
on the map : 



m 

~ ~ir '^^~ cosec 9-tgp, 



(1.30) 



where 3 is the measured angle on the map of a given projection, a 
is the corresponding angle in a location, and cj) is the latitude of 
the final point of the orthodrome. 

The distortions of directions and distances on this projection 

are great. In this connection, it is impossible to use a protractor 

to measure the directions and a scale to measure the distances on 
the map without corresponding corrections. 

A central polar projection is used for constructing gnomonic 
systems, while the regularity in the distortion of directions is 
used for calculating the nomograms of the orthodrome direction. 

The gnomonic system and the nomogram of the orthodrome direc- 
tion can be used for the graphic (approximate) calculation of the 
length of the orthodrome , the coordinates of its intermediate points . 
and the direction. The loxodrome and other lines of position of 
the given projection are represented by complex curves. 

The property of orthodromicity of a central polar projection 
has been used for the publishing of oblique central projections 
which have been used at radiogoniometric points in civil aviation. 
The middle of the base (the middle of the orthodrome distance be- 
tween two radiogoniometers) was taken as the point of tangency of 
the plane of the figure of such maps. In this case, the coordinates 

37 



of the position of the aircraft are very easily defined as the in- 
tersection of two straight orthodrome bearings (lines) extended 
from the radiogoniometers . 

Maps of the differential rangefinding (hyperbolic) system of 
long-range navigation (DSLN-1) are made on such a projection, since 
the spherical hyperbola on the projection is also expressed by a 
hyperbola . 

Equally spaced azimuthal (central) projection 

This projection is constructed by calculating and transforming 
conventional meridians (radii) to full size, equal to the principal 
scale transferred from the globe. The projection is used only for 
the publication of special small-scale maps (1:40,000,000), which 
are used as reference maps for measuring distances from a central 
point on the map . 

Usually a large administrative or aviation center, from which 
it is necessary to know the shortest orthodrome distance in any di- 
rection to a given point on a map, is chosen as the point of tan- 
gency of the plane of the figure of the projection. In such a pro- 
jection, for example, a map is constructed with the point of tan- 
gency at Vnukovo Airport , with circles plotted at equal distances 
from the airport. The geographic meridians and parallels are repre- 
sented by complex curves. This does not allow the directions to be 
measured . 

Stereographic polar projection 

The center of projection in a stereographic polar projection 



/43 



a) Plane of 
the Figure 




Fig. 1.28. Stereographic Polar Projection: (a) Position of the 
Point of Projection; (b) Appearance of the Projection. 



38 



iiiiimii 1 1 III 



is separated from the point of tangency of the plane of the figure 
by two radii of the globe at the point B (Fig. 1.28, a). 

Here the angle 6 = 90° - <() , while the angle 

6 90-? 



An equation of the projection can be derived from equations of 
the elements shown in Figure 1.28. 

R = X; 

6 

The meridians in the projection are straight lines which di- 
verge radially from the pole (Fig. 1,28, b), and from the point of 
tangency of the plane of the paper at an angle equal to the differ- 
ence in longitude: 6 = AX. 

The parallels are concentric circles, whose radii are propor- 
tional to the tangent of the latitude. 



/44 



The special scale along the parallel is determined by the equa- 



tion 



dS 



map 



dp 



2Rfltg — 



''^qlobe -'^''V Rd(9(y^9) '• 



Here (j) = 90° - 9 , while after integration 



1 



: sec • 



26 



cos2 • 



(1.30) 



The special scale along the parallel is determined by the equa- 



tion 



dS map prfX 



2RiS~ 



''•^globe 



rdX R cos <f 



but cos (j) = cos (90 - 6) = sin 6, so that 

e 



sine 



= sec2_. 



(1.31) 



1 . e 



m — n— sec2 



9 J 90-<p \ 

Y=sec2(-^). 



39 



II 

n iiiiiii II iiiiiiiniiiiiinii 



■iin iniiiiii iiiiiiii I Mil 



Therefore, this projection is isogonal (m - n) , but not equally 
spaced {m ^ 1 and n ^ 1) or equally large imn ^ 1). 

On maps of a stereographic projection, a circle drawn on the 
globe is represented by a circle on the plane (map); however, the 
center of this circle does not coincide with the projection of the 
center of the circle on the globe. This makes the projection in- 
effective for use in rangefinding systems, since lines of equal 
length will be represented by eccentric circles. 




The maximum distortion of lengths at 70° latitude does not ex- 
ceed 3% (3 km in 100 km), whereas if the plane of the figure is in- 
tersected (for example, at 70° latitude), the distortion of the 
lengths at the poles does not exceed 3% (and at 60° latitude, 4%). 

The orthodrome on maps of a stereographic projection has an /45 
insignificant bend toward the equator and is constructed in prac- 
tice as a straight line. 

The loxodrome is represented by a logarithmic spiral. It is 
possible to continue it (just as in conic projections) along the 
flight angle, which is measured in the middle of the part of the 
straight line joining the control (rotating) points of the flight 
path . 

In determining the position of an aircraft by means of an air- 
craft radio compass, a correction for the angle of convergence of 
the meridians is allowed according to the formula 

where X^ and A^ are the longitude of the radio station and the air- 
craft, respectively. 

On maps of a stereographic projection, in order to facilitate 
determining directions in the polar regions according to a sugges- 
tion by V. I. Akkuratov, a supplementary system of "arbitrary" mer- 
idians (Fig. 1.28, b) parallel to the Greenwich meridian (A = 0°) 
and perpendicular to it (A = 90°) is plotted. Then the true Green- 
wich flight angle will equal: 

TFAcr = TFAar ± A^ 

where TFA^r is an arbitrary flight angle measured from a direction 
perpendicular to the Greenwich meridian (A = 90°); A^ = 90° is the 

i+0 



location of the route (part of the path) to the east of the Green- 
wich meridian; and A^ = 270° is the location of the route (part of 
the path) to the west of the Greenwich meridian. 

Nomenclature of Maps 

At the present time, a map with a scale of 1:1,000,000 (1 cm = 
1 km) which is executed in an international projection is considered 
the basic topographical map of the world. As described above, each 
sheet of this map encompasses a territory within the limits of ^■° 
of latitude and 6° of longitude. This has made it possible to com- 
pile an international designation for the sheets of maps. 

For the purpose of quickly choosing a given sheet of a map, 
each of them bears a designation of its rank in a definite system. 

This designation is called international map nomenclature . 



Th 



le sheets are situated in rows along parallels which run f 
lator to a latitude of 8"+° . There are a total of 21 rows 
imisphere. Each row is designated by a letter in the Lati 
;t: A, B, C, D, E, F, G, H, I, J, K, L, M, N, 0, P, Q, R, 
laps for latitudes greater than 84° are constructed in per 
■e projections). 



Each sheet of a row has an ordinal number from 1 to 60. Count- 
ing of the sheets begins from the 180th meridian and proceeds from 
west to east. The map sheets referring to the prime (Greenwich) 
meridian from the east have the ordinal number 31. Thus, columns 
of map numbers are obtained. 

To choose the necessary map sheet, it is necessary to know the 
approximate coordinates of the point for whose region the sheet is 
selected . 



/46 



69° E 



For example: the point coordinates latitude 50° N, longitude 



Let us divide the latitude of the point by 4, and we will ob- 
tain the necessary row of map sheets: 5 v '+ > 12. Therefore, the 
map sheet is located in the thirteenth row, which is designated by 
the letter M. 

Let us divide the longitude of the point by 6 , and we will ob- 
tain: 69 V 6 > 11. The ordinal number of the sheet will then be 
30 + 12 = 42. 

For convenience in selecting map sheets, composite tables have 
been constructed. These tables are executed on small-scale maps 
with a straight, equally spaced cylindrical projection by ruling 
the indicated map every 4 degrees in latitude and every 6 degrees 
in longitude, with corresponding designations showing the rows and 
columns of ordinal numbers of the maps (Supplement 1). 



41 



In addition, on the face of each map sheet is a diagram showing 
how the given sheet fits' to the adjoining one (Fig-.- 1.29). The 
sheet on which this diagram is drawn fits in the middle and is 
shaded . 

Sheets of maps with larger scales 
have standard schemes qf arrangement with- 
in the limits of a sheet with a scale of 
1:1,000 ,000. 

For example, a map sheet with a scale 
of 1:1,000,000 contains 4 map sheets with 
a scale of 1:500,000, which are designated 
by letters of the Russian alphabet: A, 
B , C , and D . 

By an analogous method, the division 
of a map sheet with a scale of 1:1,000, 
000 into sheets with larger scales is car- 
ried out. Roman and Arabic numerals are 
used for their designation. Here the map 
nomenclature retains the designation of the sheets in the initial 
division, beginning with a scale of 1:1,000,000 and up (Fig. 1.30). 

The nomenclature of map sheets with small scales (1:2,000,000, 
1:2,500,000, and 1:4,000,000) is not international and is established 
when they are printed in accordance with the regions for which they 
are published and in accordance with the dimensions of the map sheets. 




Fig. 1.29. Scheme for 
Splicing Map Sheets 
with an International 
Pro j ect ion . 



Maps Used for Aircraft Navigation ' /M-? 

Depending on the nature of the tasks to be fulfilled, it is pos- 
sible to divide maps into several groups according to their scales. 

1) Maps with detailed orientation, with scales of 1:500,000 
and up, are used in civil aviation during flights for special pur- 
poses (geomagnetic mapping and photography, chemical treatment of 
areas, searching for small objects in the execution of special tasks, 
"joining" of radio engineering projects in airport regions, compi- 
lation of diagrams for piercing clouds, and for other purposes). 

2) Flight maps with scales of 1:2,000,000, 1:1,000,000, and 
1:500,000 are used in civil aviation as basic flight maps. Crews 
of light aircraft and helicopters at comparatively low speeds use 

maps with scales of 1:1,000,000 and 1:500,000, while crews of high- /48 
speed aircraft use maps with scales of 1:2,000,000 and 1:1,000,000. 

3) Aircraft maps with scales of 1:4,000,000, 1:3,000,000, 
1:2,500,000, and 1:2,000,000 are used in civil aviation for general 
orientation and plotting of position lines with the aid of radio- 
engineering and astronomical facilities. For these purposes, crews 
of light aircraft at low speeds and helicopters use maps with only 
the last two scales. 



42 



h) Special maps with scales of 1:40,000,000 and up (to 
1:2,000,^000), with special emphasis on different purposes of appli- 
cation: lines of equal distance from definite points, a hyperbolic 
system, azimuths from radio-engineering installations, etc. are used, 
These include maps with reference materials of smaller scales: maps 
with time zones, magnetic declinations, composite tables of map 
sheets , etc . 



//-4/ 




t'ltOOOO 



1-200000 



PZOOOOO 



1:500000 (N-it1-8) 



Fig. 1.30. Scheme for Dividing a Map Sheet with an International 

Pro j ect ion . 

Also, special flight maps with scales of 1:1,000,000 and 
1:2,000,000 with plotted and marked flight routes are published for 
civil aviation. As a rule, they are compiled on oblique cylindrical 
or oblique conic projections, with the least distortions of angles 
and lengths along the route. The orthodrome on such maps is prac- 
tically a straight line. 

The contents of a map depend on its scale, the aerographlc fea- 
tures of the regions for which it is compiled, and the purpose of 
the map . 



43 



On maps of all scales , the following are drawn in some kind of 
detail : 

(a) relief; 

(b) hydrography (seas, rivers, lakes); 

(c) populated points; 

(d) network of railroads, highways, and country roads; 

(e) vegetation or ground cover (large forests, meadow, swamp, 
sand, desert, etc.); 

(f) isolines of magnetic declinations and magnetic anomalies. 

The legends of the indicated elements are usually executed on 
the maps at the lower edge of the sheet. 

On maps, a relief is expressed by three methods: 

1) It is expressed by isolines of equal height on the surface 
of the relief (horizontals), i.e., lines formed at the intersection 
of a relief with horizontal planes which are situated one above the 
other, with height intervals depending on the scale of the map; the 
height of the horizontals above sea level is designated by numbers. 

2) It is expressed by layered coloring; a special color desig- 
nated on a special (hypsometric) scale on the lower edge of the map 
is assigned to each interval of relief height. 

3) It is expressed by brown shading, i.e., by special coloring 
with thickening of brown in the highest areas of the relief and the 
steepest slopes. This use of color gives a natural, volumetric idea /^9 
of the nature of the relief. 

In addition to the above methods of representing relief on maps, 
marks of command heights (which exceed neighboring heights), with an 
indication of the height of these points above sea level, are shown. 

Hydrography is shown on maps by a blue color. Its detail de- 
pends on the scale and purpose of the map. 

Populated points , depending on the scale of the map and the 
areal dimensions of the points , are represented by contours or con- 
ventional symbols in accordance with the point's dimensions or its 
population . 

In lightly populated areas, all populated points are designated. 
On small-scale maps of densely populated areas, some of the points 
are omitted. The number of points drawn depends on the scale of 
the map and the population density of the area. 

The detail of the highway network depends on its density, the 
vegetative or ground cover, and the scale of the map and its purpose. 

Besides the above general contents of maps, specially prepared 



414 



flight maps represent a navigational situation, i.e., the arrange- 
ment of radio-engineering facilities for aircraft navigation, posi- 
tion lines of aircraft, and special markings for navigational meas- 
urements and calculations are shown. 

On some forms of specially prepared maps (map-diagrams), some 
of the elements of the general contents are omitted or simplified 
for the purpose of a more detailed and graphic representation of 
the navigational situation. 

6. Measuring Directions and Distances on the Earth's Surface 

Orthodrome on the Earth's Surface 

In the practice of aircraft navigation at the present time, an 
orthodrome direction is the main and most widespread direction. 

In order to explain all the problems connected with measuring 
moving angles, distances, and coordinates in flight along an ortho- 
drome, let us examine an orthodrome on the Earth's surface (Fig. 
1. 31) . 

An orthodrome, in general, lies at an angle to the Earth's 
equator and intersects it at two points, the distance between which 
(along the arc of the equator) is equal to 180°. Only the equator, 
which likewise appears as an orthodrome, is an exception. 

In Figure 1.31, a and b, line XqMi is the arc of the equator, 
line \qM is the orthodrome examined by us; points \q and Xq + 180 
are the points of intersection of the orthodrome with the equator; 
Pj^AqP^ is the meridian of the point of intersection of the ortho- 
drome with the equator; P^MMiP^ is the meridian of the point M on 



/50 





Fig. 1.31. Orthodrome on the Earth's Surface: (a) Position of the 
Orthodrome on a Sphere; (b) Relationship between Longitudes and Lat- 
itudes of Points on the Orthodrome. 



^5 



the orthodrome ; 90° - ag is the angle between the plane of the equa- 
tor and the plane of the orthodrome ; X is the londigude of the point 
M ; (j> is the latitude of the point M ; -B , +B are points on the ortho- 
drome of maximum latitude, which are called vertex points. 

Let us erect a normal to the plane 
of the equator at point M-^ (see Fig. 
1.31, b) and extend it to an intersection 
with the vertical of point M on the or- 
thodrome (point M2 ) . It is obvious that 
the triangle M1M2 will be a right tri- 
angle. Here M1M2 will be the tangent 
line of the angle ({' • 

Let us drop from points M^ and M2 , 
perpendiculars to the aperture axis of 
the orthodrome with the equator XgO. One 
of them will lie in the plane of the 
equator, the second in the plane of the 
orthodrome ; both will converge at one 
point on the aperture axis (point N). 




Fig. 1.32. Determining 
Distance on an Ortho- 
drome . 



It is obvious that line MiN will be 
a line of the sine of angle X, while an- 
gle M1NM2 will be the aperture angle of the plane of the equator 
with the plane of the orthodrome (90° - mq). Here the triangle 
NM1M2 will also be a right triangle. 

Thus, for point M and for any point on the orthodrome, the 
following equation will be valid: 

tgy 



/51 



tgOO'-Oo)- 



slnX 



or 



tg oo =i ■ 



sin X 
tg<P 



(1.32) 



Formula (1.32) is valid only for cases when the point Xq is 
the point of origin of the longitude. When the longitude of the 
point is not equal to zero, the longitude of the point Xq must be 
subtracted from the longitude of the point M(Xj^), i.e., the refer- 
ence system of longitudes must be reduced to this point. Then 



'g<»o= 



sin ( K„ — Xq) 
tg9« 



(1.32a) 



In the future, for the sake of simplicity, we will consider 
the longitude of Xq equal to zero. 

It is possible to determine the moving azimuth according to 

the formula 

tgo = tgdosecXsectp, (1.33) 



46 



Considering that tg cxq = — r- , it is possible to reduce (1.33) to 

the form: ^ ^ 



tga=^tg\cosec<f 
or ctga= ctgXsirt?. J (1.33a) 



'1 



Formulas (1.33) and (l.33a) are obtained by differentiation 
of (1.32). 

Since the ratio — r = tg an = const remains valid for every 

tg ^ ^ " ^ 

length of an orthodrome , it is obvious that the elementary differ- 
ence quotient sin X and tg <j) will also be constant for every length 
of an orthodrome and will equal: 

rfsinX 

:= tg oo = const. 



dtg<f 
Therefore, it is possible to write (1.32) in the form: 



dtgf d<f 

df 

whence cosX d\ 

sec2? ' d<f ~*^'^ 

or 

dif cos X cos3 9 

On the Earth's surface, the linear scale of longitude is equal /52 
to the linear scale of latitude multiplied by the cosine of lati- 
tude. Therefore, the tangent of the moving azimuth of the ortho- 
drome will be expressed by the derivative ^^ , divided by the co- 
sine of the latitude: ^j^ "-^ 

tga== "^ = 'g°° 

COS If cos X C0S2 <j) 

sinX 
or, considering that tgao= ^„ > 

we arrive at (1.33a): tg a = tg X cosec ?. 

In the practice of aircraft navigation, it is usually neces- 
sary to deal v/ith two points on the Earth's surface. With the ex- 
ception of special cases, neither of them is on the equator. 

Formulas (1.32) and (1.33) can be used only in those cases when 
the point of intersection of an orthodrome with the equator (i.e., 

^■7 



the longitude of a point on the orthodrome , the width of which is 
equal to zero) is known. 




Fig. 1.33. Elements of a Spherical Triangle. (a) Triangle on a 
Sphere; (b) Relationship of Angles and Sides of a Spherical Triangle, 

Let us derive an equation which makes it possible to determine 
the coordinate X of a given point on an orthodrome on the basis of 
the coordinates of two known points on it. 

Let us assume that we have two arbitrary points on the Earth's 
surface with coordinates <i>iAi and <|)2A2. We will take the differ- 
ence of the longitudes of these points as AA(AA - X2 " ^i)- Then, 
according to (1.32a), 

sin (Xi -^ Aq) sin [(Xi — Xp) + ii\] 



Transforming the right-hand side of this equation, we obtain: /53 



sin (X; — Xq) ^ sin (X, — Xq) cos AX + cos ((X{— Xp) sin AX 



Dividing both sides of the equation by sin (X^ 

ing by tg 4>2i "^ will have: 

tg 92 stif (X] — Xq) cos ax + cos (X; — Xq) sin AX 

tg?i ~ sln(X, — Xp) ~ 

= cos AX + ctg (Xj — Xp) sin AX, 



Xq) and multiply- 



from which 



ctg (Xi — Xp) = tg <P2 ctg ipi cosec AX — cfgAX. 



(1.34) 



Equation (1.34), which makes it possible to determine the 
longitude of the point of intersection of an orthodrome with the 
equator (Xg), is very important. Knowledge of this coordinate 
makes it possible to calculate easily all the remaining elements 
of the orthodrome . 



48 



Having substituted the value X in (LS^l-) for the value (Aj - 
Xq), as before, and substituting into (1.33a) the value ctg X from 
(1.34), we obtain the following equation for a point with the coor- 
dinates <i)i Xi : 

ctg o = tg ij!2 cos 9i cosec AX — ctg AX sin <pi. (13 5) 

Formula (1.35) is usually used for calculating the azimuth of 

an orthodrome at the initial point of the straight-line segment of 

the path vfhen there is no necessity for determining the remaining 
elements of the orthodrome. 

In general, it is better to solve (1.34) independently, and 
then find the solution by substituting X into (1.32) and (1.33). 

Simple transformations of (1.32) reduce to formulas which make 
it possible to determine the coordinates of intermediate points on 
an orthodrome: 

tg<p = slnXctgoo, (1.36) 

slnX=tg<i>tgai. (1.36a) 

Given the arbitrary value of a point coordinate on the ortho- 
drome <J) or X , it is possible to obtain the value of the second coor- 
dinate of this point on the basis of these formulas. 

The formulas from (1.32) through (1.36), given by us, make it 
possible to determine the initial and moving azimuths of the ortho- 
drome, and also the coordinates of its intermediate points. 

In order to determine the length of the orthodrome or distances 
along it (.S) let us derive equations which connect the coordinates 
of the points of the orthodrome with its length. 

In Figure 1.32, the triangles ONMi and ON1M2 are similar. The 
straight line ON is a line of the cosine of the arc X, while ON' is 
a line of the cosine of arc S. 

The hypotenuse of triangle ONMi is equal to the radius of the /54 
Earth, vihile the hypotenuse of triangle ON1M2 is the line of the 
cosines of arc (j) . 

Therefore, cos 5 = cos X cos?. (1.37) 

Equation (1.37) makes it possible to determine the distance 
from the starting point of the orthodrome to any of its points with 
knovfn coordinates . 

If the initial point of the orthodrome and the coordinates of 
any two points along it are known, the distance (S) between the 
latter is determined as the difference between the distances to 
the initial point: 

49 



^\fl = 52 — Si. 

If the coordinates of the starting point are not known, and 
the necessity for determining the other elements of the orthodrome 
(besides the distance between the two points) is lacking, then the 
indicated distance can be determined by the formula 

cos 5 = slntpi sin<f2+ costpiCos<p2Cos 4X. / -, 30') 

Formula (1.38) is not derived from simple geometric ratios. 
For its derivation, it is necessary to use the spherical triangle 
{P^Mg^M-^) (Fig. 1.33a). 

Let us join points ?vr^a and ^j^ by verticals with the center of 
the Earth 0. Let us draw tangents to the arcs P^^a ^"^ ^N^b 3"^ "the 
point Pjf up to the intersection with the indicated verticals at 
points M^i and M-^i (Fig. 1.33, b). We will obtain two plane tri- 
angles Pf^^^al^bl ^^'■^ O^al^bl with "the common side Mg^iM-^i . Obviously, 

^ai^bi=<'^N^ai^^ + <'PN%i>'^-2PN^al^N^bi,cosMa^PN%^ 

At the same time, 

M^:^Mi^^ = (0M^^)^+(0M^^)-20M^^0Mi,^oos M^^ OM^i (1.39) 

Since^ai ^■'-'1 ^^ the common side of the triangle, the left- 
hand side of the first equation is equal to the right-hand side of 
the second. 

Taking the radius of the Earth as equal to 1, from the right 
triangles OP-^M^i and ^^N^bl "® f ind ; 

%^a, = tg;b; Pj^M,.^ ~ tg a; OM^^ = sec b; 
Ml, ^^ sec a; L A^^.P^Mi^jP; L M^pM^ -=p. 

Substituting the indicated values into (1.39), we obtain: 
tg2b+tg2a — 2tgbtgacosP = sec2fl + sec2b— 2secasecbcosp; 
sec2a= 1 + tg2fl; sec2b=14-tg2b 

Therefore, 2tgatgbcosP = 2 — 2?ecasecbcosp. (1.40) 

>« 1 ^ • T • •!_ j.i_ • J ^ /■ T ,.r^\ -L cos a cos b , . . 

Multiplying both sides of (l.UO) by we obtain: 

sin «sin bcosP = cosacosb — cosp 

ojP cosp = cos a cos b + sin a sin b cos P. (1.41) /5[ 

Formula (1.4-1) is the first basic formula of spherical trigo- 
nometry and is widely used in aircraft navigation with the use of 
astronomical facilities (the remaining formulas of spherical trigo- 
nometry are given in Supplement 2). 

50 



In our case , 

LP = AX; Lp = Sabi L b'=f= 90° — ijij; Lfl = 90° — ?,. 

i.e., (1.41) has the form: 

cos 5 = sin <f 1 sin 92 + cos <i>j cos 92 cos AX. 

When determining point coordinates of the orthodrome , there is 
the same necessity to solve the inverse problems according to the 
knovin orthocromic distance (S) . 

For this let us return to Figure 1.32, in which it is obvious 
that the line MN i is the line of the sine of the arc S, while line 
MM2 is equal to MN i cos ag. At the same time, MM is the line of 
the sines for arc (j) . Therefore, 

sin If = sin 5 cos do (1.4-2) 

Formula (1.42) makes it possible to determine the coordinate 
(j) along the traversed orthodrome distance from the initial point. 
The coordinate A in this case is determined according to (1.36a). 

slnX = tgytgoo. 

Thus, we have an analytical form of all the necessary trans- 
formations for determining the elements of the orthodrome on the 
Earth's surface. However, in the practice of aircraft navigation 
it is sometimes more convenient to apply other formulas which de- 
termine separate elements of the orthodrome. 

For example, if the coordinates of two points on the Earth's 
surface and the orthodrome distance between them S are known, the 
azimuth of the orthodrome (a) at the starting point can be deter- 
mined by the formula 

cos tpo sin AX / -, ,, ^ \ 

sina= ~ . (1.43) 

sins 

Formula (1.43) can be transformed to determine the distance 
between points at a known azimuth: 

cos 92 sin AX 
sin 5— — . M U3n1 

It is obvious that both formulas are obtained from the equation 

sin 5 sin o = cos 92 sin AX, 

which in turn is derived by means of Figure 1.34, where line BB i 

is a perpendicular dropped from point B to the plane of the equa- /56 

tor, and is a line of the sine of the latitude of this point, while 



51 



line B1A2 is a perpendicular dropped from point B^ to the plane of 
the meridian which passes through point A. Obviously, 

A^Bi = cos tp2 sin 4X, 

Let us erect another perpendicular to the plane of the equator 
at point A2I we will then obtain plane A1A2B1B perpendicular to the 
plane of the equator and the plane of the meridian of point A . 





Fig. 1.34. Determining Special 
Elements of an Orthodrome . 



Fig. 1.35. Determining the Ini- 
tial Azimuth or the Vertex of 
an Orthodrome . 



If we rotate the indicated plane around the" line BAi in such a 
way that it remains perpendicular to the plane of the meridian of 
point A up to the moment when line A2A1 becomes perpendicular to 
the vertical of point A, the distance BA^ will not change. In this 
case, straight line BAi will be the line of the sine of the arc AB , 
while its length will be determined by the formula 



^,5 = 



A.,B 



'2^1 



sin a 



from which it follows that 

sin S sin a = cos <(2 sin AX. 

By a similar method, the initial angle of the orthodrome or 
the latitude of the vertex point is determined if the azimuth of 
the orthodrome at any point on the Earth's surface is known. 

In Figure 1.35, arc Xq^b is the equator; arc \qMB is the ortho- 
drome; line OP-^ is the axis of the Earth; OPq is the axis of the 
orthodrome; M is a point on the Earth's surface; and B is the ver- 
tex point. 



52 



Let us erect a perpendicular P^O to the vertical of point M 
from the center of the Earth so that it is located in the plane of 
the meridian of point M. Let us also draw a plane parallel to the 
plane of the equator through the poles P^ , Pq, and Py . The angles /57 
PqP^O and PyiP-^0 will be right angles, since lines PqPn and Pf^Pj^ lie 
in a plane perpendicular to P^O. 

Angle PqPmO is also a right angle, since the plane of triangle 
PqPj^O has a slope to the axis perpendicular to P-^0 and parallel to 
PqPj^ . It is obvious that angle P^OP-^ is equal to the latitude of 
the vertex point, while angle P]i[OP]j is equal to the latitude of 
point M. Therefore, 

OPf^^ OPi cos 98 = 0P„ cos <p„; 
OPu^OPo sin a, 

whence cos ?a == sin Oo = cos <?„ sin a. (1.1^4) 

Formula (1.44) is used to find the latitude of the vertex point, 
vjhich also appears as a complement of the initial angle of the or- 
thodrome up to 90°. The longitude of point M relative to- the start- 
ing point of the orthodrome in this case can be determined by (1.35a). 

Vfith a known azimuth of the orthodrome at any point on the 
Earth's surface, the coordinates of its starting point can be ob- 
tained directly according to (1.33a), from which it follows that 

tgX = sin9tga, (1.45) 

Then it is not difficult to determine the initial angle of the or- 
thodrome . 

In some cases, in order to calculate the elements of the ortho- 
drome, coordinates of the vertex point rather than of the starting 
point are used. In these cases, the functions of the otQ angle are 
replaced by inverse functions of the latitude of the vertex point 
which are equal to them, just as functions of longitude are, since 
these angles differ by 90° . 

For example, (1.36a) has the form: 

COS?lB=tg<pCtg<fBl 

vrhile (1.45) has the form: 

ctg Xg = sin <f tg a. 

To explain the procedure for determining all the elements of 
an orthodrome, let us examine (as an example) an orthodrome which 
passes through two points on the Earth's surface with these coor- 
dinates: Ml = latitude 60° N, longitude 30° E; % = latitude 80° N, 
longitude 40° E. 

53 



First we shall carry out the general solution of the problem 
of finding the elements of an orthodrome . For this we shall use 
(1.34). Substituting into this equation the functions of the coor- 
dinates of the points Mi and M2, we obtain: 

ctg (h — Xo\ = tg 80° ctg 60°cosec 10° — ctg 10°; 
ctg(X,-Xo) = -~^ -5/n = 13.228; 
X = (Xj - >^) = 4°18.'; Xo = 25°42'. 

The initial azimuth of an orthodrome, according to (1.32), 
will be : 

tg oo = sin X ctg <p. 

Let us determine it on the basis of the coordinates of point /5i 

Ml : 

tg Oo = sin 4°18' -ctg 60° = 0.574-0.075 =■ 0.0433; 

do = 2°29'. 

According to (1.33a), the moving azimuth of the orthodrome (a) 
for point Mi is equal to 

ctg a = ctg 4°18' -sin 60° = 13.228-0.866 = 14.455; 

= 5°. 

The distance from the starting point of the orthodrome to 
point Ml, according to (1.37), equals 

cos 5i =- cos 4°1 8' - cos 60° = 0. 9972 -0.5 = 0. 4986; 
5, = 60°4', 

while the distance to point M2 , according to the same formula, is 

cos 52 = cos I4°18'-cos 80° = 0.969-0. 1736 = 0.1682; 
52 = 80°17'. 

The distance between Mj and M2 is then defined as the differ- 
ence between the distances to the starting point: 

S = 52 — 5i = 80°17' — 60°04' = 20°13'. 

Coordinates of any intermediate point can be determined accord- 
ing to (1.36) or (1.36a) . 

For example, the longitude of point M2 according to its lati- 

sin X2 = tg 80° tg 2°29' = 5.671 -0.0433 = 0.246; 
X= (Xj — Xo) = 14°15'; Xj = 39°57'. 



54 



Thus, all the necessary elements of an orthodrome are easily 
determined . 

Let us now assume that we had to determine only the azimuth of 
the orthodrome at point M^ . For this we will use (1.35): 

0.5-5,671 
ctg a = '^ ^^gg -0.866-5.671 = 11.4225,' 

a = 5°. 

Knowing the azimuth of the orthodrome at one of its points 
makes' it possible to determine the distance to any point by using 
(l.H3a), or in our example: 

.„ - 0.1736-0.1736 

sin 5 = — =; 34'ifi- 

0.0872 -".>5*oo. 

5 = 20°13'. 

Using the azimuth of the orthodrome at one point, it is possi- 
ble to determine the latitude of the vertex point or the Initial 
azimuth of the orthodrome according to (1.414-). 

For our orthodrome, using point Mi, we obtain: 

sin Oo = cos 60° sin 5°"= 0.5-0.0872 =^Q.0436; 
Oo = 2°30'. 

After this, the intermediate points of an orthodrome are easily /59 
determined . 

Thus, it is possible to determine the elements of an orthodrome 
beginning with the distance between two points according to (1.38), 
changing to the moving azimuth according to (l.M-3), and then to the 
latitude of the vertex point according to ( 1 . M-M- ) . 

Orthodrome on Topographical Maps of Different Projections 



Let us examine an 



/I A Sin If 



orthodrome on maps of a simple equally spaced 
cylindrical projection, which essen- 
tially represents a geographical coor- 
dinate system on a scale of angles. 




I— 4A- 



Fig. 1.36. Elementary 
Segment of an Orthodrome 
on a Map of a Cylindri- 
cal Projection. 



To explain this, however, let us 
draw on the Earth's surface an elemen- 
tary normal cone at some latitude; we 
shall examine it on the above projec- 
tion ( Fig . 1.36). 

As is already known, the radius 
of this cone when unrolled equals: 

po = ^ ctg <po 



55 



or, taking the radius of the Earth as 1, 

Po = ctg <Po- 

According to (1.20) the scale of the projection along the paral- 
lel is equal to: 

1 
n = ■ 



COS<j> 



In order to draw our unrolled cone on a cylindrical surface, 
it is necessary to straighten the cone first and then extend it. 
Obviously, the segments of the meridians remain straight lines dur- 
ing straightening of the cone, but they must be unrolled together 
with the surface elements to an angle equal to AX sin (j) . 

Let us now draw a straight line AB in the east-west direction 
on an elementary cone. 

During straightening of the cone, the indicated straight line 
will acquire a curvature, the radius of which will be equal to the 
radius of the unrolled cone (r), but curved in the opposite direc- 
tion. Therefore, 

During extension of our cone along a parallel to a scale nz , 
each of its elements (including elements of our straight line) will 

undergo an extension equal to ~ . Therefore, the radius of the /60 

^ ^ cos (j) 

straight-line element will increase and will equal: 

ctgv 

r, o = = cosec tp. 

''>"• cos <f 

As is evident, the straight-line element, situated along the 
parallel (in general, in a direction perpendicular to the axis of 
the cylinder) acquires a curvature. The straight-line element sit- 
uated in the direction of the axis of the cylinder does not acquire 
a curvature. Therefore, if the straight-line element is situated 
at an angle to the axis of the cylinder, its radius of curvature 
will equal : 

Ctg<j> 

''a,b. — ; = cosec 9 cosec a.. 

-^'^^ cos <f sin a ^ (1.46) 

In geometry, the curvature of a curve is considered to be a 
value inverse to the radius of the curvature. Therefore, the curv- 
ature of our element will equal: 

1 

=sin<iisina. 

'■a.b, (1.47) 



56 



An orthodrome on the Earth's surface does not have its own 
curvature of each element of an orthodrome on a map in a normal, 
equally spaced cylindrical projection will be expressed by (1.47), 
from which it is obvious that the maximum curvature of the ortho- 
drome will be observed at its vertex points, while the starting 
points, i.e. , the points of intersection of the orthodrome with the 
equator (Fig. 1.37), will appear as points of inflection. 

Thus , the orthodrome on a map 
of an equally spaced cylindrical 
projection has a form reminiscent 
of a sine curve. This curve is 
the graph of the ratio of the coor- 
dinates of the orthodrome with a 
known initial azimuth (a). 




Fig. 1.37. Graph of an Ortho- 
drome in a Cylindrical Pro- 
j ect ion . 



As a result of the nonisogon- 
ality of an equally spaced projec- 
tion, the slope of a tangent to 
the curve of the orthodrome does not reflect its directly moving 
azimuth, with the exception of the azimuth at starting points. 

The moving azimuth of an orthodrome along a curve can be deter- 
mined if we consider the relationship between the scales m and n at 
the investigated points. With equal scales, the dip angle of the 
tangent to the curve is determined by the formula 



tgct = 






n ^ - riQ sec (() or n j^ = n^ sec (J). There- 
fore , the actually moving azimuth of the orthodrome in an equally 



In our case, the scale 
, the actually moving a: 
spaced cylindrical projection is determined by the formula 



/61 



dX 

tg a = cos <f. 

d<( 

It is obvious that in an isogonal normal cylindrical projec- 
tion, the orthodrome will also have a shape reminiscent of a sine 
curve. However, as a result of the extension of the scale along 
the latitude (n - riQ sec <)) ) , the amplitude of this curve will be 
increased. The more it is increased, the smaller the initial azi- 
muth of the orthodrome will be, and the greater the latitude of the 
vertex points . 

In contrast to an equally spaced projection, in this projection 
the dip angle of the tangent to the curve will correspond to the 
moving azimuth of the orthodrome at any point ,, since the scales 
along the longitude and latitude are equal to : 

m = n = sec (f. 



57 



Thus, the orthodrome in a cylindrical projection has the form 
of a curve which is convex in the direction of the increase in the 
scale of the projection. This feature of the orthodrome is common 
to all projections which have an increase in one direction. 

Let us cite a brief analysis of the bend of the orthodrome with 
a varying map scale, in accordance with the general case. 

Let us assume that we have a spherical trapezoid which is rep- 
resented on a map in the form of a rectangle (Fig. 1.38). The length 
of any parallel (^x^ °^ 'the trapezoid is equal to its length on a 
rectangle divided by the scale of its representation. The scale of 
representation of the meridians in any part of the rectangle is 
equal to one . 

During extension of the trapezoid into a rectangle, each 
straight-line element on its surface acquires a curvature. 



fx rfcp '. r^ 



Therefore, for an equally spaced cylindrical projection 

1 d cos y 



''x df 



= — sin <j>. 



Since — = 0, for a straight line passing at an angle to the 
■^ A 
meridian we will have 

1 
— = — sin 9 sin a. 

r 

A minus sign shows that bending occurs in the direction of a 

decrease in the value of — or in the direction of an increase in 

n 
the scale . 





/62 



Fig. 1.38. Conical Trapezoid 
Represented in the Form of a 
Rectangle . 



Fig. 1.39. Bending of an Or- 
thodrome in the Directions 
of Scale Increases. 



58 



Let us now assume that we have a projection, a change in the 
scale of which occurs in two principal directions [for example, 
simultaneously in the north and east (Fig. 1.39)]. 

It is obvious that a straight line AB , passing at an angle to 
the meridian with a change in the scales in two principal directions, 
will simultaneously undergo bending in opposite directions, i.e., 
its component curvatures will be subtracted: 



rfip 



sin a ■ 



dk 



For the general case , 



R 

dz 



sin a — 



dx 



Therefore, the orthodrome on maps constructed with tangential 
cylindrical projections will have convexity: 

a) at latitudes greater than the latitude of a parallel which 
is tangent to a geographic pole; 

b) at latitudes lower than the latitude of a parallel which 
is tangent to the equator (Fig. 1.40). 




■ Con i c straight 1 
"0 r thod reme 



ne 



Fig. 
Tange 

the s 
stant 
limit 
gle t 
be a 
maps 
recti 
able , 
both 



1.40. Orthodrome on 
ntial Conic Projecti 



cale alo 
, while 
s of eac 
o the me 
wavy lin 
with a s 
on will 

while i 
parallel 



ng t 
the 
h ma 
ridi 
e ar 
cale 
be s 
n th 
s an 



he latit 
scale al 
p sheet . 
an withi 
ound a s 
of 1:1, 
o insign 
e places 
d the or 



a Map o 
on . 

ude in 
ong the 
There 
n the 1 
traight 
000 ,000 
if icant 

where 
thodrom 



f a 



In intersecting conic 
projections, the ortho- 
drome has the same form as 
in tangential projections. 
Here , its point of inflec- 
tion is situated on the 
middle parallel between 
the parallels of inter- 
section . 



/63 



thes 

Ion 

fore 

imit 

pri 

, de 

tha 

sepa 

e wi 



IS 

in 
je 
e maps r 
gitude h 
, an ort 
s of one 
ncipal d 
viatlons 
t they a 
rate map 
11 have 



Of 
the o 
an in 

ction . 

emains 

as sma 

hodrom 
sheet 

irecti 
from 

re pra 
sheet 

breaks 



spec 

rtho 

tern 

As 

pra 
11 c 
e dr 

on 
on . 
the 
ctic 
s ar 



ial 

drom 

atio 

is 
ctic 
hang 
awn 
this 

How 
prin 
ally 
e sp 



inte 
e on 
nal 
know 
ally 
es i 
at a 

map 
ever 
cipa 

unn 
lice 



rest 

maps 
pro- 
n , 

con- 
n the 
n an- 
will 
, on 
1 di- 
otice- 
d. 



59 



As we have already shown, the orthodrome in a central polar 
projection is expressed by a straight line. However, its moving 
azimuth, with the exception of the directions 0, 90, 180, and 270° 
cannot be determined by simple measurements on a map, but demand 
the introduction of corrections according to (1.29) and (1.30). 

In a polar stereographic projection, the orthodrome is also a 
nearly straight line. However, to determine its azimuth, it is 
necessary to use general equations of an orthodrome on the Earth's 
surface . 

Loxod rome on the Earth's Surface 

The loxodrome direction at the present time is used only to 
determine the mean path angle of flight on short segments of a path 
by the use of magnetic compasses. 'With the use of magnetic com- 
passes, not a geographic but a magnetic loxodrome direction is used, 
This leads to a bending of the flight path which does not lend it- 
self to precise analytical descriptions. 




As we already know, a toxodrome is a line on the Earth's sur- 
face which joins two points and intersects the meridians at a con- 
stant angle. 

In general, a loxodrome is a spiral line which goes to the 
Earth's poles. As a result of this, it has curvature not only in 
a vertical plane, but in a horizontal plane as well. Meridians, 
the equator, and parallels which are also loxodrome lines, expressed 
in the first two cases by a great circle and in the last case hy a 
small circle on the Earth's surface, are the exception. 

The curvature of a loxodrome in a horizontal plane increases 
sharply with an approach to the Earth's poles. As a result, it is 
not used at all for flights in polar latitudes. 

Let us determine the curvature of a loxodrome, its extension. 



/61+ 



60 




a Map of a Conic 



and its deflection as compared to the orthodrome direction at a 
given latitude <() . 

The maximum curva- 
ture of a loxodrome at 
a given flight altitude 
will occur when the 
flight is in an easterly 
or westerly direction, 
and it will vanish in a 
flight to the north or 
south . 

Let us assume that 
a flight at altitude ((> 
occurs in an easterly 
direction. In this 
case, the angle of turn 
of the loxodrome from point A to point B will be equal to the angle 
of convergence of the meridians (6) between these points (Fig. 1.41). 

B = _(Xb — X^)sin9. 

Its length (5) from point .4 to B will be 

•S = (Xb — X^)cos<p, 

where A^ and Ag are the longitudes of points A and B and (f> is the 
mean latitude between points A and B. 

The radius of curvature of the loxodrome i^/^x^ can be determined 
as the ratio of the length of part (5) to the angle of turn (6). /65 
If we take the radius of the Earth as 1, then 



Fig. 1.41. Loxodrome on 
Pro j ection . 



'■x=— =ctgv. 



(1.48) 



The part of the loxodrome which runs along the meridian does 
not have a horizontal curvature. Therefore, if the loxodrome passes 
at an angle to the meridian, the radius of its curvature at any 
point will equal: 

r = r-^ cosec a = ctg 9 cosec a. 

(1.49) 

Example : Determine the radius of curvature of a loxodrome 
passing at an angle of 30° to the meridian at a latitude of 45°. 



Sol ution 



r = y?3 ctg 45° cosec 30° = 2/?3 = 12742 km 



where i?3 is the radius of the Earth 



61 



The curvature of the loxodrome in a horizontal plane creates 
some lengthening of the straight-line parts of the path. The later- 
al deviations from the line of the orthodrome direction may turn 
out to be very significant here. 

In Figure 1.42, the straight line AB is the orthodrome; arc AB 
is the loxodrome ; 6 is the angle of turn of the lo:kodrome from point 
A to point B. The length of the straight line is 

t 
^B = 2/?sln — - . 

while the length of the arc is 

AB = m. 

Lengthening of the path along the loxodrome (hS) is determined 
by the formula 

AS = /?8-2y?sin-|-. ^^^^^^ 

Example : Determine the lengthening of the path along the loxo- 
drome passing through points A and B on the Earth's surface, with 
the following coordinates: A: latitude 55° N, longitude 38° E; 
B: latitude 55° N, longitude 68° E. 

Since the latitude of the starting and end points is the same, 
the direction of the loxodrome coincides with the Earth's parallel 
at a latitude of 55° . 

The radius of curvature of the loxodrome will be: 

/• = r^ = /?3 ctg 55° = 6371 -0.7002 = 4461 km 

The angle of turn of the loxodrome is determined by the formula 
— 6 = (Xa — X,) sin % 6 = ~ 30° sin 55° = — 30°-0,8192 = 24.576°, 



Then 



sin ~ = sin 12°17' = 0.2127. 



Substituting the value of the radius of curvature and the an- /66 
gle of turn of the loxodrome into (1.50), we obtain: 

24^576:4461__2^^gj 2127 ='15.6 km 
•^ " 57,3 

From this example, it is obvious that at middle latitudes, 
with flight paths up to 2,000-3,000 km long, the curvature of the 
loxodrome creates relatively small lengthenings of the path (in our 



62 



example, less than 1%); however, in approaching the polar latitudes, 
lengthening of the path will increase , together with a decrease in 
the radius of curvature of the loxodrome . 



Significant lengthenings of the path along the loxodrome occur 
at middle latitudes with very long distances between points on the 
Earth's surface. For example, at a latitude of U-0° , with a distance 
of 11,000 km between points, lengthening of the path along the loxo- 
drome can exceed 4,000 km, i.e., more than 30%. 

In Figure 1.42, it is obvious that 
with a constant radius of curvature of the 
loxodrome, its greatest discrepancy with 
respect to the orthodrome (deflection) will 
be observed at half the path between points 
A and B. 




Loxodrome 

Fig. 1.42. Radius of 
Curvature of a Loxo- 
drome . 



Here , 



&Z = R- 



■ R cos -—- 



or 



iZ^RU-coSYJ- (1.51) 



In the example analyzed by us , 

AZ = 4461 (1—0,9771) = 102,6 km 



Thus, the discrepancy between the loxodrome line of the path 
and the orthodrome, even at comparatively small distances between 
points on the Earth's surface, will be very substantial. This is 
the basic cause of the limitation of the length of the loxodrome 
segments of the path. 

In the practice of aircraft navigation, since the loxodrome 
direction of flight is used only in limited path segments, the azi- 
muth of the orthodrome (a), measured on the central meridian between 
the starting and end points of the segment is taken as the loxodrome 
direction of the flight. 

This angle can also be determined on the basic of the approx- 
imate formula 



tgo = 



X, — X 



f2 — ?! 



COSl^ 



av 



(1.52) 



/67 



The length of the loxodrome segment of the path (S) is deter- 
mined by the formula 

T2 — 91 
cos o (1.53) 



5 = 



".2 •—A I 

sin a 



(1.54) 



63 • 



Formulas (1.5M-) and (1.52) are approximate and have a simple 
geometrical interpretation. 

Formula (1.53) is derived analytically. 

Considering that the loxodrome intersects the meridians at a 
constant angle, the ratio remains constant: 

dS 1 



d<f cos a 



from which 



5= ! \d:f^^iZUt2^, 

COS a i ^ cos a 

In the majority of cases, in calculating the distance along 
the loxodrome, it is more advantageous to apply (1.53). However, 
with loxodrome directions close to 90 or 270°, the values ^2 ~ ^\ 
and cos a simultaneously approach zero. This leads to large arith- 
metic errors in calculation and ultimately to an ambiguity in the 
solution. In these cases, it is more advantageous to use (1.54), 
the errors in which will be negligibly small, since a small differ- 
ence in the latitudes between the points means that the mean cosine 
of the latitude becomes practically equal to the cosine of the mean 
latitude . 

Example : Determine the loxodrome direction and the distance 
between points A and B on the Earth's surface, the coordinates of 
which are: A: latitude 56° N, longitude 38° E; B : latitude 68° N, 
longitude 47° E. 

Solution: According to (1.38), let us find the direction of 
the loxodrome : 

,g o = /*^~^ cos 62° = 0.35Z1; a = 19°24'. 
68 •"" 56 

Let us determine the loxodrome distance according to (1.53): 

5 = 111,1-^^=^, = 1413 km 
cos 19°24' 

Loxodrome on Maps of Different Projections /6E 

A loxodrome has the appearance of a straight line only on maps 
of a normal isogonal cylindrical projection. 

Oh maps of normal isogonal conic and azimuthal projections, 
the loxodrome is a curved line intersecting the meridians at a con- 
stant angle a. Therefore, knowing the direction of the loxodrome 
in order to draw it on a map it is sufficient at the starting point 

64 



to plot this direction up to the intersection with the next meri- 
dian, where the indicated direction must be extended to the next 
meridian in line. Continuing our plotting to the final point, we 
will obtain a broken line very close to the loxodroifie . 

On maps with nonisogonal projections, the loxodrome will have 
a variable angle to the meridians, which depends on the ratio of 
the scales 



tgam=«go— T 



(1.55) 



where a is the angle of intersection of the loxodrome with the meri- 
dian at a location; a^ is the angle of intersection of the loxo- 
drome with the meridian on a map; n and m are the scales of a map 
at a given point along the principal directions east-west and north- 
south, respectively. 

For example, on maps with an equally spaced normal cylindrical 

projection, where H. = sec (j) , 

m 

tg "07= *g ° 5^<= ■?' 

i.e. , the loxodrome will have a curvature in the direction of a 
pole, whereas it has a natural curvature in the direction of the 
equator . 

General Recommendations for Measuring Directions and Distances 

Orthodrome directions and distances for straight-line segments 
of a path of more than 1200-1500 km in all cases must be determined 
by analytical means, independently of the scales and map projections 
used. With a length of the path segments of more than 2000 km, the 
intermediate points of the orthodrome must also be determined in 
such a way that the distance between them does not exceed 800-1000 
km . 

On short path segments (up to 1200-1500 km), the methods of 
determining directions and distances depend on the scale and pro- 
jection of the maps, as well as on the means and methods of air- 
craft navigation used. For example, in using precise automatic 
navigational devices, it is always advantageous to use analytical 
forms to solve these problems. 

It is possible to carry out direct measurement of distances 
and directions on maps by means of a scale and protractor, with the 
length of the path segments being not more than 1500 km if these 
maps are executed on an international polyconic projection and have 
a scale of 1:1,000,000 or 1:2,000,000 (the latter within the limits 
of one (or, in extreme cases, two) adjoining sheets). 



/69 



65 



We must note that good results in measuring directions and dis- 
tances can be obtained on route maps constructed on oblique cylin- 
drical or oblique conic projections when the flight direction coin- 
cides with or is located close to the axis of the route map. How- 
ever, in directions at an angle to the axis of the route map, the 
results of measurements are significantly worse than on maps with 
an international polyconic projection. 

In using maps constructed with all other projections, only the 
analytical form of determining distances and directions, with calcu- 
lation of intermediate points along the orthodrome after every 200- 
300 km of the path, must be applied. 

The loxodromic flight direction can be measured directly only 
on maps with an isogonal normal cylindrical projection. Here, seg- 
ments of distances up to 300-M-OO km on this projection can be meas- 
ured by means of a varying scale located on the edge of the map. 

On maps in other projections, generally speaking, there is no 
need to measure and plot the loxodrome line of the path in parts of 
mDre than 300-100 km. 

Since the loxodromic flight direction in short path segments 
is used as the mean orthodrome direction, it is considered equal to 
the orthodrome as indicated by the mean meridian between the start- 
ing and end points of the path segment. 

In view of the fact that in short segments of the path the lox-. 
odrome line does not show significant deviation from the orthodrome 
as a rule , it is not plotted on maps but is considered coincident 
with the direction of the orthodrome. 

7. Special Coordinate Systems on the Earth's Surface 

In the practice of aircraft navigation, rectangular and geo- 
graphic coordinate systems are insufficient, and it is necessary to 
use at least three or four coordinate systems simultaneously. 

Actually, elements of aircraft movement are examined in a mov- /70 
ing rectangular coordinate system. The center of a rectangular 
system moves in one of the surface coordinate systems which is con- 
nected with the given flight path, which in turn is determined in a 
geographic coordinate system. 

The indicated order of the connection of the coordinate systems 
is minimal. For some purposes, it is advantageous to examine air- 
craft movement relative to the airspace, i.e., a supplementary co- 
ordinate system whose center shifts in the moving rectangular system. 

With the use of gyroscopic devices as well as astronomical 
ones, it is necessary to use a universal (stellar) coordinate sys- 
tem. The use of radio-engineering navigational facilities is con- 



66 



nected with the use of a whole series of special types of surface 
coordinates by which the position of the aircraft on the Earth's 
surface is determined. 

Let us examine the most important surface coordinate systems 
used in aircraft navigation. 

Orthodromic Coordinate System 

.The orthodromic coordinate system for calculating the path of 
an aircraft is the one most widely used at the present time. 

In this system, the direction of the straight-line path segment 
(Fig. 1.43) is taken as the main axis X. The line perpendicular to 
the Z-axis and also situated in the plane of the horizon is the sec- 
ond axis , Z . 




Fig. 1.43. Orthodromic Coordinate System, 



In Figure 1.43, angles a^ and a2 are the directions of the 
first and second straight-line segments of the path, measured from 
the meridians of their starting points. Points Oj and O2 are the 
starting points of the segments, the coordinates of which are de- 
termined in the geographic coordinate system. The orthodrome dis- 
tances O1O2 and O^O-^ are the lengths of the straight-line segments; 
the angle TA-^ is the angle of turn of the orthodromic coordinate 
system at point O2 • 

Since an aircraft moving above the Earth's surface in a given 
direction has only small random deviations from the given flight 
path (as a rule, not more than 20-30 km), it is possible to take 
the spherical surface of the Earth within the area of the possible 
deviations of the aircraft from the J-axis of the orthodrome system 
as a cylindrical surface. Then the unrolling of the cylinder gives 
us a rectangular system XZ on a plane. 

Let us assume that an aircraft moves from point O^ at a small 
angle to the OxXj axis equal to ij; - a^, and covers a distance S. 



/71 



67 



The coordinates of the aircraft at point M^^ are determined by 
the equations: 

^a=5cos(.l/-ai); | 

2'^=5sln(<j;-ai). J (1. 56) 

Measuring the Z^ coordinate constitutes checking of the path 
of the aircraft according to distance, while measuring the Z^ coor- 
dinate constitutes checking of the path according to direction. 

Periodic measurement of the Z^ and Z^ coordinates make it pos- 
sible to determine all the basic elements of aircraft movement; for 
example : 

a) Direction of aircraft movement ( i|j ) : 

2a2~^ai 
^ = arctg- -— + '^. (1.57) 

where Xg^ , Z^^^ are coordinates of the aircraft at the first point, 
^aa » ^a2 ^^® coordinates of the aircraft at the second point; 



b) Speed of aircraft movement along a given flight path {W) 

'^a2~-^a2 
W= (1.58) 



where t is the flying time of the aircraft between points Ja^ sn 



d 



^a2 



c) Remaining flying time to point O2 



Xrem 

where ^pgm ~ '^l'^2 " -^a • 

d) Necessary flight direction for arrival at point O2 : 

Z 

4, = ai— arctg-^r: — . (1.60) 

rem 

Formulas (1.55) to (1.60) are entirely obvious and do not re- 
quire special derivations or proofs. 

To refine the coordinates of the aircraft in the orthodrome 
system, we can use correction points (CP), visual or radar land- /72 
marks on the Earth's surface, locations of ground radio facilities, 
etc. (Fig. 1.44). 



Translator's note: arctg = cot"-'-. 
68 



If the correction point is observed from an aircraft at an 
angle to the given route, at a distance from the aircraft equal 

to i? , the coordinates of the aircraft 
will be determined by the formulas : 



Zop 




Fig. 1.44. Determining the 
Orthodromic Coordinates of 
an Aircraft from a Correc- 
tion Point . 



Xg^=Xcp-Rcos 



n 



(1.61) 



During flight over the correc- 
tion point, i.e., when this point is 
observed at an angle equal to 90° to 
the flight path, (1.61) is simplified 
and takes the form: 



X^=X 



cp 
cp- 




m(x d) 

' rem' 




Fig. 1.45. Transfer of the Next Stage in a Course to an Orthodromic 
Coordinate System: (a) with the Aircraft Position on the Path of 
the Given Course; (b) with Deviation of the Aircraft from the Path 

of a Given Course. 

The simplicity of the geometrical transformations and the na- 
tural perception of the coordinates of the aircraft in an orthodrome 
system, both of the path covered by the aircraft and of the devia- 
tion allowed from the given path, make it the most acceptable coor- 
dinate system for a given flight path. 

In high-speed aircraft (as a result of a large turning radius), 
in order to emerge without deviation at the next stage of the or- 
thodrome path, it is necessary to consider the linear advance to 
the angle of turn {TA) , This transfer is connected with transform- 
ations of the coordinates of the aircraft from the orthodrome sys- 
tem of the preceding stage to the system of the following stage. 

In Figure 1.45, a, point M located on the flight path of the 
preceding stage of flight is the point of the beginning of turn for 
arrival at the flight path of the following stage. Obviously, the 
coordinates of this point in the system of the following stage will 
be equal to 



/73 



69 



Z2 



A2 = Areincos'rA; \ 
= A* rem sin T A. ) 



(1.62) 



In general, when the coordinate Z at the beginning of the turn 
is not equal to zero, i.e., if the aircraft is not located strictly 
on the given flight path when beginning the turn, the transforma- 
tion of the coordinates must be carried out according to the fol- 
lowing formula (Fig. 1.45, b): 



X2 = Zsinyn — Xrem cosTA; 
Z2 = Zcosyn+A' rem. sin 



TA; I 
TA. ) 



(1.63) 



In the process of turning, the coordinates of the aircraft are 
measured in the system of the following part of the flight in which 
their calculation after turning is carried out. 

The orthodrome system examined 
by us is sometimes called the stage 
orthodromia coord'inate system. In 
some instances , a rectangular co- 
ordinate system is used for flight 
over an area, e.g., for maneuver- 
ing of an aircraft in the region 
of an airport and for special-.pur- 
pose flights, etc. In these cases, 
the direction of the meridian at 
the point of origin of the coor- 
dinates or some other direction 
[for example, the direction of the 
take-off -landing zone at an alr- 
Z-axis, and a rectangular coor- 







Cx,z,) 


















/ 














N 
























{X^i 


'.) 


























k(Y«) 




- 








/ 





















(x,z,-o) 

1 












z 















Fig. 1.46. Rectangular Coor- 
dinate System for Flight 
over an Area. 



port (Fig. 1.46)] is taken as the 
dinate system is constructed from this. 

The flight is carried out along the given coordinates of the 
points of the route [for example, along the coordinates of the be- 
ginning of each of the four turns in the rectangular maneuver of 
making an approach to land at an airport (X.^Z;^), (Zj^z)^ (^323), 

The limits of applicability of an areal rectangular coordinate 
system are limited by the effect of the sphericity of the Earth on 
the precision of measurements. In practice, without noticeable dis- 
tortions, such a system can be used within a radius of 300-400 km 
from the point of origin of the coordinates. 

It Is also applied with the use of navigational indicators In 
flight, when the orthodrome direction of part of the course is taken 
as the X axis . 



/74 



70 



Arbitrary (Oblique and Transverse) Spherical 
and Polar Coordinate Systems 

In the solution of navigational problems with a geographical 
coordinate system in polar regions, very significant errors arise. 

A special chapter is devoted to problems of accuracy in air- 
craft navigation. In the present section, for the purpose of illus- 
tration, only (1.36a) is examined. 

It is obvious that with the approach of the aircraft to a lati- 
tude equal to 90°, the tangent (j) will approach infinity. Therefore, 
small errors in measuring the latitude of the location of the air- 
craft will cause the errors in calculating the longitude to grow 
indefinitely. 

To avoid a loss of accuracy in solving navigational problems, 
especially by automatic navigational devices, random spherical co- 
ordinate systems are employed. 

^ Arbitrary spherical systems dif- 

fer from a geographical system by the 
fact that the poles of these systems 
do not coincide with the geographic 
poles. Therefore, in these systems 
all the analytical transformations of 
distances and directions which are 
carried out for a geographic coordi- 
nate system are justified. 

For transferring from a geograph- 
ical coordinate system to an arbitrary 
spherical system, or vice versa, it 
is necessary to derive special equa- 
tions: let us examine Figure l.M-7. 

igure 1.47, a cross section 
rth's sphere is shown. Here 
n in such a way as to pass 

poles of the geographic sys- 
s, i.e., so as to appear as 
eographic and arbitrary sys- 
t such a plane exists with 
itrary spherical system. 




Fig. 
of S 
on t 

the 

thro 

tem 

the 

terns 

any 



1.47. 
pheri ca 
he Eart 

plane o 
ugh the 
and the 
plane o 
simult 
distrib 



Trans 
1 Coo 



f the 
cent 
arbi 
f the 
aneou 
ution 



format 
rdinat 
urf ace 

cross 
er of 
trary 

merid 
sly . 

of th 



1 on 
es 



In F 
of the Ea 
section is chose 
the Earth and the 
coordinate system 
ian in both the g 
It is obvious tha 
e poles of an arb 



/75 



Let us agree that a reading of the longitude both in the geo- 
graphical and arbitrary systems will run from the indicated plane 
of intersection. Lines AB and AiBi in Figure 1.47, and the lines 
parallel to them, appear as lines of intersection with the planes 
of the equator and the parallels in the geographical and arbitrary 
systems. Point P is the pole of the geographic coordinate system; 
?! is the pole of the arbitrary system; angle 9 is a combination of 
the axes of the geographical and random systems. 



71 



Let us choose point M ( (j) i X i ) on the Earth's surface and pro- 
ject it onto the plane of the cross section (point Mi). It is ob- 
vious that OL will appear as the line of the sines of the latitude 
of point M in an arbitrary system, while LMi will appear as the 
line of the cosines of the longitude of point M of this system in 
the plane of its parallel, i.e., 

LMi = cos Xj cos ^j. 

The latitude of point M in the geographical system will equal: 

siny = 0/:cose — ijW,sine 



or 



sin <f = sin 91 cos 6 — cos Xj cos (jjj sin 6. 



(1.64) 



It is obvious that a perpendicular dropped from point M to the 
plane of intersection (point Mi) will be the line of the sine of 
the longitude in the arbitrary system in the plane of the parallel 
of this point; at the same time, the line of the sine of the longi- 
tude in the plane of the parallel of the point in the geographic 
coordinate system will be 

sin X = sin Xj cos <pi sec <f . 



from which it follows that 

MMi ■-= sin Xi cos 91 = sin X cos <p. 

Formulas (1.64) and (1.65) make it possible to determi 
coordinates of a point in a geographical coordinate system 
ing to its coordinates , known in the arbitrary system under 
condition that the plane coinciding with the axes of both s 
is taken as the initial meridian. After solving the proble 
cording to (1.64-) and (1.65), it is necessary to introduce 
rection into the X coordinate equal to the longitude of the 
pole in the geographic coordinate system. 



(1.65) 

ne the 
accord- 

the 
ystems 
ms ac- 
a cor- 

Pi 



Since the principles of construction of spherical and geograph- 
ic coordinate systems are identical, for the solution of the reverse 
task (transferring from the geographic system to the arbitrary one), 
it is sufficient to drop the subscripts in the functions of the co- 
ordinates of (1.64) and (1.65) whereever they occur and to add them 
where they are absent : 

sin fi = sin ip cos 6 — cos X cos 9 sin's; 
sin Xj = sin X cos 9 sec f j. 



Formulas (1.64) and (1.65) were given with a consideration of the 
flattening of the Earth at the poles, i.e., the Earth was taken as 
a sphere with a mean radius . 



/76 



72 



Position Lines of an Aircraft on the Earth's Surface 

Thus far, we have examined coordinate systems on the Earth's 
surface as systems which connect the position of an aircraft with 
the Earth's surface during its movement in a given direction. 

In aircraft navigation, it is often necessary to determine the 
elements of aircraft movement according to consecutive coordinates. 
It is obvious that means and methods for measuring the coordinates 
of an aircraft are necessary for this purpose. 

Usually the two-dimensional surface coordinates of an aircraft 
are determined separately according to two lines of the aircraft's 
position measured at different times or according to two lines 
measured simultaneously. In some cases, it is sufficient to deter- 
mine one line of the aircraft's position. 

The geometric locus of points of the probable location of an 
aircraft on the Earth's surface is called the posit-ion tine of an 
aircraft . Similar groups of aircraft position lines are called a 
family of position lines. 

For example, if the latitude of the location of an aircraft 
is determined by astronomic means based on the elevation of Polaris, 
the parallel on which the aircraft is located will be a position 
line of the second family. 

Let us assume that the longitude of an aircraft was determined 
simultaneously on the basis of the altitude of a star, the azimuth 
of which is equal to 90 or 270°. The longitude obtained by such a 
method is a position line of the second family. 

Direct measurement of the geographic coordinates of an aircraft 
is possible only by astronomic means, and not in all cases. 

In determining the location of an aircraft by optical or radio- 
metric means, the families of position lines generally do not coin- 
cide either with the grid of geographic coordinates or with the 
given flight direction. 

At the present time, there are several types of coordinate 
systems which are used as families of aircraft position lines in 
the application of radio-engineering and astronomic facilities of 
aircraft navigation. They include the following: 

1) A two-pole azimuthal system^ in which the radial lines 
{bearings) diverging from two points on the Earth's surface with 
known coordinates are families of position lines. 

2) Polar or azimuthal range- finding system^ in which the bear- 
ings from a point on the Earth's surface with known coordinates are /77 
the first family of position lines of this system, and concentric 

73 



■■■■■■■ ■■■■ I 



circles at equal distances from the indicated point are the second 
family . 

3) L-lnes of equal azimuths (LEA), which are position lines 
relative to known points on the Earth's surface, at each of which 
the azimuth of a known point retains a constant value. 

^) Difference-range finding (hyperbolia system) ^xn which each 
family of position lines is bipolar; a constant difference of dis- 
tances to the poles of the system is preserved on each position 
line . 

5) Over-alt range finding (eltiptioal) system^ in which the 
family of position lines is bipolar; a constant sum of the distances 
to the poles of the system is preserved on the position lines. 

6) aonfoaat hyperbotia-eZtiptiaat system^ in which the fam- 
ilies of position lines are ellipses and hyperbolas confocal with 
them . 

From the above list of coordinate systems, it is evident that 
each has arisen from the nature of the navigational values measured 
by the devices used. The indicated values are called navigational 
parameters . 

For^example, for a hyperbolic system the difference in dis- 
tances serves as a navigational parameter, and in an azimuthal sys- 
tem (or for lines of equal azimuths) the azimuth serves as a navi- 
gational parameter, etc. 

In evaluations of the accuracy of navigational measurements, 
considering that the intersecting segments of the position lines 
of any system can be assumed to be straight-line segments in the 
region of the location of the aircraft, the concept of a unified 
coordinate system is sometimes introduced for the purpose of study- 
ing the general properties of all the above systems , including the 
geographic and orthodrome systems. 

In studying these coordinate systems , it is necessary to con- 
nect each of them with the geographic system for locating the in- 
termediate points of the position lines, in order to plot them on a 
map. In addition, it is necessary to know the analytical form for 
determining the coordinates of an aircraft in a geographic or ortho- 
drome system on the basis of known parameters of navigational sys- 
tems without plotting position lines on the map, as is done in auto- 
matic navigational devices. 

Bipolar Azimuthal Coordinate System 

Bearings for an aircraft, i.e., orthodrome lines diverging 
from two points on the Earth's surface with known coordinates, are 
position lines in the azimuthal coordinate system. 



74 



Let us assume that we have two points Oi and O2 on the Earth's 
surface (Fig. 1.48). 

If the map being used has been executed on a projection having 
the properties of isogonality and orthodromicity , e.g., on an inter- /78 
national projection, the indicated position lines on the map can be 
taken as straight lines originating at points 0^ and 02- 

However, satisfactory accuracy 
in determining the coordinates of 
an aircraft at the intersection of 
the bearings as straight lines on 
a map is preserved at comparatively 
small distances and only on maps 
with an international projection. 

In general, for the precise 
plotting of position lines on a 
map, let us consider the points 0^ 
and O2 as poles of an arbitrary 
spherical coordinate system. Let 
us consider the distances S from 




Fig. 1.48. Bipolar Azimuthal 
Coordinate System. 



these points to any point on the Earth's surface M as complements 
of the latitude of point M in these coordinate systems, up to 90°: 

5i = OiAf = 90° — <pi; S2 = O2M = 90° — <(2. 

In this case, the coordinates of point M in the geographical 
system are determined according to (1.64) and (1.65). 

Taking the meridian of point 0^ as the prime meridian of the 
geographic system, Oj as the azimuth a for the longitude in the 
spherical system, and a value of 90° for Si as the latitude in this 
system, let us obtain (in the geographical coordinate system) 

sin <p = cos Si cos 6 — sin Oj sin Sj sin 6, 

where 

O = 90°-<fo,; 

sin X = sin Oi sin Si sec 9. 

Given the definite value Si and substituting different values 
for ai , e.g., greater than 1°, formulas let us use the given formu- 
las to find the coordinates of the points of intersection of the 
azimuthal lines with the circle of equal distance Si in the geograph- 
ic system. 

Given another value for Si and having carried out the same 
operations with a^, we will obtain the coordinates of the points of 
intersection of the azimuthal lines with a circle of equal distance 
having this radius. 

Continuing to increase Si to a full radius of operation of a 



75 



navigational device, let us obtain the coordinates of the interme- 
diate points of the azimuthal position lines, running from pole Oi 
in the geographical coordinate system, with the longitude changed 
to the value Xqi • Introducing a correction in the values of the /79 
longitudes of the intermediate points for the indicated value Xq , 
let us obtain the longitudes of these points from the prime meridian 
of the geographic system. On a map of any projection, by joining 
the points obtained by lines running from point 0^, we will obtain 
position lines of the first family. 

In this way, it is possible to obtain the family of position 
lines from point O2 , taking it as the pole of the second arbitrary 
spherical coordinate system. 

Let us now determine the coordinates of point M in the geo- 
graphical system, based on known azimuths measured at points 0^ and 
O2, without recourse to the plotting of position lines. Let us 
first solve this problem in the spherical system of one of the poles 
of a navigational device, e.g., O2 (see Fig. 1.4-8), taking the azi- 
muth of point Oi as the prime meridian. 

According to (1.64) and (1.55), the coordinates of point M in 
this system are determined by the equations : 

sin ^1 =i sin 92 cos 8 — cos Xj cos <(2 sin 6, 

where is the angular distance of O1O2; 

sin Xj = sin X2 cos (p2 sec'^j , 

where A1X2 are respectively ai , a2 . 

From (1.65) it is evident that 

slnX; _ cos 92 8lnX2 cos 91 

8inX2 ~ C0S91 °'' .slnXj "" cosipj ' (1.66) 

Substituting into (1.52), instead of cos cjjj its value accord- 
ing to (1.64), we obtain: 

sinXo 
— , y =tg<P2COs9 — cosX2Sln6 

or 

tg <P2 = sin X2 cosec Xj sec 6 + cos X2 tg 8. (1.67) 

Since the azimuth of point M in the O2 system is considered 
known, we obtained both coordinates of point M in this system. 

For transferring to the geographical coordinate system, it is 
again possible to use (1.64) and (1.65), considering as angle 9 
the value ^^2 > ^^"i ^s the prime meridian the longitude of the point 
O2 . 

76 



It is obvious that here it is necessary to introduce a correc- 
tion into the X2 coordinate for the value of the azimuth of point 
Oi from point O2 » i.e., the corrected value of X2 will equal: 



X2j,= X.2 + Oq^. 



(1.68) 



It is also obvious that after transforming the coordinates into /80 
geographical ones, it is necessary to introduce a correction into 
coordinate X2 for the longitude of point O2 • 



>^ + ^0.' 



(1.59) 



Formulas (1.6'+) and (1.65) also make it possible to Implement 
a transfer from a spherical system with pole O2 "to the orthodrome 
system. This is necessary for determining the position of an air- 
craft relative to a given flight path. Actually, it is possible to 
consider the orthodrome system as a spherical system if we measure 
the X- and Z-coordinat es not as linear but as angular measures, 
i.e. , we take the J-coordinate as A and the Z as <^ . 

In this instance, it is advantageous to take the ^-coordinate 
of point O2 as the prime meridian and the value 90° - Z of this 
point as the angle G • The coordinates of point M in the ortho- 
drome system will then equal: 

sin A' = sin 92 cos 6 — cos X2 cos <P2 sin 6; 
^1n Z = sin X2 cos lyj sec X. 

If the Xo-coordinate , not equal to X02 > is taken as the prime 
meridian, then after transforming the coordinates according to 
(1.64) and (1.65) a correction equal to X02 is introduced in the 
X[^ coordinate. 

Goniometric Range- F i nd i ng Coordinate System 

The goniometric range-finding system is the most convenient 
system for conversion to the geographic or orthodrome system. 

Since direction and distance are measured simultaneously in 
this system, for conversion to the geographical system it is suffi- 



cient to use (1.64) and (1.65), taking the value 90° 
angle and Xqi for the prime meridian, 



'01 



for 



In this instance the value 90° - 5 is considered the latitude 
of the point M in the coordinate system with pole Oi, while the azi- 
muth of point M is considered as the longitude. In the geographical 
system, the coordinates of the point M will equal: 

sin <f = sin ipi cos 6 — cos Xj cos <pi sin 8; 
sin X = sin Xi cos fi sec tp. 

After transformation, it is necessary to introduce a correction 



77 



to the coordinate X for the longitude of point Oi : 



(1.70) 



The conversion to the orthodromic coordinate system is imple- 
mented in the same manner as was done in the bipolar azimuthal sys- 
tem after solving (1.67). 

If the radius of action of the boniometer range-finding coor- 
dinate system is small (on the order of 300-1+00 km), it is possible 
to disregard the sphericity of the Earth in converting to the or- 
thodrome system and the problem of transfer is considerably simpli- 
fied (Fig. 1.49). 

In Figure 1.49, it is evident that with known values of R and 

a in the goniometer range-finding system, the coordinates of point 

M in the orthodrome system can be determined according to the fol- 
lowing formulas : 






(1.71) 
(1.71a) 



/81 



where i|j is the direction of the orthodrome segment of the path rela- 
tive to point Oi. 

Bipolar Range- F i nd i ng (Circular) Coordinate System 

In a bipolar range-finding system (Fig. 1.50), the distance to 
two points on the Earth's surface with known coordinates is a meas- 
ured navigational parameter. 




Fig. 1.4-9. Conversion of Polar 
(Goniometer-Range-Finding) Co- 
ordinates to Orthodromic Coor- 
dinates . 




Fig. 1.50. rsipolar Range-Find- 
ing Coordinate System. 



The indicated distance is usually determined according to the 
time of passage of radio signals from the aircraft to the ground 
radio-relay equipment and back to the aircraft. 



78 



In Figure 1.50, it is evident that the task of determining the 
coordinates of an aircraft in a circular system is double-valued. 
The point of intersection of the circles of equal distance to the 
poles Oi and O2 is considered the location of the aircraft. Since 
there are two such points for any pair of circles , additional signs 
are used for choosing the actual point, e.g.: 

a) Provisional aircraft position at the moment of measurement. 

b) Tendency toward a change in distance during flight in a 
definite direction. 

In Figure 1.50, it is obvious that in flying from north to 
south, the distances i?x and i?2 will decrease at point M and increase /82 
at point Ml . 

On maps with different projections, circular position lines 
will have a different appearance. Usually they are plotted on maps 
on an oblique central and international projection on which the ap- 
proximate form of the circles is preserved. 

For plotting the indicated lines on a map with any projection, 
it is necessary to determine the coordinates of their intermediate 
points. This problem is solved in the same way as for bipolar azl- 
muthal systems , with the sole difference being that after determin- 
ing the coordinates of intermediate points , the latter are not 
joined by radial position lines, but by circular lines. 

In converting from a circular to a geographic or orthodrome 
system, it is necessary first to determine the coordinates of point 
M in the spherical system relative to one of the poles of the cir- 
cular system. 

Considering the line lO 2 as the initial meridian of this sys- 
tem, the latitude of point M in the system 1 according to (1.64) 
will be 

sin.yj = sin i(>2 cos 6 — cos A2 cos tp2 sin 8, 

where 9 is the angular distance between points 0^ and O2 ; <(> 1 > <t'2 
are 90° - Ri and 90° - i?2 , respectively. 

Carrying out simple transformations, we obtain: 



cos Xo = 



sin 92 cos 6 — sin ^^ 
cos f2 sin 



(1.72) 



Formula (1.72) makes it possible to determine the X-coordinate 
in the O2 system. Since tl^e ^-coordinate in this system is deter- 
mined directly as 90° - i?2 » it is possible to consider the problem 
solved . 



79 



The conversion to the geographic or orthodrome system is im- 
plemented by the same means as in the azimuthal and goniometer 
range-finding systems. 

Lines of Equal Azimuths 

Lines of equal azimuths (LEA) are a family of aircraft position 
lines which converge at one point on the Earth's surface, on each 
of which the azimuth of the known point retains a constant value 
(Fig. 1.51). 

For finding the location of an aircraft , it is used along one 
line of equal azimuths of two families , as is done along two bear- 
ings in an azimuthal bipolar system. 

Lines of equal azimuths were widely used in the period when the 
radiocompass (aircraft radiogoniometer), measuring the distance from 
the aircraft to the ground radio station, was the most refined nav- 
igational facility. 

Along with lines of equal azimuths , a method of determining 
the coordinates of an aircraft by plotting bearings from a radio 
station to an aircraft (taking account of the convergence of the 
meridians between them) has become widespread. 

An advan''"age of the lines of equal azimuths , in comparison with 
bearings for an aircraft, is the fact that the solution of the prob- 
lem of determining an aircraft's coordinates is independent of its 
location, whereas in order to plot bearings it is necessary to know 
the approximate coordinates of the aircraft for calculating the con- 
vergence of the meridians . 

In examining lines of equal azimuths, there is no sense in de- 
riving an analytical form of transformations for converting to the 



/83 





T 7 T 

Equator 



Fig. 1.51. Line of Equal Azi- 
muths (LEA) . 



Fig. 1.52. Determining the 
Coordinates of Intermediate 
Points of an LEA. 



80 



geographic or orthodromic coordinate system. Let us limit ourselves 
to an examination of the means of calculating intermediate points 
for plotting them on a map in order to make it possible to determine 
the coordinates of an aircraft according to the intersection of the 
lines of equal azimuths of two families on a map with any projection, 

In Figure 1.52, one of the lines of equal azimuths of a family 
converging at point M is shown. At this point, the orthodromes in- 
tersecting the equator at different angles oq, j coo* etc. converge. 

According to (1.32), it is possible to find the longitude Xg 
of the points of intersection of a family of orthodromes with the 
equator, given the values of the initial angles oq . 



According to (1.UM-), 



cos <p, = COS <p sin a. 



Or since cos 



sm a 



> 



cos w = 



sin Op 
sin a ' 



(1.73) 



Formula (1.73) makes it possible to determine the latitude of 
a point on any line of the family of orthodromes which converge at 
point Af , where the azimuth of point M is equal to the given value 
of a . 



/Qi\ 



The longitude of the indicated point can be determined accord- 
ing to (1.36a) by substituting into it the given initial angle of 
the orthodrome and the latitude obtained from (1.73). It is obvious 
that the longitude obtained will be measured from the starting 
points of the family of orthodromes. Therefore, to reduce it to 
the geographic system, it is necessary to introduce a correction 
for the longitude of the indicated initial points. 

Having solved this problem for every value of oq with given 
values of a, let us obtain the intermediate points of the family 
of lines of equal azimuths . 

The problem of determining the coordinates of intermediate 
points on lines of equal azimuths of the second family, whose plot- 
ting on a map yields a grid of intersecting aircraft position lines, 
is solved analogously. 

D i f f erence-Range- F i nd i ng (Hyperbolic) Coordinate System 

The circular range-finding system of aircraft position lines 
examined earlier is used with comparatively small distances from 
the ground radio-engineering equipment to the aircraft, since the 
sending and radio-relaying of radio signals to the aircraft over 
great distances involves technical difficulties . 



81 



The technical solution of the problem is greatly simplified if, 
instead of relaying aircraft radio signals, we send simultaneous 
radio signals from two ground radio-engineering installations, with 
their subsequent reception by the aircraft. 

However, in this instance it is advantageous to measure not 
the absolute distances from the ground installations to the aircraft, 
but only the difference .in distances to them. 

The system of position lines for the difference in the distances 
to two points on the Earth's surface is called the differenae-range- 
finding or hyperbolia system. 

The geometric locus of points, the difference in whose distances 
to two given points (foci) is a constant value equal to 2a (Fig. 
1.53), is called a hyperbola. 

The distance along the focal axis from the point of intersec- 
tion of the focal and conjugate axes to the peak of the hyperbola 
is the value "a". It is possible to designate hyperbolic aircraft 
position lines on a map by doubling the value of "a" as an ordinal / 
number. The distance along the focal axis from the focus to the 
intersection with the conjugate axis is designated by the value "e". 

To determine the position of an aircraft, two families of 
hyperbolic position lines constructed from three points forming 
pairs of focal axes are usually used. At each of these points, 
ground radio-engineering installations for synchronous transmission 
of radio signals are established. 



In order to plot hyperbolic position lines on a map with any 
projection, the intermediate points of hyperbolas in the spherical 
system of one of its foci, e.g. Fi, are determined first. Here the 
direction F \F 2 is taken as the initial meridian of this system. 

Bearing in mind the fact 
that the latitude of any point 
in the Fi system equals 90° - Si 
and 90° - S^ in the ^2 system, 
the value S2 = ^i + 2a, and 
taking the distance F iF 2 as the 
angle 9, it is possible to write 
(1.6H) in the following form: 

cos (5i + 2a) = cos Si cos 2c — cos Xj sin 5i sin 2c, 

Hence , 






V 


>1 


a 


u 




fO 


w 


B 


•H 


•H 


X 


bO 


m 


(0 




6 





Fig. 1.53. Difference-Range- 
Finding (Hyperbolic) Coordinate 
System . 



1 cos Sy co s 2c — cos (Si + 2a) 

COS A| — .—__„ , ^^ ._ ^_ 

sin 5i sin 2c 



(1.74) 



82 



Given the definite values of S as circles of equal radius and 
changing the values of 2a, it is possible to determine the value 
of X of all the hyperbolas of the family at points of intersection 
with the indicated circles. 

For conversion to the geographical coordinate system, the 
intermediate points are recalculated according to (1.64) and (1.65), 
after which they are plotted on a map and joined by smooth lines. 

Hyperbolic coordinate systems are usually used in the applica- 
tion of radionavigational devices with a large effective radius. 
Therefore , the automatic conversion of the hyperbolic coordinates 
to geographical or orthodromic coordinates is advantageous. 

The problem indicated is solved comparatively easily when all 
three foci of the hyperbolic system are situated on one orthodrome 
line (Fig. 1.54). 



According to (1.74J, 

, _ cos ^1 COS 2c J — cos (5] + 2ai) 
sin Sisln2ci 



cog ^1 cos 2c; — cos (Si + 202) 
sin Si sin2c2 



Expanding the value of the cosines of the sum of the angles 
and carrying out a reduction, we obtain: 



cos 2ci — cos 2fli + tg 5i 8ln 2gi _ cosgCj — co3:^g2 4- tg-^i sin 2^2 
sin 2c, ~ 8in2c2 



Multiplying both sides of the equation by sin 2ci'sin 2^2 and 
rearranging the terms, we obtain: 



/86 



cos 2*1 sin 2c2 — cos 2c2 sin 2 c, — sin 2c2 cos 2ai + sJn 2ci cos 203 =f 
= tg 5i (sin 2ci sin 2^2 — sin 2c2 sin 2«,^ 



or 



„ _ _sln 2c2 (cos 2ci — cos 2a i ) — sin 2ci (cos 2C2 — cos 2fl2) 
sin 2ci sin2a2 — sin 2c2 sin 2ai 



(1.75) 



The task is simplified even more if the distances FFi and FF2 
which are chosen are identical, i.e. 2ai = 2C2' In this case 



cos 2^2 — ,cos 2fli 
*^ '"" sln2«3 — sia2ai ' 



(1.75a) 



Formulas (1.75) are used for determining the coordinates of an 



83 



aircraft in a spherical system with the pole at point F, bearing 
in mind that ^ = 90° - Si. 

The A-coordinate with a known value of Si is easily determined 
on the basis of il.T^). For conversion to the geographic or ortho- 
drome system, the same formulas (1.6U) and (1.65) are used. 

The problem of conversion to the spherical (and consequently, 
to the geographic coordinate system) if the foci of the hyperbolic 
system are not located on one orthodrome line (Fig, 1.55), is much 
more complicated to solve. 

It is obvious that (1.74) can be reduced to the form: 



cosXi = 



cos Si cos 2ci — cos Si cos 20] + sin Si sin 2ai 
sin5i sin 2^1 



Carrying out simple transformations, we obtain: 

cos Xj = ctg Si ctg 2ci — ctg 5i cos 2ai cosec 2cj + sin 2«i cosec 2ci 



or 



ctg 5, 



cos Xi — sin 2ai cosec 2ci 
ctg 2ci — cos 2«i cosec 2ci 



(1.76) 



In Fig. 1.55 it is evident that in the FF2 system Aj^ = Xi + 3; /87 
therefore, the following equation is valid: 



cteS — ^°^ ^^' + ^^ ~ ^'" 2^2 cofe c 2C; 
ctg 2c2 — cos 2^2 cose^2c2 



(1.76a) 



Designating the second terms of the numerators of (1.76a) by 
X and the denominators by I and reducing to a common denominator, 
we obtain : 




Zr, F 2c, 



Srf!a^ 




Fig. 1.54. Conversion of Hyper- 
bolic Coordinates to Spherical 
Coordinates (Special Case) 



Fig. 1.55. Conversion of Hyper- 
bolic Coordinates to Spherical 
Coordinates (General Case). 



8 4 



OP (cosX, — Ai)K2 = [C0S(X, +p)-JV2]K, 

Vq cos Xj — K2A'i = Kj cos Xj cos ? — Kj sin Xj sin p — K1X2. 

Rearranging the terms, replacing sin X, by /I - cos'^X , and 
squaring both sides of the equation we obtain: 



or 



cosX, (Vj — cospK,) — Jfir2 + A^r, = — Vl — C082X, slnpKi 

cos2Xi ( y| ;^ 2yi K2 cos p + J1) + 2cos X, (Aijy, + Xi K2) X 
X (K2- n cosP) + (ATay, - ;Vi}'2)2- sinapKl 



Thus, the coordinate Xj is determined by the solution of a 
quadratic equation 



(;y2>"l-JflK2)(K2-C0SpKi) 

cos X. = ± 

K? — 2yiK2C0sp+y^ 



[/ -(>'?-2yiK2 



cos ? + yl) [(X2 Ki — ^1 r2)'' — 8iii2 p kJ] 



where 



Kf-2KiK2COsp+ yj 

Xj = sin 2ai cosec 2ci; 

X2 = sin 2^2 cosec 2c2; 
Yi = ctg2ci — Cos2aj cosec 2cj; 
^2 = ctg 2c2 — cos 2a2 cosec 2c2 . 



There is no sense in substituting the indicated values Xi , 
^2, ^1 5 and Y2 into (1.77), since this hampers its solution greatly. 
In practice, it is easier to determine the numerical values of these 
magnitudes first, on the basis of the known values of 2ai, 2a2, 
2ci, and 222, and to substitute them into (1.77). 

Knowing the coordinate X makes it possible to determine easily 
the coordinate <)) 1 , e.g., according to (1.76), keeping in mind the 
fact that (j) 1 = 90° - ^i , and then to convert to the geographic or 
orthodrome system using (1.64) and (1.65). 

Overs 1 1 -Range-F i nd i ng (Elliptical) Coordinate System 

Hyperbolic navigational systems are the most easily implemented 
technologically of all the range-finding systems. However, from 



85 



the point of view of use in flight, they are geometrically disadvan- 
tageous . 

Both families of position lines are divergent, and at distances 
exceeding 2c from the center of the system are practically directed 
along the radii of this center. This leads to an increase in error 
in determining aircraft coordinates with an increase in the distance 
from the center . 

In addition, with an increase in distance, the angle of inter- 
section of the hyperbolic lines of the two families decreases. 
This also lowers the accuracy of determining the aircraft coordinates 

Combination of hyperbolic position lines with elliptical lines 
turns out to be more advantageous (Fig. 1.56). 

It is known that the geometrical place of points, the sum of 
whose distances to two given points (foci) is a constant magnitude 
equal to 2a, is called an ettipse , The distance along the major 
axis from its intersection with the minor axis to the top of the 
ellipse, i.e., its semi-axis, is considered to be the value "a", 
in this case. The distance from the intersection of the axes to 
the foci is considered to be the "e" value. 

If, in addition to the difference in distances to the foci, 
the distance to one of them is measured, it is easy to implement 
the hyperbolic-elliptical system of position lines. 

Actually, if one distance Si is known and the difference in 
the distances hS = 2a^ , obviously the distance to the second focus 
is 

2ae = 2Si + 2ah 



where a^ is the major semi-axis of the ellipse and a-^ is the parame- 
ter a of the hyperbola. 

Obvious advantages of the 
hyperbolic-elliptical system 
include the following: 

(a) There is an absence 
of divergence in the second 
family of position lines. The 
elliptical position lines are 
closed, so that the accuracy of 
determining the aircraft's 
position along them does not 
decrease with an increase in 
distance . 



A \ ' \ ;' /> 

\ \ A j ^ ^ \' ^ I 1 



Fig. 1.56. Hyperbolic-Ellipti- 
cal Coordinate System. 



(b) Orthogonality of the 



/89 



86 



position lines appears at any point of the system. In a confocal 
hyperbolic-elliptical system, the position lines intersect only at 
a right angle . 

(c) Two foci, instead of the three for a hyperbolic system, 
are sufficient for the construction of a confocal hyperbolic- 
elliptical system. This leads to a simplification of the transfor- 
mations during conversion to a geographic or orthodrome system. 

However, in spite of the advantages of a hyperbolic-elliptical 
system indicated above, a wide distribution was not obtained. This 
was 'connected with great technical difficulties in measuring distance 
to points on the Earth's surface at distances exceeding straight- 
line geometric visibility of the object from flight altitude. 

The above problem is solved by keeping on board the aircraft 
a reference frequency (quartz-crystal clock) which permits syn- 
chronization of the transmission of radio signals from the ground 
with reference signals on board the plane. It is therefore possible 
to determine the travel time of the signals. 



Hence 5 strict stabilization of the reference frequency on 
board the aircraft is the main task for the technical implementation 
of hyperbolic-elliptical systems. 

To plot elliptical lines on a map with any projection, the 
intermediate points are determined according to the same formulas 
as the family of hyperbolas. For example, in (1.74), considering 
the second distance in it (.S2) not as the sum Si + 2aj^ , but as the 
difference 2ag - Si , 



cos X.J = 



C OS 5} CO S 2c — cos (2a — Si) 

ain .?. cin 9/- ' 



sln5i sin 2c 



(1.78) 



where Xi is an angle with the vertex at point Fi 
major axis of the ellipse. 



measured from the 



Given the different values of ^i and determining the values of 
Xi for each of them with a constant value of 2a, we shall obtain 
intermediate points of an ellipse in a spherical system Fi. 

Changing the value of 2a and performing these operations with 
Si, we obtain intermediate points of the next elliptical position 
line, etc. Recalculation of intermediate points is implemented in 
the geographic system according to (1.64) and (1.65) as in other 
cases analyzed by us. 

In the hyperbolic-elliptical system, the conversion to the 
geographic or orthodromic coordinate system is very simple. 

In fact, the value of Si and the parameter 2a in this system 



87 



are measured. Therefore, (1.78) is useful for the problem of 
calculating the spherical coordinates of an aircraft along measured 
parameters and for a subsequent transfer to the geographic or 
orthodrome system. 

8. Elements of Aircraft Navigation /90 

Aircraft flights are carried out in airspace. The physical 
composition of airspace, as well as the speed and direction of its 
shift relative to the Earth's surface, exert a substantial influence 
on the trajectory of aircraft movement in a geographic or orthodromic 
coordinate system. 

Until recently, the direct measurement of the speed and direc- 
tion of aircraft movement relative to the Earth's surface was a 
problem. At the present time, this problem has been solved. How- 
ever, it is not advisable to install the complex and expensive 
equipment which measures the indicated parameters on all aircraft. 

In solving navigational problems, parameters of aircraft move- 
ment relative to the airspace are usually measured to the greatest 
extent possible, and then additional parameters of the movement of 
the aircraft which are connected with movemert in airspace are 
found . 

Summarizing the measured parameters of aircraft movement , the 
value and direction of the speed vector of the aircraft relative to 
the Earth's surface are found. 

The parameters of aircraft movement with which we must concern 
ourselves in carrying out aircraft navigation are called elements 
of aircraft navigation. 

Elements of aircraft navigation are divided into three groups 
which determine the direction, speed, and altitude of flight. 

Elements which determine Flight Direction 

The basic element which determines the direction of aircraft 
movement in airspace is called the aircraft course. 

The aircraft course (generally designated by y) is the angle 
between the direction of a meridian on the Earth's surface and the 
direction of the longitudinal axis of the aircraft in a horizontal 
plane . 

Usually it is considered that the airspeed vector of an air- 
craft in the plane of the horizon coincides with the direction of 
the longitudinal axis of the aircraft, although this is actually 
not entirely true. Therefore, an understanding of the course often 
coincides with an understanding of the direction of the flight air- 
speed vector . 



88 



Depending on the reference system chosen, the following special 
varieties of aircraft courses can be distinguished: 

(a) A tvue aouTse (TC) is measured from the northern end of 

a geographic meridian which passes through the point of intersection 
of the Earth's surface with the vertical of the aircraft. The lat- 
ter is usually called the position point of the aircraft (PA): 

(b) The orthodrome course (OC) is measured from the northern 

end of a geographic meridian of the starting point of a rectilinear /91 
(orthodrome) segment of the path or from another conditionally 
chosen (reference) meridian along which the zero point of the 
course-reading scale is established. 

(c) The magnetic course (MC) is read from the northern end 
of the magnetic meridian which passes through point PA, 

In addition to these varieties of aircraft courses, there is 
another concept, the compass course (CC), i.e., a course based on 
the responses of a compass. In textbooks on aircraft navigation, 
the concept of compass course has included only magnetic compasses, 
but we have broadened this concept to include all methods of 
measuring an aircraft course with an Instrument . 

Aircraft courses are measured by three different methods, i.e., 
stabilization of the zero reading of the compass along the meridians: 

Magnetic course, by means of magnetic systems. 

True course, by means of astronomical systems. 

Reference course, by means of gyroscopic devices. 

All of these methods have instrumental errors or a deviation 
designated by A^,. Individual components of errors in the course 
devices are components of the deviation. 

Any of the three types of aircraft courses can be obtained 
from responses of a course device, allowing for its deviation, e.g., 



OC 
TC = 
MC 



-CC„rth+ ^o\ 
= CCastr + ^c > 
= CCmag + ^c J 



(1.79) 



In the general case , 

Y = CC + Ac 

As a correction for any measurement , the value A^ is considered 
positive when the compass underestimates the value of the measured 
magnitude, and negative when the compass readings are too high. 

89 



In the future, when we study the relationship among the three 
types of aircraft courses, we will consider that the value of each 
has been corrected for the deviation of the device. 



The interrelationship between magnetic and true flight courses 
is established with the least difficulty, since the^e courses are 
measured from the meridians which pass through point PA (Fig. 1.57a). 

In Fig. 1.57a the northern geographic meridian is designated 
by Pjj , the direction of the magnetic meridian by P^. 

Since the magnetic meridian is shifted to the left relative 
to the geographic meridian, the magnetic declination at the given 
point is negative. With a positive deviation, the magnetic merld- /92 
ian is shifted to the right relative to the geographic meridian. 

From the figure, it is evident that 



MC=TC-Am 
TC=MC + A 



M- J 



(1.80) 



In the case of Af^ , the value is negative; therefore, the abso- 
lute value of the true course turns out to be less than the magnetic 
course . 

Converting from the true or magnetic course to the orthodrome 
course (or vice versa) is more complex (Fig, 1.57, b). 




Fig. 1,57. Interrelationship of Aircraft Courses; a: Magnetic 

and True; b: True and Orthodrome. 

The direction of the geographic meridian passing through point 
PA is designated by P^', the direction of the reference meridian 
^r . m . is shown by a dotted line which intersects the geographic 
meridian at angle 6 . Therefore 



TC = OC- 
OC = TC + 



:;) 



(1.81) 



90 



The value 6 , the angle of convergence of the meridians , is 
considered positive when the direction of the geographic meridian 
at point PA is proportional to the reference meridian extended 
clockwise (to the right) and negative when the geographic meridian 
is shifted to the left. 

On small path segments (500-600 km), the angle of convergence 
of the meridians is approximately equal to: 

S =Ur.m.- ^PA) sin cp^^ (1.82) 

In general, for converting from the orthodrome course to a 
true course or vice versa, it is necessary to determine the longi- 
tude of the starting point of the orthodrome of each rectilinear /93 
path segment, e.g., on the basis of (1,34), or, if the azimuth of 
the orthodrome is known at the starting point of the path segment, 
on the basis of (1.33a). 

The angle of convergence of the meridians between the starting 
point of the path section Mi and any moving point M on a section, 
according to (1.33a), will be equal to 

B = o — «! = arctg (tg >. cosec f) — arctg (tg X-j cosec (pi), (1.83) 

where A and Xi are measured from the starting point of the ortho- 
drome . 

If the longitude of any other point on the Earth's surface, 
e.g., the point of take-off of the aircraft, is taken as the refer- 
ence meridian, the angle of convergence of the meridians will be 
determined as the sum: 

6 = Si + 62 + . . . 6 ^ 



where & i ; &2 ■ • • sltb the angles of convergence of the meridians 
between the starting and end points of the preceding path segments 
determined on the basis of (1.83); 6^ is the angle of convergence 
of the meridians from the starting point to the moving point of the 
last path segment (on the basis of the same formula). 

The angle of convergence of the meridians calculated in this 

way allows conversion from a true course to an orthodrome course 

and vice versa at any flight distance with any number of breaks in 
the path. 

For conversion from an orthodrome course to a magnetic course 
and vice versa, (1.80) and (1.81) are used, from which it follows 
that 



91 



OC =P1C +A« + 8; 



MC 



OC 



+ 4« + 5; 1 

-A„-8. ) 



(1.84) 



The sum of the magnitudes Aj^ + 6 is taken as the overall cor- 
rection for conversion from a magnetic to an orthodrome course and 
vice versa, and is designated by A.... Then (1.84) assumes the 
form : 

OC = MC + A; 
MC = OC-A. 



The dvlft angle is the second element determining the direction 
of aircraft movement . 



In an aircraft, the angle between the airspeed vector and the 
groundspeed vector in a horizontal plane is called the drift angte 
(Fig. 1.58). In general, the drift angle is designated by the 
Latin letter a. In those instances when special designations for 
courses are used in the solutions of navigational problems the 
drift angle is designated by the Russian letters for DA. 

In Fig. 1.58. OP^ is the direction of the meridian at point 
PA; OT is the direction of the airspeed vector and the longitudinal 
axis of the aircraft; OW is the direction of the groundspeed vector 
relative to the Earth's surface; u is the wind speed vector. 



/94 



The drift angle of an aircraft is considered positive when the 
groundspeed vector (vector of aircraft movement relative to the 
Earth's surface) is further to the right of the longitudinal axis 
of the aircraft and negative if it is further to the left. 

The angle between the northern end of the meridian and the 
groundspeed vector or the vector of the speed of the aircraft rela- 
tive to the Earth's surface is called the flight angle (FA). The 
general designation for the flight angle is i|) . 

The flight angle, like the course of 
the aircraft, can be measured from the 
reference meridian, the geographic meridian, 
and a magnetic meridian passing through 
point PA. 

Special values of the flight angles 
have the following designations: 




Fig, 1.58. The Path 
Angle of Flight. 



(a) The orthodrome flight angle is 
OFA , with obligatory indication by a sub- 
script of the longitude of the reference 
meridian. For example, OFAi^q = 96°. 



92 



(b) The true flight angle is TFA. 

(c) The magnetic flight angle is MFA. 

In Figure 1.58, it is evident that the flight angle is generally 

or in special cases : 



OFA = OC 
TFA = TC 
MFA = MC 



+ DA>) 
+ DA|> 
+ DA;) 



(1.85) 



The interrelationship between the special values of the flight 
angles and the method of conversion from one special value to 
another corresponds completely to the interrelationship between 
special values of aircraft courses: 

OFA = TFA + 6=MFA+ A; 
TFA = OFA - 5 = MFA + Am; 
MFA = OFA - A = TFP - A^ . 



In determining the direction of aircraft movement relative to 
the Earth's surface, it is sufficient to know the course of the air- 
craft as the angle between the direction of the meridian and the 
lateral axis of the aircraft, and the drift angle as the angle be- 
tween the lateral axis of the aircraft and the direction of its 
movement. These elements, together with elements of flight speed, 
make it possible to determine approximately the speed and direction 
of the wind at flight altitude . 

For precisely determining the wind at flight altitude, it is 
necessary to separate the part of the drift angle of an aircraft 
caused by the wind. It is obvious that to do this it is necessary 



/95 




Fig. 1.59. Moment and Control 
Force with Assymetry of Engine 
Thrust . 




Fig. 1.60. Lateral Glide 
with Transverse Roll. 



93 



to determine the direGtion of the airspeed vector of the aircraft 
as the course and glide of dynamic origin, arising in flight. 

There are several causes of lateral glide in aircraft during 
flight. The basic causes are the following. 

1. Assymetry of the Engine thrust or Aircraft Drag (Fig. 1.59) 

Let us assaime that with symmetrical drag, one of the engines 
has a somewhat greater thrust than the other. The difference in 
thrust AP will produce torque in the aircraft relative to the ver- 
tical axis, i.e., the course of the aircraft will change. 

For stabilizing the flight direction, a moment must be applied 
to the empennage of the aircraft which is equal in magnitude and 
opposite in direction to the moment of thrust, i.e. 



where AP is the assymetry of the thrust; F^ is the control force; 
Lj is the arm of thrust assymetry (from the axis of the engine to 
the axis of the aircraft); and L^ is the arm of control (from the 
center of the empennage area to the center of gravity of the air- 
craft ) 



/96 



The lateral force which causes gliding of the aircraft will 



be: 



F^ = 



APLr 



(1.86) 



In an analogous manner, the force which causes gliding of an 
aircraft with assymetry of drag arises. 

In this instance, the moment of rotation of an aircraft causes 

excess drag on one wing of the aircraft. The distance from the 

lateral axis of the aircraft to the center of its application is 
the arm of this force. 

2. Allowable Lateral Banking of an Aircraft in 
Hor i zon ta 1 Flight. 

With allowable lateral banking (Fig. 1.60), the horizontal 
component of the lift will appear: 

^,= 018 p. 

where G is the weight of the aircraft and 6 is the angle of lateral 
banking . 

For example, with a flying weight of the aircraft of 75 t, the 
allowable banking in horizontal flight, equal to 1°, causes a 



91+ 



lateral component of lift >« 1.3 t. 

3 . Cor i o 1 i s Force 

During flight in the Earth's atmosphere, as a result of the 
diurnal rotation of the Earth's surface, a lateral Coriolis force 
acts on the aircraft : 



/\-, = 2<i>e BTm sin <(, 



(1.87) 



where (Og is the angular velocity of the Earth's rotation; W is the 
speed of the aircraft relative to the Earth's surface; m is the 
mass of the aircraft; and (j) is the latitude of the point PA. 

k. Two-dimensional Fluctuations in the Aircraft Course 

During two-dimensional rolls (without banking), an aircraft 
(as a result of inertia) tries to maintain the initial direction of 
movement. This causes lateral gliding of the aircraft V^, equal to; 



V^= V sin At, 



(1.88) 



where V is the speed of the aircraft relative to che airspace, and 
Ay is the magnitude of the change in the aircraft's course. 

The indicated lateral gliding of an aircraft gradually dies 
down as a result of acceleration caused by the lateral airflow 
over its surface . 



5. Gliding During Changes in the Lateral Wind Speed 
Component at Flight Altitude 



/97 



This type of gliding arises as a result of the inertia of the 

aircraft . First , lateral airflow over the aircraft or something 

similar (gliding in airspace) will appear, followed by a change in 
the direction of aircraft movement. 

From the above five examples of lateral aircraft gliding, 
constant lateral forces are the causes of the gliding In the first 
three cases, while abrupt beginning and gradual diminution of 
gliding are the causes in the last two cases. 

The magnitude of stable gliding with a constantly acting 
lateral force can be determined according to the formula 



or 



* - V c,Sp 



(1.89) 



95 



where Z is the operative lateral force; Cg is the coefficient of 
lateral drag of the aircraft; S is the area of the longitudinal 
section of an aircraft with a vertical plane; and p is the mass 
density of the air flight altitude. 

To calculate gliding in flight, it must be integrated and 
converted to angular glide (agi): 



•^V=^' 



(1.90) 



where V^ is the lateral component of the airspeed and V^ is the 
longitudinal component of longitudinal speed. 

The direction of the airspeed vector is determined by the 
formula 



Tv =T + «gJ 



(1.91) 



The drift angle of an aircraft, whose cause is the action of 
the wind at flight altitude, will be: 



« = 'f — Tk = 4' — Tf — flgl 



(1.92) 



As we have already said, determining the gliding of an air- 
craft in airspace is necessary only for precise measurements of 
wind speed and direction at flight altitude. For the purposes of 
aircraft navigation, there is no need to separate out the causes of 
lateral aircraft movement. 

Elements Which Characterize the Flight Speed of an Aircraft 

The flight speed of an aircraft is measured both relative to 
the airspace surrounding the aircraft and relative to the Earth's 
surface . 

Measuring the speed of aircraft movement relative to the air- 
space is significant both from the point of view of flight aero- 
dynamics (stability and control of the aircraft) and from the point 
ov view of aircraft navigation. 

It is known that the lift of a wing, the drag of an aircraft, 
and the stability and controllability of an aircraft depend on the 
square of the airspeed. 

For example , at flight speeds which are significantly less 
than the speed of sound, the drag of an aircraft is determined by 
the formula 



/98 



96 



2 

where Q is the lateral drag of an aircraft, o^ i^ ''^^e drag coef- 
ficient, S is the maximum area of the lateral cross section of an 
aircraft, and p is the mass air density at flight altitude. 

The value- -^—^ — characterizes the aerodynamic pressure of the 
atmosphere on the surface of an aircraft , 

All the aerodynamic characteristics of an aircraft are deter- 
mined relative to this value. 

In determining the aerodynamic characteristics of an aircraft, 
the- aerodynamic pressure (and therefore the speed of flight) reduce 
to conditions in a standard atmosphere, i.e., to flight conditions 
near the Earth's surface, with an atmospheric pressure of 760 mm Hg 
and an ambient air temperature of 15° C. Therefore, speed indicators 
which measure airspeed on the basis of aerodynamic pressure are 
calibrated according to the parameters of a standard atmosphere. 

With an increase in flight altitude, air density decreases. 
To preserve aerodynamic pressure at flight alti'tude, it is necessary 
to increase flight airspeed, although responses of the airspeed 
indicator which measure airspeed on the basis of aerodynamic pres- 
sure remain constant . 

Flight airspeed which is measured on the basis of aerodynamic 
pressure and which influences the aerodynamics of the flight of the 
aircraft is called aerodynamia speed (^aer^* 

It is necessary to consider, however, that with an increase in 
flight speed, especially in approaching the speed of sound, aero- 
dynamic speed does not completely correspond to the aerodynamic 
characteristics of an aircraft which are determined under the condi- 
tions of a standard atmosphere and which are inherent in flight 
speed. This is becuase the factor of air compressibility begins to 
exert an influence. To ensure safe pilotage of the aircraft in 
these instances, a corresponding correction is introduced into the 
indications of aerodynamic speed. 

For the purposes of aircraft navigation, it is necessary to /99 
know the actual speed of an aircraft in space. 

The actual airspeed, which we shall call simply airspeed (V) , 
can be obtained from aerodynamic speed by introducing corrections 
for the change in air density with flight altitude and temperature: 



V- V, 



'aeti-^»^H + AV^, + AV^c^p. 



97 



where ^^er ^^ ^^^ aerodynamic speed; AV^ is the correction for 
speed as a result of flight altitudes; hV^ is the correction for 
speed as a result of air temperature; and A7g „„„ is the correction 
for speed as a result of air compressibility 



cmp 



Correction for flight altitude is of basic i,mportance . Cor- 
rection for air temperature is significantly smaller and is intro- 
duced only in those cases when the air temperature at flight alti- 
tude is significantly different from the temperature calculated for 
this altitude . 

At the present time, there are devices which indicate flight 
airspeed directly, taking altitude into account. Corrections must 
be introduced in the responses of these devices only for instru- 
mental errors of the devices and (in individual cases) for a dis- 
crepancy between the actual air temperature and the calculated 
temperature at a given altitude . 

In published textbooks on aircraft navigation, airspeed has 
been classified as "indicated" (measured on the basis of aerodynamic 
pressure) but true, and as "indicated , corrected for methodological 
and instrument errors." 

Since there are now devices which measure both these speeds, 
each of them is "indicated". In addition, increasing airspeeds 
have required the introduction of corrections in aerodynamic flight 
speed. This has caused a new classification of air speeds. 

The speed of aircraft movement relative to the Earth's surface * 
is called flight groundspeed (.W) . 

Flight groundspeed can be measured directly by means of Doppler 
or inertial systems , determined by sighting along a series of land- 
marks on the Earth's surface, and also calculated on the basis of 
flying time between two landmarks on the Earth's surface. In 
addition, groundspeed can be determined by adding the airspeed and 
wind vectors , if the wind speed and direction at flight altitude 
are known . 

Navigational Speed Triangle 

The interrelationship of the elements of flight direction and 
speed in the chosen frame of reference of aircraft courses is 
clearly illustrated by a navigational speed triangle. 

In Fig. 1,61 a navigational speed triangle is shown for a /I 
general case, i.e., independently of the meridian which is used as 
the basis for measuring an aircraft course. 

Straight lines OPjj and OiP^ in the figure show the direction 
of the meridian at point PA; V is the airspeed vector; W is the 
groundspeed vector; y is the course of the aircraft (C), a is the 



98 



drift angle (DA); \p is the flight angle (FA), 6 is the direction 
of the wind vector relative to the meridian for reading the air- 
craft course; 6^ is the flight angle of the wind (WA) read from the 
given line of the path; and 6y = a + 6^ is the course wind angle 
(CWA), read from the longitudinal axis of the aircraft. 

A speed triangle can be solved graphically by construction of 
vectors on paper or by a mechanical apparatus, using a special de- 
vice (a wind-speed indicator which is a combination of rules, dials, 
and hinges with movable and immovable joints). 

A speed triangle is solved analytically on the uasxs of a 
known sine theorem. From Figure 1.63, it is clear that in the given 
case the sine theorem will have the form: 



sing 
U 



sin 8,,, 



sinb- 



(1.93) 



From (1.93) the value of the drift angle and the flight ground- 
speed are easily determined on the basis of known values of the air- 
craft course, airspeed, and the speed and direction of the wind. 



Now, let us define the path angle of the wind 

8^ = S-^/. 



(1.94) 



The drift angle of an aircraft according to (1.93) is deter- 
mined from the formula 



slna= — sill 8^ 



(1.95) 



The value of the flight groundspeed is then easily determined 



Vsln8, 

W= ■■ I , 

sln5,„ 



(1.96) 




Fig. 1.61. Navigational 
Speed Triangle 



These problems are especially 
simple to solve with slide rules having 
a combination of sine logarithms with 
a logarithm scale of linear values. 

In this case , combining the log- 
arithm of the sine of the wind angle 
with the logarithm of the airspeed, we 
obtain directly: 



/lOl' 



99 



Igsinfl — Iga = IgsinS^— Ig K= IgsinB^ — Ig'W 

or on scales of navigational rulers with the designations used with 
special values for aircraft courses. 



D A __W A DA+WA [ Igsln 

y vy igsp 



(1.97) 



To determine wind speeds and directions at flight altitude on 
the basis of known values of airspeed, groundspeed, and drift angle, 
let us use Figure 1.62. 




that 



From the figure, it follows 

0D= V^ cos DA; 
OiD= V sin DA; 
DM = OM — OD= W—Vcos DA. 



Fig. 1.62. Determining the 
Angle and Speed of the Wind 
with Known Values of the 
Groundspeed and Drift Angle 
of the Aircraft . 



Therefore , 

OiD VsinDA 

tg W A = -^— = — 

DM W— KcosDA 



(1.98) 



The flight angle of the wind determined in this way permits 
the further solution of problems on the basis of the sine theorem 
[Equation (1.93)]. 

With small drift angles (practically up to 10°), cos DA !=a i, 
i.e., it is possible to consider in approximation that 



tgWA=- 



Ksln DA 
W -V 



(1.99) 



For solution on a slide rule, (1.99) is reduced to the form: 

tg WA sin DA 

V ^ w-v' 

r^tgWA-lg V=lgslnDA— lg(W^— K) 



Translator's note; Ig = log. 



100 



or on a slide rule. 



DA. WA 
W—V V 



Ig sin Ig tg 



Igsp 



After finding the flight angle of the wind, the value of the 
wind speed is determined on the basis of the sine theorem. 

Elements Which Determine Flight Altitude 

The flight altitude of an aircraft (H) is measured from a 
special initial level of the Earth's surface. The initial level 
for measuring flight altitude is chosen depending on the purposes 
for which it is measured. 

For example, in order to distribute the counter and incidental 
movements of aircraft in airspace (flight echelons), the initial 
level for measuring the altitude on each aircraft must be general. 
To ensure the safety of flights of individual aircraft at low alti- 
tudes, it is desirable that the flight altitude be measured from 
the surface of the relief over which the aircraft is flying. In 
making an approach to land at an airport, flight altitude is meas- 
ured from the level of the landing point. 

Usually 2 or 3 kinds of altitudes are measured at the same 
time. Therefore, it is necessary to classify them and to establish 
a relationship between them. 

At the present time, the following kinds of altitudes are dis- 
tinguished (Fig. 1.63): 



/102 




Level!r=760 mm Hg 



Fig. 1.63. Interrelationship of Different Systems for Measuring 

Flight Altitude 



101 



(a) Absolute flight altitude (^abs^ ^^ measured from the mean 
level of the Baltic Sea in the same way as the height of a relief 
on the Earth's surface. 

(b) Relative flight altitude (^rel) is measured from the /103 
level of the take-off or landing airport, 

(c) Tvue flight altitude "^tr" i^ measured from the surface 
of the relief over which the aircraft is flying. 

(d) Conventional barometria altitude "Hy^^" is measured from 
the conventional barometric level on the Earth's surface, where the 
atmospheric pressure is equal to 760 mm Hg . 

Absolute, relative, and true flight altitudes are determined 
by barometric altimeters with correction of their readings for 
instrumental and methodological errors . The latter can also be 
measured by radio altimeters and aircraft radar equipment or deter- 
mined by aircraft sighting devices. There is a relationship be- 
tween the three indicated altitudes which makes it possible to 
switch from one kind of altitude to another. 

Conventional barometric altitude is measured by barometric 
altimeters without considering methodological errors. Therefore, 
it has no direct connection with the first three kinds of altitudes, 
and at a high flight altitude it can be distinguished from the abso- 
lute altitude closest to it by 900-1000 m. 

The main advantage of a conventional barometric altitude is the 
convenience of using it for echeloning flights according to altitudes 
when the important thing is not the precise measuring of altitude 
but only the preservation of safe altitude intervals between 
neighboring echelons. The latter condition is satisfied, since if 
we permit two aircraft to meet in one region and at one altitude, 
the methodological corrections in these aircraft will be identical. 
Therefore , such a meeting cannot occur if aircraft maintain dif- 
ferent altitudes based on instruments. 

From Figure 1.63 it is evident that true flight altitude is 
distinguished from absolute flight altitude by the height of the 
relief over which the aircraft is flying, and from relative alti- 
tude by the height of the relief above the airport level from which 
relative altitude is measured: 



tr 
tr 



= H 

= H 



abs 
rel 



H 
AH 



r; ) 



(1.100) 



where H.^ is the altitude of the relief above sea level; AH^ 
height of the relief above the level of the airport. 



is the 



102 



Relative altitude is distinguished from true altitude by the 
height of the relief, while it is distinguished from absolute alti- 
tude by the height of the airport above sea level: 



^rel 
^rel 



= fftr + 
= •^abs ■ 






r; \ 

^air •' 



(1.101) 



Finally, absolute flight altitude can be determined on the 
basis of the values of true or relative flight altitude: 



/lOit 



^abs 
^abs 



--H 
■■Ht 



tr + ^r; I 

tr + ^aivf 



(1.102) 



Calculating Flight Altitude in Determining Distances 
on the Earth's Surface 

In measuring directions on the Earth's surface, flight altitude 
does not exert a direct influence on the value of the measured 
angles or on the accuracy of the measurements. 

Actually, by direction on the Earth's surface we mean direction 

of the line of intersection of the horizon plane with the plane of 

a great circle (orthodrome) which joins two points on the Earth's 
surface . 

Since the vertical at any of these points on the indicated 
line lies in the plane of a great circle, flight altitude does not 
exert an influence on the direction of the orthodrome and therefore 
on direction on the Earth's surface. 

In measuring distances on the 
Earth's surface, flight altitude can 
play an important role and can lead to 
large measurement errors if we do not 
allow for errors in flight altitude 
(Fig. 1.64). 

In the figure, straight lines OAi 
and OBi are verticals of the position 
of an aircraft at points A and B. 

Obviously the distance S between 
points ^1 and Si at flight altitude is 
greater than distance S between points 
A and B on the Earth ' s surface : 

Fig. 1.6U. Calculating 

Flight Altitude in 

Determining Distances . 




103 






Re+H 



(1.103) 



whence 



-(-a 



jLS 

or '45 = 5 — 



(1.104) 



where i?„ is the radius of the Earth (equal to 6371 km) and H is the 
flight altitude. 

Each kilometer of flight altitude lengthens the path between /105 
points on the Earth's surface by a value expressed in percent: 



MOO 
6371 ' 



i 0.016%. 



For example, at a flight altitude of 10 km, a distance on the 
Earth's surface equal to 3000 km lengthens to the value 



3000-100,016 
6371 



= 4,8 KM, 



The indicated lengthening of the path of the aircraft does not 
exert a substantial influence on the time of the aircraft flight 
along the path. The influence of flight altitude on determination 
of the position of the aircraft by rangefinding and, especially, 
hyperbolic devices turns out to be more substantial. 

Let us assume that a rangefinding device is located at point 
A on the Earth's surface, while the aircraft is located at point 
Bi, at flight altitude. 

As is evident from Figure 1.66, the distance from the ground 
radio-engineering apparatus to the aircraft R along a straight line 
will equal AB\, while the distance along the Earth's surface S is 
equal to AB . 

Let us drop a perpendicular from point A on the Earth's surface 
to point D on the vertical OBi . Obviously, 

AB\ = Am + DB\, 



since 



ABx = R; 

D = /JySin 

DBi = Rq— /?gCOs S + H, 



AD = /JySin 5; 



l0t^ 



Then __^ 

/?=K«|sln2S + (/?e-/?e<:os 5 + //)». ^^ ^^^^ 

With STaall angular distances 5 (up. to &° along the arc of the ortho- 
drome), when B sin S ?« 5 , while cos S pa 1 , (1.105) takes the form: 



R = Vs^+f^- (1.106) 

Figure 1.66 can likewise be used for determining the maximum 
distance of geometrical visibility of objects on the ground from on 
board the aircraft, or of an aircraft from the Earth's surface. 

It is obvious that with maximum visibility, line ABi must be 
tangent to the Earth's surface, i.e., it is located in the plane of 
the horizon. In this case, angle OABi will be a right angle. 



Therefore , 



OA2 + AB\ = OBl 



/105 



d2 



%+{ffe^SS)2 = (RQ + Hyi. 



Expanding the right-hand side of the equation, we obtain: 

Considering that at distances up to 600-700 km, R^igS rs S, and 
disregarding the value H^ as negligibly small in comparison with 
2RqH , we obtain the approximate formula 

S = V2Rjf. (1.107) 

Substituting in (1.107) the value of the radius of the Earth 
(6371 km) we obtain: 

5= Vm42H =113 Vff. 

Bearing in mind that as a result of the refraction of light 
or radio waves in a vertical plane, the distance of geometrical 
visibility increases approximately by 8%, the practical result will 
be: 



105 



S . =172 Vh. 

VIS 



(1.108) 



Formula (1.108) determines the limits of applicability of 
(1.101+) or (1.106). Since we have agreed to consider cos 5 = 1 and 
R sin 5 = £■ up to 5 = 6° , which on the Earth's surface corresponds 
to 666 km, it is obvious that at flight altitudes up to 25 km it 
is always possible to use (1.106). 

It is necessary to use the precise formula (1.104) at distances 
of more than 700 km. This is possible at flight altitudes exceeding 
25 km. 




Fig. 1.65 



Calculating Flight Altitude in Determining the Path 
Length of Electromagnetic Wave Propagation 



Let us pause now to discuss the influence of flight altitude 
on the accuracy of measuring distances at very small ranges, i.e., 
in cases when long radio waves capable of traveling around the 
Earth's surface are used (Fig. 1.65). 

In the figure , ground radio engineering equipment is located 
at point A on the Earth's surface; the aircraft is at point B at 
altitude H. Line AB is the curve of propagation of a radio wave 
front . 

If we conditionally move the Earth's surface to the right by 
a value equal to H/2, then the line of radio wave propagation be- 
comes concentric with the Earth's surface and will have a radius of 
curvature i?i = R^ + H/2. 

Therefore, the increase in distance from point A to point B 
can be considered as a lengthening of the orthodrome at a flight 
altitude equal to H/2, i.e.. 



/.10 7 



AS = Si — S = S-;^- 
2/?e 



106 



Elements of Aircraft Roll 

It is known that the radius of aircraft roll in airspace at a 
given banking 3 equals: 

V2 



*tgP 



If a flight is carried out 
rolling of an aircraft through 



^ 



with a counter or incidental wind, 
an angle of 90° involves an increase 
or decrease in the mean radius of 
roll of an aircraft relative to the 
Earth's surface (Fig. 1.66), 




u,'-Z50 km/hrs 
J 



u,'0;y'eoo 'k'm/hT 



u, - +250^ m /h r 

Fig. 1.66. Deformation of 
the Roll Trajectory in the 
Presence of Wind. 



craft from the original flight 
will be a deviation opposite to 



In fact , for a change in the 
direction of the groundspeed vector 
of an aircraft by 90° , with a shift 
from the plane of incident wind to 
a lateral plane, it is necessary to 
execute a roll of an aircraft to the 
right or left through an angle of 
90° + DA, and in changing from the 
plane of incident wind to a lateral 
wind to a lateral plane through an 
angle of 90° - DA. 

During rolling of an aircraft 
in airspace , in the first instance 
there will be a deviation of the air- 
direction; in the second case, there /lO i 
the original. 



Exampte . Let us examine the roll of an aircraft through 90°, 
with a flight airspeed of 600 km/h and with a counter and incident 
wind speed of 250 km/h (70m/sec). 

According to (1.6), the radius of the aircraft in airspace 
with banking of 15° will be 



1672 



9,81 »g 15» 



= 10 500 m 



The drift angle at the end of rolling through 90° will have 
the following value : 

250 
DA=arc,g- = 23''. 



107 



Therefore, in the first case it will be necessary to turn the air- 
craft through 113° , and in the second case through 67° . 

The angular velocity of roll at F = 600 km/h (167 m/sec) and 
R = 10,500 m will be 

^ K-57,3 167-57,3 „„^ , 



Let us determine the additional shift of the aircraft as a 
result of wind during rolling in the first instance: 

70 in/sec-113° 
0,9deg/sec 



and in the second instance : 

70-67 

Obviously, during roll (in the first case through 113° and in 
the second case through 67°) the movement of the aircraft in direc- 
tion X will not be identical, since 

Bjc = R sin yP. 

Then the general path of an aircraft in direction X will equal 
In the first case , 



In the second case , 



Rjc = 10,5 sin 113° + 9 = 18,5 km 



/?^= 10,5 sin 67° — 5 = 4,5 km 



Let us now determine the lateral shift of the aircraft R^ 
during roll: In the first case. 



and in the second case 



/?« = /?+■/? sin 23° =14,5 km 



Rz = R — R sin 23' = 6 k m. 



For comparison, let us examine the roll of an aircraft through 
90°, with a radius calculated not on the basis of airspeed, but on 
the basis of groundspeed: 

B7= V± Ux. 



108 



In the first case, the radius of rolling is 



/109 



„ 2372 

'^^ 9,81 tg 15° =21 km 



and in the second case 



/? = 



972 



9,81 tg 15' 



— =3,5 km 



Let us compile a table with the results obtained: 



Roll 
parameters 



X 

z 



K;(=+250km/h 



10,5 


10,5 


10.5 


21 


10,5 


18,5 


10,5 


14,5 



«^=— 250 km/h 



10,5 
3,5 
4,5 
6 



From the table , it is evident that the results of the calcu- 
lations carried out on the basis of the groundspeed are much closer 
to the actual results than calculations on the basis of airspeed. 




Calculations o 
basis of groundspee 
high as 200-300 km/ 
a new line of fligh 
ried out with an ac 
km. Some inaccurac 
only in the lateral 
ever, this is not o 
nif icance , since th 
deviation coincides 
line of flight . 



f roll on the 
d with winds as 
h, when entering 
t , will be car- 
curacy of 1-2.5 
ies arise , but 

direction. How- 
f practical sig- 
e direction of 

with the new 



Fig. 1.67. 
Aircraft to 



Approach of an 
a Given Line 

with the Presence of an Ap 

proach Angle . 



With a decreas 
of roll, the trajec 
according to the gr 
closer to the actua 
aircraft roll. The 
future we will proceed from flight groundspeed in c 



e in the angle 
tory calculated 
oundspeed comes 
1 trajectory of 
ref ore , in the 
alculating roll 



In aircraft navigation, including maneuvering before landing, 
it is necessary to solve three types of problems, taking into ac- 
count the roll trajectory. 



109 



1. Combination of Roll with a Straight Line 

Let us assume that an aircraft is approaching a given line of 
flight at a definite angle (Fig. 1.67). 

It is obvious that the angle of roll of the aircraft for fol- 
lowing along the given line is equal to the approach angle (a). 
Let us determine the distance (Z) from the given line on which it 
is necessary to begin the roll so that the roll trajectory will be 
joined with the given line. 

In Figure 1.67, it is evident that this distance is equal to: 



/llO 



or 



Z = R — R cos o 
Z = /?(l— coso). 



(1.109) 



Example . An aircraft approaches a given line of flight with 
a groundspeed of 900 km/h at a 25° angle. Determine the lateral 
distance from the line of flight at which it is necessary to begin 
a roll for a smooth approach to the line. 



Sol ution 



R = 



2502 



= 26,5 km 



9,81 tg 15° 
Z = 26 ,5(1 — cos 25°) = 2, 46 km 



2. Combination of two rolls 

If, during flight along a given flight line, a deviation from 
it occurs and it is necessary to approach the given line by the 
shortest trajectory, an approach maneuver is used which is a combi- 
nation of two rolls (Fig. 1.68). 




Fig. 1.68. Approach of 
an Aircraft to a Given 
Flight Line with a Paral- 
lel Flight Line. 



Since the value of Z in this case 
is considered known, while the radius 
of roll is determined on the basis of 
the groundspeed and the given banking 
in the roll, it is necessary to deter- 
mine the value of the angles ai = a^ 
of the combined rolls. 

It is obvious that in this case , 
in each of the two combined rolls , 
the aircraft approaches the flight 
path by a value Z/2; therefore. 



110 



-- = '?(1 — COSo), 



whence 



'^°^"='-^- 



(1.110) 



For example , let us say that an aircraft having a groundspeed 
of 900 km/h has deviated from a given flight path by 5 km; to make 
the approach, it is necessary to execute two combined rolls with 
banking of 15° to angles up to 25°. 

3. Linear prediction of roll (LPR) 

Let us examine two solutions to problems, with a consideration 
of the roll trajectory of an aircraft which includes one rectilinear 
part of the path. 

Linear prediction of roll is calculated in instances of a /111 
break in the flight path at turning points in the route (Fig. 1.69). 

In the figure , TPR is the turning point in the route and TA is 
the turn angle of the flight path equal to the roll angle of an 
aircraft ( RA ) . 

As is clear from Figure 1.69, the radius of roll of an air- 
craft, at its beginning and end, is directed perpendicular to the 
preceding and following orthodrome segments of the path. The lines 
0-TPR form the bisector of the angle of roll. 

Thus, we have two identical rectangular triangles with vertex 
angles equal to RA/2. The linear prediction of roll (LPR) is the 
line of tangency of the roll angle, divided in half: 



LPR 



TPR 





Fig. 1.69. Linear Prediction 
of Roll of an Aircraft (LPR). 



Fig. 1.70. Linear Lag of Air- 
craft Roll (LLR). 



Ill 



i 



LPR=/?tg 



RA 



(1.111) 



Example: Determine LPR with a flight groundspeed of 900 km/h 
and an angle of turn to the new flight path of 40° for banking in 
a roll of 15° . 



Solution. 



R 



2502- 



=26,5 km 



9,81 tg 15° 
LPR=26,5-tg20° = 9,6 km 



Linear predictions with roll angles from to 150° are given 
in Table 1.1. 



km/hr 



400 
500 
600 
700 
800 
900 



R. M 



4600 
7'^ 
10.600 
14 700 
18500 
23500 



Prediction with roll angles 
fromO to 150° ," km" 



15° 



30° 



45° 



60° 



75° 



90° 



0,6 


1,2 


2,0 


1.0 


2,0 


3.1 


1.4 


2,8 


4,2 


1.9 


3,8 


6,0 


2.4 


4,8 


7.7 


3,1 


6,2 


9.7 



2,7 3,5! 4,6 
4,3 5,7| 7,3 
6,l| 8,210,6 
8,3 11,0 M, 4 
10,7]l4,o'l8,5 

13,5 18,023,5 

I 



105' 



120' 



135° 150= 



'roll 
to 90° 
sec 



6,0 8,011,0,15,0 
9,7 12,8,18,027,5 
13,8 18,3'25,5]40,0 
19,0 25,035,0 52,0 
24.032,0'42, 070,0 
30,040, o'58,087,0 



'65 
82 
100 
116 
132 
148 



/li: 



In some cases, the necessity for flight above the TPR with the 
flight angle of the following part of the path can arise (Fig. 1.70) 
e.g., in flights of different kinds for testing aircraft and ground 
navigational equipment. In these cases, instead of linear predic- 
tion, linear lag of roll (LLR) is calculated, while the roll is 
carried out in the direction opposite to the turn of the new flight 
path by the angle 

RA=360°— TA 

In Figure 1.72, it is clear that the LLR is a line of the 
tangents of the turn angle of the flight path divided in half, i.e., 
with the same turn angles , the formula for the LLR remains the same 
as for the linear prediction of roll: 



LLR =«tg- 



RA 



112 



J 



CHAPTER TWO 
AIRCRAFT NAVIGATION USING MISCELLANEOUS DEVICES 
1. Geotechnical Means of Aircraft Navigation 

Geotechnical means of aircraft navigation constitute a por- 
tion of the navigational equipment of an aircraft which has an 
autonomous character and is used under all flight conditions , 
independently of the use of other special devices such as those 
employing radio engineering or astronomy, for example. 

Such devices include those which measure the aircraft course 
airspeed, and flight altitude, as well as devices for automatic 
solution of navigational problems. 



/113 



high 

phy s 

the 

red 

dent 

is b 

engi 

emat 

the 



Geo 
ly d 
ical 
fiel 
rang 

on 
eing 
neer 
i cal 
syst 



technic 
i verse 

fields 
d of el 
e , etc . 
the phy 

carri e 
ing or 

basis 
em for 



ievices for aircraft navigation are based on 
--• — .--T-- jr-„ -.-T-^ iiao of natural geo- 




Aircraft navigation using only geotechnical devices can be 
carried out in cases when it is possible to check the navigation- 
al calculations (even periodically) by determining the locus of 
the aircraft by other means or visually. 

Historically speaking, the development of radio-engineer- 
ing and astronomical means for aircraft navigation has been directed 
toward a solution of only one problem, namely, the determination 
of the aircraft coordinates on the Earth's surface, which proved 
a necessary adjunct to the geotechnical means of aircraft navi- 
gation in flight under conditions when the ground was not visible. 



In recent years, there has been a development of the radio- 
engineering, astronomical, and astro-inertia.l systems for solving 
problems which are inherent in geotechnical devices for aircraft 
navigation, i.e., measurement of the aircraft course, airspeed, 
turn angle, altitude, etc. 



/114 



113 



2. Course Instruments and Systems 

Course instruments are intended for determining the position 
of the longitudinal axis of an aircraft in the plane of the hor- 
izon or (what amounts to the same thing) for measuring the course 
of the aircraft. 



It is necessary to know the aircraft course in order to determine 
both the flight direction and the position of the aircraft relative 
to orientation points on the ground. 

As we have mentioned above, there are several systems for 
calculating the aircraft course, and the selection of the system 
of calculation is governed both by the requirements of aircraft 
navigation and by the technical possibilities for equipping the 
aircraft with the corresponding instruments. 

sent time, there are no course instruments which 
sfy the requirements of aircraft control under all 
eref ore , aircraft usually are fitted with several 
e instruments operating on different principles 
rent systems of calculation; each of them is used 
tions which are most favorable for it. In some 
struments are combined into complexes , called course 
the operation of the individual instruments is closely 
makes it possible to exploit the positive qualities 
in actual operation. 



At th 


e pre 


completely 


sati 


conditions 


Th 


different 


cours 


and using 


diffe 


under the 


condi 


cases , these in 


systems , w 


here 


related . 


This 


of each of 


them 



Methods of Using the Magnetic Field of 

D i rect i on 



the Earth to Determine 



Directions on the Earth's surface can be measured most ac- 
curately by astronomical methods. However, this requires opti- 
cal visibility of the sky, complex and accurate apparatus, and 
tedious calculation. Directions on the Earth's surface can be 
determined more simply and in many cases quite reliably by using 
the magnetic field of the Earth. 

The magnetic field of the Earth (Fig. 2.1) is characterized 
by the following parameters at every point on its surface: 



(H); 



(Z) 



(a) Directionality of the horizontal component of the field 



(b) Directionality of the vertical component of the field 



(c) The direction of the plane in which the vectors H and 
Z lie relative to the geographic meridian at the given point. 

The plane in which the vectors H and Z are located is called /115 
the plane of the magnetic meridian. The angle between the planes 



im- 



of the magnetic and geographic meridians is called the magneti.Q 
deatination and is represented by A^^ . 

The points on the Earth's surface at which the magnetic mer- 
idians intersect are called magnet-io -poles. Obviously, the hori- 
zontal component of the magnetic field is lacking at the magnetic 
poles, while the intensity of the vertical component reaches its 
maximum value . 




Fig. 2.1. Magnetic Field 
of the Earth. 

is the resultant vector of E 



The magnetic poles 
do not coincide with the 
ones . The coordinates o 
Magnetic Pole are 7'4-°N a 
those of the South Magne 
are 68°S, 143°E (as of 1 

The device of a fre 
magnetic pointer mounted 
of the magnetic meridian 
to determine direction o 
surface. Therefore, at 
on the Earth's surface t 
be a reliable indication 
parameters which charact 
magnetic field of the Ea 

The total intensity 
netic field of the Earth 
and Z. Consequently, 



j2 _ 



H' 



of the Earth 

geographi c 
f the North 
nd 100°W; 
tic Pole 
952) . 

ely rotating 
in the plane 
is used 

n the Earth's 

every point 

here will 
of the three 

erize the 

rth . 

of the m a,g 
(vector T) 



(2.1) 



The oersted (Oe) is the unit of measurement for the total 
intensity of the magnetic field, as well as the intensity of its 
components; in other words, it is the intensity of a field which 
interacts with a unit magnetic pole with a force of one dyne. 

The limits of change in the intensity of the components in 
the magnetic field of the Earth are the following: 

(a) Horizontal : from zero in the vicinity of the magnetic 
poles to a maximum at the magnetic equator (0.4 oersteds in the 
vicinity of Indonesia); 

(b) Vertiaal: from zero at the magnetic equator to 0.6 oer- 
steds in the vicinity of the magnetic poles. 

A smaller unit of intensity, the gamma (y)? is used for very 
precise magnetic measurements; it is equal to one hundred thou- 
sandth of an oersted. 

The angle which characterizes the inclination of the vector 



115 



of total intensity of the magnetic field of the Earth to the plane 
of the true horizon is called the magnetic ■Lnotination "6". 



/116 



larctg^ 



(2.2) 



Charts of the magnetic fields are prepared for convenience 
in using the magnetic field of the Earth to determine directions 
on the Earth's surface. 

A chart of magnetic inclinations is extremely important for 
aircraft navigation. Lines joining points on the Earth's surface 
which have the same magnetic declina-^ion are called isogonios . 
They are printed directly on flight and large-scale geographic 
maps . 

To determine the true course, the magnetic declination deter- 
mined from the chart at the locus of the aircraft (with its sign, 
as a correction) is entered in the readings of the magnetic compass. 

Figure 2.2 shows a map of the World with the magnetic declina- 
tions entered on it; the isogenics are shown as they appear on 
the Earth's surface. The positive isogenics on the chart are marked 
by solid lines, while the negative ones are marked by dashed lines. 

All of the isogenics meet at the magnetic poles of the Earth, 
and the compass readings (and consequently the magnetic inclin- 
ation) change by 180° when passing through the magnetic pole. 

In addition, the isogenics also meet at the geographic poles, 
since the directions of the magnetic and geographic meridians are 
opposite between the magnetic and geographic poles, but coincide 
after passing through the pole, i.e., the declination changes by 
180° . 

The map of the World showing the magnetic declinations has 
the isogenics only for the normal magnetic field of the Earth. 
In addition to this normal field, there is also an anomalous field, 
caused by the magnetization of the soil in the upper layers of 
the Earth. Regions and areas of changes in the declination in 
such regions are marked on large-scale charts. 

The reliability of operation of magnetic compasses and the 
magnitude of the errors in their readings depend on the intensity 
of the horizontal component of the magnetic field of the Earth. 
Errors in the readings of compasses, particularly when the air- 
craft is rolling, depend only on the intensity of the vertical 
comonent . 

The lines on the Earth's surface which connect points with 
the same intensity of the horizontal or vertical components of 
the magnetic field are called isodynamic lines. 



115 



■■ IHIII III IIIIIIIHII !■ 



Figure 2.3 shows a map of the World with the isodynamic lines 
for the horizontal component of the Earth's magnetic field, while 
Figure 2.h shows those for the vertical component. 



/117 




Fij 



2. 2 



World Chart of Magnetic Declinations. 



117 



/Ill 




Fig. 2.3. World Chart of Isodynamic Lines for the Horizontal Com- 
ponent of the Earth's Magnetic Field. 



118 



/119 




Fig. 2.4. World Chart of Isodynamic Lines for the Vertical Compo- 
nent of the Earth's Magnetic Field. 



Only general (outline) charts of isodynamic lines are used /120 
in aircraft navigation. These lines do not appear on flight charts. 

Lines on the Earth's surface which connect points with the 
same declination of the magnetic field are called {.soatines . Form- 
erly, outline maps of isoclines were used jointly with charts of 
isodynamic lines showing the total intensity of the magnetic field 
to determine the errors of magnetic compasses. At the present 
time, these charts are no longer used, since it is better to use 
the isodynamic lines of the horizontal and vertical components 
of the magnetic field. 



Variations and Oscillations 



the Earth's Magnetic Field 



There are several hypotheses regarding the origin of the mag- 
netic field of the Earth, but none of them has been adequately 
proven as of the present time. Possible factors in the formation 
of the magnetic field are the subsurface and ionospheric electrical 
currents, as well as the magnetic induction and magnetic hysteresis 



119 



of the soil, composing the structure of the Earth's sphere. 

Even if these factors are not primarily responsible for the 
formation of the magnetic field of the Earth, they are in any case 
important influences on its structure and stability. 

An analysis. of the isolines of the intensity of the components 
in the magnetic field of the Earth and the magnetic declinations 
reveals that their configuration is determined both by general 
laws of the distribution of magnetic forces in the field of a mag- 
netized sphere, as well as by local disturbances in the general 
structure of the field. Therefore, the stationary magnetic field 
of the Earth is assumed to consist of a sum of fields: 

(a) The field of the uniform magnetized sphere; 

(b) The continental field, related to the nonunif ormity of 
the relief and the structure of the internal layers of the Earth; 

(c) The anomalous field, related to the existence of depos- 
its of magnetic material in the upper layers of the Earth's core. 

As systematic observations of the structure of the magnetic 
field of the Earth have shown, it does not remain strictly sta- 
tionary but undergoes constant changes. Changes or variations 
in the magnetic field of the Earth have a diverse nature. 

Annual or constant changes in the magnetic field of the Earth 
are called seoulav variation. These variations constitute the 
difference between the average annual values for the elements of 
the Earth's magnetism. The causes for the annual variations are 
changes in the components of the stationary field with time, i.e., 
the magnetic moment of the Earth and the continental field. 

The annual variations in the declination at middle latitudes 
reached 10-12', and up to 40' at high latitudes; therefore, when 
using charts of magnetic declinations, or isogenics on flight charts, 
it is necessary to consider the period when they were made. If 
the chart of magnetic declinations is obsolete, changes must be /121 
made when using it for the variation in the declination during 
the time which has passed since the chart was made. The desired 
correction is determined from special charts of the secular vari- 
ations of the magnetic field of the Earth. The isolines of equal 
secular variations in declination on a chart are called isopors . 

In addition to the slow systematic changes in the magnetic 
field of the Earth, there are also periodic and even chaotic changes 
which are related to the so-called internal field of the Earth, 
the main cause of which is ionospheric currents. These are esti- 
mated periodically or are disregarded entirely. 



120 



Magnetic Compasses 

The magnetic compass is the simplest course device; in most 
cases, it is sufficiently reliable though not sufficiently accu- 
rate . 

However, a simple magnetic compass with a freely turning mag- 
netic needle is not suitable for use on board an aircraft, since 
its readings would be inaccurate and unstable. Various kinds of 
interference would influence the operation of the compass during 
flight, including: 

(a) Movements of the aircraft relative to its axis; 

(b) Vibrations produced by the operation of the engines and 
by the movement of the aircraft through the air; 

(c) The effect of the magnetic field of the aircraft, which 
would cause deflections of the magnetic needle from the plane of 
the magnetic meridian, i.e., compass deviation. 

Obviously, a magnetic compass which is intended for use on 
an aircraft must have devices for compensating the interference 
mentioned above. 

The simplest form of an aircraft magnetic compass is the inte- 
grated compass, i.e., one in which the course transmitter (sensi- 
tive element) and the indicator are combined in a single housing. 

Of the large number of types of magnetic compasses which have 
been devised as of the present time, the one most used nowadays 
is the "KI" (an abbreviation for the historic name of the magnetic 
compasses which were devised in the past for fighter aircraft). 
Compasses using other systems are called distance-magnetic or gyro- 
magnetic compasses. 

Any integrated aviational magnetic compass consists of the 
following main parts (Fig. 2.5): 

The bowl or container 1 of the compass, filled with a damp- 
ing fluid to decrease the oscillations, usually liqroin; on the 
bottom of the bowl is a pivot support for the movable part of the 
compass, with a damping spring and pivot bearing made of agate; 



/122 



The movable part of the compass, consisting of 
is a combination of a magnetic system (H-shaped magnet), a floa 
to reduce the weight of the card and reduce the friction on the 
bearing, a needle pivot, and a rotating scale for the readings, 
mounted on the magnetic system; 



card 2 , which 
float 
e 



Chamber 3, compensating for thermal expansion and contraction 



121 



of the damping fluid; the expansion chamber is located above the 
bowl and is connected to it by holes of very small diameter. This 
allows air bubbles to escape from the bowl into the chamber and 
permits the fluid to flow back and forth with expansion and con- 
traction. It also prevents it from splashing in the bowl as the 
airplane moves ; 

A device ^ for getting rid of deviations of the compass, which 
contains several bar magnets pressed into drums which rotate in mutu- 
ally perpendicular planes with the aid of screws. Rotation of 
the drums permits them to be set to a position where the magnetic 
field of the bar magnets compensates for the magnetic field of 
the aircraft acting on the compass card. 




Fig. 2.5. Combined Magnetic Compass: (a) Cross 
Section; (b) External View. 

The design of the magnetic compass described above reduces 
the effect of interference with its operation to a considerable 
degree and the compass readings are quite stable. Nevertheless, 
magnetic compasses (especially integrated ones) have a number of 
shortcomings which prevent the course from being calculated under 
certain conditions. The most important of these shortcomings are 
the following: 

(1) A limitation of the choice of mounting location for the 
compass aboard the aircraft; the integrated compass must be lo- 
cated in a place which is suitable for determining the course, 
and therefore close to other instruments and moving parts for con- 
trolling the aircraft, which produce a large and varying devia- 
tion of the compass; 

(2) The impossibility of using the compass when the aircraft /123 
is turning. When the aircraft makes a turn, several factors act 

on the compass card to move it from its customary position: the 
pressure of the damping fluid on the card, the action of centri- 
fugal force on the southern, somewhat elongated portion of the 
card, as well as a change in the structure of the magnetic field 
of the aircraft while turning. The deviations of the compass card 



122 



from the plane of the magnetic meridian when the aircraft is turn- 
ing are particularly noticeable when the aircraft course crosses 
the northern and southern directions. These deviations are called 
the northern and southern turning errors . 

The instability of the structure of the magnetic field of 
the Earth at the locus of the aircraft and its changes with time 
are the major shortcomings of using magnetic compasses of all types 

Deviation of Magnetic Compasses and its Compensation 

The cause of magnetic compass deviation is the presence of 
parts on board the aircraft which are made of materials exhibiting 
magnetic properties . Some of these parts have a constant magnetic 
field. Parts of this kind are called hard magnetic iron. Another 
group of parts are magnitized under the effect of the magnetic 
field of the Earth and are called soft magnetic iron. 

According to Coulomb's law, the force (F) of the interaction 

c masses (m) is inversely proportional to the distance 
— f ■^\ 



of magnetic mciaae 

between them (p). 



^ = -^- (2.3) 



Therefore, the deviation of the magnetic compass increases 
very sharply with the approach of its sensitive element to parts 
which have high magnetization. 

According to the principle of independence of the action of 
forces at a given point in the aircraft, it is possible to sum 
the magnetic fields coming from individual parts of the aircraft 
and to subject them to the equivalent effect of a single magnetized 
bar located at a certain point. However, if we take into account 
the diverse nature of the action of the hard and soft magnetized 
iron on different courses and during different motions of the air- 
craft, it is better to subject this field to the equivalent action 
of bars which have a constant and varying magnetization. 

Let us assume that the equivalent bar of hard magnetized iron 
is located horizontally and coincides with the direction of the 
longitudinal axis of the aircraft (Fig. 2.6). 

With a magnetic course of the aircraft equal to zero, the 
vector F of the field intensity of the bar coincides in direction 
with the horizontal component of the magnetic field of the Earth /12H 
H, which does not produce any deviation of the compass card from 
the plane of the magnetic meridian. 

_In the case of aircraft courses equal to 90 or 270°, the vec- 
tor F of the field intensity of the bar is located at right angles 



123 



to the vector H, producing maximum deviation of the card from the 
plane of the magnetic meridian. 

Hence , when the aircraft is turning around its vertical axis 
through 360°, the resultant vector (P-j-) of the hard magnetic iron 
and the magnetic field of the Earth will coincide at two points 
with the direction of the magnetic meridian, and will be at a maxi- 
mum distance from it at two other points . Devtat-ton of this kind 
-is aalled semiairoutar, i.e., it has zero value with every 180° 
rotation of the aircraft (Fig. 2.7). 

Finally, it cannot 
be expected that in 
the general case the 
equivalent bar of hard 

Hi " \ 1 I >" magnetic iron will 

coincide in direction 
with the longitudi- 
nal axis of the air- 
craft. However, this 
does not alter the 
nature of the semi- 
circular deviation, 
but only shifts the 
graph of deviation 
relative to the course 
scale of the aircraft 






Fig. 2 
a Bar 



F 



6. Deviation of Compass Card by 
f Hard Magnetic Iron. 



by an 
craft 



angle which is equal to that between the axis 
and the axis of the equivalent bar. 



of the air- 



Semicircular deviation of a magnetic compass can be compen- 
sated easily. To do this, it is sufficient to make a bar of hard 
magnetic iron and place it near the compass installation in such 
a way that its field is opposite to the direction of the field 
of the equivalent bar of hard magnetic iron. 

Let us now assume that there is no hard magnetic iron aboard 
the aircraft, but a field of soft magnetic iron is located hori- 
zontally and contributes to the action of the equivalent bar, coin- 
ciding in direction with the longitudinal axis of the aircraft. 

The essence of the effect of the soft magnetic iron on the 
compass readings consists in the fact that the bar, which is lo- 
cated in a certain position relative to the magnetic field of the 
Earth, is not magnetized in the direction of the field but along 
the length of the bar. 



/125 



ula 



The magnetization of the bar can be expressed by the form- 



B = pHcos a , 



(2.1+) 



124 



where B is the magnetic induction, y is the magnetic permeabil- 
ity of the bar, H is the intensity of the magnetic field, and a is 
the angle between the direction of the intensity vector of the field 
and the direction of the bar. 



■ -^ e, n <>* ^ i 



s — *■ 



/ 


— 





\ 


\ 


















) 


V 


y 





MC 



90 



m 



270 



360 



Fig. 2.7 



Graph of Semicircular 
Deviation . 



On courses 
in this case, the 
of the equivalent 
cides with the di 
of the horizontal 
nent of the vecto 
sity of the magne 
of the Earth (a = 
the magnetic indu 
the bar is maximu 
no compas 



will be 
ation . 



and 180° , 
direction 
bar coin- 

rection 
compo- 

V of inten- 

tic field 
0); although 

ction of 

m , there 

s devi- 



wil 
its 
on 
in 
2 . 8 



In changing 
course from to 
from to 270° , t 
induction of the 
decrease, but the 
between the vecto 

1 increase. It is obvious that the deviation will th 
maximum at a course of ^5 or 315° and will reach zero 

courses of 90 and 270°. A similar change in deviation 

the flight sectors from 90 to 180° and from 180 to 270 

). 



the aircraft 

90° or 

he magnetic 

b ar wi 11 
angle 

rs H and B 

en reach 
once again 
will occur 

° (Fig. 



It is clear in the figure that the deviation from the soft 

magnetic iron during one complete turn of the aircraft around the 

vertical axis passes through zero four times, i.e., it has a quar- 
ternary nature . 

The action of one magnetic bar of soft iron clearly illustrates 
the quarternary nature of the alternating magnetic field of the 
aircraft. In practice, however, with the exclusion of rare cases, 
the alternating magnetic field of the aircraft cannot amount to 
the effect of one bar of soft magnetic iron. 

In fact, if we take two bars of soft iron and locate them at /126 
90° to one another, the resultant vector of induction of the bars 
will coincide with the bisectrix between them (Si = B 2) only in 
the case when the intensity vector of the magnetic field (Hi) coin- 
cides with the bisectrix of the angle between the bars (Fig. 2.9). 
In all other cases, the induction vector will approach the axis 
of the bar which is closer to the intensity vector of the magnetic 
field. 

If we consider the action of one bar, th'e vector of magnetic 
induction will change in value but will always coincide with the 



125 



axis of the bar. This essentially explains the existence on the 
aircraft of both semicircular and quarternary deviation as well as 
deviations of higher order. 



5 y^K i y^-=x, f 



. s -^ 



% N * 



i 

vy ^2 



MQ 




" 90 m 270 

Fig. 2.8, Fig. 2.9 . 

Fig. 2.8. Graph of Quarternary Deviation from Soft Magnetic Iron, 

Fig. 2.9. Magnetic Induction of Crossed Bars of Soft Iron. 



In addition, if we disregard the deviation of higher order, 
the deviation from soft magnetic iron cannot be eliminated by using 
a suitable bar of soft iron, since it will also be magnetized like 
all other parts of the aircraft and will not lead to a reduction 
but rather to an increase of the deviation. 

Equatizing the Magnetic Field of the Aivovaft 

The cause of magnetic compass deviation on board an aircraft 

is generally a lack of coincidence between the resultant components 

of the magnetic field of the aircraft with the vector of intensity 
of the Earth's magnetic field. 

When the aircraft rotates around its axis, the alternating /127 
magnetic field of the aircraft not only rotates along with it, but 
simultaneously changes in magnitude and sign. Therefore, in order 
to determine the magnitude and sign of the deviation for various 
aircraft courses, it is advisable to express its field components 
in the form of forces acting along the axes of the aircraft. 

Obviously, the magnitude of these forces (with the exception 
of the components made of hard magnetic iron) will vary with changes 
in the magnetic course of the aircraft {yy[). 

Depending on the nature and character of the action of the 
components of the magnetic field on the sensitive element of the 
compass, we can divide them into three groups: 



126 



(1) Components of the magnetic field of the Earth along the 
axes of the aircraft; their designations coincide with the desig- 
nations for the aircraft axes X, ¥, Z. The resultant vector of 
these components is Y. 

(2) The components of the magnetic field of the aircraft made 
of hard magnetic iron have the designations: P along the X-axis 

of the aircraft; Q along the Y-axis, and E along the Z-axis. 

(3) Components of soft magnetic iron of the aircraft. As 
follows from what has been said above, they cannot be viewed as 
a simple part of the resultant vector along the axes of the air- 
craft . 

For convenience in mathematical operations, these components 
lead to an equivalent effect of nine bars of soft magnetic iron, 
of which three bars coincide with each of the axes of the aircraft. 
This means that each of the three bars which coincide with a given 
axis of the aircraft is magnetized by a component of the magnetic 
field of the Earth which is located only along some one axis of 
the. aircraft . 

Equivalent bars a, bj e are located along the Z-axis of the 
aircraft; bar a is magnetized by the component of the magnetic field 
of the Earth X, bar h by component Y, and bar a by component Z. 

Equivalent bars d, e, f are located along the J-axis , and bars 
Qj h, k are located along the Z-axis; they are magnetized by the 
same components of the vector T. 

The contribution of the magnetic field of the soft iron in 
the aircraft to the equivalent effect of nine bars acquires phys- 
ical significance in_summing the magnetic induction of the compo- 
nents of the vector T, along the axes of the aircraft. 

For example, the J-component of the magnetic field of the Earth 
acts on bars Uj d, g, and the resultant induction from these three 
bars shows how the vector of the magnetic field from the soft mag- 
netic iron of the aircraft IX would be located if the components 
of the magnetic field of the Earth Y and Z were equal to zero. 

In other words , the equivalent bars are equivalent to the vec- 
tors of division of the magnetic induction from the components of 
the magnetic field of the Earth along the axes of the aircraft (Table 
2.1). 



In summing the magnetic forces along the axes, we obtain the 
equations for the magnetic field of the aircraft: 



/128 



■?' = ?+ QJ^'dX+ fy 4- fZ; 
Z' = Z + R + ^+Ty+kZ: 



(2.5) 



127 



These fields will be used as a basis for deriving formulas 
for the deviation of magnetic compasses on an aircraft. 







TABLE 2.1 






axis of 


Resultant forces 


the 
aircraft 


T 


^ 


IX 


J mY 


nZ 


OX 


X 


P 


aX 


bY 


cZ 


oy 


Y 


Q 


dX 


eY 


fZ' 


oz 


Z 


R 


gx 


hY 


kZ 



The sum of the vectors X'^ Y' and Z' gives a total vector T' 
acting on the sensitive element of the compass. 

Deviation Formulas 

In the equations of the magnetic field of the aircraft, the 
constant terms are only the components of the field of the hard 
magnetic iron, P, Q, R. However, to calculate the deviation in hori- 
zontal fligh_t, we can consider that the magnetic induction Z from 
the vector T is constant along the vertical axis of the aircraft 
(terms cZ , fZ, kZ) . 

In addition, horizontal flight will not involve the third 
equation in (2.5), determining Z'. 

If we also consider that the sum of the vectors X and I con- 
stitutes the horizontal component of the magnetic field of the Earth 
H, 

X= H cos 7; 
i/=//slnT, 



the first two equations in (2.5) can be rewritten to read as fol- 
lows : 



A"' = // cos 7 -)- aH cos 7 — ft// sin 7 -h c^ + P; ] 
f^' = //sin7 + cf//cos7 — e//sln7+/Z+ v/, J 



(2.6) 



where y is the magnetic course of the aircraft. 

The vectors X'y I' are the components of the magnetic field 
along the longitudinal and transverse axes of the aircraft at the 
locus of the compass . 

The magnetic compass deviation (6) is expressed by the angle 

between the direction of the horizontal component of the magnetic 

field of the Earth H and the horizontal component of the total mag- /129 
netic field on the aircraft H' (Fig. 2.10). 



128 



Obviously, tg6 is equal to the ratio of the projection of vec- 
tor R' in a direction perpendicular to the magnetic meridian fl", 
to its projection on the magnetic meridian H"' : 



H" X' sin f + r cos -1 
^ H" A^'cosf— y'sin^ 



(2.7) 



If we substitute into Equation (2.7) the values of X' and Y' 
from Equation (2.6), and also reduce similar terms, replacing the 
values sinycosY, sin^y and cos^y by their obvious homologues % sin^Y 
% (I-cos^y) and h. (1+cos^y)j we will have: 

-^ // + (cZ + />) sin Y + (/Z + (?)C0S7+^^ // sin 2t -t- 
d +b 



tgB = 



(2,8) 



H + 



a + e 



H+(cZ + P) cos Y — (/2 + (?) sinY + 



. a'— e d + b 

+ -T— // cos 2y— — r— // sin 2y 



The terms in Equation (2.8), with a coefficient equal to unity, 
have a constant character, i.e., they are independent of the air- 
craft course at a given magnetic latitude. The terms which have 
the coefficients 2 sin y and 2 cos y have a quarternary character. 
The terms with coefficients sin y and cos y have a semicircular 
character . 



All of the forces designa 
of Equation (2.8) are directed 
meridian while those in the de 



ted by values lo 

at an angle of 
nominator coinci 



cated in the numerator 
90° to the magnetic 
de with it . 



The force 



d 



H is independent of the aircraft course; it is 



proportional to the horizontal 
the Earth and is directed at a 
ian. This force is related to 
of the aircraft by the magneti 

a fun 
the 1 
nate 




component of th 
n angle of 90° t 

the magnetizati 
c field of the E 
ction of the mag 
ocus of the aire 
this force by Aq 

The force cZ+P i 
longitudinal axis of 
the result of the Ion 
of the field from the 
P and the induction f 



e magnetic field of 
o the magnetic merid- 
on of the soft iron 
arth and varies as 
netic latitude of 
raft. We will desig- 
\E. 

s directed along the 
the aircraft; it is 
gitudinal component 
hard magnetic iron 
rom the vertical 



Fig. 2.10. Deviation of Magnetic Compass 
Aboard an Aircraft. 



129 



■component of the magnetic field of the Earth, This force is desig- 
nated BqXH and changes with the magnetic latitude of the aircraft /130 
location only in accordance with the first term. The projection 
of the force on the normal to the magnetic meridian is proportional 
to the sine of the magnetic course of the aircraft. 



The force fZ+Q is d 
nature and character of 
along the transverse axi 
jection on the normal to 
the cosine of the magnet 



The forces 



a - b 



an 



iron on the aircraft, ma 
The former is designated 
the double course of the 
dicular to the double co 

The force H+^^ H 
is in the denominator, a 
of the magnetic meridian 



esignated by Cq}^H, and is analogous in the 

its changes to the force BqXH, but is directed 

s of the aircraft. Consequently, its pro- 

the magnetic meridian is proportional to 
ic course of the aircraft. 

/J t J-, 
d are related to the soft magnetic 

gnetized by the magnetic field of the Earth. 
DqXH and coincides with the direction of 
aircraft; the latter is EqXH and is perpen- 

urse of the aircraft. 

s designated XH . In Equation (2.8), it 
nd therefore coincides with the direction 



If we substitute into Equation (2.8) these designations for 
the forces and divide the numerator and denominator by XH , the latter 
will give us 

^^_ A + BoSin1-i-C„ cos t + Dq sin 2'y + £o cos 2^ 

I + B0COS7 — Cosln7 + £>ocos27 — £oSln2-j " (2.9) 

Expression (2.9) is called the point-deviation formula, and 
the coefficients Aq^ Sqj Cqj Dq and Eq are the point coefficients 
of deviation. 

The point-deviation formula is inconvenient to use, so it has 
been simplified for practical purposes. 

Since it is almost always necessary to select a place for mount- 
ing the compass on the aircraft where the deviation does not exceed 
8-10°, we can let tg 6 = 6 . 

The denominator of Formula (2.9) can be expressed in the form 
of a binomial: 

[1 + (Bo cos f — Co sin 7 + Do cos 27 — Eo sin 2f)]-i = (1 + a)->. 

We know that with a < 1, the expansion of the binomial gives 
the converging series: 

(1 +a)-i = l— a + a2 — a3. .. 

For practical purposes, we can limit ourselves to the first 



130 



two terms of the series 



(l+a)-iwl-«, 



so that Equation (2.9) assumes the form: 



t = (Ao f BoSlnT+ Qcos^ + £)os1b27 + £oC08 27)(l — BqCOSt + 
+ Co sin T — Do cos if + Eg sin z^). 



(2.10) 



Having carried out the multiplication of the multipliers, re- /131 
duced the similar terms, and carried out simple trigonometric conver- 
sions. Equation (2.10) assumes the form: 



8 = i4 + B sin 7 + Ccos 1 + Z) sin 2 7 + £ cop 27 + Z' sin 3t + O cos 3f + 
+ //sin4-j + A'cos47... 



(2.11) 



Here the coefficients A^ B, C, D, E have a somewhat different value 
than in the point-deviation formula: 



Bl 



r2 
^0 



A = A<i\ B = Bo + AoCo; D = Do+-T-+-r-+ A^o', 



C — Co — j4oBoi ^ = ^0 — BqCq — AolJo, 



The coefficients of deviation of higher orders, i.e., propor- 
tional to the sines and cosines Sy , ^-y » • • • , can be disregarded, 
since they are much smaller than any of the first five coefficients 
Then Formula (2.11) assumes the form: 



8 = vl + B sin -f + C cos Tf + /> sin 2-r + £ cos 2^, 



(2.12) 



where A is the coefficient of constant deviation, 5j C are the coef- 
ficients of semicircular deviation, and D^E are the coefficients 
of quarternary deviation. 

Formula (2.12) is called the approximate formula of deviation, 
and its coefficients are the approximate deviation coefficients. 
However, it is completely satisfactory for practical applications, 
especially if we recall that other factors are acting on the compass 
which are very difficult to allow for. 

Calautation of Approximate Deviation Coefficients 

We will assume that we know the deviation of a magnetic com- 
pass at eight symmetrical points: 0, 15, 90, 135, 189, 225, 270 
and 315° . 

According to Equation (2.12), the deviation at these points 
must have the values: 

8o = yl + BsinO° + C cos 0° + Z> sin 0* + BcosO°. 



131 



Since sinO° = 0, cosO° - 1, then 6 q = A + C + E ; 
la = A-\-B sin 45° + Ccos 45° + Z> sin 90° + £ cos 90° 

or, if we consider the values sin90° = 1, cos90° = 0, 

645 = i4 + B sm 45° + C cos 45° + />. 



Similarly, we can obtain a system of equations for the devi- 
ation of the eight points: 



/132 



60 = A ■(- C + £; 

\i = A + B sin 45° + C cos 45° + D; 

V = A + B — £,•< 
hh = A "-f B sin 45° — C cos 45° — D; 
\^ = A~C + E; 

ha.b = A — B sin 45° — C cos 45° + D; 
S270 = A — B — E; 
8.11s = -4 — B sin 45° + C cos 45° — D. 



(2.13) 



Summing Equation (2.13), we obtain: 

*0 + S16 + %0 + 6186 + 6,80 + *22S + ^70 + 63,6 = 8i4 



or 



i=0 



consequently , 



A = 



8 



To find the approximate deviation coefficient B, we multiply 
each of the Equations (2.13) by the coefficient at B, depending 
on the aircraft course. Then, keeping in mind the fact that sinH5° 
= cos^+S", the equations for 60 and &iqq become zero and the remain- 
ders assume the form: 

8,5 sin 45° = A sin 45° + B sln2 45° + C sln2 45° + D sin 45°; 

68o = A + B — B; 

Sigj sin 45° = A sin 45° + B sin2 45° — C sln2 45° — D sin 45°; 
—8225 Sin 45° = A sin 45° + B sin2 45° + C sln2 45° — D sin 45°; 
—8270 = A + B + B; 
— B315 sin 45° = A sin 45° + B sin2 45° — C sin2 45° + D sin 45°, 

In summing the six remaining equations, the sum of the terms 
containing coefficient A becomes zero, since three of them have 
a plus sign and the remaining three, symmetrical to the first, have 
a minus sign. 

The sum of the terms containing coefficient B is equal to 



132 



J 



/ — 9 

2S + 4S sin^ 45°, but since sin^ i+5° -^~5") ~ 'K^ this sum will be 
equal to 4S . 

The sum of the terms containing coefficient C , as well as the /133 
sum of the terras containing coefficient D, is equal to zero. 



Cons equently , 



S8,-ri=4B 



<=0 



or 



B = — -^B/sln^,. 



Similarly, we can find the formulas for determining the re- 
maining three coefficients: 



o 

C"= —^5/ COSY/; 

8 

£» = — ^6/cos27,; 

8 

^=—^8,- COS 2a;,. 



i = 



(2.1M-) 



J 



Change -in Deviation of Magnetie Compasses as a Function of 
the Magnetic Latitude of the Locus of the Aircraft 

The deviation of a magnetic compass determined for a given 
point on the Earth's surface, does not remain fixed for other points, 
but changps depending on the magnetic latitude of the locus of the 
aircraft . 

uDviously, a change in deviation cannot take place as a result 
of changes in the magnetic induction of soft magnetic iron from 
the horizontal component of the magnetic field of the Earth. 

By the same token, the induction from the component producing 
the deviation will change in the same proportion with a change in 
the horizontal component of the magnetic field of the Earth, as 
the principal directional position of the compass card. Consequently, 
the compass deviation remains constant. 

The deviation from magnetic induction of the horizontal com- 
ponent of the field of the Earth has a constant and quarternary 
character : 



133 



MH=^H; £><^//=«-=i//; s^ff^t±lff. 

Hence, we reach the conclusion that the constant and quartern- 
ary deviation at various magnetic latitudes remains constant. 

Essentially, the change in the deviation with a change in the /134 
magnetic latitude is the result of the influence of hard magnetic 
iron and partially as a result of induction with soft magnetic iron 
from the vertical component of the Earth's magnetic field. This 
takes place because the magnitude of the vectors Pj Q remains constant 
with a change in the directional vector H. Consequently, with an 
increase in the magnetic latitude, the semicircular deviation must 
increas e . 



In addition, with an increase in the magnetic latitude, the 
induction of the soft magnetic iron from the vertical component 
of the Earth's field increases with a simultaneous decrease in the 
directional force H. However, if we consider the predominant influ- 
ence on the aircraft produced by the hard magnetic iron, we can 
consider in approximation that the semicircular deviation is inversely 
proportional to the horizontal component of the magnetic field of 
the Earth. 



Bo = 



cZ +P 



Co = 



fZ+Q 

XH ' 



which gives the following for the approximate coefficients of de- 
viation B and C: 



^2 = Si 777 •" Cj = C, -J- , 

"2 "2 



(2.15) 



where B i 3 Ci, Hi are the approximate coefficients and the horizontal 
component of the Earth's field at the point where the deviation 
is measured; Sjj (^23 ^2 ^^^ "the same values at a point with a dif- 
ferent magnetic latitude . 

With known coefficients B + C, the semicircular deviation at 
a given point on the Earth's surface can be determined by the form- 
ula 



H\ H 

B = B -— sin T + C -7- cos 7. 

"2 Ho 



(2.16) 



Eliminat-ton of Deviation in the Magnetic Corn-passes 

Modern magnetic compasses are fitted with a device for com- 
pensating only semicircular deviation, resulting from hard magnetic 
iron . 



134 



In addition, by a suitable rotation of the compass housing 
in its mountings, we can compensate for the constant component of 
deviation along with the adjustment error of the compass. 

Elimination of quarternary deviation by magnetic means encoun- 
ters considerable technical difficulty. Therefore, if we keep in 
mind the relatively low value of the quarternary deviation relative 
to the semicircular deviation, as well as its constant value at 
various latitudes, we will not be able to get rid of the latter 
but will enter it on special graphs for compass correction. 

Modern remote control magnetic compasses have devices for me- 
chanical compensation of deviation of all orders. 



/135 



The device for compensating semicircular deviation consists 
of a system of four cylinders mounted in pairs, with permanent magnets 
installed in them (Fig. 2.11). 




When the magnets are tilted (Fig. 2.11, c), the horizontal 
component of their field appears, and can be set so that it is equal 
but directed opposite to the magnetic field of the aircraft (hori- 
zontal component), located along its transverse axis. The maxi- 
mum effect of the small magnets will be observed when they are in 
the horizontal position (Fig. 2.11, b). 



135 



II 



The cylinders for compensating deviation at courses of 90 and 
270° are mounted in the transverse axis of the aircraft in such 
a way that the small magnets can be used to compensate for the compo- 
nent of the magnetic field of the aircraft which is directed along 
its longitudinal axis . 

The rotation of the longitudinal and transverse cylinders is 
accomplished by means of special handles made of diamagnetic mater- 
ial . 

To determine and get rid of deviations, the aircraft is placed 
on a specially prepared stand, made of concrete (for heavy aircraft ) /136 
but without a metal core. 

The stand must be of sufficient size so that aircraft of any 
kind can be rotated in a circle and the distance from the stand 
to other aircraft and metal structures is at least 200 m. 

The accuracy of the setting of the aircraft on a given course 
for determining and getting rid of deviation can be checked in one 
of the following two ways : 

1. Direction finding of landmarks from on board the aircraft. 

In the center of the area where the aircraft is to turn, a magnetic 
direction finder or theodolite is mounted on a stand so that the 
indicating dial is located exactly in a horizontal position, and 
the zero reading on the dial coincides with the direction of the 
magnetic meridian. For this purpose, these instruments are fitted 
with a bubble level and orienting magnetic needle. 

Then two or three distinct and prominent landmarks on the hor- 
izon are selected (towers and chimneys are best for this purpose), 
and their magnetic bearings (MB) are determined with the aid of 
a sight, rotating on the dial used for determining the bearings. 

The landmarks should be located as far as possible from the 
area so that the shifting of the aircraft from its center during 
rotation will not produce any noticeable changes in the bearings 
of the landmarks. For light aircraft, this distance should be at 
least 2-3 km; for larger aircraft with a greater radius of turn 
on the ground, it should be at least 5-6 km. 

After determining and recording the magnetic bearings of the 
landmarks, the aircraft is mounted on the stand. The direction 
finder is placed in front of or behind the aircraft at a distance 
of 20-100 m, depending on the length of the aircraft, exactly along 
its longitudinal axis so that the forward and rear points on the 
axis of the aircraft will be projected on the sight, e.g., the centers 
of the nose and keel. Then the dial on the direction finder is 
set to the magnetic meridian, and the direction of the longitudinal 
axis of the aircraft is measured, and its initial course is set. 



136 



It is necessary to recall that the minimum distance for the 
direction finder from the aircraft is limited by the effect of the 
aircraft on the magnetic needle of the deviation direction finder, 
and the maximum distance is set by the length of the aircraft, since 
at a distance of more than 100 m, with an aircraft which is not 
very long, this method will be insufficiently precise. 

After the direction finder has been moved to the aircraft, 
the magnetic needle is fixed and set so that one of the selected 
landmarks (Fig. 2.12) appears at a course angle ( CA ) equal to 



CA 



MBL - Mc 



(2.17) 



where MBL equals the magnetic bearing of the landmark and MC is 
the initial magnetic course of the aircraft. 



If the above condition is satisfied, the zero point on the 
direction finder dial will coincide exactly with the longitudinal 
axis of the aircraft . 



/137 



To set the aircraft on definite courses, a table of course 
angles for landmarks for each aircraft course is compiled. 

For example, if the deviation has been determined at eight 

points, but the selected landmarks have magnetic bearings of 115 

and 328°, then the course angles for the courses which we require 
will have the values shown in Table 2.2. 



TABLE 2.2 



MC 


Cal^(MBL= 


,,^Q Cal (MBT-= 
-115^ 2 3280) 


CC 


Ak 





115 




32(5 


358 


+2 


45 


70 




283 


42 


+3 


90 


25 




238 


91 


—1 


135 


340 




193- 


133 


+2 


180 


295 


148 


178 


+2 


225 


250 




103 


224 


+ 1 


270 


205- 




58 


271 


—1 


315 


160 




13 


313 


+2 



When using this table, the sight of the direction finder is 
set to a given course angle for a landmark and the aircraft is then 
turned until the axis of the sight lines up with the direction of 
the selected landmark. It is clear that the aircraft is then set 
precisely on the desired course. 

The second landmark is an extra one in case the first is ob- 
structed by some part of the aircraft such as the empennage or wing, 



137 



The method of setting an aircraft on course by the method de- 
scribed above for obtaining the course angles of landmarks is the 

most precise and reliable one, 
especially since a fixed area 
can be set up at an airport for 
correcting devia;tions and doing 
other work to set the bearings 
of landmarks and compiling tables 
of course angles for given air- 
craft courses . 

However, this method is not 
always practicable. In some cases, 
it may be impossible to select 
suitable landmarks, and in other 
cases the visibility may be inad- 
equate for them to be seen. In 
some aircraft, there may be diffi- 
culty in fastening the direction / 
finder on board the aircraft in 
clear field of vision for observ- 




Fig. 2.12. Determination of 
Aircraft Course by the Course 
Angle of a Landmark. 

a place where there would be ; 
ing the landmarks . 



2 . D F rec 

This method is 



an 



or 



j.ii^^ ...^uwwv^ -- used in 

craft on courses of 0, ^5 , 



t i on finding of 

J -• _ cases when it is impossible to set the 

by the 



tai 1 

air- 



aii aircraft from the nose «. ,.>.... 

when it is impossible to set the air- 
90°, etc., by the method described above. 

In this case, the aircraft is set each time(e.g., according 
to the readings of the magnetic compass) to a given course. Then 
the direction finder is located along the extension of the longi- 
tudinal axis at a distance of 20-100 m from the aircraft, depend- 
ing on the type of the latter; the correcness of the setting of 
the aircraft on course is then determined as in the first case before 
mounting the direction finder on board the aircraft. It may be 
necessary to turn the aircraft for a secondary check. 

This method is less convenient than the first, since it is 
necessary to shift the direction finder for each course, set it 
exactly along the extension of the aircraft axis, adjust the zero 
on the dial along the magnetic meridian, and make the dial level, 
in addition to measuring the distance to the aircraft. Under unfav- 
orable conditions aboard the aircraft, this operation may have to 
be repeated after moving the aircraft. The advantage of this method 
is its independence of the existence of landmarks, meteorological 
visibility, and peculiarities of aircraft design. 

Semicircular deviation of magnetic compasses is corrected and 
eliminated at four basic points: 0, 180, 90 and 270°. 

It is clear from (2.13) that semicircular deviation at the 
and 180° points is equal in value, but opposite in sign, and ex- 
pressed by the maximum value of coefficient C. Deviation from 



138 



coefficient B is equal to zero on these courses. 

However, all of these courses are subject to the action of 
a constant deviation in the coefficient A and quarternary deviation 
E in addition to the semicircular deviation. This means that the 
values of the constant and quarternary deviation are equal in value 
and sign. 

Consequently, if the deviation on course 0° is set to zero 
by turning the cylinder of the deviation- correcting apparatus with 
the marking "N-S", the semicircular deviation will be compensated 
for and the constant and quarternary deviation will simultaneously 
be compensated for. It will change with the same sign to a course 
of 180°, where its value doubles. Therefore, after setting the 
aircraft to a course of 180°, it is necessary to set the deviation 
not to zero, but to half the rotation of that cylinder, and in the 
reverse direction. 

Hence, the semicircular deviation from coefficient C can be 
eliminated completely and precisely without disturbing the constant 
and quarternary deviations . 



Analogously, by turning the cylinder of the deviation- corre ct- 
ing apparatus with the marking "E-W", it is possible to reduce the 
deviation to zer-o for a 90° course and by half for a 270° course, 
which completely gets rid of the semicircular deviation from coeffi- 
cient B without disturbing the constant and quarternary deviations. 



/139 



TABLE 23 



MC 



Deviation 
Shown 



Up to 






12 





180 


+4 


+2 


90 


+7 





270 


-2 


—1 



-s 



— 


N7 


_ 




















L^J 












^^.,..— .^ 




s 


y 




/ 




( 


■A 


/- 






\^ 


/ 






V 


y 








— 










- - 


— 















«5" 



30 135 m 22S 270 



JI5 



*5 



-5 



Fig', 2.13. Graph of Deviation of a Magnetic 
Compass . 



139 



The operation with semicircular deviation is described in a 
special table (Table 2.3). 

Obviously, the remaining deviation at these points will be 
equal to +2° for courses of and 180° and -1° for courses of 90 
and 270° . 

After getting rid of the semicircular deviation, the aircraft 
is set to courses at 4-5° intervals and the remaining deviation is 
measured. An example of the recording is shown in Table 2.2. 

After summing the remaining deviation for eight courses (Graph 
5, Table 2.2) and dividing the sum by eight, we obtain the value 
of the constant deviation 

A 2+3-1+2+2+ 1 -1+2 

A = -^— = + 1^25°. 

The bowl of the compass must be set in its mounting to this 
value. If we disregard the value of 0.25° produced by turning the 
bowl of the compass through 1°, the remaining deviation for the 
eight courses will have a value of +1, +2, -2, +1, +1, 0, -2, +1 
so that the graph of the corrections can be compared with the read- 
ings of the compass (Fig. 2.13). 

If the aircraft is intended for use on flights at magnetic 
latitudes where there will only be small changes, this will mark 
the end of the work with deviation. 

In preparing for long distance flights , with considerable changes 
in the magnetic latitudes, the coefficients of the semicircular /I^j-O 
deviation B and C must also be found with determination of their 
changes with magnetic latitude. 

In this case, the coefficient B will be equal to: 



B = 



+ 1 sin (P + 2 sin 45° — 2 sin 90° + I sin 135° + 1 sin 180° + 
+ — 2 sin 270° + 1 sin 315° 



+ 1.4 — 2 + 0.7 + + + 2 — 0.7 



„-i- ^ 0.35, 



and coefficient C will be 



l + l,4_0^0,7+ l.t-0_ 4.0,7 

C =■ ■ . = 0,00. 

4 



140 



Gyroscopic Course Devices 

Regardless of the fact that measures have been employed for 
a long period of time which are directed toward increasing the accur- 
acy of readings and the stability of operation of integrated mag- 
netic compasses , their shortcomings have not been completely over- 
come . 

In addition, magnetic course devices are difficult to use in 
a flight along an orthodrome for long distances , due to the complex- 
ity of the calculation of the magnetic declination as it changes 
along the route. 

All of this has made it necessary to seek new ways of devis- 
ing course instruments and systems which will satisfy the require- 
ments of aircraft navigation at all stages and all conditions of 
flight. 

The first steps in this direction were made by the remote con- 
trol magnetic compasses, containing a magnetic transmitter (a sensi- 
tive element) located at any convenient point in the aircraft, 
whose readings were transmitted by means of special potentiome trie 
transmitters to dials mounted in the cockpit. 

This made it possible to mount the compass in the pilot's 
field of vision and ensure optimum conditions for operation of the 
compass from the standpoint of deviation. However, there were still 
considerable shortcomings in the operation of the compass, such 
as instability of the readings with movement of the aircraft and 
the impossibility of using it when the aircraft was turning. 

In addition, the reliability of operation of the compass de- 
creased, since the potentiometric connection with reliable contacts 
produced an additional delay in the turning of the sensor card to 
a significantly greater degree than was the case for the rotation 
of a freely moving card on its bearing in an integrated compass. 

The next steps in increasing the accuracy and reliability of / m-1 
operation of course devices was made by the gyroscopic semicompasses 
and magnetic course sensors linked with gyroscopic dampers. This 
made it possible to use the course instruments while the aircraft 
was turning and to achieve stability of course readings under any 
flight conditions. Analysis of the induction course sensors, free 
of friction during turning of an aircraft , significantly increased 
the reliability of magnetic compasses. 

However, the greatest reliability and accuracy in course meas- 
urements for aircraft has been achieved by the building of complexes 
of course instruments (course systems), combining the operation 
of gyroscopic, magnetic, and astronomic sensors. The principle of 
these systems is a stable and prolonged maintenance of the system 



I'll 



M 



for estimating the course, with a gyroscopic assembly having peri- 
odic correction of the readings by means of a magnetic or astro- 
nomical sensor, or input of corrections manually as desired by the 
crew . 

Fvino'ipte of Operation of Gyroscopic Instruments 

The gyroscope is a massive balanced body, rotating around its 
axis of symmetry at a high angular velocity. 

Gyroscopes are usually made in a form such that they have rel- 
atively low weight and small size, yet have a maximum inertial moment 
which is reached relative to the basic mass of the gyroscope as 
far as possible from the center of rotation within the given dimen- 
sions of the gyroscope. 

Let us recall that the inertial moment J in mechanics is the 
product of the mass times the square of the distance to the axis 
of rotation: 

' (2.18) 

where rj is the distance from the mass to the axis of rotation. 

For a' complete cylinder, which constitutes the basic mass of 
a gyroscope (Fig. 2. 14-), the inertial moment is 



■(-i-^?) 



(2.19) 



The gyroscope has two interesting properties which are used 
in a number of devices for pilotage and navigation: 

(1) Axial stabitity 3 i.e., the ability to maintain the di- 
rection of its axis of rotation in space in the absence of moments 
of external forces tending to change this direction; 

(2) Axiat precession of rotation under the influence of mo- 
ments of external force, i.e., a slow rotation of the axis in a 
plane which is perpendicular to the applied force, with maintenance 
of the direction in the plane of the application of the force. 

The first property of the gyroscope is usually used for sta- 
bilizing the directions of the axes of the coordinates for deter- 
mining the required values, the banking of the aircraft, the angle 
of pitch, and the course. The second property is used to set the 
axis of the gyroscope in the desired position, e.g., to the vert- 
ical of the locus of the aircraft, to the plane of the true hori- 
zon, for compensation of the apparent rotation of the axis due to 
the diurnal rotation of the Earth, etc. In addition, the property 
of precession is sometimes employed in devices which integrate the 



/142 



142 



action of the forces with time 
navigation devices. 



in the construction of inertial 



To explain the principles of operation of gyroscopic devices, 
let us consider the physical significance of the two properties 
of a gyroscope mentioned above. 




— r, -X ^ — - 



For the 
assume that 
is located a 
the axis of 
we shall sel 
at some poin 

Let us 
of a force F 
has been til 



Fig. 2.14. 



Gyros cope 
Rotor . 



Obvious 
of the eleme 
does not change when it passes through 
motion of the element at points A, Ai 
ference remain parallel. The tangents 
at the point C and diametrically oppos 
equal to Acj) . 



sake of simplicity, we shall 
the mass of the gyroscope 
long the circumference around 
rotation (Fig. 2.15), and 
ect an element of this mass 
t on the circumference . 

assume that under the influence 
, the axis of the gyroscope 
ted to an angle Acji . 

ly , the direction of rotation 
nt of mass of the gyroscope 

points A and B, since the 
and B, Bi tangent to the circum- 

to the direction of motion 
ite to it are at an angle 



Consequently, at these points there arises a difference in 
the velocities 



A7 = 7 sin A(() . 



(2.20) 



The greater the angular velocity of rotation of the gyroscope 
and the radius of the ring, the greater will be the circumferential 
speed of the element of mass and the magnitude of the vector A7. 



Obviously, the reaction 

duce resistance to the vector 

and Ci , i.e. , the forces F„ and Fp, 

_ P ^ 1 



of the mass of the gyroscope must pro- 
velocity change at the points C 



arise at these points, directed 



opp 
axis 



'■ ' ' _ p Pi ■" ' 

osite to vector A7 and producing the precession of the gyroscope 
s . 




It is easy to see that the inertial forces directed against 

the external force will be exactly equal to the latter, so that 

no rotation of the axis of the gyroscope in the plane of the action 
of the external force will be observed. 



/143 



143 



The precession rate of the gyroscope can be determined easily 
if we know the moment of inertia of the rotor and the moment of 
the applied external force. 

A change in the moment of inertia of the gyroscope with time 
will be proportional to the moment of the external force 



il 



whence 



dt dt i u • 



M_ 



(2.21) 



(2.22) 




where U equals the moment of the external force , J equals the mo- 
ment of inertia of the gyroscope, w is the angular velocity of gyro- 
scope rotation, and to ^ is the angular velocity of precession. 

By the change in the moment 
of inertia of the gyroscope, 
we mean here the change in the 
direction of the vector of iner- 
tia . 

At the same time, the rotation 
of the axis of the gyroscope 
through 180° produces an opposite 
motion of all points on the rotor, 
which amounts to a braking of 
the gyroscope from its initial 
angular velocity to zero, with 
a subsequent speeding up in the 
opposite direction to the same 
angular velocity. 
Fig. 2.15. Precession of a 
Gyroscope Axis. 

Begvee of Freedom of the Gyroscope 

By degrees of freedom in mechanics, we mean the directions 
of free motion of a body which is not limited by connections of 
any sort. For example, an object sliding along a given line (rail) 
has one degree of freedom; an object moving in any direction in 
a plane has two degrees of freedom, and an object which is moving 
in three dimensional space has three degrees of freedom. 

Besides the degrees of freedom of linear motion, there are /lUU 
also degrees of freedom of rotational motion of a body around its 
three axes . 

Hence, a completely free body has six degrees of freedom. 



14M- 



The rotors of gyroscopes in navigational and pilotage instru- 
ments have supports which limit their linear motion in a certain 
direction relative to the axes of the aircraft, so that when we 
are talking about the degrees of freedom of a gyroscope we are rer 
f erring only to the degrees of rotational motion. 

A gyroscope is considered to be free if all three degrees of 
rotational motion are free (Fig. 2.16). 



The first degree of 
of a gyroscope is the ro 
of its rotor around the 
bearings A^Ai. If these 
are tightly fastened to 
of the machine, as is do 
example for the flywheel 
ery , the gyroscope will 
one degree of freedom. 
if these bearings can mo 
an axis perpendicular to 
ings BjSj), then there w 
two degrees of freedom. 

If bearings B,Bi ca 
have the freedom to move 
still another (third) ax 
pendicular to B ,B i (bear 
the gyroscope will have 
of freedom and its axis 
set readily to any direction in space. 




Fig. 2.16. Gyroscope with 
Three Degrees of Rotational 
Freedom . 



free dom 
tation 
axis in 

bearings 
the body 
ne for 
s in machin- 
have only 
However , 
ve around 

A,Ai (bear- 
ill be 



,n also 

around 
is , per- 
ings C,Ci), 
three degrees 
can be 



As we can see from Figure 2.16. the degrees of freedom of the 
gyroscope are ensured by pairs of bearings and (with the exclusion 
of the first) rotating frames. 

A gyroscope usually has two rotating frames, internal and ex- 
ternal. In course gyroscopic instruments, the internal frame, to- 
gether with the rotor and the bearings of the gyroscope, serves 
to set the gyroscope axis in the plane of the true horizon. The 
same frame contains a sensitive element for correcting the gyro- 
scope axis for this plane. The internal frame of the gyroscope 
along with the rotor and sensitive element for correction are called 
the gyro assembly. 

The external frame ensures free motion of the axis of the gyro- 
scope in the plane of the horizon; from its position in the unit, 
we can get an idea of the direction of the gyroscope axis relative /145 
to the axis of the aircraft, or vice versa, thus making it possible 
to determine the aircraft course. 



145 



B-lveotion of Fveaession of the Gyroscope Axis 

The direction of the^ precession of the gyroscope axis under 
the influence of the moment of external forces can be seen in Figure 
2.15. 

For a rapid and error-free determination of the direction of 
the precession of the gyroscope axis, we use the concepts of "pole 
of the gyroscope" and "pole of the external force", and use the 
rule of the right-hand screw. 

For example, in observing the rotation of a gyroscope which 
is turning clockwise as viewed from the top (turning the screw in- 
ward), the pole of the gyroscope will be considered as being located 
at the lower end of its axis (Points P and Pi); with left-hand ro- 
tation of the gyroscope, at the upper end of the axis. Analogously, 
with a right-hand direction of the moment of external force, the 
pole of the moment is considered as being directed along the screw, 
in its rear portion as shown in our diagram (Point C^ ) . With a 
left-hand direction of the moment of external force, its pole is 
located in the front part of the picture (Point C). 

The precession of the gyroscope is always directed in such 
a manner that the pole of the gyroscope attempts to reach the pole 
of the external force by the shortest path. 

In our diagram, the lower end of the gyroscope axis will tilt 
backward, and the upper one forward, i.e., if we look at the draw- 
ing from left to right, the axis of the gyroscope will rotate clock- 
wis e . 

Apparent Rotation of Gyroscope Axis on the Earth *s Surface 

A freely moving gyroscope, with an ideally stabilized external 

and internal support and the lack of noticeable friction in the 

bearings , tends to keep the position of the axis of rotation of 
the rotor in space. 

On the Earth's surface, however, due to the diurnal rotation 
of the Earth and partially due to the curvilinearity of its motion 
around the Sun, there arises an apparent rotation of the gyroscope 
axis in the vertical and horizontal planes. 

The apparent rotation of the gyroscope due to the motion of 
the Earth around the Sun is expressed as a slight deviation of the 
rotation of the gyroscope axis from the apparent diurnal rotation 
of the Earth, as a result of the fact that the Earth makes a com- 
plete rotation around the Sun along its orbit in the course of a 
year. This conditional rotation amounts to a total of about 1/365 
of the apparent rotation of the gyroscope due to the diurnal rota- 
tion of the Earth. Hence, this value will not be considered in 
future . 



1^6 



Let us consider the apparent rotation of the gyroscope axis 
at various points on the Earth's surface, which appears as a result 
of the rotation of the Earth around its axis. We will assume that / m-6 
we have a freely mounted gyroscope, whose axis at the initial moment 
coincides with the vertical of the locus (Fig. 2,17, a). 

Obviously, if such a gyroscope is placed on a pole of the Earth, 
the axis of its rotation will coincide with the axis of rotation 
of the Earth and there will be no apparent rotation of the gyro- 
scope axis (position A in the diagram). 

If the gyroscope with a vertical axis is placed on some lat- 
itude (j) (position B in the diagram), its axis will be at an angle 
to the axis of rotation of the Earth, equal to 90°-(j). As we can 
see from the diagram, the apparent rotation of the gyroscope axis 
will describe a cone with an aperture angle at the vertex equal 
to 2 (90-(|)). 

In the case when the latitude of the locus is equal to zero 
(position C in the diagram), the aperture angle of the cone will 
be equal to 180°, i.e., it will turn in the plane of rotation. 

Now let us examine the case when the axis of the gyroscope 
at the initial moment is located horizontally at various points 
on the Earth's surface (Fig. 2.17, b) and coincides in direction 
with the meridian of the Earth. 

It is obvious that the axis of the gyroscope located on the 
pole (position A) will remain horizontal and will rotate in the 
plane of the horizon with the angular velocity of the Earth. The 
axis of a gyroscope located at some latitude (position B) will de- 
scribe a cone with an aperture angle equal to 2(j). The axis of the 
gyroscope located on the Equator will remain horizontal and will 
have no apparent diurnal rotation. 

It is important to note in this regard that if there is any 
kind of correcting force which acts constantly on the gyroscope 
axis in the plane of the true horizon, the angular velocity of the 
rotation of the gyroscope axis in the plane of the horizon will 
be equal to (Fig. 2.17, c): at the pole, the angular velocity of 
rotation of the Earth; at the Equator, zero; at any other point, 

(0 = fi sin (j) , (2.23) 

where fi is the angular velocity of the Earth's rotation and oi is 
the angular velocity of the apparent rotation of the gyroscope axis. 

From the examples which we have seen, it is clear that a freely 
moving gyroscope can be used to determine the position of the air- 
craft axis only in the following cases: 

(a) To determine the position of the vertical axis (banking. 



147 



II 



pitch) only at the poles; 

(b) To determine the direction of the longitudinal axis (course 
of the aircraft) only at the Equator. 



In order to render the gyroscope useful for determining the 
position of the aircraft axis at any other point on the Earth's 
surface, we used devices which compensate for the apparent rota- 
tion of the axis of the gyroscope due to the diurnal rotation of 
the Earth, as well as its own drift, which arises as a result of 
imperfect balance, friction in the bearings, etc. 



/147 




Fig. 2.17. Apparent Rotation of a Gyroscope on the 
Earth's Surface: (a) With Vertical Axis; (b) With 
Horizontal Axis; (c) With Constant Correction of the 
Axis in the Horizontal Plane . 

To keep the axis of a gyroscope constantly in the vertical 
position, pilotage devices ( gyrohorizon , gyrovert i cal ) , or in the 
horizontal position in the case of course instruments, are usually 
fitted with pendulum devices which act as sensitive elements react- 
ing to any deviations which may arise. 

The signals from these devices are converted to air currents 
in pneumatic devices and to moments of special electric motors in 
electrical devices. 



148 



Electrolytic gravitational correction (Fig. 2.18) is most widely 
used at the present time. This device consists of a bubble level 
attached to the lower part of the gyro assembly. Unlike a conven- 
tional level, its chamber is filled with an electrically conductive 
liquid (electrolyte), while on the top of the spherical surface 
are mounted four current- carrying contacts. 



When the gyro assembly is in a vertical position (Fig. 2.18, 
a), the bubble level is located so that all four contacts are cov- 
ered half-way by electrolyte, so that the moment applied to the 
frame of the gyro assembly by the correcting motor is equal to zero, 



/ms 



Fig- 
Gr avi 

and a 
the 1 
one p 
in or 
frame 



If for some reason the 

sembly varies from the verti 

the current-carrying contact 

not be uniformly covered by 

(Fig. 2,18, b), resulting in 

able distribution of current 

windings of a small motor an 

moment which is applied to t 

of the gyroscope in such a w 

the precession which is prod 

the gyro assembly to a given 

ical position. For course d 

which have a vertical extern 

horizontally located axis of the gyroscope, in order 

atter to the plane of the horizon, it is sufficient t 

air of current-carrying contacts with a gravitational 

der to regulate the moment of the forces acting on th 





2.18. Electrolytic 
tational Correction 



gyro as- 
cal position, 
s will 
the fluid 

a suit- 
s to the 
d in a 
he axis 
ay that 
uced brings 

vert- 
e vi ces 
al frame 

to correct 
o have 

leve 1 , 
e external 



Obviously, for those devices which measure direction on the 
Earth's surface, in addition to devices for correcting the axis 
of the gyroscope in the plane of the true horizon, there must also 
be other devices which compensate for the apparent rotation of the 
axis of the gyroscope in the horizontal plane due to the diurnal 
rotation of the Earth. 

Gyroscopic Semicompass 

In principle of operation, the gyros emi compass (GSC) is a gyro- 
scope with three degrees of freedom and its axis of rotation located 
in the horizontal, a vertical external frame, and a fluid gravita- 
tional corrector, attached to the gyro assembly. The rotation of 
the gyroscope rotor is produced by alternating three-phase current, 
while the correction of the axis in the horizontal position is 
achieved by an electromagnetic moment applied to the external frame. 

The gyrocompass has has a very sensitive balance and low fric- 
tion in the axes of the supports, which ensures a low intrinsic 
shift of the gyroscope (called "dvift") . In addition, in order 
to compensate for this "drift", the gyroscope is fitted in the 



149 



horizontal plane with a special balancing potentiometer and motor, 
which apply a moment to the external frame of the gyroscope in the 
vertical plane . 

This same motor is used for compensating the apparent diurnal 
rotation of the axis of the gyroscope, and is therefore fitted with 
a special latitudinal potentiometer, which regulates the moment 
of the motor in such a way that the rate of precession of the gyro- 
scope axis is equal to and coincides in direction with the rate 
of rotation of the Earth's meridian in the plane of the true hori- 
zon at the given latitude . 

By comparing the formula for the precession of the gyroscope 
axis (2.22) and the formula for the angular velocity of rotation 
of the Earth's meridian (2.23), we can determine the moment which 
is inquired to be applied to the gyroscope axis to compensate for 
the diurnal rotation of the Earth 



/149 



M = QJta sin 



(2 .21+) 




where M is the moment applied to the gyroscope axis, Q is the ang- 
ular rotational velocity of the Earth, J is the inertial moment 
of the rotor of the gyroscope in the plane of its rotation, to is 
the angular velocity of rotation of the rotor, and cji is the lati- 
tude of the aircraft's location. 

With a constant rate of rotation 

of the rotor of the gyroscope, all 

of the coefficients which enter into 

the right-hand side of (2.2i+), with . 

the exception of sin cj) , are constants . 

The latter must be regulated in flight, 

Therefore, the potentiometer which 

regulates the moment according to 

the latitude of the aircraft , as 

well as the balancing potentiometer, 

„ ,„ „ , „ , are mounted on the control panel 

Fig. 2.19. Control Panel j. ., ,„. „ ^^. 

^ ,.^„ r- rs r. • °^ "the gyrocompass (Fig. 2.19). 

of KPK-52 Gyrosemicompass . s>y t- a / 

The external frame of the gyro- 
scope is fitted with a scale for estimating the gyroscopic course 
and a selsyn- transmitter for transmitting the course to the indi- 
cators . 

The indicating dial and the selsyn- transmitter are free to 
rotate along with the external frame and can also be set with the 
aid of a motor to any angle relative to the frame. The setting 
of the indicator dial to the zero position is accomplished manually 
by turning a special handle on the control panel marked "L-R" (left - 
right), see Figure 2.19. 

Hence, the gyrocompass is a sort of "keeper" for the course 



150 



calculation set by hand: the direction of the zero setting of the 
course on the GSC remains constant in the plane of the horizon, so 
that the gyrocompass is an orthodromic course device^ and is cap- 
able of guiding a flight along an orthodrome over any distance. 
The advantage of a gyrocompass is its independence of operation 
from the magnetic field of the Earth, and consequently the fixed 
accuracy and stability, in operation at any point on the Earth's 
surface, as well as the ease of determining the course without any 
kind of methodological corrections; this is particularly important 
for automatic navigational devices 



/150 




'rors 



It is relatively easy to eliminate errors in the operation 
of the GSC, which arise in the form of "drift". For this purpose, 
the operation of the GSC is tested on the ground for a period of 
one to two hours with an attempt being made to use the rotation 
of the balancing potentiometer to set the minimum excursions of 
the needle with time from the true settings. 

If a considerable deviation of the needle from the correct 
readings of the gyroscope is noticed during flight, this can be 
corrected by shifting the latitude scale on the control panel rela- 
tive to the average latitude of the given path segment. This means 
that the degree by which the scale is shifted for each degree at 
the time that the drift occurs will be the following at various 
flight latitudes : 



Range of Latitudes 
Degrees 



- 


- 32 


32 - 


- 42 


42 - 


- 60 


60 - 


- 70 


70 - 


- 90 



Magnitude of Scale 
Deviation, Degrees 

4 

5 

6 
10 
20 



The latitude on the scale must be increased if the tendency 
of the GSC is directed toward a reduction of the readings for the 
course with time, and it must be reduced if the course readings 
increase with time. 

It should be mentioned that all shifting mentioned above with 
regard to the gyros emicompass is in reference to northern latitudes. 
In southern latitudes, the latitudinal compensations for the apparent 



151 



rotation of the axis of the gyroscope must be reversed, since the 
rotation of the meridian takes place in the opposite direction rela- 
tive to the northern latitudes. In addition, the system for intro- 
ducing corrections to the movement of the needle of the gyrosemi- 
compass must also be shifted to the opposite direction. 

Shortcomings of the gyrosemicompass include the fact that it 
is necessary to set its readings manually at the beginning of a 
flight and to make corrections en route. During flight, especially 
in rough air, this involves a certain amount of difficulty, since 
it is impossible to separate the movement of the indicator needle 
due to course variations from those motions which are caused by 
setting the course manually, i.e., the value of the course to which /151 
the GSC must be set becomes variable. 

In addition, the GSC is subject to Cardan errors during turns. 

The essence of the Cardan errors is the shift in the reading 
of the indicator dial during banking. When the aircraft is banking 
less than 8°, these errors do not have any practical significance, 
but they rapidly increase with the degree of banking and can reach 
6-8° . 

The Cardan errors have a quaternary nature . They are equal 
to zero in banking in the plane of rotation of the rotor of the 
gyroscope and in the plane of the position of the axis of its rota- 
tion. Maximum errors arise when the gyroscope axis is then at an 
angle of 45° to the plane of the banking. 

Therefore, the axis of the gyroscope can assume any position 
relative to the axes of the aircraft, and also with respect to the 
zero point on the course indicator scale, and the graph of the bank- 
ing error is "floating", i.e., its maxima and minima can assume 
any position on the indicator dial while retaining the values and 
periodicity of the errors . 

These errors automatically disappear when the aircraft comes 
out of the turn; however, they do constitute certain shortcomings 
in the pilotage of an aircraft, i.e., they disturb the correct esti- 
mation of the moment when the aircraft begins to stop banking in 
making a turn. 

Distance Gyromagnetic Compass 

The distance gyromagnetic compass (DGMC) has significant ad- 
vantages over the integrated and distance magnetic compasses, since 
it is suitable for use when the aircraft is banking at a certain 
angle and completely damps the oscillations of the magnetic card 
in flight in a turbulent atmosphere . 

The gyromagnetic compass is a combination of magnetic and gyro- 
scopic course devices, in which the role of the course sensor is 



152 



played by the magnetic transmitter and the role of the stabilizer 
of the readings is played by the gyro assembly. 

Let us consider the combined system which is presently used 
for distance gyromagnetic compasses, e.g., the DGMC-7 (Fig. 2.20). 

The basic parts of the distance gyromagnetic compass are the 
magnetic sensor, the gyro assembly, and the main course corrector. 

In addition to these main parts, the compass must be fitted 
with a power supply (not shown in the diagram), as well as compen- 
sating and regulating devices: 

(a) Compensating mechanism (combined with the gyro assembly); 

(b) Rapid compensation button; 

(c) A mechanism for compensating the remaining deviation (com-/152 
bined with the main course indicator); 

(d) Outputs for course repeaters and other indicators; 

(e) Two-channel amplifier. 

The magnetic transmitter of the compass has a card whose axis 
carries a dial for showing the course directly on the transmitter 
(it can be used to get rid of semicircular deviation), as well as 
the brushes for the wires leading to the potentiometer on the trans- 
mitter . 

The transmitter potentiometer has a three-wire circuit con- 
necting it to the gyro-assembly potentiometer, through which it 
receives alternating current from the power supply. 

The transmitter in the damping suspension is mounted in the 
aircraft at a location where there is a minimum influence on the 
cards of the magnetic and electromagnetic fields of the aircraft. 



Compensation 
Button' 



correctio 
release 



3 



r 



rrr:^_ 



magnetic' 
transmitter 



com pen 
sating 
mechanism 

f 



H I iampii 



,gyro 
assembly 



fier 




U 
■D, 



Fig. 2.20. Functional Diagram of Distance Gyromagnetic Compass 
(DGMC) . 



153 



The transmitter housing carries a device for correcting semi- 
circular deviation. If the semicircular deviation at the point 
where the magnetic sensor is mounted does not exceed 1-2°, the devia- 
tion device is not used, since in this case it would not improve 
but would rather detract from the operating conditions of the trans- 
mitter . 



The gyro assembly consists of the gyroscope with a horizon- 
tal axis and a Cardan support, which ensures three degrees of free- 
dom for the gyroscope rotation. The external frame of the gyro 
assembly rotates around the vertical axis . 

The gyroscope is set in motion by means of a three-phase motor, 
whose stator is mounted on the internal frame of the gyro assem- 
bly and whose short-circuited rotor is the rotor of the gyroscope. 

For correction of the gyroscope axis in the horizontal posi- 
tion, the lower part of the gyro assembly is fitted with a two- 
contact gravitational corrector, whose activating mechanism is a 
motor which produces a moment of force that is applied to the ex- 
ternal frame of the gyroscope and acts in the horizontal plane. 



/153 



If for some reason the axis of the gyroscope varies from the 
plane of the true horizon, the contacts of the corrector will be 
covered nonuniformly by the shifting conducting fluid, thus result- 
ing in a distribution of currents passing through the corrector. 
This in turn transmits a signal for a correcting moment of force 
to be applied to the external frame. As a result of the preces- 
sion of the gyroscope axis, it is shifted to a horizontal position, 

The external frame of the gyro assembly carries a master 
selsyn for connecting to the principal indicator of the compass 
(the pilot's indicator, PI) and a three- conductor cord for connec- 
tion to the magnetic transmitter. 

The master selsyn and cable are connected closely together 
and can rotate together with the external frame of the gyro assem- 
bly. However, they can also rotate relative to the external frame 
by means of a special coordination mechanism. 

The coordination mechanism consists of a small motor with a 
reduction gear for the s low- coordination regime, in which the rate 
of rotation of the selsyn is 1-4° per minute. 

When it is necessary to carry out a rapid coordination, the 
motor is switched to reduced reduction by means of the rapid-coord- 
ination button and a special relay. The rate of rotation of the 
selsyn in this case is raised to 15-15° per second. 

The potentiometer of the gyro assembly is firmly fastened to 
the housing. 



154 



The coordination of the magnetic transmitter with a gyro assem- 
bly is accomplished as follows (Fig. 2.21). 

The alternating current passes through contacts A and B to 
reach the potentiometer of the gyro assembly and is picked up by 
pickups 1, 2, 3 mounted on the external frame of the gyro assem- 
bly, from which it passes to the pickups on the transmitter poten- 
tiometer, la, 2a, 3a. 

It is clear from the figure that if the position of the brushes 
of the current pickups on the transmitter A^jB]^ relative to the 
current leads of the potentiometer la, 2a, 3a differs from the posi- 
tion of thfe current pickups of potentiometer A, B relative to their 
current connections 1, 2, 3 by 90°, there will be a current in the 
pickups of the transmitter. 

At the same time, between the current connection A and the 
current pickup A^ in this case, there will be a portion of the poten- 
tiometer in the gyro assembly A-1 and a portion of the transmitter 
potentiometer la-Aj, represented as a sum of the four circumfer- 
ences . Such a length of winding of potentiometer will be placed 
between current connection A and current pickup B^ (segments A- 
2 and ■2a-Bi). Consequently, a potential difference will develop 
between points A^ and B^. 

We can reach an analogous conclusion if we consider the path 
of the current from connection B to pickups A j^ and B^. 



If the position of the brushes of current 
differs from the position of connectors A and 
is not 90° (considering their 
sections), there will be a 




relationship to 
current in pickups 
rent is 
channe 1 
and the 
the coo 
The pot 
in the 
with th 
begin t 
low spe 
an equi 
rents o 



pickups Ai and B^ 
B by an angle which 
the potentiometer / 15H 
Aj and Bj. This cur-..„ 

fed to the first 

of the amplifier, 
n to the motor of 
rdination mechanism, 
entiometer brushes 
gyro assembly, along 
e selsyn- transmitter , 
o rotate at a very 
ed until there is 
librium of the cur- 
n pickups Aj and Bi. 



Fig. 2.21. Potentiometric Trans- 
mitter of Position Signal. 

me Li'd 

less of the apparent rotation of the gyroscope 
tion of the Earth and the natural changes in t 



Th 
the mas 
bly con 
agree w 
the tra 



us, the position of 
ter of the gyro assem- 
stantly shifts to 
ith the position of 
nsmitter card, regard- 
axis due to the rota- 
he gyroscope axis. 



155 



Inasmuch as the agreement of the readings of the selsyns of 
the transmitter and gyro assembly takes place at an angular veloc- 
ity which does not exceed 4° per minute, the readings of the gyro 
assembly cannot show the influence of rapid changes in the posi- 
tion of the transmitter card, i.e., the mechanism for coordination 
is a damper which averages out the readings of the compass for an 
average position of the card. 

In order that no transmitter errors be transmitted to the gyro 
assembly when the aircraft is making a turn, the DGMC complex includes 
a correction switch which automatically shuts off the correction 
mechanism of the gyro assembly from the compass card when the air- 
craft is turning. Estimation of the readings of the aircraft's 
course during turns is made with a purely gyroscopic operation regime 
of the DGMC. 

Inasmuch as the apparent diurnal rotation of the gyroscope 
axis cannot exceed 1° in four minutes of turn, while the turning 
time of the aircraft at an angle up to 90° as a rule does not exceed 
1-3 minutes, no great errors in the compass readings are produced 
during the turn and the gyromagnetic compass can be used success- 
fully for turning an aircraft at a desired angle. 

Agreement of the gyro assembly with the basic course indicator 
is accomplished by means of a master selsyn (Fig. 2.22). 



Winding AB rotates inside the housing of the master selsyn, 
allowing alternating current to flow in the windings of the selsyn 
0-1, 0-2, 0-3. Currents which are symmetrical in phase also arise 
in the windings of the slave selsyn O^-li, 0i-2x, O^-Sj. Hence, 
the magnetic field of the resultant currents of the slave selsyn 
will be parallel to the magnetic field of the supply winding AB . 
Therefore, if winding AjB^ of the slave selsyn occupies a position 
which is perpendicular to the supply winding AB , the current in 
it will be equal to zero. 



/155 



If the angle between 
windings AB and A^B^ differs 
from a right angle , there 
will be a current in winding 
A;^B]^; this current passes 
through the second chan- 
nel of the amplifier to 
a motor which turns winding 
A^Bi, with an indicator 
scale showing readings 
up to a position where 

Fig. 2.22. Master Selsyn for Trans- AB is perpendicular. 

mitting Position Signal. 

The potentiometric 

and selsyn systems, with amplification of currents and analysis 

of signals by means of small motors , give very precise agreement 




156 



of readings and transmit them with high mechanical moments and good 
damping. This permits us not only to obtain precise and stable 
readings with the compass, but also to apply an additional stress 
to the course indicators or the intermediate links. For example, 
they can be used to set the mechanical compensators for deviation, 
and to take readings from otber indicators or devices which use 
course signals. 

The device for mechanical compensation of the residual devi- 
ation consists of a circular curved strip with special bends, which 
operates by means of a lever and pinion to produce an additional 
turning of the needle on the scale for showing the magnetic course. 
The adjustment screws are mounted along the edge of the strip, usually 
at every 15°, thus making it possible to compensate for the resid- 
ual deviation practically down to zero. 

However, it is not recommended that residual deviation greater 
than 2-3° be compensated, if it is possible to get rid of it by a 
deviation device with a magnetic transmitter, for the following 
reas ons : 



(a) Not getting rid of, but compensating for, semicircular 
deviation leads to considerable changes in it, depending on the 
magnetic latitude of the locus of the aircraft. 



/156 



(b) When the aircraft is turning and the magnetic correction 
is switched off while the compass is operating in a regime of gyro- 
scopic stabilization, the mechanical compensation for deviation 
(if it is shown on the indicator) causes errors in the course readings 
in the form of overshooting and lagging, equal to the value of the 
compensated deviation, thus making it more difficult to turn the 
aircraft at a given angle . 

In addition to the mechanical compensator for the residual 
deviation, the main indicator has a declination scale whose revolu- 
tion to the value of the magnetic declination of the locus of the 
aircraft converts the compass readings from magnetic to true. 

To link it with other devices, the main indicator has both 
a master and a slave selsyn, whose indications can be transmitted 
either with the aid of the activating motors or by a direct selsyn 
connection . 

In the case of direct selsyn connection, the windings of the 
selsyn in transmitter AB and the selsyn of the indicator AjBj are 
connected in parallel with the alternating current source. In this 
case, the winding A B of the slave selsyn attempts to set itself 
according to the regulation of the magnetic field, produced by wind:: 
ings 0;ilj, 0^21, 0x3i, i.e., it automatically assumes the position 
of the power winding AB of the master selsyn. 

The direct selsyn connection has a lower sensitivity for the 



157 



matching of the selsyns and a smaller working moment, so that there 
is a reduced accuracy of transmission. Hence, it is used for trans- 
missions where there are no particularly high demands made on accur- 
acy, e.g., for pilotage course repeaters connected to the main indi- 
cator. 

Gy ro i nduct i on Compass 

In the preceding paragraph, it was mentioned that the distance 
gyromagnetic compass has considerable advantages over the integrated 
compass. However, the magnetic transmitter of this compass has 
a serious shortcoming. 



The fact is, that the magnetic moment which moves the trans- 
mitter card to the plane of the magnetic meridian is itself very 
small, and while it is sufficient for turning the floating card, 
it is frequently insufficient for overcoming the friction of the 
brushes on the current pickups, especially in flight at high mag- 
netic latitudes. Therefore, this transmitter is unstable in oper- 
ation and frequently goes out of order. 

To overcome this shortcoming, new types of induction magnetic 
transmitters have been developed; in addition to having an increased 
threshold of sensitivity, they do not have the ability to move in 
the horizontal plane (in the azimuth); consequently there are no- 
errors due to splashing of the fluid over the sensitive element or 
obstruction; they are less sensitive to the influence of accelera- 
tions when the aircraft is yawing, and the size of the transmit- 
ter is smaller. 



/157 



The operating principle of the induction- type sensitive ele- 
ment is the dependence of the value of the alternating magnetic 
induction of the core upon the presence of its constant component, 
exerted in the core by the horizontal component of the terrestrial 
magnetism . 

For example, if the core has a constant component of magnetic 
induction in the direction of the vector OA (Fig. 2.23, a), then 
in order to bring it up to complete saturation in this same direc- 
tion we will require an additional vector AB . The change in indue- 







OB-OA 



^d 



b) 



tf h- 



_J 



OA + OB 

Fig. 2.23. Induction Saturation of the Core of the 
Sensitive Element: (a) Induction Vector Coincides 
with Saturation Vector; (b) Induction Vector and 
Saturation Vector are in Opposite Directions. 



158 



tion in this case is expressed by the difference between the vectors 
OB-OA. 

As we see from Figure 2,23, b, when the magnetic induction 
is brought up to full saturation, the change in induction in the 
opposite direction will be equal to the sum of the vectors OA + 
OB . 

The transmitter of an induction compass has three sensitive 
elements, each of which is made as follows: Two parallel magnetic 
cores ■ made of permalloy (a material with a high magnetic permea- 
bility and a very low value of magnetic hysteresis) have separate 
primary windings, connected in opposite phase, and a common secon- 
dary winding around both cores (Fig. 2, 2^1, a). Alternating current /15 ! 
flows through the primary windings of the cores . 

Obviously, if the constant component of the magnetic induc- 
tion of the cores from the horizontal component of the Earth's mag- 
netic field is zero, the vectors of its change with passage of an 
alternating current through the winding will be the same in both 
c'ores , but in opposite directions, and there will be no alternating 
current in the secondary winding. 

If the cores have a constant component of magnetic induction, 
the vector of the change in magnetic induction will be greater in 
one and smaller in the other; this will produce pulses of alter- 
nating current as shown in the graph in Figure 2.24, b. The mag- 
nitude of the current pulses will be proportional to twice the value 
of the constant component of the magnetic induction of the cores. 

The sensitive elements in the transmitter are arranged in the 
form of a triangle and their secondary windings form a sort of master 
selsyn (Fig. 2.25). The rotating winding of the slave selsyn is 
connected to the amplifier and mounted in a position perpendicular 
to the resultant vector of the electromagnetic field of the slave 
selsyn by means of an activating motor with reduction gearing. 

The primary winding of this transmitter is mounted in an inter- 
mediate element between the transmitter and the gyro assembly in 
a correction mechanism which has a device for mechan compensation 
of residual deviation and is used as a correction mechanism for 
the following system. 




Fig. 2.24. Sensitive Element of Induction Transmitter; 
ing; (b) Graph of Current. 



(a) Wind- 



159 



The induction transmitters for the course are reliable and 
stable in operation, but their accuracy of operation drops when 
the transmitter is tilted to a sufficiently greater degree than 
is the case for magnetic transmitters. 

At the same time, if the tilting of the transmitter takes place 
in the plane perpendicular to the magnetic meridian, the vertical 
component of the magnetic field of the Earth, projected on the plane 
of the sensitive element, forms a magnetic induction normal to the 
magnetic meridian; the banking deviation will then be determined 
by the formula 



tg & = 



■jT sm ^ sm 
n 



(2.25) 



where i is the banking of the transmitter, 9 is the angle between 
the plane of the magnetic meridian and the banking plane of the 
transmitter, and Z^H are the vertical and horizontal components 
of the Earth's field, respectively. 



For example, with the ratio — = 3 and the angle 9 



90' 



each 



banking radius of the transmitter will produce an error of approx- 
imately 3° in the operation of the compass. 

2 
The ratio 77 = 3 corresponds (e.g.) to the latitude of Moscow /159 

and increases rapidly with an approach to the polar regions. There- 
fore, the banking errors in the induction transmitter can take on 
very significant values . 

In order to reduce the errors in the induction transmitter, 
its sensitive element is mounted on a float mounted in a Cardan 
support. The body of the transmitter is filled with fluid to reduce 
the pressure on the axis of the frame of the Cardan suspension 

(a mixture of ligroin and 
methylvlnylpyridine oil). 
The Cardan suspension ensures 
the horizontal position of 
the sensitive element during 
banking and pitching to within 
17° . 

The induction transmitter, 
like the magnetic one, is 
mounted aboard the aircraft 
in a position such that it 
is exposed to the smallest 
magnetic field of the air- 
craft and one which is as 
constant as possible; a 
deviation mechanism is mounted 
on it to record the semi- 




Fig. 2.25. Diagram Showing Con- 
nection of Elements in Sensor of 
Gyroinduction Compass. 



160 



circular deviation of the transmitter. 

However, the curvilinear trajectory of flight (although the 
radius of curvature is very great), in addition to the accelera- 
tion produced by Coriolis forces, produces a constant tilting of 
the sensitive element of the transmitter, the deviation from which 
is transmitted to the main indicator and its repeaters. 

For example, at the latitude of Moscow and an airspeed of 800 
km/hr, the tilting of the sensitive element of the transmitter due 
to the acceleration of the Coriolis forces will be equal to approx- 
imately 20', which undergoes deviation equal to 1° in a flight in 
the northerly and southerly' directions. 

The gyroscopic induction compass (with the exception of the 
induction transmitter) is built in a manner similar to that of the 
distance magnetic compass. 

Its principal components are the induction transmitter, the 
gyro assembly and the course indicator. 



In addition 
amplifiers 
rid of res^^„^^ ^ 
mechanism for rap 
from the main 



dition to the principal units, ttiere is a_Lso a. power 
, correction mechanism with a curved device for gett 
idual deviation, a connecting chamber, a button with d 
for rapid coordination, a correction switch, and repeaters 
ain course indicator. 



there is also a power supply, 

for getting /160 
a 



The correction mechanism is the intermediate link between the 
induction transmitter and the gyro assembly. The connection between 
the induction transmitter and the correction mechanism is made with 
a selsyn, while the connection between the correction mechanism 
and the gyro assembly, the gyro assembly with the main indicator, 
and the main indicator with the repeaters is made by potentiom- 
eters . 

The main indicator also has a curved device for getting rid 
of errors in the distance transmission of the course indications 
from the gyro assembly to the indicator at the factory. 

The correction switch is a two-stage gyroscope which serves 
for automatically disconnecting the gyro assembly from the correction 
mechanism; this disconnects the circuit for azimuth correction from 
the induction transmitter and disconnects the correction of the 
horizontal position of the axis of the gyroscope rotor when the 
aircraft is making turns with an angular velocity greater than 36 
deg/min . 

Disconnecting the induction transmitter during turns gets rid 
of the considerable errors which arise due to the influence of the 
vertical component of the Earth's magnetic field Z. In order to 
ensure that the gyroscope correction will not be disconnected in 
a turbulent atmosphere when the aircraft is bumping and yawing. 



161 



the correction switch has a delay mechanism which disconnects the 
correction only after 5-15 sec have elapsed following the moment 
when the aircraft reaches an angular velocity of 36 deg/min. 

The course repeaters are simple in design and consist of three- 
phase magnetoelectric lagometers whose accuracy for determining 
the course is lower than that of the main indicator. 

Despite the numerous advantages of distance gyromagnetic and 
gryoinduction compasses over integrated compasses, they do not com- 
pletely satisfy the requirements of aircraft navigation, partic- 
ularly with regard to automation of its processes , since the follow- 
ing shortcomings of compasses still persist: 

(a) The dependence of the accuracy with which the course is 
measured upon the magnetic latitude and the impossibility of using 
the instrument at high magnetic latitudes. 

(b) The difficulty of maintaining an orthodromic direction 

of flight, since the magnetic flight angles which are then obtained 
vary . 

(c) The magnetic loxodrome along which a flight can be car- 
ried out with a constant magnetic flight angle is a complex curve, 
since it depends on the intersection of meridians and magnetic declin- 
ations, which limit the length of the straight-line flight segments , /161 
along which the flight angle can be assumed constant. 

(d) Regardless of all the measures which have been taken to 
get rid of and correct for deviations, as well as the consideration, 
of magnetic declinations, the accuracy of the measurements of the 
magnetic course still remain low(within the limits of 2-3°). 

The majority of these shortcomings can be overcome by using 
gyroscopic semi compasses with high accuracy, or course systems which 
make it possible to fly in a regime using highly sensitive gyro- 
semi compasses (the GSC regime). 

Details of Deviation Operations on Distance Gyromagnetic 
and Gy ro i n duct i on Compasses 

Deviation operations on distance compasses are carried out 
using the same method as for integrated compasses, with certain 
changes necessitated by features of the design and mounting of these 
compasses , 

In several types of aircraft, the semicircular deviation at 
the point where the transmitters are mounted can be very low. In 
these cases, the deviation devices must be removed from the trans- 
mitters and all forms of deviation are compensated for by a mechan- 
ical compensator on the main course indicator or on the correction 
mechanism . 

16 2 



The compensation for the residual deviation, using a mechan- 
ical compensator, is carried out on 2H courses: 0, 15, 30, ..., 
345° , in which the aircraft is set to the desired courses , and a 
screw is turned (corresponding to the course of the aircraft) in 
order to bring the remaining deviation to zero. The graph of the 
remaining deviation on the main course indicator is not plotted. 
However, if differences in readings between the main indicator and 
its repeaters are noticed, it is necessary to plot a graph of the 
corrections for the readings on the repeaters. 

After each two intermediate settings of the aircraft on course 
(at the points 0, 45, 90, 135, 180, 225, 270 and 315°), it is neces- 
sary to mark the readings of the compass transmitter on the scale 
of the compass course on the main indicator (for induction trans- 
mitters, on the scale of the correction mechanism), and use this 
to determine the coefficients of semicircular deviation B and C. 
The form shown in Table 2.4 is recommended for convenience in deter- 
mining these coefficients. 

The coefficients are calculated according to the formulas: 



B = 



2 8/ Sin MC 



C = 



2 6/ COS MC 

(=0 



where 6. is the compass deviation on individual courses 

TABLE 2.4. 



MC ° 


8° 


sinMC 






45 




0.7 


90 




1 


135 




0.7 


180 




o' 


225 




-0.7 


270 




— 1 


315 




-0.7 



RslnMC 



cos MC 



1 

0,7 


-0.7 
— 1 
-0,7 



0.7 



/162 



8 cos MC 



The calculated coefficients must be in the form of tables, 
attached to the instrument panel along with the main course indi- 
cator. In addition to the coefficients on the table, it is also 
necessary to show the place where the deviations were corrected 
or the horizontal component of the magnetic field of the Earth at 
the point where the correction was carried out. 

Since the semicircular deviation, as well as all its other 



163 



forms, can be made by a mechanical compensator at the magnetic lati- 
tude of the point where the correction was made. Formula (2.16) 
for calculating the deviation for other magnetic latitudes assumes 
the form 



' = ^(^-0^'"^-'^t^-')"''- 



(2.26) 



Course Sys terns 

The most complete devices for measuring the course of an air- 
craft are the course systems. Course systems are combinations or 
complexes of various course transmitters mounted on the aircraft, 
with their readings displayed on general indicators . Such trans- 
mitters include the following: 

Magnetic induction (MC regime); 

Astronomical (AC regime); 

Gyroscopic (GSC regime). 

In principle, the course system consists of a combination of 
the design features of a gyroinduction compass, gyrosemicompass 
and astronomical course transmitter, whose operating principle will 
be discussed in the chapter devoted to astronomical means of air- 
craft navigation. 

The primary feature of the design of the gyroscopic portion 
of the course system is the presence of a third frame for the gyro- 
scope with a horizontal axis, coinciding with the longitudinal axis 
of the aircraft. The purpose of the third frame is to select the 
Cardan errors in the readings of the gyrosemicompass when the air- / 16 3 
craft is turning. 

The use of this third frame completely excludes Cardan errors 
from the transverse rolling of the aircraft, since the second frame 
of the gyroscope (with a master selsyn) will always be in a vert- 
ical position. 

The setting of the second frame of the gyroscope in a vert- 
ical position is accomplished by means of an electrical circuit 
and a mechanical device for matching it with the so-called gyro- 
vertical, mounted on aircraft for pilotage purposes. 

The second feature of course systems is the use (as a rule) 
of two gyro assemblies, a main one and a standby, which improve 
the reliability of the system and ensure reciprocal control of the 
readings . 

Figure 2.26 shows the control panel and the indicator of the 
course system. The course system operates on the main indicator 
in a regime in which the switch for the operating regime is set 
at the top part of the panel (MC, AC, or GSC). 



164 



r 



When switching the course system 
MC or AC regimes, in order to correct 
sary to press the button for rapid co 

the readi 
the readi 







After cor 
returned 

The 
for manua 
the cours 
The switc 
the panel 
switch th 
potentiom 
for the r 
Northern 
covers at 
"main" an 
for the b 
the main 



from the GSC regime to the 
the readings, it is neces- 
rrelation in order to adjust 
ngs of the gyro assembly to 
ngs of these transmitters, 
relation, the switch is again 
to the GSC position. 

pushbutton course control serves 

1 setting of the values for 

e system only in the GSC regime. 

h on the left-hand side of 

, marked "N-S", is used to 

e polarity of the latitudinal 

eter in order to compensate 

otation of the Earth in the 

or Southern Hemisphere. The 

the bottom of the panel, marked 
d "standby", cover adjustments 
alancing potentiometers of 
and standby gyro assemblies. 



Methods of Using Course Devices 
for Purposes of Aircraft Navi- 
gation 



The 



methods of using course equip 
solving powers of 



/16I+ 



ment 



Fig. 2.26. Control 
Panel of Course System, 



Panel of Co 

if flight, the remainin 
.nd a given route (air 



depend upon the resolving powers of 
the complex of course devices mounted 
on the aircraft, the presence of other 
equipment for purposes of aircraft navi- 
gation, and also on the distance, geograph- 
ic and meteorological conditions of 
flight. 

While the meteorological flight 
nditions along a given route (path) 
ange in the course of time and can 



COiiu-LLj-uiia cixuiig a g±ve 
change in the course of ^^...^ ^ 
ary depending on altitude and distance 
"'""■' ^"' a given type of aircraft 



„ conditions for _ 
route) remain constant 



In discussing the methods of using course devices in flight, 
the constant conditions listed above can be divided into three groups 

(1) The aircraft is equipped with an integrated or distance 
gyromagnetic (induction) compass. Flights are carried out over 
long or medium distances without significant changes in magnetic 
latitude. The equipment for constant measurement of the airspeed, 
drift angle, and automatic calculation of the path are lacking on 
the aircraft. 



165 



II 



(2) The aircraft is fitted with a distance gyromagnetic or 
gyroinduction compass and a gyrosemicompass or course system of 
average accuracy. Flights are made over long distances with consid- 
erable changes in magnetic latitude. There is no equipment for 
automatic measurement of the drift angle or airspeed, or calculat- 
ing the flight according to these parameters on board the aircraft. 

(3) The aircraft is fitted with a course system of high accur- 
acy, as well as devices for automatically measuring the drift angle, 
the airspeed, and calculating the path. Flights are made at any 
geographical latitude and for any distance. 

Methods of Using Course Devices Under Conditions Inclu- 
ded in the First Group 

Under the conditions in the first group, i.e., when flights 
are being made over short distances in aircraft which have simple 
navigation equipment, the following methods are used to prepare 
the calculated data and use the course devices in flight. 

In preparing for a flight, the route of the flight to be made 
is entered on a flight chart. If the flight chart is one which 
is in an international or diagonal cylindrical projection, the 
straight-line portions of the flight between the turning points 
along the route are plotted as straight lines by means of a ruler. 
When using charts which are plotted with an isogonal cylindrical 
projection (Mercator), the straight-line portions of a flight whi ch /165 
is very long are plotted as a curved line on the basis of the inter- 
mediate points along the orthodrome , calculated by analytical means. 




If the indicated correction is more than 3° in the straight- 
line portion of the flight, this segment is divided into two, three 
or more parts and the flight path angle is determined for each. 
This is usually not done by simple division of a straight line into 
equal parts, but by selecting characteristic orientation points 
along the section of the route, the flight between which can be 
made at the constant flight path angle. 

If we consider the low accuracy of the indications of the 



166 



magnetic compasses in a relatively short length of flight segment 
for a flight with a given flight path angle, the latter are deter- 
mined not by analytical means, but by simple measurement of the 
direction of the segment on the chart by means of a protractor. 

Measurement of the loxodromic flight path angle can be made 
relative to the meridian which intersects the segment at a point 
which is closest to its center, considering the magnetic declin- 
ation of this point. However, to increase the accuracy of the meas- 
urements, it is recommended that it be done at two points, at the 
beginning and end of the segment, considering the average declin- 
ation of these points. 

Obviously, in the first case the magnetic flight angle of the 
segment will be 



MFA = a 



m 



Mn,' 



while in the second case 



MFA = 



2 



where a 



b ' 



are the azimuths of the orthodrome at the begin- 



ning, the middle, and end, respectively. 

An advantage of the second method is the double measurement 
of the angles and the averaging of the declinations, since the accur- 
acy of two measurements and the averaging of their result is always / 166 
higher than the accuracy of a single measurement. 

For the first group of conditions, it is possible to have some 
simplified preparation for the course equipment of the aircraft 
for the flight. Since the flights are made with relatively low 
measurements of magnetic latitude, there is no need to determine 
the coefficients of semicircular deviation B and C or to consider 
their changes during the flight. 

If the deviation is compensated by a mechanical compensator, 
it is assumed to be zero during the flight. In considering the 
residual deviation, a value is assigned to it as shown on the graph. 

During the flight, the course of the aircraft is checked so 
that its value together with the drift angle of the aircraft will 
be equal to a given magnetic flight path angle of the flight seg- 
ment . 



MFA. 



MC -t- US 



MFA 



g' 



On the other hand, since the magnetic course of the aircraft is 
equal to the compass course, it is necessary to add the compass 
deviation : 



167 



MFA^ = CC + Ac + US = MFAg. 

Froblems 

1. The direction of a flight segment measured along the aver- 
age meridian is equal to 48° ; the magnetic declination in the middle 
of the segment is +7°. Determine the given magnetic flight path 
angle of the segment. 

Answer: mfa = 41° . 

2. The direction of a flight segment measured along the in- 
itial meridian is equal to 136°, 132° at the final meridian, with 

an initial magnetic declination of +7° and a final one of +5°. Deter- 
mine the MFA„. 

Answer: MFA = 128°. 

3. The given magnetic path flight angle of a segment is equal 
to 84°, the drift angle was equal to -6°, the deviation of the mag- 
netic compass is +4°. Determine the required compass course for 
following the flight lines . 

Answer: cc = 86° . 

4. The compass course of an aircraft is equal to 54°, the 
compass deviation is +3°, the drift angle is +6°. Determine the 
actual flight path angle. 

Answer: MFA^ = 63° . 

Methods of Using Course Vevioes Under Cond-itions of 

the Second Group 

When flights are being made over long distances using distance 
gyromagnetic and gyrosemi compass es or course systems, but without 
any automatic course calculation, the use of course instruments 
in flight and preparation of charts for a flight is accomplished 
by devices which are somewhat different from those which are recom- 
mended for the conditions of the first group. 

The most important of these devices is the plotting of the 
orthodromic course along the straight- line segments of the flight 
with a gyrosemicompass or a course system in the "GSC" regime, with 
periodic correction of the gyroscope course by means of a magnetic 
or astronomic transmitter. 

As a rule, in flights over long distances, the flight chart /167 
is one with a scale of 1 : 2 , 000 , 000 on the international projection. 
If a straight line within the limits of one sheet of this map, with 
distances ip to 1200-1500 km, can be assumed with insignificant 
error to be an orthodrome, then when two or more sheets are combined 

168 



and the route does not run along a meridian or when sheets of this 
chart are used separately at great distances, the orthodrome must 
be located along points which are determined by calculation. When 
splicing two adjacent sheets along the meridian, the orthodrome 
has a significant break in it, and in this case (when it crosses 
the adjacent sheets) a straight line cannot be taken as the ortho- 
drome . 

On the charts of all other projections, except the central 
polar and special route maps in a diagonal, cylindrical projection, 
when the line of the tangent (cross-sectional strip) of the cylin- 
der coincides with the axis of the route, the orthodrome is calcu- 
lated analytically and plotted on the chart according to the calcu- 
lated intermediate points . The distances for the sections of the 
orthodrome are also determined by analytical means. 

The orthodromic flight path angles of the route segments under 
these conditions are measured or calculated analytically relative 
to the initial meridian of each flight segment. If the straight- 
line segments of the flight have a very short length, the flight 
path angles calculated from the initial meridians of the segments 
can be applied to the system relative to the selected reference 
meridian (Fig. 2.27) according to the following formula: 



OFA 



TFA + 6 = TFA + (A, 



ref-^init-' sxn 



■•m 5 



where 6 is the angle of convergence between the reference and initial 
meridians of the segment. 



Since the condition for the second group assumes flights over 
long distances with considerable changes in the magnetic latitudes, 
the preparation of the magnetic compasses must be made with a consid- 
eration of determination of the changes in the semicircular devi- 
ation during the flight. 



second 
ual de 
not pi 
sary t 
and C: 



ourse devices intended for flights under conditions of the 

group have devices for mechanical compensation of the resid- 
viation. Therefore, the graph of the deviation for them is 
otted. However, in getting rid of the deviation, it is neces- 
o determine and write down the coefficients of deviation B 



B 



i=0 



C = 



2 h cos 7/ 

1 = 



It is then necessary to write down the intensity of the horizon- /16! 
tal component of the Earth's magnetic field at the point where the 
deviations were corrected. 



169 



To calculate the changes in the semicircular deviation during 
flight, the corrections for the magnetic course at different seg- 
ments of the route must be determined when preparing for a flight. 
They are determined for a number of points along the flight path, 
on the basis of the magnetic flight angles of the route at these 
points with a frequency such that the difference between two adja- 
cent corrections along a straight line path does not exceed 1° and 
after each turning point on the route. 



In fact, the changes in the semi- 
circular deviation at correspond- 
ing points along the route will differ 
only slightly from the calculated cor- 
rections, since the course which is 
followed will be prepared with a consid- 
eration of the drift angle of the air- 
craft. However, the errors which arise 
in this process will be small and can 
be disregarded. 

During the flight, the gyrosemi- 
compass or the course system is cor- 
rected for the magnetic or astronomical 
transmitter when flying along the refer- 
ence meridians or the turning points 
of the route (TPR). If the correc- 
tion is made on the basis of the mag- 
netic transmitter, then the main indicator will have the required 
correction entered on its dial. This correction is equal to the 
sum of the magnetic declination and the change in the semicircular 
deviation along the magnetic latitude . 




Fig. 2.27. Calculation 
of Flight Path Angles 
from Reference Meridian, 



For correction, the course system is switched to the "MC" re- 
gime and the button is pushed to match the readings . The system 
operates for a period of 1-2 min in the slow coordination regime 
and is then switched to the "GSC" regime . 

In this manner, the systems are corrected for the astronom- 
ical transmitter. Having determined the latter on the basis of 
the coordinates of a star and the locus of the aircraft, the system 
is switched to the "AC" regime, the coordination is carried out, 
and then switched back to the "GSC" regime . This means that at 
the turning points of the route, no corrections are required on 
the scale of the declinations. 

The correction of the gyrocompass is made in the same manner, 
except that the course is set on the gyrosemicompass not by com- 
paring the readings of the transmitters , but by manual setting on 
the basis of the readings of the magnetic or astronomical trans- 
mitters . 

After correction, the flight is carried out with an orthodromic 



170 



course up to the next turning point of the route or reference merid- 
ian . 

When it is necessary to make a correction for the orthodrom- /169 
ic course between two reference meridians, the correction is set 
on the main indicator and is equal to: 

for the magnetic transmitter, 

A = A + (X ^-X,,^) sin <}) ; 
M ref MC m 

for the astronomical transmitter 

^ = ^^ref-^MC^ ^^^ *>"• 

Then the readings are matched in the manner described above. 

Prob lems 

1. The east longitude of the reference meridian is 40°, the 
north latitude of the reference point is 52°. The coordinates of 
the setting point of the route are: longitude 43°, latitude 54°. 
The true flight path angle of the segment at the starting point 
is 67°. Determine the orthodromic flight path angle calculated 
from the reference meridian. 




rse 



Answer: 



+3.5°. 



3. The east longitude of the reference meridian is equal to 
70°, the north latitude of the reference point is 58°. The air- 
craft is located at the point X - 76° , cj) = 60° ; the magnetic declin- 
ation of the location of the aircraft is equal to +11°, while the 
correction for the change in the semicircular deviation Ag^ = +2°. 
Determine the correction for the readings of the magnetic compass 
for correction of the orthodromic course. 



Answer : 



+ 8' 



171 



Methods of Using Course Devices Under the Conditions of 

the Third Group 

The third group of conditions for using course devices refers 
to flights in aircraft which are fitted with precise course sys- 
tems, apparatus for automatic measurement of the airspeed of the 
aircraft, the drift angle, and automatic calculation of the flight 
path of the aircraft. 



The 
aircraft 
with corr 
located s 



conditions of the third group assume a prolonged autonomic 
navigation with no visibility of the ground or over water, 
section of the aircraft coordinates only at individual points 
located significant distances apart. This places particularly strict 
requirements on the accuracy of the plotting of the orthodrome on 
the charts, the determination of the flight path angles, and the 
retention of systems for calculating the aircraft course, since 
the course is a basis for the automatic calculation of the flight 
in terms of direction. 

From the theoretical standpoint, a more precise and conven- /170 
ient form for using the course devices under conditions of the third 
group is the following: 

In preparing the flight charts for each orthodrome section 
of the flight between the turning points on the route, regardless 
of their length, we determine the conditional shift in the longitude 
(X ), i.e., the difference between the longitude calculated from 
the point where the given orthodrome intersects the Equator (Aq) 
and the geographical longitude (A): 

^s ~ ^0 ~ ^ • 

Here, the orthodromic longitude Ag is determined for the start- 
ing point of each segment by the formula 

ctg \ = tg<P2 ctgVi cosec AX — ctg AA. 

After determining the change in the longitude, the longitude 
of any point along the route can be converted easily to the ortho- 
dromic system, thus making it possible to determine relatively easily 
all of the required elements of the orthodrome for these points: 

(a) The azimuth of the point of intersection of the orthodrome 
with the Equator (ag) 



tgao = 



Sin \ 
»g"Pi 



(b) The coordinates of intert ermediate points for plotting 
the orthodrome on the map: 



172 



tg<p( 



sin Xo, 
fg«o 



(c) The initial. 



drome 



intermediate, and final azimuths of the ortho- 



«ga/ = 



Sin <fi 



(d) The distance to any point along the orthodrome iS^) from 
the point of its intersection with the Equator 



cos S . 



cos A , cos (|)^ 



(e) The distance between any two points along the orthodrome 
as a difference in the distance from the point of intersection with 
the Equator 



1-2 



= So ~ S 1 




In this case, the path angle of the first orthodromic flight /171 
segment is considered to be equal to the azimuth of this segment 
relative to the meridian of the airport from which the aircraft 
took off. The path angles of all subsequent segments are obtained 
by combining the orthodromic flight angle (OFA) of the previous 
section with the turn angle ( TA ) of the line of flight at the turning 
points along the route (Fig. 2.28): 



OFA- 



1 ' 



OFA„ = ai + TAi + TA2...TA„_i 



The turn angles along the line of flight are found as the dif- 
ferences of the azimuths of the orthodrome, intersecting at the 
turning points of the route, determined according to the formula 



tga,= 



tgX, 



0/ 



sin fj 



Obviously, the latitude of the turning points will be common 
for the two orthodromes; for one it will be final, for the other 



173 



it will be initial. As far as the longitude is concerned, it is 
determined on the basis of the geographical longitude of the turning 
point of the route, considering the shift in longitude of the prev- 
ious and subsequent segments. 

When flying above a continent , 
the best method of correction 
for the orthodromic course under 
conditions of the third group 
is to introduce corrections into 
the course as a result of calcu- 
lations of the aircraft path. 



)TA 
OFAi=ab'+/IX 1 





Fig. 2.28. System for Calcu- 
lating Path Angles by Combi- 
ning the Turn Angles along 
the Flight Path. 



For example, if the readings 
of the calculating devices on 
board the aircraft at both the 
initial and final points indicate 
that it is on the line of flight, 
but has undergone a lateral devi- 
ation AZ during the flight, then 
obvious ly 

tgAT = -, 



where Ay equals the error in the readings of the orthodromic course, 
and S is the length of the control section of the flight. 

The sign of the correction to the compass reading coincides 
with the sign of AZ. 

With positive values of AZ, (a shift from the line of the de- 
sired flight to the right), the readings of the compass will be 
reduced and the correction must be positive; in the case of devi- 
ation to the left, the correction must be made with a minus sign. 



of 
more 



when determination of the correctness /172 
-^^^ "ath in terms of direction is 

must be made 



the 
d 



In flights over water, when de 
the calculation of the aircraft path in terms of direction is 
e difficult, the correction of the gyroscope course must be m 
by astronomical methods. This means that the difference between 
the orthodromic and true courses at any point will be equal to t 
difference between the orthodromic path angle of the segment and 
the running azimuth of the orthodrome at a given point; 

OC - TC = OFA - a. 

If the positive difference of the courses turns out to be greater 
(or if it is negative, turns out to be smaller) than the differ- 
ence between the path angles , the reading for the orthodromic course 
will be increased and it will be necessary to reduce it manually 
by the course detector. When the readings of the orthodromic course 
are low, it must be increased. 



174 



In this manner, but with reduced accuracy, the orthodromic 
course can be corrected magnetically: 



OC 



(MC+ ..) = OFA 

M 



For the conditions of the third group, the preparation of the 
magnetic compasses must be carried out according to the rules given 
above for the conditions of the second group. However, the use 
of magnetic transmitters for correction of orthodromic course during 
flight is limited to cases when the readings of the orthodromic 
course cannot be checked on the basis of the results of calcula- 
tions of the path or by means of astronomical course transmitters. 

The meteorological conditions of a planned flight, especially 
over long distances, call for careful preparation of all course 
equipment on the plane, since it may become necessary to use devices 
for measuring the courses which belong to all three groups of condi- 
tions . 

3. Barometric Altimeters 

The principal method of measuring flight altitude for naviga- 
tional purposes is the barometric method. It is based on the meas- 
urement of the atmospheric pressure at the flight level of the air- 
craft . 

For special purposes, such as aerial photography or aerial 
geodesic studies, as well as for signaling dangerous approaches 
to the local relief when coming in for a landing under difficult 
meteorological conditions, electronic devices for measuring alti- 
tude are used, which are more accurate in principle than the baro- 
metric method. However, they are not widely employed for navigational 
purposes because they are used only for measuring the true flight 
altitude. On the basis of the barometric method of measuring alti- 
tude, it is the law of change of atmospheric pressure with increase 
in height which means that the calibration of the altimeter dial 
must be made on the basis of the conditions of the international / 17 3 
standard atmosphere. 

The conditions of the standard atmosphere are as follows: 

(a) The pressure at sea level is equal to 750 mm Hg , or 1.0333 
kg/cm^ . 

(b) The air temperature at sea level is +15° C with a lin- 
ear decrease for flight altitudes up to 11,000 m of 6.5° for each 
1000 m of altitude. Beginning at 11,000 m, the air temperature 

is considered constant and equal to -56.5°. 

To understand the operating principle of the barometric al- 
timeter, let us recall the familiar equations from physics which 



175 



describe this state of gases and the conditions of their change. 

Thus, according to the Boy le-Mariotte law, with isothermal 
compression (i.e., fixed temperature), the pressure of a gas changes 
in inverse proportion to its volume so that the product of the volume 
times the pressure remains constant: 



pv = const , 

where p is the pressure of the gas and u is the volume of the gas 
at temperature t. 

According to the Gay-Lussac law, heating a gas by 1° C at con- 
stant pressure causes the gas to expand to 1/273.1 of the volume 
which it occupied at zero temperature: 



V — Vo — 



Vo 



273.1 



i. 



where Vq is the volume at zero temperature and the same pressure. 

By combining the Boy le-Mariotte and Gay-Lussac laws, we obtain 
the state equation of a gas: 

^''^ 2^ (' + 273.1). 

This equation is known as the Clapeyron equation. The temper- 
ature (t +273.1° C) is called the absolute temperature iT) , i.e., 

calculated relative to absolute zero (-273.1° C) -'- , and the constant 

p y 
value of — 2— 2_ is called the qas constant. 

273.1 

A gram molecule of any gas (gram mole, or simply mole), i.e., 
the number of grams of a gas which is equal to its molecular weight, 
always occupies exactly the same volume (22.1+1 liters) at zero temper- 
ature and a pressure of 1 atm. 



The gas constant for one mole of gas is called the universal 
gas constant (i?): 

PoVo 



/IT 



R = 



273. 1 * 



With P = 1 atm, V = 22.'+l liters. 

The Clapeyron equation for one mole of gas in this case as- 
sumes the form 



-'- This value is usually assumed to ba approximately 273° in cal- 
culations , 



176 



pv = RT . 
The numerical value of the universal gas constant is 

1.033-22410 



R = 



273.1 



^84,8 kg/cm( degrees/mole ) 



In technical calculations, the weight of the gas is usually 
expressed in kilograms. Therefore, we do not use the universal 
gas constant but rather the characteristic gas constant 

M 

where M is the number of grams of gas per mole, or its molecular 
weight . 

Then 

pv = BT. 

The constant B for air is 29.27 m/degree. 

By using the gas constant, we can find the weight density of 
air (y) at a given pressure p and absolute temperature T. 



y 



BT' 



Let us define an area on the Earth's surface measuring 1 cm^ , 
and erect a vertical column on it which extends upward to the limits 
of the Earth's atmosphere (Fig. 2.29). 

Obviously, the drop in pressure with increased altitude to 
the distance i\H at a certain height will be equal to: 

1 ^j. 



or 



'p ~ BT 



(2.27) 



By using Equation (2.27) and the altitude temperature gradient , /175 
we obtain the so-called barometric formula 



/'// = /'o^l-^'>/Ki; 



(2.28) 



where Tq is the temperature on the ground under standard conditions 
equal to 281° K, and tgp is the vertical temperature gradient. 



177 



Formula (2.28) is obtained from Formula (2.27), switching to 
infinitely small values: 



d£_ _dH 
p ~ BT 



(2.27a) 



Integrating (2.27a) and keeping in mind that ffj^ 



we obtain: 



0~ gr 



B, 



or 



'H H 

f rfp 1_ P dH 

p, s-^ 



. Ph 1 , 
In — = ^r— Ig 



Po Bt 



'gr- 



gr 



1 
1— £-//) gr 



-^ETl 




Fig. 2.29. Column of Air on 
Earth's Surface. 



Solving Equation (2.28) for H, we obtain the standard 
metric formula for the troposphere: 



the 
hypso- 



'grL VPo/ J 



(2 .29 ) 



Substituting into Formula (2.29) the numerical values of Tq, 



t gr-p and B, we obtain: 



// = 44 308 



1 — 



\Po) 



(2.30) 



We can use Formula (2.30) to calculate the hypsometric tables 
which relate the flight altitude up to 11,000 m to the atmospheric 
pressure; these tables are used to adjust and correct altimeters. 

Under the conditions of a standard atmosphere, the air tem- 
perature at altitudes greater than 11,000 m is considered to be 
constant, so that the barometric formula for these altitudes can 
be written as follows : 



1^^ I //—1 1000 



Pn 



BT„ 



(2.31) 



We obtain Formula (2.31) by integrating Equation (2.27) for /176 
11,000 m and consider T^ equal to Tui 



178 



III nil ■■■ ■ II mil 



Pa 11 



or 



Pa 

In — = ■ 
Pn 



BTn. 

//— llflOO 
BTn 



Solving Equation (2.31) for H, we obtain the standard state 
formula of the hypsometric table (Table 2.5) for altitudes greater 
than 11,000 m. 



H=nOOQ + BTi 



' Ph 



(2.32) 



H. M 



—500 

500 
1000 
1500 
2000 
2500 
3000 
3500 
4000 
4 500 
5000 
5500 
6000 

6 500 

7 000 

7 500 
8000 

8 500 

9 000 
9 500 

lOOOO 







TABLE 


2.5. 








Ph. 


T„. °K 


' a, 
m/sec 


H. M 


Ph> 


T„. °K 


a, 

m/sec 


806 .^2 


291.25 


342.1 


10 500 


183.40 


219,25 


297.2 


760,2 


228.00 


340.2 


11000 


169.60 


216.50 


295.0 


716.0 


284.75 


338.3 


I200O 


144.87 


216.50 


295.0 


674,1 


281.50 


336.4 


13000 


123.72 


216.50 


295.0 


634.2 


278.25 


334.4 


14000 


105.67 


216.50 


295,0 


596.2 


275.00 


332,5 


15000 


90.24 


216.50 


295.0 


560.1 


271.75 


330.5 


16000 


77,07 


216.50 


295,0 


525.8 


268.50 


328.5 


17000 


65.82 


216.50 


295.0 


493,2 


265.25 


326.5 


18000 


56,21 


216,50 


295,0 


462.2 


262.00 


324.5 


19 000 


48.01 


216.50 


295.0 


432.9 


258.75 


322.5 


20 000 


41.00 


216.50 


295.0 


405.1 


255.50 


320,5 


21000 


35.02 


216.50 


295.0 


378.7 


252,25 


318,4 


22000 


29,90 


216.50 


295.0 


353,8 


249.00 


316.3 


23 000 


25.54 


216.50 


295,0 


330.2 


245,75 


314.3 


24 000 


21.81 


216.50 


295.0 


307.8 


242.50 


312.2 


25 000 


18.63 


216.50 


295,0 


286.8 


239,25 


310.1 


26 000 


15.91 


216.50 


295.0 


266.9 


236,00 


308.0 


27 000 


13,59 


216.50 


295.0 


248.1 


232.75 


305.9 


28 000 


11.60 


216.50 


295.0 


230.5 


229.50 


303.7 


29 000 


9.91 


216.50 


295.0 


213.8 


226.25 


301.6 


30 000 


8.46 


216.50 


295.0 


198.2 


223.00 


299.4 











Note: The table for adjusting and correcting the barometric 
altimeters is given in abbreviated form. The value a represents 
the speed of sound at flight altitude under standard conditions, 
given in the fourth column of the table. 



Substituting the value of B and T 
to the log ten (In N= 2.30259 IgN), this 

.Pn 



//=11000+ 14600 Ig^ 
Ph 



= 216.5°, and shifting 
formula assumes the form: 

(2.33) 



179 



i I 



Formulas (2.30) and (2.33), suitable for compiling hypsomet- /111 
ric tables and calibrating altimeters, are not completely suitable 
for calculating the methodological errors in the altimeter, related 
to a failure of the actual air temperature at heights from zero 
to the flight altitude of the aircraft to agree with the conditions 
of the standard atmosphere. 

Since the accuracy of altitude measurement is affected by the 
air temperature not only at the flight altitude but at all inter- 
mediate layers from the one on the ground up to that at the flight 
altitude, it is better to use the formula which relates the flight 
altitude not to the temperature gradient, but to the average temper- 
ature of the column of air which we have selected, and to use this 
to calculate a hypsometric table for adjusting and correcting baro- 
metric altimeters. This formula has the form: 



^=^W"5- 



(2.34) 



Formula (2.34) is obtained by integrating Equation (2.27a) 
at a constant average temperature: 

"h h 

dp 1 



Sf^'SK^"' 



p> 



whence 



av p^ 



If we consider that 



^v- 



= 273 + (^,= 273(1 +2^j]. 



and the value B = 29.27, by using the 
from natural logarithms to the log 10 
form: 



coefficient 
Formula ( 2 



for transition 
34) assumes the 



ff =. 18 400 Il + -^]l„l2.. 
[ ^273.1) ^Ph 



This formula is known as the Laplace formula. 

Description of a Barometric Altimeter 

The sensitive element in the barometric altimeter is a cor- 



rugated manometric 
box has two _ 
one of which is 
a 



oj.cj.»= c j.=;.i.= i^ .- j-w the barometric altimeter is a cor- 
metric (aneroid) box 1 (Fig. 2.30), made of brass. The 
rigid points (on the top and bottom corrugated surfaces), 
is fixed or tightly fastened to the casing of the appar- 
the other is movable. 



:>ne or which is rixed or tightly 
itus , while the other is movable 



180 



In principle, the aneroid box can be either evacuated or filled /17 i 
with a gas . 

Usually, the space within the box is filled with a gas to a 
pressure such that when the box is heated, the thermal losses of 
its elastic properties will be roughly compensated by an increase 
in gas pressure within the box when it is heated. 

The casing of the altimeter is hermetically sealed and con- 
nected by a nipple to a sensor of the atmospheric (static) pres- 
sure . 




Fig.. 2.30. Schematic Diagram of Barometric 
Altimeter . 

When the aircraft is located at sea level, the aneroid box 
is compressed to the maximum degree, since the atmospheric pres- 
sure acting on it has a maximum value. 

With a gain in altitude, the atmospheric pressure in the cham- 
ber decreases and the aneroid box expands due to its elastic prop- 
erties, shifting its movable center (with bimetallic shaft 2) up- 
ward . 

As it moves, the center displaces rod 3, which in turn acts 
through a lever U to convey a rotary motion to shaft 5. 

Shaft 5 carries a toothed sector 6 with a counterweight, fitted 
with a cog wheel 7, which transmits the movement to the pointer 
through another gear. 

Thus, the motion of the center of the box is used to indicate 
the flight altitude on the scale of the instrument. 

In addition to the parts listed above, the kinematic portion 
of the instrument includes elements intended for regulating the 



181 



11 



instrument and adjusting the backlash in the transmission mechan- 
ism . 

1. Zero-point bimetallic compensator. This device is intended /179 
for compensating the temperature changes in the elastic properties 

of the box for zero altitude. If the atmospheric pressure in the 
casing of the instrument is set to zero altitude, but the temper- 
ature of the box increases, the loss of elastic properties of the 
material in the box creates additional compression, causing the 
indicator needle to shift from the zero altitude reading. The bi- 
metallic strip bends as the temperature changes, due to different 
coefficients of linear expansion for the two materials of which 
it is made. By rotating the strip in its socket, it is possible 
to set it in a position such that the deflection in the direction 
of the shift of the center of the box will exactly correspond to 
the additional travel of this center, but in the opposite direc- 
tion. Then rod 3 remains in place and the indicator needle will 
not move from the zero position. 

2. The regulating mechanism of the device. This consists 
of strip 4 and an adjustment screw. 

Turning the screw pushes the strip away from rod 5, changing 
the arm of the lever. This is used to regulate the angular veloc- 
ity of rotation of the shaft, i.e., the transmission ratio of the 
apparatus. The transmission ratio of the rotation of the shaft 
is set so that the readings of the needle correspond to the atmo- 
spheric pressure in the casing of the apparatus. 

3. High temperature compensator. When the elastic properties 
of the box change due to the effect of temperature, this not only 
causes an additional compression at zero altitude but also changes 
the amount by which its center moves with a change in altitude. 

For compensation of this error, strip ^■ is of bimetallic construc- 
tion. When the instrument is heated, and the travel of the center 
of the box increases, the end of the strip bends away from the shaft, 
thus reducing the transmission ratio for rotating shaft 5 and compen- 
sating for the increase in sensitivity of the box. 

It is important not to confuse the instrumental temperature 
errors of the instrument, which are compensated by the zero and 
altitude bimetallic compensators, with the methodological temper- 
ature errors in the altimeter. 

The instrumental errors are related to the temperature in the 
casing of the instrument, acting on the properties of the mater- 
ial from which the sensitive element is made, and can be overcome 
by compensators . 

The methodological errors which are related to the nature of 
the changes in pressure with flight altitude can only be corrected 
by special formulas. The building of a compensator for methodo- 

182 



logical errors is impossible, since in the general case the temper- 
ature of the casing is not equal to the average air temperature 
from zero altitude up to the flight altitude of the aircraft. 

In order to increase the accuracy of the altitude readings , /180 
altimeters are made with two pointers . This means that the aneroid 
box is made double, increasing the travel of the movable center 
by a factor of two. Between the toothed sector and the axis of 
the pointer, there are additional gears which increase the trans- 
mission ratio of the mechanism several times. The main pointer 
of the instrument makes several revolutions; the number of revolu- 
tions of the pointer is equal to the change in altitude in thou- 
sands of meters. 

In addition, there is a pressure scale 8 for setting the al- 
timeter readings relative to a desired level. 

The altimeter mechanism, along with the axis of the main pointer, 
is rotated within the housing by means of a rack and pinion 9, consist- 
ing of a driving gear 10 and a driven gear 11. Thus, the main pointer 
of the instrument can be set to any division on the scale. 

Simultaneously, by means of driving gear 10, the pressure scale 
8 is set in motion, which can be used in conjunction with the main 
scale to determine the pressure at the level at which the flight 
altitude is calculated. 

In the VD-10 and VD-20 altimeters, a movable ring is mounted 
around the main scale; it is rotated by means of a rack and pinion 
and driving gear 10 at an angular velocity equal to the rate of 
turn of the mechanism. It is used for shifting a movable index 
along the circular scale of the instrument, and can be set to the 
barometric altitude of the airport where the landing is to be made. 
This serves the same purpose as the pressure scale. However, the 
latter can only be used over a range of pressures from 670 to 790 
mm Hg , while the movable index can be set to any airport altitude. 

In cases when pressure scales are not sufficient for airports 
located at high altitudes, the pressure at the level of the air- 
port is not measured aboard the aircraft, but rather the baromet- 
ric altitude of the aircraft is used for setting the movable index 
of the altimeter. 

Errors in Measuring Altitude with a Barometric Altimeter 

The errors in measuring the flight altitude with barometric 
altimeters can be divided into instrumental and methodological errors: 

Instrumental errors. These are related to incorrect adjustment 
of the altimeter, friction (wear) in the transmission mechanism, 
as well as temperature effects on the material of the sensitive 
element. The errors from so-called hysteresis are particularly 



183 



important, i.e., the residual deformation of the sensitive box with 
changes in flight altitude of the aircraft over wide limits. 

In addition, instrumental errors include errors in sensing /181 
the static pressure, related to dynamic flight of the aircraft. 

Methodological errors. In the barometric method of measur- 
ing altitude, these include errors in correspondence of the initial 
atmospheric pressure, the pressure along the flight route, and the 
average air temperature with the calculated data. 

Under flight conditions encountered in civil aircraft, method- 
ological errors in measuring altitude in approaching aircraft are 
extremely rare, so that these errors do not disturb the mutual posi- 
tion of the aircraft and are not taken into account. However, they 
do have significant value in determining the safe flight altitude 
above the relief, as well as in making special flights (for pur- 
poses of aerial photography, e.g.). 

In practice, the baric stage at low flight altitudes (the dif- 
ference in altitude which corresponds to a drop in pressure of 1 
mm Hg) is considered roughly equal to 11 m. However, at flight 
altitudes of 20,000 m, the baric stage is equal to 155 m, i.e., 
14 times greater than on the ground. 

The increase in the baric stage with flight altitude, as well 
as the errors in measuring static pressure due to aerodynamic proc- 
esses, complicate a precise measurement of the barometric altitude 
at great altitudes and high speeds . 

In a flight according to a table of corrections, it is rela- 
tively easy to compensate for instrumental errors in the apparatus, 
related only to its regulation. Consideration of all other instru- 
mental errors presents greater difficulty, so that all measures 
are usually taken to reduce them to a minimum by carefully prepar- 
ing the apparatus, selecting the point of calibration, and design- 
ing the static pressure sensor. 

Methodological errors in altimeters are estimated by deter- 
mining the true altitude of the aircraft above the relief for special 
purposes and in calculating safe flight altitudes above the relief. 
Changes in atmospheric pressure along the flight route, relative 
to sea level, are calculated in baric stages, so that the lowest 
flight altitude oscillates as follows: 

Aff = Ap-11. 

For example , if the pressure measured at sea level at the point 
where the altitude is measured differs from 760 mm Hg to 15 mm, the 
methodological error in measuring the altitude from the level of 
760 mm will be 15-11 = 165 m. 



184- 



Hence, if the corrected pressure is greater than 760 mm Hg , 
i.e., equal to 775 mm in our example, the readings of the altim- /182 
eter will be reduced and the correction will have to carry a plus 
sign, while if the corrected pressure is lower than the calculated 
pressure, it will have a minus sign. 

Methodological errors in the altimeter, which arise due to 
a failure of the actual mean air temperature to coincide with the 

ational 

^ w^ w„ -_ (3_ . _„ „„__... ---ig from 

the fact that the instrument indicates a flight altitude on th 



a rai±ure or rne acTuax mean air xemperaxure to coinciae wiT:n 
calculated temperature, are accounted for by means of a navig 
slide rule, a description of which is given below. Proceedin 
the fact that the instrument indicates a flight altitude on the 
basis of the calculated mean temperature of the air, and the cor 
altitude must be determined on the basis of the actual altitude, 
the equation reads as follows: 



rrected 



Lows 

H. _^ = BT In— i^ ; 

mst av . c . p 



Po 

H = BT In , 

corr av . a . p ' 



where T ^ is the average calculated temperature and T ^.^ ^ ^ ^ is 
the average actual temperature . 



Whence 



Therefore , 



H 



H 



inst 



T 



corr T av . a , 

av. c . 



7-0 +r„ 



av . a . ■ 



Ig H 



corr 



= 'g — :^ — + ig 7^ 



(2.35) 



av . c . 



where Tq and Tjj are the temperatures on the ground and at flight 
altitude, respectively. 

By using Formula (2.35), we can calculate the scales of the 
navigational slide rule NL-10 for making corrections in the readings 
of altimeters for air temperature up to altitudes of 12,000 m. 

For altitudes above 12,000 m, the corrected altitude is found 
by the formula 



'H 



H - 11,000 = -sf 

corr T 



H 



(H . j_ - 11,000) , 
mst 



where Tjj and Tjj are the actual and calculated temperatures at 
a c 

the altitude . 



185 



The navigational slide rule for these altitudes is also pro- 
vided with logarithmic scales according to the formula 



lg(H -11,000) = lgT„ + Ig 

corr ° H ° 

a 



H. ^-11,000 
mst ' 



216.5° 

4. Airspeed Indicators 



(2.36) 



/183 



The flight of an aircraft takes place in the medium of air, 
so that a simplest and easiest method from the technical standpoint 
for measuring airspeed would be to measure the aerodynamic pres- 
sure or so-called velocity head of the incident airflow. 

For purposes of aircraft navigation, it is better to measure 
the speed of the aircraft relative to the surface of the ground, 
since the air mass practically always has its own movement rela- 
tive to the latter. At the present time, there are radial and iner- 
tial methods of measuring the speed relative to the ground, but 
the measurement of airspeed does not lose its significance even 
in the presence of such equipment. 

The fact is that the stability and maneuverability of an air- 
craft depends on the airspeed. In addition, the operational regime 
of the motors on the aircraft and the fuel consumption depend on 
the airspeed. 

The operating principle of airspeed indicators is based on 
a measurement of the aerodynamic pressure of the incident airflow. 

The relationship between the rate of motion of a liquid or 
gas and its dynamic and static pressure was first established by 
the St. Petersburg Academician Daniel Bernoulli (1738), working 
with incompressible liquids or gases (Fig. 2.31). 

According to the principle of inseparability of flow, the prod- 
uct of the speed of an air current (7) multiplied by the cross sec- 
tional area of a tube iS) must be uniform everywhere within its 
cross section. Consequently, in a narrow part of the tube, the 
speed of the flow must be greater than in a wide section. 

In the general case, if the tube is not horizontal, a mass 
of gas m enters the tube during a time At which introduces an energy 
consisting of three components: the potential energy of the gas 

mgh 



the kinetic energy 



rnVJ 



and the work of influx into the tube 



186 



where g is the acceleration due to the Earth's gravity, h is the 
difference in the gas levels , and p is the gas pressure inside the 
tube. 



These components determine the energy of the gas flowing out / 18^■ 
of the tube. Therefore 



— — +:PvSxV^U + mghx ■■ 



mVi 



+ P^z^t^f + "*S'^2- 



The product SVht is the volume of fluid flowing through the 
cross section of the tube in a time At. Therefore, dividing the 
mass into the volume gives us the density (p), which is 




&^ = 



Fig. 2.31. Flow in a Tube with Varying Cross 
Section . 



If the tube through which the current is flowing is horizon- 
tal, hi = h2, therefore 



P^? 



+ Pi = 



PVl 



+ P2. 



(2.37) 



i.e. , the sum of the dynamic and static pressures at any point in 
the tube remains constant, since the dynamic component is propor- 
tional to the gas density (fluid density) and the square of the 
speed of flow . 



For adiabatic compression, i.e., when the process takes place 
with compression of the gas (air) without exchange of heat energy 
with the surrounding medium, which almost always can be considered 
valid for high speed events, this equation takes the form: 



^^4-^+^, + ^, = g. + ^ + ^,4-^,. 



(2 .38) 



where y is the unit weight (weight density) of the gas, V is the 
internal (thermal) energy of the gas, and E is the potential energy 
of the gas . 



187 



Therefore, a change in the rate of airflow during flight due 
to the flow being retarded is usually negligible; the component E 
can be considered constant and may be omitted from the equation. 
Then each of the remaining terms of the equation, if we multiply 
them by mg , will characterize the component energy included in a 
unit mass of gas flow: V'^/2g equals the kinetic energy of the flow 
(for a unit mass m7^/2), p/y is the energy of the pressure, and 
U is the thermal energy. 

For measurements of airspeed, we can use sensors which allow /185 
us to separate the dynamic air pressure from the static pressure. 

Figure 2.32 shows the operation of an air pressure sensor (Pitot 
t ub e ) . 

In the cross section of the airflow, the speed Vi will cor- 
respond to the airspeed, and the pressure pi will correspond to 
the static pressure of the air at flight altitude . 



.^^ 



^ 



/Z_ 



/' 



^p. 



3^p 



total 

st 



Fig. 2.32. Air-Pressure Sensor (Pilot Tube). 

(1) Static Pressure Pst; (2) Total Pressure Ptotal' 

Within the limits of the opening in the sensor for total pres- 
sure, the rate of flow will be equal to zero (the critical current 
or current of complete braking). 

Obviously, at this point the pressure p2 will correspond to 
the total pressure (the velocity head plus the static pressure), 
and Equation (2.38) acquires the following form for this case: 



2^ 11 



total 



( 2. 39) 



Let us consider that for airspeeds up to M-OO km/hr the com- 
pression of the air can be disregarded, i.e., the values y and U 
are constants. Then Equation (2.39) assumes the form: 



PtotalPst 



Y 



(2.40) 



E 



where y^^ is the unit weight of gas at a given altitude. 

Since y^ = Pn? (where p„ is the mass density), the difference 



188 



between the total and static pressures (velocity head) will be equal 
to 



p7^ 



^total ~ ^st 



whence 



V = 



V 



2(Ptotal-Pst) 



(2.41) 




This drop causes movement of the top of the box, which can 
transmit its movement by means of a system of gears similar to the 
mechanism in a single -pointer altimeter, eventually moving a pointer 
on an axis to show the airspeed on a scale which is graduated in 
kilometers per hr. 

Formula (2.'+l) can be used to describe airspeed indicators 
for low speeds, such as the US-350. If we introduce the weight 
density to this formula in the form 



whence 





st 



(2.H2) 



It is clear from this formula that in order to determine 
the true airspeed, it is necessary to know not only the value of 
the velocity head, but also the atmospheric pressure and the temper- 
ature of the air at flight altitude . 

The airspeed, which is measured only on the basis of the veloc- 
ity head, is called the aerodynamic or indicated speed. In view 
of the fact that calibration of the speed indicator is made for 
flight conditions at sea level at standard temperature and air pres- 
sure, during flight under these conditions the indicated speed will 
be equal to the true airspeed. Under other conditions, however, 
the indicated speed must be converted to the true airspeed. 

At high altitudes and speeds, the difference between the air- 



189 



speed and the indicated speed becomes so significant that it becomes 
difficult to use the latter for navigational purposes. In addi- 
tion, for airspeeds above 400 km/hr , it becomes necessary to take 
the compression of the air into account as well. Therefore, for 
aircraft operating at high altitudes and speeds, a combined speed 
indicator "CSX" has been developed, which measures both the indi- 
cated and true airspeed. 

In terms of its design, this indicator differs from the usual 
speed indicators in that the speed is measured in two ways: 

(a) The first method consists of the conventional system for 
indicating speed and is used to measure the indicated airspeed (the 
large pointer on the dial); 

(b) The second system incorporates a special compensator for 
changes in air density with altitude by means of a system of gears /187 
and is used to measure the airspeed. 

The compensator is an aneroid box, which changes the length 
of the arm of a control lever, increasing the latter's mechanical 
advantage when the atmospheric pressure (as well as the density 
of the air at flight altitude) is reduced, and vice versa. 

It should be mentioned that in the case of high speed aircraft, 
the sensor for total pressure is usually separated from the static 
pressure indicator, so that it is possible to select the most suit- 
able position for mounting them on the aircraft. This means that 
the role of the static pressure indicator is played by openings 
which are made on the lateral surface of the fuselage of the air- 
craft and are linked to the instrument itself by tubing. 

In addition to the details of design described above, the reg- 
ulation of the systems in the CSI are made by taking the compres- 
sion of the air into account when the flow is retarded in the detec- 
tor for total pressure. 

Therefore, compression of the air on braking will be accom- 
panied by heating, and therefore by an increase in its internal 
energy . 

The relationship between the internal energy of the gas , its 
pressure, and weight densities is expressed by the formula: 



U- 



1 £_ 



(2.43) 



where K = -^ is the ratio of the specific heata iifthe gas when it 
is heated, with retention of constant pressure and constant volume. 



190 



For air, this coefficient is K = 1,4. 

By substituting the value U into Formula (2.39), we can change 
it to read as follows: 

YL,Pst,__L_ P^t__ ptota l 1 /'tota l 
2^ fi ^- 1 ■ -ri T2 K-l ' 72 • 



2^ 71 ^-1 72 ^-1 



(2.44) 



After making some simple conversions, 

V^_ K /PtotAlpst, 
2^ f<-l\ 72 7J 

taking Pst/^i out of the parentheses, we will have: 

£=^ ^fet/Ptotal7i._,y (2.45) 

2*? A:-1 7i \Pst T2 / 

For the adiabatic process, there is an equation which is known /18i 
as the Mende leyev-Clapeyron equation: 

Pi _ P2 

,K K • 

Til T2 

from which we obtain for our case 

1 



T2 V /'total 



'^total 

Substituting the value Y1/Y2 in Formula (2.45), we obtain 



2^ ^-1 ■ 7^ LI Pst ) i 



Assuming that y^ = y^y , so that Pst/Yi = ^^H ' "® ^^'^ rewrite this 
equation in the form: 



"=i^^«^«P°^^'=^' +')'-.] 



and finally obtain the formula which can be used to calibrate the 
combined speed indicator by the airspeed in the channel for subsonic 
airspeeds : 

K = /^^BrJptot-l::&t^,p;i, 

'^"^ "-^ -"st / 1 (2.46) 



191 



The temperature at flight altitude (T^) is assumed to be stan- 
dard according to the flight altitude (or Pst^' i.e., up to 11,000 
m, Tg = 288°-6.5°fl, and above 11,000 m Tg = 216.5° K (-56.5° C). 

To calibrate the airspeed indicator, it is necessary to know 
the pressure in its manometric box and in the housing of the appar- 
atus, corresponding to the pressure in the sensors of total and 
static pressure under the given flight conditions. Therefore, (2.46) 
is solved relative to the pressures and assumes the form: 



Ptotal'^st _r. , (/C-1)F2 W_ 
p ^ L "^ 2KgBT J 



st 
or, if we insert the numerical values of Kj g^ B, 

Ptotal"Pst r, , _ZL?-5 



(2.47) 



st 



r V2 -13.5 

L "*" 20607J '' 



— 1. 



(2 .47a) 



As we have already pointed out, (2.46) is valid for subsonic 
airspeeds. At speeds which exceed the speed of sound, the flow /189 
of the particles differs from their flow at subsonic speed. 




We know that the rate at which sound travels (a) in air de- 
pends only on the temperature of the medium and is expressed by 
the formula 



a = yfUgBT. 

In other words, if ^ = 9.81 m/sec^, and the coefficient for air 
is equal to 1.4, while B - 29.27 m/degree , 



a = /4127-=20.3y'r m/sec. 

The ratio of the airspeed to the rate of propagation of sound 
in air is called the Maoh number: 

M = I. 
a 

If we replace Kg BT in Formula (2.46) by a^ , we will have the 
expression for M (Mach number) for subsonic airspeeds: 



= |/_^r(itQialZ^t_/x'_ll. (2.48) 



M 



192 



II ■■■■II ■ III ■■■ 



The latter formula indicates that in order to determine the 
Mach number, it is necessary to know only the velocity head and 
the static pressure at flight altitudes. There is no necessity 
to measure air temperature for this purpose. 

For subsonic airspeeds, the relationship between the total 
pressure, the static pressure, and the Mach number is expressed 
as follows : 

^+1 1 M ^^ 



P total- -^st ^ (K+IW'' /_J_)^ ^-^ ,— _ 1. (2.1+9) 



In this formula, if we replace M^ by its value as obtained 
in Equation (2.48), we will obtain the formula for calibrating the 
airspeed indicator for supersonic airspeeds: 

AT+l 1 K 

^ total- ^st l_2_J- U -1/ UgBT) 

^ ■ — • — = I — 1 . 



^st 



[l^l-K^-') (2.50) 



If we substitute the numerical values of K for air, equal to /19 
1.4, in Equation (2.50), we can convert it to the simpler form: 

^otal- ^st _ _J66,7K' . 

Errors in Measuring Airspeed 

Errors in measuring airspeed, like those involved in measuring 
flight altitude, can be divided into instrumental and methodolog- 
ical ones. Instrumental errors include those which are related 
to improper adjustment of the apparatus and instability of its oper- 
ation with changes in the temperature of the mechanism in the device. 
In addition, instrumental errors also include errors in sensing 
dynamic and especially static pressures with sensors which depend 
on the mounting location on the aircraft. 

Instrumental errors are corrected by correction charts, which 
are compiled when the apparatus is tested, taking into account the 
errors in indicating the static pressure for a given type of air- 
craft . 

Methodological errors include those involving failure of the 
actual air temperature at flight altitude to correspond with the 



19 3 



calculated temperature for combined indicators of speed, and with 
the temperature and pressure for other speed indicators. 

Strictly speaking, the methodological corrections which must 
be taken into account in converting the indicated speed to the air*- 
soeed, are not instrument errors, since the indicated speed has 
its own independent value. However, from the navigational stand- 
point, it is convenient to consider them methodological errors. 

In aircraft navigation, it is possible to use both the single 
pointer dial for indicated speed (Type US-350 or US-700), as well 
as the combined indicator (Type CSI-1200) and others, so that the 
methods of calculating the methodological errors can be viewed sepa- 
rately . 

It should be mentioned first of all that the dials of speed 
indicators are calibrated to take into account the compressibil- 
ity of the air for a true airspeed equal to the indicated speed. 

In fact, at high altitudes, the true airspeed is almost always 
much greater than the indicated speed, so that it is necessary to 
consider that there is an error in the difference between the com- 
pression of the air at the actual and calculated airspeeds: 

A7 = AF -A7 

comp comp.a. comp.c. 



There are special, precise formulas for determining the cor- 



rections for LV 



comp 



for use with indicators of instrument speed 



at subsonic and supersonic airspeeds, and they are used to draw 
up a table of corrections (Fig. 2.33); we will limit ourselves to 
discussing only the simple approximate formula 



/191 



AV ^-LpndW.^,_i\ 
comp- 12 1^100 ) \P„ ')• 



(2.51) 



where l^ind is the indicated airspeed and AF^^j^p 
for the indicated speed, 



is the correction 




After making the corrections in the indicated speed for the 
compression of the air, conversion of the latter into airspeeds 
is done on an navigational slide rule. 



194 



Ill II II 



since the dial of the indicated airspeed is calibrated by the 
formula 



V =11 / ^^^otai-/'H ) 



and the airspeed is 






^true=l/ ^ ^^^"' 



then if we divide the second formula by the first we will obtain; 



f Ph To 



^true=^instl/ -^-:^- (2.52) 



Let us substitute into Formula (2.28) the following values: 
tgp = 6.5 deg/km, Tq = 288° K, and B = 29.27. We will then obtain 

p„=Po(\— 0.0226/y)^-2^ , 

and if we let the value p^ be substituted into Formula (2.52), we 
will obtain: 



instF 7-0 (1-0,02: 



^true = ^instF ^o " d - 0fi22mf'^^ 



'^ Vue= 'S ^inst'^T'^(^^^ + '^)~T 'g 288- 2.628 lg( 1-0 ,0226//). (2.53) 

According to Formula (2.53) we can convert the logarithmic 
scales of navigational slide rules for converting the indicated 
airspeed into the true airspeed. 

Calibration of the combined speed indicator on the basis of 
the true airspeed is performed by taking into account the compres- 
sibility of the air over the entire range of the scale. The methodo- 
logical error in the reading is related only to the differences 
between the actual air temperature and the calculated temperature 
at the flight altitude. 

Since the airspeed, as shown by a combined speed indicator /192 
under standard temperature conditions , is expressed by the formula 



195 



/ 



St •' 

and the corrected value for the airspeed at flight altitude in ac- 
cordance with the actual temperature is 



F 



corr 



flg7-//a[( ^total ^st 4-lU -l], 

^st 



if we divide the second formula into the first, we will have 



7 =7 1/^^^a 

corr CSI y ~f 



'Ho 



or 



/ 273 + f„ 
^corr'^CSI K 288 -0.0065// iAst 



(2.54) 



After looking up the logarithm of the latter, we will obtain a 
formula which can be used to construct the logarithmic scale on 
the NL-IOM for a combined speed indicator: 



-^S^corr = lg^CSI+T'S^^^^+'^>-T'2^^^^-'''^^^inst) 



(2 .54a) 



Relationship Between Errors in Speed Indicators and 

Flight Al t i tude 

In describing the errors in barometric altimeters and airspeed 
indicators, instrumental errors of aerodynamic origin are found, 
which are related to errors in recording the static pressure by 
the air pressure sensors. 

Experience has shown that aerodynamic errors in the speed in- 
dicators due to incorrect recording of the dynamic pressure are 
negligibly small by comparison with the errors in incorrect re- 
cording of static pressure. This is explained by the fact that it 
it immensely easier to measure the pressure of a retarded airflow 
with a sensor that is aimed into the airflow, than it is to select 
a location on an aircraft for a static-pressure sensor, such that 
the latter will not be distorted by the airflow over the body of 
the aircraft. 

In connection with the fact that the static pressure from 
the sensor is transmitted simultaneously to the hermetic chambers 
of the speed and altitude indicators, there must be a mutual rela- 



196 



tionship between the errors in the measurement of altitude and /193 
speed owing to errors in recording the pressure. 

At the same time, the velocity head according to which the 
dial of the speed indicator is calibrated is equal to 

p ^ -p ^ = P^ . (2.55) 

'^total '^st 2 

Since the errors in measuring the velocity head are equal to 
the errors in measuring the static pressure , then 

An = P-fl. A(f2). (2.56) 

'^ St 2 

Under standard conditions, Pq = 0.125 kg/sec^/m^. The static 
pressure is usually given in mm Hg . The specific gravity of mer- 
cury is 13.6, so that the pressure of 1 kg/cm^ would equal 10,000/ 
13.6 = 735 mm Hg . 

On the other hand, since the parameter p has m'* in the denomi- 
nator, the pressure expressed by (2.56), relative to an area of 
1 m^ , must be divided by 10,000 to determine the value for 1 cm^ , 
so that we finally obtain 

735 125 
V=+= •■ ,„' A ( V2) ==-0.0048 S(V2). 

St 2-10000 V / • \ ' 

Example: At an indicated speed of 396 km/hr (110 m/sec), at a 
flight altitude of 5000 m, the aerodynamic correction for the speed 
indicator is 36 km/hr, (lOm/sec). Find the aerodynamic error in 
the altimeter. 

Solution: 

Vst = 0.0048(1102— 1002)= 0.0048-2100 = 10.8 mm Hg . 

According to the hypsometric table, the baric stage at a 
flight altitude of 5000 m is equal to 18.5 mm Hg ; hence, the aero- 
dynamic component in the altimeter error is 

hH = - 10 . 8-18. 5 = - 200 m. 

Formula (2.56) is an approximate one, but it yields sufficiently 
accurate results up to an indicated airspeed of 400 km/hr. The 
altimeter error can be determined more precisely if we know the 
dynamic pressure and take into account the compression of the air 
at different instrument readings . 

Table 2.6 shows the velocity head at various indicated air- 
speeds, and can be used to determine the aerodynamic corrections 
of the altimeter. The third column in Table 2.6 shows the mano- 



197 



metric stage, i.e., the change in pressure with change in airspeed 
by 1 km/hr. If we multiply the aerodynamic correction of the speed 
indicator by the manometric stage and then use the hypsometric 
table, it will be easy to determine the aerodynamic correction for 
the altimeter for a given flight altitude. 



mst 



P -P 

total st 



TABLE 2.6 
Y 



for 1 km/hr 



inst 



P -P 

total St 



/19it 



50 


0.89 




100 


3.57 




150 


8 




200 


14.3 




250 


22.37 




300 


32.4 




350 


44.27 




400 


58.25 




450 


74.23 




500 


92.35 




550 


112.7 




600 


135.7 





0.054 


700 


188,3 




0.089 


800 


252 




0.126 


900 


322 




0.162 


1000 


418 




0.2 


1100 


522.8 




0,24 


1200 


645.8 




0,28 


1300 


787.2 




0.32 


1400 


947,2 




0.36 


1500 


1125.4 




0,41 


1600 


1317.6 




0,46 


1700 


1525,7 




0.53 


1800 


1748.8 





Ap 
km/hr 

0.65 

0,8 

0,96 

1,04 

1,23 

1,42 

1,6 

1.78 

1,92 

2.08 

2.23 

2,4 



Ay 



comp 

no 
m 
wo 
so 

so 

70 

so 

50 

uo 

30 
20 

m 















^.-- 


7— 


























[^ 


f 

V 


r 


/; 


\ 




















// 




/ 


^ 




< 




\^ 








- -■ - 




/ 


^ 






4/'' 


^ 


j^ 










/ 




lM. 


"V 


t^ 


'^/^ 


<^ 




\ 








M' 


s/^ 




/ ^ 


"Si 


{'>. 


\ 


\ 




rA 








i 


vh 




^ 




V 


/J 


\ 


y 


\ 




\ 


\ 






iv 




/A 




V 


<V/' 


/ 


V 


r^ 




^ 


-\ 






'-h 








(/^ 


tv 


\ 


1 


V 


\ 


\ 


\ 


\— L- 


A 


^ 


y t 




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jT 


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d 


% 


,\ 


\ 


\ 


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^ 


Z 


y 


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\ 


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\ 


^ 


k^ 


[--^ 








\ 




\ 


\ 


\ 


\ 



400 SOO 800 1000 KBO 1100 KOO ZOOQ V . 



xnst 



Fig. 2.33. Graph of Corrections for Air Compression, 



19 8 



5, Measurement of the Temperature of the Outside Air 

Measurement of the temperature of the outside air during flight 
is necessary first of all for determining the true values of the 
airspeed and flight altitude . 

The thermometer for measuring the outside air temperature is /19 5 
a remote-controlled instrument, i.e., its sensitive element is mounted 
outside the cabin of the aircraft and i'.s exposed to the airflow, while 
the indicator is mounted on the instrument panel in the cockpit. 

At the present time, electric thermometers are used for meas- 
uring the outside air temperature, and their operating principle 
is based on the changes in electrical conductivity of materials 
depending on their temperature. 

A schematic diagram of such a thermometer is shown in Figure 
2.3M- and consists of an electrical bridge made of resistors. 

If the arms of the bridge 1 and 1^, as well as 2 and 2^, have 
.the same resistance when connected in pairs, no supply voltage will 
flow through bridge AB and consequently through the temperature 
indi cator . 




One arm of the bridge (2^) 
is made of a material which has 
a high thermoelectric coefficient, 
and is mounted on the surface 
of the aircraft to be exposed 
to airflow . 

Depending on the temperature 
of arm 2]^, its resistance changes, 
thus affecting the amount of current 
which passes through bridge AB 
with the temperature indicator 
connected to it. 



Fig. 2.34. Schematic Diagram 
of Electric Thermometer. 



Thermometers of this kind, 
when used at low airspeeds, indi- 
cate the temperature with an accuracy of 2-3°. However, at high 
airspeeds, due to drag and adiabatic compression of the airflow 
on the forward section of the sensor, the latter is subjected to 
local heating that creates methodological errors in measuring temper- 
ature . 



For an exact determination of the methodological errors of 
this thermometer, we will require a sensor with complete braking 
of the airflow, as is the case in sensors used to measure the total 
pressure in airspeed indicators. 

If we keep in mind that y = p/BT, (2.'44) can be changed to 



199 



read as follows : 






where Tx is the temperature of the retarded flow. Therefore 

K-1 



/196 



T„= T„- 



2KgB 



V2 



(2.57) 



If we substitute in (2.57) the values K - 1.4 and B = 29.27, we 
will obtain 



^T=^^t = 



VI 



2000 • 

where 7 is the velocity, expressed in m/sec, 



a) 



b) 



^-^^^^ m^^^^ ^^^^^^^^ 



-<<^^^^^^^^^^^^ ^^^^^^^^^:-^^^ 



Fig. 2.35. Sensors for Electric Thermometer for 
Measuring Outside Air Temperature. (a) TUE ; (b) TNV , 



Since the conversion coefficient for changing from m/sec to 
km/hr is 3.6, for a speed expressed in km/hr 



^t■- 



V2 



V2 



2000-3.62 



26 000 



(2.57a) 



Practically speaking, it is highly unsuitable to use thermom- 
eters for measuring outside air temperature which have complete 
retardation of airflow, since in this case the sensor will not be 
exposed to the flow and this will result in a high thermal inertia 
of the thermometer, i.e., rapid changes in temperature during flight, 
which could take place at high flight speeds, would not be detected 
by the thermometer. 

For sensors which are exposed to the airflow, the coefficient 
of drag is within the limits of 0.5 to 0.85. The TUE and TNV ther- 
mometers in use at the present time have coefficients of drag which 
are nearly the same (approximately 0.7). The scale of corrections 
for the thermometer for measuring outside air temperature (TUE), 
located on the navigational slide rule, can be used with sufficient 
accuracy for the TNV thermometers as well. 

The sensor of the TUE thermometer is in the shape of a rod 



200 



with a winding on the surface, covered by a cylindrical housing 
(Fig. 2.35, a). When a flow of air passes through such a sensor, 
it is heated on one side . 



/197 



The sensor of the TNV thermometer is made in the form of a 
de Laval nozzle. The sensitive element is located in the narrowest 
portion of the nozzle (Fig. 2.35, b) and the air flows symmetrically 
over it. Therefore, this sensor has less thermal inertia and gives 
more accurate readings in different flight regimes . 

6. Aviation Clocks 

The measurement of time plays an extremely important role in 
aircraft navigation, since the calculation of the path of the air- 
craft on the basis of the component airspeed and time is involved 
in almost all navigational equations. This means that an increase 
in the airspeed places increased demands on the accuracy of the 
measurement of time. It is especially important to have an exact 
determination of the moments of passage over control checkpoints, 
i.e., in this case, the exact measurement not of elapsed time but 
of time segments between the moments when the aircraft is passing 
over landmarks . 

There are also factors which demand high accuracy in deter- 
mining the time and the exact operation of aviation clocks . For 
example, the coincidence of the flight plans of individual aircraft, 
communication with the tower, and especially in astronomical calcu- 
lations, where an error in calculating the elapsed time of 1 min 
could produce an error in determining the aircraft coordinates of 
27 km. 

The operating principle of all existing devices for measuring 
time is their comparison with the time required for some standard 
event to occur. In this case, the standard event is the period 
of oscillation of the balance wheel of a clock (a circular pendu- 
lum). All of the remaining mechanism of the clock acts mainly as 
a mechanical counter of the number of oscillations of the pendu- 
lum . 

However, it exerts a considerable influence on the accuracy 
of operation of the clock; when the main spring of a clock is wound 
completely, the clock runs somewhat faster, and when the spring 
has run down the clock runs slower. The most important role in 
measuring time is played by the accuracy of adjustment of the actual 
period of oscillation of the pendulum. 

We know that the period of oscillation of a body around its 
axis (torsional oscillation) is related to the deformation of the 
body as determined by the formula 



'-'Yi-' 



201 



where T is the period of oscillation of the body around the axis, 
J is the moment of inertia of the body, and D is the modulus of 
torsion , 

The product of the modulus of torsion times the angle through 
which the body rotates ( (j) ) is the torsional moment: 

M = D^. 

The period of oscillation of a balance can be adjusted both /19 8 
by changing its moment of inertia (for which purpose adjusting screws 
are located along its outer circumference), or by changing the mod- 
ulus of torsion. 

The moment of inertia of the balance wheel is changed by screw- 
ing the adjusting screws symmetrically in or out along the entire 
circumference, in order not to disturb the balance of the pendulum. 
This means that a portion of the mass is brought closer to or moved 
further away from the center of rotation of the balance. 

The modulus of torsion is adjusted by means of a hairspring; 
the balance wheel is adjusted by changing the free length of the 
hairspring, for which purpose a movable stop, which acts as a regu- 
lator, is mounted near the point where the hairspring is fastened. 

It should be mentioned that many factors affect the precis- 
ion with which a clock operates, but the most important ones are 
temperature and magnetic effects. Therefore, a number of measures 
are taken to exclude these factors. 

The balance wheel of an accurate clock is usually made of bi- 
metallic material and divided along the plane of the diameter. 

When the temperature falls and the flexibility of the hair- 
spring increases (the modulus D increases), one-half of the balance 
expands and its ends move further away from the center of rotation, 
thus compensating for the temperature error in the clock. 

The harmful effect of magnetic fields on the accuracy of clocks 
can usually be overcome by using diamagnetic parts in the balance 
wheel, hairspring and escapement, or else the entire clock mechan- 
ism is placed within a shielded housing made of iron alloy. 

Special Requirements for Aviation Clocks 

In addition to the general requirements for clock mechanisms 
(high accuracy, compensation for temperature and magnetic effects), 
aviation clocks have additional requirements placed upon them: 

(a) Protection against vibration and shock, so that the clocks 
on an aircraft must be mounted in special shock mountings . 



202 



(b) Ensuring reliable operation under conditions of low tem- 
perature; for this purpose, aviation clocks are usually fitted with 
electric heaters. 

(c) Reliability and accuracy of operation under various con- 
ditions. The hands, numerals, and principal scale divisions are 
made larger and covered with a luminous material to permit their 
use during night flights. 

(d) The possibility of measuring simultaneously several time 
parameters. This means that several dials are usually driven by 
the mechanism. 

Aviation clocks of the ACCH type (aviation clock- chronometer /199 
with heater) are made to satisfy all the conditions listed above. 

The elapsed time is indicated on these clocks by a main dial 
with a central pointer. To calculate the total flight time or the 
flight time over individual stages, there is an additional scale 
in the upper part of the clock. The start of the clock hands is 
marked on this scale, while the time when they stop as well as the 
resetting to zero are accomplished by pushing a button on the left- 
hand side of the clock housing. This same button, when pulled out, 
is used to wind the main spring of the clock. 

Below the "flight time" scale, there is a pilot light which 
is used to signal the following by means of a special shutter: 

(a) Start of mechanism: red light. 

(b) Stop mechanism: the light is half red and half white. 

(c) Pause: white light. 

To measure short time events , the clock is fitted with a sweep 
hand (thin central pointer) and an additional scale at the bottom 
of the apparatus where the minutes are counted. The sweep hand 
is started, stopped and held by pressing a button on the right- 
hand side of the housing. 

In addition to the ACCH, the aviation chronometer 13 ChP is 
currently in use. It employs a potentiometri c circuit; the version 
fitted with indicators is the 20 ChP. This chronometer, especially 
intended for purposes of astronomical orientation, is operated by 
remote control and consists of three main indicators: 

(a) An elapsed-time indicator whose readings are always linked 
to the chronometer at the transmitter. 

(b) Two time indicators for measuring the altitude of lumin- 
aries; their readings are also connected to the chronometer at the 
transmitter, but at the moment of measurement of the altitude of 



203 



the luminary by means of a sextant, a stop signal is sent to one 
of them and the time of measurement is noted. 

After the reading is made, the minute hand is set to the elapsed 
time according to the readings of the first dial by pushing the 
button. Each time the button is pressed, the hand moves forward 
one minute. The sweep second hand lines up with the readings of 
the transmitter immediately after the indicator is switched on. 

These dials do not have any hour hands . The time in hours 
is determined by readings from a Type ACCH clock. 

7. Navigational Sights 

At the present time, nav.i gational sights are used only for 
special purposes such as aerial photography. They are not used 
in passenger aircraft. 

There are several types of navigational sights, which differ /200 
in their design. However, all are intended for measuring the course 
angles of landmarks (CAL) and their vertical angles (VA). 

The course angle of a landmark is the angle between the lon- 
gitudinal axis of the aircraft and the direction of the landmark. 
The vertioat angle is the angle between the vertical at the point 
where the aircraft is located and the direction of the landmark. 

The sight can be used to solve a great many navigational prob- 
lems related to determination of the locus of the aircraft and the 
parameters of its motion. 




Fig. 2.36. Determining the Value (a) of Aircraft Bera- 
ing; (b) of the Distance from a Landmark to the Aircraft 

Verti cal . 

1. Determination of the locus of the aircraft in terms of 
the course and vertical angles of the landmark (Fig. 2.36). In 
this Ccse, the true bearing from the landmark to the aircraft is 
(Fig. 2.36, a) 

TEA = TC + CAL ± 180° , 



204 



while the distance from the landmark to the vertical of the air- 
craft (Fig. 2.36, b) is 

5 = ff tg VA, 

where TBA is the true bearing from the landmark to the aircraft, 
TC is the true course of the aircraft, CAL is the course angle of 
the landmark, H is the flight altitude, and VA is the vertical angle 
of the landmark . 

Obviously, if the aircraft course is determined by a magnetic 
compass, in order to solve this problem we must also add to the 
readings of the compass the corrections for the deviation of the 
compass of the magnetic declination of the locus of the aircraft. 



TC 



CC + A + 
c 



M' 



The correction for the deviation of the meridians between land- 
marks and the locus of the aircraft in this case is not taken into 
account, since the measurement of the vertical angles can be made 
satisfactorily up to 70-75°, i.e., at distances which do not exceed /20 1 
three to four times the flight altitude. 

In solving this problem, it is particularly important to know 
the true flight altitude above the level of the visible landmark, 
since errors in determining the distance will be proportional to 
the errors in measuring the flight altitude. Therefore, the readings 
of the altimeter must be subjected to corrections for the instru- 
mental and methodological errors and the elevation of the landmark 
above sea level must also be taken into account if measurements 
are not being made in a level location. 



2. Determination of the location of an 
the bearings from two landmarks (Fig. 2.37). 



aircraft in terms of 
In this case. 



IPSi = TC + CALj ± 180° ; 
IPS2 = TC + CAL2 ± 180°. 

The position of the aircraft is determined by the intersec- 
tion of bearings IPSj^ and IPS2 on the map. If the direction finding 
is made over great distances, especially in the polar regions, the 
measurements of the bearings must include a correction for the dis- 
placement of the meridians. 

An advantage of this method is its independence of flight al- 
titude, and consequently, of the nature of the local relief. 

However, this method requires careful measurement of the course 
-■ngle of the second landmark, since the aircraft may move consid- 
erably away from the line of the first bearing during a prolonged 
measurement . 



205 



a 



i ng 



3 

to 



Determination of the drift angle of the aircraft accord- 
risual points. To determine the drift angle by this means, 
it is set 'at a course angle of 180° and a zero vertical angl 




Fig. 2.37. Determining the Po- 
sition Line of an Aircraft by 
Two of its Bearings. 



With an exact maintenance 
of the course, by the pilot, 
observing, the directions of 
visual points and turning the 
sight to keep it parallel to 
the course chart, the sight 
is set in the direction in which 
the aircraft is moving. The 
■drift angle of the aircraft 
is then calculated on a special 
s cale . 

This method is used for 
low flight altitudes, i.e., 
with rapidly changing visual 
landmarks . 



using a 

urement 
aircraft 
in terms 
the pilo 
the cros 
in these 
the pilo 
vertical 
20° at h 
visual p 



k. Determination of the 
drift angle of an aircraft by 
backsight. The essence of this method lies in the meas- /20 2 
of the course angle at which visual points recede from the 
. After setting the sight, as in measuring the drift angle 

of the location of visual points (CAL = 180°, VA = 0), 
t waits until the characteristic visual point appears in 
s hairs of the sight at the position of the bubble level 

cross hairs. Then, keeping the aircraft strictly on course, 
t waits until the landmark leaves the cross hairs in the 

plane at an angle of 40-50° at average altitudes or 15- 
igh altitudes. Then, by turning the sight, he matches the 
oint with the course marking and calculates the drift angle. 



5. Determination of the drift angle of an aircraft by sight- 
ing forward. In measuring the drift angle by sighting forward, 
the sight is set to the zero course angle and a visual point is 
selected on the course chart, which preferably lies at a vertical 
angle of 45 or 26.5°. In this case, with VA = 4-5°, the distance 
to the landmark will be equal to the altitude, while at VA = 26.5° 
it will equal half the altitude: 




206 



S^ = H tg VA. 

The drift angle of the aircraft is determined as the ratio 
of its initial distance to its final distance: 



tg US = 



Sz tgVA2 

57 "" tgVAi 



At drift angles on the order of 10° , the tangent US can be 
replaced by its value, while the tangent VU2 can be replaced by 
the value of the lateral deviation (LD): 



US 



LD 



tgLDi 



or, with an initial value of VAj: 
VAi=26 . 5° , US = 2 LD. 



45°, US = LD; with an initial 



All three of these methods described above for determining 
the 'drift angle are used in locations which have many landmarks, 
i.e., where it is easy to pick out a visual landmark at the desired 
visual angle . 

6. Determination of the ground speed of the aircraft by means 
of a backsight. To determine the ground speed by this method. The 
sight is set on the course angle scale to 180° , and to zero on the 
vertical angle scale. The bubble in the level is set at the inter- 
section of the cross hairs. 



Having selected the characteristic point as it passes through /20 3 
the intersection of the sight, the sweep second hand is started 
and the pilot waits until this point has moved to a vertical angle 
of 35-40° 




where H is the flight altitude and t is the time measured by the 
sweep second hand. 

7. Determination of the drift angle and the ground speed of 
the aircraft from a landmarl< located to the side. This method is 
used in the case when it is desired to measure the drift angle and 
the ground speed and the pilot has only one landmark at his disposal 



207 



which is not located along the line of flight of the aircraft. Being 
careful to keep the aircraft strictly on course, he looks through 
the sight at the landmark and waits until its course angle is equal 
to 45 or 315° , depending on whether it is to the left or right of 
the flight path of the aircraft. 

At a course angle for the landmark of US or 315° , the vert- 
ical angle of the landmark is measured and the sweep second hand 
is started. 




Fig. 2. 38. Determining the 
Drift Angle and Ground Speed 
by a Landmark Located to the 
Side. 



Leaving the 
vertical angle i 
tion, the sight 
the motion of th 
its fixed positi 
chart . At the b 
CAL = 90° + US) , 
the landmark wil 
then will increa 
sequently, the 1 
first move away 
the intersection 
and will then ag 
it. At the mome 
mark is at the i 
the cross hairs , 
hand is stopped 
angle of the Ian 
lated . 



setting of the 
n the same posi- 
is rotated to follow 
e landmark , noting 
on on the course 
eginning (up to 

the distance to 
1 decrease, but 
se again. Con- 
andmark will at 
to one side from 

of the cross hairs 
ain begin to approach 
nt when the land- 
ntersection of 

the sweep second 
and the course 
dmark is calcu- 



If CALi = 45°, the bisectrix of the triangle OAB (Fig. 2.38) 
will be located at the course angle, which is equal to: 



CAL 



M-5°+CAL2 



bis 



while the drift angle of the aircraft will be equal to CAL2jj_g-90° , 720'+ 
so that 



US 



CAL2-135° 



If CALi = 315° , 



CAL 



315° + CAL; 



bis 



US = CAL^ . -270° 
bxs 



or 



US = ^^^p-^^5° 



208 



At points 1 and 2 the distance from the aircraft to the land- 
mark is equal to 

S. = So = H VA. 



Consequently, the distance between points 1 and 2 is deter- 
mined by the formula 



1-2 



OE7 ^ WA • CALi+CALp 
2H tg VA sm S; ^ 



Clearly, the reason for the change in the course angle of the 
landmark from CAL i to CAL2 , was the shift of the aircraft from point 
to point Oi, so that 

Si_2 = OOi. 

Consequently, the ground speed is 

W = ^^^ . 
t 

The majority of navigational problems which we have discussed, 
which are solved by means of mechanical or optical sights, can be 
solved using the radio devices which are installed nowadays aboard 
modern turboprop and jet aircraft, which will be described in the 
next chapter. 



209 



8. Automatic Navigation Instruments 

In Section 2 of Chapter I, it was mentioned that in the 
general case, all the elements of a flight regime are not strictly 
fixed, with the exception of the extreme points of deviation from 
a given trajectory. Therefore, the crew of an aircraft must con- 
stantly deal with average values of measured navigational elements 
(average course, average speed, average wind, etc.). 

If all the elements which have been mentioned had a constant 
given value, the practical problems of aircraft navigation could 
be solved quite simply and the question of automating the processes 
of aircraft navigation would be superfluous . 



/205 




The simplest device used for automating the computation of the 
aircraft path in terms of the changing values of navigational param- 
eters and times is the automatic navigational device, which has been 
devised on the basis of the general features of aircraft navigation. 

At the present time, the navigation indicator Type NI-50B, is 
widely used. We shall now discuss its design and the method of its 
application . 

The NI-50B navigation indicator is an automatic navigation device 
which calculates the path of the aircraft on the basis of signals 
from sensors for the course and airspeed, taking into account the 
measured wind speed during flight. In addition, the indicator can 
be used to determine the wind parameters at the flight altitude. 

Calculation of the path of the aircraft with the use of the 
NI-50B can be performed both on the basis of orthodromic systems 
of coordinates for s traigh t- line flight segments, as well as in a 
rectangular system of coordinates with any orientation of its axes . 

Without going into the details of the design of the instrument, 
let us examine its schematic diagram, purpose, and operating prin- 
ciples of the individual parts , as well as the ways in which the 
system as a whole can be employed. 

The navigation indicator consists of the following parts: auto- - 
matic speed indicator, control unit, automatic course-setting device, 
wind indicator, and device for calculating the aircraft coordinates 
(Fig. 2.39). 



210 



The automatic speed control consists of a device which converts 
the pressure from the sensors of total and static pressure Into elec- 
trical signals, corresponding In value to the airspeed of the air- 
craft, according to Formula (2.'+7a) 



P -P 
total st 

^st 



~['^ 2060 7 ) 



The automatic speed control has two horizontal manometrlc boxes. 
One of them (aneroid 1) is used to measure the static pressure, while 
the other Is used to measure the aerodynamic pressure 2 as the differ- 
ence between Ptotal ^^'^ Pst • /206 

Both boxes are connected by means of linking mechanisms to po- 
tentiometers 3, which regulate the current ratio in the balancing 
circuit, according to the ratio of the dynamic pressure to the static 
pres s ure . 

It is clear from Formula (2.47, a) that the ratio of the dy- 
namic pressure to the static pressure is not linearly related to 
the airspeed of the aircraft. In order to develop electrical signals 
which are proportional to the airspeed, the control unit contains 
an automatic speed control mechanism. This mechanism consists of 
a magnetic signal amplifier M- , coming from the automatic speed con- 
trol, activating motor 5, and a potentiometer 6 with a special pro- 
file, which levels out the nonlinearlty of the signals from the auto- 
matic airspeed control. Thus, the turn angle of the axis of the 
potentiometer of the analyzing mechanism becomes proportional to 
the airspeed . 



automati c 
speed control distributor unit 



magnetic 
amplifier 

wind sensor 




coordinate calculator 



automatic 
course control 



Fig. 2.39. Schematic Diagram of Navigational Indicator. 



211 



By means of a second potentiometer, connected by its axis of 
rotation to the activating mechanism, sends out electrical signals 
which are proportional to the airspeed, in the form of a DC volt- 
age . 

The automat-io course control is intended to distribute the sig- 
nals which are proportional to the airspeed, along the axes of the 
coordinates for calculating the path. 

Let us assume that we must make a flight over a path segment 
with the orthodromic flight angle ij; (Fig. 2.4-0). 

If the aircraft is now to fly with an orthodromic course y, /207 
the airspeed must be divided into two components: 

K^= Vcos(7-i;); 
Vt= K sin (1 — 4/). 

It is clear that if there is no wind at the flight altitude, 
these components of the airspeed must be multiplied by the flight 
time to give us the change in the aircraft coordinates during this 
time : 

A^= K^Af;. AZ= V^M. 



V /^ 




-Vcosd' 




^Vcosa, 



6-v 



- Vsinoi 



Fig. 2.40. 



Fig. 2.41. 



Fig. 2.40. Distribution of the Airspeed Vector along the Coordi- 
nate Axes . 

Fig. 2.41. Sine-Cosine Distributor. 

The division of the course signals by the axes of the coord- 
inates in the automatic course control is accomplished by means of 
a sine-cosine potentiometer (Fig. 2.41). 

The sine-cosine potentiometer consists of a circular winding 
with power supplied to it at two diametrically opposite points. 



212 



Two pairs of pickups slide along the coils; they are located at right 
angles to one another. 



Obviously, if we say 


that the 


the one in which one pair 


( cos ine ) 


and the second (sine) will be 


locat 


then the maximum current will 


flow 


while that through the second 


pair 


pickups from zero to 90° , 


the 


curre 


drop from maximum to zero 


and 


that 


from zero to the maximum. 


However , 


the pickups will not take 


place ace 


laws , but proportionately 


to the an 



zero position of the pickups is 
coincides with the supply leads 
ed at an angle of 90° to them, 
through the first pair of pickups 
will be zero. By turning the 
nt in the cosine pickups will 
in the sine pickups will increase 

the change in the current in 
ording to the sine and cosine 
gle of rotation of the pickups. 

In order for the law of change of currents to approach the sine- 
cosine, the winding of the potentiometer is given a profile or is / 20 i 
fitted with special regulating shunt resistors. 

Rotation of the pick-up shoes of the potentiometer is involved 
in figuring the course of the aircraft which is arriving from a course 
system or other course instrument. 

In order to apply the components of the aii 
ing system for calculating the aircraft coordina uea , tne c±rcu±ax' 

tentiometer is made movable and can be mounted in 



of the airspeed to the receiv- 



:: ui i u X- u cix l; u X d L J- ii^ L 11 e d J- x' u X' d ± L t^ u u X' u X ii a t e s , thc circular 
finding of the potentiometer is made movablf 
-,•„„ -K,, „„-,„„ „4: -, p3q]<; and pinion, 

:_n -I- JT __n-..n_-^,- j^ ^ ^ pOSltion 



in the automatic 



winding or the potentiometer is made movable and cai 
any position by means of a rack and pinion, located 
course control, and a special scale for calculating 

The angle for studying the system of coordinates for calculat- 
ing the path relative to the meridian from which the aircraft course 
is measured is called the chavt angte. In the majority of cases, 
the chart angle is made equal to the orthodromic path angle of the 
path s egment . 



Hence, by applying to the winding of the sine-cosine potentiom- 
eter a voltage which is proportional to the airspeed, we obtain signals 
at the outputs of the potentiometer which are proportional to the 
component of the airspeed along the axes of the coordinates V^ and 

For a precise regulation of the navigational indicator as a 
whole, these signals are calibrated manually by means of a poten- 
tiometer (see Fig. 2.39, Position 8), located in the control unit. 

The wind sensor has a schematic similar to that found in the 
automatic course control, with the exception that the voltage which 
is proportional to the windspeed is analyzed directly at the sensor 
by means of a potentiometer (see Fig. 2.39, Position 9) and is set 
by manually turning knob "w" so that the setting of the pick-up shoes 
on the sine-cosine potentiometer agrees with the wind direction. 

Thus, we have three set parameters on the wind sensor: the 



213 



^ 



wind speed (u), the wind direction (6), and the chart angle ( i(j ) . 

It is clear that the difference between angles 6 and i|^ gives 
the path angle of the wind. As a result, we obtain signals at the 
output of the sine-cosine potentiometer which are proportional to 
the component of the wind speed along the axes of the coordinates 
for calculating the path . 

The outputs of the sine-cosine potentiometers of the automatic 
course control and the wind sensor are connected in series, so that 
we obtain signals at their common outputs which are proportional 
as folllows 

Vjc + Ujc= Kcos (7 — ij/) + tt^^os AW 
Vz-\-Ut= 7 sin (y --'}') + a sin AW 



i.e., signals which make it possible to calculate the path of the 
aircraft with time, considering the manual setting of the wind value 
for the flight altitude. 

The oooTdvYiate oateulatoT consists of two integrating motors /209 
that work on direct current (see Fig. 2.39, Position 10), whose speed 
of rotation strictly corresponds to the magnitude of the signals 
coming from the automatic course control and the wind sensor. The 
revolutions of the motors are summed by two counters, whose readings 
are shown on a scale which is graduated in kilometers of path cov- 
ered by the aircraft along the corresponding axes . 

A pointer marked "N" shows the path of the aircraft along the 

J-axis, i.e., along the orthodrome , while a pointer marked "E" shows 

the travel along the Z-axis, or the lateral deviation from the desired 
line of flight. 

The names of the pointers ("N" and "E") were given because at 
a chart angle equal to zero, the pointer "N" will show the path 
traveled by the aircraft in a northerly direction from the start- 
ing point while the pointer "E" shows travel in an easterly direc- 
tion . 

To set the pointers of the counter to zero (at the starting 
point of a route) or to the actual coordinates of the aircraft when 
correcting its coordinates, there is a special rack and pinion which 
is used to turn the "N" pointer when it is pushed inward and to turn 
the "E" pointer when it is pulled out. 

9. Practical Methods of Aircraft Navigation Using 

Geotechnical Devices 

Flight experience shows that in addition to a knowledge of the 
devices for determining each of the elements of aircraft navigation, 
successful completion of a flight, means that it is necessary to 



214 



obtain and use the measured values, i.e., to master the devices used 
for aircraft navigation prior to automation. 

These devices do not depend on systems of measuring flight angles 
and aircraft courses, since they have limited fields of application. 
In addition, in describing them, it is necessary to recall that the 
readings of navigational devices contain all necessary corrections. 
Therefore, in the formulas which have been found to be necessary, 
we have used the common designations for navigational parameters. 

Under practical conditions of aircraft navigation, an impor- 
tant role is played by the pilots' calculating and measuring instru- 
ments. However, in many cases, instead of using these instruments, 
approximate calculations are performed mentally. Approximate mental 
extimates can be used to advantage in all cases when the problem 
can be solved more precisely by means of calculating instruments 
in order to avoid any chance gross errors. 

Methods of approximate (yet sufficiently accurate for practical 
purposes) estimation of navigational elements in flight without the 
use of calculating and measuring instruments are called pilots' vis- 
ual estimates. The rules for pilots' visual estimates will be given 
later on in the description of the suitable methods of aircraft naviga- 
tion. 



Takeoff of the Aircraft at the Starting Point of the Route 



/210 



The starting point of the route (SPR) is the first control land- 
mark along the flight path from which the aircraft will travel along 
the route at a given path angle \p . 

The final point on the route (FPR) is the last control land- 
mark along the route, from which the maneuver to land the aircraft 
begins . 

Regardless of the fact that the path angle of the flight is 
usually reckoned from the airport from which the aircraft took off 
up to the SPR, as well as from the FPR to the airport where it is 
to land, these values have significance only for general orientation 
in the vicinity of the airports . 

In connection with the fact that the first turn of the aircraft 
after takeoff is made after the aircraft reaches a certain altitude 
(200 m, e.g. ) and that many factors influence takeoff conditions 
(such as atmospheric pressure, wind speed and direction, flying weight 
of the aircraft, etc.), an exact determination of the location of 
the beginning and end of a turn is usually difficult. Therefore, 
the path angle and the distance from the first turn to the SPR has 
a variable nature and cannot be determined exactly. 

Methods of bringing the aircraft to the initial point on the 
route differ somewhat from the general methods of aircraft navigation 
along the flight route . 



215 



The basic difference between the methods of aircraft navigation 
involved in bringing an aircraft to the SPR, and the aircraft navi- 
gation along the route, is that in the first case we do not have 
a strictly determined path angle for the flight and can reach the 
given point from any direction, i.e., in the given case the navi- 
gation is made in a polar system of coordinates. In the second case, 
we have a given line of flight, and the aircraft navigation takes 
place along a straight-line orthodromic system of coordinates. 

In Figure 2.42, a, we see that the flight path angle from the 
center of an airport in the direction along the SPR and the short- 
est line for the aircraft's path to the SPR after takeoff and gain- 
ing altitude until the first turn are at right angles. 

Since the line of flight is not constant when the aircraft reaches 
the SPR, the problem involves bringing the aircraft to a given point 
with the minimum number of changes in the course, or (in other words) 
along the shortest path. 



Practically speaking, visual control of an aircraft to bring 
it to the SPR is done as follows . 




With the proper selection of the course to the SPR, i.e., when 
the lead angle (LA) is equal in value to the drift angle of the air- 
craft, the landmark will be observed at a constant angle to the axis 
of the aircraft, CAL = const (Fig. 2.42, b). 



In this case, it is necessary to continue the flight along the 
previous course until the SPR is passed or (in high-speed aircraft) 
until there is a linear lead on the turn. 

If the drift angle turns out to be less than the lead which 
has been taken (Fig. 2.42, c), a slipping of the landmark will be 
observed from the direction of the longitudinal axis of the aircraft. 
In this cascj, the aircraft must be shifted in the direction of the 
landmark so that- its course angle turns out to be less than the initial 
one . 

The slipping of the landmark in the direction of the longitud- 
inal axis of the aircraft (Fig". 2.43, d) indicates that the lead 
which has been taken is less than the drift angle, and the aircraft 
must be turned away from the landmark so that its course angle is 
greater than the initial one. 



216 



Thus , the course to be followed by the aircraft is set visually 
when the SPR is located along a straight line. This problem is 




SPR 



Fig. 2.42. Lining Up an Aircraft with the SPR: (a) Path 
Angle (ip ) and Shortest Distance (S); (b) Aircraft Course 
Chosen Correctly; (c) Aircraft Course must be Increased; 
(d) Aircraft Course Must be Decreased. 

best solved when there is a navigation level on board, by using the/212 
so-called method of half corrections. This method involves the fol- 
lowing: if the lead which has been taken turns out to be greater 
or less than the required one, it then changes in the required direc- 
tion by half of the initial lead which was taken. If this turns 
out to be insufficient, it is changed again by half of the initial 
value until the course angle becomes stable or the sign of the cor- 
rection must be changed to the opposite. 

Reverse correction is made by one-fourth of the initial lead, 
and if this is insufficient or too much, a correction is made which 
is equal to one-eighth of the initial lead. It is not usually neces- 
sary to break down the corrections more than eight times , since the 
value of the correction will then be no more than 1-1.5°, which is 
no longer of practical importance for visual aircraft navigation. 

In the absence of a sight aboard the aircraft, the course angles 
for the SPR are determined by visual observation; to solve this prob- 
lem, the pilot requires a certain degree of experience which is 
gained in the course of the training of flight cruise in actual 
flight or in special training devices, as well as in practice flights. 



217 



Selecting the Course to be Followed for the Flight Route 

The course to be followed by the aircraft along the flight 
route not only must be set so the aircraft passes over certain control 
landmarks in the proper order, but must also ensure that the flight 
takes place exactly according to the given line of flight- 
There are three principal methods of selecting the course to 
be followed: 

(a) When deviations occur from the line of a given path ( LGP ) 
during the flight, 

(b) At a landmark along the line, 

(c) In the direction of the landmark points. 

The most universal and widely used method is the first one. 
This method involves the following: after flying over a certain 
control point, the calculated course to be followed along the given 
line of flight is determined as follows 



y = ^ 



'calc ' 



which the aircraft follows until the first characteristic point 
along the flight path. 

If, at the moment that it is flying over this point, the air- 
craft turns out to be on the given line of flight, the course is 
then considered to be sufficiently correct. 

If the aircraft has undergone some shift to the right when 
it passes over this point, the linear lateral deviation from the 
desired line of flight is determined and the required correction 
is found for the course of the aircraft: 



tg Ay = 



LLD 



where LLD is the linear lateral deviation and S^ is the distance /213 
covered . 

Example : An aircraft has flown from a control landmark for 
a distance of 36 km and has deviated 3 km to the right of the desired 
path. Determine the required correction in the course (Fig. 2.43): 



Sol uti on. 



tgAf = 



1 



36 12 ' 

A7 = -5°. 



218 



To reach the desired line of flight, it is usually pecessary 
first of all to make a double course correction (in our case, 10°), 
and then (when the aircraft has covered a distance equal to the 
base of the measurement, or is traveling along the line of the desired 
path) the lead in the course is reduced by a factor of two, leav- 
ing a correction in the course which is equal to the set angle of 
drift. 

If the closest turning point in the route (CTR) is located 
at a distance which is smaller than the base of measurement, then 
in order to attain it, correction must be made in the course for 
the distance covered for the travel parallel to the line of the 
desired path and over the distance covered, in order to reach the 
desired path at the moment when the next control landmark is being 
passed . 

Let us say that in our example the distance to the next land- 
mark is still 30 km; the correction for the remaining distance will 
be equal to : 

tgA,rem=^ = ^ 

Aj^— 6°. 



V 



-^ 



•Sc 



"^ 



lXd" 



Fig. 2.43. 
Corre ct i ons 
Followed . 



Determination of 
in Course to be 



Since the correction for the 
distance covered was equal to - 
5° , the total correction for the 
course in order to get the air- 
craft to the CTR must be equal 
to -11°. 

The problem is solved similarly 
when the aircraft has wandered 
to the left of the desired path, 
but with the difference that the 
correction in the course to be 
followed is positive in this case. 



In solving problems in determining the desired corrections 
in the course to be followed, we preferably use methods involving 
visual observation by the pilot without the use of any calculat- 
ing instruments or tables. In the opposite case, while the pilot 
is solving the problems, the aircraft will cover a considerable 
distance, thus complicating the realization of the desired solu- 
tions . 



/214 



The first method of pilot's visual estimation in this case 
will be the visual estimation of the lateral drift from the line 
of flight. 



219 



charac 
verse 
of the 
tance 
with a 
altitu 
tude . 
Interm 
by vis 
ical a 
is app 



f an ai 
teristi 
is dete 

point 
from th 

vertic 
de , whi 

These 
ediate 
ual obs 
ngle is 
roximat 



rcraft i 
c point , 
rmined b 
is close 
e point 
al altit 
le at an 
angles a 
values o 
ervation 
roughly 
ely equa 



s tra 
the 

y the 
to 2 

whi ch 

ude o 
angl 

re us 

f ver 
and 
equa 

1 to 



veling 
distan 

verti 
6.5°, 

is eq 
f 45° , 
e of 6 
ually 
tical 
interp 
1 to 5 
1. 5 fl 



to the side of the above mentioned 
ce from it by flight along the tra- 
cal angle . When the vertical angle 
the aircraft is located at a dis- 
ual to half the flight altitude; 

the distance is equal to the flight 
3.5° it is twice the flight alti- 
determined by visual observation, 
angles and distances are determined 
elation. For example, if the vert- 
5° , then the distance to the point 
ight altitudes . 



This method, with sufficient training, gives a very high 
accuracy for determining the location of the aircraft relative 
to a given point along the route, and consequently, with respect 
to the line of flight (on the order of 0.1 H) at vertical angles 
up to 65°. At very large angles (grater than 65°) from the vert- 
ical of the aircraft, the errors in distance will be greater 
and this method cannot be used. 

The second method of visual estimation by the pilot which 
is used in solving this problem is the mental calculation of the 
required course corrections following linear lateral deviation 
(LLD) . 

For convenience in metal calculation, one radian is assumed 
to be 60° rather than 57.3, but this does not introduce any consid- 
erable errors (the maximum error in angles up to 20° does not exceed 
1°). 

This allows the required correction to be made in the course 
in terms of the approximate ratio of the lateral deviation to the 
distance covered: 



.LD/5 


Ay. deg 


LLD/5 


A7, deer 


LLD/5 


-^T. de& 


1/60 


1 


1/12 


5 


1/6 


10 


1/40 


1.5 


1/10 


6 


1/5 


12 


1/30 


2 


1/8 


7 


1/4 


15 


1/20 


3 


1/7 


8 


1/3 


20 


1/15 


4 











These ratios are easy to remember if we know that in order 
to obtain their required correction it is adequate to divide the 
number 60 into the distance covered, when the lateral deviation 
is taken per unit of measurement. 

Obviously, if this method for course correction is employed 
and the aircraft does not reach the desired point along the line 



220 



of flight, so that there is still some lateral deviation, the lateral 
deviation and the distance from the point at which the course was / 215 
last changed can be used to correct the course. 

Selection of the course to be followed according to a land- 
mark along the route can be used in the case when the flight takes 
place along a straight- line portion of a railway or highway and 
means that the crew must change the course of the aircraft so that 
it follows this linear landmark. After changing the course by 
an additional turning of the aircraft , the crew returns to the 
desired course and travels in the desired direction once again. 

The selection of the course to be followed on the basis of 
orientation landmarks is a variety of the latter method. 

In this case, the course is selected so that the closer of 
two selected landmarks along the line of flight constantly (up 
to the moment that the aircraft flies over it) remains in a line 
with the further landmark. After passing by the closer landmark, 
the aircraft follows the desired course or choses the next land- 
mark, located beyond the second one, and continues its flight along 
this line . 

Change in Navigational Elements During Flight 

The majority of navigational elements (course, altitude, speed) 
are determined in flight on the basis of indications of the corre- 
sponding instruments, with introduction of corrections for instru- 
mental and methodological errors. 

Automatic radio devices, based on the Doppler principle, make 
it possible to make measurements directly (during flight) of such 
elements as the drift angle and the ground speed. 

Other methods of aircraft navigation do not permit direct 
measurement of the latter two elements, so that in order to de- 
termine them it is necessary to use various pilotage techniques. 

In the absence of sights, the drift angle of the aircraft 
can be determined as follows. 

Let us suppose that we are traveling along a given route and 
that a control landmark on this route has been passed. After 15- 
20 min of flying time, we select another landmark by which we test 
the correctness of the course which has been selected. If no lat- 
eral deviation of the aircraft occurs on this segment, it means 
that the aircraft course has been properly set, i.e., the drift 
angle is equal in value to the previous course, but has the oppo- 
site sign 

a = \ii - y , 



221 



where a is the drift angle of the aircraft, y is the aircraft course, 
and ^ is the given flight path angle. 

It is not always possible, however, to correctly set the course 
to be followed. 



If a lateral deviation of the aircraft from tjie line of the /216 
desired path arises in our flight segment, the course to be fol- 
lowed will be incorrect and the actual flight angle will be 



"I'll) = tl-a + arctg 



AZ 



where AZ equals the deviation of the aircraft from the LGF , and S 
is the length of the segment over which the drift angle was meas- 
ured . 

The angle of deviation of the aircraft from the line of the 
desired flight path arctg AZ/S is considered to be negative if 
the aircraft deviates from it to the left, and positive if it devi- 
ates to the right. As in the method of selecting the course, this 
angle is determined by methods of visual estimation by the pilot. 

In the case of improper selection of the course to be fol- 
lowed, the" latter can be determined as the difference between the 
actual flight angle and the course being followed: 

" = Y* — T = +3 + arctg — — — f . 

It is much easier in flight to determine the ground speed 
of an aircraft: the same landmarks are used for this purpose as 
those used for determining the drift angle of the aircraft. To 
do this, it is sufficient to determine the times when the aircraft 
flies over the first and second landmarks, after which the ground 
speed is determined by the formula 

^ = 1^ 

where S is the distance between the landmarks and t is the flying 
time between the landmarks . 

The division S/t is done as a rule on scales 1 and 2 of a 
navigational slide rule (Fig. 2.HM-), with the exception of those 
cases when the flying time is less than 60 min. For example, 6, 
10, 12, 15, 20 and 30, or even 40 and 48 min are possible. In 
these cases, the groundspeed will be equal to 105, 6S , 5S , 45', 3S , 
2S , 1.5S and 1.255', respectively, and is easily determined men- 
tally by multiplying the distance between the landmarks by one 
of the numbers given above. 



222 



h-jL 



=^ 



\ dlstance(kiii) 



A^|yii[\Hi|ij|('l\i|i|MWM il ll|IIM 



turn 
angles 



i n 'i M' iiii ii ii i i timiuii i i iiii i i ii i iii i ii i ii ij. i ijm iii Mt iii i i i ii i[ ii | ^ 



t t 

is ' aV-" A / / 



40 BO 00 70 I 






(in in1n or sec) 



a e 7 a 9 [io] 



je 



inir 



xiSfl 






■ -t, i'-f. ■- Y i "-" 'I 1 1 I I I.I I I I I ll l lll l ll|l l lllllll|llllllilll lll lll ll ! »l » l H^- T H. ^^ ttnedn nr ot- ntn) 

06^ ^^ J* ^ 2' ^' 4' ^8' a- r a' r ^' ip- ac ao* . 4o- y\ eo* eo' to* _t^"9ents r ^p. 



?;^ Ml"l | l ' "*"^ l' » ' «^'!«'AI' ' 'tMM I . ' Jm l «, ' | l ,1 I l,' i ' ; ' ,M , HAM , l, 



ilili l i M ilililll/ MMMM ,li' l\W>l l ll)Ml i '^yil)^'J , Vai ) i i U i,li i i LL ii |^^ ^ 






a 8 7 B 9 10 



i 



turn radius 



altitude 6 



40 00 BO 70 80 90 100 \ 100 300 300 400 BOO tOO 700 SOO BO0 1000 



Vj^"* ! 1 p I ' I I i l I t ■ . . t I I I I i I I I r I i I i-i I I'umiuiirnimiiiiiiMiiiiiNiiiiimmiii iipiiiimiii i \ i I i I i r i i i t i u m n m m mniiu 



6 30 \ \ ( 



&^ 



\ 




for heights above Uemoeratu 
12,000 m 



At*"« ruE »' ? *^ ^ /•i'»*M"i!f**!)V;*'i*i'i*jrB'ji*jrii*«'«*4i'ir' 



7 ^ • • 10 It la 

"''■inst'";: jneasuredaltjtjjd. 
cted aititudel and^speed 



|^Wi»co*a(i(t,'t»J| 



^^^!Mihll:fM.==^^^i-JM; 




To~^l 



A 



Fig. 2.44. Scales on Navigational Slide Rule NL-IOM, 



to 

K) 
CO 



H 



To measure the ground speed as well as the drift angle, it 
is desirable to select distances between landmarks which are no 
less than 50-70 km apart. Over short distances, in order to avoid 
gross errors, it is necessary to determine and mark down very exactly 
the time that the aircraft passes over the control landmarks . 

Measuring the Wind at Flight Altitude and Calculating /218 
Navigational Elements at Successive Stages 

The principal factor which complicates the processes of air- 
craft navigation at flight altitude is the wind. With availabil- 
ity of exact data regarding its direction and speed, all problems 
of aircraft navigation can be solved by a combination of general 
methods of aircraft navigation independently of the visibility 
of terrestrial landmarks. 

When the aircraft has on board only the most general devices 
for aircraft navigation, the problem of determining the wind at 
the flight altitude as well as the drift angle and the ground speed 
can be solved if terrestrial landmarks are visible. 

The wind at flight altitude does not remain constant but is 
constantly changing with time and especially with distance. In 
order to be able to prepare the navigational data for the next 
stage of flight, it is necessary to determine the wind at the very 
end of the preceding stage and even in this case, the data on the 
wind which are obtained are obsolete to a certain degree and are 
not completely satisfactory for the needs of calculating. 

Under the conditions when an aircraft is flying along an air 
route, there are three navigational parameters which basically 
determine the speed and direction of the wind at flight altitude: 
the airspeed (F), ground speed iW) , and the drift angle for a given 
course . 

The wind calculated on the basis of these parameters will 
not be reckoned from the meridian of the locus of the aircraft 
(LA) but from the line of flight of the aircraft. 

The calculation of the path angle of the wind (AW) is car- 
ried out on the navigational slide rule by means of a key (Fig. 
2.45, a) . 

Example: W = 360 km/hr; V = 320 km/hr; drift angle = +8°. 
Determine the wind angle . 

Solution : (Fig. 2 .45 , b) . 

Answer: AW = 48° . 

If we know the wind, it is easy to determine its speed by 
means of a key which is marked on the rule (Fig. 2.46, a). For our 

224 



example, see Figure 2.4-5, b. Answer: 



60 km/hr. 



The direction of the wind relative to the meridian of the 
locus of the aircraft (LA) is determined by the formula 

6 = AW + ij^. 

If the flight is made with magnetic flight angles , the wind 
direction is obtained relative to the magnetic meridian of the 
LA. This direction is also used to calculate the navigational 
elements in the next stage of the flight. 

Information on the speed of the wind and its direction is trans- 
mitted from the aircraft to ground stations, also relative to the 
magnetic meridian of the LA, and is used for controlling the flight 
of the aircraft. 

The angle of the wind for the next stage of the flight is /219 

AW = 6 - \p , 

where 6 is the wind direction and (p is the flight path angle of 
the next stage of the flight. 



a) 

® Sin US 



;^AW® ■ 



b) 





«•© 



® 



w-i' 



® 



ill 



no 



Fig. 2.4-5. Calculation of the Path Angle of the Wind on the 
Navigational Slide Rule: (a) Key for Determining the Wind 
Angle; (b) Determining the Wind Angle. 

The values for the ground speed and drift angle of the air- 
craft for the next stage of the flight are calculated on the navi- 
gational slide rule by means of a key (Fig. 2.47, a). 

Let us assume that the flight in the preceding stage was made 
with a MFA = 38°, in the next stage with an MFA = 56°, and with 
an airspeed of 320 km/hr. The data obtained on the wind at the 
preceding stage are AW = 48°, u = 60 km/hr. 



a) 

©US 



pm 



b> 
® 



iS' 



©" 



® 



50 



ilO 



Fig. 2.46. Calculation of the Wind Speed on the Navigational 
Slide Rule: (a) Key for Determining the Speed; (b) Deter- 
mination of the Speed. 



225 



The direction of the wind relative to the meridian of the 
LA is 

6 = 48 + 38 = 86° , 

while the angle of the wind for the next stage of the flight is 

AW = 86 - 55 = 30° . 
a) b) 

© us: .aV US+AW ® 5.5' JO' 35,5' 



® ^ 



w 



® 



60 



izo no • 



Fig. 2.47. Calculation of the Drift Angle and Ground Speed 
on the Navigational Alide Rule: (a) Key for Determining the 
Drift Angle and Ground Speed; (b) Determination of the 
Drift Angle and Ground Speed. 



The value of the groundspeed and the drift angle for the next 
stage of the flight are also determined by means of the naviga- 
tional slide rule (Fig. 2.47, b), i.e., 

\1 - 370 km/hr ; US = +5.5°. 

The values of the drift angle can be used to determine the 
calculated course to be followed in the next stage of the flight. 
In our case , 



Y = 5i 



5 . 5 



52.5° . 



If the flight is made with orthodromic flight angles, then 
in order to calculate the navigational elements for the next stage 
of the flight it is unnecessary to convert the wind angle to its 
direction relative to the meridian of the LA. In this case, the 
wind angle for the next stage of the flight is determined as the 
difference between the wind angle of the preceding stage of the 
flight and the angle of turn in the route (Fig. 2.48): 



AW; 



AW- 



TA 



In our example, AWi = 48°, TA = 56-31 
= 30° . 



= If 



and AW2 = 48- 



However, in order to transmit information regarding the wind 
to ground stations, it is necessary to determine the wind direc- 
tion relative to the meridian of the LA. 

Obviously, the true wind direction at the point LA is 

■^true = AW + a. 



/220 



226 



where a is the azimuth of the orthodrome at the point LA; the maj 
netic direction of the wind is 



«M = AW + a - Aj^ 



Consequently, if the calculation of the orthodromic path angles 
is made from the reference meridian, then 

6 = AW + (A -X ^)sin(() - A„. 

M LA ref av M 



Example: \q-^-70°, Xla=85°, 
5°, i|j = 38°, AW = 48°. 



av 



:52' 



Solution 

6 



true 



The true wind direction is 
= 48 + 38 + 15-0.8 = 98° , 



and the magnetic wind direction is 




= 48 + 38 + 15 -0 .8 + 5 = 103° 



Fig. 2.4-8. Determination 
of the Wind Angle in a 
Successive Flight Stage. 



Calculation of the Path of the Aircraft and Monitoring 
Aircraft Navigation in Terms of Distance and Direction 

In the preceding paragraphs, we have discussed the methods 
of placing the aircraft on course, determining the navigational 
elements during flight, and calculating them for the following 
stages of the flight. 




Therefore, it becomes necessary to use continuous calcula- 
tion of the aircraft path in terms of time at certain periods, 
when it becomes necessary to check the aircraft path with respect 
to distance and direction. 

Calculation of the aircraft path is always done with prev- 
iously calculated parameters (the calculated course and ground speed. 



227 



calculated time). At the same time, all the values and moments 
of change in the aircraft course are determined, which make it 
possible to determine the calculated position of the aircraft by 
plotting and thus to determine the additional errors in aircraft 
navigation . 

Calculation of the path of the aircraft means that after the 
last identified landmark has been left behind, the crew aims the 
aircraft toward the next landmark during a certain period of time 
which is used to fix all the values of the actual course of the 
aircraft . 

If the proper landmark has not been sighted when the sched- 
uled time has elapsed, due to meteorological conditions, the calcu- 
lated time for flying over this landmark is determined, and the 
aircraft is set to the next phase of calculated flight on the basis 
of the previous values for direction and velocity of the wind. 

Thus, calculation of the path (flight on the basis of prev- 
iously determined data) can continue until the conditions for visual 
orientation improve. However, it is necessary to keep in mind 
that the accuracy of aircraft navigation then decreases contin- 
uously due to the accummulation of errors with time, as well as 
in connection with the obsolescence of the data on the wind, meas- 
ured prior to the last reliably sighted landmark. 

When the conditions for visual orientation improve, the crew 
takes measures to check the path of the aircraft in terms of dis- 
tance and direction. 



To check the path in terms of distance, linear landmarks are 
usually employed, which intersect the route of the flight at an 
angle close to 90°. 

Five to ten minutes before the calculated time for flying 
over these landmarks, depending on the flying time according to 
the previously calculated data and the speed of the aircraft, the 
pilot carefully begins to examine the landscape, looking for the 
landmark; at the moment that he flies over it, the approximate 
position of the aircraft is determined relative to distance and 
time . 



/222 



When flying over a control landmark, the pilot also tries 
to determine the lateral deviation of the aircraft from the desired 
path on the basis of additional features of the landmark (curves 
in rivers, tributaries, road junctions, populated areas, forest 
outlines , etc . ) . 

Having determined the point of intersection of the landmark, 
the pilot projects it along the line of the desired path, fixing 
the position of the aircraft (in terms of distance at the moment 
that it flies over the landmark)and the direction. 



228 



At the present 
when the ground is n 
tional equipment. L 



time, aircraft us 
ot visible are fi 
ight planes (whic 




side of acute angle 



Fig. 2.49. Lead Tow 
Angle of the Travers 
mark . 

than the calculated 
craft can be aimed a 
linear landmark. 



ard the Acute 
e of a Land- 



time for flying p 
t a control landm 



ed 

tte 

h f 

occ 

Ion 

the 

ori 

the 

occ 

on 

in 

tio 

int 

by 

2.4 

Ian 

fie 

as t 

ark 



for Ion 
d with 
ly at 1 
asional 
g dista 
condit 
entatio 
less , w 
ur , the 
the bas 
the cou 
n of an 
ersecti 
a linea 
9) . In 
dmark m 
Id of V 
it . A 
which 



g di 
spec 
ow a 
ly r 
nee 
ions 
n ar 
hen 

fli 
is o 
rse 

acu 
on w 
r la 

thi 
us t 
i ew 
f ter 
line 



stanc 
ial r 
Ititu 
equir 
fligh 

for 
e poo 
such 
ght c 
fas 
and t 
te an 
ith t 
ndmar 
s cas 
appea 
s omew 

this 
s up 



e fl 
adio 
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ed t 
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r . 

case 
an b 
ligh 
he d 
gle 
he r 
k (F 
e, t 
r in 
hat 
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with 



ights 

naviga- 

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the 



In individual cases, when the pilot does not recognize the 
terrain over which the aircraft must fly, after the conditions 
for visual observation have improved, the crew sets the aircraft 
on course to fly toward the next control landmark, and the pilot 
makes an estimate on the chart of the aircraft flight in terms 
of airspeed, fixed course, and flying time with these courses from 
the last recognized landmark. 

The point obtained has a calculated wind vector during the 
flight time, after which the pilot compares the chart with the 
location in the following manner: 

(a) In the region of the end of the wind vector (the most 
probable position of the aircraft); 

(b) In the vicinity of a calm point; 

(c) In terms of the wind vector direction from its begin- 
ning to end, with a continuation of the wind vector 1.5 to 2 times 
and turning it to the left and right at angles up to 90° from the 
calculated direction; 

(d) Turning the wind vector (extended 1.5 to 2 times) in 
the remaining semicircle. 

Naturally, these operations must be carried out with constant /223 
change of the calm point, depending on the direction of the move- 
ment of the aircraft . 

If the location of the aircraft cannot be determined in this 
manner, other measures must be taken to find landmarks, such as 



229 



the location of a characteristic linear or large-area landmark 
(lake-, sea), and also by making inquiries from the ground, etc. 

Use of Automatic Navigational Devices for Calculating the 
Aircraft Path and Measuring the Wind Parameters 

To a considerable degree, automatic navigational devices sim- 
plify the work of the pilot in calculating the path of the air- 
craft and in measuring the wind parameters at flight altitude. 

These devices are mounted on high-speed passenger aircraft 
which have complete radio navigational equipment, thus considerably 
increasing the effectiveness of their use. 

Such devices, which are based on the general methods of air- 
craft navigation, can be used in straight-line systems of coord- 
inates at any orientation of their axes . 

The direction of the axes of the coordinates is selected by 
the pilot depending on the conditions for which the system is being 
used. For example, for flying along a route, it is most advan- 
tageous to combine the axis of the system OX with the directions 
of the straight-line segments of the flight, i.e., to calculate 
the path in an orthodromic system of coordinates in stages. 

To carry out special operations in this region, e.g., at test 
sites for radio navigational systems for short-range operation, 
the axis OX is combined with the average meridian of the flight 
area (magnetic or true), depending on which system for calculating 
the flight angles is being used to make the flight. 

In preparing to land and maneuvering in the vicinity of the 
airport, the axis OX coincides with the axis of the landing strip 
at the airport, etc. 

In all cases when an automatic navigational device is being 
used, a rectangular system of coordinates should be applied to 
the flight chart in the given region, parallel to the axes of the 
system OX and OZ . 

Parallel lines are drawn at 20 mm intervals, so that on charts 
with a scale of 1;1, 000, 000 this corresponds to 20 km, while on 
those with a scale of 1:2,000,000 it is M-O km, etc. For this purpose, 
a special stencil is included in the set of navigational instru- 
ments for the NI-50B indicator. 

In using an automatic navigational device with orthodromic 
coordinates in stages, no additional devices are needed other than 
the general navigational divisions of the chart. 

During flight, the apparatus is connected to a source of direct 
current, and the chart angle on the automatic course control is set 



230 



in accordance with the selected system for calculating the air- 
craft to coordinates. The windspeed and direction are set on the 
wind sensor on the basis of the results of measurements during 
the preceding flight segment. 

If the navigational indicator is used with an orthodromic 

system of coordinates in sections, the setting of the chart angle 

and wind is made at the end of the preceding stage of the flight 

before flying over a turning point in the route (TPR). On the 
coordinate calculator in this case, the pointer "N" is set to a 

value equal to the linear lead for the turn (LLT) and pointer "E" 
is set to zero . 



/224 




At the moment when the aircraft 
emerges from the turn on the new 
course (Fig. 2.50), the alternating 
current is connected to the instru- 
ment and the indicator begins 
to calculate the flight path. 

At small turn angles in the 
line of flight (up to 30°), the 
turn trajectory of the aircraft 
is very close to TPR. In this 
case, the two pointers on the 
indicator should be set to zero, 
and the mechanism switched on 
when the TPR is passed as the air- 
craft is turning. At the beginning 
of the straight- line segment of flight, if possible, it is neces- 
sary to mark the established coordinates of the aircraft on the 
computer as the aircraft passes over a given landmark. 



Fig. 2.50. Transition to an 
Orthodromic System of Coor- 
dinates in a Successive 
Flight Stage. 




Constant knowledge of the aircraft coordinates facilitates 
both visual and radial orientation. However, aircraft coordinates 
obtained on the basis of a computer will not always correspond 
precisely with the actual coordinates, since the speed and direc- 
tion of the wind during flight change over the distance covered. 



The navigational indicator also makes it easier to determine 
the wind parameters at flight altitude. This is done as follows: 

At the end of a stage in the flight, the aircraft coordinates 
are recorded with the computer (Point B in Figure 2.51, a) and the 



231 



actual location of the aircraft is determined visually or by means 
of radionavigational devices (Point B^). These Points BBi deter- 
mine the vector of the change in the wind at flight altitude for 
the flight time of a given stage of flight. 

The problem of determining the wind vector in this case can 
be solved easily on a flight chart. To do this, a reverse line 
must be drawn from Point B and the length of the wind vector is set 
on the sensor (u -t) during the flying time from Point A to Point 
B (Point 0). Then the vector of OB will constitute the vector 
of the calculated wind (and OBj, the actual wind) at flight alti- 
tude . 



/225 




Fig. 2.51. Measurement of the Wind by Means of a Navi- 
gational Indicator: (a) Wind-Change Vector; (b) Wind Vector, 

In order to obtain the value of the wind in km/hr, it is suf- 
ficient to divide the length of the vector OBi by the flying time 
between Points A and B, expressed in hours. 

The problem of measuring the wind can be simplified if we 
consider that the wind at the sensor is zero for the flight stage, 
i.e., we introduce the value of AW = 0, U = into the wind sensor. 
Then Point B will be the indication of the coordinates of the air- 
craft at the end of the flight stage, while Point Bj will repre- 
sent the actual coordinates (Fig. 2.51, b). Consequently, vector 
BB^ will be the wind vector for the flying time in this stage. 

The use of the navigational indicator in rectilinear coord- 
inates for flights in a given region is not different in principle 
from using it in orthodromic coordinates and stages. However, 
the important advantage of the orthodromic system of coordinates 
is then lost, i.e., the relationship of the coordinates to the 
checking of the path for distance and direction. Therefore, the 
position of the aircraft in this case can be determined only in 
terms of the coordinates of the network superimposed on the chart. 

The rectangular system of coordinates can be extended over 
a relatively small area (on the order of ^+00 x 400 km), since the 
effect of the sphericity of the Earth begins to show up in large 
areas . 

In conjunction with this, in the case of flights by a coord- 
inate system, it is not necessary to set a new chart angle for each 



232 



change in the line of flight and to describe the coordinates of 
the aircraft in a new system for calculation, which to a consid- 
erable degree compensates for the loss of those advantages which 
we have in the orthodromic system of coordinates in stages. 

Details of Aircraft Navigation Using Geotechnical Methods 
in Various Fligiit Conditions 

The conditions for aircraft navigation using geotechnical 
devices are determined primarily by the presence and nature of 
landmarks, as well as by their contrast relative to the surround- 
ing terrain. 

The best landmarks for visual aircraft navigation are lin- 
ear ones (large rivers, railways and highways, the shores of large 
bodies of water). Lakes, large and small populated areas, char- 
acteristic mountain peaks, etc., are also good landmarks, while 
grain elevators, water tanks, churches, industrial enterprises, 
etc., can be used for flights at low altitudes. 

For aircraft navigation in an area which is poor in landmarks, 
we can use separate sighting points on the Earth's surface in the 
form of spots, individual trees, foam on the surface of the water, 
etc. Such points are not landmarks, since it is impossible to 
determine their location on a flight chart, but they can be used 
to measure the drift angle and the ground speed when there is a 
sight on board and also make it possible to increase the accur- 
acy of aircraft navigation during flight between control landmarks. 



/226 




The visibility of all landmarks, with the exception of il- 
luminated populated areas, is considerably decreased at night, 
especially when the Moon is not out. Therefore, populated areas 
are the principal landmarks at night; their appearance at night 
can differ from their appearance in the day. 

An important factor which determines the conditions for air- 
craft navigation is the stability of operation of magnetic com- 
passes. Conditions of aircraft navigation without the use of gyro- 
scopic compasses are unfavorable in the polar regions, as well 
as low altitudes in the vicinity of the magnetic anomalies. 

The flight altitude also has a significant influence on the 
aircraft navigation conditions. In clear weather, optimum conditions 



233 



for visual orientation exist at heights on the order of 1000-1500 
m , since at this altitude the angular velocity at which the land- 
marks go by is small, all of their details can be seen clearly, 
and the field of view of the crew covers a very large area, which /227 
is important in comparing the charts with the landscape. 

However, these altitudes can only be used wh-en there is a 

small amount of clouds along the flight route. In cloudy weather, 

flights are made at lower altitudes, as low as the relief of the 
terrain will allow. 

At low altitudes, the conditions for visual orientation are 
worse, since the angular velocity with which the landmarks go by 
increases and the area which the crew of the aircraft can scan 
is reduced. 

An increase in the flight altitude (above 1.5 km) in clear 
weather have a small influence on the conditions of visual orien- 
tation, but at great heights the visual visibility of landmarks 
(depending on weather conditions) is much worse than at low and 
medium altitudes . 

The selection of scales and chart projections for making a 
flight depend primarily on the altitude and speed of the flight. 
At low altitudes, it is best to use charts with a large scale of 
1:500,000 or 1:1,000,000. At high altitudes and high speeds, it 
is best to use charts with scales of 1:2,000,000 and 1 : i+ ,00 , 000 . 

For flights along routes which are very long, charts are used 
which are made up of projections showing the properties of ortho- 
dromicity (the orthodrome on the chart has a shape close to a straight 
line), i.e., charts in the international or transverse cylindrical 
projection. For the polar regions, charts with tangent stereo- 
graphic projection are used. 

10. Calculating and Measuring Pilotage Instruments 

Purpose of Calculating and Measuring Pilotage Instruments 

Pilotage calculating and measuring instruments are intended 
for the following: 

(a) Measuring distances and directions on flight charts. 

(b) Calculating navigational elements both in preparing for 
flight and when completing it. 

(c) Calculating methodological errors in the readings of 
navigational instruments (the readings of the airspeed, altimeter, 
and outside air thermometer) . 

(d) Calculating the elements of aircraft maneuvering. 



23H 



Measurement of distances on flight charts is made by means 
of a special navigational slide rule. A feature which distinguishes 
this slide rule from conventional slide rules is the presence of 
several scales for measuring distances on charts with different / 22 1 
scales . 



Measurement of directions on flight charts is made by means 
of navigational protractors, made of transparent material. 

The protractors are simultaneously used as triangles, which 
make it possible to make certain constructions on flight charts 
and diagrams (laying out the traverses of landmarks, parallel shift 
of lines , et c . ) . 

Calculations of navigational elements, corrections to nav- 
igational devices, and elements of maneuvering are presently car- 
ried out with the aid of navigational logarithmic slide rules, 
the best modification of which is the navigational calculating 
slide rule NL-IOM. 

In addition, to calculate certain navigational elements, we 
can use special devices for setting up the speed triangle (wind- 
speed indicators). However, due to the improvements in navigational 
calculating slide rules, they have a very limited application. 




Thus, the operations described above involving numbers can 
be applied to the summing of the segments of a scale on the ruler, 
which simplifies calculation to a considerable degree 



The scales of the navigational slide rule NL-IOM (see Fig. 
2.44) are grouped so that one side is used for solving problems 
in determining navigational elements of flight as well as maneu- 
vering elements, while the other side is used for calculating the 
corrections for the readings of navigational instruments . 

In addition, the upper beveled edge of the ruler (Position 
17) carries a scale divided into millimeters, which can be used 
to measure distances on the map. 

The scales on the ruler 1 and 2 are intended to determine 
the ground speed from a known distance covered in a given time, 
or from a given distance at a known ground speed and time. 



235 



Therefore 



so that 



W =: '^ and S = Wt. 
t 



IgW = \gS - Igt and IgS = IgW + Igt 



Scale 1 is the scale of logarithms of distances in kilometers /229 
or flight speeds in km/hr; scale 2 is a scale of logarithms of 
flying time in min or sec up to the rectangular index marked 100 and 
beyond, in hrs or min. 

The principle of solving problems by determining the airspeed 
over a given distance at a given time is as follows : 

Let us say that an aircraft has covered a distance of 165 
km in 12 min and that we must determine the ground speed in km/hr. 

We set the marking on the slider to the 165 position on the 
distance scale; by moving the adjustable scale 2, we set division 
12 on it opposite the marking on the slider. We can then read 
off the distance covered by the aircraft in minutes of flight oppo- 
site the number 1 at the beginning of the scale: 

IgW km/min = Ig 165 - Ig 12 = Ig 13.8. 

However, since 1 hr is 60 min, the speed in km/hr would be 
equal to 

IgW km/h = Ig 13.8 + Ig 60 = Ig 825 

or 

W = 825 km/h. 

By combining the first and second effects, we obtain 

IgW = Ig 165 - Ig 12 + Ig 50, 

i.e., in order to solve the problem, it is sufficient to set the 
number 12 on scale 2 opposite the number 165 on scale 1 and oppo- 
site number 60 on scale 2, which is marked with a triangular mark- 
ing, and then to calculate the ground speed from scale 1 (Fig. 
2.52, a) . 

The problem is solved analogously if the flight time is meas- 
ured in sec. In our example, it will be 720 sec: 

IgW = Ig 165 - Ig 720 + Ig 3600 = Ig 825; 

W = 825 km/h. 



236 



The ground speed in this case is calculated from scale 1 oppo- 
site the number 3600 on scale 2 (the number of seconds in 1 hr), 
marked with a circular index. 

To determine the distance at a given groundspeed and flying 
time (5 = Wt) , the logarithms of these numbers are added: 



lg5' 



leW + let 



On the rule, the triangular or circular index on the movable 
scale 2 is set opposite the known ground speed on scale 1. The 
index marking on the slider is set opposite the given flying time 
on scale 2, after which the position of the indicator on scale 
1 shows the distance covered in this time. 



/230 



Example . W - 750 km/hr, t =1 hr and 36 min 
tance covered. 



Find the dis- 



solution. See Figure 2.52, b. 

Ig 5 = Ig 750 + Ig 1 h 36 min= 1200; 

Answer. S = 1200 km. 



a) 



ies 



S15 



b> 
® © 



750 



tzaa 



12 



® ® 



1 h 36 min 



Fig. 2.52. Calculation on the NL-IOM: (a) of the Ground 
Speed; (b) of the Distance Covered on the Basis of Ground 

Speed and Time . 

Let us apply the keys to NL-10 for solving problems in deter- 
mining the ground speed and distance covered on scales 1 and 2: 

(a) To determine the ground speed for a distance covered 
in a known time (fig. 2.53, a or 2 . 5 M- , a). 

(b) To determine the distance covered from the ground speed 
and time (Fig. 2.53, b or 2 . 5 1+ , b). 



a) 
O 



Skm 



b). 
© 



® 



t sec 



"T 



5 KM 



w 



0- 



t man 



Fig. 2.53. Keys for Determining the Ground Speed on 
the NL-IOM, on the Basis of the Distance Covered and 

the Time. 



237 



I 



Movable scale 3 with the signs of the logarithms , which is 
the same (up to 5°) as scale 4 for the logarithms of the tangents 
and is also divided into scales 3 and 4, along with the fixed scale 
of distances or altitudes 5, which essentially repeats scale 1, 
are all intended for working with trigonometric functions. 

The majority of problems which are solved on these scales 
are based on the properties of a right triangle, so that the value 
of the sine of 90° and tangent 1+5° (scales 3 and H), whose loga- 
rithms are equal to zero, are marked on the rule by a triangular 
index . 

If the problem is solved from a known leg, e.g., determin- 
ing the error in the course on the basis of the distance covered 
and the linear lateral deviation (Fig. 2.55), we use scales U and 

5 on the rule . 

Z 
tg^Y=-^0^ lgtgAY=5,IgZ-.lg^. 

The key to solving this problem is shown in Figure 2.56. 

In the case when the hypotenuse of the triangle is known, 
the problems are solved by using scales 3 and 5. For example, sup- 
pose we wish to determine the location of the aircraft in ortho- /231 
dromic coordinates (-Sfg, Z^) on the basis of known coordinates of 
a landmark (^-[_, Zj), the distance and direction of which have been 
determined by means of a radar located on board the aircraft (Fig. 
2.57) . 

It is clear from the figure that the orthodromic coordinates 
of the aircraft will be equal to 

Jg^ = X-L - i? cos e ; 

Zg = Z-|_ - fl sin e , 

where E is the distance to the landmark and 9 is the path bear- 
ing of the landmark (the angle between the given line of flight 
and the direction of the landmark). 



a) b) 

(D .V 5 ® 

^ ^ ® h 



tv 



mm 



Fig. 2.54. Keys for Determining the Distance Covered on 
the Basis of the Ground Speed and Time, Using the NL-IOM. 

The difference in the coordinates of the landmark in aircraft 
are represented by X and Z, respectively, and are found on the 
logarithmic rule (Fig. 2.58, a, b), 

238 



In aircraft navigation, a number of problems are solved which 
are connected with the distances and directions (e.g.), the check- 
ing of a course in terms of the distance covered, determination 
of the position of the aircraft by using methods of visual and 
radar measurements, and many others. The essence of the solution 
of these problems is obvious from the examples given. 



© 



tgat 



® 



X 



Fig. 2.55. 



Fig. 2.56 



Fig. 2.55. Determination of the Course Error from the Change in 
the Lateral Coordinate. 

Fig. 2.56. Key on the NL-IOM for Determining the Aircraft Course 
Error . 



For cases when the angles measured are greater than right 
angles, the sine scale 3 is numbered backwards, so that sin 180-a 
- sin a, for example: 

Ig sin 135° = Ig sin 1+5°. 

Scales 3, 4 and 5 can be used to solve special problems of 

oblique-angled triangles, e.g., the solving of speed triangles. 

The key for solving this kind of problem is given on the right- 
hand side of the scale 3. 



/232 



The theorem of signs, well known from trigonometry, deter- 
mines the relationship between the angles and lengths of the sides 
of oblique-angled triangles. In the case where the speed triangle 
is used (Fig. 2.59), this theorem has the form: 



sin US sin AW sin(AW+US) 



V 



W 



(2.58) 




Fig. 2.57. Determination 
of the Orthodromic Coor- 
dinates of the Aircraft . 



It is obvious that the relation- 
ship of Equation (2.58) is equiva- 
lent to the following: 

IgsinUS - lgu= lgsinAW-lg7 = 
lgsin(AW+US) - Igf/, 

which is expressed by the key on 

the navigational rule (see Fig. 2.60, 

a). 



239 



a) 



vi) 



Stnm-B) 



&X 



b) 



Stng 



Q) 



tl 



:SL 



Fig. 2.58. Keys for Determining the Aircraft Coordinates 
on the NL-IOM; (a) Z-Coordinates ; (b) Z-Coordinates . 

Example. MFAg = 35°, l^true = ^0° km/hr, 6 = 85°, u = 60 km/hr. 
Find the drift angle of the aircraft and the groundspeed. 

Solution. In our example, the wind angle is 



AW = 



MFAg = 85 



35 = 50°. 



Having set the slider indicator to the division represent- 
ing 400 km/hr on scale 5, and also having lined up the 50° divi- 
sion on the logarithm sine scale 3 with the same slider indicator, 
we obtain the drift angle equal to 6.5°, and a ground speed of 
440 km/hr (Fig. 2.60, b). 

This key for solving speed triangles is suitable for deter- 
mining speed and drift angle of an aircraft at known wind param- 
eters. However, it is not suitable for determining wind param- 
eters 'in measuring the drift angle -and the ground speed. 



This problem can be solved as follows. 

Let us say that on the basis of measurements, we know the 
airspeed of an aircraft, the ground speed and the drift angle, /233 
and we want to find the speed and direction of the wind (w) at flight 
altitude (see Fig. 2.59). 

It is clear from the diagram that the running component of 
the wind at flight altitude is 



u„ = W - V cos US , 



(2.59) 



while the lateral component is 



V sin US = {W-V cos US) tg AW 



(2.60) 



If we consider that the drift angle of the aircraft rarely 
exceeds 15° , and the cosine of the angle of drift is practically 
always close to unity. Formula (2.60) can be written as follows: 



te AW = 



7 sin US 
W - V ' 



However, since 



240 



then we have the ratio 



V sin US - ({/-7) tg AW, 



sin US _ tg AW 
\J - V ~ V 

which can be used as a key on the slide rule (Fig. 2.61, a). 





Fig. 2.59. Navigational 
Speed Triangle. 



Example . 
520 km/hr; US 
angle . 



V 



. L w c w J. a, o. 

ree but with 
^ key which 



= 450 km/hr; W = 
+10° . Find the wind 



Solution. The difference be- 



solution. The difference b 
tween the ground speed and airspeed (,W-V) is equal to 70 km/hr. 

If we set this value on scale 5 opposite 10° on scale ^■ (Fig. 

2.61, c), we will find the wind angle to be equal to 48°. The wind 

speed is found with the aid of a key which is described in the sine 
theorem (Fig. 2.61, d). 



Answer. 



105 km/hr. 



The fixed scale on the ruler 6, like scale 5 , is a scale of 
logarithms of linear values, but the scale is twice that used on 
first five scales _. - 



ms of linear values, but the scale is twice that used on 
t five scales. Therefore, when comparing any of the fir 
les to the fixed scale, a number is obtained on the latt 
garithm is equal to half the logarithm of the numbers of 
t five s cales . 



is as caL_ _ _ 

that used on 

f the first 

er 



the 

five s ca 

whose logax--L L 11111 xo c^u 

the first five scales. 

Example . In setting the marker of the slider to the number 
400 on scale 5 or 1, this marker shows half the log of 400 on the 
sixth scale, which corresponds to the square root of 400 or 20. 

If the desired number is set on scale 6, we will obtain num- 
bers on scales 5 and 1 whose logarithms are equal to twice the loga- 
rithm of the given number, thus corresponding to that number raised 
to a power of two. 

The turn radius of the aircraft with a given banking angle 
(g), as we know, is determined from Formula (1.6). 



/234 



i? = - 



g tgf 



241 



Therefore, the problem of determining the turn radius is solved 
by means of scales H, 5 and 5: 

Igi? = 21gF - Igg- - Ig tgg. 

Therefore, in solving this problem, it is necessary to have 
the logarithm of the square of the speed and to set it on scale 
6. The logarithm of the tangent of the banking angle is calculated 
with the aid of scale ^■ . 



a) 
©US 



©" 



AW AW+US 



w 



b) 

(Dsy 



® 



so 



50- 



56.5 



m 



Ada 



Fig. 2.60. Calculation on the NL-IOM: (a) Key for Solv- 
ing the Navigational Speed Triangle; (b) Solution of 
Navigational Speed Triangle . 

If we consider that in order to determine the turning radius, 
the airspeed of the aircraft must be expressed in m/sec and not 
in km/hr, as we did on scale 6, and also that it is necessary to 
take into account the acceleration due to gravity g, we have a marking 
i? on scale 4 which corresponds to the logarithm of the number 

5 = 0.00787, 

3^62.9,81 



i.e.. Formula (1.6) assumes the form: 

0,00787^2 



^ = - 



tgP 



or 



lg;? = 21g K+ Ig 0, 00787 -igigp, 



which corresponds to the key for the navigational slide rule which 
was shown in Figure 2.62, and which is found at the beginning of 
the third scale of sines. 

The last scale on the slide rule NL-IOM is the scale la, which 
is intended to determine the turning time (tp) of the aircraft at 
a given angle ( UT ) at a known turning radius (i?) and flight speed 
(F). This scale is a scale of logarithms for the arc of the circum- 
ference, relative to the radius of turn of the aircraft. 



Obviously, the turning time of the aircraft at the given angle /235 
will be 

IkR UT 



'" = ■ V 



360 



(2.61) 



242 



In this formula, the value 2tt/360 is a constant multiplder . 

In order not to have to calculate it each time, scale la is set 

to the value of the logarithm of this multiplier at the left-hand 
side . 



After dropping this multiplier. Formula (2.61) assumes the 



form : 



<p = 



RUT 



or 



ig'p = ig>? + igUT-ig v^. 



a) 
© 5inUS 



w-c 

c) 

© w- 



© 



70 



tgAW^ 



4/' 



isa 



© tgUS 


tg AW 




V 


® ,0- 


',1' 



© 



m 



iSO 



Fig. 2.61. Calculation on the NL-IOM: (a,b): Keys for 
Determining the Wind Angle; (c,d): Determining the 

Angle and Speed of the Wind. 

which can be expressed on the rule scales by a key shown in Fig. 
2.62, b . 

Example. R = M- . 5 km, V = HOO km/hr, UT = 90°. Find the turn- 
ing time of the aircraft . 

Solution. See Fig. 2.62, c. Answer. *_ = 6 4- sec. 

On the back of the rule are scales for making methodological 
corrections in the readings of navigational instruments (altimeters, 
airspeed indicators, outside-air thermometers). 

Adjustable scale 7, with a movable diamond-shaped index and 
the adjacent scales (fixed scale 8 and movable scale 9) are intended 
for making corrections in the readings of barometric altimeters 
in case the actual mean air temperature does not agree with the 
calculated temperature obatined when adjusting the apparatus. These 
corrections can be made with the formula 



Igff 



corr 



1 H ^ ^ mst 
Ig 2 ^ T 



243 



According to this formula, the adjustable scale 7 is a scale 
of logarithms T q+T u/ 2 . For convenience in use, the logarithms 
Tq + Tjj/2 on the rule are marked tg+t. The arithmetic effects of 
converting temperatures from the centigrade scale to the absolute 
scale and their division in half are taken into consideration in 
the design of the scales in such a way that it is not necessary 
to make them each time during the flight. 



/236 



a) 



B 



b) 








40 



V/Sff 
90 (fa) 







Y-W 

I r 



UT 



© 



ip 



© V 



Si 



Fig. 2.62. Calculation on NL-IOM: (a) Key for Determin- 
ing Turn Radius. (b) Key for Determining Turn Time. (c) 
Determination of Turn Time. 

Fixed scale 8 (corrected altitude) is simply a scale of log- 
arithms of altitude, while the movable scale 9 (instrumental alti- 
tude) is a scale of logarithms of altitude, divided by the aver- 
age calculated temperature obtained for each altitude, i.e., 



H 



lg7 



inst 



av . c , 



^ mst 



IrT 



av . c . 



The key for solving problems by introducing methodological correc- 
tions to the readings of the altimeter are shown in Figure 2.63, 
a . 

Example. The flight altitude according to the instrument is 
^inst = 6000 m; tjj = -35°. Find the flight altitude corrected for 
the methodological error. 

Solution. The actual temperature for a zero altitutde is de- 
termined from the temperature gradient equal to 6.5 deg/km: 



to = t^ + 6.5 



km 



■35 + 6.5-6 = +4° , 



so that ta+i 



O'^'^H 



•31° . 



If we set this temperature value on the slide rule (Fig. 2.63 



b), we will obtain H 



corr 



5.74 km, 



To introduce corrections in the readings of the altimeter at 
flight altitudes above 12 km, we use movable scale 10 with the adja- 
cent fixed diamond-shaped index, as well as the adjacent scales: 
fixed scale m for the corrected altitude and speed, and fixed 



21+4 



scale 15 for the instrumental altitude and speed. 

Corrections to the readings of the altimeter at flight alti- 
tudes above 12 km are made by Formula (2.36). 

Expressing the altitude in km, this formula can be written /237 
as follows: 

lgiH^^^^-11) = IgT^ - Ig 216.5 + Ig(ff^^^^-ll). (2.62) 

av 

Adjustable scale 10 is a scale of logarithms ( Ig3'jy^^-lg216 . 5 ) . 
Scales 14- and 15 are scales of logarithms (H-ll km), so that they 
are simple, unique logarithmic scales -on which we can carry out 
multiplication and division of numbers, but with additional numbers 
which are shifted by 11 km to calculate altitude. 

a) b) 

Fig. 2,63. Calculation on NL-IOM: (a) Key for Introducing 
Methodological Correction in Altimeter Reading. (b) Deter- 
mination of Correction for Measured Flight Altitude. 

In accordance with Formula (2.52), the key for introducing 
corrections in the readings of the altimeter at flight altitudes 
above 12 km is shown in Figure 2.64 a. 

Example, ^inst ~ 1^ ^^'■' '^H = -50°. Find ^corr* 

Solution. See Figure 2.64, b. Answer: ^corr ~ 14,400 m. 

Note. Since the altitude of the tropopause at middle latitudes 
is not exactly at an altitude of 11 km, but can change within limits 
of 9-13 km, after solving the problem by means of the key shown 
in Figure 2.63, b, the flight altitude must be corrected for the 
additional correction AH - 900 + 20 {tQ+tj^) which is shown on the 
rule at the right-hand side below scale 14. 

a) b) 

®(^ "corr® ® -so' M,« ® 



^ ''■inst® ^ " ® 

Fig. 2.6tt. Calculation on NL-IOM: (a) Key for Intro- 
ducing Correction in Flight Altitudes above 12,000 m; 
(b) Determination of Correction for Flight Altitude 

above 12,000 m. 



21+5 



The methodological corrections due to the failure of agree- 
ment of the actual air temperature with a calculated value are made 
to calibrate the speed indicator (type "US") with the aid of Formula 
(2,53). 

In accordance with this formula, the scale IM- on the ruler for 
log7^jP^g and scale 15 for log V'j^jjg-i- are purely logarithmic scales 
of linear values. Adjustable scale 11 (temperature for speed) is 
a scale of logarithms 

ylg(273" + <//). 

while adjacent to it is fixed scale 12 (instrument altitude alti- /238 
tude in km) with a scale of logarithms 

-— lg288 + 2,628 lg(l -J- 0,0226//). 



The key for introducing corrections in the readings of the speed 
indicator "US" is shown in Figure 2.65, a. 

Exam-pie. t^ = -30°, ^inst = '^ ^^' ^inst - ^^° km/hr. Find 
the airspeed. 

Solution: See Figure 2.65, b. Answer: 638 km/hr. 
® t^ V,coTr ® -3S' SJ8 ® 



® "'xn^X ''inst ® 



45P 



QS) 



Fig. 2.65. Calculation on NL-IOM: (a) Key for Introducing 
Correction in Readings of Type "US" Speed Indicator; (b) 
Determination of Correction for Reading of Type "US" Speed 

Indicator . 

For speed indicators of type "CSI", the corrections given above 
are found by Formula (2.54 a). 

It is clear from this formula that fixed scale 11 and movable 
scale 15 (for ^inst^ ^"^ ■'-^ (for ^corr^ will be the same for the 
speed indicators of types "US" and "CSI". 

Instead of fixed scale 12, we can scale 13 on speed indicators 
of type "CSI", which is a scale of logarithms 



1 



=- lg(288 - 0.0065 H. ^) 
2 ^ mst 



The key for introducing corrections in the readings of these 
indicators is shown in Figure 2.66, a. 



21+6 



Example. H = 10 km, tjj - -45°, Vqqj = 800 km/hr. Find the 
corrected airspeed. 

Solution: See Figure 2.66, b. Answer: 808 km/hr. 

Rule scale 16 is set up according to the formula 

1/2 



M^K 



26000 



and is used for introducing corrections into the readings of the 

thermometer for the outside air, type "TUE". This same scale can 

be used at subsonic airspeeds, and the error will not be greater 
than 1-2° for the type "TNV" . 

In practice, the front side of the navigational slide rule 
NL-IOM can be used to solve a number of other problems, the keys 
for whose solution are directly dependent on the nature of the prob- 
lem. 

One example of such a problem is the determination of the de- /239 
viation angle of the meridians between two points on the Earth's 
surface . 

The angle of deviation of the meridians can be determined by 
Formula (1.82). 

Obviously, this problem can be solved on a ruler by means of 
a key shown in Figure 2.67, a. 

a) b) 

® t„ Vco:^'g) @ -f SOS ® 

Fig. 2.66. Calculation on NL-IOM: (a) Key for Introducing 
Correction in Reading of Type "CSI" Speed Indicator; (b) 
Determination of Correction for Reading of Type "CSI" Speed 

Indicator . 

The scales on the back of the ruler can be used to solve some 
other problems. For example, movable scales 14 and 15 are the ones 
most suitable for multiplication and division of numbers. 

Scale 14 is marked off with the following values: AM (Ameri- 
can statute mile, equal to 1.63 km); NM (nautical mile, equal to 
1.852 km), and foot (equals 32.8 cm). These markings are used for 
rapid conversion of measurements from one system to another. 



247 



a) h) 



i (A,-A,l (g) JIS S8S 

Fig. 2.67. Calculation on Nl-IOM: (a) Key for Deter- 
mining Angle of Deviation of Meridians; (b) Conver- 
sion of the Length of the Arc of the Orthodrome into 

Kilometers . 

Example . Convert the length of the arc of the orthodrome 5°16' 
to kilometers . 

Solution. 5°16' = 316 NM (nautical miles). 

Having set division 100 on scale 1^ on the navigational slide 
rule opposite 316 on scale 15 (Fig. 2.67, b), we obtain the answer 
(585 km) . 

On scales 14 and 15, by using the settings of scales 11 and 

12, we can solve problems in determining the Mach number at a known 

airspeed and air temperature at a flight altitude, or determine 

the airspeed at a given Mach number and air temperature. 

Therefore, the speed of sound in air is found by the formula 



M = 



a = 20, 3 1/273° + %, 
V' true V^true 



<* 20,31/273° + % 



or /240 

'g^ = 'gVue'220,3— |-lg(273°.+ %). ^^^^^^ 

Scales 14 and 15 are scales of log V, fixed scale 11 is the 
scale of 1/2 log (273°+*^), and fixed scale 12 is a scale of 2.628 
log (1-0.02265), which is movable relative to scale 11 to the value 
l/21og288. 

H 1- 



@ J,25 277,5 M UM @ 

Fig. 2.68. Determination of Mach Number on NL-IOM, 



248 



In order to get log20.3 from the value 2.628 log (l-0.0226ff), 
it is important to replace H by a value of 3.25 km. Therefore, 
if we find an altitude of 3.25 km and set it on fixed scale 12, 
we will obtain the key for solving the problem with a certain M 
number (Fig. 2.68). 

Obviously, the value M = 1 corresponds to the airspeed (in 
km/hr) which is equal to the speed of sound. 

To determine the speed of sound in m/sec, it is necessary to 
set the value of 0.2775 (1/36) on scale 15. If we use the rectang- 
ular index with the marking of 1000 for M = 1, then division 0.2775 
will correspond to the number 277.5. 



Note. In general, for converting zero altitude, correspond- 
ing to 1/2 log288, to the value of log20.3, it is necessary to shift 
it to the right to the value 2.51 km, and the functions of scales 
14 and 15 in the key shown in Figure 2.100 will change places. Then 
Formula (2.63) will be valid. 




The fact that the numbers 2.51 (with a shift to the right) and 
3.25 (with a shift to the left) are not equal is explained by the 
fact that zero altitude under standard conditions does not corre- 
spond to zero temperature but to +15°. Therefore, to make zero 
temperature match division H - 3.25, the scale must be moved by 
an amount such that it lines up with the marking H = -2.51 km. 



249 



CHAPTER THREE 

AIRCRAFT NAVIGATION USING RADIO-ENGINEERING DEVICES 

1. Principles of the Theory of Radi onavi gational 

Instruments 

Geotechnical methods of aircraft navigation, although they /2'4l 
form the basis of the complex of navigational equipment on an air- 
craft, do not permit a complete solution of the problems of air- 
craft navigation when there are no terrestrial landmarks or when 
the latter are invisible. 

The principal reason for this is the variation of the wind 
at flight altitude, which means that the flight cannot be maintained 
for a significant period of time without checking the distance and 
direction of the path being followed. 

Astronomical means, however, are not always helpful in deter- 
mining the location of the aircraft, since the heavenly bodies are 
just as invisible as terrestrial landmarks when flying in clouds 
or between cloud layers. In addition, in order to determine the 
location of the aircraft, it is necessary to see at least two luminar- 
ies in the sky simultaneously, which is not always possible under 
normal flight conditions . 



sary to seek new methods of reliably 

anv Dhvsical and geograph- 

dence upon meteor- 
t of radio-engin- 



All radio-engineering devices for aircraft navigation use the 
properties of the propagation of electromagnetic waves in the Earth!s 
atmosphere to varying degree.s . 

We know that the phase velocity of the propagation of wave 
energy in dielectric media is 




Cl= ,r 



V^" 



250 



where c i is the rate of propagation of electromagnetic waves in 
the medium, a is the rate of propagation of electromagnetic waves 
in a vacuum, y is the magnetic permeability, and e is the dielec- 
tric constant. For a vacuum, p=e = 1. 



/242 



In addition to the phase propagation rate of electromagnetic 
waves, there is also a group propagation rate of electromagnetic 
energy . 

In a vacuum, the phase and group propagation rates for elec- 
tromagnetic waves are the same in all cases. 

In dielectric media, especially in solids, liquids, and (to 
a much smaller degree) gases, the phase propagation rate depends 
on the frequency of the oscillatory process. This is explained 
by the inertia of the dielectric medium, i.e., the dielectric perme- 
ability of the medium depends on the oscillation frequency. 

The dependence of the phase propagation rate upon the oscil- 
lation frequency is called dispersion. If the waves propagate in 
an electromagnetic medium with different frequencies, their phase 
rate may not be the same. In this case, the total energy of the 
waves will be maximum at those points in space where the phases 
of the waves are closest to coincidence. In addition, there will 
be points where the total energy of all the waves will be equal 
to zero, i.e., where the positive phases of the waves will be bal- 
anced by the negative ones. 

The points with maximum total energy are called centers of 
wave energy. The rate at which the centers of wave energy move 
in space is the group rate of the waves . 

The group rate of propagation of electromagnetic waves in space 



'^g-r- 



c, — 



rfc. 



where Cgp is the group rate, oj is the average spectral frequency, 
and Ci is the average phase rate of the spectrum. 

It is clear from the formula that with positive dispersion, 
the group rate of the waves exceeds the phase rate of their prop- 
agation . 

Wave Polarization 

Figure 3.1 is a graphic representation of a propagating elec- 
tromagnetic wave in the horizontal plane as a function of the vertical 
open circuit. 



251 



In this case, the vector of the electrical field, and there- 
fore the displacement currents, will coincide with the direction 
of the dipole of the circuit (dipole open antenna). The plane of 
the vector of the magnetic field coincides with the horizontal plane. 

Obviously, the electromagnetic wave is a transverse wave, i.e. , /2H3 
the amplitudes of the oscillations of the electric and magnetic 
fields are located at right angles to the direction of wave propa- 
gation . 

The direction of the plane of oscillation of the electric field 
is called the vector of wave polarization. In our sketch, we have 
electromagnetic waves with a vertical polarization vector. 

In receiving electromagnetic waves, it is important to be sure 
that the direction of the dipole of the receiving circuit coincides 
with the vector of wave polarization. In this case, the oscillations 
of the electric field and the axis of rotation of the magnetic field 
coincide with the direction of the dipole, and both of these factors 
will bring the electromagnetic force (and consequently the conduc- 
tivity currents) to the receiving antenna. 



isophasal circles 




Fig. 3.1. Propagation of an 
Electromagnetic Wave from a 
Verticle Dipole. 

to the transmitting antenna, 
ization vector of the waves. 



If the waves are vertically 
polarized and the receiving antenna 
is located in a horizontal position, 
no emf will be produced in the 
dipole . 

With the dipole in a horizontal 
position, the electromagnetic waves 
reaching the antenna will have 
a horizontal vector of polariza- 
tion. In this case, the receiv- 
ing antenna must be horizontal; 
in addition, the direction of the 
antenna in the horizontal plane 
must be perpendicular to the line 
. , it must coincide with the polar- 



The circles in Figure 3.1 join points in the horizontal plane 
which have identical phases for the electromagnetic waves. These 
circles are called isophasal. 

From the viewpoint of the receiving antenna, the isophasal 
circles (and the isophasal spheres in the propagation area) are 
the directions of the wave front. 



252 



Propagation of Electromagnetic Oscillations in 
Homogeneous Media 

In order to make use of the principles of design of various 
transmitting and receiving radio navigational instruments, it is 
necessary to become acquainted with the characteristics of the prop- 
agation of electromagnetic oscillations in inhomogeneous conduct- 
ing and nonconducting media. 

Electromagnetic wave processes in dielectrics constitute the /244 
conversion of the potential energy of the electrically deformed 
medium to the kinetic energy of displacement currents and vice versa 
(the kinetic energy of the field into the potential deformation 
of the medium ) . 



ation is not 



)f dielectric materials, polari z,ci l j-uu j-s u^ 
because wave energy is prop 
^+-nr^-n= A decrease 



For the majority of dielectric m 

related to absorption of wave energy, .. _ . . _ __ 

agated practically without losses in all directions. A decrease 
in the oscillation power with distance takes place due to the fact 
that the wave energy fills an increasingly large volume, which (as 
we know) is proportional to the cube of the radius of the sphere 
whi ch it f i lis . 



we know) is proportion 
which it fills 

Significant losses in wave energy can occur in solid dielec- 
trics with polar molecules. In this case, the polarization is not 
related to elastic deformation but to the motion of molecules, which 
causes a conversion of wave energy into heat. 

In conducting media, the electromagnetic waves carry alter- 
nating conductivity currents. This means that conductors always 
undergo absorption of wave energy and its conversion to heat. 

Thus, the propagation of wave energy in media, exhibiting both 
electronic and ionic conductivity, is practically possible to a 
slight depth which depends on the conductivity of the medium and 
the frequency of the oscillations. The higher the oonductivity 
of the medium and the greater the frequenoy of osoillatiorij the 
shatlower the depths to which the oscillations witt propagate . 

Since the propagation rate of electromagnetic waves depends 
on the dielectric and magnetic permeability of the medium, and the 
electronic or ionic conductivity of media can be assumed to be a 
very high (approaching infinity) dielectric permeability, the concept 
of optical density of media has been introduced. 

The minimal optical density (equal to one) is possessed by 
a vacuum (where the propagation rate of the waves is equal to c) . 
The optical density of all other dielectrics is greater than unity. 
In ideal conductors, the optical density is equal to infinity (the 
propagation rate of electromagnetic waves is equal to zero). 

In portions of a medium with varying optical density, electro- 



253 



magnetic oscillations change the direction of their propagation. 
The change in direction of propagation of electromagnetic waves 
on the surfaces of particles of the medium with different optical 
density is called refraation of rays. In addition, under certain 
conditions , there is reflection of waves from the surfaces of the 
sections. The coefficient of reflection depends on the difference 
between the optical densities of the media, the frequency of the 
oscillations, and the angle of incidence of the wave. 

When the path of a wave (propagation direction) runs from a /245 
less dense medium to a more dense one, with a certain angle of inci- 
dence to the surface, there may be no separation of the reflected / 
wave. Such an angle is called the angle of total intevnal vefteo- 
tion of the denser medium. If the medium with the greater optical 
density is a conductor, irreversible absorption of wave energy may 
take place in it (conversion of wave energy into heat). 

With the gradual change in the optical density of the medium, 
there is a continuous refraction (bending) of the line of propaga- 
tion, called Tadiorefractton. 

The optical inhomogenei ty of a medium characterizes the prop- 
agation characteristics of waves of different frequencies in the 
Earth's atmosphere. 

All harmonic oscillations in a medium are characterized by 
an oscillation frequency ( o) ) and an amplitude oscillation (£") . 

If we say that the amplitude oscillation is the maximum value 
of the intensity of the electrical field, then at any fixed point 
the oscillation process will satisfy the expression: 



where 



E = Eq sin(a)t + (j) ) , 
is the initial phase of oscillation 



The derivative of the field intensity with time will charac- 
terize the magnitude of the displacement current 



dis 



dE 
-dt 



■eE r\Msin(.b)t + (^ ) . 



while the second derivative will express the acceleration of the 
displacement current 

J^. = eEnixi^ cos ( wt + d) ) . 
dis ^ 

The distance between the two closest points in space which 
lie along the line of propagation of the wave front, in which the 
wave phase is identical, are called the wavelength (A), which is 
equal to Cj/u 



254 



Electromagnetic waves can be subdivided into four groups on 
the basis of their propagation characteristics in the Earth's atmo- 
sphere . 

1. Long waves, from 30,000 to 3000 m (10-100 kHz). These 
waves have a surface type of propagation. Conducting media such 
as the Earth's surface and the upper ionized layers of the atmo- 
sphere have a deflecting effect upon them. 

2. Medium waves, from 3000 to 200 m (100-1500 kH25) have a 
complex type of propagation. In the day, when the ionized layers 
of the atmosphere are lower, the type of propagation is superficial 
as in the case of long waves; at night, the medium waves have both 
a surface and spatial type of propagation. 



3. Short waves J, from 200 to 10 
spatial type of propagation. 



(1500-30,000 kHz) have 



/246 



4. Ultra-short waves, less than 10 m, have a radial type of 
propagation. They can be reflected from conducting layers on the 
Earth's surface, but only under certain conditions can they be re- 
flected from the ionized layers of the atmosphere. Therefore, it 
is thesfe waves which are used within the limits of geometric visi- 
bility of objects. The resistance to these waves on the Earth's 
surface is insignificant. 

From the point of view of electrical conductivity and relief, 

the Earth's surface has a complex nature which depends on the time 

of year and weather conditions. The ionized layers of the atmo- 
sphere also have a varying nature. 

The ionized D layer, which is closest to the Earth's surface, 
is only observed in the daytime and depends on the time of year, 
time of day, and geographical latitude; it may appear at heights 
from 50-90 km. This layer has an effect on the propagation of long 
and medium waves. The critical frequency of the layer is . M- MHz 
(750 m). Waves with frequencies higher than the critical are not 
reflected from the layer. 



at a 

reta 

this 

to 

of m 

boun 

hour 

mum 

meas 

and 

in t 



Above 
heigh 
ins it 
layer 
.9 MHz 
edium 
dary w 
s , the 
ioniza 
uremen 
conseq 
he hor 



this is the E layer, whose ionization maximum is reached 
t of 120-130 km. This layer is the most stable one and 
s effect both day and night. The critical frequency of 

with maximumi llumination is 4.5 MHz; at night it drops 
this layer has a maximum effect on the propagation 
and intermediate waves (the short waves at the spectral 
ith the medium waves). During the evening and morning 

layer changes its parameters so that the surface of maxi- 
tion decreases. This leads to errors in radionavigation 
ts , since it reverses the vector of the wave polarization 
uently the direction of propagation of the wave front 
izontal plane. 



255 



The third ionized layer (.F) is the most unstable one both in 
terms of time of day as well as season of the year. Its average 
height is 270-300 km. During the daytime in summer, this layer 
divides into two parts (Fi and F 2) • In addition, the F layer shows 
some shifting in homogeneities, which make it difficult to predict 
the propagation conditions for electromagnetic waves. The F layer 
has an influence on the propagation of short waves. 

It should be mentioned that the medium and short waves are 
reflected both from the ionized layers of the atmosphere as well 
as from the Earth's surface, so that they may undergo multiple reflec- 
tion. 

All of this combines to give us the complex picture of the 
propagation of electromagnetic waves in the Earth's atmosphere, 
which must be taken into account in radionavigat ional measurements. 

The peculiarities of propagation of electromagnetic oscillations 
in a conducting feeder channel in receivers and transmitters include /2M-7 
the following. 

Unlike constant and low-frequency alternating currents, high 
frequency currents propagate mainly along the surface of a conductor, 
since the reaction of the magnetic field within the conductor is 
greater than on its surface (the skin effect). This causes all 
high frequency conductors to be constructed with an eye toward increas- 
ing the surface, e.g., tubular and multiple-filament (stranded wire). 

However, these measures are insufficient for waves in the cen- 
timeter range. It is much better to use hollow conductors for these 
waves, called wave guides (Fig. 3.2), 



■I 




R% 



Fig. 3.2. Propagation 
Electromagnetic Waves 
Along a Wave Guide. 

the surfaces of which i 
the vertically polarize 
walls of the box and be 
opposite wall also with 
reflection of the waves 
lations . In this case, 
along which the waves w 
any resistance. 



In a h 
propagation 
the vector 
— in the dire 
vector of p 
of of the wave 
gation dire 
are the s am 
type wave g 
s measured in whole 
d wave (striking th 
ing reflected from 
a whole number of 
will take place in 
the box-type wave 
ill propagate thems 



omogeneous medium, the 

rate of the field along 
of the wave polarization, 
ction of the perpendicular 
olarization, in the plane 

front, and in the propa- 
ction of the wave front 
e . Therefore , in a box- 
uide , the distance between 

numbers of half waves, 
e top and bottom internal 
them) will strike the 
half waves. Consequently, 

resonance with its oscil- 
guide will act as a channel 
elves practically without 



If the distance between the walls is equal to a whole, odd 
number which is one-quarter of the wavelength, then (as is easily 



256 



seen) the reflections from the walls of the wave guide will take 
place each time in opposite phase with the oscillations. In this 
case the wave guide will have infinite resistance, and the wave 
energy will not be propagated in it. In Figure 3.2, the vector 
of wave polarization must be turned 90° to accomplish this. 

Principles of Superposition and Interference of Radio Waves 

The principle of superposition is applied to wave processes, 
i.e. 5 each of the wave processes acts independently of other processes 
which are taking place in the medium or circuits. 

At the same time, the results of different processes can be 
summed by means of a simple superposition of oscillation vectors. 
If the vectors of two coherent (coinciding in frequency) processes 
such as the oscillations in the intensity of a field or displace- / 2 M-8 
ment currents, are equal in amplitude and coincide in phase, the 
total amplitude of the oscillations will be doubled. Under these 
conditions, if the oscillations are in opposite phase, the total 
amplitude of the oscillations will be equal to zero and no method 
will suffice to detect the presence of the wave processes involved. 

Summing of the results of the processes in opposite phase is 
called wave -intevfeTenoe . The case in which the result of summing 
of the oscillations is equal to zero is called total -IntevfevenGe . 

The properties of interference of radio waves are widely em- 
ployed and r adionavigat ional devices both in receivers and trans- 
mitters, especially in measuring the direction of an object. 

Principle Characteristics of Rad i on a v i ga t i on a 1 Instruments 

The principle characteristics of transmitting radi onavigat i onal 
instruments are the following: 

(1) The radiated power, characterizing the operating range 
of the system . 

(2) Accuracy and stability of the frequency structure, as 
well as synchronization of special navigational signals. 

As far as the antenna arrays are concerned, which incorporate 

certain characteristics for radiation of signals, we will discuss 

them under the heading of "Principles of Operation of Concrete Naviga- 
tional Instruments". 




signals are combined and an intermediate frequency is produced which 

is equal to the difference between the frequencies given above. 

In such devices, further amplification of the signal is carried out 

with a constant, lower frequency, which makes it possible to use 

amplifier devices with very high coefficients of amplification, 

as well as to ensure a high selectivity of the receiver. 

Usually, receiving radionavigational instruments fulfill two 
functions: (a) reception and amplification of the signals from 
a transmitter (b) separation and indication of measured navigational 
parameters . 

The basic characteristics of receiver navigational instruments 
are the following: 

(1) Sensitivity of the receiver which characterizes the pos- 
sible receiving range for signals from a transmitter. 



(2) Selectivity of the reception; this parameter is usually 
obtained by narrowing the frequency band which the receiver will 
pass, which usually characterizes the freedom from noise of the 
re ceiver . 



/2H9 



(3) The accuracy with which the navigational parameters are 
selected and recorded. 

Operating Principles of Radionavigational Instruments 

In accordance with the laws of propagation of electromagnetic 
waves in space, it is possible in principle to measure the follow- 
ing parameters of electromagnetic waves: amplitude, phase, fre- 
quency, and transmission time of the signal. 

According to the principle of technical operation, radionav- 
igational devices are divided into amplitude, phase, frequency and 
time devices . 

In addition, with a mutual exchange of radio signals between 
objects which have relative motion to one another, changes in the 
frequency characteristics of the signals occur which are known as 
the Doppler effect, which is used to build automatic airspeed indi- 
cators and devices for measuring the drift angle of an aircraft. 

Measurements of the parameters listed above for electromag- 
netic waves from the navigational standpoint make it possible to 
determine the following navigational elements: 



(a) The direction of the object, by means of goniometric sys- 



tems ; 



(b) The distance to an object, by means of rangefinding sys- 



tems ; 



258 



(c) The difference or sum of the distances to the object: 
hyperbolic or eliptical systems ; 

(d) Speed and direction of movement of the aircraft: auto- 
matic Doppler meters for ground speed and drift angle. 

For convenience of application, in many cases the navigational 
systems are compensated for measuring two navigational parameters 
simultaneously. For example, there are the goniometric-rangef inding 
systems, difference-rangef inding instruments, etc. 

The panoramic radar located on the ground and on the aircraft 
are goni ometri c-range finding devices with a single unit of navi- 
gational equipment. 

In studying methods of applying radionavigational systems , 
it is a good idea to classify them according to the principles by 
which the navigational parameters are measured. Therefore, the 
further subdivision of the material will be made on the basis of 
these principles. 



Radionavigational devices can also be subdivided into auto- 
matic and non- automati c . Non- automati c devices, when they consist 
of systems of ground control and apparatus aboard the aircraft, 
are called navigational systems. Automatic devices are called auto- 
matic nav-igat-ionat systems when the operation of several types of 
navigational devices is combined organically on board the aircraft. 
For example, the automatic Doppler system for aircraft navigation, 
which consists of a Doppler meter for the drift angle and the 
ground speed, course devices, and the automatic navigational instru- 
ments . 



/250 



2. GONIOMETRIC AND GON I OMETRI C-RANGE FI NDI NG SYSTEMS 

The goniometric radionavigational systems are the simples 
ones from the standpoint of technical requirements, and are th 
fore those which are most widely employed at the present time. 



lest 

ere- 




Fig. 3.3. Reception 
of Electromagnetic 
Waves by a Frame 
Antenna . 



In the majority of these systems, 
the amplitude method of measurement is 
employed, based on the interference of 
electromagnetic waves. This principle 
serves as the basis of the operation of 
ground and aircraft-mounted radio direc- 
tion finders, which are also called radio 
compas ses . 

Let us imagine a frame-type receiving 
antenna, located in a field of outwardly 
directed radio waves (Fig. 3.3). If the 
frame antenna is located relative to the 
transmitter so that the direction^, of the 



259 



propagating waves will be perpendicular to the plane of the frame, 
the left and right vertical sides of the frame will be on the same 
isophasal circle. In this case, the high-frequency currents which 
are conducted in the sides of the frame will agree in phase and 
will consequently be directed toward one another. This gives complete 
interference of the oscillations of the currents in the frame, and 
there will no reception of signals from the transmitting station. 



If the frame is turned around the ver 
so that the direction of the plane of the 
direction of the transmitting station, the 
be at different isophasal circles, maximal 
device. Thus, the currents in the vertica 
undergo a certain phase shift which will g 
of the signals from the station. The maxi 
will be observed in the case when the dist 
is equal to half the wavelength. Then the 
ical sides of the frame will be in opposit 
tudes will be added. However, this requir 
of cases) cannot be fulfilled, since the a 
too unwieldy; therefore, we use that part 
obtained with a phase shift through a smal 
angle. In these cases, the receiving fram 
turns and a radio receiver with very high 
The vector diagram of the reception direct 
will have the form of a figure eight (Fig. 



tical axis through 90° , 
frame coincides with the 

sides of the frame will 
ly distant for the given 
1 sides of the frame will 
ive maximum reception 
mum effect of the frame 
ance between its sides 

currents in the vert- 
e phase and their ampli- 
ement (in the majority 
ntenna device becomes 
of the effect which is 
1 (frequently very small) /251 
e is supplied with many 
sensitivity is employed, 
ionality of the frame 

3.4) . 



The greatest accuracy in range finding is "obtained with min- 
imum reception, while at the maximum the change in amplitude of 
the received signal is obtained by turning the frame at a slight 
angle. Therefore, range finding by means of a frame is always done 
with minimum reception or audibility of the signal. 




)a 



Fig. 3.4. Fig. 3 . 5 

Fig. 3.4. Diagram of Reception of a Frame Antenna. 

Fig. 3.5. Edcock-Type Antenna. 



260 



trom 

not 

nent 

f ram 

and 

base 

are 

for 

ment 



The 
agnet 
only 

of t 
e . I 
me cha 
d rad 
equiv 
verti 
ioned 



recei 
ic wa 
the V 
he po 
n add 
nical 
io ra 
alent 
cally 
abov 



ving frame antenna has the shortcoming that when elec- 
ves are being propagated through space, it picks up 
ertically polarized wave but also the horizontal compo- 
larization vector in the top and bottom sides of the 
ition, the frame antenna with its large dimensions 

rotation is inconvenient to use. Therefore, ground- 
ngefinding installations use special antennas which 
to a frame type in the characteristics of reception 
polarized waves, but are free of the shortcomings 
e; they are called Edcock antennas (Fig. 3.5). 



The picture shows one pair of Edcock dipoles with the coil 
of a goniometer between them. Obviously, in open dipoles, no inter- 
ference will be observed when they are located on one isophasal 
circle. However, the difference in potential at the ends of the 
goniometric coil will be equal to zero, since they are connected 
to symmetrical points on the dipole . If the dipoles are located 
on different isophasal circles, then the phase shift will disturb 
the potential equilibrium at the ends of the coil and a high-fre- 
quency current will pass through it. 

A similar pair of dipoles is mounted in the plane perpendic- 
ular to the first pair. 

The high-frequency current in the goniometer coils will depend /252 
on the direction of the transmitting station relative to the crossed 
dipoles . 

In a goniometric instrument, in addition to the two fixed dipole 
coils mounted at an angle of 90°, there is a movable searching coil, 
connected to the input circuit of the receiver. 

If the searching coil is placed in the resultant field of the 
fixed coils, the reception will be maximum; when a coil is placed 
at an angle of 90° to the resultant field, reception will be min- 
imal . 




receiver 



Fig. 3.6. Inclusion of an 
Open Antenna for Solving 
Ambiguity of Reception. 



The horizontal wires connecting 
the antenna dipoles are located as 
close as possible to one another, 
so that the electromotive force con- 
ducted in them from the horizontal 
component vector of polarization 
will be in the same phase, and their 
total interference will appear at 
the inputs in the goniometer coils. 
Therefore, the antenna does not pick 
up component waves with horizontal 
polarization, thus considerably reducing 
the range finding error for waves 
in space . 



261 



The reception characteristics of the frame antenna (includ- 
ing the Edcock type) have two signs, i.e., we have two maxima and 
two minima of audibility, so that it can be used to determine the 
direction line on which the transmitting and receiving objects are 
located, but does not solve the problem of the sides of the mutual 
position of the objects (see Fig. 3.U). 

To solve the ambiguity of reception with radio rangefinding 
instruments, an open antenna with an externally directed (circular) 
reception characteristic is used in addition to the frame antenna 
(Fig. 3.6). 

The phase of the high-frequency current in the open antenna, 
depending on the reception direction, will coincide with the phase 
of one of the sides of the frame receiver and will be in opposite 
phase with the currents in the second side of the receiver. As 
a result, the current amplitudes of an open antenna will be added 
to one-half of the figure eight of the frame antenna and will inter- 
fere with the other half of the figure eight (Fig. 3.7, a). 

In combining the characteristics of the frame and open antennas, 
we obtain a total characteristic which has the form of a cardioid. 
If we connect the open antenna and turn the frame antenna through 
90° clockwise, the maximum reception shown in Figure 3.7, b will 
shift to the upper part of the picture while the minimum will shift 
to the lower part (Fig, 3.7, c). This corresponds to one side of 
the minimum of the frame receiver being transferred to the maximum, /253 
and the second remaining minimum. 





Fig. 3.7. Diagram of Directionality of a Frame Antenna 
Combined with an Open Antenna. 

Let us suppose that we have defined a line (bearing) on which 
the transmitting and receiving points are located at minimum aud- 
ibility. After connecting the open antenna and turning the goni- 
ometer coil through 90°, we can determine the direction of the trans- 
mitter. If the audibility of the signals increases sharply, the 
transmitter is located in the direction of the upper part. If it 
remains as before or changes slightly, the transmitter is located 
at the opposite side. 



262 



The principles described above for finding the direction of a 
transmitter are used in ground radio direction-finding installations. 
In this case, the transmitter is the radio on board the aircraft. 

Ground radio direction-finders in principle can operate at 
all wavelengths. The most widely used radio rangefinders operate 
on short and ultra-short waves . 

The position of the aircraft can be determined by means of 
the ground radio rangefinder in terms of the minimum audibility 
of the signal from the transmitter located on board. In addition, 
visual' indicators are mounted on the ultrashort wave (USW) range- 
finders, such as cathode-ray tubes. 

In this case, the frame of the direction-finder or the gon- 
iometer coil is set to rotating rapidly, and the scan of the cathode- 
ray tube is synchronized with it. The amplitude of the scan is 
related in magnitude to the amplitude of the received signals in 
such a way that at minimum reception the maximum amplitude of the 
scan is observed. Then, on a scale which is marked along the periphery 
of the tube face, we can determine the direction of the aircraft 
in terms of the position of the maximum deflection of the scan. 

With a relatively low density of air motion, the ground radio 
rangefinders are a sufficiently effective and precise method of 
aircraft navigation. An advantage of ground radio rangefinders 
is the lack of a need to mount special radio equipment on the air- 
craft. The radio rangefinders and receivers which are used for / 2 5 M- 
receiving signals from ground direction-finding points mainly have 
other purposes, and their use for navigational purposes is not related 
to the increased complexity and weight of the equipment on board. 

At the same time, however, the use of ground radio direction- 
finders has a number of serious shortcomings, which have led to 
a search to find new ways of radionavigational control of flight. 

The most important of these shortcomings are: 

(a) Lack of a visual indicator on board the aircraft to show 
its position, thus reducing the ease of aircraft navigation. 

(b) A small capacity for the ground installations; at the 
same time, the radio direction-finder can only operate with one 
aircraft, which is clearly inadequate when there are a great many 
flights . 

Aircraft Navigation Using Ground-Based Radio D i rec t i on- F i nde rs 

The use of ground-based radio direction-finders can be used 
to solve the following navigational problems: 

(a) Selection of the course to be followed and flight along 



263 



the straight-line segments of a route, at the beginning or end of 
which radio direction-finders are located. 

(b) Control of the aircraft path in terms of distance. 

(c) Determination of the aircraft location on the basis of 
bearings obtained from two ground-based radio direction-finders. 

(d) Determination of the ground speed of the aircraft, as 
well as the drift angle, direction and speed of the wind at flight 
altitude . 

Usually, the international "Shch"-code is used for determin- 
ing the bearings from on board the aircraft. The crew of the air- 
craft reports its position, gives the required code for its posi- 
tion, and presses the telegraph key of the transmitter for a period 
of 20 sec. 

In recent years, both state and local civil airlines have adopted 
USW direction finders, with visual indicators. They are oriented 
according to the magnetic meridian of the location of the USW direc- 
tion-finder, and (depending on the flight altitude) are used in 
a radius of 100-200 km as a form of trace direction-finder, report- 
ing on board the aircraft the "forward" (away) and "back" (return) 
magnetic bearings of the aircraft. If the crew of the aircraft 
requests the forward true bearing, the operator of the USW direc- 
tion-finder (supervisor) calculates the magnetic declination of 
the location of the distance finder and reports the forward true 
bearing to the aircraft . 



Distance finding by means of USW distance finders with a vis- 
ible indicator is used in the course of communication with an air- 
craft, i.e., with a depressed tangent of the connected USW trans- 
mitter on board the aircraft. 



/255 



The operator of the ground radio direction-finder, after the 
required measurements, gives the call letters of the aircraft, the 
code expression for the bearing as requested by the aircraft or 
used for USW communication, and gives the magnitude of the bearing 
in degrees . 

The code expressions for the bearings in the international 
Shch code have the following meanings (Fig. 3.8): 

ShchDR: magnetic bearing from distance -finder to the aircraft, 
or forward bearing. 

ShchDM: magnetic bearing from the aircraft to the distance- 
finder (measured relative to the local meridian of the location 
of the distance-finder), or reverse bearing. 

ShchTE: true bearing from the distance-finder to the aircraft, 
or the forward true bearing. 



264 



ShchGE: azimuth of the aircraft at 
trol distance-finding station. 



a distance from the con- 




Fig. 3.8 
Bearings 



hGE 
hTE and S^ 



Code Expressions for 
1 the Shch-code. 



ShchTF: location of 
aircraft (coordinates or 

Due to the small eff 
radius of the USW rangefi 
they are not grouped into 
tance-f inding nets like 1 
or medium-wave stations, 
operate independently, an 
not give the location of 
aircraft . 

Seteot-ton of the Cours 

be Followed and Control o 

D-treation 



the 
link) . 

G ctive 
nders , 

dis- 
ong 
but 
d do 
the 



e to 

f Flight 



The selection of the course 
to be followed and flight along a s tr aight- line path segment are 
accomplished by means of periodic inquiries and determinations of 
the forward or reverse bearings of the aircraft (ShchDR or ShchDM). 

If the radio distance-finder is located at the starting point 
of a flight segment (flight from the distance-finder), then the 
ShchDR bearings are requested. When the aircraft is passing pre- 
cisely over the ground radio distance detector and follows a constant 
course for a certain period of time, the first bearing of the air- 
craft after passing over the dis tance - finder can be used to deter- 
mine the drift angle (Fig. 3.9). Usually in this case the ShchDR 
bearing will be equal to MFA , so that 



US = ShchDR 



MC . 



If the ShchDR does not correspond to the given flight path 
angle for the path segment, then the aircraft is put on the desired 
line of flight after determining the drift angle and the course 
to be followed is set so that the total of the course and the drift 
angle of the aircraft will equal the given path angle. 



/256 



It should be kept in mind that in the general case ShchDR is 
not equal to MFAg , since the former is the orthodromic bearing meas- 
ured at the starting point of the segment and the MFA is the loxodroraic 
path angle measured relative to the mean magnetic meridian: 



ShchDR 



MFA 



where Aw^^^ is the magnetic declination at the midpoint of the seg- 



ment , A 



Ml 



is the magnetic declination at the location of the radio 



distance-finder, and 



6^^ is the deviation of the meridians between 



265 



the initial and middle points on the path segment 



^ fc _^^^ 




\ VJ- -"Time of outward 

-— 1'^S \^ bearing _ 




Fig. 3.9 



Fig . 3 . 10 . 



Determination of the Drift Angle After Flying Over a 
Radio Distance-Finding Station. 



Fig. 3.9. 

f 

Fig. 3.10. Path Segment Between Two Radio Distance-Finding Stations 



In principle, orthodromic control of the path for a loxodromic 
flight is inconsistent, because in practice the course to be fol- 
lowed in a loxodromic system of path angles is selected so that 
flight takes place along the orthodrome. 

In order to maintain the given flight direction over the path 
segment with sufficient accuracy, it is necessary to note that at 
each bearing (ShchDR or ShchDM) the aircraft will be located on 
an orthodromic line of the given path and will therefore maintain 
this bearing . 

Let us explain this by a concrete example. 

We will assume that we must make a flight from a point A(A - 
105°, Af/j = -1°) to a point B(A = 115°, Aj^ = -7°) and return (Fig. 
3.10). The magnetic flight angle of the segment is 95 or 275°, 
while the average latitude of the segment is 52°. 

Obviously, for flight in an easterly direction from the dis- 
tance-finder, located at point A 

ShchDR - MFA+A,, -A„ -6 = 95-1++1-4 = 88°. 
M M 1 av 
av ^ 

For flight in a westerly direction, the ShchDM from this dis- 
tance-finder must be equal to 268°. 

For a flight in an easterly direction, the initial course of /257 
the aircraft must be set not on the basis of MFA = 95°, but from 
ShchDR = 88°. In the opposite case, the aircraft slowly begins 
to deviate from the line of the desired bearing at an angle of 7°. 



266 



Analogously, for the point B (ShchDR = 282°, ShchDM = 102°), 
the initial course must be set 7° greater than one would conclude 
on the basis of the MFA. 

Of course, it is impossible to make flights with a constant 
MFA at distances at which the magnetic direction of the flight changes 
by 14° , This example is given only to illustrate the geometry of 
the process. It would be more accurate to divide this segment into 
four parts 150 km long with the following flight angles: MFA^ = 
90°, MFA2 - 93°, MFA3 = 97°, and MFA4 = 100°. In the first two 
segments, we must use a distance finder which is located at Point 
A (ShchDR = 88°), while for the latter we must use the distance 
finder- at Point B (ShchDM = 102°). 

This division of the flight segment into parts for the case 
of a flight according to a ground distance finder is an approxima- 
tion of the initial MFA to the ShchDR of the initial distance finder, 
while the latter is approaching the ShchDM of the range finder located 
at the terminus of the flight. 

In the orthodromic system of calculating flight angles, the 
distance between the OFA and bearings ShchDR and ShchDM from one 
of the distance finders will be constant in value and will depend 
only on the meridian selected for calculating the path angles. In 
the special case when the reference meridian coincides with the 
meridian where the range finder is located, OFA will differ from 
ShchDR only in the magnitude of magnetic declination for the loca- 
tion of the distance finder: 



OFA - ShchDR + A, , 

M 



fore , in an or1 



1 uc j.ci wi-c , J.11 an orthodromic system of calculating flight angles, 
the course to be followed by the aircraft changes more rarely and 
to a much lesser degree than in a loxodromic system, but all elements 
of aircraft navigation, including the speed and wind direction, 
are determined more accurately. 



are determined more 



There 
the c 

to a mucn xesser aegree xnan 1 
of aircraft navigation, includ 
mined more accurately 

For selecting a course and maintaining the flight direction 
of an aircraft in terms of ground radio distance- finders , the method 
of half corrections is used, which consists of the following: 

Let us say that at a point position of the aircraft on the 
line of a given path, the latter is on course with a certain anti- 
cipation of drift. 

After a certain period of time, on the basis of the bearing 
obtained from the distance finder, it is found that the aircraft 
is shifting from the line of flight toward the direction of the 
wind vector. This indicates that the correction in the course which 
has been taken is insufficient. Therefore, it is necessary to return 
the aircraft at an angle of 10-15° to the given line of flight, 
and the previously employed lead in the course to be followed is 
doub led . 



267 



If in this case the aircraft begins to shift from the line 
of flight toward the side opposite the wind vector, then after the 
second aiming of the aircraft along the given line of flight, it 
is necessary to make a correction in the course which is halfway 
between the latter and the former. If the deviation takes place 
along the direction of the wind vector, the correction in the course 
must be increased. 

In addition, if the deviation of the aircraft from the line / 2ZQ 
of desired flight takes place, the difference between the latter 
and the former corrections is divided and added to the course with 
a positive or negative sign, depending on the direction of the air- 
craft deviation. 

The placing of the aircraft on the desired line of flight by 
selecting the course with all the deviations mentioned is oblig- 
atory only in a flight from the distance-finder along a forward 
bearing (ShchDR). In a flight toward a radio distance-finder along 
a reverse bearing (ShchDM), the aircraft must follow the line of 
the desired path only in the case when it is going beyond the limits 
of the established trace. With small deviations (by distances from 
the radio distance-finder of up to 200 km within the limits of 1- 
2°), it is sufficient to select the course to be followed by the 
same method of half corrections relative to the last ShchDM (reverse 
bearing), without going to the desired line of flight each time. 

The method of half corrections is the general one used for 
flight toward the radio dis tance- finder and away from it. However, 
in practical use, there are considerable differences between flight 
toward the distance-finder and away from it: 

(1) In a flight from the radio distance-finder, the drift 
angle can be measured at the beginning of the segment, while in 

a flight toward the distance-finder it can be determined only after 
selecting the course to be followed with a stable ShchDM. 

(2) In a flight from a radio distance-finder, the course to 
be followed by the aircraft must change in the direction opposite 
the change of the bearing: ShchDR increases, and the course must 
also decrease, and vice versa. In a flight toward a radio distance- 
finder, the change in the course must take place in the direction 

of the change in bearing: ShchDM increases, the course must be 
increased, and vice versa. 

(3) As we have already pointed out, a flight away from a radio 
distance-finder in all cases must be made strictly along the given 
bearing, while in the flight toward a radio distance-finder (within 
certain limits) it is permissible to select the flight to be fol- 
lowed according to the last stable bearing. 



268 



Path Controt in Terms of Distance and Determinat'lon of 

the Aircraft ' s Location 

For the purposes of controlling the path in terms of distance, 
as well as determining the location of the aircraft, we can use 
the true bearings from the ground radio distance-finder to the air- 
craft (SchTE) . 

For checking a flight in terms of distance, we usually select 
the control landmarks along the flight route and determine their 
precalculated bearings from the radio distance-finder located to 
the side of the aircraft route (Fig. 3.11). 

Three to five minutes before the aircraft reaches the control 
landmark, a series of "forward true" bearings are requested (ShchTE). 
When the bearing of the aircraft becomes equal to the calculated / 2 59 
one, the passage of the control landmark is noted. 

By using long- and medium-wave radio direction-finders, the 
location of the aircraft is determined from bearings of two or three 
mutually related ground radio direction-finders, one of which is 
the command station. 



ftrue 

,ShchTE=140° 




Upon request from the crew 
of an aircraft, with regard to 
the azimuth and distance from 
the command distance-finding sta- 
tion (ShchGE), the aircraft measures 
its distance simultaneously from 
two (three) distance measuring 
stations, while auxiliary distance 
finders report the measured bear- 
ing to the command distance station. 



Fig. 3.11. Previously Calcu- 
lated Bearing of a Landmark. 



The operator of the command 
radio dis tance - finding station 
uses a special plotting board 
to determine the true bearings 
of the aircraft with the aid of movable rulers with their centers 
of rotation at the points where the radio dis tance -finding stations 
are located; having measured the distance to the aircraft (the points 
of intersection of the bearings), the operator transmits to the 
crew of the aircraft its position (the true direction and distance 
from the command radio dis tance- finding station). 

If the crew of the aircraft desires to obtain data regarding 
the location of the aircraft in different forms (e.g., geograph- 
ical coordinates or relationship to some landmark), they must ask 
for the ShchTF bearing from the command radio distance-finding sta- 
tion . 



269 



Determination of the Ground Speed, Drift Angle, and Wind 

The ground speed of an aircraft can be found by using ground 
radio distance-finders as well as other non-automatic radionaviga- 
tional devices during flight on the basis of the distance covered 
by the aircraft between two successive indications of its position 
(LA): 

W = - . 

The successive landmarks for the LA are the points at which the 
aircraft passes over previously calculated bearings along the route 
or locations for the aircraft marked on a map which were obtained 
from the command distance-finders upon request of bearings ShchGE 
or ShchTF. 

The drift angle can be determined in three ways with the aid 
of ground radio distance-finders: 

(1) The difference between the "forward" bearing (ShchDR) /260 
and the course of the aircraft after passing over the radio dis- 
tance-finding station: 

US = ShchDR - MC; 

(2) By the difference between the path angle of the flight 
and the course of the aircraft after selecting a stable "forward" 
bearing (ShchDR) or "reverse" ShchDM: 

a = ^ - y , 

where a is the drift angle of the aircraft, ^i is the path angle 
of the flight, and y is the course of the aircraft. 

(3) On the basis of the path angle and the mean course of 
the aircraft between successive indications of the PA {A and B): 



secona ana rnira mernoas give exacT resuxTs on±y in 
of the path segment, i.e., when crossing the meridi; 
to which the path angle of the segment is measured, 
ning and end of the segment, the errors are maximum, 

In the orthodromic system of calculating path angles and courses, 
the accuracy of determining the drift angle is approximately the 
same for all three methods . 

The speed and direction of the wind at flight altitude is 



270 



determined with the aid of ground radio distance-finders in two 
ways : 

(1) According to the ground speea of the aircraft, the air- 
speed, and the drift angle. To solve this problem, we can use a 
key oi the navigational slide rule for determining the wind angle 
(Fig, 3.12, a) and for determining the wind speed (Fig. 3.12, b). 



of t 

firs 

bear 

aire 

lati 

the 

on t 

the 

the 

on t 

time 

ampl 



(2) 
he a 

t iQ 

ing 
raft 
on i 
calm 
he c 
aire 
calm 
he b 
ove 
e : 



By 
i re r 

cati 
is m 

is 
s ma 

pos 
hart 
raft 

poi 
asis 
r a 



the 
aft 

on o 
arke 
f lyi 
de ( 
itio 

wit 

in 
nt a 

of 
give 



differ 
on the 

f the a 
d on th 
ng from 
accordi 
n of th 
h simul 
terms o 
nd the 
the She 
n path 



ence 
flig 

ircr 
e fl 
the 
ng t 
e ai 
tane 
f th 
s eco 
hGE 
segm 



between 
h t chart. 

aft on th 
ight char 
first lo 
o the ave 
rcraft is 
ous reque 
e ShchGE 
nd pos it i 
b earing , 
ent . Let 



the ac 

This 
e basi 
t . Du 
cation 
rage c 

deter 
St of 
or Sh c 
on of 
is the 

us CO 



tual 

met 
s of 
ring 
, th 
ours 
mine 
the 
hTF. 
the 

win 
ns id 



and calm 

hod means 

the Shch 

the time 

e calm pa 

e , airspe 

d, and al 

second po 

The vec 

aircraft , 

d vector 

er the fo 



coord i nates 

that the 
GE or ShchTF 

that the 
th calcu- 
ed and time ) , 
so entered 
sition of 
tor between 

determined 
for the flight 
llowing ex- 



After 2 4- min of flight between two successive locations, the 
wind vector is equal to 14-0° in direction and 30 km in magnitude. 

If we divide the modulus of the wind vector by the flight time /261 
in hours (0.4), we will get the wind speed 



u 



30 :0 .4 = 75 km/h . 



The first method of determining the wind is the one most widely 
employed. However, on large passenger aircraft with automatic navi- 
gational indicators on board (e.g., NI-50), by means of which auto- 
matic quiet calculation of the aircraft path can be carried out, 
the second method is the most suitable and precise. When this is 
done, it is no longer necessary to plot the wind vector on the flight 
chart . 



a) 



® a.nUS 
^^ 



tgAwi) 



b) 

® Sin us 



® 



Sin AW 



resting point 



6i 



Xaw 

77-^LA 



Fig. 3.12. ■' Fig. 3.13. 

Fig. 3.12. Keys for Determining the (a) Wind Angle and (b) Wind 
Speed on the NL-IOM. 

Fig. 3.13. Determination of the Wind by the Difference in the Coor- 
dinates of the Calm Point and the Location of the Aircraft. 



271 



It is clear from Figure 3.13, that 

AZ ^ AZ 

tgAW = -T-rr ; Ut = — : — T-rr 
* hX smAW 



where 

AZ = Z^ . - Z ; 
LA rp 

LA rp 

If we know the distance of the orthodromic coordinates of the 
location of the aircraft and the calm point, this problem is easily 
solved on the navigational slide rule using the following key: 

For determing AW (Fig. 3.1M-, a), and for determining ut (Fig. 
3 . 14, b ) 



) b) 

® tg AW y ® s.nAV^ ^ 

(T) &i AX /iz 



ut 

Fig. 3.14-. Determination of (a) Wind Angle and (b) Wind Speed 
on the NL-IOM. 

Example: MFA = 110°; A^^ = -7°; AZ = H-0 km; AZ = 20 km; t - 
15 min. Find the direction and wind speed at flight altitude. 

Solution: AW = 26° (Fig. 3.15, a) ut - 45 km (Fig. 3.15, b). 

45 KM 45 km . „„ , 
u = — — :- = —— - = 180 «^/ hr 
lomin 0,25 t^p 

To determine the wind direction relative to the meridian of /262 
the aircraft's location, the calculation of the given path angle 
of flight should be aipplied to that meridian, and then the wind 
angle should be added. 

a) b) 



Q) 20 «" (J) 



10 «i 



Fig. 3.15. Determination of (a) Wind Angle and (b) 
Wind Speed on the NL-IOM. 

In a flight with magnetic path angles, we will have approx- 
imately 



272 



or m our case 



6„ = MFA + AW 
M 



6„ = 110 + 26 = 136° 
M 

"5 = fi.+A.. = 135-7 = 129° 
M M 



Automatic Aircraft Radio D i s tance- F i nders ( Rad i ocompasses ) 

Automatic aircraft radio direction-finders ( radiocompasses ) 
are very widely employed. Aircraft with piston engines use them 
as a reliable, operative, and highly precise method of aircraft 
navigation. Large passenger aircraft with jet engines, for a number 
of reasons, cannot make such effective use of radiocompasses, but 
they continue to use them successfully along with other more pre- 
cise means of aircraft navigation. 




The 



accuracy of distance-finding for ground radio stations 
with the aid of radiocompasses is somewhat lower than the accur- 
acy of distance finding for aircraft with ground radio distance- 
reasons : 



with the aid of radiocompasses is somewhat lower than 1 
acy of distance finding for aircraft with ground radio 
finders, which can be explained by the following three 



an- 



(1) Stationary radio dis tance -finders can have special 

s which are equivalent to frame-type antennas but are fi-cc 

f the frame ; on air- 



tennas which are equ..- » a j.= aj l. .. ^ j.j.a.,--_ -^ ^ - 

f the effect related to the horizontal sides „^ ^..^ ^^^... 

---c^ ■' ' ' ' -"""-■i-T^ *-„ ,-„„j-^TT „,,_u antennas due to their 



craft , it is no 
unwieldiness 



t possible to install su 



ch 



(2) The bearing of an aircraft is measured with the aid of 
ground radio dis tance -finder s directly from the direction of the 
magnetic or true meridian, passing through the radio distance-finder 

at a fixed setting of the antenna system relative to the vertical; /263 
in distance-finding with ground radio stations by radiocompasses 
located on board aircraft, the error in the bearing includes the 
errors in measuring the aircraft course; in addition, the accur- 
acy of distance measurement is reduced due to the longitudinal and 
transverse banking of the aircraft. 

(3) Errors in distance-finding due to the effect on the prop- 
agation of electromagnetic waves over the relief of the surround- 
ing medium to a certain degree is taken into account in measuring 

the distance of aircraft with the aid of ground radio distance-finders 
(by means of preliminary test flights and the recording of a curve 
of radio deviation). 



273 



The considerable difference in flight conditions does not permit 
us to solve this problem for radio compasses located on board an 
aircraft. On the average (with a probability of 95%), the errors 
in locating aircraft with ground radio distance-finders in flat 
country is 1-2°, and 3-5° in the mountains. The errors in meas- 
uring the distances with the aid of radiocompasses in flat areas 
is 3-5°, and can reach 10-15° in mountainous areas, especially at 
low flight altitudes . 

Accordingly, the practical operating range of a ground radio 
distance-finder with satisfactory results of tracking is 300-400 
km (except for those which work on USW, where the operating range 
is determined by the s traigh t- line geometric visibility). 

A satisfactory accuracy in determining the bearings of radio 
stations with the aid of on-board radiocompasses is obtained at 
distances up to 180-200 km. Nevertheless, radiocompasses have found 
increasingly broad application for purposes of aircraft navigation, 
and are more popular than the ground radio distance-finders due 
to their considerable autonomousness and the ease with which they 
can be employed. 

For purposes of increasing operativeness , as well as forming 
a reserve and ensuring reliable operation of radiocompasses, two 
sets of them are used in most aircraft. 

The basic control system for an on-board radio compass con- 
sists of the following: 

(a) Frame antenna with mechanical device for rotating it and 
a mechanism for compensating radio deviation. 

(b) Open antenna, 

(c) Superheterodyne receiver with a device for commutation 

of the phase of the frame antenna and an electrical device for turning 
the frame antenna (tracking system). 

(d) Indicator of course angles of radio stations. 

(e) A shield for the remote control of the radiocompass . 




21^ 



which is equivalent to a drop in the wavelength of the received 
signal and therefore an increase in the phase shift between the 
sides of the frame. To increase the magnetic and dielectric per- 
meability of the medium, the effect of the frame will increase. 




low 
frequency generator 



Fig. 3.16. Diagram of Amplitude Modulator at the Output of 
the Radiocompass Receiver. 

In contrast to the amplitude ground radio direction-finders 
which we have discussed thus far and which belong to the "E" type 
(carrier-wave amplitude), radiocompasses presently use the method 
of amplitude modulation of the received signals (direction-finder 
type "M" ) . 

The essence of the method is that • r e cept i on of the signals 
takes place simultaneously with an open and a frame antenna, with 
the phase of the frame antenna being constantly switched by the 
low-frequency generator. This means that an amplitude-modulated 
signal is obtained at the input of the receiver. 

A simplified diagram of the amplitude modulator at the input 
of the receiver is shown in Figure 3.16. 

The control grids of L^ and L2 receive a negative voltage u„q , 
so that when the low-frequency generator is turned off, these tubes 
will be closed and the signals from the frame antenna will not be 
passed. 



When the low-frequency generator is turned on, tubes Lj and 
L2 open alternately, and the signal from the frame antenna reaches 
the input of the receiver in phases which are separated by 180°, 
and when these are combined with the signals from the open antenna, 
they undergo amplitude modulation. 



/265 



275 



r 




Fig, 3.17. Zero, Positive 
and Negative Modulation. 



Obviously, depending on the direction of the radio station 
(Fig. 3.17), with a fixed radiocompass frame, the amplitude modu- 
lation can be positive (Position 1), zero (Position 2), and nega- 
tive (Position 3). 

The tracking system at the output of the receiver is designed 
so that the frame of the radiocompass rotates in the direction which 
will produce a zero modulation of the signal, 

A diagram of the output section 
of the receiver is shown in Fig- 
_ ure 3.18. 

^W^ \ I yi "" ^ reference voltage on the anodes 

of tubes L^ and L2 , formed previously 
in the positive half-periods of 
the rectangular pulse, is supplied 
to the switching circuit of the 
frame antenna at the input of the 
receiver. If the input signal is 
modulated by the frame signal, the 
average anode current of one of 
the tubes will be greater than that 
of the other. This produces a disturb- 
ance of the balance of the bridge 
circuit in the magnetic amplifier, 
made of permalloy cores , and a current passes through the rotor 
winding of a small motor. The stator winding of the motor is con- 
stantly supplied with a voltage which is shifted 90° in phase by 
capaciitor C, from a low-frequency generator which supplies the 
bridge circuit. 

The motor will continue to rotate until the direction of the 
radio station is no longer perpendicular to the frame of the radio- 
compass, and the modulation of the signal of the open antenna by 
the frame becomes zero. 

In the case when the frame antenna is turned toward the radio 
station in the opposite plane, the phase of the frame changes by 
180°, In this case, in the presence of modulation, the rotation 
of the frame will take place not in the direction of reduction, 
but initially in the direction of increase of modulation, thus causing 
the frame to turn through 180°, In this manner, the readings from 
the radiocompass are all given the same sign, 

A block diagram of the radiocompass is shown in Figure 3.19. 

The controls for the radiocompass are mounted on a special 
control panel. Usually, the radiocompass has three operating regimes 
(besides the "off" position), so that a selector switch is mounted 
on the panel. 



276 



I. Tuning. In this regime, only the open antenna of the radio /266 
compass is connected. A special vernier on the control panel is 
used to tune the device to the frequency of the ground radio sta- 
tion, either by ear or by a visual tuning indicator. When tuning 
by ear, reception takes place in the "telegraph" regime, i.e., the 
second heterodyne of the receiver is turned on to convert the inter- 
mediate frequency of the receiver to sound. In the telegraph regime, 
the call letters of the radio station are also heard, if the station 
is transmitting on a non-modulated frequency. 



^\r- 




r^if 



I reference 
voltage 

JUirL o 





H 

o 

Hi 
U 

a 

0) 

bD 

>> 
O 

a 

3 

cr 

0) 

u 
o 



Fig. 3.18. Diagram of Output Section of Radiocompass 
Receiver . 

I I . Compass regime. In this regime, both the open and frame 
antennas of the radiocompass are connected. In this case, the track- 
ing system of the receiver turns the frame antenna depending on 
the direction of the radio station and the direction of the radio 
station is shown on an indicator (course angle or bearing). 



III. Frame regime. In this regime, only the frame antenna 
of the radiocompass is connected, and the bearing of the radio sta- 
tion can be determined with minimum audibility of its signals in 
the telegraph regime. The rotation of the frame is carried out 
by means of a special pushbutton switch on the control panel with 
the label "left-right". The reading of the bearing in this case 
has two signs . 

Recent models of the ARK-11 automatic radiocompass do not diffe 
in their principle of operation from the operating principle describ 
above for the ARK-5 radiocompass, but they have several design featu 
and advantages : 



/267 

r 

ed 
res 



277 



fa 



(a) Complete electrical remote control. 

(b) Possibility of setting the apparatus to nine previously 
selected channels (frequencies) in the range from 120 to 1340 kHz 
and switching from one receiver channel to another by means of an 
automatic pushbutton switch, located on the control' panel . There 
is also a provision for smooth manual setting over the entire oper- 
ating range of the radiocompass (with the tenth button depressed). 

(c) Increased noise stability of the receiver. 

(d) Possibility of operation in combination with a non-con- 
trolled antenna of open type with a low aerodynamic resistance and 
a low cperating altitude (on the order of 20 cm). 



,frame> 



M 



deviation | 
compensator 



NK 



input 
device 



selsyn 
trans- 
mitter 




rearing 
{indicator 



receiver 



output 
device 



control 
panel 



1 



Fig. 3.19. Functional Diagram of Radiocompass. 

The control panel of the ARK-11 differs in design from that 
of the ARK-5. In addition to the "off" position, there are four 
operating regimes. The first three regimes are the same as described 
above. The fourth regime "Compass II" is a spare and is used in 
the case of intense electrostatic noises when the usual distance- 
finding methods become unstable. 

In the "Compass II" regime, instead of the open antenna, a 

second frame antenna is used, mounted on a common frame-antenna 

block, perpendicular to the basic frame and forming a unit with 
the basic frame. 

The reference signal in this case reaches the input of the 
receiver not from the open but from the additional frame antenna, 
which is less sensitive to noise. However, the additional frame 
antenna which has the same properties as the main antenna, changes 
the phase of the reference signal by a further 180° when it is 
turned through 180°, so that both positions of zero reception of 
the main frame antenna will be positions of stable equilibrium, 
and consequently it is possible to have an error in determining 
the course angle of the radio station of 180° . 



278 



The control panel of the ARK-11 has a toggle switch for nar- /268 
row and wide frequency bandpass: "wide-narrow". In the "narrow" 
position, the extraneous noises in the earphones are reduced and 
the desired radio station can be heard more clearly. 

Other control units on the ARK-11 panel (subrange switch, knobs 
for coarse and fine setting, toggle switches and buttons) have the 
same markings as in the ARK-5. 

Eadiooompass Deviation 

Conditions for directional reception of electromagnetic waves 
on an aircraft are not favorable and depend on the direction of 
propagation of the wave front in both the horizontal and vertical 
planes . 

If the reception of signals from a ground radio station is 
being made at considerable distances which exceed 5-6 times the 
flight altitude, the vertical component of the vector of propagation 
of the wave front has less of an effect on the reception conditions. 
In this case, we can use a compensated curve of radio deviation, 
which is a function only of the course angles of the radio station. 

The reason for the radio deviation is a reflection of elec- 
tromagnetic waves from the surface of the aircraft or their re- 
reflection from individual parts of the aircraft. Since the radio 
compass frame is mounted in the plane of symmetry of the aircraft 
X-Z, the deviation at course angles zero and 180° is close to zero. 

The transverse plane of the aircraft Y-Z is also close to the 
plane of symmetry, so that the deviation at course angles 90 and 
270° is not great and passes through zero at course angles close 
to it. 

The maximum asymmetry of the aircraft takes place relative 
to the directions 45, 135, 225 and 315°. Therefore, the radio devi- 
ation at these course angles reaches a maximum. 

Hence, the curve of radio deviation has a quarternary appear- 
ance (Fig. 3.20) with extreme values AP = + 12 to 25° depending 
on the type of aircraft . 

Radio deviation is compensated by a mechanical compensator 
located on the axis of rotation of the frame antenna. The compen- 
sator has a control strip which produces an additional revolution 
of the axis of the master selsyn by means of a special transmis- 
sion. The required shape is given to the control strip by means 
of 24 compensating screws to set the readings for the radiocompass 
at 15° intervals on the scale from zero to 360°. 

Before the first determination of radio deviation, the com- 
pensator is usually neutralized, i.e., each of the screws is unscrewed 

279 



to such a position that the control strip has a shape with the 

correct curvature and the additional r-"^ation of the axis of the /269 

master selsyn is equal to zero at all course angles. 

To determine radio deviation, a ground radio station is se- 
lected (preferably at a distance of 50-100 km from the airport) 
and the true bearing is measured as accurately as possible on a 
large-scale chart (usually 1:500,000), and then the magnetic bearing 
of this radio station (MBR) is determined. 




CAR 



Fig. 3.20. Graph of Radio Deviation. 

By means of a deviation distance finder, magnetic bearings 
of one or two separate landmarks (MBL) are measured from the center 
of the area where the radio deviation will be plotted, in the way 
which was described in Chapter II, with a description of their devi- 
ation of the magnetic compasses. If the area for the deviation 
operations at the aerodrome is constant, the MBL will be known earlier. 

The aircraft is then rolled out on the runway. The deviation 
distance finder is installed in the aircraft in a line 0-180° exactly 
along its longitudinal axis, and the course angle of the landmark 
(CAL) is calculated to get rid of installation errors in the radio- 
compass. The corresponding CAR = 0: 



CAL 



MBL 



MBR, 



In the deviation distance-finder, the level line is set to the cal- 
culated CAL, and the aircraft is turned until the sight line of 
the distance-finder coincides with the direction of the selected 
landmark. In this case, the longitudinal axis of the aircraft will 
be lined up exactly with the radio station (CAR = 0), and the mag- 
netic course of the aircraft will be equal to the magnetic bearing 
of the radio station as measured on the chart (MBR). 

Turning on the radio compass and setting it to the desired 
radio station, the reading of the radiocompass is taken ( RRC ) . If 
RRC is not equal to zero, we will have the installation error of 
the frame: 



280 



est 



CAR 



RRC 



Then, without turning off the radiocompass , it is necessary 
to loosen the fastening screws which hold the frame to the fuse- 
lage and then (by turning the base of the frame) adjust it until 
the indicator points to 



/270 



RRC = CAR = , 

after which the frame is re-fastened to the fuselage. 

The remaining installation error, if RRC is not equal to zero 
after the frame has been fastened down, can be compensated for imme- 
diately either by the navigator or the pilot by turning the body 
of the selsyn relative to the indicator scale. 

After compensating for the installation error, the radio de- 
viation is determined successively at 24 RRC ' s at 15° intervals. 
To do this, it is necessary to set the sight line of the deviation 
distance-finder along the longitudinal axis of the aircraft to 0- 
180°, loosen the dial of the deviation distance-finder and move 
it so that the line of sight 0-180° passes through the selected 
landmark, and then fasten the scale of the distance- finder once 
again. In this case 

CAR = and MBR = 0. 

The deviation dis tance - finder mounted in this manner makes 
it possible to calculate the course angles of the radio station 
(CAR) on the scale dial by turning the aircraft to any angle. 

Consequently, if we turn the aircraft according to the indi- 
cations of the radiocompass to a RRC = 15°, and then to 30, 4-5, 
60°, etc., successively (setting the sight system of the deviation 
dis tance - finder each time to a selected landmark), we can calcu- 
late the CAR immediately from the scale on the dial. 

Thus, in each reading of the radiocompass, we determine the 
actual course angle of the radio station and can write the radio 
deviation as follows: 

A = CAR - RRC. 
r 

Compensation of radio deviation is performed after it has been 
determined. To do this, the graph of radio deviation is plotted 
and the extreme values of the graph are divided into three equal 
parts to avoid sharp bends in the strip, after which two intermed- 
iate graphs of radio deviation are plotted. 

The compensator is then removed from the axis of the frame; 
by turning the proper screws, compensation is made for the radio 



281 



deviation in terms of the first intermediate graph, calculating 
the correction made in the selected portion of the radio compass 
by means of a special pointer on the compensator. Then the devi- 
ation is compensated by the second intermediate graph, and finally 
by the curve of radio deviation 



Compe 



compensation for radio deviation by all three graphs is per- 
formed in an order such that after each introduction of a positive 
correction there is a correction of equal magnitude but negative, 
i.e., with a mirror image of the course angles. Usually, the order 
of compensation is selected as follows: 0, 15, 345, 30, 330, 45, 
315, 60, 300, 75, 285, 90, 270, 10 5, 255, 120, 240, 135, 225, 150, 
210 , 165, 195 and 180° . 

After compensation for radio deviation, the compensator is 
mounted on the mechanism of the frame; the aircraft is turned and 
the deviation distance-finder is used to check the correctness of 
the operations which have been carried out. If any errors in compen- 
sation are discovered, the radio deviation is compensated once again 
by an additional turning of the screws corresponding to the readings 
of the radiocompass . 

In addition to the method described above for correcting radio 
deviations on the ground, there are others. For example: 

(a) Determination of the magnetic course of an aircraft by 
distance-finding at the tail (nose), as described in Chapter II, 
and the calculation of course angles of the radio station on the 
basis of it. 

(b) Range finding of a radio station which is visible from 
the airport (e.g., a distant power radio station). 

In aircraft where the frame antenna of the radio compass is 
mounted below the fuselage, determination of radio deviation on 
the ground is impractical, since the reflection of electromagnetic 
waves from the surface of the ground causes a distortion of the 
electromagnetic field. In these aircraft, the radio deviation is 
determined in flight. 



/271 




going away from it 



To save time, the flight can be carried out over a 24-angle 
route, i.e., practically along a course which crosses the straight- 



282 



line flight for 20-30 sec for each recording of the readings of 
the radio compass and course. However, in this case, it is neces- 
sary to determine the location of the aircraft at each point being 
measured and to enter it on a chart so that when the data is analyzed 
it will be possible to determine the bearing of the radio station 
from the point at which the reading was taken. 

In fact, the course angle of the radio station (CAR) at the 
moment when the recordings are made is determined by the formula 



CAR = TBR 



TK, 



and the radio deviation of the radio compass is determined as the 
difference : 



A = CAR 
r 



RRC 



Compensation for radio deviation is made after the aircraft 
lands in the same way as after determining it on the ground, but 
without checking the accuracy of the work which has been carried 
out, since this would require repetition of the flight. 

Aircraft Navigation Using Radiocompasses on Board the Aircraft / 212 

Radiocompass es on board the aircraft make it possible to solve 
the same navigational problems as ground radio distance-finders. 

(a) Path control in terms of direction and selection of the 
course to be followed during flight toward the radio station and 
away from it . 

(b) Measurement of the drift angle after flying over the radio 
station . 

(c) Checking the path for distance by measuring the distance 
to a radio station located to the side. 

(d) Determination of the position of the aircraft by obtain- 
ing bearings from two radio stations. 

(e) Determining the drift angle and the groundspeed from suc- 
cessive positions of the aircraft, as well as the wind parameters 
at flight altitude. 

The solution of these problems by means of a radiocompass mounted 
on board the aircraft is very similar in principle of solution to 
the ground radio distance-finders, especially if the indicator for 
the course angles of the radio compass is combined with the course 
indicator of the aircraft and thus shows the reading for the bear- 
ing (Fig. 3.21). 

The figure shows the course indicator for the navigator. 



283 



combined with the bearing indicators of two radio compasses (USh- 
M). The course of the aircraft is measured on the inner, movable 
scale of this indicator (relative to a triangular mark on the outer 
scale), while the course angles of the radio stations are indicated 
on the outer fixed scale according to the position of the point- 
ers of the radiocompasses . On the inner, movable scale, opposite 
the arrows, it is possible to calculate the bearings of the radio 
stations, while the other ends of the pointers can be used to show 
the bearings of the aircraft . 

However, this method is practical only for use with one radio- 
compass, since the total correction is then shown on the scale of 
deviations, and is effective only for one radio station 

M 

where 6 is the difference between the meridian of the aircraft loca- 
tion and the meridian of the radio station, and Aj^^ is the magnetic 
declination of the location of the aircraft. 

Obviously, in the general case this correction will be dif- 
ferent for different radio stations . 




The necessity to make corrections for the deviation of the /273 
meridians is one of the principal shortcomings of radiocompasses. 
This shortcoming to a certain degree can be reduced by using an 
orthodromic system foi' estimating the path angles and courses of 
the aircraft. In this case, the need to introduce corrections is 
no longer applicable, if the radio station is located on the refer- 
ence meridian for computing the path angles, and in any case the 
correction remains constant if this condition is not observed. 

The use of combined indicators considerably simplifies the 
operations related to the use of radio compasses mounted on board 
aircraft. Therefore, the methods of using them for navigational 
purposes must be viewed as non-recorded indicators of course angles 
of radio stations, assuming that in the combined indicators, the 
addition and subtraction of the angles according to those same rules 
is carried out automatically. 

It is clear from Figure 3.22 that the magnetic bearing of the 
radio station (MBR) and the true bearing of the radio station (TBR) 
are added from the course of the aircraft in corresponding systems 
of calculation and the course angle of the radio station: 

281+ 




Fig. 3.21. Combined Indicator for Course and Course 
Angles of a Radio Station. 

TBR = TC + CAR; /27^ 

MBR = MC + CAR. 

Similarly, in the orthodromic system of calculating courses 

OBR = OC + CAR. 

where OBR is the orthodromic bearing of the radio station and OC 
is the orthodromic course. 

In determi- ing the location of the aircraft, the true bear- 
ings of the aircraft (TBA) are plotted on the flight chart from 
gi'ound radio stations. 

TBA = TBR + 6 + 180° , 

where 6 is the angle of convergence of the meridians. 

In calculating the true bearing of the aircraft , ±80° are added 
if the TBA has a numerical value less than 180° and subtracted when 
the value of the bearing exceeds 180° . 

Consequently, in calculating courses from the true meridian 
of the aircraft's location, where 



285 



TBA = TC + CAR + 6 + 180° , 



6 = (A -X )sln<j) 

r , s . a ^av 



and in calculating the course from the magnetic meridian 

TBA = MC + A., + CAR + 6 + 18*0° . 
M — 

It is assumed that the LA is determined with the aid of mag- 
netic compass deviation. 

In the orthodromic system of calculating courses, 

TBA = OC + CAR +6 + 180°, 

r . m . r . s . — 

where ^-p .m.v . s . ^^ '^^^ angle of convergence of the meridians (ref- 
erence and radio station) which is equal to ( ^r . s . ~ ^r . m . ) ^ i'^'i'av • 

Let us consider the means of solving problems by means of radio- 
compasses located on board aircraft, taking the rules mentioned 
above into account. 




Obviously, a flight along a 
given line of flight from a radio 
station or to a radio station 
must take place with a constant 
true bearing of the aircraft. In 



/275 



radio station. 



a flight from a raaio station, 
this bearing is equal to the azi- 
he orthodrome relative 



Fig. 3.22. 
Station 



Bearings of Radio 



and Aircraft 



nis Dearing is equal to rne azi- 
luth of the orthodrome relative 
o the meridian of the radio sta- 
tion. However, if the flight 
takes place in a direction toward 
the radio station, then the true 
bearing of the aircraft differs 
from the azimuth of the ortho- 
drome of the meridian of the radio 
station by a value equal to 180° . 

3ath in terms 



of d 
rd a radio station and away from a r 
- "^ ^' ■ •■ ■ bearing of 



wa 

parison 

value 



of the actual true 




For example, in calculating the course from the magnetic merid- 
ian of the aircraft's location, we must satisfy the equation: 



286 



'init 



MC + A„ + 
M 



6 + CAR; 



o J- = MC + 
ref 



M 



+ 6 + CAR + 180° , 



where ainit i^ "the true bearing of the aircraft and a^ef is the true 
bearing of the radio station. 

In using a combined indicator for the bearing, the total cor- 
rection A = Af^ + 6 can be entered on the scale of declination. Then 
it is necessary to satisfy the condition that a^n^-j- = TBA and that 
apef = TBA +_ 180° . 

The true bearing of the aircraft is calculated from the other 
end of the indicator pointer, i.e., in flight toward a radio station, 
the direct reading of the pointer on the radiocompass must be equal 
to the final azimuth of the orthodrome , while in flight away from 
a radio station the other end of the pointer must show a reading 
equal to the initial azimuth of the orthodrome. 

Inasmuch as the total correction (A) over the length of the 
path segment will change constantly, it is necessary to determine 
this correction in each measurement in order to take into consider- 
ation other indicators or entering the declinations of the combined 
indicator on the scale. This poses considerable difficulty in using 
the radiocompass in flight. 

The problem is simplified considerably in an orthodromic sys- 
tem of calculations for the aircraft course. In this case 



TBA 



OBA + 6 



r . m . r . s . 



where OBA is the orthodromic bearing of the aircraft. 

The correction 6-p m r s . -^ ^ constant for all s tr aight -line 
path segments; after setting it on the scale of deviations, the 
TBA can be read off immediately on the indicator over the entire 
straight- line path segment. 

Note. If a radio station is located at the starting point 
of the route (SPR) with a reference meridian, the flight can be 
made directly along the orthodromic bearing of the aircraft with 
a zero correction on the declination scale. Correction for devi- 
ation of meridians is only valuable when the radio station is located 
to the side of the flight route or the meridian of the radio sta- 
tion does not coincide with the reference meridian. 



As in the case of using a ground radio direction-finder in 
selecting the course to be followed along straight-line segments 
of a path by means of a radiocompass, it is necessary to be guided 
by the following rules : 



/276 



287 



(a) In a flight from the radio station, with a drift of the 
aircraft to the right, the bearing is increased and the course to 
be followed must be reduced; with a decrease in the bearing, the 
course must be increased. 

(b) In the case of a flight toward the radio station, the 
course must be increased when the bearing increases and decreased 
if the aircraft bearing decreases . 

In a flight along a course determined by a radio direction 
finder, if the course to be followed is selected on the basis of 
stable bearings ShchDR or ShchDM, the course is considered to have 
been selected by using the radiocompass on board if the course angle 
of the radio station remains constant. 

For example, in a flight toward a radio station at a constant 
course of the aircraft, an increase in the course angle of the radio 
station corresponds to a drift of the aircraft to the left (the 
TBA increases). In order for the aircraft to follow a constant 
bearing, the course of the aircraft must be increased. When the 
lead in the course is equal in value to the drift angle of the air- 
craft, the course angle of the radio station will remain constant 
and equal to the drift angle both in value and in sign. 

In selecting a course, the same method of half correc tion is 
us ed . 

Example . It is necessary to make a flight toward a radio sta- 
tion with a orthodromic path angle of 82° . Let us assume that the 
drift of the aircraft will be to the left within limits of approx- 
imately 10°. Select a course to be followed by using the method 
of half correction. 

In this case, after flying over the turning point in the route, 
it is necessary to assume an orthodromic course of 92°. The course 
angle the of radio station will then be equal to 350°, which is 
equivalent to a numerical value of -10°. 

In a flight with a course of 92°, if the course angle of the 
radio station is increased (i.e., if we acquire the values of 351, 
352, 353° in succes sion), it is necessary to place the aircraft 
on the line of flight and to take a lead of 15° (course equals 97°, 
CAR equals 345° ) . 

Let us assume that this lead turns out to be too great; then 
the CAR begins to decrease, taking on values of 344, 343, and 342° 
Then, after a second placing of the aircraft on the path, it is 
necessary to take an intermediate lead in the course of 12-13° (CAR 
equals 348-347°). If the CAR is to be stable, it is necessary to 
ensure that the orthodromic bearing of the radio station is equal 
to the orthodromic path angle and to continue the flight with the 
selected course. 



288 



The course to followed in a flight away from a radio station 
is selected in the same manner, the only difference being that when 
the CAR increases, it should not be reduced but increased further. 



In using other indicators, the selection of the course is also 
accomplished by means of stable path angles of radio stations . How^ 
ever, in order to control the path of the aircraft in terms of direc- 
tion, it is necessary to determine on each occasion the bearing 
of the aircraft or the radio station by summing the course and course 
angles of the radio station, taking into account the deviation of 
the meridians of the radio station and aircraft and the magnetic 
declination at the point where the aircraft is located. 



It should be mentioned once again that in a flight toward a 
radio station, the selected stable course angle of the radio sta- 
tion is always equal to the drift ^ngle of the aircraft , regard- 
less of whether the aircraft is located on the line of the desired 
flight or as a slight deviation from it. For example, a stable 
TAR = 350° corresponds to a drift angle of -10° . In flight away 
from the radio station, the drift angle is always equal to a stable 
CAR minus 180° . 



1211 



The drift angle of the aircraft can be measured directly after 
flying over the radio station. 

At the same time, after flying over the radio station with 
any constant course, the course angles of the radio station will 
be stable, so that 



US 



CAR 



180° 



However, in the majority of cases the drift angle is deter- 
mined during flight away from a radio station as a difference between 
the magnetic (true or orthodromic) bearing of the aircraft and the 
magnetic (true or orthodromic) course of the aircraft: 



US 



MBA 



MC , 



US = TBA 



TC , 



or 



US = OBA 



OC , 



where OBA is the orthodromic bearing of the aircraft and the OC 
is the orthodromic course of the aircraft. 

In this case, we can simultaneously determine the side to which 
the aircraft deviates (left or right) by comparing the given path 
angle and the determined range of the aircraft in the system of 



289 



coordinates being used;' the aircraft acquires the given line of 
flight according to the calculated course angle of the radio station, 
while on the line of flight corrections are made to the course which 
are e'qual to the average, angle of drift. 

The monitoring of the aircraft path in terms of distance by 
means of the radiocompass is accomplished with previously calcu- 
lated bearings of the lateral radio station 




In approaching a control landmark, the readings for the course 
and the course angle of the radio station are observed. At the 
moment when the sum of the aircraft course and the course angle 
of the radio station become equal to the previously calculated bear- 
ing (in combined indicators, the bearings of radio stations become 
equal to the previously calculated value), the moment for flying /27 8 
over the landmark is determined. 

With the aid of a radiocompass located on board, it is also 
possible to determine the location of the aircraft on the basis 
of true bearings from two radio stations. However, the accuracy 
of determining the aircraft location by this method, involving consid- 
erable difficulty in the process, is insufficiently high. There- 
fore, the method is not widely employed in aircraft navigation, 
being used only for determining approximate aircraft coordinates 
in finding lost landmarks . 

The essence of the method is the following: when two radio- 
compasses are on board, one is set to the frequencies of two ground 
radio stations, located no more than 180-200 km from the aircraft. 
It is desirable when doing this to ensure that the bearings of these 
radio stations cross at an angle close to 90°. 



If the indicators of the r adiocompasses do not agree, the course 
^of the aircraft, the course angles of the two radio stations, and 
the distance-finding time must all be described simultaneously for 



a given moment of time, 
aircraft are determined; 

TBAx 

TBA2 



Then the approximate true bearings of the 



= MC + A[^ + CARi +_ 180° ; 
= MC + Af^ + CAR2 +_ 180° . 



290 



I nil I I ■■ 



I III iiiHiiin 



The bearings which have been obtained are plotted on the flight 
chart from the meridians of selected radio stations by means of 
a protractor and scale rule. 

Having thus determined the approximate position of the air- 
craft, we can find its true bearing by introducing the precise value 
of the magnetic declination and making corrections to the deviation 
angles of the meridians of the radio station and the aircraft loca- 
tion. These corrected bearings are again plotted on the chart to 
give a precise position of the aircraft at the moment of direction 
finding . 

If only one radiocompass is mounted on board the aircraft, it 
is necessary to consider its path when determining the location 
of the aircraft for the time between the moments of direction finding, 
and this is done as follows (Fig. 3.23). 

After determining the average bearings of the aircraft, the 
latter are plotted on a chart, and then the flight path of the air- 
craft is obtained from the point of location of the first radio 
station for the time between the measurements of the course angles 
of the radio station in a direction which coincides with the course 
of the aircraft. A line is drawn through the point which is obtained, 
parallel to the first bearing up to the intersection with the line 
of the second bearing, defining the position of the aircraft at 
the moment of direction finding for the second radio station. In 
addition, the true bearings of the aircraft are found in the same 
way as in the case of two radiocompas ses . 



The lab or ious nes s of the process of determining the position 
of the aircraft is considerably relieved if the flight is made with 
orthodromic courses, but the indicators of the radio compasses must 
match. In this case, the angles of convergence of the meridians 
of the radio stations with the reference meridian for calculating 
the course are determined beforehand. 



/279 



lans 



At t 
for 



he time of measurement, the 
the first and then the seco 




t'^t 



Fig, 3.23 
the Posit 
the Beari 
stions 



Diagram for Locating 
ion of an Aircraft from 
ngs of Two Radio Sta- 



angle of deviation of the merid- 
nd radio station are entered 
on the scale of deviations 
of the indicator in succession. 
They are calculated from the 
readings of the opposite ends 
of the pointers of the radio 
compasses and are designated 
as TBSi and TBS2. The bear- 
ings obtained are final and 
no corrections are required. 

As we have already ment 
when using radiocompasses mo 
on board aircraft, the drift 



led , 



mgle o: 



IS 



291 



determined from the stable course angles of the radio stations or 
measured after flying over a radio station. 

The ground speed of the aircraft is determined by checking 
the flight in terms of distance by means of radio stations located 
to the side of the course or from the moments when the aircraft passes 
over radio stations. The latter method is not accurate, especially 
when flying at high altitudes, due to the error in the readings 
of radiocompasses when flying over radio stations. 

The determination of the wind at flight altitude is accomp- 
lished on the basis of the ground speed of the aircraft, the air- 
speed, and the drift angle by the same methods as for ground radio 
distance-finders. The method of determining the wind by using the 
successive positions of the aircraft, obtained by distance measure- 
ment from two radio stations , usually is not used due to the inad- 
equate precision of the determination of the aircraft location. 

Special Features of Using Radioaompas ses on Board Aircraft 

at High Altitudes and Flight Speeds 

High altitudes and flight speeds cause deterioration of the 
conditions for using radiocompasses aboard aircraft for purposes 
of aircraft navigation. 

The use of radiocompasses and the observation of all rules 
for retaining accuracy of distance finding is a laborious process, 
so that the increase in flight speed, calling for operativeness 
of navigational calculations , creates difficulties in using radio- 
compasses on board the aircraft. 

This shortcoming can be largely overcome by using combinations /2 80 
of bearing indicators, especially in the orthodromic system of calcu- 
lating aircraft courses . 

In addition, another shortcoming of aircraft radiocompasses 
which operate on medium and short waves, due to the increased speed 
of flight, is the effect of electrostatic noise on their operation. 

At high airspeeds, especially in clouds and in precipitation, 
a considerable electrification of the aircraft surfaces occurs. 
Static electricity, emitted at pointed portions of the aircraft 
(including open antennas) creates noise and radio interference in 
the frequency range at which radiocompasses operate. Despite the 
measures which are taken to prevent the charges from flowing by 
using special discharge devices, as well as shielding the open antennas 
of the radiocompasses , this shortcoming can be overcome only partially 
and manifests itself in very difficult flight conditions. 

High flight altitudes have an effect mainly on the accuracy 
of operation of radio compasses and especially on the accuracy of 

292 



b b, 
A _L c £,, I c 



l\ 



/ 



Fig. 3.24. Operation of 
an Open Antenna When Fly- 
ing Past a Radio Station, 



determining the moment when the air- 
craft flies over a radio station. 

The decrease in the accuracy 
of operation takes place due to the 
change in the nature of radio deviation 
at different angles of deviation of 
the propagation vector of radio waves. 
The latter changes within wide limits 
when the aircraft approaches the loca- 
tion of a radio station. 

A diagram of the appearance of 
errors in determining the moment when 
the aircraft flies over a ground radio 
station is shown in Figure 3.24 where 
there is a picture of the electrical 
field radiated by an open vertical 
antenna on a ground radio station. 



At large distances from the radio station, the electromagnetic 
wave is vertically polarized. However, there is a space near the 
radio station and above it where the polarization shifts to the 
horizontal , then back to the vertical but in opposite phase. 

Let us assume that an open antenna of the radiocompass is tilted 
backward (position a. Fig. 3.24) and the aircraft is approaching 
the rad_io station at a high flight altitude in the direction of 
ve ctor V . 



Obviously, at position a the antenna will have zero reception. 
The reception of the antenna will then increase, but in a phase 
which is opposite to the reception up to the point a. This leads / 2 81 
to a rotation of the radiocompass frame by 180° until the aircraft 
passes a radio station. Then, after passing the station, the phases 
of both the frame and open antennas change almost simultaneously 
( at the point a;^ ) . 

Thus, the change in the readings of the radio compass by 180° 
takes place until the moment when the aircraft passes over the radio 
station (at point a) and only the oscillation of the needle will 
be observed from then on. 



Fig. 3.25. Equivalent of an 
Open Antenna on Board an Air- 
craft . 




293 



s 



■ When the antenna is tilted forward (position c), the oscil- 
lations of the radiocompass needle will begin at point c, while 
the passage by the radio station with rotation of the needle through 
180° will be noted at point c^, i.e., there will be a delay in markinj 
the passage. 

For a strictly vertical antenna (position b),'the movement 
of the needle through 180° can take place prematurely. Then the 
pointer can make a reverse turn and again show the passage by the 
radio station at point bi- 

It should be mentioned that an equivalent to the open antenna 
of the radiocompass in terms of its inclination in the vertical 
plane is the resultant combining the upper or lower points of the 
antenna with the electrical center of the aircraft, constituting 
its grounding or counterweight (Fig. 3.25). 

In this figure, point 1 is the top of the open antenna, point 
2 is the receiver and point 3 is the electrical center of the air- 
craft . 

Obviously, straight line 1-3 is the equivalent of the inclin- 
ation of the open antenna, which is forward in this case. Thus, 
the setting of the open antenna above the fuselage in its forward 
section causes a delay in the reading of the moment when the radio 
station is passed. 

Mounting of the antenna in the same position with respect to 
the center but below the fuselage leads to a preliminary reading 
of the moment when the aircraft passes the radio station. The oppo- 
site picture is observed when the antenna is mounted behind the 
electrical center of the aircraft. 



is above or below the elec- 
ase an advance or delay 
these deviations depend 



Iso be a double 



The best place to mount the antenna is a 
trical center of the aircraft, but in this case an advanc 
in the readings is observed; in some cases, these deviati 
on the height and speed of flight, but there can also be 
reading involving both an advance and delayed indication. 

This system for the creation of errors in measuring a flight 
only approximately reflects the reasons for these errors. In prac- 
tice, they will depend both on the angle of pitch on the aircraft 
and on the accuracy with which the passage of the aircraft over 
the radio station is determined. 

For example, if the aircraft is passing a radio station to 
the side, then obviously there will not be an indication of pass- 
age with movement of the needle through 180°, but a deterioration 
in the passage over the radio station, i.e., errors in determin- 
ing the passage of the traverse of the radio station will be prac- 
tically non-existent. 



/282 



294 



Usually, the exact passage of an aircraft over a radio sta- 
tion occurs only in special and exceptional conditions. Therefore, 
in practice there is always a consideration of the effect of passage 
with the effect of error, which does not make it possible to con- 
sider the magnitude of the delay advance in marking the passage. 

Depending on the type of aircraft and the flight conditions , 
these errors can occur within limits equal to 1-3 flight altitudes, 
excluding the case of exact determination. However, exact deter- 
mination can occur at distances which exceed the flight altitude 
of the aircraft, beyond the limits of a zone with horizontal polar- 
ization, i.e., at very considerable deviations of the aircraft from 
the given line of flight. 

Details of Using Radioaompasses in Making Maneuvers in 
the Vicinity of the Airport at Which a Landing is 

to be Made 

The maneuver of approaching for a landing usually begins at 
a relatively low flight altitude (1200-4000 m) with a gradual reduc- 
tion of the airspeed. Therefore, the effects related to height 
and flight speed in this case are considerably reduced, 




Of course, if there is a tendency to drift in the aircraft 

course, the corresponding corrections must be made in the readings 

of the course angle of the radio station which are equal in magni- 
tude and sign to the lead which has been taken. 

The effectiveness of using radioaompasses in the vicinity of 
airports where landings are to made is increased also by the fact 
that the flight is made at short distances from ground radio sta- 
tions, which gives relatively small linear errors in determining 
the position of the aircraft in view of the errors already committed 
in measuring the course angles of the radio station. In addition, 
it is no longer necessary to calculate the magnetic declination 
and the deviation angles of the meridians . The accuracy of air- 
craft navigation in the vicinity of the aerodrome, using radiocom- 
passes, is considered to be quite satisfactory in all stages of 
the maneuver with the following exceptions: 

(a) Determination of the starting point of the maneuver by /2 83 



295 



flying past the power radio station, if the maneuver is beginning 
at a high altitude. 

(b) On a landing strip, where it is necessary to have very 
high accuracy of flight along a given trajectory for bringing the 
aircraft in for a landing. 

U 1 t ra- Shortwave Goniometric and Gon i omet r i c- Range Finding 

Sys terns 

As we have mentioned, radiocompasses have significant advan- 
tages over ground radio distance-finders with respect to uninter- 
rupted visual information on board the aircraft regarding its posi- 
tion. This means that they have been very widely employed and are 
installed in practically all types of aircraft as a rule in a double 
set. In addition, there are a number of important shortcomings 
for radiocompasses mounted on board aircraft, which reduce the accur- 
acy and feasibility of aircraft navigation. 

In addition to the errors caused by the effect of the local 
relief, which affect all systems for short-range navigation, radio- 
compasses have the following shortcomings: 

(a) Unfavorable conditions for directional reception of elec- 
tromagnetic waves on board the aircraft (radio deviation of an unstable 
nature ) ; 

(b) An increase in the errors in distance finding due to inac- 
curate measurements of the aircraft course; 

(c) The necessity to consider the deviation of the meridians 
and magnetic declinations when using magnetic compasses to deter- 
mine bearings ; 

(d) The effect of static noise in the range of received radio 
frequencies at high airspeeds; 

(e) The effect of flight altitude on the accuracy of meas- 
uring the range and determining the moment of flying over the radio 
stations . 

In addition to these shortcomings, radiocompasses are subject 
to a general disadvantage of goniometric systems: the need to plot 
bearings on the chart from two ground points to determine the loca- 
tion of the aircraft. 



Therefore it seems natural to try to build devices for short- 
range radionavigation which would have the advantages of the radio- 
compasses mounted on aircraft but would not have the shortcomings 
from which they suffer. 

Such devices are the goniometric and goniometric-rangef ind- 
ing systems which operate on ultra-shortwaves . 

296 



A common feature of these systems is the directional radiation 
of electromagnetic waves by ground instruments and their directional 
reception on board the aircraft. This feature, together with the 
range of waves employed, gives three very important advantages for 
navigational systems : 

(1) It frees the system from radio deviations on board; 

(2) The bearing of the aircraft becomes independent of the /284 
aircraft course, magnetic declination, and deviation of meridians; 

(3) It sharply increases the freedom of the system from static 
and atmospheric interference . 

There are several types of goniometric directional radio bea- 
cons and receiving devices to carry aboard aircraft, which operate 
on ultra-short waves. 




Figure 3.26 shows the schematic diagram of a radio beacon with 
a rotating directional antenna. The generator of low frequencies 
produces a frequency which is synchronized with the rotation of 
the directional antenna. On the axis of rotation of the antenna 
is a special disk, which generates the reference signal related 
to the position of the rotating antenna. The reference signal passes 
through the modulator and transmitter to reach the open antenna 
of the radio beacon. 



I 



trans- 
mitter 



call 



signal 



modu- 
lator' 



modu 

'lation 

suppres' 

SOT- I 



voltage 

indicator 
for refer 



lence phas 




M 



low 

frequency 

generator 



r= 



disc— generator 
of reference signals 



Fig. 3.26. Schematic Diagram of a Radio Beacon 
With Rotating Directional Antenna. 



297 



From the transmitter, the signal reaches the rotating antenna 
through a modulation suppressor, so that the amplitude of the signal 
radiated by the antenna in any given direction depends only on the 
position of the antenna relative to this, direction. Consequently, 
the signal from a directional antenna is modulated by a low fre- 
quency whose phase relative to the relative signal i's shifted through 
an angle equal to the azimuth of the aircraft. 

Thus, two signals reach the aircraft on the carrier frequency 
in addition to the call-letter signals: 



(1) Reference signal for beginning the reading. 



/285 



(2) The signal from the directional antenna, whose amplitude 
maximum coincides with the moment when this antenna crosses the 
line to the aircraft . 

The receiver on the aircraft has three channels (Fig. 3.27): 

(1) Channel for picking up the call letters from the beacon 
( earphones ) ; 

(2) Reference- channel ; 

(3) Azimuthal voltage channel. 



I 



receiver- 



earphone 
channel 



JO 



reference 

voltage 

channel 



azimuthal 

voltage 

channel 




phase 
discrim- 
inator 




zero indicator 



Fig. 3.27. Apparatus for Goniometric System 
Aboard an Aircraft. 

The indicator mechanism is usually based on measurement of 
the phase ratio of the reference and azimuthal signals with low 
frequency by the compensation method, i.e., the phase of the refer- 
ence signal is changed by an automatic phase shifter so that it 



298 



coincides with the phase of the azimuthal signal. In this case, 
the signal on the phase discriminator will be equal to zero while 
the bearing indicator on the aircraft will act as the pointer of 
the phase shifter. 

For flight along a given bearing, the phase shifter is set 
to a given position, so that the signal on the phase discriminator 
(and therefore on the zero indicator device) will be equal to zero 
only in the case when the aircraft is located exactly along the 
line of the desired bearing. 

The goniometric system for short-range navigation, operating 
on ultrasonic waves, in the case when the characteristic of the 
rotating directional antenna has a sharply pronounced maximum (which 
usually is achieved by using reflex reflectors), can be built on 
the principle of time ratios rather than phase ratios. 

In this case, the reference signal, when the rotating direc- 
tional antenna passes through zero reading, has a pulsed character, 
and the equipment on board must include a generator of a reference 
frequency as well as special delay devices to determine the bearing 
of the aircraft in time between the moments when the reference and 
azimuthal signals are received. 



/286 



The geometry of navigational applications of USW beacons with 
directional radiation it exactly the same as the use of ground radio 
distancef inders . All the problems of aircraft navigation such as 
selection of the course to be followed, monitoring of the path for 
distance and direction, determination of the location of the air- 
craft from two beacons, measurement of the drift angle and ground 
speed, measurement of the wind at flight altitude, etc., are solved 
in exactly the same way as for ground radio distance-finders. In 
flight away from a radio beacon, the rules for flight along the 
ShchDR bearing are observed, while in flight toward a radio beacon 
it is the rules for bearing ShchDM which are followed. 

If the flight is made using a zero-indicator instrument, the 
method of selecting the course in flight from the beacon and toward 
the beacon with a corresponding switch in the mode of operation 
of the receiver leads us to only one type: the pointer of the zero 
indicator shows the direction of deviation of the aircraft from 
the LGF . 

The main difference between using USW beacons and ground radio 
distance-finders is only that in order to determine the location 
of the aircraft from two bearings using a radio direction-finder, 
the plotting of the bearings is done by the operator of the com- 
mand distance finding station, while in the case of USW beacons 
it is done by the crew of the aircraft. 

Nevertheless, goniometric USW beacons have a much wider range 
of application than ground radio-distance-finders and aircraft radio- 



299 



compasses, thanks to the constant indication of bearings on board 
the aircraft . 

The instrumental accuracy of goniometric USW navigational sys- 
tems is higher than for ground radio distance-finders . The prac- 
tical accuracy under average conditions of application is also somewhat 
higher or equal to the accuracy of distance-finders. However, in 
using radio distance-finders, it is possible to consider to a certain 
extent the influence of the local relief on the radius of appli- 
cation, which cannot be done for USW beacons. In this respect, 
the USW beacons have less favorable operating conditions than ground 
radio distance-measuring stations. 

The operating range of a USW system S is limited by the limits 
of direct geometric visibility from the ground beacon to the air- 
craft with an insignificant increase caused by radio refraction. 

It is determined by the approximate formula 

S = 122 /H. 

However, if there are some obstacles along the path of the 
propagation of the radio waves (e.g., mountain peaks), they will 
appear insurmountable for USW. 

From the standpoint of navigational applications, it is very /2 8 7 
advantageous to combine the operation of a goniometric USW system 
with range finders. 

Rangefinding USW navigational systems are usually of impulse 
type (Fig. 3.28). 

The aii'craft transmitter sends out impulses of ultrashort waves, 
which reach the receiver aboard the aircraft at the same time as 
a reference signal. 



V 



re- 
ceiv- 
er* 



transH 
mittei: 



trans-l 
mitter^ 



re'- 
c elv- 
er" 



^ 



indicator 



Fig. 3.28 
System . 



Diagram of Long-Range Navigational 



300 



The ground receiver receives pulses of wave energy emitted 
by the aircraft, amplifies them and sends them out again through 
a transmitter into the ether, to be received by the aircraft. 

The range indicator aboard the aircraft has a generator of 
standard frequencies, a frequency-divider circuit, and a delay line 
for the reference pulse to measure the time required for the signal 
to pass from the aircraft to the ground beacon and back to the air- 
craft . 

While the signal is traveling, the duration of the delay in 
the reference pulse prior to its combination with the received signal 
determines the distance to the ground beacon, which is usually used 
as a visual indicator of the azimuth of the aircraft (the direct- 
reading instrument for distance and azimuth, DRIDA). 

The combination of azimuth and distance readings makes it very 
easy to solve the problems of aircraft navigation, especially if 
the beacon is mounted at the starting or end point of a straight- 
line flight segment. In the latter case, the crew of the aircraft 
has a constant supply of direct data regarding the position of the 
aircraft relative to the line of flight in terms of direction and 
distance . 

When the ground beacon is located to the side of the path to 
be covered by the aircraft, the problem of determining the aircraft 
coordinates is solved analytically, or very simple calculating devices 
are used to convert the polar system of coordinates for the posi- 
tion of the aircraft into the orthodromic system. 

One type of such device is the computer which is installed 
for zero indication of the position of the aircraft on the line 
of the path to be traveled during flight in a given direction. 



Let us assume that we have a straight-line path segment from 
point A to point B (Fig. 3.29). 



/288 



If we are given the path angle of the segment (^), measured 
relative to the meridian for calculating the bearings (magnetic 
or true meridian of the location of the ground beacon), and we know 
the azimuth of the end point of the segment (-^fin^ ^s well as the 
difference from it to the beacon (^fin^» "the shortest distance from 
the beacon along the line of flight (i?s) (disregarding the spher- 
icity of the Earth), can be determined by the formula 



R - R^. sin (i|) 
s f m 



f m 



or for any point lying on the line of flight 

R = R . sin (.xp- A .) . 
SI ^ 

In other words, the given line of flight is the geometric locus 



301 



of points for which 



i? .sin( i|;-i4 . ) = const = i? _ . sLnX^-A . ) 




Fig. 3.29. Diagram Showing Operation of Computer 
for Zero Indication of Path Line. 



Thus, having set the path angle of the segment ( i|/ ) on the 
calculator, the distance from the beacon to the end point of the 
segment (i?fin) ^^^ "the azimuth of the end point (^fin)^ we can f.ind 
the navigational parameter i?g which corresponds to the position 
of the aircraft exactly on the line of flight. 

If it turns out in the course of the flight that ifg is greater 
than the given value, then in the example shown in Figure 3.29 the 
aircraft deviates from the LGF to the left, and the pointer of the 
zero indicator also moves to the left. With a deviation from the 
LGF to the right, the arrow of the zero indicator will move to the 
right . 

When a flight is made in a direction which is opposite to that 
shown in Figure 3 . UO , the value sin (^A^) and consequently i?g , will 
have a negative sign, so that when the aircraft moves to the left 
of the line of flight the pointer of the zero indicator will also 
move to the left regardless of the fact that the absolute distance 
i?g in this case decreases, and vice versa. 

The calculating device for zero indication of the given line 
of flight is very simple and makes it possible to solve only one 
problem, i.e., to select the course to be followed by the aircraft 
for a flight along a given line of flight, in a manner similar to 
that for a flight from a radio beacon or along the ShchDR bearings, 
using the method of half correction. 

It is better to solve the problem or to use computers to solve 
it using the computation of numerical values of orthodromic coord- 
inates of the aircraft, working on the basis of the indications 
from the DRIDA: 



/28' 



302 



Za^Z^ + RsiniA-iiJ. 

In this case, the angle of shift of the aircraft relative to 
the line of flight is determined very simply as the ratio of the 
change in the coordinate Z to the distance covered between the points 
of two measurements (-^cov^- 

■^cov 

Example . The distance covered between measurement points is 
equal to 60 km. The coordinate Z varies from zero to +4- km. Find 
the required correction in the course for traveling parallel to 
the line of flight. 



Solution 



A4< = arctg-— - = 4°, 



We will assume that it is necessary to travel along this line 

of flight for another 60 km so that the correction in the course 

must be -8° , but at the moment when the coordinate Z becomes zero 

(and if this takes place as we have calculated at 60 km) the course 
will have to increased by 4°. 

If the juncture with this course takes place earlier or later, 
then it is necessary once again to determine the angle of shift 
of the aircraft (A(|;) and to move the aircraft to the right by this 
angle. For example, if we have an initial shift from the desired 
path of 4 km, the aircraft will reach the line of flight in 80 km, 
so th at 



Aij; = arctg 



80 



In the orthodromic system, the problem of checking the path 
for distance and determining the ground speed is solved simply 



For lack of a calculator, the problems in finding the angle 
of shift of the aircraft and checking the path for distance and 
direction, can be solved analytically by means of a navigational / 29 
slide rule. In addition, these same problems can be solved by plot- 
ting on the chart the indications of the azimuth and distance of 
the aircraft as obtained from the beacon. 

On a flight chart which has an indication of the given line 
of flight, two points based on the bearings and distances from a 
ground beacon are plotted every 15-20 min. On the basis of the 

303 



positions of these points relative to the line of flight, we can 
determine their orthodromic coordinates X and Z. It is then easy 
to solve the problems in determining the angle of shift of the air- 
craft (AiJ;), and also the drift angle and the required angle for 
turning the aircraft , the ground speed as well as the wind param- 
eters at flight altitude. 

Details of Using Goniometr-ia-Range Finding Systems at Differ- 
ent Flight Altitudes 

A special feature of ultrashort waves is their ability to be 
reflected from the interfaces of media with different optical densi- 
ties 5 and especially from conducting media in a more sharply pro- 
nounced form than is the case for waves of shorter frequencies . 



In addition, at short wavelengths, the interference which arises 
with combination of oscillations shows up more rarely than in the 
case of long waves, since the small difference in the path of the 
coherent waves in the case of short wavelengths gives a considerable 
shift in their phase. 

Let us say that the antenna 
of a ground transmitter of a gon- 
iometric or range- finding system 
is mounted ax a certain altitude 
above the surface of the ground (point 
A in Fig. 3,30). 

The electromagnetic wave at 
the receiver point B will be prop- 
agated along two paths: 




Fig. 3.30. Diagram Forma- 
tion of Lobes of Maximum 
Radiation . 



(a) Along the straight line 



AB 



(b) Along the broken line ACB with reflection at point C off 
the Earth's surface. 

It is clear in the diagram that straight line AjB is equal 
to the broken line ACB, since the angle of incidence of the wave 
is equal to the angle of reflection. 

Let us draw line AAp in such a way that triangle ABA2 is an 
isosceles triangle. 

Obviously, line AjA2 will represent the path difference of 
the rays in the straight and reflected waves. 

The reflection of radio waves involves a phase shift in the /291 
wave which depends on the optical properties of the reflecting med- 
ium. A purely mirror reflection changes the phase of a wave by 
180°. With a small differene in optical densities of the media. 



304 



when the propagation of the reflected wave takes place along a curve 
with a dip in the reflecting medium, the phase shift can take place 
differently. Let us say that upon reflection, the phase of a wave 
remains fixed. Then the resultant of the direct and reflected signals 
at the receiving point B will have a maximum when the path differ- 
ence of the beams has a value which is an even whole multiple of 
the half wave : 

AS = 2*-—; ii. = 0, 2, 4, . . .2n 

and a minimum if k is an odd multiple of the half-wave: 

x=l,3,5. . .(2rt— 1). 

Thus, there will be an interference pattern for the propaga- 
tion of radio waves in the vertical plane with maxima and minima 
of directionality of the radiation characteristic (Fig. 3.31). 

A change in the phase of the 
wave with reflection from the Earth's 
surface causes corresponding changes 
in the distribution of the maxima 
and minima of the characteristic 
of directionality, but the total 
structure of the interference pattern 
will be similar to that shown in 
the diagram. 




^■»"^«~r53rr;rr 



Fig. 3.31. Multilobe Radi- 
ation Characteristic of 
Electromagnetic Waves. 



The interference pattern of 
shading in the directions of radi- 
of radio waves by objects on the Earth's surface, as well as the 
altitude at which the antenna is mounted above the Earth's surface, 
introduce considerable corrections in the possible range of recep- 
tion of ultrashort waves. 

The operating range of a system, expressed by the approximate 
formula S = 122/H, is maximum at a sufficient power of the trans- 
mitter and sensitivity of the receiver, if the aircraft is located 
in the lobe of the maximum of directionality. However, at certain 
heights and distances, there can be "dips" in audiblity, when the 
aircraft passes through regions of radio shadow or interference 
minima. In addition, special features using USW goniometric-range 
finding devices at high flight altitudes arc related to their range- 
finding sections. 

Rangefinding instruments can be used to measure not only the /292 
horizontal but also the sloping distance from- the aircraft to its 
radio beacon (Fig. 3.32). Therefore, 

s,~ Sjf cos e 

or 



305 



n 



In the special case when the aircraft is passing above the 
radio beacon 

5jj--=0; 5h = //. 

Let us suppose that an aircraft 
is flying along a given route with R^ 
= 10 km, at an altitude which is also 
equal to 10 km with the use of a type 
i?£^sin( ip-j4£ ) = const calculating device. 
With ^-A = 90°, distance R must be equal 
to 10 km, i.e., the aircraft must deviate 
from the given course and pass over 
the radio beacon. The height errors 
in goniome tric-rangef inding devices 
have some important shortcomings in 
their use in the shortrange applications 
and especially in maneuverings in the 
vicinity of an airport. 




Fig. 3.32. Sloping and 
Horizontal Distance to 
Radio Beacon. 



Consideration of altitude errors is very important due to the 
rapidity with which the aircraft passes over the beacon, when the 
errors in measuring the distance change so rapidly that it becomes 
impossible to enter corrections without using special calculating 
devi ces . 

Therefore, the use of goniometric-range finding instruments 
for navigational measurements usually limits the distance from the 
beacon to 3-^■ flight altitudes, i.e., it defines an effective zone 
around the beacon with this radius. 



For example, at a flight altitude of 12 km, the radius of the 
inoperative zone thus defined must be equal to approximately 50 
km . 

Fan-Shaped Gon i ome t r i c Radio Beacons 

The possibilities of aircraft radio compasses are increased 
considerably by using fan-shaped goniometric beacons (Fig. 3.33). 

The picture shows the schematic diagram of a radio beacon. 
The two outermost antennas are set to some wavelength and the power 
for them is in opposite phase. The total characteristic of the 
three antennas gives the multilobe picture of radiation as seen 
in Figure 3.34-. The number of lobes depends on the ratio of the 
length of the base line between the end antennas to the wavelength, 
and their direction depends on the ratio of the phases in the outer 
and inner antennas of the radio beacon. 



/293 



306 



■^■11 ■■■mill 



With a change in the phase of the middle antenna by 180° , the 
positions of the lobes shift to their mirror images (the solid and 
dotted lobes in Fig. 3.3'+), while the points where the dotted and 
solid lobes intersect become axes of equal signals . 



n 




phase shifter 
for t QpO 



t 



transmitter 



main phasej— ' 
shifter 




Fig. 3.33 



Fig. 3.34. 



Fig.' 3.33. Fan-Shaped Radio Beacon. 

Fig. 3.34. Radiation Characteristic of a Fan-Shaped Radio Beacon. 

During the periods between commutations, if we transmit short 
and long signals in the forms of dots and dashes in an overlapping 
pattern, signals of only one type will be heard within the edges 
of the solid lobes (e.g., long signals), while within the limits 
of the dotted lobes, only short signals will be heard. In zones 
of equal signals (near the axes of intersection of the lobes), one 
will hear a continuous tone. If we then smoothly change the phase 
ratio in the end antennas, the lobes will begin to rotate, e.g. , 
to the right, and the phase ratios will change in the reverse direc- 
tion: each of the solid lobes will change places with the dotted 
lobe to the right of it, and each dotted lobe will change place 
with the solid lobe to the right of it. 

Let us assume that an aircraft is located at Point B (see Fig. 
3.34), i.e., within the limits of a dotted lobe, near the right- 
hand limit of the solid lobe, with each operating cycle of the beacon 
beginning after a pause in radiation. 

In this case, at the beginning of a cycle and after the pause, 
several fading dots will be heard, then a continuous signal, and 
finally a long series of dashes. 

If the aircraft is located in the middle of the lobe, the series 

of dots will be equal in length to the series of dashes. At a point/294 

which is close to the right-hand limit of the dotted lobe, the series 

of dots will be longer than the series of dashes. 



307 



A similar picture for the audibility would be obtained when 
the aircraft is located within the limits of the solid lobe with 
the sole difference being that at the beginning of the cycle the 
dashes would be heard, and the dots would be heard only after the 
continuous tone . 




Thus 
of an air 
to plot t 
on a char 
location 
signals i 
bearing 
also to 
them by 
as thin 
is a mult 
of signal 



, to obtain the bearing 
craft, it is sufficient 
he orthodromic lines 
t according to the 
of the axes of equal 
n order to obtain the 
f the aircraft, and 
ivide the angles between 
rthodromes as well 
ines in a ratio which ■ 
iple of the number 
s in the cycle . 



Fig. 3. 
Lines f 
Basis o 
Beacons 



35 . 
or a 
f Fa 



3° . 
(Fig, 



Th 
3 



IS n 
. 35) 



With a sector width between 
the axes of equal signals of 
15° and 60 signals per cycle, 
each signal will correspond 
to 15 min of angle . If the 
sector between the axes of equal 
signals is then divided into 
five parts , the thin orthodromic 
lines will diverge at angles of 
arrow sector will contain 12 signals of the same type 



Grid of Position 
n Aircraft on the 
n-Shaped Radio 



For example, if an aircraft is located at the sector of points 

on the first thin line to the right of the axis of equal signals, 

then 12 dots will be heard which will fade into a continuous tone, 

after which there will be 48 dashes. On the second line there will 

be 24 dots and 36 dashes, 36 dots and 24 dashes on the third, etc. 
At the limit of the sector (the axis of equal signals), a total 

of 60 dots and 60 dashes will be heard. 

Note. Practically speaking, if we consider that part of the 
signals (dots and dashes) are mixed with the continuous tone, the 
number of audible signals will be less than 60, so that after counting 
them the number of audible signals should be taken subtracted from 
60 , then divided in half and added to the number of signals of both 
types that were heard. 

If the aircraft is located between the thin orthodromic lines /29 5 
plotted on the chart, then the line of the bearing of the aircraft 
can easily be found by interpolation of the distance between the 
plotted lines . 

Fan-shaped beacons make it possible to determine very accurately 



308 



the position lines of an aircraft. To do this, with the aid of 
a radiocompass or by generally calculating the path of the aircraft, 
it is necessary to determine the approximate position of the air- 
craft with an error which is no greater than the width of one sector, 
Then, having listened to the operating cycle of the beacon with 
the radiocompass, or with the coherent receiver, we can determine 
the position of the aircraft in the sector. 

A similar method is used to determine the second line of posi- 
tion of the aircraft, using the second fan-shaped beacon, whose 
family of position lines intersects the lines of the first beacon. 
In order not to take into account the' shift of the aircraft during 
the time between the taking of bearings from the two beacons, it 
is desirable to listen to the operating cycles of the two beacons 
simultaneously using two members of the crew who are using two radio- 
compasses or one radiocompass and the coherent radio receiver. 

The accuracy of distance finding with the aid of fan-shaped 
beacons during the daytime is no worse than 0.1-0.3°. Under the 
most unfavorable conditions for distance measurement (in twilight 
when working with the space wave, or at the boundary for the use 
of surface waves), the errors can reach 3 and sometimes 5°. In 
a further zone of distance measurement, and also the short-range 
zone, with operation on a surface wave, the errors do not exceed 
.5-1° . 

The operating range of a fan-shaped beacon during the daytime 
reaches 1350 km on dry land and 1750 km above the sea. At night 
above dry land, this figure is 740 km and above the sea, 950 km. 




Unlike the radio beacons with non-directional and omnidirec- 
tional operation, which are mounted as a rule at the turning points 
of air routes , flight along the bearing line of a fan-shaped beacon 
is only a very rare case. Therefore, the principal method of air- 
craft navigation using fan-shaped beacons is determining all navi- 
gational elements including the wind parameters at flight altitude 
by successive measurements of the LA. 

This method is the most suitable one for fan-shaped beacons 
because the location of the aircraft can be determined in this manner 
with a sufficiently high accuracy. 

It is often desirable to carry out aircraft navigation during /296 



309 



V 



flight using fan-shaped beacons with conventional distance-finding 
from radio stations. For example, in a flight toward a radio sta- 
tion or away from a radio station, it is desirable to use bearings 
from fan-shaped beacons for checking the path for distance and deter- 
mining the ground speed. 

3. DIFFERENCE-RANGEFINDING (HYPERBOLIC). NAVIGATIONAL 

SYSTEMS 




The azimuth lines of position are divergent because as the 
range of operation of a system increases, increasingly high require- 
ments are imposed on the measurement accuracy, while beyond the 
limits of direct geometric visibility it is very difficult to retain 
directionality of transmission or reception due to the effect of 
local relief and especially the ionized layers of the atmosphere. 

The situation is somewhat better as far as the circular p.osi- 
tion lines are concerned. Circular lines do not diverge, so that 
the requirement for accuracy in determining them remains constant 
at all distances . 

In addition, the linear error in determining the position of 
the aircraft in a . goniometric system is proportional to the sines 
of the angles of the propagation errors: 

^Z = SslnAA. 

In range finding systems, these errors are proportional to the cosines 
of the angles of the propagation errors: 

A5 = 5(l — cosA^). 

At small angles, on the order of 6° , the cosine is practically 
equal to unity. Therefore, the errors in determining the distance 
are usually many times less than the errors in the azimuthal shift 
(Fig. 3.36). 

We can see from the figure that the linear error in determin- 
ing the direction CC^ - -SsinAA, and the linear error in distance 
is A5 = ABC - AC n S'Cl-cosAA). 

However, the technical achievement in measuring distance over 
long distances is much more complex than that in measurement of 
the azimuth . 



310 



I nil IB II ■HIIIW I IB I 11 I ■ 



As we saw in the case of USW systems, distance is determined 
by retranslation of signals from on board the aircraft by a ground 
beacon and their reception back on board the aircraft. This method, 
which is relatively easily accomplished at short distances, turns 
out to be unsatisfactory over long distances for use on medium and 
long waves . 



/297 




Fig. 3.36. Errors in Measuring 
Bearing and Range with Reflec- 
tion of Electromagnetic Waves 
from Obstacles: A: Location 
of Ground Radio Beacons; B: 
Point of Mirror Reflection of 
Radio Waves; C: Location of 
the Aircraft (Actual); Ci: 
Measured Position of the Air- 
craft ; AA : Angular Error in 
Propagation . 

time with the signals from the 
quency , we can determine the di 



The best method of measuring 
long distances at the present 
time is the maintenance of a 
calibration frequency on board 
the aircraft. The generator 
for the calibration frequency 
is set at the frequency of a 
ground transmitter and retains 
a given frequency for long periods 
of time by means of special 
stabilizing elements. 

By means of these special 
timing devices, the calibra- 
tion frequency can be converted 
to a lower frequency which is 
synchronous with the signals 
of the ground beacon. If we 
take the signals from the ground 
stations and compare them in 
generator of the calibration fre- 
stance to these radio stations. 



However, this method has not been widely employed due to the 
complexity involved in keeping a highly stable reference frequency 
on board the aircraft, although it offers considerable promise in 
future . 

It is simpler to solve the problem of determining the posi- 
tion line of the aircraft on the basis of the distance between the 
distances to two ground radio stations. In this case, there is 
no necessity for a strict synchronization of the operation of the 
ground installations with those on board. Only the transmission 
of signals from the ground stations must be synchronized. The air- 
craft generator for the calibration frequency in this particular 
case acts only as a central measuring gauge to determine the time 
intervals between the moments of reception of the signals from the 
two radio stations 



ix-i-ons . 

lization of the operation of the apparatus at ground 
IS can be achieved incomparably more easily than syn 
chronization of a ground apparatus with one aboard an aircraft, 
since the distance between ground stations remains constant, thu 
allowing us to use a synchronizing device for two or three stati 
together. In addition, ground installations are not limited by 



311 



size and weight restrictions, not to mention the apparatus on board. 

The methods of measuring the difference in distance to ground /29 8 
radio stations can involve either time (pulse) systems or phase 
systems. Each of these methods has its own advantages and disad- 
vantages . 

An advantage of the phase methods is the higher instrumental 
accuracy of the measurements, but in this case the result of meas- 
uring is obtained ambiguously, i.e., there may be several isophasal 
paths simultaneously with different distances to the ground radio 
stations, which differ in magnitude and are multiples of the length 
of the measured wave. On each of these paths, the result of meas- 
urement is the same and must be used as a measure for determining 
the pathway along which the aircraft is traveling. 

The pulse methods of measuring distance have somewhat less 
instrumental accuracy, but their results are more definite. 

Of course , it should be mentioned that for long-range navi- 
gational systems, the instrumental accuracy of measurement which 
can be attained at the present time both by the pulse and phase 
methods is sufficiently high so that their errors are many times 
less than other systematic errors which are related to conditions 
of propagation of electromagnetic energy. Since the errors in oper- 
ation of the systems under propagation conditions of radio waves 
are practically the same for both pulsed and phase systems, the 
advantage of phase methods of measurement may be restricted only 
to short distances from ground radio stations (in the short-range 
zone of effectiveness). 

Operating Principles of Differential Rangefinding Systems 

Differential rangefinding systems of aircraft navigation consist 
of two pairs of synchronously operating ground radio stations and 
a receiving-indicating apparatus aboard an aircraft. For purpose 
of reducing the amount of ground equipment for the system, one of 
the transmitting radio stations (the master) is made common for 
two pairs so that the system can include three ground radio sta- 
tions . 

The operation of the two slave stations is synchronized with 
the master station by synchronizing signals sent out by the master 
station . 

Let us begin by examining the geometry of the operation of 
one pair of ground radio stations (Fig. 3.37). 

Two ground radio stations are located at points F^ and F2. 
The line connecting points Fj and F2 will be considered as the focal 
line of the base, while the points Fi and F2 are the foci of the 
system . 



312 



Let us assume that at point M there is an aircraft which is 
receiving signals from radio stations F^ and F2. At the beginning, 
the aircraft will receive a signal from the first radio station 
and then from the second. The difference in the distances from 
the aircraft to these radio stations is determined by the differ- 
ence in time between the arrival of the signals in the pulse system 
or by the difference in modulation of the phases in the waves re- 
ceived from the two radio stations in the phase system. 



/299 



1 


\\\ 






I4"J 




^ 


w 


\ 




Hr 


C 


^ 


% 






ill 


/■f. 



Fig. 3.37. Hyperbolic Sys- 
tem of Position Lines. 



We know that the 1 
is the geometrical locu 
the difference in whose 
to two given points is 
value, is called a hype 
given points, to which 
are measured, are calle 
of the hyperbola. Cons 
knowing the difference 
tances to the two radio 
we can always plot the 
line of the position wh 
craft is located. 



ine which 

s of points , 

distance 
a constant 
rbola. The 
the distances 
d the foa-i 
equent ly , 
of the dis- 

stations , 
hyperboli c 
ere the air- 



The hyperbolic line with a difference in distances equal to 
zero becomes a straight line perpendicular to the focal axis and 
dividing the distance between the foci of the system in half (see 
Fig. 3.37). This line is called the imaginary axis of the hyper- 
bola. 

The distance along the focal axis of the family of hyperbolas 
from the foci to the imaginary axis is called the parameter c. 

It is obvious that the difference in distances from the foci 
of the hyperbola to any point along its branches is equal to twice 
the distance along the focal axis from the imaginary axis to the 
peak of the hyperbola. This distance is called parameter a. Accord- 
ingly, the difference in distances from any point to the foci of 
the hyperbola is always equal to 2a. 

The maximum density of hyperbolic lines of position is found 
along the focal axis between the foci of the system, where the dis- 
tance between the peaks of the hyperbola is equal to the differ- 
ence in parameters a. 

The magnitude of the value 2a is measured by navigational param- 
eters of the system so that the accuracy in determining the lines 
of position of the aircraft depends on the accuracy with which this 
parameter is measured. Consequently, an error in determining the 
position line of the aircraft on the focal axis is equal to the error 
in measuring parameter 2a, divided in half. 

As we see from Figure 3.37, the family of hyperbolas is divided 



313 



1 



by a family of position lines. At distances from the center of 
the system which exceed 2e, the hyperbolas practically become straight 
lines, whose direction coincides with the direction of the radii 
extended from the center of the system. Thus, the hyperbolic system 
is converted into a goniometric one. 



However, the density of the lines of position, in this case 
will not be equal along the circumference, as is the case in purely 
goniometric systems . At a given distance from the center of the 
system, the maximum density of position lines will be found at the 
imaginary axis of the hyperbola, gradually decreasing along the 
circumference as they approach the focal axis. The density of posi- 
tion lines at a distance greater than a on the focal axis becomes 
so small that the system becomes unsuitable for determining the 
location of the aircraft. 



/300 




Fig. 3.38. Effective Area of Hyperbolic 
Navigational System. 

The master station of the second hyperbolic pair can be lo- 
cated along the extension of the focal axis of the first pair. In 
this case, the angle of fracture of the base (3) is equal to zero. 

If the master station of the second pair is not located on 
the focal axis of the first pair, there is a definite fracture of 
the base (Fig. 3.38). 

The angle of fracture of the base creates a more favorable 
condition for intersection of the position line in that region of 
application of the system toward which it is directed, since the 
angle of intersection of the hyperbolas in this case approaches 
a right angle and therefore the accuracy in determining the locus 
of the aircraft is increased when two position lines intersect. 

However, this involves a decrease in the quality of the condi- 



314 



The complex of equipment for a hyperbolic navigational sys- /301 
tem aboard an aircraft usually consists of the following: a non- 
directional receiving antenna, a matching block for the antenna 
with d. receiving device, a receiver, and an indicator. 

The matching block serves to produce parameters of the receiv- 
ing antennas when signals are received from ground radio stations. 

Signals received by the antenna are transmitted to the indi- 
cator for measurement of the navigational parameter. The indicator 
has a generator for a calibration frequency, which produces stan- 
dard signals for purposes of measurement, and a number of frequency 
dividers which are required for forming electronic markings on the 
reading scales, as well as repetition frequencies for the scan on 
a cathode ray tube, synchronized with the transmissions of signals 
from ground radio stations. 

The signals which are received pass to the scan of the cath- 
ode ray tube, where the operator controls their size according to 
the amplification of the receiver. The synchronization of the scan 
on the screen is then regulated with the frequency of the received 
pulses so that the latter remain fixed on the screen. 

The operator then mixes a reference (selecting) signal from 
the generator with the signal from the master station, which is 
achieved by the intermittent introduction of small distortions in 
the generator for the calibration frequency, so that the pulses 
of the signals begin to move across the screen. The motion of the 
pulses stops when the signal of the master station coincides with 
the reference signal of the generator (usually a rectangular base 
at the beginning of the scan). 

To measure the time difference between the arrival of the signals 
and the signal from the slave station, a selecting pulse is given 
which is related to the delay in scanning of the reference signal, 
after which the indicator is switched to the reference regime and 
the reading is taken on the electronic scale. 

In some types of receiver indicators, the recording of the 
reading is made on a dial with two or three scales (for different 
scanning rates), for example, beginning with thousands of micro- 
seconds, then hundreds and finally tens, with interpolation up to 
units of microseconds. This provides increased accuracy of readings 
due to the many-fold increase in the scale of the indicator. 



315 



In systems with automatic tracking of the signals from ground 
stations , the time intervals between the moments of arrival of the 
signals are calculated on mechanical counting dials, whose rota- 
tion is related to the delay mechanisms for the selecting pulse. 
The reference signal from the generator is then reinforced, together 
with the signal from the master station, by an automatic frequency 
adjustment of the calibration generator. 

Thus, there is an automatic tracking of the signals from the 
radio station and a constant numerical indication of the output 
navigational parameter of the system, and the difference in dis- 
tances from the aircraft to the ground radio stations is expessed / 30 2 
in microseconds of radio wave propagation. 

In phase systems, by means of distributing elements in the 
calibration generator, its phase is matched with the phase of the 
signals from the master radio station, after which a phasometer 
is used to measure the phase difference between the calibration 
generator and the slave radio station, and the position line of 
the aircraft is determined from this difference. 

As we have already pointed out, if the difference in distances 
to the radio stations includes several periods of the modulating 
frequency of the ground stations, the determination will be ambig- 
uous . 

The solution of the ambiguity of this estimate can be accom- 
plished by several methods . 

(1) An initial setting of the coordinates of the aircraft 
with automatic tracking of the radio station signals. In this case, 
using known coordinates of the aircraft (e.g., on the basis of the 
visual determination of the aircraft location), the indicator is 
set by hand to show the isophasal line on which the aircraft is 
located. If constant tracking of the radio station signals is then 
carried out, completely reliable readings of the position line will 
be obtained. 

A shortcoming of this system is the necessity to relate the 
aircraft to the local terrain on the basis of the initial reading 
of the hyperbolic coordinates. In addition, during flight, there 
may be readings of other isophasal lines, due to interference, which 
can be determined and corrected only by a repeated relation of the 
aircraft to the local terrain by means of other methods . 

(2) By modulation of the carrier frequency of the ground radio 
stations at very low frequencies (with long modulating waves , consid- 
erably increasing the possible difference in distances from the 
aircraft to the radio stations). In this case, at a low frequency 
phase, the rough position of the isophasal line of a carrier fre- 
quency or the frequency of the second modulation with a small, long 
period can be determined. 



316 



(3) By using several carrier frequencies for the ground radio 
stations, the isophasal line can be considered to be determined 
if it is simultaneously on the isophasal lines for all frequencies 
at which the measure ment is carried out (usually three frequen- 
cies, since two will be inadequate in some cases). On adjacent 
isophasal lines, for each frequency used, the isophasal lines of 
other frequencies will not coincide with the readings of the phaso- 
meter . 

Navigational Applications of D i f f e ren t i a 1 - Rangef i nd i ng Systems 



Dif f erential-rangef inding navigational systems , like fan-type 
beacons, are intended primarily for determining the locus of the 
aircraft on two position lines. Therefore, the principal method 
of aircraft navigation using these systems is the determination 
of the navigational elements on the basis of a series of determin- 
ations of the LA. 



/303 



By recording and plotting on a chart a series of points for 
the locus of the aircraft, recording the time at which they were 
passed, and using a scale ruler and protractor to measure the dis- 
tances between them on the chart, as well as the distance from the 
first recording of the LA to the second, it is easyi to determine 
the speed and flight angle of the aircraft. 

4' = "1 ,2! 
s, 



'1,2 



where a ^ _ 2 is the azimuth of the second recording of the Lh from 
the first and Si 2 is the distance between the recordings of the 

LA. 

The drift angle of the aircraft is determined as the distance 
between the actual flight path angle and the average course of the 
aircraft over the segment between two successive recordings of the 
LA: 

a = ^li - Yav 

With a known groundspeed and drift angle, taking the airspeed 
into account as well as the course to be followed, the wind param- 
eters at flight altitude can be determined with the aid of a naviga- 
tional slide rule. 

In special conditions, when the flight direction coincides 
with one of the branches of the hyperbolic flight lines , the flight 
can be made along the latter. To do this, it is sufficient to main- 
tain a constant reading for the calculator of hyperbolic coordinates 
of one pair. The family of position lines for the second pair in 
this case is used to monitor the path for distance. 



317 



{^ 

M 



Monitoring the path for distance by means of the readings of 
one of the counters can be used in the case when the aircraft navi- 
gation in terms of direction is carried out using two devices, e.g. 
the USW bearing of a goniometric system or a fan-type beacon. 

To increase the feasibility of using dif f erential-rangef ind- 
ing systems, the hyperbolic coordinates can be converted to ortho- 
dromic or geographical ones (see Chapter I, Section 7). 

In some hyperbolic systems of aircraft navigation, e.g., that 
of Decca and Dectra (England), simplified methods of automatic plot- 
ting of the aircraft course on a special chart use the movement 
of a pen in mutually perpendicular directions. For this purpose, 
special charts are made on which the hyperbolic lines of the first 
and second family are laid out at right angles. Naturally this 
results in distortion of the contours of the terrain on the chart, 
as well as the scale and geographic grid, and the line of flight 
of the aircraft is also bent. 



/304 



Such a method of recording has a number of shortcomings (e.g., 
in relation to the calculation of orthodromic coordinates for the 
air craft), but it is very easy to achieve from the technical stand- 
point and its shortcomings are considerably reduced if the path 
of the aircraft has markings for distance. 

Methods of Improving Differential Rangefinding 
Navigational Systems 

The design of hyperbolic systems contains elements whose im- 
provement leads to a conversion of the system to a hyperbolic-range- 
finding or hyperbolic-elliptical system. 

Such elements include the standard frequency generators aboard 
the aircraft. When these generators operate in a highly stable 
regime, the reference signals from these generators can be kept 
so precise that it becomes possible to measure distances to one 
of the ground radio stations. To do this, it is sufficient to combine 
the phases of the frequencies of the generator aboard the aircraft 
and the ground radio station with an initial distance setting (e.g., 
the takeoff point of the aircraft). Further changes in distance 
can be determined by the deviation of the phases of these frequencies 
or by the deviation of the pulse signals, if the system is oper- 
ating in a pulse regime. 

Measurement of distance in connection with one pair of hyperbolic 
position lines makes it possible to considerably improve the accur- 
acy with which the locus of the aircraft is determined over long 
distances, and one pair of g"?ound radio stations will suffice for 
measurements. However, the conditions for measurement between the 
foci of the system near the focal axis will remain unfavorable (Fig. 
3. 39) . 



318 



It is more advantageous in this case to use the hyperbolic 
network of position lines (see Chapter I, Section 7). 

However, since we know the difference between the distances 
to the two radio stations as well as the distance to one of them, 
it is easy to determine the sum of the distances to these radio 
stations, e.g., if 



so that 



^2 > Si and Aa- = ivj — Sj, 
S2 = Si + AS 



and 



Si + S2 = 2Si + M>, 



Similarly, for the case when S2 < Si , 



S, + S2 = 2Si 



■ AS. 



Therefore, in order to obtain the number of the hyperbola, 
it is sufficient to use the difference in distances, while to obtain 
the number of the ellipse, we must double the distance to one of 
the radio stations and add the difference in distances with the / 30 5 
corresponding sign. 

One great advantage of the hyperb oil c- e llipti cal network is 
the orthogonality of the intersection of the position lines at any 
point in the field which is involved. On individual sheets of the 
chart, the hyperbolic-elliptical network has the appearance of a 
nearly rectangular grid with noticeable curvature of the position 
lines only in the vicinity of the foci of the system. 

erbolic-range- 
ly a hyper- 
em , a sys tem 
e accuracy 
nates of the 
ed over long 
d several fold, 

range of appli- 
is also consid- 
h the use of 
nd radio sta- 




When using a hyp 


finding (and especial 


bolic-elliptical syst 


of position lines) th 


with which the coordi 


aircraft are determin 


distances is increase 


so that the practical 


cation of the system 


erably increased, wit 


only one pair of grou 


tions . 



Fig. 3.39. Combination of 
Hyperbolic and Rangefinging 
Navigational Systems. 



It should be mentioned, however, 
that a serious obstacle to the devel- 
opment of systems of long-range 
navigation for use on high-speed 
aircraft is the low noise sta bility 



319 



of operation of these systems, since only very long waves can be 
used for navigation over long distances . 

4. AUTONOMOUS RADIO-NAVIGATIONAL INSTRUMENTS 

In recent years, there has been a considerable increase in 
the use of radio navigational instruments which are housed completely 
aboard the aircraft and operate without the need for ground facil- 
ities . Such instruments are called autonomous radio-navigatvonat 
instruments or, if their operation is combined with some other naviga- 
tional equipment aboard the aircraft, autonomous navigational sys- 
tems. These include aircraft navigational radar, Doppler systems 
for aircraft navigation, and radio altimeters. 

All autonomous radio-navigational instruments operate on ultra- 
short waves, since they have a very high (practically complete) free- 
dom from interference during operation (not counting artificial 
interference) . 

Doppler meters for measuring the ground speed and drift angle 
of the aircraft measure the motion parameters of the aircraft directly 
relative to the Earth's surface, which clearly differentiates them 
from all existing forms of navigational equipment, especially with 
regard to problems of automation of aircraft navigation and pilot- /306 
age of aircraft. 

Aircraft Navigational Radar 

Aircraft navigational radar is a very flexible and effective 
method of aircraft navigation during flight over land or sea close 
to coastal regions . 

In terms of the geometry of their use, aircraft radar devices 
can be included among the goniometri c-rangef inding systems. However, 
in comparison to the goniometric-rangef inding navigational systems, 
they have a number of tactical advantages: 

(1) The high saturation of ground landmarks makes it possible 
to select the most suitable ones for measurement in navigation. 

(2) The lack of errors in determining the bearings of land- 
marks from the radio deviation of both the aircraft itself and the 
local relief, something which affects all non-autonomous naviga- 
tional systems . 

(3) The possibility of visualizing ground landmarks with the 
purposes of determining ground speed and drift angle to a better 
degree than with optical methods . 

( M- ) The possibility of identifying dangerous meteorological 
conditions in flight (thunderstorms, powerful cumulus and cumulonim- 
bus clouds ) . 



320 



(5) The high accuracy and ease of the measurements using only 
one operational frequency. 

At the same time, the navigational use of aircraft radar has 
several shortcomings: 

(a) The bearing of the aircraft can be used only as a basis 
for measuring the aircraft course, thus lowering the accuracy of 
distance findings. 

(b) A certain amount of experience is needed for correct recog- 
nition of ground landmarks and the possibility of errors in deter- 
mining a landmark, since they are not labelled. 

The operating principle of radar is based on the ability of 
electromagnetic waves at high frequencies to be reflected from objects 
located along their propagation path (from the interface between 
media with different optical densities). 

To obtain a panoramic image of the terrain, a rotating or scan- 
ning antenna is used to cover a certain sector, so that its posi- 
tion must be synchronized with the position of the scanning beam 
on the screen of a cathode ray tube. In addition to synchronizing 
the direction of the antenna with the scanning direction of the 
beam, it is also necessary to ensure that the beginning of the scan 
is synchronized with the moment when the USW pulses are omitted from 
the antenna transmitter 




Thus, the radar screen shows the following: 

(a) The direction of the object on the basis of the antenna 
position at the moment of emission and reception of the signal. 

(b) The distance to the object on the basis of the time re- 



321 



quired for the signal to travel between the moment when it is emit- 
ted to the moment when it is received. 

(c) The nature of the object, on the basis of the brightness 
of the scanning beam at the point where the reflected wave is received, 



indicacor 




-\ 



receiver- 
transmitter 



modulator-l^ont^^ 
panel 



J 



][ 



antenna 



antenna 
mechanism ' 



VWrv/N/- 



generator of 
standard freaue'ncy 
and dividers 



Fig. 3.40. Diagram of Aircraft Radar. 

The radar screen has a long afterglow so that when the antenna 
has made a complete revolution, the screen still shows a trace of 
all the irradiated objects on the Earth's surface which are located 
in the field scanned by the radar. 

The main section of the radar, controlling the operation of 
the entire system, is the standard-frequency generator with fre- 
quency dividers for forming distance markings and a signal-trans- 
mission frequency synchronized with the saw-tooth scanning image 
on the screen (Fig. 3 . M-0 ) . 

The signals from the standard-frequency generator reach the /308 
modulator, where they are converted to high-voltage rectangular 
oscillations of a special length. The high-voltage pulses from 
the modulator pass to the transmitter magnetron, where high-fre- 
quency groups are generated according to the pulse length. 

The high frequency reaches the antenna through a wave guide 
and is radiated into space. 

At the same time, in synchronization with the pulses of high 
voltage which are sent to the transmitter, the scanning generator, 
forms a saw-tooth voltage which controls the scanning beam on the 
screen. The scanning rate depends on the steepness of the slope 
for the saw-tooth waves. At a low scanning rate, a fine image scale 
is obtained as well as long-distance detection of objects. When 
the scanning rate is increased, the scale of the image decreases 
proportionately with the distance covered by the radius of the screen. 
The control of the scanning rate is achieved with the aid of a switch 
on the control panel. 



322 




The resolving power of the radar in terms of azimuths is a 
function of the sharpness of the directionality of the antenna beam, 

The azimuths of ground landmarks can be determined immediately 
by the position of the antenna (and therefore by the scanning line 
on the screen), and the antenna mechanism is fitted with a selsyn 
mechanism for tilting the indicator. The azimuth reading is made 
on a scale located along the periphery of the screen. 

To measure the distance to a landmark, pulses from the fre- 
quency divider are sent to the receiver (and therefore to the scan- 
ning beam). These pulses increase the brightness of the beam at 
certain distances from the center of the screen, forming circular 
distance markings. 



When using the radar on different scales the distance mark- 
ings are shifted to different distance intervals. For example, 
with a scale of 10 km for the radius of the screen, the markings 
are usually 2 km apart; when using scales from 10 to 100 km, the 
markings are 10 km apart; at a scale of 200 km, they are 20 or M-0 

Vm ^ rt ;:! rti- _ 



Now let US follow the path of the high-frequency pulses from 
the transmitter to the object and back again, and see how they control 
the brightness of the scanning beam. 



The high-frequency pulse passes through the wave guide to the 
radiating horn of the antenna, after which it is shaped into the 
required directional diagram for radiation by means of a reflec- 
tor. Usually, the directionality of the antenna in the horizontal 
plane is made as sharp as possible. To do this, it is necessary 
for the phase of the beam when emerging from the antenna to remain 
constant over its entire perpendicular cross section (Fig. 3.41), 
i.e. , the reflection in this plane must have a shape such that the 
wave path from the horn radiator to the surface of the reflector 
and along its chord of emergence is uniform. 

The characteristic of directionality of the radiation in the 
vertical plane must be such that the illumination ot the terrain 
from the vertical of the aircraft is as uniform as possible over 
the entire effective radius of the radar. To do this, it is neces- 
sary to have the maximum amount of wave energy transmitted at small 



/309 



323 



angles to the plane of the horizon, i.e., over the maximum range, 
and to have the smallest amount of energy radiated along the vert- 
ical of the aircraft. Such a aharaateT'ist-ho -is catted the cosecant- 

square , i.e., the reflectors in the vert- 
ical plane are given a shape such that 
the amount of energy radiated into space 
is roughly proportional to the square 
of the cosecant of the angle of the plane 
of the horizon to the propagation direc- 
tion . 




In some types of radar, an acicular 
characteristic of directionality is employed, 
i.e., one which is sharpest in both the 
horizontal and vertical planes, combin- 
ing it with the cos esant-square in the 
vertical plane, e.g., by a scanning cycle. 
This is achieved by using specially shaped 
reflectors with a telescoping deflector 
or by sending energy to the antenna by 

different wave guides for the acicular and cosecant-square antenna 

characteristics of the radar. 



Fig. S.M-l. Radar An- 
tenna for Use Aboard 
Aircraft . 



Antennas with cosecant-square characteristics are used for 
circular-scan radar, mounted below the fuselage of the aircraft. 
Antennas with combined radiation are used for sector-scan radars 
and are mounted in the nose of the fuselage to cover only the area 
ahead of the aircraft. In this case, the radar screen is made with 
the center displaced so that the maximum area of the screen can 
be used . 

Usually, the tilting of the antenna in the vertical plane (and 

therefore the characteristics of directionality of the radiation) 

is adjusted manually by means of a special electrical device and 

a switch on the control panel of the radar. 



The transmitting antenna of the radar acts simultaneously as 
a receiving antenna, since the directional characteristics of the 
antenna are reversed, i.e., used both for emitting and receiving 
the wave energy . 



/310 



In order for the pulses of wave energy emitted from the trans- 
mitter not to return immediately to the wave guide of the receiver, 
special arresters are used which block the wave energy from enter- 
ing the receiver at the moment when the transmitter is operating. 

The transmission frequency of the pulses of wave energy from 
the transmitter is set so that the time intervals between them are 
not shorter than those required for propagation of electromagnetic 
waves to the most distant object at a given operating range for 
the radar and for its return to the aircraft. When using the radar 
at large-scale settings, the decrease in the pulse duration is 



32^■ 



accompanied by an increase in the transmission frequency, thus pre- 
serving the average power of the transmitter. Hence, the recep- 
tion of the reflected signals takes place in the time intervals 
between the pulses of wave energy emitted by the transmitter. 

The radar receivers have special vacuum devices (klystrons 
for generating high frequency) which play the same role as heter- 
odynes in conventional receivers. 

The signals received by the antenna are mixed with the fre- 
quency of the klystron; an intermediate frequency is produced which 
then goes on (after detection and amplifi cation ) to control the bright- 
ness of the scanning beam. 

In addition to the special features of the radar which we have 
discussed above, the receiver has additional circuits and control 
units. In particular, to allow the frequency of the klystron to 
be changed, there is an automatic frequency adjuster (AFA), etc. 

For improved contrast of the image on the screen, in addition 
to the devices for adjusting the overall amplification of the receiver, 
the operator of the radar can use a separate signal amplifier which 
operates at high and low levels. This makes it possible to dis- 
tinguish shaded or illuminated objects on the Earth's surface as 
desired. For example, to examine populated areas, the high-level 
signals are increased and the low-level signals are reduced (by 
decreasing the brightness of the background of the screen). To 
pick out rivers and lakes, the low-level signals are increased, 
thus impr-oving the visibility of shaded objects against a brighter 
general background. 



It should be mentioned that for the formation of high-frequency 
pulses by the transmitter, very high voltages must be produced in 
the modulator; this means that at high altitudes (i.e., at low atmo- 
spheric pressure), there may be flashovers in the wiring of these 
units. Therefore, these units (including the wave guides of the 
transmitter) are hermetically sealed and the required pressure is 
maintained in them by a special pump or by systems for pressurizing 
the aircraft cabin. 



/311 



Indicators of Aircraft Navigational Radars 

The aircraft radar is an autonomous goniometr i c-range- find- 
ing and sighting device, so that its indicator must be made so that 
all required navigational measurements can be performed satisfac- 
torily with it. 

Circular indicators are the ones which are of greatest inter- 
est from the navigational standpoint (Fig. 3.42). 

The center of this indicator, marking the 'position of the air- 
craft against the panorama of the field of vision, coincides with 



325 



the center of the screen. Around the edge of this screen is a scale 
of bearings, which can be rotated manually; in the upper part of 
it is a course marking which shows the position of the longitud- 
inal axis of the aircraft. The scale of bearings is set to its 
own divisions by means of a "course" ■ rack and pinion device, for 
setting the course of the aircraft by the course markings, accord- 
ing to the readings of the course instruments . 

The sighting lines of the indicator are marked on the protec- 
tive glass of the screen, which can be rotated by means of "sight" 
rack and pinion. For convenience in sighting, three movable points 
for longitudinal sighting lines are provided, and one transverse 
sighting line is provided for indicating traverses when flying over 
landmarks . 



When the radar is operating, 
circular distance markings appear 
on the screen, and the deflection 
of the luminous course lines of the 
aircraft may also be included. 

In the lower part of the indicator 
unit, in addition to the "course" 
and "sight" adjustments, there are 
other controls: "scale illumination", 
"beam scan focus", "beam brightness 
adjustment", and some types of indi- 
cators also have a "vertical and hori- 
zontal centering of scan". 

Thus, the circular screen of 
the radar can be used to measure bear- 
ings precisely or determine the course 
angle of a landmark, its distance, 
as well as the provisional line of 
motion of the landmark for purposes of determining the drift angle /312 
and the ground speed on the basis of the traverse of the flight 
over the landmark . 




Fig. 3.M-2. Indicator for 
Radar with Circular Screen 



Sector-type radar screens have somewhat fewer possibilities 
(Fig. 3.43) . 




Instead, the screen is fitted with a system of divergent lines 
for the course angle of the landmark (CAL). The determination of 
the bearings in this case is made by adding the course angle of 
the landmark to the course of the aircraft by the formula: 



326 



TBL = TC + CAL; 
TBA = TC + CAL +_ 180° + 6, 

It is very difficul 
possible on these indica 
the moment of flying ove 
of landmarks . 

Instead of visualiz 
of landmarks, the soluti 
problems on these indica 
often accomplished by a 
measurements of the LA o 
exception to this is con 
marks which move across 
the immediate vicinity o 
marking, and can be used 
the drift angle by the p 
their shifting, using th 

course angles and the ground speed when passing ove 

markings on the screen. 




Fig. 3.43. Indicator 
for Sector-Type Radar, 



t and not always 
tors to determine 
r the traverses 



ing the movement 
on of navigational 
tors is more 
succession of 
n a chart . An 
stituted by land- 
the screen in 
f the course 

to determine 
arallelism of 
e lines of the 
r the distance 



Nature of the Visibility of Landmarks on the Screen of an 

Aircraft Radar 

For purposes of aircraft navigation using aircraft radar, the 
following landmarks can be used: 

1. Large populated areas and industrial enterprises. The 

visibility and outlines of these landmarks depend on the number 
and location of metal structures and coverings in the object. Popu- 
lated areas and industrial enterprises appear as bright spots on 
the screen, as a rule, with sharply bounded outlines. This means 
that the outlines of the landmarks coincide closely with their out- 
lines on a chart or as they are seen by visual observation, as groups 
of structures with non-metallic coverings show up much less clearly / 313 
and are visible from shorter distances than metal structures and 
coverings . 

Populated areas show up most clearly with maximum amplifica- 
tion of the high-level signals and a minimum amplification of the low- 
level signals . 

2. Rivers and lakes. During the summer, these landmarks are 
visible as dark areas and spots whose outlines match those of the 
landmarks against the a lighter background of the surrounding ter- 
rain. In the winter, when these bodies of water are covered by 

a smooth layer of ice, only the river valleys are seen, especially 
against forested areas . Ice packs on rivers can be seen in the 
form of bright spots against a darker background of snow covered 
banks. Rivers and lakes can be distinguished by amplifying the 
low-level signals to increase the brightness of the entire background 



327 



I 



of the screen. Then the dark objects will be observed as still 
darker areas against the light background. 

3. Mountains. These landmarks appear on the radar screen 
in a form which is very close to their natural one, i.e., as they 
appear to visual observation. Mountains can be distinguished by 
a suitable selection of signal amplification at both high and low 
levels . 

h. Forested areas. Landmarks of this type can only be seen 
clearly in winter, against a general background of snow-covered 
surface, by amplifying the low-level signals; in summer, against 
a background of vegetation and cultivated areas, forests are seen 
very dimly and cannot be used as landmarks. 

5. Highway and railway bridges. These landmarks show up espec- 
ially well against the background of large rivers. The railways 
themselves show up clearly only when there are embankments or steel 
structures for supporting catenaries for electrified railways. 

In summer, the development of powerful cumulus and cumulonim- 
bus clouds shows up very clearly on radar screens. Areas which 
are dangerous for flight (with a large-droplet structure, and there- 
fore with intense turbulence and high intensity of electrical fields) 
appear on the screen in the form of bright spots with diffuse edges. 

These storms can be distinguished very well with maximum ampli- 
fication of high-level signals and minimum amplification of low- 
level signals. Amplification of low-level signals reduces the con- 
trast of the images of these dangerous storms, but areas of radar 
shadows begin to appear, which are very clear on the screen and 
are characteristic signs of storm clouds. 

In observing terrestrial landmarks and clouds in which there 
is thunderstorm activity, it is necessary (besides adjusting the 
amplification level of the receiver) to choose the proper inclin- 
ation of the radar antenna. As a rule, landmarks which are located 
close to the aircraft are observed with an increased inclination 
of the antenna downward, while those further away (and storm clouds ) /31'4 
are viewed with a slight inclination downward or with the antenna 
aimed upward, depending on the flight altitude and the viewing range. 

The inclination of the antenna can be selected to provide the 
optimum clarity of the images of the landmarks on the screen. 

Use of Aircraft Radar for Purposes of Aircraft Naviga- 
tion and Avoidance of Dangerous Meteorotogiaat Phenomena 

Aircraft radar can be used to solve all problems of aircraft 
navigation, beginning with the recognition of landmarks over which 
the aircraft is flying and ending with measurement of all basic 
elements of aircraft navigation. 



328 



For recognition of terrestrial landmarks, it is desirable to 
use operating scales of the radar which coincide with the scales 
of flight charts . 

With an indicator screen radius of 55 mm, an image scale of 
1:1,000,000 produces a range of 55 km on the screen. This oper- 
ating scale for the radar is most suitable when using maps with 
a scale of 10 km to 1 cm. 

Hence, when using charts with a scale of 1:2,000,000, one must 
use a radar scale of 100 km; 110 km is possible, if the design of 
the radar permits 



The 



harpest distinction of radar landmarks is obtained by 
proper selection of contrast in the image by using var- 



using the proper selection of contrast in the image by using var- 

thighandlowlevels,adjust- 
o the proper angle, and settin 



ious amplifications of the signals a_ ---„.. _ 

ing the inclination of the antenna to the proper angle, and setting 
the beam brightness on the screen. 

The location of the aircraft can be determined very accurately 
in terms of the bearing and direction from a point landmark. Point 
landmarks in this case can be the centers of populated areas, charac- 
teristic features of the shores of rivers and lakes, individual 
mountain peaks, etc. 




In using sector-type radars, the bearing of the aircraft is 
obtained by adding the aircraft course and the course angle of the 
landmark, as is done when using aircraft radio compasses with non- 
integrated indicators. 

As in the case when USW rangefinding systems are used, the 
measurement of distances with an aircraft radar means that the radar/315 
measures not the horizontal but the oblique distance (OD) of the 
landmark. Therefore, when measuring distances to landmarks, which 
are less than five times the flight altitude (ff), the measurement 
must include a correction AR , which always has a negative sign: 



A/? = — (VOD2 —H2~ R); 
;?=OD_A/?, 



where OD is the oblique distance , 
is the horizontal distance. 



H is the flight altitude , and R 



329 



If the oblique dista 
altitude (the correction 
the oblique distance), th 
zero. This is also refle 
a dark spot appears in th 
sharp limit for the begin 
of image formation is sep 
a distance which is equal 
scale. This spot is call 
uring the true altitude o 



nee to the landmark is equal to the flight 
for the flight altitude becomes equal to 
e horizontal distance will be equal to 
cted in the panorama of the image , when 
e middle of the indicator screen with a 
ning of image formation. The beginning 
arated from the center o£ the screen by 
to the flight altitude on the scanning 
ed the attimetrat and is used for meas- 
f flight above the local relief. 



TABLE 3.1, 



Oblique 
distance 

KM 



O 
10 

15 
20 
25 
30 
35 
40 
45 
50 



Flight f altitude . km 
3 i 4 I 5 I 6 I 7 I 8 I 9 I 10 I 11 I 12 



0^0 
0,0 
0.0 
0,0 
0,0 
0,0 
0,0 
0.0 
0,0 
0,0 



0,5 
0.0 
0,0 
0.0 
0,0 
0.0 
0,0 
0.0 
0.0 
0.0 



1.0 
0,5 
0.0 
0.0 
0.0 
0,0 
0.0 
0,0 
0.0 
0,0 



2,0 
1.0 
0.& 
0,0 
0,0 
0,0 
0,0 
0,0 
0.0 
0^ 



rre 


ctior 


LSI, KM 






V 


5.0 
















1,5 


2,0 


3,0 


4.0 


6,0 


10,0 


• 


— 


I.O 


•1.5 


2.0 


2,5 


3^0 


4,0 


5,0 


6»0 


0.5 


1,0 


1.5 


2.0 


2,5 


3,0 


3.5 


4,0 


0,0 


0.0 


0,5 


1,0 


1.5 


2,0 


2.5 


3.0 


0.0 


0.0 


0.0 


0.5 


1,0 


1,5 


2.0 


2,5 


0,0 


0,0 


0.0 


0.0 


0,5 


1,0 


1,5 


2,0 


0,0 


0,0 


0,0 


0,0 


0.0 


0,5 


1,0 


1,5 


0.0 


0,0 


0,0 


0.0 


0,0 


0,0 


0,5 


1.0 


0,0 


0,0 


0,*0 


0,0 


0,0 


0,0 


0,0 


0,0 



For making corrections in the measured distances for the flight 
altitude, we can use Table 3.1. 

The location of the aircraft can be determined by means of 
aircraft radar and directly in stages of orthodromic coordinates. 
To do this, the scale of bearings on the indicator must be set not 
to the course of the aircraft, but to the lead angle (LA) on the / 316 
course of the aircraft relative to a given orthodromic path angle 
of flight or drift angle. 

The sighting device can then be used to determine the path 
bearing of the landmark (PEL). For example, with LA = y-^ = -10°, 
the bearing scale must be set to 350° opposite the course marking; 
with a course angle of 40°, the path bearing of the landmark (PEL) 
will be equal to 30°; with a negative drift angle, and therefore 
a positive lead angle, such as 10°, e.g., the bearing scale must 
be set to 10° opposite the course marking. 

Knowing the path bearing and the distance to a landmark (i?) , 
we can very simply determine the orthodromic coordinates of the 
aircraft : 



330 



X = X - i?cosPBL = X - i?sin(90°-PBL) ; 
Z = Z - /?sinPBL. 

These formulas are different from (1.71) and (1.71a) only in 
the sign of the second terms on the right-hand side . This is explained 
by the fact that when we are using goniometric-rangef inding systems , 
the direction is reckoned from a ground beacon to the aircraft, 
while in this case it is reckoned from the aircraft to a ground 
landmark . 

Example . The radar landmark has orthodromic coordinates Jj^ 

= 250 km; Zj_ = 80 km and is observed with a path bearing of ^0° 

as a distance of 125 km. Find the coordinates of the aircraft X 
and Z. 

Sol uti on : 

A'= 250 — 125-sin 50° = 250 — 96 = 156 km. 
Z = 80— 125-cos40'' = 80 — 82 = — 2k^. 

Thus we have found that the aircraft is located at a distance 
of 156 km from the last PBL, 2 km to the left of the LGF , without 
resorting to a plotting of the bearings on the flight chart. 



In solving this problem, it 

is very convenient to use the cal- 

(S) 82 SB '^^ culating navigational slide rule. 



(D y '0' ^ 



To do this, the triangular 
Fig. 3.1+4. Determination of index on scale 4- is set to the 
Orthodromic Coordinates of distance of the landmark along 
an Aircraft on the NL-IOM. scale 5. The slider indicator 

is then set on scale 3 to the mark- 
ing which corresponds to 90° -PBL and PBL, while the values R sin 
(90° -PBL) and R sin PBL are set on scale 5 (Fig. 3.44). 

After this, there remains only the calculation of these values 
from the coordinates of the landmark and the determination of the 
aircraft coordinates. 

The problem is considerably simplified when the path bearing 
of the landmark is equal to 90° (flight over the traverse of the 
landmark). Then 

X = X^; Z = Z^ - R. 

It should be mentioned that the determination of the aircraft / 317 
coordinates when flying over the traverse of a landmark is advan- 
tageous, since in this case the errors in measuring the path bear- 
ing of the landmark have absolutely no effect on the accuracy of 
determination of the lateral deviation of the aircraft from the 

331 



line of flight. This is very useful for monitoring the path in 
terms of direction and correcting the course of the aircraft by 
using autonomous Doppler measurements of the ground speed and drift 
angle . 

This method of determining the orthodromic coordinates of an 
aircraft is also suitable for use with sector-type radars. In this 
case, the path bearing of the landmark is determined by the formula 



PEL 



CAL + LA 



This problem can then be solved in the same way as for circ- 
ular-screen radars. However, on sector-type screens as a rule, 
it is not possible to determine the markings of the traverse of 
flights over landmarks. Therefore, for an accurate control of the 
path, taking into account the reduced accuracy of direction finding, 
due to the lack of sighting lines, it is necessary to image the 
landmarks at course angles which are as large as possible. 



The ground speed of 
mined most easily with th 
measurements of the LA, ( 
dromic coordinates, when 
the aircraft on a chart, 
scribed above. However 
speed on the basis of sue 
aircraft, is insufficient 
angle of an aircraft. Th 
ment to be made along a g 
the drift angle quite fre 
of successive measurement 
a large base for measurem 



the aircraft and drift angle can be deter- 
e aid of aircraft radar by using successive 
locus of the aircraft), especially in ortho- 
it is not necessary to plot the locus of 
The essence of this method has been de- 
the method used for measuring the ground 
cessive measurements of the locus of the 
ly practical for measurements of the drift 
e fact is that for an accurate measure- 
iven path, it is necessary to determine 
quently and rapidly, so that the method 
s of the locus of the aircraft requires 
ents . 



In some cases, it is advisable to use other methods for deter- 
mining the ground speed (e.g., if visual points lie in the field 
of vision of the radar which do not allow the position of the air- 
craft to be determined) since they do not appear on the chart. How- 
ever, they are suitable for determining the drift angle and the 
ground speed by visual methods. 

There are several methods of determining the drift angle and 
the ground speed by visual means. Let us discuss several of them 
which are most often employed: 

1. Measurement of the drift angle of an aircraft on the basis 
of the secondary Doppler effect. The directionality of the charac- 
teristic of radiation from an aircraft radar in the horizontal plane 
is made as narrow as possible. The narrower the beam for the prop- 

■ -■ - - -^ - "■ ^ netic waves, the better the resolving power 

^.•--, j.-._^_^_-__ / ^icular to the radius 



s made as narrow as possible. ihe narrower tne beam ror tne prop- 
.gation of electromagnetic waves, the better the resolving power 
if the radar in a tangential direction (perpendicular to the radius 
if the scan). However, in order to produce a very narrow charac- /31i 
:eristic of radiation, we must use an antenna reflector on the radar 
'hich has very large dimensions. Therefore, the practical width 



which has 
of 

332 



The widening of the characteristic of directionality within 
these limits is undesirable in principle for surveying the terrain, 
but can be used very advantageously for measuring the drift angle 
by the so-called secondary Doppler effect. The essence of this 
method is the following. 

Let us say that we have stopped the rotation of the radar an- 
tenna at a certain angle to the direction of the aircraft's motion 
(Fig. 3.45). 




Fig. 3.45. Creation of the 
Secondary Doppler Effect. 



In the picture, we can see 
the reflection of the electromag- 
netic waves from the elementary 
area S which we have selected. 

The high frequency reflected 
from the Earth's surface, when re- 
ceived aboard the aircraft, will 
not be equal to the frequency radi- 
ated by the radar, but will have 
a certain positive or negative fre- 
quency shift which is called the 
Doppler effect. 



Let us also note that the Doppler effect is proportional to 
the cosine of the angle between the direction of the aircraft's 
motion and the direction of the wave propagation (i.e., g-a). Angle 
3 here represents the course angle of the antenna position of the 
radar, while the angle a represents the drift angle of the aircraft. 

For the sake of simplicity, let us consider the Doppler effect 
only for two extreme limits of the beam with a common character- 
istic of radiation directionality, the left-hand beam is marked 
L and the right-hand beam R in our diagram. 

The solution of the characteristic will be represented by the 
angle 6 , so that 






where /p is the Doppler frequency. 

Thus, we see that the Doppler effect on the left-hand edge 
of the beam is greater than on the right-hand side, so that the 
frequency received by the antenna from the left-hand side of the 
beam will be somewhat higher than that from the right. 

The frequencies of the left (L) and right (R) boundaries of 
the beam will be combined in the receiver and produce an intermediate 



333 



frequency as follows 



/319 



/. 



^D^ 



f^^. 



which will amount to amplitude modulation of the received signal. 

Now let us say that the direction of the antenna coincides 
with the direction in which the aircraft is moving, i.e., 3=cc . Then 
the Doppler frequencies of the left and right sides of the beam 
will be uniform in value and proportional according to the cosines 
6/2: 

/pL~ + cos y;/d^ cos Y-. 

and the amplitude modulation from the edges of the beam will be 
abs ent . 

In actuality, there will be a very low-frequency amplitude 
modulation owing to the difference in the Doppler frequencies of 
the edges of the beam relative to the effect of the center of the 
beam (the bisectrix of the radiation characteristic), but due to 
the very small difference between the cosines of the angles, the 
beat frequency will be very low (expressed in Hertz), while the 
visual effect of the beat is maximum. 

With circular rotation of the antenna, the beating of the fre- 
quencies is not noticeable to the eye, since each of the luminous 
points is rapidly crossed by the scanning beam and appears on this 
screen as an individual point with subsequent afterglow. 

A slight impression remains of the secondary Doppler effect 
in a fixed antenna, when its direction differs considerably from 
the direction in which the aircraft is moving, since the flicker- 
ing of the points in this case takes place at high frequencies and 
is blurred by the afterglow on the screen. 

If the direction of the antenna slowly approaches the direc- 
tion in which the aircraft is moving, the luminous points all begin 
to flash at a reduced frequency and increased amplitude. A slow 
but bright flashing of the luminous points on the screen indicates 
a coincidence of the direction of the antenna with the direction 
in which the aircraft is moving. The drift angle of the aircraft 
is determined by the position of the scanning lines on the screen 
with naximum secondary Doppler effect. 

Measurement is performed best of all with a large-scale oper- 
ation of the radar (20 km for the screen radius), using a scanning 
delay of 20 km. It is then necessary to make a corresponding ampli- 
fication in the receiver for the common amplification channel, in 
both the high and low signal levels, as well as the corresponding 
inclination of the antenna. 



334 



One advantage of the method of determining the drift angle 
of the aircraft according to the secondary Doppler effect is its 
high accuracy. With a little experience in selecting the receiver /320 
amplification and the angle for tilting the antenna, measurements 
can be made literally within several seconds. 

Several types of sector-type radars have a special operating 
regime and an additional indicator for measuring the drift angle 
according to the secondary Doppler effect. 

2. Measurement of the 
drift angle and ground speed 
by sighting points near the 
course. If a clearly visible 
point is located near the line 
of flight of the aircraft on 
the radar screen, the ground 
speed and the drift angle of 
the aircraft can be measured by 
the movement of this point. 




To a 



void gross errors in 
T1+ riiio to altitude 



Fig. 3.^6. 



± \^ a.v\_/_i.vj. ^j.v.'Ous c;xi\_/j.o 

measurement due to altitude 
errors, the sighting of the 
points must be made at distances 
from 60 to 30 km. At the moment 
when the point being observed 
crosses the 60 km marking, th« 



The drift angle is calculated directly from the bearing scale 
with negative drift angles being calculated as added to 360°. To 
determine the ground speed, the correction for flight altitude for 
a distance of 30 km is added to the length of the base, taking into 
account the correction for a distance of 60 km as equal to zero. 
Thus, at flight altitudes of 8-10 km, the length of the base turns 
out to be equal to: 

At a height of 8 km, 30.5 km; at a height of 9 km, 31 km; at 
a height of 10 km, 31.5 km. 

The ground speed is determined by means of a navigational slide 
rule (Fig. 3,4-7, a). 

Let us say that at a flight altitude of 10 km the time required 
to fly along the base between the 60 and 30 km markings is 2 min 
and 15 sec (Fig. 3.47, b). The ground speed in this case is 840 
km/hr . 



335 



This method can be used with sufficient accuracy for measuring 
the drift angle of the aircraft. The accuracy of determination 
of the ground speed is obtained with a low and therefore very small / 321 
measurement base. Thus, e.g., at an airspeed of 900 km/hr, the 
error in measuring the flight time on the baseCwhich amounts to 
M- sec) produces an error in measuring the ground speed of about 
30 km/hr. In addition, at large drift angles, when the vector of 
the motion of the target point does not agree with the radius of 
the screen, errors arise in determining the measurement base from 
the distance markings on the screen. 



© 



w 



© t 



sec 



© 

1 — - u 

© ® 



S1,S 
-I — 



2 min 1 5 sec 



g4ff 



Fig. 3.47 



Fig. 3.47. Determination of Ground Speed 
of a Point Near the Course Indicator on a 
Radar Screen. 

Fig. 3.48. Determination of Drift Angle 
and Ground Speed by Means of a Right 
Triangle . 




Fig. 3.4i 



3. Determination of the drift angle of the aircraft and the 
ground speed by means of a right triangle. This method is more 
convenient and precise in comparison to the sighting of the motion 
of a landmark near the course. In addition, the use of the right 
triangle method makes it possible to select more freely the land- 
marks on the radar screen in order to track them. 

The bearing scale of the radar is set to zero opposite the 
course marking, after which the course angle of the landmark is 
measured with the sight, its distance on the circular markings is 
observed, and the timer is switched on. Leaving the sighting instru- 
ment in a fixed position, the operator observes the motion of the 
landmark across the screen. At the moment when it crosses the per- 
pendicular line on the sight (Fig. 3.48), the timer is switched 
off, the distance to the landmark is determined by the circular 
markings, and the flight time along the base is calculated. 

Corrections are then made in the first and second measurements 

of the oblique distance for the flight altitude; angle a between 

the position of the sighting line and the direction of the move- 
ment of the landmark is then determined as follows: 



tgo = 



R2_ 



and the length of the measurement base is : 



336 



coso sin (90 — «) 

This problem is easily solved on a navigational slide rule /322 
(Fig. 3.49). 

a ®- T ?- b I J"-" f- 

Fig. 3.H9. Keys for Determining the (a) Acute Angle of the 
Triangle and (b) Measurement Base on the NL-IOM. 

The drift angle is determined as the difference between the 
first course angle of the landmark and the angle a (Fig. 3.49, a), 
while the ground speed is determined as the length of the base rela- 
tive to the time required to cover the distance (Fig. 3.49, b). 

Example . At a flight altitude of 10 km, the course angle of 
a landmark was initially equal to 8° at OD^ = 60 km. The oblique 
distance at the moment when the landmark crosses the transverse 
line in the sight was 23 km. The flight time along the base was 
5 min and 35 sec. Find the drift angle of the aircraft and the 
ground speed. 



Solution. The correction for the flight altitude for the 
distance will be considered as equal to zero. The correction f 
the second distance (OD = 23 km, H = 10 km) is equal to 3 km, s 
that the horizontal distance is HD2 = 20 km. 

On the navigational slide rule, we find the angle a = 18.5° 
(Fig. 3.50, a) and the length of the measurement base is S = 60 
km (Fig. 3.50, b ) . 

a ^ ^^ ¥ b -^— f^ ^ 



first 
or 
so 



(D 80 60 (£) SO 



(?) M 830 
-t 1 — 



® 



4 min 3 5 sec 



Fig. 3.50. Determination of (a) the Acute Angle of a Tri- 
angle, (b) the Base and (c) the Ground Speed on the 

NL-IOM. 



Therefore 



US = 8 - 18. 5° = -10 . 5° . 



337 



The ground speed (W) is therefore equal to 830 km/hr (Fig. 
3.50 , c) . 

k. Determination of the ground speed and drift angle of an 
aircraft by double distance finding using a sighting point with 
equal oblique distances. 

This method is the most precise of the methods which we have 
discussed which use sighting of landmarks. However, it calls for 
the maximum time for measurement and calculation. 

When a highly visible point shows up in the forward part of 
the screen, the crew waits until it reaches one of the circular 
distance markings (Fig. 3.51). At the moment when this point crosses 
the distance marking, the timer is switched on and the course angle 
of this point is measured. The crew then waits until this point 
moves across the screen and crosses the same circular distance mark- 
ing at the rear of the screen. At the moment when it crosses it, 
the timer is switched off and the course angle of the point is meas- 
ured once again. 



Since in this case HDi = HD2 , the line of motion of the point 
(from A to B) is perpendicular to the bisectrix between CALi and 
CAL25 i.e., if the point moves to the right of the course line of 
the aircraft, the drift line of the aircraft is determined by -the 
formula 



/323 



US 



CAL]+CAL2 
2 



90° , 



and if the point moves to the left of the course line 



or 



US = CALi+CAL? 



270< 



To determine the ground speed, a correction for flight alti- 
tude is made in the oblique distance at points A and B and the length 
of the measurement base is determined by the formula 



^„ . CALy-CAL 
S = 2i?sin ^-r 



L 



If HDi = HD2 exceeds five times the flight altitude, the cor- 
rection for altitude is considered to be zero. 

Example. H = 10 km, HDi = HD2 = 60 km; CALi = 32°; CAL2 = 
152°; the flight time along the base is 8 min and 15 sec. Find 
the drift angle and the ground speed. 



338 



Solution 



US 



32° + 152° 1 



-90° = + 2°; 



152 32 

5 = 2-60 sin = 120 sin 60°; 



By using a navigational slide rule, we can solve the latter 
equation and find the ground speed (Fig. 3.52). 



5 <=^ 105 KMi 

IT" 765 km/hr. 




CAL 



® ^P 



© W5 



7S'S 

is- 



Fig. 3.51. 



® W5 m Q) 8 min 1 5 sec 

Fig. 3.52. 
Fig. 3.52. Determination of (a) the 
Measurement Base and (b) the Ground 
Speed on the NL-IOM. 



Fig. 3.51. Determination of the Ground 
Speed and Drift Angle by Double Dis- 
tance Finding of a Landmark at Equal 
Oblique Distances. 



We should mention that in solving problems in determining the / 324 
drift angle of an aircraft by the four methods enumerated above, 
the bearing scale of the radar may be set to the aircraft course 
rather than zero, e.g. , according to the orthodrome . Then the course 
angles in all the formulas will be replaced by the bearings of the 
landmarks, and the result of the solution will not be the drift 
angle but the actual flight angle of the aircraft. 

Autonomous Doppler Meters for Drift Angle and Ground Speed 

Autonomous meters for ground speed and drift angle of an air- 
craft, based on the Doppler effect, offer broad perspectives for 
automation of the processes of aircraft navigation and pilotage 
of aircraft. 



^ 



8 



Fig. 3.53. Diagram of Formation of Doppler Fre- 
quency With a Moving Object. 



339 



eters of an aircraft 



Continuous measurement of the motion parame 
makes it possible to use simple integrating devi 
automatic calculation of the aircraft path in time. In addition, 
a constant knowledge of these parameters makes it possible to regu- 
late them in such a way that the aircraft follows a given flight 
trajectory with a minimum number of deviations. 



All other radio devices for aircraft navigation make it possible 
to determine only the locus of the aircraft. The motion parameters 
of an aircraft can be determined only discretely for individual 
path segments 5 using the navigational devices described above. 

As we pointed out at the beginning of Chapter One, the flight 
regimes of an aircraft are almost never stable, with the exception 
of the end points of curves along separate parameters. A strictly 
stable flight regime for all parameters simultaneously is never 
encountered. Therefore, automatic or semi-automatic calculation 
of the path on the basis of motion parameters measured over individ- 
ual segments is a very approximate and unreliable method. 

The operating principle of Doppler meters is the following. 

Let us say that we have a moving source of electromagnetic 
oscillations at a high frequency A and a fixed object B which reflects 
these oscillations (Fig. 3.53). 



If the source A remains fixed relative to object B, then after 
a period of time which is required for the electromagnetic waves 
to travel from point A to point B, electromagnetic oscillations 
will be set up in the latter at the same frequency as those emit- 
ted by the source. 

When the source of oscillations moves toward point B, each 
successive cycle of oscillations is emitted somewhat closer to this 
point; its propagation time to reach point B is somewhat less than 
in the preceding cycle, so that the moments at which the oscillation! 
arrive at point B can be compared. 

Let us call the wavelength of the source X, and the propaga- 
tion rate of electromagnetic waves a. With a fixed source, the 
frequency of the oscillations (/) both at the point of emission 
and at the point of reflection of the waves will be equal to 

With a movable source, the number of oscillations reaching 
point B per unit time is increased by the number of wavelengths 
contained in the distance covered by the aircraft in that same unit 
time , i.e., 

c-yw c w 



/325 



340 



the 



The increase in the frequency W/X, produced by the motion of 
source, is called the Doppler frequency (f-Q). 



Similarly, the oscillation frequency at the point of reflec- 
tion will decrease if the source recedes from the reflection point 
for the electromagnetic waves. 

Doppler meters work on the same principle of signal transmis- 
sion as aircraft radars, i.e., frequencies are received that have 
been emitted by aircraft sources after their reflection from the 
Earth's surface. Therefore, a double Doppler frequency is received 
which arises along the path of electromagnetic waves, from the air- 
craft to the reflecting surface and along the reverse route from 
the reflecting surface back to the approaching or receding aircraft. 

There are three ways of separating the Doppler frequency in 
receiving signals aboard an aircraft: 

(1) The internal coherence of the signals, when the received 
frequency is combined within the receiver with a frequency radi- 
ated by the source, as a result of which there is a beating of the 
double Doppler frequency; 

(2) External coherence, when the receiving antenna picks up 
signals which have been reflected from the ground as well as sig- 
nals radiated by the transmitting antenna through the external med- 
ium ; 

(3) Autocoherence of the signals; in this case, the frequen- /3 26 
cies of signals reflected from the Earth's surface in the forward 

and backward radiation of the re cei ving- transmi tt ing antenna are 
combined in the receiver without the frequency radiated by the antenna. 
Since the oscillation frequency is increased by 2 f-Q relative to 
the preceding beam, and decreased by the same value for the follow- 
ing beam, the beat frequency will be equal to four times the Dop- 
pler frequency. 

If we agree to call the Doppler frequency the beat frequency 
separated in the receiver as a result of superposition of the signals, 
then for the cases of internal and external coherence we will have 



/t 



2W_ 
X 



and for the case of autocoherence we will have: 

„ _ W 
•^ D X ■ 

In principle , Doppler meters with internal and external coher- 
ence can be made with a single-beam antenna, but with autocoher- 
ence a minimum of two beams is required. In practice, as we will 
see later on, it is convenient to use antennas with three or four 
beams. Recently, the most widely employed type is the Doppler meter 
with four-beam antennas . 



341 




Fig. 3. 54. Projection of the 
Ground-Speed Vector on the 
Direction of the Radiation 
of Electromagnetic Waves. 



Since the characteristics 
of directionality of the antennas 
of Doppler meters in the general 
case do not coincide with the 
vector of the ground speed of 
the aircraft, it is necessary 
to consider the' actual Doppler 
frequencies separated in the receivers 

Usually, the slope of the 
antenna beams of the meter is 
selected so that the areas of 
their reflection from the Earth's 
surface are not too far from the 
aircraft, i.e., the power of the 
transmitter is used most effectively. 
The slope angle of the beam rela- 
tive to the horizontal plane is 
called the angle 6 (Fig. 3.54). 



Obviously, when the beam is inclined relative to the plane 
of the horizon, the Doppler frequency will be proportional not to 
the modulus of the ground speed vector, but to its projection in 
the direction of the antenna beam. For example, for a meter with 
internal coherence , 

•^D - "T =°^ ®- 

On the other hand, the ground speed vector of the aircraft 
can be divided into two vector components: 

oir= oWi + Wi r. 

The vector WiW is directed perpendicular to the antenna beam, 
and therefore the Doppler effect is not produced. The vector 



./327 



OWi = OWcos 



is effective 



In addition to the fact that the antenna beam is set at a cer- 
tain angle to the vertical plane, the antenna beam is usually directed 
at a certain angle to the longitudinal axis of the aircraft in the 
horizontal plane. For example, with a four-beam antenna, the longi- 
tudinal axis of the aircraft is the bisectrix of the angles betweeh 
the directions of the forward and rear beams of the antenna (Fig. 
3.55). The angle between the longitudinal axis of the aircraft 
and the direction of the antenna beam in the horizontal plane is 
called the angle 3. 



Hence, in receivers with internal and external coherence 
separated Doppler frequency 



the 



342 



■ Hini II 11 



^D = 



2r 



cos 8 cos (P — «)t 




Fig. 3.55. Diagram of the 
Positions of the Beams from 
an Antenna on a Doppler 
Met er . 



verse rolling of the aircraft 



where a is the drift angle of the 
aircraft . 

In the special case where the 
drift angle of the aircraft is ab- 
sent, the Doppler frequency for 
each antenna beam will be the same 



1W 
f =— — cosBcosp. 



Three- and four-beam antennas 
are desirable because they make 
it possible to compensate automatically 
for errors in measurements which 
arise with longitudinal and trans- 



At the same time, in cases when single beam or two-beam an- 
tennas are used, they must be placed on gyros tabili zing devices. 
In the opposite case, longitudinal or transverse rolling of the 
aircraft will change the slope angle of the antenna 9, thus leading 
to a change in the Doppler frequency. 

In the case of a four-beam antenna, the longitudinal or trans- 
verse rolling of the aircraft produces a change in the slope angle 
of one pair of beams in a positive direction and changes the oppo- 
site pair in the negative direction by the same magnitude. If angles 
6 are then located on an approximately linear section of the cosine /32i 
curve, the frequency shift of the opposite antenna beams will be 
opposite in sign but approximately the same in magnitude, which 
can also be used for compensating roll errors in the system (Fig. 
3.56, a) . 




''')CX'\ 



■f. 



Fig. 3.56. Shifts in the Doppler Frequency with 
Tilting of the System: (a) Change in the Cosines 
of the Angles; (b) Frequency Shift. 



343 



For example, in the case of receivers with autocoherence , when 
the Doppler frequency increases in the front right-hand beam and 
decreases by the same magnitude in the rear left-hand beam, the 
beat frequency of one pair will simply be retained. 

In systems with internal and external coherence, turning of 
the antenna leads to doubling of the frequency spectrum of the oppo- 
site beams (Fig. 3.56, b). 

With a horizontal position of the longitudinal and transverse 
axes of the aircraft , the Doppler frequency in the forward and oppo- 
site rear beams will be the same. Tilting the system shifts the 
frequency spectrum of one of the beams forward, and that of the 
opposite beam backward to the same extent. However, if we add up 
these frequencies with time and divide them by the measurement time, 
the average frequency will turn out to be equal to the frequency 
of the horizontal position of the axis of the aircraft. 

Doppler meters which are presently in use can be divided into 
four types, depending on the regime of emission and reception of 
signals : 

1. Pulse meters. In transmitters, these meters produce high- 
frequency pulses in the same manner as is done in aircraft radars. 
Reception of reflected signals takes place in the intervals between 
pulse emission. In order to separate the Doppler frequency, auto- 
coherence by beam pairs is employed. 

A shortcoming of this method is the presence of "dead" alti- 
tudes, i.e., when the reflected signals arrive at the moment coin- /329 
ciding with the emission of pulses, and not in the intervals between 
them. In addition, when flying over mountainous terrain, the distance 
to the Earth's surface according to opposite beams of the antenna 
may not be the same, thus leading to a failure of the arrival of 
reflected signals to coincide for these beams and producing a disturb- 
ance of their coherence . 

Another shortcoming of pulse meters is the need for high volt- 
ages to drive the magnetron in analyzing the high-frequency pulses, 
thus necessitating a hermetic sealing of the transmitter units and 
subjecting them to a certain pressure. 

2. Meters using continuous radiation of high frequency. In 

this case, the high frequency is radiated continuously by the trans- 
mitter. Reception of signals is accomplished with a separate antenna, 
having a certain by-pass coefficient with the transmitting antenna. 

The reflected signals are combined in the receiving antenna 
with the frequency produced by the transmitting antenna, so that 
the beat frequency is separated out in an external coherence system. 

The advantages of this method are the independence of the re- 



3 44 



iiiii~i ■■■■■II I 



ception conditions for the signals of flight altitude and the local 
relief. In addition, devices of this type worked at relatively 
low powers in the receiver mechanism. 

A shortcoming of this method is the need to have separate an- 
tennas for transmission and reception of the signals. 

3. Meters with continuously pulsed radiation. These meters 
employ a constant regime of generation and transmission with pulsed 
emission of a portion of the high frequency into the antennas by 
means of commutating devices. The reflected signals are combined 
with the frequency developed by the transmitter in the intervals 
between the moments when the high-frequency segments are emitted 
(internal coherence). This means that it becomes possible to use 
part of the advantages of continuous emission (operation at rela- 
tively low voltages in the transmitter circuits) and the pulse systems 
(reception and transmission with a single antenna). 

However, the shortcomings still remain which afflict pulsed 
meters, i.e., the presence of "dead" altitudes and the effect of 
the relief on reception conditions. In addition, there are also 
difficulties in using these meters at low flight altitudes, since 
at a very short signal path, the moments of transmission and recep- 
tion are practically impossible to separate. 

h. Meters with frequency modulation of signal transmission. 

Frequency modulation of signal transmission can be used either in 
a pulsed or continuous -puis ed regime of operation for the meter. 



If the tran 
frequency at low 
of reception of 
tainous regions 
signals, when th 
sion of the sign 
to the combinati 
signals will be 
emission, since 
greater than the 
this case , thee 
tainous regions . 



smission of high-frequency pulses at a constant 

altitudes makes it possible to superpose the moments 
signals on the moments of emission, while in moun- /330 
there may be disruptions in the coherence of the 
ere is a change in the frequency of the transmis- 
als, and only a portion of them will contribute 
on with the radiation moments. The majority of 
received in the intervals between the moments of 
the duration of the intervals is made sufficiently 

duration of the pulses. To a certain degree, in 
ffect of disruption of coherence is reduced in moun- 



The best properties are exhibited by continuous -puis ed meters 
with frequency modulation of signal transmission, since in this 
case all positive qualities of the continuous and pulsed systems 
are employed. However, the shortcomings of the pulsed systems remain, 
including difficulty in making measurements at very low flight alti- 
tudes . 

Of the types of Doppler meters which we have discussed, the 
ones which are currently used most widely are the meters with con- 
tinuous radiation and continuous -puis ed meters with frequency modu- 
lation of signal transmission. 



31+5 



1 t 

\\/\/\/\7\ 



wwv/wv 



T 



A^\>^W\M' 




Fig. 3.57. Doppler-Meter Antenna: (a) Waveguide 
Lattice; (b) Diagram of Beam Formation. 



In the first types of Doppler meters, beginning with the single 
beam versions, reflector- type antennas were used with an isosceles 
directional characteristic. Recently, antennas of the "waveguide 
network" type have come into use. 

The principle of operation of these antennas is the follow- 
ing (Fig. 3.57, a). 

Imagine a rectangular lattice, made up of waveguides, to one 
corner of which an electromagnetic wave of high frequency is applied. 

On the upper walls of the transverse waveguide in this lat- 
tice, there are slots for emission of wave energy into space. 

The wave energy, propagated along a transverse waveguide, emerges 
through the slits with a certain shift in time from one slit to /331 
the next, so that there is an interf f erence of the waves emerging 
from the slits, as is the case in spaced antennas (Fig. 3.57, b). 

The direction of the radiation maximum and the isophasal lines 
are located at an angle to the surface of the waveguide. 

Since the electromagnetic energy propagated along the longitud- 
inal wave guide reaches the next transverse waveguide also with 
a shift in time, a similar picture of interference with a tilting 
of the isophasal line also takes place along the waveguide lattice. 
As a result, an isophasal surface is formed above the waveguide 
lattice, having a slope in the direction of its diagonal toward 
the corner opposite the corner at which the electromagnetic waves 
enter the lattice. Consequently, one of the beams will be formed 
along the diagonal of the antenna. 

If wave energy is also transmitted from the diagonally oppo- 
site corner of the lattice, two oppositely directed antenna beams 
can be formed simultaneously. 



31+6 



Flat, multi-beam antennas, especially when fixed in position, 
are very useful, since they can be placed below the radio-trans- 
parent housing flush with the skin of the aircraft and do not produce 
any additional aerodynamic resistance during flight. 

Schemattc Diagram of the Operation of a Meter with Contin- 
uous Radiation Regime 

The high frequency processed by the transmitter passes through 
a commutation device to the transmitting antenna, where the beams 
for propagation of electromagnetic waves are formed in pairs. The 
commutating device is connected to the counter of the meter, to 
separate the frequencies of the first and second pairs of beams 
(Fig. 3.58). 

A portion of the wave energy radiated by the transmitter reaches 
the receiving antenna, where it is combined with the received signals 
reflected from the Earth's surface, so that the Doppler frequency 
of the given pair of beams can be separated. 

The separated Doppler frequency, after amplification, passes 
to the calculating device, at whose output is an indicator for the 
ground speed and drift angle of the aircraft. 

As we already know, for the four-beam antenna of a Doppler 
meter with internal coherence, the separated frequency by beam pairs 
will be equal to: 

(a) For the first pair. 



f 2W 

^Di = -Y- 

(b) For the second pair, 



cosO cos (P + a); 



/332 



/ 



Do = - 



2W 



cosOcos(P — fl). 



For a Doppler meter, the sign of the angle is not important, 
but its absolute value is. Therefore, we can simply assume that 
with a positive drift angle, the drift angle in the right-hand pair 
of beams will be calculated from the angle 3, while in the left- 
hand pair these angles will be added. With a negative drift angle, 
the calculation of the angles will be performed in the left-hand 
pair of beams, and combined in the right. Therefore, the last two 
formulas given above can be written in the form 



JD^ = Ji51_ cos e COS (p + a); 
f = -— — cos 6 cos (p — a), 

•I T\ - A 



347 



■ 



where the sign shows that the formulas change places for the 
left and right-hand pairs of beams when the sign of the drift angle 
changes . 




signals f'true signa ls y 



computer 



indicator ^,9 



nii 



ei* 



receiving 
antenna 



_i.'ai)tnmatic 
2 |navigationa!„ 
* device '^ 



receiver 



IC 



commur- 

tation 

device- 



3 [ 




counter X,7 



trans 
mitter i" 




transmitting 
antenna 



Fig. 3.58. 
Meter . 



Functional Diagram of A Doppler 



Note, Since the pairs of beams are diagonal to the wave guide 
lattice, and each of them contains a left and right-hand beam relative 
to the longitudinal axis of the aircraft , the left or right-hand 
pair of beams here is referred to as a pair whose leading beam is 
directed to the left or right of the longitudinal axis of the air- 
craft . 

Obviously, in the case of a fixed antenna on the aircraft (Fig. 
3.59), the first problem for the calculating device of the Doppler /333 
meter is the determination of an angle at which 



fr 



f 



D2 



(P + a) \^ 



cos(p — a) ; 



Since the angle g is a constant value, and the frequencies /q , 
f-Q2 s^s variable, the solution of a problem of this kind does not ^ 
present any significant difficulties. The desired angle a is the 
drift angle of the aircraft. 

The second problem for the computing device is the determin- 
ation of the ground speed (W) with a previously known drift angle 



348 



and Doppler frequency for a pair of beams: 



W = 



w 



^D,* 



COS e COS (P + a) 



cos 6 cos (P — a) 




Fig. 
Doppl 
the R 
Beams 



3.59. D 

er Frequ 

ight- an 

of an A 



ifferece in 
encies of 
d Left-Hand 
nt enna . 



tion of the aircraft motion 



/ 



D 



/ 



D' 



The calculated drift angle of 
the aircraft and the ground speed are 
transmitted to the visual indicator 
of these parameters and also to the 
automatic navigational device for inte- 
gration of the aircraft path in time. 

The problem of the calculating 
device of the Doppler meter is sig- 
nificantly simplified by mounting a 
movable antenna on the aircraft. In 
this case, the direction of the antenna 
is set so that the Doppler frequen- 
cies of both antennas /p and /pi will 



1 2 

be the same, i.e., the bisectrix of 
the beams will coincide with the direc- 



= ^D.- 



2W 



cos 008 p. 



Then the drift angle of the aircraft is determined by the course 
angle of the antenna setting, and the ground speed is found by the 
formula : 



(a) With internal and external coherence 

W = 



^D^ 



(b) With autocoherence 

W-- 



2 cos 8 cos p 



4 cos e cos P' 



This means that all coefficients entered into the formulas (with 
the exception of /p ) are constants while f-Q is a variable quantity. 

We should mention that during flight above the ocean, Doppler 
frequencies from pairs of beams in a Doppler meter are somewhat 
lower than above dry land at the same airspeeds. This is caused 
by peculiar features of the reflection of electromagnetic waves 
from the surface of the water. 



/33^■ 



In flight above dry land, if the conditions for diffuse reflec- 
tion of the waves are approximately the same over all areas in contact 
with the Earth's surface, and the maximum amplitude coincides with 



349 



the center of the beam at the maximum of the radiation character- 
istic, then the reflection conditions above a watery surface will 
depend to a considerable extent upon the angle of incidence of the 
beam. Therefore, the leading edge of the beam will have a sharper 
angle of incidence (and therefore a lower signal amp,litude), while 
the trailing edge of the beam will strike more obliquely and have 
somewhat greater amplitude. Consequently, the maximum amplitude 
of the signals shifts from the center to a region of lower Doppler 
frequency (see Fig. 3.54). 

To compensate for errors in the operation of the meter above 
water, the circuit is designed to include a calibration element 
which is switched on from the control panel by turning a switch 
from the "land" position to the "sea" position. 

Over a smooth watery surface (with a swell less than a scale 
value of one), the potential of the reflected signals becomes inad- 
equate to ensure operation of the meter, and the latter then is 
turned off by switching the automatic navigational device to memory 
operation . 

The channel of the Doppler frequency receiver is fitted with 
a filter intended to damp out all parasitic frequencies produced 
by other electronic devices mounted aboard the aircraft which cou'ld 
disturb reception of reflected signals from the Earth's surface. 
The filter must have a narrow passband within the region of Doppler 
frequencies of the received signals. 

If the frequency of a carefully adjusted filter differs con- 
siderably from the midpoint of the range of Doppler frequencies 
being employed, it begins to introduce errors in the measurement 
of the Doppler frequency, shifting it toward the point of fine tuning 
of the filter. Therefore, filters are used with automatic tuning 
for the frequency of the signals employed. 

Use of Doppter Meters for Purposes of Aircraft Navigation 

Doppler meters for ground speed and drift angle are very effec- 
tive in aircraft navigation. The following problems can be solved 
directly by using a Doppler meter: 

(a) Maintainance of a given direction of flight along an ortho- 
drome or loxodrome, automatically if desired. To do this, it is 

only necessary that the sum of the course (y) and drift (a) angles /335 
of the aircraft be constantly equal to a given flight path angle (^) : 

ip = y + a; 

(b) The calculation of the path of the aircraft in terms of 
distance can be solved on the basis of the ground speed and time: 

S = ¥t. 



350 




In view of the above, as well as the relative simplicity of 
automating aircraft navigation on the basis of Doppler measurements, 
the latter are practically impossible to use without combining them 
with automatic navigational instruments. 



Automatic navigational instruments connected to Doppler meters 
calculate the aircraft path with time in an orthodromic or geographic 
system of coordinates. 

To calculate the path of the aircraft in an orthodromic sys- 
tem of coordinates, the navigational devices are connected to trans- 
mitters of the orthodromic course (a gyro assembly for the course 
system, operating in the GSC regime). The automatic system includes 
a transmitter of the flight angle or (as it is usually called) the 
given chart angle (GCA). 

The signals for the drift angle of the aircraft, obtained from 
the meter, and the course signals of the aircraft, obtained from 
the course system, are combined and their sum compared with a given 
path angle fed into the transmitter. 

If the sum of the course and the drift angle of the aircraft 
is equal to the given path angle of the flight (Jj = y + a, the ground 
speed is directed along the J-axis: W = W^i W^ = 0. 

If this equation is not satisfied, the vector of the ground 
speed is divided into two components: 

UTt'^ ITcos (t + « — "l*); 
Wt= Wsln{i + a-^). 



The vector components obtained for the ground speed along the 
axes of the coordinates are integrated over time and calculators 
are used to find the running values of the aircraft coordinates X 
and Z . 



/336 



Calculation of the aircraft path and geographic coordinates 
can also be done directly on the basis of the signals from the Dop- 
pler meter and the course calculator. However, to do this it is 
necessary to have an exact knowledge of the true course of the air- 



351 



craft and to express the division of the ground-speed vector of 
the aircraft according to the formulas: 



dt 
dk 
dt 



sin {f + a) 



= W 



cos T 



To ensure operation of the gyroscopic transmitter in a regime 
of true course 5 in addition to the moment which compensates for 
the diurnal rotation of the Earth 

0)^ = Q sin cf, 

it is necessary to add the moment which compensates for the change 
in the true course with time due to the eastern or western compo- 
nent of the ground-speed vector of the aircraft: 



'•r= 



Wsia if + a) sin y 
cosy 



= Wsln(,t + a)tg<f. 



However, calculation of the aircraft course by this system 
cannot be considered adequate for three reasons: 




(2) The constant dependence of the operation of the course 
system on the operation of the Doppler meter and a calculating device 
introduces inaccuracies into the aircraft navigational elements . 

For example, when the Earth is not visible, a flight can be made 
over dry land; however, if the aircraft then begins to travel over 
a smooth watery surface, the reflected Doppler signals will not 
only introduce errors into the accuracy with which the path is calcu- 
lated with time, but will also incorporate errors in the operation 
of the course system. 

(3) The errors which appear in the calculation of the air- 
craft course at the points of correction of its coordinates cannot 

be used directly for correction of the aircraft course, as can easily 
be done in an orthodromic system of coordinates. 

A more logical calculation of the geographic coordinates of 
the aircraft would involve the orthodromic system of aircraft naviga- 



xne aircrarT wouxa involve xne orxnoarouii c sysxeui or aircrai 
tion, based on a constant conversion of the orthodromic cour 
the aircraft to the true course on the basis of the running 



inates of the aircraft 



se of 
running coord- 



^true = «''=»g^^+^* 



tgV 
siny/ 



ort 



352 



where 



Ay = Y - ^ 
ort ort ort 



The true course for the aircraft obtained in this manner can 
be used to calculate the geographic coordinates of an aircraft as 
was shown earlier; it can also be used for correcting the ortho- 
dromic course by astronomical means. 

The advantages of a method of this kind are the independence /337 
of the true course from the ground speed and its automatic correction 
along with the correction of the aircraft coordinates. 

However, we should mention that the calculation of the air- 
craft course and geographic coordinates should really be replaced 
by a constant conversion of the running orthodromic coordinates 
into geographic ones, e.g. , by Formulas (1.64) and (1.65): 



sine 



geog 

s inX 



sind) cosG-cosA ^sin( 
ort ort 



= smX cosd) seed) 
geog ort ort geog 

In this case, the geographic coordinates will always agree 
strictly with the orthodromic ones, so that there will be output 
parameters from only one integrating device and automatic correc- 
tion in the second system with correction of coordinates in one 
of them . 

In general, the geographic coordinates are not of much inter- 
est as far as aircraft navigation is concerned. However, they are 
important for ensuring accurate operation of navigational trans- 
mitters (latitudinal correction of course systems, analysis of gyre 



For purposes of aircraft navigation, automatic navigational 
devices are much more dependable for calculating the path of the 
aircraft in orthodromic coordinates. 

In addition to the basic regime of operation by signals from 
a Doppler meter, automatic navigational devices as a rule have an 
operating regime with "memorized" navigational parameters. 



The regime for operating by "memory' 
one of the following two versions. 



can be incorporated in 



1. By "memorizing" the last values of the ground speed and 
drift angle of the aircraft. In this version, in the case when 



353 



there is an interruption in the arrival of Doppler signals for some 
reason, (e.g., when there are no waves in a flight over water), 
the path can be calculated by "memory" for a period of 15-20 min, 
only under the condition that the flight direction and airspeed 
have been recorded. With a changing flight regime for the aircraft, 
calculation by "memory" leads to large errors, since the ground 
speed and drift angle change on a new course or with a change in 
other parameters. 



this vari 
potentiom 
eters : 



Then, if 
the path 
of the wi 
the wind 
of the fl 
redis trib 
is not di 



By "memorizing" wind parameters at flight altitude. In 

ety, the calculating device is provided with special "memory" 
eters, which constantly set the value of the wind param- 

Uji = Wcos (Tf + a — iji) — V cos (7 — ij<); 

Ug=W3ln(T + a — ^)~ Vsin(Y — 4;). 

the signals should not be received from the Doppler meter, 

of the aircraft can be calculated by comparing the vector 

nd speed along the axis of the system of coordinates with /33 i 

vector components added to it. If the given path angle 

ight then changes , the components of the wind vector are 

uted among the coordinate axes and the calculation regime 

s turbe d . 



However, in both the first and second methods of "memorizing" 
navigational parameters, no provision is made for an exact calcu- 
lation of the aircraft path during a long period of time, since 
the wind parameters change with distance . In these cases , the navi- 
gational mechanism is used for calculating the path of the aircraft 
on the basis of discrete data obtained by measuring the ground speed 
and drift angle, e.g., by means of a aircraft radar or some other 
device, as is done (e.g.) when using the navigational indicator 
NI-50B. 



In some types of navigational instruments, inertial or astro- 
inertial instruments are used as memory devices. 




Inertial navigational devices are gyros tabi li zed platforms 
on which accelerometers and special gyroscopes are mounted which 
integrate the accelerations of the aircraft with time along the 
axes of the reference system. 



In the case when the motion of the aircraft along one or two 



354 



axes takes place with acceleration, a moment is applied to the axes 

of the gyroscope which is proportional to these accelerations, so 

that precession of the gyroscope axes takes place, i.e., there is 
integration of accelerations with time. 



Since 







dt 



and 



0'' 

where a^ and a^ are the accelerations along the corresponding axes, 
we can use the position of the gyroscope axes to get an idea of 
the components of the aircraft speed along the axes of the coord- 
inates . 

The components of the ground speed along the axes of the ref- 
erence system can be integrated in turn with time by means of a 
navigational instrument. 

In an operating Doppler meter, the position of the axes of 
the integrating gyroscopes can be corrected by signals from this 
meter. In the case when the Doppler information does not arrive, 
the inertial device can be used for a long period of time to retain 
"remembered" values of the components of the speed along the axes 
of the coordinates, correcting them for any accelerations that arise 
in the way of wind changes, as well as in changes in the flight 
regime . 



Aircraft navigation using Doppler meters and automatic nav- 
igational instruments becomes extremely simple and practical, but 
very careful preparations for flight and exact measurements of the 
coordinates of the aircraft at the correction points are required. 
An exact measurement of the aircraft course is extremely important 
in this regard. 



/339 



On the other hand, the fact that the crew is constantly aware 
of the ground speed, the drift angle of the aircraft, and its coord- 
inates makes it possible to maintain a given flight trajectory for 
long periods of time according to the indications of the instru- 
ments. To do this, it is sufficient that the sum of the aircraft 
course and the drift angle be constantly equal to the given path 
angle, and that the Z-coordinate of the aircraft be equal to zero. 

It is particularly easy to solve problems in aircraft navigation 
if the readings of the aircraft course and the drift angle are obtained 
from the indicator in the form of a sum, i.e., as the actual path 
angle of the aircraft flight. It is then sufficient to pilot the 
aircraft so that with Z equal to zero, the flight angle will actually 
be equal to the given one. 



355 



In a case when the path angle of the flight is not maintained 
precisely and the Z-coordinate of the aircraft is not equal to zero, 
or, if the improper operation of a system has caused the aircraft to 
deviate from the given flight path as revealed by correction of 
its coordinates 5 the path angle of the flight is set so that the 
aircraft approaches the given line of flight at an angle of 3-5°. 
When the Z-coordinate decreases to zero, the path angle of the flight 
becomes equal to the given value. 

The aircraft can be placed on the given line of flight by using 
the autopilot. For this purpose there must be a calculating unit 
aboard the aircraft for relating the Doppler meter with the auto- 
matic navigational device and an autopilot which solves the simple 
prob lem : 

AZ+kh^ = Cj 

where AZ is the lateral deviation from the line of flight, Ai|; is 
the angle of approach to the line of flight, and k is the selected 
coupling factor. 

The aircraft is then steered so that a lead in the path angle 
of the flight is taken when the aircraft deviates to a certain degree 
from the given line of flight with a certain coefficient. Then, 
in the presence of lateral deviation, the aircraft will automat- 
ically move into the line of flight, decreasing its lead as it ap- 
proaches the latter. 

Certain difficulties in aircraft navigation when using Dop- 
pler meters with automatic navigational devices are encountered 
in converting the computer to calculate the path in orthodromic 
coordinates of the previous stage, at the turning points along the 
route. The methods of conversion to the new system of calculation 
of coordinates is shown in Chapter II, Section 9. However, when 
using Doppler meters, it is better to set the aircraft coordinates 
to the reference system of the previous stage before beginning the /3140 
turn of the aircraft. For example, with Z^ = , Zi = -LLT. 



X2 = -LLT cos TA; 
Z2 = LLT sin TA . 



c 

c 

I 

b 

se 

a 

m 




djusts itself according to the path angle of the 
ent, and it can be used to calculate the aircraft 



the reference system of this segment. 



coordinates in 



356 



The transition of the aircraft to the next orthodromic seg- 
ment of the path is accomplished by the indications of the second 
calculator, after which the first calculator is cleared and set 
for the next path segment. 

As we have already pointed out, in the case of double calcu- 
lators, their readings are mutually related, i.e., they are con- 
verted according to the formulas : 

X2 - X^cosTA-Z isin TA ; 
Z2 - XisinTA+ZjcosTA . 

Therefore, in correcting the coordinates of the aircraft on one 
of these computers, a correction is automatically made in the air- 
craft coordinate in the reference system of the next stage. 

Thus, at each turning point along the route, the aircraft makes 
a turn in a previously prepared and corrected system of coordinates 
for the next stage of flight, thus completely getting rid of any 
undesirable features of the transition which might occur if only 
one calculator were used. 

Preparation for Flight and Correction of Errors in Aircraft 
Navigation by Using DoppZer Meters 

Aircraft navigation using Doppler meters for measuring the ground 
speed and drift angle of an aircraft can be done very simply and 
rapidly. However, the required accuracy for aircraft navigation 
when using these devices can only be achieved with very careful 
preparation for flight, as well as careful correction for errors 
in aircraft navigation which arise during flight. 

When using Doppler meters, there may be errors in measuring 
the following elements in aircraft navigation due to errors in the 
transmitters : 

(a) Measurement of the aircraft course; 

(b) Measurement of the drift angle and ground speed; 

(c) In the programming of the given path angle and the dis- /3^1 
tance of the flight stages; 

(d) In the integration of the aircraft flight along the axes 
of the coordinates by the automatic navigational device. 

The accuracy with which the aircraft course is measured is 
of extreme importance for aircraft navigation when using Doppler 
meters and is closely related to the proper programming of path 
angles for each flight stage. This is explained by the very high 



357 



requirements for accuracy in determining path angles in preparing 
for flight. 

Preparation for flight using Doppler meters must be carried 
out properly according to the third group of conditions in Chapter 
Two, Section 2. 



For each flight segment, all parameters of the orthodrome must 



be determined, beginning with X 



dis 



dis 



'0 1- 



1 5 



'g ^1 = tg <f2 ctg ?i cosec AX — cfg AX. 

It is then necessary to determine the original azimuth of the 
orthodrome ag by the formula 

sInXoi 

'8"" — ::; — ' 

tgTl 

and then the coordinates of the intermediate points on the ortho- 
drome for plotting them on the chart: 

slnXn 
tgy/ = ^. 

tg«o 

The distance between the turning points along the route along 
the orthodrome can be determined by the formula 

cos Si = cos Xq cos <fi. 

If we introduce into this formula the coordinates of the initial 
and final points of the flight segment, and (when necessary) any 
intermediate points, we can find the distance to those points from 
the starting point of the orthodrome. The distances between the 
points are determined by calculating the distances from the start- 
ing point of the orthodrome to them. 



For programming the flight path angle we determine the azi- 
muths of the orthodrome at the beginning and end of each segment 
according to the formula 

tgo, = — — S-. 
sin ft 

If it is proposed that we use astronomical methods for cor- 
recting the aircraft course in flight (e.g., in flight over water 
or terrain which has no identifying landmarks), the course correc- 
tion points are marked and the azimuths of the orthodromes at the 
correction points are determined by this formula. 



/3 42 



Reference points are selected for correcting the coordinates 
of the aircraft during flight. Usually these are landmarks which 
show up clearly on radar or places where goniometric-rangef inding 
installations are located. Then the orthodromic coordinates of 



358 



these points are determined, and the ground goniometric-rangef inding 
instruments are used to determine the azimuths of the orthodromic 
segments of the path on which these devices will be used, relative 
to the meridians on which the ground beacons are located. 

The given path angle for the first flight segment is considered 
equal to the azimuth of the orthodrome at the starting point of 
this segment. The path angles of all subsequent path segments are 
considered to be equal to the sum of the path angle of the prev- 
ious segment plus the angle of turn in the path at the turning point 
on the route . 

Doppler meters have relatively low errors in measuring the 
drift angle of an aircraft, so that they can be compensated for 
in the total by the errors in aircraft course. 

In general, besides the errors in measuring the drift angle, 
depending on the operating regime of the meter, the height and speed 
of flight, which have a more or less constant character, there are 
errors which have a fluctuating nature (oscillations in the meter 
readings from the average value). The principal reason for fluc- 
tuations is the varying conditions of reflection of electromagnetic 
waves from the Earth's surface. 

When some point is encountered which reflects electromagnetic 
waves well, in the ellipse of reflection from the Earth's surface, 
the maximum of the amplitude of Doppler frequency is first displaced 
forward (for a rear beam, backward); then, as the point passes through 
the ellipse of reflection, the maximum of the amplitude shifts toward 
the average Doppler frequency and then backward, into a region of 
lower frequencies. 

Thus, there is first a positive "firing" of the Doppler fre- 
quency, then a leveling off, and finally a negative "firing". For 
the rear beam, the "firings" of frequencies take place in reverse 
order . 

The periods of fluctuating oscillations are short and depend 
on the time required for the reflecting points to pass through the 
ellipse of reflection. Practically speaking, they are located within 
the limits of 3-6 sec, so that they can be smoothed out to a consid- 
erable degree by selecting the proper rate of analysis for the read- 
ings of the drift angle and ground speed. 

As far as the calculations of the aircraft path for distance 
and direction are concerned, the fluctuating oscillations do not 
have any noticeable effect on it, since after 3-5 min of flight 
the integral value of the positive fluctuations becomes equal to 
the integral value of the negative fluctuations . 



The process of navigational exploitation of autonomous Dop- 
pler systems for aircraft navigation can be employed for adjust- 
ing the system itself, i.e., in correcting the aircraft coordinates 



/343 



359 



manually or automatically it is possible to determine and compen- 
sate simultaneously the systematic errors in the operation of the 
system as a whole. 

In fact, if the aircraft (at the starting point of a flight 
segment) is located precisely on the desired flight line, but its 
Z-coordinate is equal to zero, and we keep the coordinate Z equal 
to zero during all subsequent stages of the flight, the aircraft 
will have to remain on this line constantly. If this is not the 
case, an error will crop up in the calculation of the aircraft path 
with respect to direction, i.e., a certain angle will develop between 
the given and actual flight path angles of the aircraft. 

It is most likely that under the conditions of precise deter- 
mination and setting of a given path angle on the transmitter, an 
error in calculation will arise as a result of improper measure- 
ment of the aircraft course, since gyroscopic devices can show drift 
in their readings with time. Therefore, the total correction which 
is required for proper calculation of the path should most logically 
be made in the readings of the course instrument. 

If a portion of the errors in calculating is not related to 

the operation of the course instrument, then their contribution 

to the course errors will not make the accuracy of aircraft naviga- 
tion any worse . 

The latter statement is valid for a complex of instruments 
which permit calculation of the path in terms of direction, but 
it is theoretically not completely valid for instruments intended 
for distance finding of landmarks for the purpose of making correc- 
tions in the aircraft coordinates . 

Nevertheless, if we consider that the total error in measuring 
the drift angle and calculating the paths in terms of direction 
with an automatic apparatus is no more than 0.2 to 0.3° as a rule, 
we must recognize that correction of the aircraft course by the 
results of calculating the path is much more accurate than correct- 
ing it by any other me thods, including astronomical ones. 

During flight, the actual aircraft coordinates are determined 
by the distances B and the path bearings of the landmarks (PBL), 
selected for this purpose by the formulas: 

(a) In the measurement of aircraft radars 

X = Xy-E cos PBL ; 
Z = Zy-R sin PBL. 

(b) In the measurement of goniometric-rangef inding systems: 

X=X„ + /?cos(/l — 4i„); 
Z = ^^ + /?8ln(i4 — 4<„), 



360 



where ^m and Zx, are the orthodromic coordinates of a ground bea- 



'M 



con . 



Obviously, the formulas for the aircraft radar and the gon- 
iome tric-rangef inding systems are invariable. The difference in 
the signs of the second terms on the right-hand sides is explained 
by the fact that the bearing of a landmark is obtained with the 
aid of an aircraft radar but the bearing of an aircraft relative 
to a ground beacon is obtained with the aid of a goniometric-range- 
finding system. 



/3^^■ 



At the moment when the distance and path bearing of a land- 
mark or aircraft are determined from a ground beacon , the indicator 
readings for the aircraft coordinates are recorded. After deter- 
mining the actual coordinates of the aircraft by means of a naviga- 
tional slide rule 5 they are compared with the coordinates on the 
indicator recorded at the moment of distance finding, and the errors 
in calculating the coordinates are found: 

act calc"^ 

AZ = Z ^ - Z ^ , 
act calc-" 

where '^^Q-t- and ^aQ-t are the coordinates of the aircraft on the basis 
of the measurement results and -^calc and Zl^alc s^r-e the coordinates 
of the aircraft according to the readings on the calculator. 

The corresponding corrections are then entered in the read- 
ings of the running orthodromic coordinates of the aircraft on the 
calculator . 

The characteristic feature of the solution of these problems 
is the lack of a need to fix the time of measurement of the air- 
craft coordinates and the introduction of corrections in the readings 
of the calculators when they change, as is necessary when using 
all other radi onavi gat i onal instruments. 

This feature is completely characteristic for Doppler systems. 
The relationship to time here is maintained only with selection 
of regimes of speed for reaching checkpoints at a given time. In 
measuring the aircraft coordinates and all other elements of air- 
craft navigation, the time need not be taken into account. 

Let us examine further the methods of getting rid of syste- 
matic errors in calculating the aircraft path and primarily the 
measurements of the aircraft course with the use of Doppler meters. 

For a precise determination of the errors in measuring the 
aircraft course, we need to determine the actual coordinates of 
the aircraft at at least two successive points, with the measure- 
ment base on the order of 200-300 km. 



361 



At the first point, the actual coordinates of the aircraft 
are determined and the readings of the calculator are corrected. 
At the second point, the actual coordinates of the aircraft are 
determined once again and the error in calculation is found, which 
has been accumulated during the flight time along the base from 
the first to the second measurement point. 

If we consider the error in the readings of the calculator 
at the first point to be equal to zero (since they have been cor- 
rected), the error in measuring the course is determined by the 
formula 



Af = arctg 



•^1,2 



where AZ2 is the error in calculating the aircraft coordinates at 
the second point, and Xi 2 i^ "'^^e length of the measurement base 
between points 1 and 2. 



/31+5 



This problem is easily solved on a navigational slide rule 
(Fig. 3.60). 

By using a Doppler meter, it is possible to find not only the 
errors in measuring the aircraft course, but also the nature of 
their accumulation with time. 



(t> . y 



® 



a 2, 



v-z 




Fig. 3 .50 



Fig. 3.61 



Fig. 3.50. Determination of the Error in Measuring the Course on 
the NL-IOM. 

Fig. 3.61. Determination of Gyroscope Deviation on the Second 
Measurement Base. 



a 
© 



JS 



ISi 



®"- 



b 



® 



l(,S 



(sg'iff ) 



_3_ 



/70 



22S 



Fig. 3.62. Use of the NL-IOM to Determine (a) Degree of Deviation 
of the Gyroscope and (b) Orthodromic Coordinates of the Aircraft 

from Ground Radio Beacons . 



362 



As we already know, the deviation of a gyroscope with time 
can be compensated by a suitable shift of the latitude on the compen- 
sator for the diurnal rotation of the Earth. If the deviation of 
the gyroscope is significant (2-3 deg/hr), it can be determined 
by changes in the errors in calculating the path on two adjacent 
bases, preferably of the same length (Fig. 3.61). 

With considerable deviations of the gyroscope axis, the path 
of the aircraft turns out to be curvilinear if the Z-coordinate 
recorded on the calculator is equal to zero, and the final error 
in measuring the course will be greater than this average error 
which appears on the first base at the initial value (Ayd) divided 
in half. 

Therefore, after introducing the corrections in the readings 
of the course instrument, an error remains in the measurement of 
the course which is equal to this value. 

If the measurement is repeated on the adjacent base, of approx- 
imately the same length, the error found in the measurement of the 
course will consist of two values: 

(a) The error in the initial setting, equal to ky^/2; 

(b) The average error due to the deviation of the gyroscope 
on the second base, also equal to hy^/2. 

Thus, the error which is found will constitute the magnitude /3H6 
of the gyroscope deviation during the flight time along the second 
base . 

In order to determine the magnitude of gyroscope deviation 
per hour of flight, it is sufficient to divide the error which has 
been found into the flight time on the second base. 

Example. The flight time of an aircraft on the first and sec- 
ond bases is 20 min each. On the first base, an error in measuring 
the aircraft course was found and compensated for. However, on 
the second base the error in measuring the course turned out to 
be equal to 1°. Find the magnitude of gyroscope drift per hour 
of flight. 

Solution: 



Ay. 



1° 



0.33 hr 



3 degrees/hr 



If the deviations of the gyroscope are small (0.5-1°), they 
cannot be found by measurement from a second base. However, in 
this case, it is not necessary to shift the latitudinal potentiometer 
in making compensation. It is sufficient to correct the readings 
of the course periodically (along with the correction of the air- 
craft coordinates) by the results of the measurements on one base. 



363 



When necessary, a Doppler meter can be used to determine the 
small-scale variations in the gyroscope (0.5-2 deg/hr). To do this, 
both the aircraft coordinates and error in measuring the course 
are determined on the first base. 

On subsequent bases, only the aircraft coordinates are deter- 
mined and corrected. On the last base, the error in the aircraft 
course is again determined. The error which is found will consti- 
tute the deviation of the gyroscope from the moment of the end of 
the first to the end of the last base. 

At the same time, the remaining error at the end of the first 
base is equal to t\y^/2, while the error found at the end of the 
last base, (e.g.) the fourth, is equal to: 



Ay, 



Ay 



+ Ay-, + AYj + 
d2 d3 



Ay. 



i.e. , if the last base is equal to the first, the error which will 
be found by measuring the course will be equal to the deviation 
of the gyroscope in the second, third, and fourth bases. Thus, 
the flight time will be sufficient for showing up even small-scale 
deviations of the gyroscope per hour of flight. 

Since the navigational use of Doppler meters does not pose 
any difficulties, but the detection of errors in calculating the 
path and measuring the aircraft course are much more difficult, 
we would like to conclude by providing several examples of how to 
determine these errors. 

1. The last correction of aircraft coordinates was made at 
the point X = 156 km, where the error in the Z-coordinate was found 
to be zero . 

The flight then continued with maintainance of the Z- coord- 
inate on the computer equal to zero. At the point X = 330 km, accord- 
ing to the readings of the computer, the actual coordinates of the 
aircraft were determined on the basis of a radar landmark, having 
the coordinates X 1= 375 km, Z^ = 61 km. The polar coordinates /347 
of the landmark were as follows: PEL = 54°, i? = 72 km. Find the 
errors in calculating the coordinates in measuring the aircraft 
course . 



Solution: 

following : 



by using a navigational slide rule, we find the 



R cos PEL = 42.5 km; 
H sin PEL = 58 km. 

Consequently, the actual coordinates of the aircraft are as 
follows: X = 375-42.5 = 332.5 km; Z = 61-58 = 3 km, while the errors 
in calculating the coordinates are: AX = +2.5° km, AZ = + 3 km. 



364 



The error in measuring the aircraft course is 

3 



A7 = arctg 



332,5—156 



= 0°58'. 



In this case, the aircraft deviated to the right from the given 
path, so that the readings of the course instrument were reduced 
an/ it was necessary to make a correction equal to +0°58' or approx- 
imately + 1° . 

2. After correcting the coordinates in the aircraft course, 
considered in the first example, the aircraft traveled along a base 
equal to 180 km with a ground speed of 850 km/hr. 

A second check of the aircraft coordinates revealed that the 
error in measuring the course was +0°35'. The flight was made within 
latitudinal limits of 50-60°. Find the degree of deviation of the 
gyroscope per hour of flight and the required shift in the lati- 
tudinal compensator to get rid of it. 

Solution: The flight time of the aircraft along the base is 
equal to 13.5 min. The length of the second base is approximately 
equal to the first base, so that the deviation of the gyroscope 
along the second base is equal to the error found by measuring the 
cours e . 

The deviation of the gyroscope was found by means of a navi- 
gational slide rule (Fig. 3.62, a). 

Answer: the deviation of the gyroscope per hour of flight 
amounts to 154' or 2°3M-'. 

At latitudes of 50-60°, for each degree per hour of deviation 
in the gyroscope, it is necessary to shift the latitudinal compen- 
sator by 6°. In our example, the gyroscope deviated in the direc- 
tion of a reduction of the course indication so that the latitude 
on the compensator had to be set to the value: 6° x 2.6 = 15.6°. 
After the desired change in the setting of the latitudinal poten- 
tiometer is made, the deviation of the gyroscope should cease com- 
pletely . 

3. For correcting the aircraft coordinates, a goniometric- 
rangefinding system is employed. The orthodromic coordinates of 
a ground radio beacon are: 

X„ = 187 km; Z„ = 142 km. 

The flight angle of an orthodrome segment, measured relative 
to the meridian of the point where the beacon is established, is 
equal to 64°. Find the orthodromic coordinates of the aircraft 
if its azimuth (A) is equal to 24° and R = 225 km. 



365 



1 . e 



Solution: 

^-^^ = 320°; 
cos 320° = cos 40° = sin 50°; 
sin 320° = — sin 40°. 

By using a navigational slide rule, we find (Fig. 3.62, b), 

» 

R cos 320° = 170 km; 
/? sin 320° = — 145 K^. 

Consequently, the orthodromic coordinates of the aircraft are 

^•=187 4- 170 = 357«*; 
Z= 142— 145 = — 3/«^. 



5. PRINCIPLES OF COMBINING NAVIGATIONAL INSTRUMENTS 



/31+8 



In Chapters Two and Three of the present work, we discussed 
the complexes of navigational instruments, which make it possible 
in one way or another to automate the processes of aircraft naviga- 
tion or measurement of individual navigational parameters. 

The first navigational complex is the course system. 

The basic principles of combining individual transmitters into 
a course system is the combination of the readings for purposes 
of automatic mutual correction (the MC , AC, GSC regimes), and also 
to combine the readings of individual instruments to improve the 
navigational values, constituting the sum of individual elements. 
For example: OBR = OC + CAR. 

The second complex is the navigational indicator NI-50B, in 
which there is a course transmitter, a transmitter of the airspeed, 
and a manually-set wind transmitter. 

The most complete of these complexes is the autonomous Doppler 
system of aircraft navigation, which works in conjunction with course 
transmitters and an automatic navigational device. 

Thus, the basic reasons for combining navigational instruments 
are the following: 

(a) Comparison of readings for purposes of mutual correction, 

(b) Combination of readings for purposes of automatic summation. 

Combination of individual transmitters into navigational sys- 
tems not only makes it possible to solve navigational problems auto- 
matically or semi- automati cally , but also makes it possible to realize 
their solution for automatic pilotage of an aircraft along the given 
trajectory. An example of such a realization is the automatic pilot- 



366 



age of an aircraft on the basis of signals from Doppler meters with 
automatic navigational Instruments. 



These complexes generally involve autonomous 


navig 


ational in- 


struments: transmitters 


for the course, airspeed. 


and 


drif t angle . 


The only exception is the 


aircraft radiocompass , w 


hose 


readings are 


combined with the reading 


s of course instruments t 


o obt 


ain hearings 


However, this is a result 


of a peculiar feature on 


the 


use of radio 


compasses (for obtaining 


the bearing it is necessary to 


add the 


course angle of the radio 


station to the aircraft 


course ) . The use 


of simple combinations of 


navigational systems sue 


h as 


ground ra- 


dars , radio distance-find 


ers , externally directed 


gonio 


metric and 


goniometric-rangefinding 


systems , fan-type beacons 


, and 


hyperbolic 


systems cannot be combine 


d satisfactorily. 







The first characteristic of combined navigational systems and 
aviational sextants is that they are intended only for determining 
discrete values of aircraft coordinates. Therefore, they can be 
used in navigational complexes as sources of information which dup- 
licate the results of automatic calculators of the aircraft path, 
i.e., only for purposes of correcting previously obtained navigational 
parameters . 

The second feature of these devices is that with a relatively 
high accuracy of coordinate measurement for the aircraft, they can- 
not be used to determine the first derivatives of these coordinates 
with time. Let us illustrate this with a concrete example. 

Let us say that some navigational instrument, taking its instru- 
mental errors into account (for electronic devices, considering the 
conditions for propagation of electromagnetic waves, and for astron- 
omical ones, the accelerations of the level of the aircraft) make it 
possible to determine the successive coordinates of an aircraft with 
an error which does not exceed 1 km, so that the error in measure- 
ment can change in value and sign. 

In this case, the error in determining the direction of the 
aircraft motion on the basis of two successive measurements may have 
a maximum value of 

AZ 2 

With a measurement base of 30 km (approximately 2 min of flight 
in a jet aircraft), the angular error in the measurements can reach 



367 



_2_ = _!_«4°. 
30 15 

If the measurements are made more frequently, (e.g.) oftener 
than each minute of flight, the error in measuring the direction can 
reach 8° . 

With continuous measurement of the aircraft coordinates , the 
numerator in our example can retain its value , but the denominator 
will tend toward zero, i.e., the error in determining the direction 
of flight or (what amounts to the same thing) the first derivative 
of the Z-coordinate with time, will be equal to infinity. 

We can reach an analogous conclusion for the case of determin- / 350 
ing the ground speed of an aircraft (the first derivative of the X- 
coordinate with time) by continuous measurement of it at a succession 
of points where the LA is measured. 

This example shows that communication and astronomical naviga- 
tional systems can only provide a rough pilotage of the aircraft 
along a given trajectory. With a very precise measurement of the 
aircraft coordinates along the route (with errors no greater than 
200-300 m) and a very careful damping of the readings (averaging 
for time), automatic pilotage will take place with variations of 
the course within limits of 5-6°, i.e., 5-10 times greater than 
would be obtained by the results of measuring the drift angle by 
a Doppler meter. 

The only exception to this is the pilota-ge of an aircraft 
using strictly stabilized zones of landing beacons, where the errors 
in determining the deviations from a given trajectory are measured 
in several meters. Under these conditions, the pilotage of an air- 
craft can take place with variation of the course within limits of 
1-2° with a very precise maintainance of the general direction of 
flight. 




etric-range- 



make the spherical conversions. 

It is somewhat simpler in this regard to use goniometric-r 
finding methods for short-range navigation and aircraft radars. 
Due to the limited radius of their operation, the polar coordin 
of these devices can be converted into orthodromic ones by solv 
simple equations for plane representations, using simple calcul 
ting devices with low accuracy. 



368 



Combined nav 
combined into nav 




369 



CHAPTER FOUR 

DEVICES AND METHODS FOR MAKING AN INSTRUMENT LANDING 

SYSTEMS FOR MAKING AN INSTRUMENT LANDING 

Landing an aircraft under conditions of limited ceiling and 
meteorological visibility in the layer of the atmosphere near the 
ground is the most complicated and difficult stage of the flight. 
Even under favorable meteorological conditions, a proper landing 
of the aircraft requires considerable attention and experience on 
the part of the crew. 



/351 




@^^ 



Fig. 14.1. Setting the 
Aircraft Course for 
Lining Up with the 
Runway • 



Experience has shown that in order 
to land any kind of aircraft, it is neces- 
sary that it be located exactly on the 
landing path at a certain distance from 
the touchdown point (Fig. U . 1 ) . It is 
also necessary that the course followed 
by the aircraft be selected so that the 
vector of the ground speed is directed 
along the axis of the landing and take- 
off strip (LTS). 



However, it is not desirable to land an aircraft with a lead 
in the course being followed in order to compensate for the drift 
angle, since this causes considerable lateral stresses on the air- 
craft undercarriage when it begins to taxi along the runway. There- 
fore, the longitudinal axis must be lined up with the LTS immediately 
before landing, by making a flat turn without banking. Then (since 
the turn was flat) the aircraft will keep the desired direction 
of motion relative to the Earth's surface for a short period of 
time, showing lateral deviation relative to the air mass flowing 
over it. 

This shift gradually dies out, eventually turning into a drift 
angle relative to the new course of the aircraft. Therefore, the 
selection of the approach angle must be made several seconds (no 
more than 5 or 7) before landing the aircraft. 



It should be mentioned that the correct selection of an air- 
craft course while keeping it simultaneously on the given trajec- 
tory for landing poses considerable difficulties for the crew in 
preparing to land. 



/352 



370 



In cases when the aircraft is not lined up with the runway, 
it is necessary to carry out a maneuver which will bring it on to 
the axis of the LTS , and which involves a considerable loss of time 
and also a loss of distance along axis LTS (Fig. 4.2). 




fWTOTTI 



Let us say th 
along the LTS axis 
reached a point at 
deviation is equal 



at in a flight 
, the crew has 

which their lateral 

to Z. 



Fig. 4.2. S-Shaped Maneu- 
ver for Lining Up the Air- 
craft with the Runway . 



Obviously, in order to line 
up the aircraft with the LTS axis 
in the most economical fashion and 
without any remaining deviation rela- 
tive to the LTS axis, it is neces- 
sary to turn the aircraft toward 
the runway through a turn angle ( TA ) 
of a magnitude such that the lateral 
deviation of the aircraft from the 
LTS axis is reduced by a factor of two. Then the aircraft must 
be turned by the same amount in the opposite direction but through 
an angle such that the trajectory along which the aircraft is tra- 
veling when it emerges from the turn coincides with the LTS axis. 

In order to avoid loss of the selected direction of the ground 
speed vector while making the turns, i.e., shifting the aircraft 
after placing it on the landing course, the turns made by the air- 
craft must be coordinated as much as possible. 

It is obvious from Figure 4.2 that the magnitude of each of 
the two coordinated turns for bringing the aircraft on to the runway 
can be determined by the formula 

Z = 2(i?- cosTA) = 2i?( 1-cosTA) , 



wh ence 



cosTA 



Z 
2i? 



where i? is the turning radius of the aircraft with a given bank- 
ing and airspeed during the turn. 

Obviously, while the aircraft is making the maneuver to land, 
it must travel through a path along axis LTS 



X 



2i?sinTA. 



Example . When an aircraft is descending and is lined up with 
the runway on the desired course and with a horizontal ground speed 
of 280 km/hr, there is a lateral deviation from the LTS axis equal 
to 60 m. 



371 



Find the angles of the combined turns of the aircraft with 



/353 



a given banking of 8° and the path of the aircraft along the descent 

path during the completion of the maneuver. 

Solution. The radius of the turns made by the aircraft are 

found by using a navigational slide rule (Fig. 4.3), which gives 
the answer 4-500 m. 



cosTA 



60 



4500 



.9867 ; 



sinTA 



TA = 9°20 ' 



. 1625 ; X = 2 4500 . 1625 



1463 m, 



ANSWER: R = 4500 m; TA = 9°20'; X - 1463 m, 



However, we must take into account the fact that the desired 
aircraft path in lining up with the runway must be chosen on the 
basis of the assumption that the turns are made with a constant 
banking angle, i.e., with a stable turn regime. At the same time, 
there is a delay in the maneuver produced by the reaction of the 
crew and mainly due to the inertia of the aircraft when entering 
and emerging from the turns . 



© (^ 



tsaa 

Fig. 4.3 



280 




Fig. 4.4 



Fig. 4.3. Using the NL-IOM to Determine the Turning Radius of an 
Aircraft . 

Fig. 4.4. Landing Profile for a Jet Aircraft. 

As special tests have shown, the delay in the maneuver occurs 
primarily along the descent path and has practically no influence 
on the desired magnitude of the angles of the combined turns. This 
is explained by the fact that when the aircraft is entering and 
emerging from a bank at the beginning and end of the maneuver, the 
axis of the aircraft practically coincides with the axis of the 
LTS and the aircraft has practically no lateral velocity at these 
points . 

As far as the movement of the control surfaces when making 
the turns is concerned, the time required to move them is approx- 
imately two times less than the time required for the aircraft to 
enter and leave the turn, so that the lateral component of the air- 



372 



craft speed at turn angles up to 12° has a magnitude less than one- 
fifth of the longitudinal velocity. 

The delay time in the maneuver depends on the square of the 
horizontal velocity of the aircraft. At glide speeds of 280 km/hr, 
the delay time is equal to 4.5 sec of flight time on the average, 
or 350 m of the aircraft's flight along the LTS axis. This means 
that in our example, the required travel of the aircraft in lining 
up with the runway is equal to approximately 1800 m. 

At the same time that the course is b.eing selected which must 
be followed in order to make the landing, the crew must begin some 
distance away from the landing point to set up the desired descent /35M- 
trajectory in the vertical plane (Fig. H.4). 

In Figure 4 . M- , Point A is the point of transition from hor- 
izontal flight along the landing path to the descent regime of the 
aircraft . 

Point B is the point where the landing distance begins, which 
is also called the critical point for safe transition to making 
another pass. After this point has been passed, a second attempt 
at landing cannot be made, so that the aircraft must make a final 
selection of the aircraft course before this point is reached and 
the deviation of the aircraft from the given trajectory (upward 
and downward) must not exceed certain limits. Before this point 
is reached, a decision must be made either to make the landing or 
circle around the airport once again. 

After the starting point for the landing distance has been 
passed, the crew carefully observes the altitude. To do this, a 
leveling point C is selected along the approach to the airport (this 
is a conditional designation for the point where the descent tra- 
jectory of the aircraft crosses the Earth's surface), toward which 
the further descent of the aircraft is aimed. 

With proper descent and a constant pitch angle of the aircraft, 
this point is projected at a constant level on the cockpit window. 
If the approach is being made too rapidly, this point shifts upward 
on the glass, and if the aircraft is coming in too slowly it moves 
downward . 

Before reaching Point C (at an altitude of 8-15 m, depending 
on the type of aircraft) the aircraft levels off and then lands 
at Point D. 

The descent trajectory of the aircraft in the vertical plane 
is called the glide path. The aircraft is kept on a fixed glide 
path by selecting the proper angle of pitch for the aircraft and 
the correct amount of power to the engines. This process is much 
simpler in principle than the selection of the course to be fol- 
lowed by the aircraft, since it does not require maneuvering but 



373 



only the proper setting of the pitch angle and the levers which 
control the motors. However, it complicates landing as a whole 
because both processes must be carried out simultaneously while 
a given horizontal airspeed is being maintained. 

Unlike all other navigational devices, the systems used in 
making an instrument landing are intended specially for keeping 
the aircraft on a given descent trajectory before landing in the 
horizontal and vertical planes. 

The proper operation of these devices and the maneuverabil- 
ity of the aircraft determine the minimum permissible distance from 
the LTS at which the aircraft can be piloted by instruments or by 
instructions from the ground, with correction of any errors that 
may occur after changeover to visual flight. The more precisely 
the desired trajectory is maintained by instruments, the closer 
the transition to visual flight will lie to the landing point and 
the lower the altitude at that point. 

The limits within which an aircraft can be piloted by instru- /355 
ments without the airport being visible and with no terrestrial 
landmarks in sight which could show approaches to the airport is 
called the weather minimum for landing the aircraft. 

At the present time, there are three principal types of sys- 
tems for making instrument landings: 

(a) A simplified landing system which involves lining up the 
aircraft with radio stations. 

(b) A course-glide landing system. 

(c) A radar landing sytem. 

A necessary complement to each of these systems is the sys- 
tem of landing lights at the airport. 

Simplified System for Making an Instrument Landing 

The complex of devices in the simplified system for making 
an instrument landing on the basis of information from two master 
radio stations includes the following: 

(1) Two master radio beacons, located on the LTS axis, whose 
standard designation is the short-range master station (SRMS), located 
1000 m from the end of the LTS, and the long-range master station 
(LRMS), located 4000 ra from the end of the LTS. 

(2) Two USW marker beacons with a narrow vertical propaga- 
tion characteristic for electromagnetic waves, located on the same 
sites as the LRMS and SRMS. 



371+ 



(3) The lighting of the approaches to the LTS and its out- 



line 



(4) The complex of aircraft radio navigational and pilotage- 
navigational equipment as a whole . This includes : 

(a) One or two radio compasses, 

(b) A marker receiver, 

(c) Course control of the aircraft, 

(d) A barometric altimeter, 

(e) A radio altimeter for low altitudes, 

(f) An airspeed indicator, 

(g) A gyrohorizon, 

(h) A vertical speed indicator (variometer). 




a very limited application j.wx j^^j-j^^^^^ ^^ 
its use is very simple from the standpoint of 
which must be taken into account. 



errors 



the methodological /356 



In particular, we shall acquaint ourselves with the operating 
principles of the following pieces of equipment: marker devices, 
radio altimeters for low altitudes, the gyrohorizon and variometer. 

Marker Devices 



In order to make a landing with the simp 
very important to know (admittedly, at separa 
tance remaining until the end of the runway. 



landing with the simplified system, it is 

te points) the dis- 



As we know, aircraft 



radio compasses do not permit a precise 
nt when an aircraft flies over the control 
e to the special characteristics of the 



determination of the moment when an aircraft flies 

radio station; this is due to the special character __ ^_^„ „_ ^.._ 
operation of the open antenna aboard the aircraft. To solve this 
problem, marker beacons and aircraft marker receivers have been 



Marker radio beacons are transmitters with a directional trans- 
mission characteristic vertically upward, sometimes with a slight 
deviation toward the LTS so that the limit of the directional char- 
acteristic of the radiation is located to one side of the LTS and 
as close as possible to the vertical. In this case, an aircraft 
which is flying over the beacon towards the LTS will receive the 
signals from the marker transmitter at the moment when it is exactly 
above the beacon. 

For purposes of recognition, the transmission from the marker 



375 



beacon is not continuous but in the form of frequent short pulses 
(SRMS) or longer, less frequent signals (LRMS). These signals are 
heard aboard the aircraft for a period of 3-6 sec after it has flown 
over the vertical limit of the radiation characteristic and before 
it crosses the second, deflected limit of the characteristic. 



A still simpler device is the aircraft marker receiver. It 
is set to one frequency which is the same for all beacons . There- 
fore , it is very simple in design, has small dimensions, and requires 
no attention for use except to be switched on and off. 

When used in a complex together with course-glide devices, 
the marker receiver is turned on by a switch which is combined with 
the course-glide equipment, so that the crew does not have to inter- 
fere in its operation at all. In many cases, the marker receiver 
is combined with the switch for the radio compasses, the purpose 
being to ensure a low consumption of electrical energy, and allow 
stability and high reliability in the operation of this receiver. 

The marker receiver is connected to a light signal (a red light 

on the instrument panel in the cockpit marked "marker") and to a 

device which gives a simultaneous sound signal by means of a bell. 

Thus, when the aircraft flies over the marker, the lamp flashes 
and a series of short rings is heard. 



Low-Altitude Radio Altimeters 



/357 



At the present time, low-altitude radio altimeters based on 
the principle of frequency modulation are the ones most widely em- 
ployed . 



4. 5 



A schematic diagram of such a radio altimeter is shown in Figure 




oscillation 
'counter. 



indicator 



TTT^frmisT^f^f^^T^sx 



Fig. 4.5. Diagram of Low-Altitude Radioaltimeter , 



376 



The radio altimeter transmitter has a modulating device which 
produces a saw-tooth wave. For this purpose, we can use (e.g.) 
a variable membrane capacitor with mechanical oscillation of the 
membrane . 

The frequency of the signals 
reflected from the ground and 
picked up by the receiving antenna 
has the same saw-tooth character- 
istic, but is shifted in time 
by a value t , required for the 
electromagnetic waves to travel 
from the transmitting antenna 
to the ground and back again to 
the receiving antenna (Fig. 4.6). 




Fig. 4- . 6 . Frequency Char- 
acteristic of Radioaltimeter , 



It is clear from the figure that the frequency difference be- 
tween the emitted and received waves at any moment in time (with 
the exception of the segments between the extreme values of the 
frequency characteristic) will be strictly linear with respect to 
the flight altitude. For a complete retention of the linearity, 
these segments can be cut out by cutting off the receiving section 
with a n-shaped voltage at the end points of the emitted frequency. 

The emitted and received frequencies are combined in the bal- 
ancing detector, where a low frequency is formed which is propor- 
tional to the flight altitude. 

Following amplification, the low frequency is converted to /35 8 
rectangular oscillations which are calibrated both in terms of ampli- 
tude and duration. Thus, the counting circuit will receive pulses 
which are of uniform magnitude, and whose number per unit time will 
depend on the flight altitude. 

The number of calibrated pulses is summed and fed in the form 
of a direct current to the indicator, whose pointer shows the alti- 
tude in meters. 

In the simplified landing system, the radio altimeter plays 
only an auxiliary role as an indicator of a dangerous approach to 
the ground, since its readings depend upon the nature of the relief 
and cannot be used for checking the rate of descent. To set up 
the descent trajectory of the aircraft, barometric altimeters are 
us ed . 

In more complete landing systems , the radio altimeter can be 
used to give a trajectory value as well, but only in the last stage 
of descent before landing above a given final area of safety adjoin- 
ing the LTS. 

Since those landing systems which ensure descent of the air- 
craft by instruments until the point where the landing distance 



377 



begins use the radio altimeter only to signal a dangerous approach 
to the ground, we can exclude them for convenience from the group 
of basic pilotage instruments located in the center of the field 
of vision of the pilot, and use audible signals. If an aircraft 
is making a descent and reaches the limit of permissible altitude 
above the ground, the audible signal warns the crew of the neces- 
sity to terminate descent. 

Gyrohortzon 

The artificial indicator of the position of the horizon rela- 
tive to the axis of the aircraft ( gyrohori zon ) is a common pilot- 
age instrument, intended for piloting the aircraft when the true 
horizon is not visible. However, it is very important in guiding 
the aircraft along a landing trajectory, where it is used for main- 
taining a desired landing trajectory. 

In principle, the design of the gyrohorizon is simpler than 
that of the gyrosemicompass , e.g., unlike the latter, the gyrohor- 
izon has a vertical axis of rotation for the gyroscope, and a grav- 
itational correction device suspended from the bottom of the gyro 
assembly. This serves to keep the gyroscope axis constantly vertical 
in the aircraft. 

The external frame of the gyrohorizon is located horizontally, 
while its axis of rotation coincides with the longitudinal axis 
of the aircraft. Therefore, we can immediately determine the exist- 
ence and magnitude of a lateral rolling of the aircraft by the posi- 
tion of the external frame relative to the axis of the aircraft. 

For this purpose, a silhouette of the aircraft has been pasted 
on the glass which covers the dial, and a horizontal strip which 
moves up and down imitates the position of the visible horizon. 

Figure 4.7 shows the schematic diagram of the gyrohorizon. 




Fig. 4- . 7 . Diagrams of Gyrohorizon: (a) Kinematics; (b) Indicator, 



378 



Gyro assembly 1, with a vertical axis of th 
and the gravitational correction device 2 mounte 
are suspended in the horizontal external frame 3 
bly bearings 4. The carrier for the horizon lin 
to the casing of the gyro assembly and displaced 
relative to the horizontal axis of the gyro asse 
direction of the aircraft's flight from the forw 
instrument). The axis of the line is fastened a 
external frame, also along the flight direction 
Therefore, when reducing the angle of pitch of t 
strip of the gyrohorizon 7 moves upward, remaini 
horizontal axis of the gyro assembly. When the 
the horizon line moves downward as the true hori 



e gyroscope rotor 
d at the bottom, 

on the gyro assem- 
e 5 is fastened 

somewhat forward 
mbly (along the 
ard part of the 
t the front to the 
of the aircraft, 
he aircraft, the 
ng parallel to the 
pitch angle increases, 
zon does . 



During lateral rolling of the aircraft, the casing of the gyro- 
horizon (along with the silhouette of the aircraft) rotates rela- 
tive to the bearings of the external frame 9 in the direction in 
which the aircraft is rolling, which provides an indication of the 
rolling of the aircraft relative to the horizontal strip. For esti- 
mating and maintaining given longitudinal and lateral rolling of 
the aircraft, a scale is located between the outer frame and the 
horizon line and shows scale divisions for estimating the magnitude 
of the rolling in degrees. 

The gyrohorizon, fitted with the kinematic system described 
above, can be used within limited degrees of longitudinal and trans- 
verse rolling of the aircraft. Obviously, the rear bearing of the 
outer frame, i.e., the one located between the outer frame and the 
scale, must be mounted on a support in the unit. This support acts/360 
as a pivot for the lever supporting the horizon line, e.g., when 
the aircraft rolls over on one wing. 

In the case of considerable changes in the pitching angle of 
the aircraft (e.g., in a Nestrov loop), a support will hold the 
lever for the strip in a notch on the outer frame. 

The projection of one of these supports limits the degree of 
freedom of the gyroscope, thus leading to a "dis-location" of its 
indications, and a very long period of time is required to readjust 
them by gravitational correction. 

To ensure "nondis location" of the operation of the gyrohor- 
izon, the gyroscopic section protrudes outside the housing of the 
instrument, i.e., constitutes a separate gyroscopic instrument, 
a gyrocompass without a limited degree of freedom. The readings 
of the gyrovertical are transmitted to the horizon indicator by 
means of master and slave selsyns. 

We should also note that gyrohorizons or gyroverticals are 
transmitters which indicate longitudinal and lateral rolling for 
the operation of autopilots, acting as transmitters of turn angles 
of the aircraft in the horizontal plane, in which gyroscopic semi- 
compasses are used. 



379 



Yaviometev 

A vavlometev is a device which measures the rate of vertical 
descent or climb of an aircraft. 



The operating principle of a variometer is based on the decel- 
eration of a current of air which equalizes the pressure inside 
the body of the unit with the external static pressure. This means 
that when vertical movement occurs, a pressure drop develops within 
the body of the unit and in the static tube (Fig. 4.8). 



capillary 



b: 



cxxr 



• pcco 



Fig. 4.i 
iome ter . 



from static 
pressure 
sensor 

Diagram of Var- 



The pressure from the static 
pressure intake passes directly into 
the manometric chamber of the instru- 
ment. Within the body of the instru- 
ment, this pressure passes through a 
capillary opening, i.e., with retar- 
dation. Therefore, when the aircraft 
gains altitude, the pressure in the 
unit will be somewhat higher (when 
the aircraft descends, somewhat lower) 
than inside the manometric chamber. 
This pressure drop is proportional 
to the vertical speed of the air- 
craft . 



To measure this drop, the variometer is fitted with a transmit- 
ter mechanism, similar in principle to the mechanism of the altim- 
eter or speed indicator. The indicator scale is graduated directly 
in terms of vertical speed, as expressed meters/sec. 

Angle of Slope for Aircraft Glide /361 

The proper selection of an angle of slope for gliding is very 
important for all instrument landing systems, and especially for 
the simplified systems guided by master radio stations, both from 
the standpoint of making a safe landing and the meteorological 
minimum at which a landing can be made. 

When making an approach to land, it is very important that the 
flight altitude (^) correspond to the remaining distance (iS) to the 
point where the aircraft touches down: 

E = S tge 
rem 

where 9 is the glide angle. 

A simplified system of instrument landing makes it possible 
to determine the remaining distance to the landing point only when 
passing over the LRMS and SRMS. 

The point at which the aircraft begins to descend from the 



380 



altitude established for circling above the field is determined by 
calculating the time, and is therefore insufficiently exact. A 
descent between the LRMS and SRMS is also made by calculating the 
path of the aircraft with time, but this calculation takes only a 
short period of time and is performed after a certain point has 
been passed; it is therefore more accurate. 

According to the standards adopted in the USSR, the flight 
altitude for circling over an airport (for aircraft with gas tur- 
bine engines) has been set at 400 m; for piston-engine aircraft, 
it is 300 m. In both cases, however, the true flight altitude above 
the local terrain surrounding the airport must be no less than 
200 m. This altitude reserve is retained even when coming straight 
in for a landing, until the beginning of descent in the designated 
gli de pattern . 





From 


the moment 


when descent begins 


in a gli 


aircraft 


passes over 


a certain marker (LRMS), the 


ab 


ove the 


relief is ] 


kept at a minumum of 


150 m. 


th 


e LRMS, 


and before 


reaching the SRMS, the heigh 


ab 


ove the 


terrain is 


reduced from 150 to 


50 m. D 


however , 


it is necessary to keep in mind 


the fact 


a 


possibl 


e premature 


loss of altitude , in 


case of 


St 


rong head wind. ¥• 


or this reason, it is 


conside 


fl 


ight altitude between the LRMS and the 


SRMS (fl 


ab 


ove the 


SRMS) must 


be at least 50 meters above 


in 


the vicinity, beg. 


inning at half the di 


stance b 


an 


d SRMS 


and extending to the point where 


the SRM 



de , and until the 
altitude reserve 

After flying over 

t of the aircraft 

uring this maneuver, 
that there may be 
an unexpected 

red that the minimum 

ight altitude 

the heighest point 

etween the LRMS 

S is locate d . 



These same altitude reserves are maintained even when using 
more complete landing systems, although in this case the given glide 
path for the aircraft is defined in space and the probability of a 
premature descent is sharply reduced. In this case, however, the 
basic method for checking the proper descent is the measurement of 
the barometric altitude when flying over the marker points, thus 
guaranteeing safety of flight in case the landing instruments 
aboard the aircraft or on the ground should malfunction. 



/362 



In cases when the approaches to an airport are free of ob- 
structions, the angle of slope in the glide path is set equal to 
2°40'. The flight altitude relative to the level of the airport in 
this case is set at 200 m above the LRMS and 60 m above the SRMS. 



Typical Maneuvers 



Landing an Aircraft 



Simplified systems for bringing an aircraft in for a landing 
are used at airports with a low traffic density, where the installa- 
tion of complex landing systems would not be justified. Conse- 
quently, it is difficult to know in advance whether these airports 
will have provision for radar control, to set up the approach and 
landing pattern on command from the ground. Hence, the approach 
for landing is made with the same devices which are used in landing 



381 



the aircraft along a straight line. For this reason, a successful 
accomplishment of the maneuver under these conditions will be assured 
if the starting point for the maneuver is one of the marker points 
of the system. 



70sec.^""''>O 




M-1200 








H = J900-t?00 



Usually a LRMS is use 
this purpose, since at the 
ity of airports , it is the 
control facility at the ai 
There are then three possi 
ways to bring the aircraft 
the starting point for the 
neuver : 

(1) An approach of th 
craft to the LRMS, with a 
angle close to the landing 

(2) An approach to th 
with a path angle nearly p 
dicular to the landing cou 

(3) An approach of the aircraft to the LRMS, with a path 



H'2800 



Fig. 4.9. Large and Small 
Rectangular Landing Patterns 



d for 


ma] or- 


mam 


rport . 


ble 


to 


ma- 


e air- 


path 


course 


e LRMS, 


erpen- 


rse . 


angle 



nearly the reverse of the landing course. 

In directing the aircraft toward the LRMS at path angle close 
to the landing course, the approach for landing can be made along 
a more or less straight- line course (Fig. 4.9). 

A large rectangular route is covered in this case, if the 
aircraft approaches the airport at a great altitude (for aircraft 
with gas turbine engines, this is 3900 to 4200 m), and an additional 
length of time is required for the aircraft to descend before land- 
ing. 

In this case, in making the approach to the LRMS, the aircraft 
makes a turn to the path angle for landing (in the following, the 
path angles will be referred to as magnetic), at which the aircraft 
descends to 2800 m (relative to the pressure at the level of the 
airport where it is landing). 

At an altitude of 2800 m, the double turn begins (first and 
second turns without a straight line between them) at 180° with a 
descent to 1200 m. Flight then continues with a magnetic path 
angle (MPA) opposite to the landing angle, with descent to the 
altitude set for circling over the airport. 

In aircraft with gas turbine engines , limits have been set 
for the horizontal airspeed with the undercarriage lowered. There- 
fore, in a flight with a MPA opposite to the landing angle, the 
flight altitude for circling the field is maintained for 5 to 6 km 
until the LRMS is passed, so that at the moment when it actually 
is passed, the speed of the aircraft in horizontal flight can be 
cut to the speed established for lowering the undercarriage. 



/363 



382 



After passing over the traverse of the LRMS, the flight 
continues opposite to the landing direction for 70 sec, prior to 
starting the third turn (usually at a flight altitude of 400 m , 
up to CAR = 120° to the right and up to CAR = 240° on the left 
straight-line paths). The undercarriage is lowered in this path 
segment . 

After a period of 70 sec flying time from the moment when the 
traverse of the LRMS is passed or until CAR-120 (240°) is reached, 
the third turn is made. Since the horizontal airspeed in the 
vicinity of the third turn is much less than in the vicinity of the 
doubling of the first and second turns, the radii of the third 
and fourth turns (with a banking angle of 15 to 17°) are then much 
less than the radius of the double turn. Therefore, between the 
third and fourth turns there is a period of straight-line flight 
which lasts 50 to 55 sec. This straight -line segment is used for 
preliminary lowering of the wing flaps before landing, and also 
acts as a "buffer", which compensates for errors in aircraft nav- 
igation in cases when the effect of a side wind in making the ma- 
neuver from the starting point until the end of the third turn has 
not been estimated sufficiently precisely. 

In these cases, the "buffer" line can be extended or shortened 
somewhat, but the last (fourth) turn must be always made on time. 

At airports where the nature of the local terrain or complex 
wind conditions render flight along a straight line at 400 m im- 
possible (for aircraft with gas turbine engines), but the estab- 
lished flight altitude is 600 or 900 m, the duration of the flight 
from the traverse of the LRMS to the beginning of the third turm 
is increased, so that after the aircraft emerges from the fourth /364 
turn it is located below the glide path established for a given 
approach direction and has a segment of horizontal flight to the 
end of the glide path which is only 2D to 30 sec long. This time 
is needed to prepare the crew for landing and for extending the 
flaps fully . 

For example, if the flight altitude along a straight -line 
course is set at 500 m, and the slope angle of the glide path is 
2°4' , the fourth turn must be executed no closer than 15 km from 
the end of the LTS, since the aircraft (at an altitude of 600 m) 
enters the glide path at a distance of 13 km from the end of the 
LTS, and 2 km are required for the horizontal flight segment before 
entering the glide path. 

Consequently, the start of the third turn under calm condi- 
tions, after passing the traverse of the LRMS, lasts 2 minutes 
and 30 seconds of flying time (at Y - 350 km/hr), with CAR approx- 
imately equal to 135° (225°). If the flight altitude along the 
straight-line path is set at 900 m, the flying time from the 
traverse of the LRMS to the beginning of the fourth turn is increased 
to 3 minutes and 30 seconds, so that it is advisable to increase 



383 



the glide path up to 4° for the purpose of shortening the time in- 
volved in making the descent. 

In cases when an aircraft is approaching an airport with a path 
angle close to the landing angle, at an altitude of 1500 m or less, 
the double first and second turns are made immediately after pas- 
sing the LRMS . The descent to circling altitude and reduction of 
speed to lower the undercarriage in this case are performed in the 
designated turn. Hence, the large rectangular flight pattern in 
converted to a small one, and the maneuvering time is shortened to 
about 4 . 5 min . 



If the aircraft approaches the airport at the altitude estab- 
lished for circling the field, the radii of all four turns are made 
approximately the same, so that in order to create the "buffer" 
line between the third and fourth turns, the first and second turns 
of the aircraft are executed in succession with a time interval 
between the end of the first and the beginning of the second turn 
which equals ^■0 sec. 

The fourth turn on the large and small rectangular patterns 
is made along the course angle. In aircraft with gas turbine 
engines , the CAR at the beginning of the fourth turn (when turning 
to the right) must be equal to 70° (and -290° when turning to 
the left). 

The landing approach for aircraft with piston engines' is made 
according to the small rectangular pattern, with different first 
and second turns, and the same time parameters between the first 
and second turns (40 sec), from the traverse of the LRMS to the 
beginning of the third turn (70 sec) (at a flight altitude of 
300 m). The fourth turn for these aircraft begins at CAR = 75 or 
285°. 



/365 



However, due to the lower airspeed along the straight-line 
segments and the smaller turning radii, the linear dimensions of 
the maneuver for aircraft with piston engines are much less than 
for aircraft with gas turbine engines. In addition, due to the 
shorter time for each turn, the total time for executing the ma- 
neuver for aircraft with piston engines is shorter (for example) 
by 1 minute . 



Jfflsec, 




Fig. 4-. 10. Landing Maneuver When 
Approaching the LTS Axis at a 
90° Angle. 



384 



When an aircraft is approaching an airport at an MPA which is 
perpendicular to the landing angle, the landing altitude for air- 
craft with gas turbine engines is usually set at 1200 m above the 
level of the airport (Fig. 4- . 10 ) . After passing the LRMS, the air- 
craft continues on a course which lasts for ^■0 sec until descent, 
and the second turn is also executed with loss of altitude. 

After completing the second turn, the flight lasts 30 sec until 
the beginning of the third turn, when the undercarriage is lowered. 

A similar maneuver is executed by piston-engine aircraft, 
with the sole difference that the flying time from the LRMS to the 
beginning of the second turn is set at no less than 1 min , since 
the airspeed of these aircraft in all stages of the landing approach 
until emergence from the fourth turn is roughly the same. 

If an aircraft approaches an airport with an MPA which is close 
to the reverse of the landing angle, the crew of a gas turbine 
aircraft travels along a small rectangular pattern with different 
sides for the first and second turns (Fig. M-.ll,a). In the case 
of aircraft with piston engines, the so-called standard turn is 
executed in this instance (Fig. 4.11,b) on the landing course. 

These maneuvers agree in terms of the magnitude and direction 
of the turns; in the former, however, there is a s t raight -line 
segment between the second and third turns for lowering the under- 
carriage, while there is a "buffer" line between the third and 
fourth turns. In addition, if the aircraft approaches the airport 
at the flight altitude for circling the field, and all the turns 
of the aircraft are made without loss of altitude (the radii of 
all the turns being the same), then between the first and second 
turns there will also be a period of flight along a straight line 
for a period of 40 sec. 



In the case of a standard turn, all four turns will be made in / 366 
succession without there being any straight- line segments between 
turns . 



-J'i5',sec. 




CAi? 



O fT!X 



Fig. 4.11. Landing Maneuver with a Course Opposite to the Landing 
Course, (a) Along a Straight-Line Path; (b) Standard Turn. 

Analogous maneuvers for approaching to make a landing can also 



385 



be made by using more complete landing systems. However, air- 
ports that have such systems, as a rule, are also equipped with 
radar devices to monitor the aircraft maneuvering in the vicinity 
of the airport. Therefore, the beginning of the landing maneuver 
need not necessarily be made at the marker point on the LTS axis, 
thus making it possible to come in for a landing along the shortest 
path from any direction. 

A small or large rectangular pattern is usually used as the 
basis for setting up a landing approach along the shortest path. 
However, it is not generally completed, usually beginning at the 
point of tangency of the entrance into the maneuver to one of its 
turns . 

Calculation of Landing Approacli Parameters 
for a Simplified System 

In the preceding section, we discussed the typical maneuvers 
for landing an aircraft when approaching the airport from any di- 
rection. The execution of these maneuvers does not pose great 
difficulty for the crew of the aircraft, since the flight is made 
with a sufficient altitude reserve and sufficient speed, while the 
demands on the accuracy of making the maneuver are not very high. 

The main difficulty lies in flying along a given descent tra- 
jectory in the glide path, due to the very high demands on the 
maintenance of flight direction, altitude, and horizontal glide 
speed, depending on the remaining distance to the touchdown point. 
In the case of aircraft with gas turbine engines , there is the 
additional need to reduce the airspeed gradually as the airport is 
approached. 

In order to facilitate the task of descending along a given /367 
trajectory to a certain degree, as well as to avoid serious errors 
in flight along the landing path, some preliminary calculations 
are made, of which the following is the most important. 

If the landing approach is made in a dead calm, the geometric 
dimensions of the maneuver (and consequently, the point where the 
descent begins along the landing path) are determined by simple 
relationships between the airspeed, time, turn radii, flight 
altitude in circling the field, and established steepness of the 
glide path . 

The calculated data for making a landing in a calm are usually 
plotted on special landing patterns, devised for each airport. 
Under actual conditions, however, it is necessary to take into ac- 
count the head wind and side wind components (for the landing 
course), which can have a very great effect on the making of a 
landing . 



386 



Calculation of Corrections for the Time for 
Beginning the Third Turn 

In preparing to land, especially with the aid of a simplified 
system, it is necessary to ensure that the aircraft emerges from 
the fourth turn onto the landing approach always at the same distance 
from the LTS. Obviously, in order to solve this problem, it is 
necessary to consider only the head-wind component for the landing 
course . 

In making an approach to land along a rectangular pattern, the 
last reliable point for determining the J-coordinate of the air- 
craft (the distance along the axis of the direction of the airport) 
is the traverse of the LRUS, while in a standard turn it is the 
passage over the LRMS with an MPA opposite to the landing angle. 

If we do not take the wind into account when coming in for a 
landing, the aircraft will enter the landing path at a distance 
from the LTS which exceeds the distance for calm conditions by 
the value 



hX = u t, 

where t is the flying time from the traverse of the LRMS to the 
emergence from the fourth turn, or from the moment when the air- 
craft passes over the LRMS until it emerges from the standard turn. 

Example : The flying time from the traverse of the LRMS to the 
emergence from the fourth turn in a calm is 4 min , divided into 
these stages : 

Traverse of the LRMS to beginning of third turn... 70 sec 

Third turn 50 sec 

Buffer line 50 sec 

Fourth turn 60 sec 

The speed of the head-wind component on the landing course is /36 i 
Ux = 15 m/sec. 

Find the value of AJ for emergence from the fourth turn. 
Solution: 

kX = 240 X 15 = 360C m. 

In order for the distance for emergence from the fourth turn 
to remain the same as in a calm, it is necessary to shorten the 
flying time from the traverse of the LRMS to the beginning of the 
third turn by a value 



At = 



3600 
V+u 

X 



387 



In our example, for an airspeed of 400 km/hr (110 m/sec) and 
a course opposite to the landing course. 



A/ = 



3600 



110+15 



3600 
125 



= 29 sec 



Thus, a flight from the traverse of the LRMS to the start of 
the third turn would last 4-1 sec instead of 70 sec. 



By combining the formulas for obtaining the values for LX and 
Ai, we finally obtain the formula for determing the value ht : 



M = - 



tu. 



V + u, 



For our example. 



240-15 ' „„ 



The problem for a standard turn is solved in the same way. 
In this case, the time t is the time from the passage over the 
LRMS in a course opposite to the landing course, to the end of the 
standard turn; the value of At is calculated from that for a calm 
in flying from the LRMS to the start of the standard turn. 

Calculation of the Correction for the Time 
of Starting the Fourth Turn 

The beginning of the fourth turn in coming in for a landing is 
usually determined from the course angle of the LRMS. For example, 
when executing a maneuver to the right: 

cigCAR=^.. 

where R is the radius of the turn made by the aircraft, and X is 
the distance along the LTS axis from the LRMS to the starting point 
of the fourth turn. 



Under the influence of a side wind, the fourth turn is begun 
earlier if the lateral component of the wind on the landing course 
is favorable between the third and fourth turns, later if this 
component is unfavorable. 

Obviously, if we take the wind into account: 

R-\-t-u^ 



ctgCAR= 



X 



/369 



where t is the time of the fourth turn and Ug is the lateral com- 
ponent of the wind speed. 

For example, with a turning radius of 4500 m and X = 12.5 km. 



388 



r-AD 4500 

etc C AF'= — 

^ 12500 

CAR =70°, 



If the lateral component of the wind appears on the "buffer" 
line and is favorable, with a speed of 10 m/sec, then for a turnin; 
time of 60 sec we will have: 

^.„ 4500 + 60-10 
ctg C AR= ^ 



12500 
CAR =68°, 

i.e., the turn must begin 2° earlier than under calm conditions. 

Cataulati-on of the Moment for Beginning Descent 
Along the Landing Course 

Under calm conditions, the distance at which the aircraft 
emerges from the fourth turn is determined by the flying time from 
the LRMS to the start of the third turn. 

For example, with 7 = 111 m/sec (M-00 km/hr), this distance 
will be : 

^-70-111 wSk^ toLRMS 

The fourth turn will be completed at approximately this dis- 
tance if a correction for the effect of the wind is made in the 
time for starting the third turn. Consequently, with a standard 
location of the LRMS, the distance from the point where the air- 
craft comes out of the fourth turn to the touchdown point is 12 km. 

The distance for beginning the descent along the glide path 
is determined by the formula 

^^ = //c,ctg6, 

where X is the distance for beginning the descent and E is the 
flight altitude for circling the field. ^ 

For example, at a circling altitude of B - 400 m and a slope 
angle for the glide path 9 - 2°40': 

X , = 400-clg2°40' = 8500 J<. 
a 

Thus , after coming out of the fourth turn at a distance of 12 /370 
km from the LTS , the aircraft must follow the landing path without 
losing altitude for a period of time 



389 



Xr -Xd 

where tjj is the time of horizontal flight along the landing path. 

For example, if the horizontal airspeed after coming out of a 
turn is 360 km/hr (100 m/sec), and the head wind'is moving at 15 
m/sec, the time for horizontal flight in our case will be 

12000 — 8500 3500 

'h = 100-15 =-ir='*' "^^ 



Under calm conditions, the time for horizontal flight in this 
case will be : 

3500 

'^ =1oo' = =^^'^^^ 

Practically speaking, the descent of the aircraft must begin 
5 to 6 seconds before this time has actually elapsed, since a 
certain period of time is required to guide the aircraft into its 
landing regime. 

Calculation of the Vevtioal Rate of Descent 
Along the Glide Path 

The vertical rate of descent of an aircraft along the glide 
path is determined by the simple formula 

Ky=W^tgO = (K^_a^)tge. 

For example, with a mean horizontal rate of descent of 290 
km/hr (80 m/sec), a head wind of 15 m/sec, and a slope angle in 
the glide path of 2°40': 

V-y = 65.tg2°40' = 3^/ sec 

The calculation of the vertical rate of descent is of partic- 
ular interest for piston-engine aircraft, whose horizontal glide 
is about 50 m/sec. 

Since the head wind can be as h5gh as 25 m/sec on landing, 
the vertical glide speed for these aircraft can change by a factor 
of 2, i.e., from 2.3 to 1.15 m/sec. 

In the case of aircraft with gas turbine engines, the ratio 
of the maximum rate of descent to the minimum rate, with the same 
steepness of glide, is 1.5. 



390 



Determination of the Lead Angle for the /371 

Landing Path ~ 

A knowledge of the approximate value of the drift angle, and 
consequently the necessary lead angle for the landing path of an 
aircraft, considerably facilitates the choice of the course to be 
followed along a given descent trajectory. 

The value of the drift angle along the landing path can be 
determined by the approximate formula 

tg US= TT-^^^ . 



In flight along a given descent trajectory, however, the hor- 
izontal airspeed, altitude, and wind are variables, so that it is 
sufficient to use the following rule in finding the drift angle: 

(a) For aircraft with gas turbine engines, at glide speeds of 
270-290 kra/hr, the lead angle is considered to be equal to 0.7° 
for each 1 m/sec of side wind. 

(b) For aircraft with piston engines, (glide speeds of 180-200 
km/hr), the lead angle is considered to be 1° for each 1 m/sec of 

s i de wind . 

For example, with a side wind along the landing path of 

8 m/sec, coming from the right, the lead angle will be: 

- for aircraft with gas turbine engines, 5.5° to the left; 

- for aircraft with piston engines, 8° to the left. 

The calculations given above for the time of starting the 
third turn, the course angle for beginning the fourth turn, the 
time for beginning the descent, the vertical rate of descent, and 
the lead angle for the landing path, must all be made by the crew 
of the aircraft before approaching the airport on the basis of 
landing- conditi on information. All calculations must be complete 
before the landing maneuver begins. 

Landing the Aircraft on the Runway and Flight 
along a Given Trajectory with a Simplified Landing System 

While making preparations for landing, the crew must prepare 
the course to be followed by the aircraft along all the straight- 
line segments of the approach pattern, with the exception of the line 
between the third and fourth turns, beginning with a calculation 
of the drift angle. 

The radiocompass must be set by the LRMS ; if there are two 
sets of radiocompasses , the second must be set by the SRMS. 

Along the line between the third and fourth turns , the course 
to be followed is always equal to the MPA of the "buffer" segment. 



391 



so that the start of the fourth turn will be determined by the CAR. 
The slight drift of the aircraft which occurs at this time, as we 
have seen, is compensated by redefining the time for starting the 
third turn. 



/372 



When the course angle of the LRMS becomes equal to the cal- 
culated value, the fourth turn is executed with a banking angle of 
15° before acquiring the calculated landing path. 

If all the calculated data are correct, the aircraft will come 
out of the turn precisely on the landing path with the desired 
course. At the moment when the aircraft emerges from the fourth 
turn, the timer is switched on to determine the time for beginning 
descent in the glide path. 



In the majority of cases, however, due to errors in the oper- 
ation of the radiocompass , improper maintenance of the course and 
air speed of the aircraft, errors in determining the side-wind com- 
ponent, and failure to bank at the proper angle when turning, the 
acquisition of the glide path by the aircraft is not accurate. 

The accuracy with which the aircraft acquires the landing 
path is determined by a comparison of the magnetic bearing of the 
LRMS with the MPA for landing. IF MC + CAR = MPA , but the com- 
bined reading of the radiocompass is MBR = MPA, , the aircraft will 



be exactly on the axis of the LTS. 



'l' 



If MBR is greater than MPA-, , the aircraft will be to the left 
of the given landing path. With MBR smaller than MPA^, the aircraft 
will be to the right of the given landing path. 

The difference between MPAj^ and MBR is called the acquisition 
error a. 

Example: MPA^ = 68°, with a calculated drift angle of +3°; 
the aircraft emerged from the fourth turn with MC = 65°, the course 
angle for the turn over the LRMS was 358°; find the acquisition 
error . 



Sol ution 



a =68 — (65 + 358) = 5°, 



i.e., the acquisition error is 5° to the right. 

For lining up the aircraft with the landing path, the course 
followed by the aircraft is usually changed by doubling the acquisi- 
tion error. In our example, the course to be followed must be re- 
duced 10°, so that the CAR of the LRMS becomes 8°; the flight is 
continued at this course until the value of the course angle in- 
creases to the magnitude of the acquisition error, i.e., becomes 
13°. 



392 



When the pointer of the radiocompass is on the 13° mark (on a 
combined indicator, a bearing of 68°), with a slight lead (no more 
than 1 to 2°), the aircraft makes another turn to the calculated 
landing path, and the CAR of the LRMS becomes equal to the calcu- 
lated drift angle of the aircraft (3° in the example). 



the 



As the aircraft continues to follow the landing path on tl" 
calculated course, the CAR will remain equal to the calculated 
drift angle if the course of the aircraft has been properly selected. 

If the CAR is increased, the aircraft will drift to the left of 
the LTS axis, and the path being followed will have to be increased 
for acquisition of the desired line of flight, and decreased later 
on, although it will remain somewhat greater than the calculated 
value (the CAR is then less than the calculated drift angle). If /373 
the CAR is then to remain constant, the course to be followed must 
be selected properly. 

Similar operations in selecting a course are carried out when 
the aircraft deviates to the right of the desired line of flight. 
These operations will have the form of a mirror image of the oper- 
ations described above, i.e. , when the CAR is reduced, it is also 
necessary to reduce the course to be followed in acquiring the de- 
sired line of flight, then increase it somewhat, but still keep it 
below the calculated value. 

In the case when the course angle of the LRMS continues to 
change, after the first operation to correct the course by acquir- 
ing the line of the given course, the operations are repeated using 
the familiar method of half corrections. 

Thus, the readings of the radi ocompasses , beginning with the 
LRMS and then the SRMS, are used to maintain the given direction 
of the descent trajectory. 

When the aircraft is calculated to have reached the point for 
beginning its descent, it is shifted to a descent regime with a 
calculated rate of descent. The vertical rate of descent is main- 
tained by observing the variometer readings and those of the gyro- 
horizon, while maintaining the established regime of horizontal 
airspeed on the basis of the instrument-speed indicator. 

The gyrohorizon must be used to maintain the vertical rate of 
descent, because the readings of the variometer are less stable than 
those of the angle of pitch of the aircraft obtained with the aid 
of the gyrohorizon indicator. The readings of the variometer must 
be averaged over the time. 

In addition, the variometer has slight delays in the readings 
with a change in the angle of pitch of the aircraft. Therefore, 
the gyrohorizon is employed to select the angle of pitch for the 
aircraft at which the average readings of the variometer are equal 



393 



to the calculated values, and this angle is maintained by the readings 
on the gyrohorizon. 

If the horizontal airspeed is then increased or decreased rela- 
tive to the given value, it is regulated by changing the thrust of 
the engines and simultaneously changing the angle of pitch slightly 
to maintain the calculated rate of descent. 

A failure to maintain the calculated settiiq^ for the glide 
path, or errors in calculations, may cause the aircraft to pass 
over the LRMS earlier at the required altitude, so that the descent 
of the aircraft is terminated and the aircraft is once again placed 
in the regime of descent at the moment it passes over the LRMS. 
However, if the given altitude has not been attained when passing 
over the LRMS, the vertical rate of descent is increased at the 
stage of the flight between the LRMS and the SRMS . 

Similarly, the descent of the aircraft is terminated if it 
reaches the altitude set for passing over the SRMS before the sound 
of the SRMS is heard, marking the location of the latter. /374 

The minimum weather for the ceiling when landing with a simpli- 
fied system, in the case of aircraft with piston engines, is not 
set any lower than the altitude for passing over the SRMS; in the 
case of aircraft with gas turbine engines, it is significantly 
higher. Therefore, the aircraft can be allowed to descend only in 
the case when the crew of the aircraft can see the lights of the 
approaches to the LTS and the end of the runway. 

Course-Glide Landing Systems 

The simplified system for landing an aircraft as described in 
the preceding section, using the master radio stations, has a 
number of important deficiencies: 

(a) The measurement accuracy of the aircraft bearing, using 
an aircraft radiocompass and course meter, is very low, so that 
it does not make it possible to land the aircraft (especially 
those with gas-turbine engines) with low weather minima. 

(b) The operation of radiocompasses during flight in clouds 
and precipitation is highly subject to atmospheric disturbances, 
thus complicating a landing with these devices as guides. 

(c) The simplified system requires constant checking of the 
position of the aircraft along a given descent trajectory in terms 
of direction only; the descent of the aircraft in a given glide 
path is accomplished by maintaining the vertical rate of descent 
of the aircraft and calculating the time, thus complicating the 
landing procedure and not ensuring safe descent under especially 
difficult conditions. 



394 



If we consider that the period of landing the aircraft with 
low ceiling and low meteorological visibility is the most difficult 
and dangerous stage of the flight, it is necessary to devise more 
complete systems of instrument landing. One such system is the 
course-glide landing system. 

The geometric essence of course-glide systems is the use of 
radio-engineering methods to define two mutually perpendicular 
planes in space (Fig. 4- . 12 ) : 

(,a) A vertical plane which intersects the Earth's surface 
along the LTS axis . 

(b) An inclined plane which represents the glide path of the 
aircraft . 

If the aircraft is in one of these two planes, the readings 
of the corresponding pointer on the indicator (direction or glide) 
must be equal to zero. 



When the aircraft moves out of one of these planes, the 
corresponding pointer shifts from zero. The shift of the pointer 
must be linear within certain limits (i.e., proportional to the 
deviation of the aircraft from the given plane). 

Obviously, the given trajectory for the descent of the air- 
craft is the line of intersection of these two planes. When the 
aircraft is on the given trajectory, both Indicator pointers must 
point to zero on the indicator. 



/375 




Fig. 4.12. Radio-Signal Planes of a Course-Glide 
Landing System. 

For the best visual determination of the position of the air- 
craft relative to a given descent trajectory, the pointers on the 
Indicator are made in the form of strips, one horizontal for glide 
and one vertical for direction. The movement of the strips then 
occurs in a direction which is opposite to the deviation of the 
aircraft from a given trajectory (Fig. "4.13). 



395 



The center of the instrument, with a silhouette of an air- 
craft shown on the scale, shows the position of the aircraft rel- 
ative to the course plane and the glide plane. Thus, for example, 
in Fig. '^.IS the aircraft is located below the given glide path 
and to the left of the LTS axis . To set the aircraft on the de- 
sired trajectory, it must be turned in the direction of the planes, 
i.e., upward (to increase the angle of pitch) and to the right. 

The indicator for the direction and glide has the traditional 
name of "Landing System Apparatus", or LAS for short. 

Ground Control of Course-Gl-ide Systems 

The principal pieces of equipment in a course-glide landing 
system are two ground beacons which form the course zone and the 
glide zone marking the given trajectory for the descent of the 
aircraft . 

Both beacons operate on meter or centimeter wavelengths. 

The antennas of the beacons that use meter waves are crossed 
horizontal dipoles (horizontal frames) in course beacons and 
horizontal dipoles in glide beacons. 

Thus, the electromagnetic waves from the beacons are horizon- 
tally polarized, which to a certain degree reduces their effect on 
the directional characteristics of the antennas on the ground 
control facilities at the airport. 



/376 




At* Aj A2 A A, A^ A^ 













' 



to transmitting device 



Fig. tt.l3. Fig. i+.14. 

Fig. 1+.13. Indicator of Course-Glide Landing System. 

Fig. 4.14. Diagram of Location of Antennas of Course Radio Beacon. 

However, the Earth's surface plays a role in the formation of 
the course zone and the glide zone by these beacons. The course 
zone then becomes multilobed in the vertical plane, with the 
major lobe being the working lobe, which has a glide angle of the 
bisectrix which corresponds roughly to the slope angle of the glide 
plane of the aircraft. The Earth's surface is of still greater 



396 



importance for the formation of the glide zone, whose slope angle 
depends on the height of the antenna above the ground. 

The involvement of the Earth's surface in the formation of the 
beacon zones imposes limitations on the possibilities of the 
beacons in terms of ensuring the accuracy with which the aircraft 
can be landed. This is especially true for the glide zone, whose 
location can change with the state of the Earth's surface (wet or 
dry ground, grass cover, snow). The accuracy of the location of 
the course zone is subject to the influence of the local relief 
and equipment located within the limits of the directional charac- 
teristic of the antennas. 

The most important of these shortcomings can be overcome to a 
great extent by employing beacons which operate on the centimeter 
wavelength, using reflecting antennas to form very narrow directional 
characteristics . 

At the present time, however, these beacons have not been 
adopted sufficiently widely and are not used in enough locations. 
Therefore, we shall give a brief description of the course-glide 
systems only for the meter wavelengths. 

In addition to the beacons, which form the course and glide 
zones, the course-glide system for landing also includes marker 
devices, whose locations can coincide with those for the markers 
in a simplified landing system. 

In foreign practice, the first (long-range) marker is located / 377 
7 km from the end of the LTS ; at a slope angle for the glide path 
of 2°30' and a circling altitude of 300 m, this marks the point at 
which the aircraft begins to descent in a glide. However, no 
significant advantages are gained by placing the marker at this 
spot, since the flight altitude of the aircraft when circling the 
field depends on the type of aircraft, while the slope angle for 
the glide path depends on the nature of the surrounding terrain. 
This means that the point for beginning the glide does not always 
coincide with the standard location of the aircraft (7 km). 

For purposes of checking for the correctness of the location 
of the glide zone, it is better to choose a marker located ^ km 
from the end of the LTS, since at this point the aircraft will al- 
ready have the selected rate of descent for following the glide 
path, and the altitude of its location will be determined more 
precisely . 

An inherent part of the course-glide landing system is also 
the lighting system for the approaches to the runways and along 
the edges of the runway itself. 

The Gourse beaaon is a transmitting device with an antenna 
system which consists (as a rule) of five or seven horizontal 
antennas (Fig. 4.14). 

397 



fl 



Antenna A has a radiation characteristic which is directed 
externally in the horizontal plane, and is powered by a transmitter 
operating without modulation on the meter wavelength. 

Antennas Ai and A2 receive amplitude-modulated frequencies from 
the transmitter, one at 90 Hz and the other at 150 .Hz. 

Antennas A^ and A^^ (as well as ^45 and A^, in some types of 
beacons) serve to regulate the directionality of the radiation 
characteristic, as well as the direction of the radio-signal zone 
of the entire system. 

The combined result of the electromagnetic oscillations of the 
entire antenna system forms the directional radiation characteristic 
of the electromagnetic waves in the horizontal plane; an example of 
this is shown in Fig. 4.15. Figure 4. 15. a, shows the shape of the 
radiation characteristic in the horizontal plane; the left side 
is tie one modulated' by the 150 Hz frequency, while the right side 
is modulated by the 90 Hz frequency. 

Along axis AB , where the radiation characteristics intersect 
the modulation frequencies of 150 and 90 Hz, the modulation depth 
of the carrier frequency by both low frequencies is the same (i.e., 
the difference in modulation depth is zero). 

When the aircraft moves to the left of axis AB , the depth of 
the modulation with the 90 Hz frequency increases and that with the 
150 Hz frequency decreases. The picture is reversed when the air- 
craft moves to the right of the axis. 

The dotted lines in Fig. 4.15 show the projections of the 
radiation lobes of the electromagnetic waves in the vertical 
plane (as shown in Fig. 4.15,b) on the horizontal plane. 

Line AB is the common axis with a difference in modulation 
depths which is equal to zero for all lobes. However, the prin- 
cipal operating lobes are the first ones, located nearest to the 
ground . 

b) 



/378 





Fig. 4.15. Ra,dtation Characteristics 
of Course Radio Beacon: (a) in the 
Horizontal Plane; (b) in the Verti- 
cal Plane . 



398 



The radiation characteristic o 
in such a way that the axis of the 
depth coincides exactly with the ax 
necessary that the difference in mo 
3-4-° of the equal-signal axis incre 
deviation. With further deviation 
up to 10°), the difference in modul 
but not in proportion to the latera 
value or decrease, but without chan 
to the left or right of the radio-s 



f the course beacon is regulated 
zero difference in modulation 
is of the LTS. Hence, it is 
dulation depths within limits of 
ase linearly with the lateral 
from the LTS axis (within limits 
ation depths must also increase, 
1 deviation; it can maintain its 
ging sign in the entire hemisphere 
ignal axis , 



The distance for possible reception of the beacon signals in 
the sector 10° from the equal-signal axis in the working lobe of 
the zone must be within the limits of 45 to 70 km. 

The gZ-ide beacon is also a transmitting device, operating in 
the meter wavelength, but at a frequency different from that of the 
cours e beacon . 

The antenna system of the glide beacon consists of only two 
antennas (an upper and a lower), mounted on a common mast. The 
upper antenna is double, as shown in Fig. 4. 16, a. 

Both the upper and lower antennas receive an amplitude -modula- 
ted frequency, but with different modulation frequencies (for 
example, 90 and 150 Hz). 

Each of the antennas, together with the ground forms an inde- 
pendent working lobe with its own modulation frequency (Fig. 4-.16,b). 
The points of intersection of the working lobes in the vertical 
plane also form a radio-signal axis AB with a zero difference in 
the modulation depth. 

Since the characteristic of the antenna directionality in the /379 
horizontal plane is rather broad, the surface with a zero difference 
in modulation depth is conical, with AB as the generatrix. There- 
fore, the glide path can be an ideal straight line only in the case 
when the antenna system of the beacon is located at the point where 
the aircraft touches down on the runway. 



trans» H 
mining 



^ 



y^ 



-y 



y 



Fig. 





Fig. 4. 17 



Fig. 4.16. Glide Radio Beacon: (a) Diagram of Antenna Location; 
(b) Radiation Characteristic. 

Fig. 4.17. Hyperbolic Trajectory for Glide Plane. 



399 



Ill Mil I I II II II III I III I 



However, the glide beacon cannot be located on the LTS axis 
or even in the immediate vicinity of the LTS, since it would con- 
stitute a flight hazard. Therefore, the intersection of the cone 
with the zero difference in the modulation depth for the glide 
beacon of the equal-signal plane of a course beacon gives a hyper- 
bolic trajectory which does not touch the ground (Fig. 4.17). 

As the aircraft approaches along the landing path toward the 
traverse of the glide beacon, the glide path begins to "float" 
above the ground, moving upward after passing over the beacon. 

Since the location and shape of the directional characteristic 
of the two antennas of the glide beacon depends on the height of 
the antennas above the ground, the characteristic and the position 
of the line of their intersection in the vertical plane is regu- 
lated by the change in the height of the upper and lower antennas 
above the ground. 

As in the case of the course beacon, the increase in the 
difference of the modulation depth with deviation from the glide 
surface upward or downward must be linear with this deviation. 
However, the curvature of the curve of the change in the difference 
in modulation depth will not be symmetric in this case, as it is 
for the course beacon. A steeper curve for the change in the dif- 
ference of modulation depth is found above the glide surface, and 
a less steep curve is found below the surface. 

The operating range for a glide beacon in a sector of +_8° from 
the LTS axis must be at least 18 to 25 km. 

Ai-para ft -Mounted Equipment for the Course-Glide 

Landing System 

The following units make up the aircraft-mounted equipmeiit for 
the course-glide landing system: 

(a) Antenna and receiver for course-beacon signals. 

(b) Antenna and receiver for glide-beacon signals. 

(c) Control panel. 

(d) Landing-system apparatus (LSA) 



/380 



The receivers of signals from the course and glide beacons 

contain essentially the same elements, with the exception of the 

AAC (automatic amplification control), which is not shown in the 
figure . 



'^ ^1 

convertori-. IFA 



detec 
1 tor 



- LFA 



(go Hz 

nfilter 



J_ 



heterodyne" 



90Hz| 
recti ( 
fier 



50Hz 
I fi lter . 




Fig. 4-. 18. Diagram of Aircraft-Mounted Glide Radio Beacon 



400 



The glide-beacon receiver uses the circuit for the reinforced 
AAC. The latter is not employed in course-beacon receivers, since 
it would add the microphone commands relayed via this beacon to the 
aircraft when the communications receivers are out of order. 

The signals from the course and glide beacons are picked up 
by the antennas and amplified by the HFA. The selection of the 
frequency channel is made on the basis of the first intermediate 
frequency by quartz returning of the heterodyne from the control 
panel. The signals are then amplified by the IFA and LFA channels, 
so that the signals pass through 90 Hz and 150 Hz filters to the 
rectifiers, then to the emergency blinker, and finally to the 
receiver ground. The indicator for the course or glide zone is 
connected in a bridge circuit between the rectifiers for the 90 
and 150 Hz signals. 

If the signals do not reach the receiver or there is some 
malfunction in the receiver blocks somewhere ahead of the 90 and 
150 Hz filters, the readings of the LSA indicator on that channel 
will be zero; if the equipment is operating properly, it means 
that the aircraft is located precisely in the corresponding zone. 
Therefore, the LSA system includes the emergency blinkers. When 
no current is flowing in the 90 and 150 Hz rectifiers, the current 
through the emergency blinker windings will not flow, and a signal 
indicating that the apparatus is malfunctioning will be displayed 
on -fhe indicator. 

The design circuit for the receivers of the signals from the / 381 
glide and course beacons includes potentiometers for electrical 
balance of the LSA Indicators. Each of the rectifiers receives 
signals which do not pass through the 90 and 150 Hz filters. The in- 
dicator pointer should then point to zero. If the balance of the 
currents in the rectifier is upset, it causes the regulating poten- 
tiometer to rotate. 

The balancing potentiometer of the receiver for the signals 
from the glide beacon is usually mounted on the receiver housing, 
while the receiver for signals from the course beacon is mounted 
on the control panel. 

For smoothing the short-period oscillations of the course and 
glide indicator of the LAS, due to local disturbances in the 
radio-signal zone, the indicator circuit contains a special sealed 
unit damping capacitors in the circuit for turning on the apparatus. 

Location and Parameters for Regulating the 
Equipment for the Course-Glide Landing System 

The radio beacon for the course zone of the course-glide 
system for landing an aircraft is mounted at a distance of 600 to 
1000 m from the end of the runway, along an extension of the axis 
of the LTS. 

401 



f 



The beacon for the glide zone is mounted to the side of the 
LTS (as a rule, to the left of the landing path), at a distance of 



r 



glide beacon 



fOOOn 



-^ 



250- 



"Y^l} 



short range imarker 



-275m/ I50-200M 



course beacon 



BOO- 1000 m 



Fig. 4.19. Diagram of Location of Ground-Based Equipment for Course- 
Glide System. 

150 to 200 m from its axis and 250-275 m from the end of the runway 

The axis of the zone of the course beacon coincides with the 
LTS axis. A controx point is chosen for measuring the parameters 
for regulating the system on the LTS axis . 

The control point is selected as a point where the antenna 
receiving signals from the glide beacon aboard the aircraft will be 
located at the moment when the aircraft touches down on the runway. 
It is considered that this point is located at an altitude of 6 m 
above the surface of the LTS, and is plotted from the location of 
the glide beacon, 75 m toward the end of the LTS (i.e., the dis- 
tance from the end of the runway to the control point is 180 to 
200 m) . 

The slope angle for the glide path is calculated from a theo- 
retical plane located 6 m above the surface of the LTS. The ver- 
tex of the slope angle of the glide path is 1he control point (CP). 

The width of the zone of the course and glide beacons is reck- 
oned from the angles of deviation from the given descent trajec- 
tory, calculated respectively from the point where the course 
beacon is located and from the control point, within the limits of 
which the strips of the landing-system apparatus deviate from the 
zero position to the limits of the scale. 

Obviously, the angle of deviation of the LSA strip depends on 
the difference in the modulation depths in the beacon zones, as 
well as on the sensitivity of the receiver aboard the aircraft. 
Therefore, the angular width of the zones of the course and glide 
beacons is regulated by the sensitivity of the receivers mounted 
aboard the aircraft, which are used as standards. 

The standards for the width of the course-beacon zone are 
set as follows : 

(a) The angular width of half the zone must be located within 
2 to 3° of the LTS axis. 



/382 



402 



(b) The linear width of half the zone at a distance of 1350 
m from the control point (1150 m to the end of the runway) must 
be equal to 150 m. An expansion of the zone from the nominal value 
to 45 m and a narrowing to 30 m is considered permissible. 

The horizontal scale of the LSA (see Fig. 4.13) from the center 
to the scale stop has 6 divisions. The first division is the white 
circle on the silhouette of the aircraft, the second is the end of 
the vane, the third, fourth, and fifth are points on the horizontal 
axis of the scale, while the sixth is the scale stop. 

The vertical scale also has six divisions, of which the second 
division here is the first point on the vertical axis of the 
apparatus . 



Each division of the horizontal scale of the LSA corresponds to 
a deviation of the aircraft from the LSA axis (relative to the point 



where the course beacon is located) within limits of 20 
25 m (+7, -5 m) from the LTS axis at a distance of 1350 
control point . 



to 30' or 
m from the 



The angle width of the zone of the glide beacon is linked to 
the slope angle of the glide path, which is determined by the con- 
ditions of the formation of the zone. The width of the zone be- 
neath the glide path is then somewhat greater than above the glide 
path . 

The standards for regulating the glide zone are the following: 

(a) The position of the upper limit at an angle to the axis 
of the zone within the limits from 0.19 to 0.21 9, i.e., approxi- 
mately 1/5 of the slope angle for the glide path. 

(b) The location of the lower limit at an angle to the axis 
of the zone within limits from 0.29 to 0.31 6 (somewhat less than 
1/3 of the slope angle for the glide path). 

Accordingly, one division of the vertical scale of the LSA in 
the upper part is equal to about 0.030 9, while in the lower part 
it is about 0.05 9, where 9 is the slope angle of the glide path. 

Landing an Aircraft with the Course-Glide System 

Setting up the maneuver for an aircraft approaching an airport 
to descend with the use of the course-glide system is performed 
according to the same rules as in the simplified system for landing 
an aircraft . 

The complement of equipment for the course-glide system for 
landing an aircraft is usually supplemented by one or two master 
radio stations with marker beacons, located in the system for 
simplified landing, which is used for setting up the maneuver for 



/383 



403 



bringing the aircraft in for a landing and to a certain degree 
reserves the course-glide system for cases of malfunction of the 
ground or airborne equipment, as well as during times when equip- 
ment is being repaired or adjusted. 

If in addition to the course-glide and master beacons, the air- 
port is equipped with radar for observing the aircraft, the maneuver 
for landing in minimum weather can be made along the shortest path 
for each landing direction and takeoff direction. 

By the same rules which govern the simplified landing system, 
preliminary calculations are carried out which ensure a simpler and 
more exact action of the crew in flight along a given descent tra- 
j e ctory . 

A portion of the preliminary calculations, such as (for example) 
the determination of the moment for starting the descent in a glide, 
cannot be done in this case if we keep in mind the fact that the 
given glide path is defined in space. The calculations of the 
drift angle of the aircraft and the vertical rate of descent along 
the landing path are of somewhat less importance in this case. 

When the maneuver for making a landing is made on command 
from the ground, the need for such calculations as the determination 
of the moment for making the third turn no longer exists. However, 
the moment for beginning the fourth turn must in all cases be 
determined by the crew of the aircraft, with the maximum accuracy 
poss ible . 

In setting up the maneuver for landing, the strips of the LSA 
can be located on any divisions of the scale and no attention need 
be paid to their readings; however, when approaching the fourth 
turn, both strips must be located on the scale stops. The strip 
for the course zone rests on the stop on the side opposite the 
direction of the maneuver, the strip for the glide zone rests on 
the stop at the top. The emergency blinkers must then be off. 

The strip for the course zone must move away from the scale 
stop during the fourth turn. The movement of the strip away from 
the stop is called deftection. 

When the fourth turn is made correctly, deflection of the strip 
for the course zone occurs at the moment when the turn angle is held 
until the aircraft acquires the calculated landing course (Fig. 
i<-.20,a). For aircraft with piston engines, this turn aisle is about 
45°; for aircraft with turbojet or turboprop engines, it is about /38M- 
30°. 

With a residual turn angle of 45° for aircraft with piston 
engines (30° for aircraft with gas turbine engines), if deflection 
of the course-zone strip does not occur, it means that the fourth 
turn is being made with a lead. 



404 



In this case, it is desirable to significantly reduce the 
banking angle during the turn or even to stop turning and follow 
the LTS axis at the residual turn angle until the LSA strip deflects 




b) 



LRMS bRMS 




Fig. 4.20. Acquisition of the Landing Path by an Aircraft 
Proper Turn; (b) With Turn Begun Late. 



(a) With 



When the course-zone strip deflects, the turn must be continued 

until the landing course is acquired. When the landing course 

is acquired, the course-zone strip must be located near the zero 
marking (center of the scale). 

In cases when the fourth turn is made with a delay (Fig. 4.20,b), 
the deflection of the LSA strip takes place earlier than 45 or 30° 
before acquisition of the landing course. In this case, the turn 
must last until the landing course and beyond, at a landing angle 
opposite to the LTS axis, depending on the magnitude of the transi- 
tion of the course strip through the center of the scale. 



For example, if the descaling occurs at the very beginning 
of the fourth turn, it is necessary to increase the banking angle 
in the turn up to 20°, and the aircraft will continue to turn to 
the opposite angle for landing (20° in aircraft with piston engines 
and 30° for aircraft with gas turbine engines). 

With less delay in turning, the opposite angle for approach 
can be within the limits of 5 to 20°. 

With reverse deflection of the course-zone strip, the aircraft 
makes a reverse turn onto the landing course, with a simultaneous 
flat turn onto the LTS axis. After the aircraft has acquired the 



/385 



405 



LTS axis, the flight continues for a time until deflection of the 
glide-zone strip takes place at a constant altitude. 

At the moment when the glide-zone strip moves away from the 
upper stop, the aircraft shifts to a descent regime with a smooth 
acquisition of the desired glide path downward. 

Dvreationat PToperties of the Landing System Apparatus 

The selection of the desired course and the vertical rate of 
descent are sources of considerable difficulty for the crew and 
require a certain degree of training. However, these difficulties 
do not arise from principles of piloting the aircraft along the 
LSA, but rather from the necessity of simultaneously observing 
several devices and instruments and selecting a flight regime in 
the vertical and horizontal planes simultaneously. 

Nevertheless, with a proper reaction of the crew to a change 
in the positions of the strips on the LAS, the landing maneuver should 
be successful in all cases and not very difficult. 

In piloting the aircraft by the LSA, two of its principal char- 
acteristics must be employed: 

(1) The indicating characterisli c , i.e., the indication of the 
position of the aircraft relative to a given descent trajectory. 

(2) The command characteristic, i.e., the ability to predeter- 
mine the actions of the crew ii selecting the flight regime. 

Inasmuch as the first property of the LSA is obvious, let us 
examine the second. 

The course and glide zones are rather narrow in space, suffi- 
ciently so that the limits of these zones can be considered 
parallel over short segments of the trajectory. 

Let us say that an aircraft at a given moment is located to 
the side of the LTS axis, and the ground speed vector of the air- 
craft does not coincide with the direction of this axis (Fig. 4.21). 
Obviously, the ground speed vector of the aircraft can be divided 
into two components: a longitudinal one W^ and a lateral one W^, 

The longitudinal component f/^ is not involved in the selection 

of the course to be followed. The principal role is played by the 

lateral or transverse component, W . 

z 

The component W „ determines the rate of motion of an LSA strip /386 
along the horizontal scale of the apparatus. With the strip fixed 
at any scale division, the component W^ is equal to zero, which 
agrees precisely with the selected aircraft course, i.e., its path 
is practically parallel to the axis LTS. 



H06 



The regulation of the LSA is set so that the change in the 
course of the aircraft (1,5 to 2°) makes the motion of the vertical 



iVx 



V 






.-_____V 


^ /^ 



-^^ 



Fig. 4.21. Division of Ground Speed Vector into Longitudinal and 
Lateral Components along the Landing Path. 

strip LSA visible to the eye. For example, in aircraft with piston 
engines, a change in the course by 2° produces a lateral shift of 
the aircraft of 2 m/sec. This means that the most dangerous region 
of flight (1200 to 1500 m to the end of the runway), the LSA strip 
crosses each scale division in 11 to 12 sec, i.e., a sufficiently 
noticeable value equal to half a scale division after each 5 to 6 
sec of flight. If a turn is made in the direction of the motion 
of the LSA strip by 2°, its motion can be halted at any division 
on the s cale . 

On this basis, the principle of selecting the course for the 
aircraft by the LSA must be the following: 

If the vertical strip is located at a significant distance from 
the center of the instrument (on the third or fourth division), it 
is then necessary that the rate of its shift to the center of the 
instrument be significant. To do this, it is sufficient to turn 
the aircraft in the direction in which the strip is moving, by 4 
to 6° . 

As the strip approaches the center of the apparatus , its rate 
of motion must be arrested by turning the aircraft 1 to 2° in the 
direction shown by the arrow. At the moment when the strip reaches 
the center of the instrument, its motion is arrested by a final 
turn of the aircraft by 1 to 2°, and the aircraft will be set on 
the LTS axis, with the course already selected. 

This method requires a very precise flight of the aircraft 
along the axis of the course zone, with periodic changes in the 
course within the limits of 1 to 2°. 

An analogous method is employed to set the vertical rate of 
descent of the aircraft, with simultaneous acquisition of the 
desired glide plane and subsequent flight along it. 

Maintenance of the descent regime of the aircraft along a 
given trajectory by the readings of the LSA continues up to the 
moment when the aircraft emerges from the clouds and makes a 



4-07 



transition to visual flight, after which a 
tude is made and the aircraft touches down 



visual estimate of alti- /387 
on the runway . 



Direat-ionat Devioes for Landing Airaraft 

Determination of the rate of shift of the strips calls for in- 
creased vigilance in observing each of them. In addition, local 
irregularities in the course zone and glide zone at individual 
points disturb the regularity of the process; this must be taken into 
consideration by the crew and carefully separated from the generally 
established tendency. 

All of this requires considerable caution and training on the 
part of the crew for making a descent along a given trajectory. 

Recently, special directional devices for piloting an aircraft 
in the course and glide zones have begun to be employed widely. 

Unlike the LSA, the directional properties of these devices are 
not expressed by the derivatives of the positions of the strips on 
the instrument with time, but directly by the positions of these 
strips . 

The most widely employed directional devices at the present 
time are those which are based on various laws of control, with an 
indication which is linked to the banking of the aircraft during a 
coordinated stable turn, or to the angle of pitch at a set rate of 
des cent . 

In pilotage of the aircraft in the horizontal plane, these 
laws represent a definite link between the course and the banking 
of the aircraft in a turn, with lateral deviation from the radio- 
signal plane of the course zone and the first derivative of this 
deviation with time. 

K^^t + ATp H KzZ + KvVz = 0, 



where Ay is the angle of approach to the landing path, 3 is the 
banking angle of the aircraft in the turn, Z is the lateral devi- 
ation of the aircraft from the zone axis, V^ is the rate of lateral 
shift of the aircraft, and K are the coefficients for the corres- 
ponding parameters. 

A similar law is employed for piloting an aircraft in the 
vertical plane: 



where V is the angle of pitch of the aircraft, Y is the deviation 
of the aircraft from the glide path in the vertical plane, and V, 
is the rate of vertical motion of the aircraft. 



y 



408 



Since the linear values Z and Y and their first derivatives 
cannot be measured directly in polar systems, their values are re- 
placed by angle values (a and AG) and their derivatives. The 
values a and Ae and their derivatives a and A6 are measured by the 
differences in modulation depths and their derivatives in the zones 
of the course and glide beacons. 



/388 



Obviously, by selecting the proper banking angle and pitch 
angle for the aircraft, the aircraft can be positioned so that both 
strips on the directional indicator are located on zero. 

The coefficients for the converted position parameters for the 
aircraft axes are selected so that whatever deviations the aircraft 
may make from the given trajectory (if the indicator strips remain 
on zero), the aircraft will still travel along the given landing 
and glide path with a predetermined trajectory (whose course depends 
upon the coefficients selected). This means that the landing 
course and vertical speed must be selected simultaneously, since 
they are required for flying the aircraft along a given trajectory. 

Hence, instead of adjusting the rate of motion of the strips 
in accordance with their motion toward the center of the instrument 
as in a normal LSA, in directional instruments the crew need only 
bring the indicator strips to the center of the instrument by chang- 
ing the banking angle of the aircraft as well as its angle of pitch; 
this significantly facilitates the task of piloting an aircraft. 

To further reduce the work of the crew, directional instruments 
are usually combined with a gyrohorizon indicator. In this case, 
the entire attention of the pilot is concentrated practically on 
the readings of only one instrument. However, directional instru- 
ments based on the rules stated above have some important short- 
comings, which to a certain degree reduce the accuracy of piloting 
an aircraft relative to piloting by the indications of an LSA. 

The proper selection of coefficients for making a turn and 
the angle of pitch of the aircraft can be made only at a certain 
distance of the aircraft from the ground beacons. During measure- 
ment of the distance, the linear width of the course and glide zones 
changes, thus leading to a failure of the system regulation param- 
eters to agree with the dynamic flight trajectory of the aircraft. 
This shortcoming can be completely overcome if the system is regu- 
lated not only by the angular deviation of the aircraft from the 
radio-signal axis, but by calculation of the distance remaining to 
the ground radio beacons: 

r=£gtgAe. 



where L^ and L 
beacons . 



are the distances to the cours-e and glide radio 



409 



Control can then be effected in a rectangular system of ccor- / 389 
dinates, and therefore with constant agreement of the regulation 
of the system with the dynamic trajectory of the aircraft's flight. 

In polar coordinates, shortcomings in the operation of the 
directional system can be eliminated by a special selection of 
converted signal coefficients (not proportional to the values of 
the signals in various sections of the trajectory) in accordance 
with the tactical characteristics of aircraft of various types. 

It should also be mentioned that in directional systems, the 
indication of the position of the aircraft relative to a given 
descent trajectory is lost. This means that on board the aircraft, 
in addition to the directional devices, there must still be a 
conventional LSA indicator, which is used as a standard to check the 
accuracy of pilotage according to the directional indicator. 

So-called paravlsual directional instruments are also beginning 
to be used nowadays; in principle, they represent a reinforcement 
of the directional properties of the LSA. 

In this case, the usual LSA indicators are located in the cen- 
ter of the field of the pilot's vision, while at the periphery of 
his vision there are imitators of the motion of the strips according 
to the first derivatives a and A9, which link the indication shown 
with the longitudinal and lateral rolling of the aircraft according 
to the laws of the design of directional instruments. 

Radar Landing Systems 

From the tactical standpoint, radar landing systems have no 
special advantages over course-glide systems; on the contrary, 
their use is less convenient, since there are no instruments a- 
board the aircraft for indicating the position of the aircraft and 
no commands for piloting it relative to a given descent trajectory. 

The accuracy with which an aircraft can be landed by means of 
radar landing systems is roughly equal to that of landing it with 
course-glide systems. Nevertheless, radar landing systems are 
widely employed, along with course-glide systems. 

The primary reason why radar landing systems have been employed 
so widely is the need for a constant check on aircraft making their 
landing approaches by course-glide systems, for the purposes of 
pointing out errors made by the crew and preventing the very danger- 
ous consequences of error. 

The second reason is the need to give the crew assistance in 
landing the aircraft if they should request it, if for some reason 
the course-glide system cannot be used. The same reasoning applies / 390 
in retaining the course-glide system in case the ground control is 
not functioning. 



1+10 



I IIHIIIIIIIII I 



I I I 



The radar landing system consists of a complex of devices 
for observing the flight of approaching aircraft (radar screen, 
USW radio distance-finder) and those actually making a landing 
(landing radar). In addition, the system includes communication 
apparatus for transmitting information and necessary commands to 
the aircraft. 

The landing radar is the heart of the radar landing system, 
so we shall pause to examine the principles of its operation. 

Unlike ground radar installations with circular screens, the 
landing radars have a sector screen, i.e., there is no rotating 
directional characteristic of the antenna, but one which scans 
(oscillates) in a certain sector. Accordingly, the scanning line 
on the radar screen also oscillates. 

The landing radar has two antennas: 

(a) The course -s e ctor antenna, with a wide characteristic in 
the vertical plane and a narrow one in the horizontal. 

(b) The glide-sector antenna, with a wide characteristic in 
the horizontal plane and a narrow one in the vertical; the scanning 
of the characteristic of this antenna takes place in the vertical 
plane . 

The scanning of the directional characteristics of the landing- 
radar antenna can be achieved either by mechanical oscillation of 
the antenna reflector or by special devices which change the phase 
of the wave along the chord of the antenna reflector, thus causing 
the plane of the wave front to oscillate (so that all the wave- 
propagation characteristics also oscillate). 

The scanning sectors of the directional characteristics of the 
antenna are made narrow; 

(a) For a course sector of 15°: to either side of the LTS axis, 

(b) For a glide sector, 9° wide: +8° upward and -1° downward 
from the plane of the horizon. 

A peculiar feature of the landing radar is the special design 
of the scanning on the course and glide screens. Thus, instead of 
the circular distance marks on conventional circular radar screens, 
the distance marks on landing radars are straight lines, i.e. , the 
delay in the distance marks is made proportional not to R, but to 
i?/cosa, where a is the angle of deviation of the scanning line from 
the axis of the scanning sector. Hence, a rectangular system of 
coordinates is formed on the screen from the polar system of 
coordinates for the aircraft. 



In addition, the radar screen has a transverse scale three 
times larger than the distance scale for the course sector and 
five times larger than that for the glide sector. This means that 
there is a corresponding relationship between the increase in the 
scale indicating the position of the aircraft relative to the given 
trajectory for the same screen radius. 

itll 



A general view of the screens of the course and glide sectors 
is shown in Fig. 4.22. 



/391 





glide path sector 



course sector 




Fig. 4.22 



Fig. 4.23 



Fig. 4,22. Landing-Radar Screen: (a) Glide Sector; (b) Course 
Se ctor . 

Fig. 4.23. Pattern on Course Screen of Landing Radar. 

The landing radar is mounted on the traverse of the center of 

the LTS , at a distance of 100 to 150 m to the side, so that the 

conditions for using it when landing at either end of the runway 
will be the same. 

In the immediate vicinity of the landing radar, there is a 
circular-scan radar for observing aircraft near and far from the 
airport . 

In setting up the landing maneuver, immediately before completin; 
the fourth turn, the short-range radar approach system is used, also 
called control-tower radar (CTR). Its screen can be used to show 
landing maneuvers for aircraft approaching from all directions. 

All turns of the aircraft are made on command from the flight 
supervisor, as are the course corrections on the straight -line 
segments between the turns, if the given flight directions are not 
maintained sufficiently accurately. 

Observation of an aircraft with the landing radar begins 
while it is making the fourth turn, using only the course sector 
s creen . 

In order to ensure that the aircraft lands precisely on a 
given descent trajectory, the required pattern is superposed on the 
landing radar screen. This pattern on the screen serves three 
purposes : 

(l) To show the given trajectory for the aircraft's descent. 



412 



(2) To provide auxiliary lines for giving commands to the 
crew of the aircraft. 

(3) To show the boundary lines for safe flight altitude and /392 
the permissible zones for landing the aircraft. 

Since the landing radar is usually used for two directions of 
landing and takeoff, and can even be used for three or four is if 
other runways intersect, the patterns for the screens are printed 
on removable celluloid sheets which can be changed when shifting 
the landing radar to a new landing direction. 

The screen for the course sector of a landing radar (Fig. 4.23) 
usually shows the following: 

1. The given landing path (axis of LTS), beginning at the end 
of the runway and extending to the limit of the screen. The follow- 
ing points are marked on this line: the beginning of descent along 

a set glide path and the locations of the LRMS and SRMS landing 
systems within the range of the master radio stations. The SRMS 
is usually fitted with a comer reflector, which produces a bright 
spot on the screen and is used in setting the radar for the given 
landing direction and as a control to check the accuracy of the 
setting of the radar after it is turned around. 

2. The lines delimiting the zone of possible aircraft landings. 

These lines are defined on the basis of the assumption that the air- 
craft, being on a course close to that for landing, can be lined 
up with the LTS axis prior to the start of the landing distance 
only in the case when 

X>2;?sln UT, 



where 



yp = arccos ( 1 



i'-w)' 



where X is the remaining distance to the start of the landing dis- 
tance, Z is the lateral deviation from the landing path, and B is 
the turning radius with a banking angle of 10° . 

The order in which these lines are plotted is the following: 

(a) Several points of deviation of the aircraft from the 
landing path are given (e.g., 30, 100, 200, 500, 1000, 2000, and 
4000 m) and the required turn angles to correct these deviations 
are determined: 

cosUT=l--^; 

(b) The required course for lining up the aircraft with the 
LTS axis is determined: 

X=2/?sinUT. 



/393 



413 



To this path, we add the distance traveled by the aircraft 
(in 4 sec for piston-engine aircraft, 7 sec for gas turbine aircraft), 
required for receiving commands and carrying out the maneuver to 
line up the aircraft with the runway. 

(c) The path obtained forthe aircraft is measured from the start- 
ing point of the landing distance (as rule, from the SRMS), and we 
obtain the minimum attainable distances of the selected points for 
the lateral deviations of the aircraft. 

By connecting the points by a smooth curve, we obtain the limit 
of the possible landing zone of an aircraft, with permissible lateral 
deviations . 

In the course of landing an aircraft, if it shows up outside 
the indicated limits, the landing cannot be allowed and the command 
is given to make another pass at the field. 

The boundary lines are usually plotted for two typical glide 
speeds of aircraft : 

rith pis ton-engines 5 200 kro/hr; 

rith gas turbine engines, 2 80 km/hr. 



for those with 
for those 



The turn radius is calculated for a coordinated turn with a 
banking angle of 10°, with the lines for starting the turn plotted 
for making a landing at approach angles of 10 and 30°. 

If the aircraft has a significant deviation from the LTS axis 
after emerging from the fourth turn, we can in principle use any 
angle of approach to the LTS axis which makes it possible to line 
up the aircraft with the landing path before the landing distance 
is reached. 

However, as experience has shown, it is simplest to line up the 
aircraft with the landing path by using only two values for the 
approach angles: 10° if the deviation of the aircraft from the 
given line path is less than 500 m, and 30 m for deviations exceed- 
ing 500 m. Then the landing-radar screen can be bounded by a 
total of two auxiliary lines for beginning the turn onto the landing 
path . 



In this case, the distance from the 
line can be determined by the formula 

Z = /?(!_ cos UT^t 



LTS axis to the auxiliary 



However, experimental data show that there is an appreciable 
delay in the aircraft's acquiring the landing path, due to the 
time involved in transmitting commands and due to the reaction of 
the aircraft and crew in making the turn. Therefore, it is better 
to plot these lines on the basis of statistical data obtained from 
experience, as determined from a large number of aircraft landings. 



/394 



4-14 



According to these data, the turn to the landing course must 
begin : 

(a) For an approach angle of 10°, in aircraft with piston 
engines, 150 m from the LTS axis (5 mm on the screen scale); for 
aircraft with gas turbine engines, it is 250 m from the LTS axis 
(8 mm on the screen scale). 

(b) With an approach angle of 30°, these distances are 450 and 
750 m, respectively (15 and 25 mm on the screen scale). 

The markings on the glide screen of the landing radar are 
shown in Fig. M- . 2 U . 




Fig. 4.24. Pattern on Glide 
Screen of Landing Radar. 

limits of these boundary lines 



In this case, the descent 
trajectory for the glide path is 
set at the airport . Above this 
glide path are two boundary lines 
for landing the aircraft; for 
aircraft with gas turbine engines 
it is 4°, and for aircraft with 
piston engines it is 5°. 

If the blip representing an 
aircraft appears above the boun- 
dary line designated for a given 
type of aircraft, the landing of 
the aircraft will be complicated. 
Therefore, when controlling the 
landing of an aircraft, it should 
not be allowed to go beyond the 



Below the established glide path, there are boundary lines for 
permissible descent of the aircraft below the glide path, i.e., the 
lines limiting the flight altitude above the local terrain: 200 m 
prior to beginning descent in a glide, 150 m before passing over 
the LRMS, and 50 m before passing over the SRMS . In addition, there 
may also be flight altitudes for circling the field, set at 300, 400 
and 500m . 

These lines are used for aircraft coming in for a landing 
according to the CGS (course-glide system). 



In the case where the blip marking an aircraft intersects one 
of these lines, further descent of the aircraft is to be considered 
dangerous and the intervention of the flight supervisor operating 
the landing radar is required. 

Bringing an Aircraft In for a Landing with Landing Radar 



/395 



The method of bringing an aircraft in for a landing with a 
landing radar is very simple and quite effective at the present 
time . 



415 



The setting up of the landing maneuver and the calculations of 
the elements of the descent is made by the same rules as in using 
the simplified or course-glide landing systems. 

The moment for starting the fourth turn is determined on the 
basis of the blip representing the aircraft on the flight super- 
visor's screen. No commands are given to the crew during the 
fourth turn . 



After the aircraft 
landing path must be fol 
landing-radar screen is 
course of the aircraft i 
craft need merely be lin 
is at an angle to the LT 
craft is not equal to th 
determine the desired co 
the angle of the blip is 
course error. For examp 
correction must be 3°. 



emerges from the fourth turn, the calculated 
lowed for 10 to 15 sec. If the blip on the 
parallel to the LTS axis, the calculated 
s equal to the landing course and the air- 
ed up with the landing path. If the blip 
S axis, the calculated course of the air- 
e landing course, but it is very easy to 
urse correction by visual inspection, since 

equal to three times the angle of the 
le , with a blip angle of 10°, the course 



Having thus determined the required correction in the course 
to be followed, the supervisor gives a command to the crew, telling 
them to acquire the desired landing path at an angle of 10 or 30°, 
thus setting the course to be followed. 

At the moment when the blip crosses the corresponding auxiliary 
line, a command is given to turn the aircraft onto the landing course, 
considering the correction given. 

In the majority of cases, when these two comm.ands are given, 
it is sufficient to line up the aircraft with the landing path on 
the desired course. If a tendency is observed during flight along 
the landing path for the aircraft to shift laterally, it can be 
corrected by commands for small changes in the aircraft course 
(by 2 or 3°), with indication each time of the course which must be 
followed . 

When the blip approaches the point where the aircraft is to 
begin its descent in a glide, a command is given to descend at a 
calculated vertical speed. If it then develops that the aircraft 
is deviating from the given glide path (either upward or downward), 
the flight supervisor corrects the vertical speed, giving new 
values for it and ensuring that the aircraft travels exactly along 
the given path. 

An advantage of the radar landing system is the relative simpli- 
city of the supervisor's task in directing the airi^raft to a landing 
and the uncomplicated actions of the crew in carrying out the super- 
visor's commands, with no previous training required. These advan- 
tages are also reinforced by the fact that the flight supervisor, /396 
who constantly watches over several aircraft coming in for a landing 



1+16 



and gives them instructions, acquires a very great amount of exper- 
ience in the course of his work, a great deal more than that which 
the crew can acquire from the landings of their own aircraft alone. 
In addition, the supervisor, in the course of his work in guiding 
one aircraft after another to a safe landing, acquires a peculiar 
"feel" for estimating the navigational difficulties on a given day 
(selection of the required vertical speed and landing course on the 
basis of his experience with aircraft that have landed earlier). 

Therefore, in practice, the accuracy of landing an aircraft 
with a radar system is no worse than with a course-glide system. 
Nevertheless, the main shortcoming of the system (a lack of indi- 
cation for the crew as to the position of the aircraft on a given 
descent trajectory) creates a certain degree of inaccuracy in 
making the landing, and in this respect the radar landing system is 
inferior to the course-glide system. 



417 



CHAPTER FIVE 
AVIATION ASTRONOMY^ 



1. The Celestial Sphere 

The shy appears to the observer as an immense hemisphere. 

The aeZestiaZ s'pher'e is an imaginary sphere of arbitrary radius,^ 
whose center is the eye of the observer (Fig. 5.1). /397 

An observer on the Earth's surface can see only the half of the 
celestial sphere which is located above the horizon, since the 
other hemisphere is located below the horizon. 

If the Earth were transparent , an observer located at any point 
on its surface would see not one but two domes which together form 
the celestial sphere. 

Specfal Points, Planes, and Circles in the Celestial Sphere 

Zenith and nadir. If a line is plotted perpendicular to the 
location of the observer (through the center of the celestial sphere), 
it will intersect the imaginary limits of the celestial sphere at 
two points (see Fig. 5.1). The point which is located above the 
observer is the zenith (Z). The opposite point is the nadir (Z'). 

True horizon. If a plane is defined through the center of the 
celestial sphere and is perpendicular to the vertical line ZZ ' , we 7398 
can call it the plane of the horizon. 

The plane of the horizon inter- 
sects the celestial sphere along 
the circumference of a great circle 
(the points NESW) which is called 
the true horizon. 

World axis. The imaginary 

line PP ' , around which the apparent 

rotation of the celestial sphere 

takes place , is called the World 

axis. It passes through the point 

of the observer, located at the 

center of the celestial sphere, 

and intersects the arbitrary limits 

^ , ^ -, ^ • -, o T. of the celestial sphere at two 

Fig. 5.1. Celestial Sphere ^ 




^This chapter was written by M.I. Gurevich 



4-18 



diametrically opposed points PP ' . The world axis is inclined to 
the horizon at an angle which depends on the latitude of the ob- 
server . 





Fig. 5.2. Vertical and 
Almucantar . 



Fig. 
Hour 



5,3. Celestial Meridian, 
Circle, and Celestial Parallel, 



Celestial poles. The points where the Imaginary world axis 
intersects the arbitrary limit of the celestial sphere are called 
the ceZesti-al poles. Point P is called the superior (north) oetes- 
tiat pole, and the opposite point P' is called the -inferior (south) 
aetestiat poZe. Only the north celestial pole is visible in the 
Northern Hemisphere, and only the south celestial pole is visible 
in the Southern Hemisphere. 

Celestial equator. The plane which passes through the center 
of the celestial sphere and is perpendicular to the world axis is 
called the plane of the eetestial equator . The great circle QEQ'W, 
along which the plane of the celestial equator intersects the celes- 
tial sphere, is called the ceZest-iaZ equator. 

The celestial equator divides the celestial sphere into 
northern (QPQ') and southern (Q'P'Q) parts. 

The plane of the celestial equator is inclined to the plane 
of the true horizon at an angle which also depends on the latitude 
of the observer. 

Vertical . The great circle on the celestial sphere whose 
plane passes through the vertical line is called the vertical. 
Every vertical passes through the zenith Z and the nadir Z'. The 
plane of the vertical is perpendicular to the plane of the true 
horizon (Fig. 5.2). 

The vertical which passes through the east and west points (E 
and W, respectively) is called the primary vertioaZ . 



/399 



LH9 



The great circle ZMZ ' of the celestial sphere, which passes 
through the zenith of the observer and a certain star (Point M, 
Fig. 5.2), is called the vertical of that star. 

Almucantar. The small circle DMD ' on the celestial sphere, 
whose plane is parallel to the plane of the true horizon, is called 
the atmucantar . 

The almucantar which passes through a given star is called the 
atmucantar of that star. 

Hour circle. The great circle PMP ' of the celestial sphere, 
whose plane passes through the world axis , is called the circle of 
declination (Fig. 5.3). Since the world axis is perpendicular to 
the celestial equator, the plane of the hour circle is also per- 
pendicular to the equator. 

The hour circle which passes through a given star is the hour 
cirote of that star. 

Celestial meridian. The vertical PZP 'Z ' , which passes through 
the celestial poles, is called the celestial, meridian (since its 
plane coincides with the plane of the meridian of the observer). 
The celestial meridian divides the celestial sphere into the eastern 
and western hemispheres. 

The north point N and south point S. The celestial meridian 
crosses the true horizon at two points, called the north and south 
■points . 

Meridian line. The plane of the celestial meridian crosses 
the plane of the true horizon to form the meridian tine. Obviously, 
the ends of the meridian line coincide with the north and south 
points. (N and S, respectively). This line is called the "noon 
line" in Russian because the shadows of vertical objects fall along 
this line at noon. 

The east point E and west point W. If we plot a straight line 
in the plane of the horizon perpendicular to the meridian line (see 
Fig. 5.3) and face north, the east point E will lie on the right at 
the point where the plane intersects the circumference of the true 
horizon, while the west point will be located on the left. 

As the figure shows , the east and west points are 90° distant 
from the north and south points. The same figure also shows that 
the east and west points (E and W, respectively) mark the points 
of intersection of the celestial equator with the true horizon. 

Celestial parallel. The small circle on the celestial sphere, /i+OO 
whose plane is parallel to the plane of the celestial equator, is 
called the cetestiat parattet (similar to the terrestrial parallels). 



420 



Diurnal circle of a star. The small circle on the celestial 
sphere, drawn through a star parallel to the celestial equator, is 
called the diurnal oiraZe of the star. 

Astronomical coordinates. As we know, in order to determine 
the location of any point on the Earth's surface, it is sufficient 
to know the two angular coordinates of this point, the latitude 
and longitude . 

In astronomy, the location of stars on the sphere is accom- 
plished by means of two angular systems of celestial coordinates: 
the apparent system of coordinates and the equatorial system of 
coordinate s . 

In each of these systems, the position of a point (star) on 
the celestial sphere is determined by two celestial coordinates. 
Let us examine the systems of celestial coordinates individually. 

Systems of Coordinates 

A'p^arent System of Coordinates 

The main circles relative to which coordinates are determined 
in this system (Fig. 5.4) are the true horizon and the meridian of 
the observer. The coordinates themselves are called the altitude 
of the star (h) and the azimuth of the star (A). 



Altitude of a star. The angle between the plane of the true 
horizon and a line from the center of the sphere to the star (angle 
M'OM, Fig. 5.4) is called the altitude of the star. The altitude 
of a star can also be measured by the arc of the vertical from the 
true horizon to the location of the given star (M'M). 

The altitude of the star is measured from to 90° (positive 

values toward the zenith from the 
^ horizon, negative values from the 

horizon toward the nadir). 

Zenith distance. Instead of 
the star, we can also use the so- 
called zenith distance of the star 
as a coordinate , measured along 
the arc ZM . 

As we can see from Figure 5.4, 
the zenith distance is the arc from 
the zenith to the location of the 
given star. It is easy to set up 
a formula to express the relation- /401 
ship between the altitude and the 
System zenith distance of a star, since 

the two add up to 90°: h + Z = 90°, 




Fig. 5.4. Horizontal 
of Coordinates. 



421 



h = 90° - Z, Z = 90° - h. Obviously, the value of the zenith dis- 
tance will be somewhere between and 180°. 

Azimuth. The second coordinate in the apparent system of co- 
ordinates is the azimuth of the star. The azimuth of a star is the 
spherical angle between the plane of the meridian of the observer 
and the plane of the circle of the vertical of the given star. 

The azimuth is calculated differently in different areas of 
astronomy: from the south point or from the north point toward the 
east and west. In aviation astronomy, the azimuth is always calcu- 
lated from the north point along the horizon in an easterly direc- 
tion (clockwise) from to 360°. We can therefore define the 
azimuth in aviation astronomy as the angle measured along the arc 
NSM ' of the true horizon from the north point through the east (the 
east point) to the vertical of the star (see Fig. 5.4), from to 
360° . 

Hence, the first system of coordinates for celestial luminaries 
is called the apparent system. The coordinates of this system are 
the altitude of the star (h) and the azimuth of the star (A). 

The altitude and azimuth will suffice completely to determine 
the location of a star on the celestial sphere. For example, the 
star M, with h = 60° and A = 240°, is indicated on the sphere (see 
Fig. 5.4). 



Equatorial System of Coordinates 

The equatorial system of coordinates is the second system of 
coordinates which is used to determine the location of a star on 
the celestial sphere. The main circles relative to which calculations 
are made in this system are the celestial meridian and the celestial 
equator . 

The coordinates in this system are the declination of the star 
(5) and the hour angle of the star (t); see Figure 5.5. 

Declination of the star. The arc of the circle marking the 
distance from the equator to the location of the given star, or the 
angle between the plane of the equator end a line from the center 
of the sphere to the star, called the deelination of the star. 



± the sphere to the star, called the aeoLinav 

Declination is measured by the arc of a circle which marks the 
istance from the equator to the location of the given star, from 

to ± 90°. If the star is located in the Northern Hemisphere, its 
eclination is considered positive, while if it is in the Southern 
emisphere, it is considered negative. 



lit; taLdX xo _Ltj'»;ciLt;u x 

declination is considered positive, w 
Hemisphere, it is considered negative 

It is clear from Figure 5.5 that if the star is located on the 
equator, its declination will be equal to zero, while the declination 
of the north celestial pole is + 90° and that of the south celestial 



422 



pole is -90°. 

Polar distance. Occasionally, instead of the declination, the 
polar distance is used as a coordinate, measured along the arc PM . /^Q^ 
The polar dvstanoe is the arc of the circle which marks the dis- 
tance from the north celestial pole to the location of the star. 

The relationship between the declination and the polar dis- 
tance is expressed by the formula 



or 



6 + PAf = 90° 
PM = 90° — 8, 
B = 90° — PAf, 



i.e. , the declination and polar distance together add up to 90°. 
Therefore the point of the south pole has a polar distance equal to 
180° . 



Hour angle of a star. The arc of the celestial equator Q'M' 
(Fig. 5.5) between the south point of the equator and the hour 
circle of a given star is called the houT angle of a star (t). 

In aviation astronomy, the hour angle is measured from the 
south point of the equator along the equator in the easterly and 
westerly directions from to 180° 



Tl 



'he western hour angle is represented by the letter W, for 
example, t = 135° W; the eastern hour angle is represented by the 
' E, for example, t = 60° E. In making calculations, the 

calculated from to 360° 



letter 

western hour a 
If the we stern 



A^^T 




/ / 




/ / 




/ / 




/ ( 


'/ 


1 -^"^ \ 




1 ^ . 




[' 1 






K 



example, t = 60° E. In making calculations, the 
ngle must sometimes be calculated from to 360°. 
hour angle is found to be greater than 180°, it is 

related to 360°, but in this case 
the result is given as an eastern 
.,- — .^ hour angle. For example, t = 265° 



M 



Fig. 5,5. Equatorial System 
of Coordinates. 



the result is given as an 
hour angle. For example, t 
W or t = 360° - 265° = 95°E. 

Right ascension of a star. 

Instead of the hour angle , it is 
sometimes more convenient to use 
another coordinate, the right as- 
cension of the star (a). The right 
ascension of a star is the angle 
as measured along the equator from 
the point of the vernal equinox 
(y) to the hour circle of the given 
star (see Fig. 5.5). 

The point of the vernal equinox 
is the imaginary point of the 



423 



intersection of the ecliptic with the celestial equator, when the 
Sun passes from the Southern Hemisphere into the Northern Hemisphere, 
The opposite point on the ecliptic is called the point of the 
autumnal equinox {9.). 

In ancient Greece, the stars were used to reckon time. The /403 
constellation Aries was located at the point of the vernal equinox, 
and was represented by the symbol (y)- Due to the precession of the 
Earth, Aries has now moved away from the point of the vernal equinox. 
This point has remained unmarked, though its name has been retained, 
and its position in the sky is determined by using some other star 
which is a fixed distance from the point of the vernal equinox. 

Right ascension is calculated from the point of the vernal 
equinox along the equator up to the hour circle of a given star in 
a clockwise direction (as seen from the north celestial pole), from 
to 360°. 

Like the hour angle of a star, the right ascension of a star 
can be reckoned in either degrees or hours, minutes, and seconds. 
This is because both of these coordinates (especially the hour 
angle) are closely related to the measurement of time. 

Thus , the equatorial system of coordinates can be used to 
determine the location of a star on the celestial sphere. 

If we know the declination and the hour angle or the right 

ascension, we can determine the location of a star on the sphere. 

For example, the star M, with 6 = +50°, t = 45°, is shown on the 
sphere (see Fig. 5.5). 

Graphic Representation of tiie Celestial Sphere 

In solving textbook problems in aviation astronomy, it is often 

necessary to sketch the celestial sphere and plot the stars on it 

according to their coordinates. Let us use a concrete example to 
study the order in which the celestial sphere is sketched. 

Example . 1. The latitude of the observer is (fi = 60°N, the 
altitude of the star h = 70°, and its azimuth A = 240°. 

Draw the celestial sphere and plot the position of the star on 
it . (Fig. 5 .6 ,a) . 

Solution. (1) Use a compass to draw the celestial meridian 
in the form of a circle of arbitrary radius. 

(2) Draw a vertical diameter (perpendicular line) and mark 
the zenith and nadir (Z and Z', respectively) at the points where 
it crosses the circumference. 

(3) Perpendicular to the vertical line, through the center of 



4-24 



the sphere, draw a large circle which will be the true horizon of 
the observer . 

(4-) Draw the world axis such that the angle it forms with the 
plane of the horizon will be equal to the latitude of the observer, 
i.e. , <j) = 60°N; mark the points where the world axis crosses the 
circumference (the north celestial pole P and the south celestial 
pole P ' ) . 

(5) At the points where the true horizon intersects the merid- 
ian of the observer, mark the north point N (close to the north 
celestial pole) and the south point S (close to the south celestial 
pole ) . 

(6) Perpendicular to the point of intersection of the celestial 
equator with the true horizon, mark the east point E (on the right, 
as viewed by someone facing north) and the west point W (on the 
left) . 

This completes the sketching of the celestial sphere. We have 
yet to plot the position of the star on the sphere on the basis of 
its coordinate data, as follows: 

(1) From the north point N, plot the azimuth of the star /HOM- 
(equal to 24-0°) along the circumference of the horizon, judging 

the angle by eye . 

(2) Through this point M', draw the circle ( semicir cumf erence ) 
of the vertical. 

(3) Along the circle of the vertical, from the plane of the 
horizon, plot the altitude of the star, equal to 70°, judging the 
distance by eye. 




S N 




Fig. 5.6. Examples of Graphic Construction of the Celestial Sphere; 
(a) At a Latitude of 60°; (b) At a Latitude of 50°. 



425 



The result of this construction will be the celestial sphere 
as seen by an observer at 60°N and the position of a star on the 
sphere according to its apparent coordinates . 

Example . 2. The observer is located at a latitude of 50°. 
Sketch the celestial sphere for this observer and plot on it the 
position of a star with the following equatorial coordinates: hour 
angle t = 130°, declination 5 = +40°. 

Solution. (1) Sketch the celestial sphere in the same order 
outlined in Example 1. 

(2) From the south point on the equator Q', proceeding along 
the circumference of the equator in a westerly direction, plot an 
hour angle t = 130° by eye (Fig. 5.6,b). 

(3) Through this point (M'), draw the hour circle (PM'P'). 

From the plane of the equator, along the hour circle, measure 
off the declination 5 = +40° and mark the location of the star on 
the sphere (point M). 

The result of this construction is the hour circle for an ob- 
server located at a latitude of (j) = 50°N; the star has been plotted 
on the sphere on the basis of its equatorial coordinates. 



Diurnal Motion of the Stars 




7405 



The reason for this apparent motion of the stars (or of the 
sky) is the diurnal rotation of the Earth on its axis from west to 
east . 

In order to facilitate a study of the diurnal rotation of the 
stars, we will assume for the sake of discussion that the Earth is 
fixed and the celestial sphere rotates on the world axis at the 
same rate that the Earth actually rotates on its axis, but in the 
opposite direction, from east to west (in other words, the way it 
actually looks to us). Since the entire celestial sphere rotates 
on the world axis, all the points (stars) located on the sphere 

426 



,J 



will turn along with it, i.e., it is clear that each star describes 
a sort of circle around the world axis. 

Diurnal parallel of a star. All of the stars rotate together 
with the celestial sphere around the world axis. From this it is 
clear that every star, fixed permanently in the sky, describes a 
circle of some size in the course of 2^■ hours. 

The circle described by a star in 24 hours in the course of 
its movement around the world axis is called the diuvnat ciTote of 
the star. This circle is also called the aetestiaZ paraZtet . 

Since the entire celestial sphere rotates around the world 
axis, it is easy to see (and important to remember) that the di- 
urnal rotation of the heavenly bodies takes place parallel to the 
celestial equator, i.e., the diurnal parallel of the star (the path 
of the star around the world axis in 24 hours) is always located 
parallel to the celestial equator. 

The magnitude of the diurnal parallel of the star depends on 
the location of the star in the sky. Obviously, stars which are 
located closer to the celestial poles (and have higher declination 
values) have a small diurnal circle. The closer a star is located 
relative to the celestial equator (the smaller its declination), 
the larger its diurnal circle will be. The largest diurnal circle 
belongs to those stars which are located on the celestial equator, 
and whose declination is zero. 

Motion of the Stars at Different Latitudes 

If we observe the diurnal motion of the stars at different 
latitudes, we will see that the sky and stars turn relative to the 
observer's horizon at different angles. This phenomenon becomes 
understandable if we recall the location of the world axis relative 
to the horizon at different latitudes. 

The world axis is located relative to the horizon at an angle /406 
which is equal to the latitude of the location. From this it fol- 
lows that the higher the latitude of a- location, the closer the 
celestial poles PP ' will be located to the zenith Z and the nadir 
Z', and the smaller the angle will be between the true horizon and 
the celestial equator. Conversely, the lower the latitude of the 
location, the further the celestial poles will be from the zenith 
and nadir, and the angle between the true horizon and the celestial 
equator will be larger. 

Figure 5. 7, a shows the angle between the true horizon and the 
celestial equator for an observer located at a middle latitude, e.g. 
50° (angle 90° - <j) = 40°). Figure 5.7,b shows the angle between 
the true horizon and the celestial equator for an observer located 
on the Equator (angle 90° - (|) = 90°), while Figure 5.7,c shows the 
angle between the true horizon and the celestial equator for an 

427 



observer located at the North or South Pole. (The angle 90° - cf) = 
0, the true horizon is parallel to the celestial equator, the zenith 
point Z coincides with the north celestial pole P, and the nadir /407 
Z' coincides with the south celestial pole P'). 

It is clear in all three figures that the angle between the 
true horizon of the observer and the celestial equator is always 
equal to 90° minus the local latitude (90 - (|) ) . 

We can draw the following conclusion from the above: the slope 
of the diurnal parallel of stars relative to the true horizon of 
the observer depends on the latitude of the observer. The higher 
the latitude of the observer, the smaller the slope of the diurnal 
parallels of the stars relative to the horizon; the lower the 
latitude, the greater the slope. 

Rising and Setting, Never-Rising and Neve r- Set t i ng Stars 

If we know that the position of the celestial equator (and 
consequently the diurnal parallels of the stars) relative to the 
true horizon of the observer depends on the latitude of the observer, 
it will be clear why some stars at a certain latitude rise and set 




b) 


eU) 




/ 




u 






K 


/ 







o 








/ 




— 1 


ol 








/ 




o 


TV 








fp\u 




o 


(1) 








VI" 








horizon] 


1 




cS 










\ 




c 










\ 




u 










\ 




p 














• iH 










>^ 


vj 


-o 








y 



Fig. 5.7. Angles Between the True 
Horizon and the Celestial Equator; 
(a) At a Latitude of 50°; (b) On 
the Equator; (c) At the Poles. 



ipir 



428 



at the horizon, others never set, and still others never rise. 

A star never sets if its declination is greater than 90° minus 
the latitude of the location, i.e., if 6 > 90° - A. 



For example, see Figure 5. 8, a. Given the latitude of the ob- 
server <l> = 60° , the declination of the star 6 - +45° . From this it 
is clear that 90° - (() = 90 - 60° - 30°. Since the declination 6 
= +45°, i.e., greater than 90° - <j) , it is clear that the star can- 
not set below the horizon of the observer. In Figure 5. 8, a we have 
sketched the celestial sphere for an observer located at a latitude 
of 60°. We mark off the declination of the star along the meridian 
of the observer (i.e., the hour circle) so that 6 = +45°, and then 
lay out the diurnal circle (diurnal parallel) of the star parallel 



/^OS 




S /V 




Fig. 5.8. Examples of Never- Set ting Stars; (a) The Star Never Sets 
Below the Horizon; (b) The Star Touches the Horizon. 



aj 








ink* 


[ tf , 


^^ 


<^\ 




^x^/ 




/" /'^v 








/ / / 




.-^ / ^\. 


p/ / / 


/ \ / 


f^y / \ 


„ \Mi// 


lux 


//A 


\.^§^S^ ' 


n 


"\,^ / 


^^--/_ 




S^A^><'/ 


^Y / 


/ 


/^j/f' 


\i/ 




y 


T< 




__,----''^^ 



5 N 



2' 



3<^^ 


"^v^^ 




--'""3^^' 


'^ yf=sn\ 


/^-jiT/ 4 


< / ^ 


)>^^C^ 


VJ^^^^^""^ ^ 


-''V^ y^ j 




^^^\/ 




—-"^ 



^-"^S 



r 



Fig. 5.9. Examples of Stars that Set; (a) Tne Star Rises and Sets; 
(b) The Star does not Rise. 



429 



to the celestial equator. As we can see from the figure, this 
circle is located above the horizon of the observer, and so a star 
which moves along this circle in the course of 2h hours will never 
set below the observer's horizon. 

The star touches the horizon, but does not go below it, in 
the case when its declination is equal to 90° minus the latitude 
of the observer, i.e., if 6 = 90- cf). 

Take Figure 5.8,b for example. The latitude of the observer 

(j) = 60°, the declination of the star 6 = +30°. From this it is 

clear that 90° - (f) = 90 - 60° = 30°. In accordance with what we 

have said, if 6 = 90° - (f , the star will touch the observer's 
horizon but will not set below it. 



In Figure 5.8,b we have sketched the celestial sphere for an 
observer located at a latitude of 60° . Along the meridian of the 
observer (i.e., the hour circle), we have plotted the declination 
of a star 6 = +30°, and have then drawn the diurnal parallel of 
this star parallel to the celestial equator. As we can see, the 
diurnal parallel of the star touches the observer's horizon, but 
does not cross it, i.e., a star moving along its diurnal parallel 
in the course of 2^- hours goes down to the horizon and then rises 
again in the course of its diurnal journey. 

A star rises and sets when its declination (in terms of abso- 
lute value) is less than 90° minus the latitude of the location, 
i.e., if 6 < 90° - (j) . 

Let us consider the following e 
observer is tj) = 30° , the declination 
Figure 5. 9, a we have sketched the ce 
located at a latitude of 30°. Along 
we have marked off the declination o 
drawn the diurnal parallel of the st 
q'). As we can see from the diagram 
course of 24 hours along its diurnal 
the horizon for a certain time (the 

parallel ) 

the rest 




xample: The latitude of the 

of the star is 6 = +4-0°. In 
lestial sphere for an observer 

the meridian (hour circle) 
f the star 6 = +4-0° and have 
ar parallel to the equator (q - 
, a star which moves in the 

parallel will be located below 
shaded part of the diurnal 
, and will be above the horizon 
of the time . 



A star never rises if its declina- 
tion is equal to or greater than 90° 
minus the latitude of the observer and 
has a sign which differs with latitude 
(the latitude Is positive and the 



Fig. 5.10 Division of the Celestial 
Sphere into Regions with Rising and 
Setting, Never-Setting and Never-Risinj 
Stars . 



/409 



430 



declination is negative, or vice versa), i.e., if - 6 >_ 90° 
or 6 = ()) - 90°. For example, the latitude of the observer i 
the declination of the star 6 = -30°. 



6 0°N, 



In Figure 5.9,b we have sketched the celestial sphere for an 
observer at a latitude of (j) = 60°N. Along the meridian of the 
observer (hour circle) we have marked the declination of the star 
6 = -30° (below the equator) and the diurnal parallel of the star 
parallel to the equator. As we can see from the diagram, a star 
which moves along its diurnal parallel will always be below the 
hori-zon and will never rise. This is completely understandable, 
since the declination of the star is negative. If the star had a 
negative declination still greater than 90 - (j) , its diurnal circle 
would be located still further below the horizon. 

Consequently, the entire celestial sphere of an observer lo- 
cated at a given latitude can be divided into three parts: 



(1) The portion of the celestial sphere with stars that never 



set . 



(2) The portion of the celestial sphere with setting and 
ris ing stars . 



(3) The portion of the celestial sphere with stars that never 



rise 



All three portions of the celestial sphere are shown in Figure 
5,10- for an observer at a latitude of 60°N. The circumference is 
the plane of the celestial meridian, ZZ ' is the vertical line of 
the observer, PP ' is the world axis. The straight line BB ' is the 
section of the plane of the celestial meridian as cut by the diurnal 
circle of the star, touching the horizon of the observer but not /'+10 
passing below it (a star whose 6 = 90° - (|) ) . This is the boundary 
of the region of stars that never set with that of the ones which 
rise and set for a given latitude of the observer. The straight 
line DD ' is the section of the plane of the celestial meridian as 
cut by the hour circle of the star in the Southern Hemisphere, 
touching the horizon but not going below it (a star whose declina- 
tion is - 6 = 90° - (j) ) . This is the boundary of the region of stars 
that never rise with that of the ones which rise and set for a 
given latitude of the observer. 



Motion of Stars at the Terrestrial Poles 

In order to get a better idea of the nature of the diurnal 
motion of stars at the terrestrial poles , let us construct a form 
of celestial sphere for an observer located at the North Pole. In 
this case, the altitude of the Pole above the horizon will be equal 
to the latitude of the observer. Since the observer is located at 
the North Pole, (j) = 90°N and consequently the altitude of the Pole 
above the horizon will be 90° (Fig. 5.11). 



t|31 



nort 

cele 

the 

This 

will 

chan 

Nort 

the 

Hemi 

hori 

with 



The 
h eel 
stial 
celes 

mean 

move 
ge as 
hern 
hor iz 
spher 
zon 5 

& < 



world 
estial 

pole 
t ial e 
s that 
paral 
the c 
Hemisp 
on , wh 
e ( and 
i.e . , 
0° wil 



axis 
pole 

P' CO 

quato 

all 
lei t 
elest 
here 
ile t 
have 
all s 
1 nev 



comci 
P coi 
incide 
r will 
stars , 
o the 
ial sp 
(havin 
he sta 
negat 
tars w 
er r is 



des wi 
ncides 
s with 
coinc 
depen 
horizo 
here r 
g posi 
rs whi 
ive de 
ith 6 
e . 



th t 
wit 
the 
ids 
ding 
n an 
otat 
tive 
ch a 
clin 
> 0° 



he vert 
h the z 

nadir 
with th 

on the 
d their 
es . Al 

declin 
re loca 
ations ) 

will n 



ical line , i 
enith Z and 
Z' , while th 
e plane of t 
ir diurnal r 

altitudes w 
1 stars loca 
ations ) will 
ted in the S 

will move b 
ever set and 



. e . , the 
the south 
e plane of 
he horizon . 
otation , 
ill not 
ted in the 

move above 
outhern 
elow the 

all those 



Motion of Stars at Middle Latitudes 

Let us examine the nature of the diurnal motion of the stars 
at middle latitudes, when < (f < 90°. 




(P'W 



Figure 
pearance of 
at a latitud 
to the incli 
axis , all st 
to the horiz 
celestial eq 
latitudes , a 
of stars ris 
the parallel 
which are lo 
these parall 
equator eith 
never set . 



5.12 shows the ap- 
the celestial sphere 
e close to 45°. Due 
nation of the world 
ars move at an angle 
on (parallel to the 
uator). At middle 

considerable number 
e and set (between /^H 
s NK and DS) , Stars 
cated farther than 
els from the celestial 
er never rise or 



Fig. 5.11. Motion of the 
Stars at the Terrestial Poles 

region of 
never setting z 

stars 

p 




-0 Z 



region „f 

never risine 
stars 




Fig. 5.12. Motion of the Stars 
at Middle Latitudes . 



Fig. 5.13. Motion of the 
Stars at the Equator. 



432 



Motion of Stars at the Equator 



Since the latitude of an observer located on the Equator is 
equal to zero, it is clear that the world axis lies in the plane of 
the horizon and coincides with the meridian line on the plane of 
the horizon, while the terrestrial poles PP ' coincide with the 
north and south points N and S, respectively. 




Culmination of Stars 



The diurnal parallel of a star crosses the celestial meridian 
at two points (Fig. 5. 14, a). These points are called the culmination 
points. The moment of passage of a given star through the celestial 
meridian is called the moment of culmination, or it is said that 
the star culminates. 

The upper auZmination of a star is the moment when the star is 
at its greatest altitude above the horizon. The lower culmination 
of a star is the moment when the star is at its lowest altitude 
above the horizon. In the case of stars that set, the lower cul- I ^'L2 
mination takes place below the horizon. 

Upper culmination of a star can take place on the southern 
portion of a meridian (between the south point and the zenith), and 
on the southern portion of the meridian (between the zenith and the 
north celestial pole), depending on the relationship between the 



a) 


^^— 


lower 


^ 


culmination'^ 


\. 


of the star 





^, upper 
r»,cuimination 
of the star 



'max' 




Fig. 5.14. Culmination of Stars on the Southern Section of the 
Meridian: (a) in the Apparent System of Coordinates; (b) in the 

Equatorial System. 



433 



latitude of the observer and the declination of the star. 

A star culminates on the southern part of the meridian (between 
the south point and the zenith) when the latitude of the observer 
is greater than the declination of the star, i.e., if ij) > 6 . 

A atar culminates on the northern part of th^ meridian (between 
the zenith and the north celestial pole) when the latitude of the 
observer is less than the declination of the star, i.e., if ^ < 6 . 

In Figure 5.14,b the celestial sphere has been sketched in 
simplified fashion, i.e., the circles of the horizon, equator, and 
parallels are not represented as circles but as diameters and 
chords. As we can see from the diagram, the latitude of the ob- 

server is greater than the declination of the star: NP > QM , i.e., 
(f) > 6 , so that the upper culmination of the star (point M') lies 
on the southern part of the meridian (between the zenith Z and the 
south point S ) . 

Let us determine the altitude of the star in this case . 



The altitude of the star (h) is the arc SM ' , but the arc SM ' 



= SQ ' 



+ Q 'Z 



M'Z, and SQ ' = 90° 



; Q'Z 



and M'Z 



By substituting these values, we will obtain h = 90° - (j) + (j; 
- ((f) - 6) or h = 90° - (|) + 5 . 

In Figure 5. 15, a the celestial sphere has also been sketched 
in a simplified form. Here the latitude of the observer ( NP ) is 
less than the declination of the star (Q'M'), i.e., <j) < 6 , so that 
the upper culmination of the star (point M' occurs on the northern 
part of the meridian (between the zenith and the point of the north 
celestial pole). Let us determine the altitude of the star in this 



/413 





Fig. 5.15. Culmination of a Star on the Northern Section of the 
Meridian: (a) Coordinates of Upper Culmination; (b) Coordinates 

of Lower Culmination. 



431+ 



case. The altitude of the star (h) is NM ' , but 



or 



I.e., 



NM' = 180° — M'Q' — Q'S 



^=180° — S — (90 — 9), 



If the star does not set, we will sometimes be interested in 
its altitude at the moment of lower culmination. 

As we see from Figure 5.15,b, MQ = QN + NM , but MQ = 6 ; QN = 
9 - (j) ; NM = h . 

By substituting the values of these arcs, we will obtain 6 = 
90° - (|) + h, so that h = (() + 5 - 90°, i.e., the altitude of the 
star at the moment of lower culmination is equal to the latitude 
of the observer plus the declination of the star minus 90°. 

Problems and Exercises 

1. What must be the declination of a star if at the latitude 
of Moscow ( ({) = 55°48') (a) it never sets, (b) it rises and sets, 
or (c) it never rises? 

Solution. (a) In order for a star never to set, we must have 
6 > 90 - (|). If we substitute the value of the latitude of Moscow 
(55°48'), we will obtain 6 > 90 - 55°48', i.e., 6 must be greater 
than +34°12'. Consequently, all stars which have a declination 
greater than +34°12' will never set at the latitude of Moscow. 
Typical stars in this category are Capella, Alioth, Vega, Deneb , 
and Polaris. /^l^ 

(a) Stars rise and set, as we know, if the absolute value of 
their declination is less than 90° - (|> , i.e., 6 < 90° - ()> i . In 
our example, 6 < 34°12'. Stars in this category for the latitude 
of Moscow are Regulus , Arcturus , Altair, etc. 

(c) In order for a star never to rise, its declination must 
be equal to or greater than 90° - (fi and varies with the latitude of 
the observer, i.e., -6 >_ 90° - <|) . In our example, 6 must be equal 
to or greater than 3M-°12'. In addition, it must also be negative 
(inasmuch as we are talking about north latitude). 

(2) Calculate which of the following stars: Aldebaran, 
Alpherants, Capella, Sirius , Procyon, Arcturus) will never rise, 
rise and set, and never set at the latitude of Leningrad ( (j) = 59°59'N). 



435 



3. Calculate the altitude of the star Dubkhe at Moscow ( (j) = 
55°U8') at the moment of upper culmination. 

4. At what altitude does the star Sirius culminate (upper 
culmination) at Leningrad? 

5. Show mathematically that all stars whose 6 > do not set 
at the Poles, while those which have 6 < never rise. 

3. The Motion of the Sun 

The Annual Motion of the Sun 

The Sun participates in the diurnal motion along with all the 

other stars . The apparent diurnal motion of the Sun is also the 

result of the diurnal motion of the Earth in rotating on its axis. 

However, the Sun also has its own so-called intrinsic motion in 
the course of a year, called the annual motion of the Sun. 

The annual motion of the Sun is difficult to observe directly. 
However, if the stars were visible in the daytime, and we were to 
observe the mutual positions of the Sun and stars for a certain 
period of time, we would see that the mutual positions of these 
bodies would change in the course of time, while the mutual positions 
of the stars and constellations in the sky would not change. 

The direction of this intrinsic annual motion of the Sun is 
opposite to the diurnal motion of the stars, i.e., from west to east. 

The annual motion of the Sun is apparent (as is the diurnal 
motion), and occurs as the result of the annual rotation of the 
Earth around the Sun. 

As we did in describing the diurnal motion of the sky and 
stars, we will consider that the Sun is moving and the Earth stands 
still. 

Due to the existence of so-called annual motion of the Sun, the 
diurnal motion of the Sun has some unusual aspects, such as: 




/415 



(c) The meridional altitude of the Sun changes constantly in 
the course of a year. 



436 



Ecliptic. In the course of its intrinsic motion, the center of 
the Sun moves along a great circle of a sphere called the ecZ-iptic 
(Fig, 5. 16, a). The plane of the ecliptic intersects the plane of 
the celestial equator at an angle of 23°27' at two points: at the 
point of the vernal equinox (y) and the point of the autumnal 
equinox (.0) . 

Tropic year. The Sun completes a journey around the ecliptic 
(through 360°) in 365.2422 mean days. 



The interval of time between successive passages of the center 
of the Sun through the point of the vernal equinox is called the 
tropic year. 

Sidereal year. In the course of its annual motion, the Sun 
makes a full rotation relative to the stars in a period of time 
somewhat longer than the tropic year (i.e., in 365.25635 days). 
This time interval, equal to the period of time required for the 
Earth to rotate around the Sun, is called the si-dereat year. After 
this interval, the Sun will have returned to its original position 
among the stars. 

Motion of the Sun Along the Ecti.pt'ia 



On 
cr OS se s 
This dat 
passes t 
equal to 
passes i 
increase 
June 22) 
called t 
is at it 



March 21, in the course of its annual motion, the Sun 
the celestial equator at the point of the vernal equinox, 
e is called the date of the vernal equinox. When the Sun 
hrough the point y, its declination and right ascension are 

zero. After March 21, the Sun continues its motion and 
nto the Northern Hemisphere, and its declination begins to 
(i.e., becomes positive). Thus, after three months (on 
the Sun is at the point K (see Fig. 5. 16, a), which is 
he point of the summer solstice . At this point, the Sun 
s highest position above the celestial equator. The 



OL'ISO' 




June 22 



Mar 21 



Fig. 5.16. Annual Motion of the Sun: (a) Motion along the Ecliptic; 
(b) Coordinates on the Dates of the Equinoxes and Solstices. 



437 



declination of the Sun at this 
ascension is 90° or 6 hours. F 
noon remains nearly constant , i 
which corresponds to the conste 
the date of the summer soZstiae 
and sets on this date will be a 
east and west points on the hor 
solstice , the Sun begins to app 
declination begins to decrease, 
sects the celestial equator at 
in the constellation Libra). 



point is +23027', and its right 
or several days , its altitude at 
.e., +23°27', so that the point K, 
nation Capricorn, has been named 
The points where the Sun rises 
t their maximum distances from the 
izon. After the date of the summer 
roach the celestial equator, its 

and by September 23 it again inter- 
the point of the vernal equinox (fi^. 



When the Sun passes through the point of the vernal equinox 
(fi_) , its declination becomes equal to zero, while its right ascension 
becomes 180° or 12 hours. 

TABLE 5.1 



Date 



Vernal equinox 
Summer solstice 
Autumnal equinox 
Winter solstice 



Occurs on 



March 21 
June 22 
September 23 
December 22 



c 


oord 


Lnate 


s 


Declination 


(6) 


Right Ascension ( a ) 


0° 






0° 


+ 23°27 ' 




90° 


or 6 hours 


0° 




180° 


or 12 hours 


-23°27 ' 




270° 


or 18 hours 



September 23 is called the date of the autumnal equinox. All 
of the events of the date of the vernal equinox are repeated on 
this date . 

After September 23, the Sun passes into the Southern Hemisphere 
and its declination becomes negative. On December 22, the Sun is 
at its lowest position relative to the celestial equator and is at /'^■17 
the point of the winter solstice (the point L, in the constellation 
Leo). This date is called the date of the winter solstice . On the 
date of the winter solstice, the Sun has a declination of -23°27', 
while its right ascension is 270° or 18 hours. The points where 
the Sun rises and sets on this date are farthest south from the 
east and west points on the horizon. 

After December 22, the Sun begins its rise along the ecliptic, 
and on March 21 it has again risen to the point of the vernal 
equinox, where its declination and right ascension are once more 
equal to zero. 

Thus, we can draw up a special table for the annual motion of 
the Sun along the ecliptic, showing its coordinates (Table 5.1; 
Fig. 5 .16 ,b) . 



438 





In the course o 


fay 


ear, as it moves through 


the sky ( 


among 


the 


stars ) , the Sun 


passes through 12 constellations, calle 


d the 


signs of the zodiac. 


The 


y have received this name 


because 


the 


majority of them bear the 


names of animals (Aries, 


the Ram; 


Taurus , 


the 


Bull, etc . ) , and 


the 


word zodn in Greek means 


"animal" . 


As the 


Sun 


moves among the 


stars 


in the course of a year. 


it is in 


the 


foil 


owing positions: 


on 


the date of the vernal eq 


uinox (March 21) , 


in the constellation 


Pisces (the Fishes); on the date of th 


e summer 


solstice June 22, in 


the 


constellation Gemini (the 


Twins ) ; 


on the 


date 


of the autumnal 


equinox (September 23), in th 


e constellation 


Virg 


o (the Virgin ) , 


and on the date of the winter 


solstice 


( December 


22) , 


in the constell 


ation 


Sagittarius (the Archer) 


• 





Diurnal Motion of the Sun 



The Motion of the Sun at the North Vole 




During the other half of the year, when the Sun has a negative 
declination, it will be below the horizon as seen from the North 
Pole. Therefore, there are six months of day and six months of 
night at the terrestrial poles. 



North Pole on the date of the summer s 
The altitude of the Sun on that date i 
ation, i.e., 23°27'. At the South Pol 
mum altitude on the date of our winter 



urn altitude above the horizon at the 

summer solstice, i.e. , on June 22. 

t date is equal to its maximum declin- 



The Sun reaches its maxim 

on the date of the summer s 

' ■" ' ' " te is equal to its maximum declm 
Pole, the Sun reaches its maxi- 
3lstice, i.e. , December 22. 




Heavenly bodies do not set if their declination is equal to 
or more than 90° minus the latitude of the observer, i.e. if 6 >_ 
90° - (j) . This situation also applies to the diurnal motion of the 
Sun. If, for example, the observer is standing at a latitude of 
76°N (between the North Pole and the Arctic Circle), then according 



439 



to the condition set forth above the Sun will not set after the 
date when its declination is equal to or more than 90° - (f> , i.e. 
more than 90-76° = +x4° . This phenomenon (to cite a specific 
example) begins on April 26. After April 26 the Sun will rise 
higher and higher above the horizon. 

The Sun reaches a maximum altitude above the horizon on the 
date of the summer solstice. After June 22, the Sun will dip 
toward the horizon but will still not set. When its declination 
is again equal to 90° - (|) , the Sun will touch the horizon. 

After August 19, the Sun's declination will be less than 90° 
- (j) , i.e. 6 < 90° - (j) and for a fixed time it will appear to an 
observer located at this latitude as a rising and setting star. 

This phenomenon will continue until the declination of the 
Sun is not equal to or more than 90° - (j) , i.e. 6 >_ 90° - cj) and will 
have a sign opposite to that of the latitude (i.e. the latitude 
is positive and the declination is negative). For an observer 
located at 76°N this phenomenon begins on November 3. Beginning on 
November 3, for an observer located at a latitude of 76°, the Sun 
will not set, since -6 > 90° 




TABLE 5.2 



/419 



















ter 


Latitude of - 
the position 


Sp 


ring 


Summer 


Autumn 


Win 


Beg. 


Duraticn 


Beg. 


Duration 


Beg . j Duration 


Beg. 


Duration 


in degrees 


date 


in days 


(fete 


in days 


date 


in days 


date 1 in days 


68 


4.1 


143 


27.V 


53 


19.VI1 


144 


lO.XII B5 


70 


17.1 


120 


17.V 


72 


28.VI1 


121 


26.XI 52 


72 


26.1 


103 


9.V 


88 


5.VIII 


104 


17.X1 


70 


74 


3.11 


88 


2.V 


102 


12.V1I1 


90 


lO.XI 


85 


76 


9.II 


76 


26.1V 


115 


19.V11I 


76 


3.X1 


98 


78 


15.11 


64 


20. IV 


127 


25.V111 


63 


28.X 


111 


80 


22.11 


51 


14.IV 


139 


31. VIII 


52 


22.x 


123 


82 


27.11 


41 


9.1V 


150 


6.1X 


41 


17.X 


133 


84 


4.111 


31 


4.1V 


159 


lO.lX 


31 


ll.X 


r44 


86 


9.1II 


2! 


30.111 


169 


15.1X 


21 


5.x 


155 


88 


14.111 


11 


25.111 


179 


20.IX 


10 


30.1X 165 


90 


19.111 





19.111 


189 


25.1X J 





25. IX 


176 



^The dates for nonsetting, nonrising or rising and setting of the 
Sun are given, taking into account the phenomenon of refraction. 



M-40 



Thus , let us sum up the diurnal motion of the Sun in the course 
of a year for an observer located at a latitude of 76° (between the 
North Pole and the Arctic Circle). 

(1) From April 26 to August 19, i.e., for 115 days, the Sun 
does not set. This period of time is called the poZ-ar summer. 

(2) From August 19 to November 3, i.e. , for 76 days, the Sun 
will rise and set daily and the period of daylight will diminish 
each day. This period of time is called the potar autumn. 

(3) From November 3 to February 9, i.e., for 98 days, the 
Sun will not rise for the observer. This period is called the 
polar winter. 

( M- ) From February 9 to April 26 i.e. , for 76 days the Sun will 
rise and set daily and the period of daylight will increase each 
day. This period of time is called the polar spring. 

The dates for the start of the seasons depend on the latitude 
of the observer. The dates given in our example are for 76°N. We 
have provided a table for the seasons as a function of the latitude 
of the observer (Table 5.2). 

Motion of the Sun above the Aratie Circle 

As we already know, the latitude of the Arctic Circle is equal 
to <|) = 66°33', or the complement of its latitude to 90° is 23°27'. 

Therefore, on the date of the summer solstice (June 22), 6 = /420 
90° - (f) , and on the date of the winter solstice (December 22), 6 = 
(j) - 90° . 

On these dates at the Arctic Circle, the center of the Sun 
touches the horizon on June 22 at the north point at the moment of 
lower culmination (point N, Fig. 5.17), and on December 22 at the 
south point at the moment of upper culmination (point S). During 
the rest of the year, the Sun will rise and set daily. The period 
of daylight will increase daily from December 22 to June 22; after 
June 22, it will decrease. 

The Sun reaches its maximum altitude above the horizon on 
June 22. It will be: h = 90- 66°33' + 23°27' ^^&°S^'. 

Motion of the Sun at Middle Latitudes 

Knovjing the maximum declination of the Sun (equal to 23°27'), 
it is not difficult to calculate the latitudes of the observer on 
the Earth's surface where the Sun will be a rising and a setting 
heavenly body during the year. 

From the conditions for the rising and setting of heavenly 

M-41 



^ 




Motion of the Sun at the Tevvestviat Equator . 



In order t 
of the Sun at t 
have sketched s 
the Sun at the 
circles are div 
standable , sine 
server located 
to the equator, 
half by the hor 
the day and bel 
year the Sun ri 
length . 



o better und 
he Equator, 
chematically 
equator. As 
ided in two 
e we already 
at one of th 

Since the 
izon, the Su 
ow the horiz 
ses and sets 



erstand the nature of the diurnal motion 
let us analyze Figure 5.18. Here we 
the diurnal trajectories (circles) of 7421 
we see from the drawing, all the diurnal 
by the horizon line. This is under- 

know that the true horizon of an ob- 
e poles is situated at an angle of 90° 
daily circles of the Sun are divided in 
n will be located above the horizon half 
on half the day, i.e. during the whole 
, and the days and nights are equal in 




Intrinsic Motion of the Moon 

The Moon, participating with all the heavenly bodies in the 
sky in the diurnal rotation of the celestial sphere, has its own 
intrinsic motion. 





Fig. 5.17. Diurnal Motion 
of the Sun Above the Arctic 
Circle . 



Fig. 5,18. Diurnal Motion of 
the Sun at the Equator. 



442 



If we observe the Moon for one night, we can easily see that 
it travels like the Sun in the sky (relative to the stars). The 
apparent motion of the Sun is the result of the Earth's motion 
around the Sun; the Moon actually moves around the Earth. 

Direotion and Rate of the Moon ' s Motion 

The Moon moves along the celestial sphere from west to east, 
i.e., in a direction opposite the diurnal motion of the celestial 
sphere . 

The great circle along which the Moon completes its motion 
around the Earth has the shape of an ellipse and is called the 
Moon's orbit. The Moon's orbit is intersected by the solar ecliptic 
at an angle of 5°08' (Fig. 5.19). The two diametrically opposed 
points at which the Moon's orbit is intersected by the solar 
ecliptic are called the nodes of the orhi-t. 

The Moon completes a full revolution along its orbit relative 
to the stars in 27.32 days. This time interval is called the 
sidereal (stellar) month. Thus, it is easy to calculate that during 
one day it moves 13.2°. Its hourly shift relative to the stars is 
approximately 0.5°. 

The motion of the Moon in its orbit may be studied in relation 
to the Sun (which has, as we know, its own motion). 

The period of revolution required for the Moon to return to a 
previous position in relation to the Sun is called the synodic /'+22 
month. It is approximately 29.5 mean solar days. 

The Moon, revolving around the Earth, accompanies it in its 
motion around the Sun. The Moon is 356,000 km from the Earth at 
perigee (the closest point to the Earth) and i+07,100 km from the 

Earth at apogee (the most distant 
P point from the Earth). 

Phases of the Moon 

The Moon, like our Earth, 
is an opaque body which illuminates 
the Earth's surface with reflected 
sunlight. The illumination of 
the Earth's surface by the Moon, 
as we can see , is not always the 
same. At different times the 
Moon is visible in the form of 
a luminous disk, in the form of 
a luminous half-disk, or a cres- 
cent. There is a time when the 
Moon is entirely invisible. The 
Moon has various phases. The 




Fig. 5.19. Orbit of the Moon, 



t+43 



periodically repeated change in the shape of the Moon is called the 
change ■in lunar phases. 

In order to explain the cause of lunar phases , let us look at 
Figure 5.20. At Point (in the orbit's center) is our Earth; at 
Points 1-8 we show the Moon in various positions in its revolution 
around the Earth, Below the figure we show the shape of the Moon 
in those eight positions for an observer located on the Earth's 
surface . 

When the Moon is in Position 1, its phase is new moon. An ob- 
server on the Earth's surface during this phase does not see the 
Moon, since its nonilluminated side is turned to the Earth and is 
located at an angular distance of not more than 5° from the Sun. 
In Position 2, the Moon is visible on the Earth in the form of a 
narrow crescent only during evening hours. When the Moon is in 
Position 3, its phase is called the first quarter and an observer 
sees it in the shape of half an illuminated disk. This phase is 
called the first quarter because at this time a quarter of the 
entire lunar surface is visible. 

During first quarter, the Moon is visible in the east at noon, 
in the south about 1800, and in the west at midnight. After first 
quarter, the illuminated part of the lunar disk begins to increase 
(Position 4); in Position 5, the entire lunar disk is illuminated. 
This is the half-moon phase. In this phase, the Moon is visible 
all night . 

After half moon, the illuminated part of the lunar disk begins/i4-23 
to decrease from the right side of the disk (in Position 5), and in 




full 
moon 



last 
quarter 



Fig. 5.20. Phases of the Moon 



44-4 



Position 7 it reaches the phase of last quarter. In last quarter, 
the Moon is visible in the east at midnight , in the south about 
0600, and in the west at noon. After last quarter, the illuminated 
part of the lunar disk diminishes even more and in Position 8 it is 
already visible in the form of a narrow crescent (on the left side 
of the disk); it then becomes invisible again, i.e., the phase of 
new moon again sets in. 

Nature of the Motion of the Moon around the Earth 



In view of the fact that the Moon, in its motion around the 
Earth, is sometimes closer to the Sun than the Earth is and some- 
times farther away, it receives acceleration from the Sun which is 
sometimes more and sometimes less than that of the Earth. As a 
result, the motion of the Moon around the Earth is complex, since 
the above factors not only change the shape and dimensions of the 
lunar orbit, but its position in space. 




The Moon, completing one revolution in its orbit, intersects 
the plane of the celestial equator twice; in the course of one 
sidereal month, its declination changes from a maximum positive 
value to a maximum negative one . 



The maximum declination of the Moon in a period of one month 
may be +28°27' and the minimum may be -28 27'. 

Location of the Moon Above the Horizon 

The location of the Moon above the horizon depends on the so- 
called waxing of the Moon (time elapsed after new moon) and the 
season . 

In the Northern Hemisphere, during the summer and the phase of 
full moon, the Moon is located comparatively low and for a short 
time above the horizon; in the winter, during the full moon, it is 
visible all night and rises rather high above the horizon. This 
depends on the declination of the Moon. In winter in the Northern 
Hemisphere, the full moons occur with a positive .declination, while 
the full moons in summer occur with negative declination. 



445 



5. Measurement of Time 

Essence of Calculating Time 

Measurement of time is the result of the presence of motion in 
universal space. Time and motion are synonymous. I'f motion ceased 
in Nature and in the Universe, there could be no discussion of time. 
It is entirely understandable that to measure time, some constant 
and uniform motion must be used. 

The rotation of the Earth on its axis (or, as a result of this, 
the apparent rotation of the celestial sphere around the world axis) 
could be such a motion. 

Special observations over a very long time interval have shown 
that the duration of the Earth's rotation around its axis has not 
changed even by a fraction of a second. This characterizes very 
well the constancy and uniformity of the Earth's rotation. 

In the practice of aviation, the following kinds of time must 
be studied and used: sidereal, true solar, mean solar, Greenwich, 
local, zone and standard time. 

S i derea 1 T ime 7425 

Time measured on the basis of the apparent motion of heavenly 
bodies (stars) is oatted sidereat time. This time may be measured 
by the hour angle of some heavenly body with respect to the meridian 
of the observer. However, for convenience, it is advantageous to 
take the hour angle not of some star but of the point of the vernal 
equinox from which the right ascension is read. 



vernal equinox t 



y 



The western hour angle of the point of the 
is called sidereal time and is represented by the letter S (S = "ty) 
in Figure 5.21. Sidereal time is equal to the 

angle and the rij 




Fig. 5.2 
the Poin 
Equinox . 



1. Hour Angle of 
t of the Vernal 



star : 



Y 



sum of the hour 
^ght ascension of a 
t . 



•Y 



Side rea 1 da 

val between two 
culminations of 
vernal equinox i 
day. The moment 
mination of the 
was taken as the 
real days . At t 
time is equal to 
days are divided 
hours , each hour 
sidereal minutes 
into 60 sidereal 



ys. The time inter- 
successive upper 
the point of the 
s called a sidereal 

of the upper cul- 
vernal equinox point 

beginning of side- 
his moment, sidereal 
00:00:00. Sidereal 
into 24 sidereal 
is divided into 60 
, and each minute 
seconds . 



446 



Sidereal time does not have a date and therefore in calculations 
when the sidereal time is more than 21+: 00, there is only a surplus 
of time above 24:00. 

Example. In calculations, there is a sidereal time of S = 
29:20. Discarding 24-:00, we obtain a sidereal time of 5:20. 

The practical application of sidereal time. Sidereal time is 
not used in everyday life, but as the basis of time signals. In 
every astronomical observatory, there are special clocks which run 
according to sidereal time. In aviation, sidereal time must be 
used in observing stars for determining the position of the aircraft 
or the position line of the aircraft. 

In the aviation astronomical yearbook, the sidereal time is 
given for each date and each hour of Greenwich mean time. Therefore, 
the sidereal time for any moment at any point on Earth may be deter- 
mined by means of the aviation astronomical yearbook on the basis / 'j- 2 6 
of Greenwich time. In everyday life, solar time rather than side- 
real time is used, since on the whole man's activity occurs in the 
daytime hours. 

True Solar Ti me 

True solar time t0 is the time measured on the basis of the 
diurnal motion of the true Sun. 

True solar time is measured by the western hour angle (t^^) of 
the center of the true Sun. 

True solar days are the time intervals between two successive 
upper culminations of the center of the true Sun. The moment of 
the upper culmination of the center of the true Sun is taken as the 
beginning of true solar days. 

At the moment of upper culmination, when the hour angle of the 
Sun is zero, the true solar time is 00:00:00. 

In proportion to the diurnal motion of the Sun, its hour angle 
increases; the true solar time also increases. 

At the moment of the lower culmination of the Sun, the true 

solar time is 12:00; when the center of the Sun is again in the 

position of upper culmination, the true solar time is 24:00. After 
this, new days begin. 

The duration of true solar days changes in the course of a 
year. This occurs because the true Sun moves during the year along 
the ecliptic, which is inclined to the celestial equator at an 
angle of 23°27' and is not a circle but an ellipse. For this reason, 
the daily shift of the Sun to the east is different on different 
days of the year. This shift is at a maximum near the solstices, 

447 



when the Sun moves parallel to the equator. On the other hand, 
near the equinoxes the shift to the east is smallest. 

The Sun lags behind the motion of the stars by either a very 
great or a very small value; the duration of true days changes all 
the time . 

In using true solar time in everyday life, it would almost be 
necessary (as a result of its nonunif ormity ) to regulate clocks 
every day, moving them ahead and back. This would be extremely 
inconvenient . 

In view of the inconvenience of calculating time on the basis 
of the true Sun, time in everyday life is calculated on the basis 
of the so-called mean Sun. 



The mean Sun is the imaginary or real Sun moving uniformly 
along the celestial equator in the same direction that the true Sun 
moves along the ecliptic, i.e., in a direction opposite to the 
diurnal motion of the celestial sphere . 

Mean Solar Time 



/427 



Time calculated on the basis of the diurnal motion of the mean 
(imaginary) Sun is called the mean solar time (m). 

Mean solar time is measured by the western hour angle (t^) of 
the mean (imaginary) Sun. Mean solar days are the basic unit of 
mean solar time. 

Mean sotar days are the time intervals between two successive 
upper culminations of the mean Sun. They are divided into 24 mean 
hours, each hour is divided into 60 minutes and each minute into 
60 seconds. The duration of mean solar days is constant. 

Mean civil time (m^). Since at the mom.ent of the upper cul- 
mination of the mean Sun its hour angle will be zero, the mean 
solar time at this moment (mean noon) will also be zero. 

In everyday life, with a 24-hour reckoning of time, this is 
very inconvenient since in this case the beginning of mean days 
comes at noon. 

Therefore, in everyday life, so-called civil time (which is 
different from mean solar time by 12 hrs . ) is used as a variety of 
mean solar time. 

Mean midnight, i.e. the mean time equal to the western hour 
angle of the mean Sun plus 12 hrs, is taken as the beginning of 
mean civil days : 



448 



where mc is the mean civil time, and m is the mean solar time. 

The plus sign is used when the mean time is less than 12 hrs 
and the minus sign is used if the mean time is more than 12 hrs. 

Example . Determine the mean civil time if the western hour 
angle of the mean Sun (mean time) is 6:00. 

Solution. mc = m + 12:00 = 6:00 + 12:00 = 18:00 

Local Civil Time 



the h 
But t 
momen 
on th 
Earth 
In Fi 
plane 
Sun a 
on th 
point 
the h 
by th 
at th 
it is 
There 
mer id 



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our 
he h 
t fo 
e Ea 
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gure 

of 
t a 
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on 
our 
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real ti 
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our ang 
r one h 
rth ' s s 
urf ace 

5 . 22 w 
the dra 
certain 
rth ' s s 
the Ear 
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gle APn 
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, at th 

will b 



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wing 
mom 
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th 's 
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Mav 
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heaven 
t) of 
nly bo 
ce var 
ill be 
ow a c 
Let 
ent : 
ce and 
surf a 
e mean 
and th 
pre s se 
he ang 
me phy 
f f eren 



olar 
ly b 
a he 
dy f 
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lar 
e le s 

poi 
Radi 

rad 
ce . 

Sun 
e ho 
d by 
le A 
s ica 
t . 



and me 
ody or 
avenly 
rom the 
in valu 
ge and 
t ial sp 

us P^A 

ius P^B 

From t 

for th 

ur angl 

the an 

PnMav i 
1 momen 



an solar time are measured by 
the point of the vernal equinox, 
body calculated at one physical 

meridians of various points 
e. For some points on the 
for others it will be small, 
here whose equator lies in the 
be the position of the mean 
is the meridian of one point / M-2 i 

is the meridian of a second 
he figure, it is apparent that 
e first point (ti) is expressed 
e for the second point (tz) 
gle BP^Mav- From this figure 
s greater than the angle BFj-iM^v- 
t the time at different 



Time calculated relative to the meridian of midnight of some 
point on the Earth's surface is called the toaaZ aiviZ time and is 
represented by T -[ . Local time may be sidereal or solar. It will 

be the same for all points lying on 
^^^^ one meridian (i.e. having the same 

geographic longitude). 

G reenw i ch T i me 

Local civil time calculated 
from the Greenwich meridian is 
called Greenwiah time and is repre- 
sented by Tqj^ . 

In astronomical yearbooks, the 
times of some celestial phenomena 
are given as well as the astronomical 
values necessary for practical 
calculations on the basis of Green- 
wich t ime . 

The relation between local 




av 



im 



Fig. 5.22. Determining 
Local Time on the Basis of 
the Mean Sun . 



4tt9 



civil time and Greenwich time. With a knowledge of the Greenwich 
time and longitude of a place expressed in time units , it is simple 
to determine the local civil time and vice versa. 

The local civil time is equal to the Greenwich time plus or 
minus the longitude of the place, i.e. 



^« = ^Qi:^W" 



Here the plus sign is used if the longitude of the place is east 
and the minus sign is used if the longitude is west. 

Exampte. 1. Greenwich time is 14:15. Find the local time 
for Moscow (Xj; = 2:28). 

Solution. Substituting the values for Tq^ and Ag in the formu- 
la Ti = Tq^ + XE, we obtain the local time T^ = 14:15 + 2:28 = 16.43. 

Example. 2. The Greenwich time is 10:42. Find the local 
time for Washington (Aw = 05:08). 

Solution. T^ = Tq-p - A^ = 10:42 - 5:08 = 5:34. 

When solving practical problems in astronomy, it is often /^29 
necessary to change local time to Greenwich time. 

^GR= ^« ± J. 

The plus sign is used if the longitude of the place is west 
(A;^) and minus if the longitude of the place is east (Ag), 

Example. 1. The local time in Ryazan (Ag = 2:39) is 1430. 
Determine the Greenwich time. 

Solution. Substituting the values for T-[_ and Ag into the 
formula Tq^ = T^ - Ag, we obtain Tq^ = 14:30 - 2:39 = 11:51. 

Example . 2. The local time in San Francisco (A^ = 8:09:44) 
is 1520. Determine the Greenwich time. 

Solution. Substituting the values for Tj_ and Ay into the 
formula Tg^ = Ti + Aw, we obtain Tq^, = 1520 + 8:09:44 = 23:09:44. 

Time difference on two meridians. It is easy to imagine that 
the difference in local time on any two meridians is equal to the 
difference in their longitudes. 

In Figure 5.23 the diameter EQ is the Greenwich meridian, the 
diameter BA is the meridian of a point on the Earth's surface, the 

450 



diameter DC is the meridian of a second point on the Earth's surface, 
Mg^ is the location of the mean Sun at a certain moment and the 
diameter KM^v is the circle of the Sun's declination. 



In Figure 5.23 it is evident that the hour angles of the mean 
Sun (tiQav and t zq ^v^ measured from both local meridians are dif- 
ferent from one another by a value equal to the difference in the 
longitudes of these meridians, since ti - t2 = X2 - ^i. 

Hence it follows that the local time on these meridians will 
differ by the difference in the longitudes of these two meridians. 

Example . Let the local time be Ti = 1204 at a point having an 
east longitude of Ag = 2:35. What is the local time at this moment 
at a point having an east longitude Xg = 4:35? 

Solution. Let us find the difference in the longitudes of 
these two points AX = 4:35 - 2:35 = 2:00. 

Let us find the local time for the point having an east longi- 
tude Xe = 4:35. Ti = 12:04 + 2:00 = 14:04. 

In solving such problems , it is important to remember that the /430 
larger the east longitude of the point, the larger will be the local 
time at this point; the smaller the east longitude of the point, 
the smaller the local time . 

Zone Time 



We have already seen that each meridian of the Earth's surface 
has its own time. If we take Khabarovsk, whose longitude is 135°5' 
(9:00:20) its local time is 5:29:48 different from the local time 
of Moscow, which has a longitude of 37°38' (2:30:32). 



e(S) 




Fig. 5.23. Time Difference 
on Two Meridians 



Sine 
surface h 
it is too 
time in e 
when movi 
would be 
move the 
for every 
1 hr for 
On the ot 
east to w 
have to b 
Therefore 
last cent 
Europe ha 
single ti 
This time 
meridians 



e each p 
as its o 

inconve 
veryday 
ng from 
ne cessar 
hour han 

degree 
every 15 
her hand 
est , the 
e moved 
, s ince 
ury , the 
ve begun 
me in th 

is meas 

of the 



omt 
wn ( 1 
nient 
life . 
west 
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of lo 
° of 
, in 
hour 
back 
the m 
coun 
to i 
eir t 
ured 
princ 



on t 
ocal 

to 
Fo 
to e 
cont 
min 
ngit 
long 
movi 

han 
cons 
iddl 
trie 
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from 
ipal 



he E 
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inuo 
ahea 
ude 
itud 
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ds w 
tant 
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s of 
duce 
tori 
the 
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arth ' s 
me , 
local 
ample , 

it 
usly 
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or 
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rom 
ould 

ly- 

the 

a 
es . 

erva- 



451 



tories of these countries. 

In France so-called "Paris time", was introduced, in Italy 
"Rome time", and in England "Greenwich time", etc. 

The introduction of a single conventional time in these countries 
did not create great difficulties , since the local time of any 
meridian of these countries (in view of their small area relative 
to the meridian of the conventional time introduced) differed insig- 
nificantly (only several minutes in all). If we take England and 
France for example , their outermost populated points (to the east 
and west) are situated in a range of 7-8° from their respective 
meridians (Greenwich and Paris): the time difference amounts to a 
total of 30 min. If we take such a country as the USSR, we know 
that the difference in the longitudes of its eastern and western 
boundaries amounts to more than 10 hrs in time. However, in pre- 
Revolutionary Russia, a common Petersburg time (the local time of 
the Pulkovo Observatory meridian) was introduced only for railroads. 
This time was 00:28:58 behind Moscow local time (local time of the 
Moscow University Observatory meridian). 

The introduction of a common time in individual countries 
partially facilitated its calculation within each country, but it 
did not solve the problem on an international scale. The problem 
of calculating time was solved most successfully after the intro- 
duction of a zone time system. 

In some countries, this system was introduced at the end of the /^3j- 
19th and beginning of the 20th century. In Russia, the zone time 
system was introduced only after the Revolution, on July 1, 1919 by 
a special decree of the Soviet Government. 

Essence of the zone time system. The entire Earth is divided 
into 24 hour zones. The outer meridians (boundaries) of each band 
are 15° of longitude (1 hr in time) apart from one another. 

The zones are numbered from west to east from the zero zone 
to the 23rd zone, inclusive. The zone included between the meridians 
7°30'N and 7°30'E is taken as the zero zone, i.e. the initial zone. 
The Greenwich meridian, which has a longitude of 0°, is the mean 
meridian of this zone. Obviously, the first zone will be located 
between the meridians X = 7°30'E and X = 22°30'E and the mean 
meridian of this zone has a longitude of 15°; the second zone is 
located between the meridians X = 22°30'E and X = 37°30'E and the 
mean meridian of this zone has a longitude of 30°, etc. 

At all the points located within the limits of the same zone, 
the common time (time of the given zone) which represents the local 
time of the mean meridian of the given zone is taken. Such a con- 
ventional time is called the zone time (T^). 

In the zero zone, the time is calculated on the basis of the 



452 



local time of the zero (Greenwich) meridian. In the first zone, 
the time is calculated on the basis of the local time of the meridian 
having a longitude of 15°E. In the second zone, it is calculated 
on the basis of the local time of the meridian having a longitude 
of 30°E, etc. Since the mean meridians of two adjacent zones are 
15° of longitude apart from one another, the difference between 
the zone time of adjacent zones is one hour. 

The number of each zone shown by how many hours the time in 
this zone is ahead of Greenwich time. For example, in the case of 
the time of the fourth zone, this means that its time is 4 hrs 
ahead of Greenwich time (Supplement 4). 

The introduction of zone time greatly facilitated the calcula- 
tion of time on an international scale, since the minute and second 
hands in all the zones indicate the same number of minutes and 
seconds. The hour hands must be moved a whole hour only when moving 
from one zone to another. When the boundaries of the hour zones 
were determined, the boundaries of states, regions and cities as 
well as natural boundaries (e.g. rivers, etc) were taken into ac- 
count. If the boundaries of the hour zones had been determined 
strictly according to the meridians, calculating the zone time (e.g. 
in Moscow, which is located on the boundary between the second and 
third zones) would have to be done on the basis of two zones: in 
the western part of the city on the basis of the second zone, and 
in the eastern part of the city on the basis of the third zone. 
I.e. the time difference in the two parts of the city would be one 
hour and in crossing the boundary the hour hands would have to be / 43 2 
moved one hour. 

In view of this, the difference between the local time of the 
outer points of this zone may be somewhat more or less than 30 min 
relative to the zone time. 

Standard Time 



In the Soviet Union, on the basis 
clocks were moved ahead one hour beg" 
Since ^^ -' - -^ - - 
c 



bince this time, the entire USSR reckc 

called standard time. Thus, the zone ^^...v, .,^^ ^..^^^--^ ^^v,,^.^ ^ „^ , 
i.e. each zone lives not on the basis of its zone time but rather 
on the basis of the time of the adjacent eastern zone. For example, 
Moscow, which is located in the second zone, lives according to 
the time of the third zone . 

The time running 3 hrs ahead of Greenwich time (time of the 
3rd zone) is called Moscow time. 

All the railroad, water, and air routes of communication in 
the Soviet Union operate according to Moscow time . 



453 



Relation between Greenwich, Local and Zone (Standard) Time 

When solving practical problems of aircraft navigation on the 
ground and in the air, it is very often necessary to convert from 
one form of time to another. 

These problems may be solved correctly only if 'the crew 
conscientiously masters the essence of calculating time. To facili- 
tate the work of the crew in solving such problems, there are 
formulas for converting time from one form to another. 

Converting Greenwich time to Mean time. Zone time (T^.) is 
equal to Greenwich time (Tq^) plus the number of the zone: 

In solving problems for the USSR, the number of the zone is 
used, taking into account the standard time, i.e. 1 hr . later. 

Example . What is the zone (standard) time in Novosibirsk when 
clocks in Greenwich show 12:00? 

Solution. On the basis of a map of hour zones or on the basis 
of a list of the most important populated points, let us find the 
number of the zone in which Novosibirsk is located. Taking into 
account that the standard hour in Novosibirsk , the clocks run ac- 
cording to the time of the 7th zone (6th zone + 1 hr ) , let us find 
the zone time on the basis of the formula 

r2= 7-j^g,+ Ar=12h+7h=19h 

Converting Zone Time to Greenwich Time. Greenwich time is /^33 
equal to the zone (standard) time minus the number of the zone 
(taking into account the standard time): 

Example . What is the Greenwich time when the clocks show 17:00 
in Krasnoyarsk? 

Solution. On the basis of a map of hour zones or on the basis 
of a list of populated points , let us find the number of the zone 
where Krasnoyarsk is located. Taking into account the standard 
time in Krasnoyarsk, the clocks run on the basis of the time of the 
7th zone (6th zone + 1 hr ) . 

Let us find the Greenwich time. Tq^ = T^-N = 17:00 - 7:00 = 
10 : 00 . 



454 



Converting Zone (standard time) time to local time. Local 
time is equal to the zone time minus the number of the zone plus or 
minus the longitude of the place (plus is used when the longitude 
is east, minus when the longitude is west): 

This formula assumes that the hour zones are counted from to 
24 eastward from Greenwich. 

Example . What is the local time in Omsk when the zone (standard) 
time there is 16:00? 

Solution. Using a map of hour zones or a list of the most 
important populated points , let us find the number of the zone 
(taking into account the standard time) where Omsk is located and 
its longitude. Taking into account the standard time in Omsk, 
the clocks run according to the time of the 6th zone (5th zone + 
1 hr); the longitude of Omsk A = 73°24'E (4:53:36). 

Let us find the local time T^ = T^ - N + Xg = 16:00 - 6:00 + 
4:53:36 = 14 : 53 : 36 . 

Converting local time to zone time (standard time). Zone 
(standard) time is equal to the local time plus the number of the 
zone (taking into account the standard time) plus or minus the 
longitude of the place (plus is used when the longitude is west 
and minus when the longitude is east). 

Example . What is the zone (standard) time in Irkutsk when the 
local time there is 18:00? 

Solution. Using a list of the most important populated points, 
let us find the number of the zone (taking into account the standard 
time) and the longitude of Irkutsk. 

Taking into account the standard time, Irkutsk is located in 
the 8th zone (7th zone + 1 hr ) ; the longitude of Irkutsk A = 104° 
18 'E (6 :57 :12) . 

Let us find the zone standard time T2; = T^ + N - Ag = 18:00 + 
8:00 - 6:57:12 = 19:02:48. 

Measuring Angles in Time Units 

Since the values of hour angles and right ascensions are used 
for measuring time, it is often more convenient to express these 

455 



values in time units rather than in degrees . Also it is often 
necessary to express the longitude of a place in time units. 

To convert hour angles and right ascensions as well as longi- / ^3^■ 
tude from degrees to hours and back again, the following equations 
must be used : 

24 hr = 360°; 1 hr = 15° or 1° = 4 min ; 1 min = 15 ' or 1 ' = 4 
sec; 1 sec = 15" or 1" = 1/15 sec. 

These equations are based on the fact that the celestial 
sphere (or the Earth on its axis) makes a complete revolution in 
24 hrs , which corresponds to 360°. 

To convert hour angles, right ascensions and the longitude of 
a place from degrees to hours, the following must be done: 

(1) Divide the number of degrees by 15 and obtain whole hours. 

(2) Multiply the remainder from dividing the degrees and ob- 
tain minutes of time. 

(3) Divide the number of minutes of arc by 15 to obtain whole 
minutes of time, which must be added to the minutes of time already 
obtained, and obtain the total number of minutes of time. 

(4) Multiply the remainder from dividing the minutes by 4 and 
obtain seconds of time . 

(5) Divide the seconds of arc by 15 and obtain an additional 
number of seconds. Add these seconds to the preceding ones and 
obtain a total number of seconds. 

(6) Discard the remainder of seconds of arc when it is less 
than 8; if it is greater than 8, consider it as 1 sec of time. 

Example, 1. Express the hour angle 163°57'35" in hours. 

Sol Ution. 150° = 10 : 00 

13° = 0:52 

45 ' = 0:03 

12 ' = 0:00:48 

00^30" = 0:00:02 

Total: 10:55:50 

To convert hour angles , right ascensions and longitude from 
hours to degrees , the following must be done : 

(1) Multiply the hours by 15 and obtain the radii. 

(2) Divide the minutes by 4 and separate out the whole degrees. 



456 



(3) Multiply the remainder of the minutes of time by 15 and 
obtain minutes of arc. 

(4-) Divide the seconds of time by 4 and separate out minutes 
of arc . 

(5) Multiply the remainder of the seconds of time by 15 and 
obtain seconds of arc. 

Example . 2. Express an hour angle of 11:27:15 in degrees. 

Solution. 11 hr = 155° 

24 min = 6° 

3 min = 45 ' 

12 sec = 3 ' 

3 sec = 45 ' 



Total : 171°48 '45" 

Time Signals / ^35 

Accurate time in astronomical observatories is determined by 
means of astronomical observations of the culmination of heavenly 
bodies . 

Special transit instruments are used for this purpose. A 
transit is mounted in such a way that its main part (the terrestrial 
telescope) is always located in the plane of the meridian. Such a 
location of the terrestrial telescope permits the observation of a 
heavenly body only at the moment when it crosses the meridian, i.e. 
at the moment of culmination. Since the Sun is at the point of 
upper culmination (crosses the southern part of the meridian) at 
true noon, it can be observed only when the hour angle of the true 
Sun is zero. Therefore, when the Sun passes through the terrestrial 
telescope of the transit instrument, the moment of true noon on the 
given meridian will be recorded. 

Knowing the precise time of the moment of true noon and com- 
paring it with the actual indication of the clock at the moment of 
observation, it is possible to check the clock and determine its 
error. In the field of vision of a transit, vertical lines are 
drawn which permit more accurate determination of the moment that 
stars cross the meridian. In astronomical observatories, the 
accurate time is determined on the basis of observing the passage 
of stars is based on the fact that at the moment that any star is 
at the meridian (we already know), the sidereal time will be equal 
to the right ascension of the star. 

Thus, these observations make it possible to determine the 
exact sidereal time. The time obtained is set on special clocks, 
which run according to sidereal time. They differ from normal 
clocks in that (by the means of a special control) they run 3 min 
and 56 sec ahead of normal clocks per day. 

457 



Once the exact sidereal time is obtained, the mean solar time 
is calculated at astronomical observatories and set on mean solar 
astronomical clocks , The time obtained will be the local mean solar 
time on the meridian of the observatory. 

When necessary it is always possible to con,vert this time to 
zone and standard time. The necessary accuracy in determining the 
true time at astronomical observatories is computed in hundredths /U36 
of a second, and therefore the process of determining accurate time 
is much more complex than we have described here. 

For accurately determining the moments of the passage of the 
stars through the meridian, the moments of the culmination of the 
stars are automatically recorded at astronomical observatories. 
With these methods, the determination of the true time is accurate 
to 2 or 3 hundredths of a second. At every observatory, there are 
accurate astronomical clocks which are manufactured on special 
order. Special care is required for these clocks, since the 
continuous changes in temperature and atmospheric pressure strongly 
affect the steadiness of the oscillation period of the clock balance 
wheels. Therefore, astronomical clocks are kept in a special room 
where a constant temperature is maintained, and they are placed 
under a hermetically sealed bell jar where a constant atmospheric 
pressure is maintained. 




Organisati-on of Time Signals in Aviation 

Time signals in aviation are organized to ensure accuracy of 
aircraft navigation. The basic task of the time signals in aviation 
is the systematic checking of clocks and the guarantee of knowing 
the accurate time at any time of day. The presence of accurate 
time is especially important when using astronomical means of air- 
craft navigation. This necessitates knowing the accurate time not 
only on the ground , but in flight . 

In order for the crew to know the correct time at any time of 
day there must be master clocks with a constant daily speed. They 
are installed in the cockpit, at the weather station, or in other 
special places. 



458 



Master clocks are used to determine the correction of other 
clocks in the periods between the transmissions of accurate time 
signals . 

Master clocks are checked and set according to the correct 
time on the basis of accurate time radio signals transmitted by 
broadcasting stations of the USSR. The correction of the clocks 
and its verification on the basis of signals is recorded in a 
special log. 

A Brief History of Time Reckoning. 

Sometimes we hear the terms "Old Style" and "New Style". 

Systems for measuring and calculating large time intervals are 
called aatendars . 

The basis for time reckoning is the tropical year, i.e. , the 
time interval between two successive passages of the Sun through 
the point of the vernal equinox. The length of the tropical year 
is approximately equal to 365 days, 5 hours, 48 minutes and 46 
seconds, but it is very inconvenient to use the tropical year for 
time reckoning, since it does not contain a whole number of days. 

Thus, for example, if we take midnight on January 1 as the be- 
ginning of one year, the second year will begin not at midnight of 
January 1 but on January 1 at 5:48:46 AM, the third year at 11:37:32 
AM, etc. We may conclude that each year the beginning of the new 
year would be shifted by 5:48:46. 

Old Style (Julian) Calendar. For Romans, the year was original- 
ly lunar and consisted of 12 lunar months. The length of the lunar 
year was 355 days, i.e., their year was 10 days shorter than the 
accepted year at the present time. With such a time reckoning, the 
beginning of the new year shifted rather quickly from one month to 
another. If, for example, we take a time interval of 10 years, 
then 100 days were accumulated during this period, i.e. , the begin- 
ning of the new year shifted by more than three months. 

To eliminate these inconveniences, approximately every three 
years the year was lengthened by one month, i.e. , this year had 
13 rather than 12 months. 

The Roman dictator, Julius Caesar, introduced a calendar re- 
form in 46 B.C. This calendar was called the Julian calendar and 
we now call it the "Old Style". The essence of it was that the 
duration of one year was considered to be 365 days rather than 355 
days. In addition, in February (which was then considered the last 
month) an extra day was included every fourth year, i.e. , that year 
had 366 days. The addition of the extra day once every 4 years 
nearly compensated for the difference accumulated in 4 years (about 
6 hr per year) and thus the constancy of the date of the vernal 



/437 



459 



equinox (March 21) was preserved, 
until the present time. 



This principle has been retained 



The extra day as we know, is now added in February: instead 
of 28 days there are 29 days in all the years which are divisible 
by 4-, e.g. 1956, 1960, 1964, etc. These years have been called 
leap years up until the present time. 

New Style (Gregorian) Calendar. The length of the so-called 
tropical year is, as we know, 365 days, 5 hours, 48 minutes and 46 
seconds. The Julian year (on the average) is equal to 365 days and 
6 hours. Thus, the Julian year is 11 min and 14 sec longer than 
the tropical year. Although this difference is small, over a large 
interval of time it may also cause the beginning of the year to 
shift. It is not difficult to calculate that in order to shift the 
date of the vernal equinox one calendar date (one day), almost 128 
years (24 hr or 1440 min, divided by 11 min 14 sec) are required. 
The gradually accumulating difference in the second half of the 
16th century amounted to 10 days. As a result, the date of the 
vernal equinox came (according to the calendar) not on the 21st, 
but on the 11th of March. 



As a result of the shift of the beginning of spring from 21 to 
11 March, the holiday of Easter (which must be close to spring) 
graduallly moved toward the summer. This greatly disturbed the 
clery, who did not want to depart from their rules. 

The Roman Pope Gregory XIII Introduced a new calendar reform 
in 1582 by decree. The essence of this reform, i.e., the transfer 
to a new style of calendar, was the following: after October 4, 15 
he ordered everyone to consider the date to be October 15 , rather 
than October 5, i.e., he ordered the 10 days accumulated over 1200 
years to be dropped so as to return the date of the vernal equinox 
to March 21. In order to avoid the accumulation of an error in 
future, it was decided that every 400 years those three days which 
differentiate the Julian year from the tropical year be dropped. 
To do this, it was decided that three leap years in every 400 years 
be considered regular years, i.e, not on the basis of 366 days but 
rather on the basis of 365 days. In order to remember this more 
easily, the centurial years in which the numbers of the century wer 
divisible by 4 were taken as leap years (for example, of the centur 
years 1600, 1700, 1800 and 1900, only the year 1600 remained a leap 
year. The others became regular years, since only 16 is divisible 
by 4). Of the centurial years, the next leap year will be the year 
2000 . 



82, 
/43S 



e 
ial 



As regards the other years (besides the centurial years), the 
calculation of the leap years remained the same as in the Julian 
calendar . 

The Gregorian calendar was gradually introduced in all civilized 
countries. In Tsarist Russia, the introduction of the new calendar 



460 



(New Style) met with great opposition. Only after the Great 
October Socialist Revolution on February 1, 1918 was the new style 
quickly introduced. Then the difference between the Old and New 
Styles has already 13 days. Neither the Old nor the New Style is 
absolutely accurate, but the Gregorian (New Style) calendar has 
less error (1 day in 3000 years). 

6. Use of Astronomical Devices 

Astronomical means of aircraft navigation permit the deter- 
mination of flight direction and the position line of the aircraft 
on the basis of the stars. The advantage of astronomical devices 
is their autonomy. Their use in flight is not related to any 
ground equipment and their accuracy is not a function of flight 
distance. The use of astronomical devices is based on the principle 
of measuring the azimuths or the altitudes of heavenly bodies. 

The position of the aircraft is determined by astronomical 
instruments from the intersection of two astronomical position lines 
(APL) or (as they are still called) lines of equal attitude. 

The astronomicat position line (line of equal altitude) is the 
straightened arc of the circle of equal altitude whose center is 
the geographic position of the star. 

The geographic position of a star (GPS) is the point on the 
Earth's surface at which the given star is observed at the zenith, 
or the projection of the star onto the surface of the Earth. The 
coordinates of the geographic position of the star represent the 
equatorial coordinates of the given star, i.e. , the latitude is 
equal to the declination of the star ( (j) = 6) and the longitude is 
equal to the Greenwich hour angle of the star (A = "'^Gr-'' This is 
evident in Figure 5.24. 



Circle of equa 
Earth's surface (""" 
tude of 

from point M with 
ca 



rrom point M wi 
it will be --"'"' 

(^TO-f-3T1i^£3 -f- 



it will be called the circle 
distance from any point on th 
the zenith distance (radius). 



juat altitude. Let an observer at point A on the 
[Fig. 5.25) at any moment of time measure the alti 
M. On the celestial sphere, if we draw a circle 
1 a spherical radius equal to the zenith distance. 
i the circle of equal zenith distances , since the 
J point on this circle to the star M is equal to 
ice ( radius ) . 



/439 



The projection of the circle of equal zenith distances (each 
point on the circle) onto the Earth is called the circle of equal 
altitude (hh') and the center of this circle is the geographic 
position of the star M on the Earth (GPS). It is called the circle 
of equal altitude because at any point on this circle, the star M 
will have the same altitude, i.e. it is observed at an angle for the 
same altitude of the star. This may be proved rather simply if we 
recall the relation between the altitude of a star (h) and its 
zenith distance (Z): h = 90° - Z. But since the zenith distance 
for any observer located on the circle of equal altitude is the same 



461 



(R of the circle 
the same . 



Z) then the height (h) for all observers will be 



Knowing this principle, we may make the reverse conclusion: 
if an observer, by measuring, determines the altitude of a star at 
a certain moment of time, then by plotting the geographic position 
of the star (GPS) with a radius equal to the zenith distance of the 
star (Z = 90° - h), the circle of equal altitude may be drawn. Ob- 
viously, at the moment of measuring the altitude of the star the 
observer (aircraft) is located on the circumference of the circle 
of equal altitude. Therefore, the circle of equal altitude is the 
circle of the position of the aircraft. 

In practice, the altitude (h) and azimuth (A) of a star are 
determined on the basis of tables at a moment of time planned 
beforehand for the point being calculated whose coordinates approxi- 
mately coincide with the location of the aircraft at the given / '^■ '/t 
moment. On the map, a straight-line segment equal to Ah is drawn 
from the point being calculated in the direction of the azimuth 
of the star. The segment APL is drawn perpendicular to the straight 
lines through its end. 

At the moment for which the calculation of the APL of the point 
being calculated was made, the altitude of the star h is measured 
by means of a sextant. In general, the measured altitude does not 
coincide with the altitude of the star at the point being calcu- 
lated. The difference between 



these altitudes is equal to the 
difference between the radii 
of the circles of equal alti- 
tude of the calculated and actual 
points of the aircraft's posi- 
tion. Since all the circles of 



zenith 
distance' 




plane of 
the horizon 




Fig, 5 .24. 
of a Star. 



Geographic Position 



Fig. 5,25, Circle of Equal 
Altitude . 



462 



iriin I lifii innni II ■ 



-I iiBiiiiia II Hill 



equal altitude of the same star at the same moment of time are con- 
centric, the APL representing various altitudes of the star are 
parallel to one another. 

Astronomical position lines which represent high altitudes 
are situated closer to the geographic position of the star, and 
vice versa. Therefore, if the measured altitude (hji,) is greater 
than the calculated altitude (h^), this means that the aircraft at 
the moment of measuring was located not on the APL of the point 
being calculated but on the APL parallel to it moved in the direc- 
tion of the star by a value Ah = hm - h^. 

If the measured altitude is less than the calculated altitude, 
the APL is moved in a direction opposite the direction of the star. 

To construct the APL on a map, it is necessary to know: 

(a) The approximate coordinates of the aircraft's position 
(the point being calculated) ^, A. 

(b) The azimuth of the star (A) for the point being calculated: 

(c) The distance between the measured and calculated altitudes 
of the star ( Ah ) . 

Determining the astronomical position lines. Before beginning 
the measurements, the electrical supply and the sextant light are 
switched on, the averaging mechanism is wound up, and the repeaters 
of the chronometer are matched with the indicator of the current 
t ime . 

On the basis of a map of the stellar sky or tables of altitudes 
and azimuths, the azimuth and then the course angle of the star are 
determined roughly. By rotating the course-angle drum, the sextant 
is set to the course angle of the star. 

By rotating the altitude drum, we bring the star into the sex- 
tant's field of vision. The sextant is set on a level base; by 
rotating the course-angle drum, the star is lined up with the bubble 
level. By rotating the altitude drum, the star is made to coincide 
with the bubble level and the averaging mechanism is connected. 
Once the pilot has been notified beforehand about the beginning of 
measurement, and the star has been accurately superposed on the 
bubble level, the averaging mechanism is switched on. When the 
averager has completed its work, a reading is taken. 

On the basis of the measured altitudes of the stars, problems 
in determining the following are solved: 

(a) One astronomical position line on the basis of the Sun /l^l 
to check the path with respect to distance and direction. 



463 



(b) Two position lines on the basis of two navigational stars 
or on the basis of one navigational star and Polaris to determine 
the position of the aircraft 

To check the path with respect to distance by means of one 
APL, stars are used whose directions are similar to the direction 
of the line of the given path. To check the path with respect to 
direction, stars are used which are situated at right angles to the 
line of the given path. 

In determining the position of an aircraft on the basis of two 
APL's, stars must be chosen so that the directions to them differ 
by an angle close to 90°. 

In calculating the astronomical position lines, the following 
auxiliary tables are used: 

(a) detached pages from an aviation astronomical yearbook 
(AAY) for the flight date; 

(b) Tables of the altitudes and azimuths of the Sun, Moon and 
planets (TAA): 

(c) Tables of the altitudes and azimuths of the stars (TAAS). 

The astronomical position line is calculated on a form like 
the following: 





Calculating the APL 


Order of 


Date Name of 


Operation 


GPA, W the star 


1 


2 


3 


1 
5 
6 
9 
1+ 


•'Moscow 
TGr 

AtGr(ASGr) 
A 




10 


tl(Si) 




3 


* 




7 


6 




13 


FB 




2 
8 


hm 

P 




1^ 


S 




15 


-r 




16 







17 


h 




11 


he 




18 


Ah 




19 
12 


^^KM 
A 




f64 







Name of 
the star 



Calculation of the astronomical position line with respect to / ^■^2 
the Sun, Moon or some planet is done in the order indicated in the 
left-hand column of the form: 

(1) The Moscow time of measuring the altitude of the star is 
recorded (Tmoscow). 

(2) The measured altitude of the star is recorded (h^). 

(3) and (4-). The latitude and longitude of the calculated 
point ( <t) and X) are recorded. 

(5) The Greenwich time of measuring is determined on the basis 
of the formula 

where (Ng; + 1) is the number of the hour zone plus the standard 
hour and is recorded on the form. 

(6) and (7) The declination of the star (6) and its hour angle 
(tQ-p) for the whole hour corresponding to the time of measurement 
are copied out of the AAY . (When using the Moon, the tgp i^ written 
for whole tens of minutes . ) 

(8) In measuring the altitude of the Moon, the parallax (P) 
is copied from the AAY. 

(9). On the basis of the interpolation table available in 
the TAA or the AAY, the correction (AtQr) for Tq^ in minutes and 
seconds is found and recorded. 

(10) The local hour angle of the star (ti) is determined by 
adding tgp, Atg-p and X. Increasing or decreasing X, it is neces- 
sary that t-| be expressed by a whole even number of degrees. If 
the western hour angle is more than 180°, its complement to 360° 
is taken. The value found for t]_ is considered the eastern hour 
angle and is recorded in the form. 

The value for the longitude of the calculated point X, written 
earlier on the form, is refined in accordance with the change 
introduced with the selection of tj. 

(11) and (12) From the TAA, the value for the altitude of the 
star at the calculated point (h^) is written, taking into account 
the correction for minutes of declination, and the azimuth A is 
recorded. If the hour angle is western, then the complement of its 
tabular value up to 360° is taken as the azimuth. 

(13) The path bearing (PB) of the star (path angle) is deter- 
mined on the basis of the formula PB = A - GPA and is written on 



465 



the form. 

(14-) (15) and (16) Corrections are written on the form from 
the pertinent tables: sextant (S), for the refraction ( -r ) and 
for the Earth's rotation (6), 

(17) The measured altitude of the star (h) is adjusted for 
corrections Nos . 8, 14, 15, 16. 

(18) The difference between the corrected value of the 
measured altitude (hjj,) and the altitude of the star at the calcu- 
lated point (hj,) is calculated on the basis of the formula 

Ah = h„ - h . 



(19) Another value for Ah is recalculated in kilometers. 
After the calculations, APL is plotted on the chart as shown above. 

The APL on the basis of stars is calculated in the same order 
as on the basis of the Sun, Moon and planets. Tables of the TAAS 
and AAY are used. In addition, instead of the Greenwich and local 
hour angles, the Greenwich and local sidereal time are determined. 

The sidereal Greenwich time (Sq^) is taken from the table 
entitled "Stars" in the AAY for the moment Tq^ . The local sidereal 
time is determined on the basis of the formula 



/41+3 



where X is the longitude of the calculated point refined with this 
calculation so that S-]_ is equal to a whole number of degrees. 

When determining the position of the aircraft on the basis of 
the intersection of the APL from two navigational stars, the meas- 
uring and recording of the time of the readings are done successive- 
ly with the shortest possible time interval. When plotting on the 
chart, the first APL is shifted parallel to itself in the direction 
of the vector of the flight speed for the distance traversed in 
this interval of time . 

The correction for the movement of the aircraft between the 
moments of the first and second measurements is also determined by- 
means of a special table applied to the tables of the altitudes 
and azimuths of stars. 

The correction for the rotation of the Earth (a) is introduced 
in the altitude of the star measured first. It is not necessary 
to shift the first APL determined, taking into account the correction 
of 6 by the movement of the aircraft in this case. 

When determining the position of an aircraft on the basis of 
stars in the Northern Hemisphere, Polaris and one of the navigational 



466 



stars situated in a westerly or easterly direction are used. Polaris 

is approximately 1° from the north celestial pole and therefore 

its height above the horizon is always roughly equal to the latitude 

of the position. This simplifies the calculating and plotting of 

APL's. 



The accurate latitude of the position of an aircraft on the 
basis of Polaris is determined by simple addition: 

Its measured altitude h,^ ; the correction of the sextant S; the 

correction for refraction -r ; the correction for the Earth's rotation 

a and the correction for the altitude of Polaris A(|> . The correction 

A(f) is given in TAAS on the basis of the value of the local sidereal 
time S^. 



The altitude of Polaris is measured later than the altitude 
of the navigational star and therefore the parallel corresponding 
to the latitude found is shifted in the direction of the flight- 
speed vector for the segment of the path traversed during the time 
interval between the first and second measurements. 



/444 



The correction due to the travel of the aircraft is also 
introduced directly in the calculated latitude by means of a table 
of corrections "D" in the TAAS. 

Astronomical Compasses 

Modern astronomical compasses are automatic devices for deter- 
mining the true course of the aircraft by the direction-finding of 
the Sun or other stars . 

Astronomical compasses of the type DAK-DB are used on aircraft. 

These astrocompas se s are mainly intended for: 

(a) Incidental determination of the true course on the basis 
of the Sun ; 

(b) Continuous measurement of the course in flight along the 
orthodrome on the basis of the Sun. 

Astrocompasses of the DAK-DB type can transmit the values of 
the true course to course system indicators, and they can also 
permit the true course to be determined on the basis of stars at 
night by means of a periscope sextant. 

Astrocompasses of DAK-DB type may be used in the range of 
latitudes from the North Pole to 10°S. Astrocompasses of a special 
type are intended for use in the Southern Hemisphere as well. They 
can operate when the Sun is not more than 70° above the horizon. 
Here the permissible error in determining the true course must not 
exceed ±2° . 



467 



An astrocompass automatically solves problems of determining 
the true course of an aircraft according to the equation: 



TC = A 



CA 



where A is the azimuth of the heavenly body and CA is the course 
angle of the heavenly body. 

The course angle of the Sun is determined automatically by 
means of a course-angle data transmitter (CAD). 

The photoelectric head is situated in a transparent case in 
the fuselage of the aircraft; by means of an electronic system, 
it is automatically oriented in the direction of the Sun and sup- 
plies an electrical signal representing the course angle (CA) to 
a computer device. 

The azimuth of the star is determined by a special computer 
whose basis is a spatial computer mechanism (spherant). When es- 
tablishing the equatorial coordinates on computers , the hour angle 
and declination of the star as well as the latitude and longitude 
of the position, the azimuth of the star, i.e. the horizontal co- 
ordinate, is given at the output in the form of electrical signals. 

The table for the Greenwich hour angles of the Sun is given in 
Supplement 5 . 



/t+45 



star 



plane of Y\ 
the horizon 




v}> 



f/^/;^ 





s 
. — -D W 



— -0 ^ 



source 



Fig. 5.26. Optical Diagram 
of an Aviational Sextant. 



A signal representing the dif- 
ference in the azimuth and course 
angle, i.e. the value of the true 
course , is fed to the indicator 
of the astrocompass. 

When using the astrocompass 
to determine and retain the ortho- 
drome course , coordinates pertaining 
to the initial point of the ortho- 
drome path line are fed into the- 
astrocompass . During flight along 
the orthodrome , the course angles 
at the initial point of the route. 
To preserve a constant value of 
the true course relative to the 
reference meridian of the beginning 
of the path, a correction on the 
basis of the flight correction 
method is automatically fed in. 
This method entails the following: 
The axis of rotation of the head 
of the CAD is vertical at the be- 
ginning of the path. Later with 
movement of the aircraft along the 



468 



orthodrome , it slopes back toward the tail of the aircraft by an 
angle equal to the arc of the traversed part of the orthodrome at 
the same time remaining parallel to the original position. The 
automatic calculation of the angle proportional to the arc of the 
traversed segment of the orthodrome is performed by the flight 
corrector, with a manual setting of the airspeed of the aircraft. 

Astronomical Sextants 

Aviational astronomical sextants are intended for measuring 
the altitudes of stars to determine the astronomical position lines 
and the position of the aircraft, as well as for measuring the course 
angles of stars. 

At the present time, periscope sextants (PS) which are adapted 
for mounting on aircraft with hermetic fuselages are the most 
common variety. 

The optical system of the PS sextant (Fig. 5.26) includes a 
cubic prism 1 for sighting stars. The cubic prism turns in a verti- 
cal plane from to 85° , with a goniometer drum to indicate alti- 
tude of a star 




The sextant has a chronometer with two independent repeaters, 
the clock mechanism of the averager of the readings and the course- 
angle transmitting selsyn. 



469 



CHAPTER SIX 
ACCURACY IN AIRCRAFT NAVIGATION 



1. Accuracy in Measuring Navigational Elements and in 
Aircraft Navigation as a Whole 

The process of aircraft navigation is directed toward a crew's 
maintaining given trajectories of aircraft movement with respect to 
direction, altitude, distance, and time. 

Since the coordinates of an aircraft and the parameters of its 
speed along the axes of coordinates of a chosen frame of reference 
are measured with definite errors, it is natural that a given 
trajectory of aircraft movement will likewise be maintained with 
some errors . 

By accuraoy of aircraft navigation is meant the limits within 
which the errors of any flight-trajectory parameter are included 
with a definite probability. 

In contrast to the accuracy of navigational devices, which 
characterizes (in the majority of cases) the errors in measuring 
one coordinate or two aircraft coordinates simultaneously, the 
accuracy of aircraft navigation depends on the conditions of imple- 
menting indicated measurements and, in some cases, on the dynamics 
of aircraft flight. 

Let us assume that an aircraft is moving in a field of constant 
wind or under conditions of calm. The direction of flight is main- 
tained on the basis of results of measuring the lateral deviation 
of the aircraft (Z) from the line of the given path at designated 
points ( Fig . 6.1). 

Points A and B in the figure correspond to the actual coordi- 
nates of the aircraft, while points Ai and Bi correspond to measured 
coordinates . 

It is obvious that on the basis of results of measurements 
(^1 and Si), the aircraft crew does not obtain an accurate notion 
concerning the direction of movement, i.e. there is an error in 
determining the actual angle of flight Ai|; . 



/4if7 



470 



In general, errors in measuring the Z-coordinate (and, there- 
fore, ijj ) will exert the same influence on the accuracy of aircraft 
navigation with respect to direction, independently of whether the 
actual trajectory of aircraft movement will coincide with the given 
trajectory or whether it is situated at some slight angle to it. 



/iJ^S 



1 



Uj 



®— — — 

6 = 



B 






Rt 



Fig. 6.1 Diagram of the Occurrence of Errors in Aircraft Navigation 

with Respect to Direction. 

However, for simplicity of argument, we will consider that on 
segment AB the actual path line of the aircraft accidentally turned 
out to correspond strictly with the given line. In this case, angle 
h^ and coordinate B\ will be magnitudes of misinformation for the 
crew which, in their graphic form, determine errors in the crew's 
actions in the flight segment BC . 

Actually, a crew located at point B precisely on the given 
path will assume that the aircraft is located at point Si . There- 
fore , for an approach to point C it will be obliged to make an ad- 
vance in the course: 



Ay = arctg ^ 



BC 



In addition, the crew will assume that an aircraft on segment 
AB did not travel parallel to the given path line, but at an angle 
Aijj , equal to 



A'^; — a ret J 



AA^ + BB 



AB 



Therefore , the total incorrect advance in the course 

^ytotal = Ai2 = Aij + A-;. 

Therefore, if the distance BC is approximately equal to AB , 
the aircraft must go not to point C but to point Ci, situated the 
following distance from point C : 

CC2 = AAi + 2BBi, 



471 



where AAi = AZi; BBi = AZ2. 

Under actual flight conditions, it is difficult to expect that 
the wind in segment BC will be the same as in segment AB . There- 
fore, if a flight is made over BC by maintaining the condition se- 
lected in segment AB , the aircraft will not appear at point C2 , but 
at point C3 , displaced from point C2 by the value of the change in 
the wind vector in segment BC with respect to segment AB relative 
to the flying time BC . 

For aircraft navigation with respect to d_lrection, only the 
lateral component of the wind change vector AU ^ will have any sig- 
nificance. Thus, the general error in aircraft navigation with 
respect to direction in segment BC is: 



AZBc = 'i^i+2A^2+.A//^^ 



/^U9 



(6.1) 



It is possible to come to an analogous conclusion by examining 
the accuracy of aircraft navigation with respect to distance if the 
condition of speed is chosen on the basis of results of measuring 
the J-coordinate at points A and B: 



Aj^Bc = AXi + 2AX2 + M-fj. 



(6.2) 



Formulas (6.1) and (6,2) determine the absolute errors in air- 
craft navigation with respection to direction and distance. In 
these formulas, only the third term on the right-hand side (hU^t 
and AU^t) is a value which depends on the length of the stage of the 
path and therefore, on flight time. Therefore, the absolute error 
grows smoothly with an increase in the length of the stage in the 
path between the control points , 

The ratio of the absolute error of a given parameter to the 
length of the stage in the path of the aircraft in which this error 
arises is called the relative error of aircraft navigation. There- 
fore, the relative error exerts an influence on the stability of 
the flight conditions of the aircraft. Let us illustrate this with 
a specific example. 

Let us assume that at the control stage in the path of an air- 
craft, with a length of 200 km, an error of aircraft navigation of 
5 km in distance and 4 km in direction has accumulated. 

The relative error in aircraft navigation with respect to dis- 
tance and direction will be: 



\x 





X " 


200 ~ 


AZ 


4 



40 



■=-2,5»/o; 



200 



50 



472 



The relative error with respect to direction characterizes the 
conditional errors of aircraft navigation: 



A(!; = arctgf 



AZ 
X 



In the following stage of flight of equal length (200 km) in 
order to balance the errors of aircraft navigation which were accumu- 
lated in the preceding stage, it is necessary: 

(a) to introduce a correction in the aircraft course equal to /U5 0| 
the error Aijj , in our case arctg 1/50 ~ 1° , and I 

(b) to change the airspeed, in our example by 2.5%. 

Let us assume now that the same error in aircraft navigation 
arose at a stage in the path about 50 km long. Then 



a;!' 


5 


X ~ 


50., ' 


AZ 


4 


" X ~ 


50 



= 10%; 



4°. 



In this case it would be necessary for us to change the air- 
craft course by 4° and the airspeed by 10% for every 50 km of the 
path, i.e. in modern aircraft, every 3-4 min of flight. 

Considering that the error in aircraft navigation Increases 
with respect to time only as a result of a change in the wind vector, 
it becomes entirely obvious that it is advantageous to choose 
control stages of flight which are very long, both from the point 
of view of the frequency of introducing corrections in the aircraft 
flight condition and in the values of the corrections being intro- 
duced . 




The necessary accuracy of aircraft navigation with respect to 
direction of the flight path is determined by thq set width of air 
routes and approach paths to airports, as well as national 



473 



boundaries . 

However, it is necessary to consider that at turning points on 
the paths, with significant turn angles for the route, the errors 
of aircraft navigation with respect to distance become errors with 
respect to direction, and vice versa. 

The accuracy of aircraft navigation during the approach of an 
aircraft landing on instruments acquires a special significance. 
The necessary length of the path of an aircraft's approach to a 
given trajectory, after changing to visual flight, depends on the 
magnitude of the aircraft's deviation from the given descent trajec- 
tory during an instrument approach for landing, and therefore on 
the weather conditions during which a landing can be made. /451 

With automatic or semiautomatic approach to landing by air- 
craft up to low altitudes (for example, up to leveling off .or landing) 
the accuracy of aircraft navigation must be such that the landing 
of the aircraft in all cases will be ensured with the execution of 
safe deviation norms with respect to the landing position and 
direction of the aircraft vector in the path. 

2. Methods of Evaluating the Accuracy of Aircraft Navigation 

In special books on the study of the accuracy of aircraft navi- 
gation with the application of navigational systems, the methods of 
probability theory (Laws of the distribution of random variables) 
are used . 

To evaluate the accuracy of aircraft navigation under practical 
conditions, it is sufficient to use only the basic conclusions of 
probability theory. Since the study of probability theory as a 
science is not the purpose of this textbook, in the majority of 
cases these conclusions will be given without proofs. 

In probability theory, variables which cannot be determined 

in advance by classical methods of mathematics, or are determined 

by methods so complex that they cannot be used for practical pur- 
poses, are considered to be random variables. 

In connection with problems of the accuracy of aircraft navi- 
gation or the accuracy of measuring aircraft coordinates by means 
of navigational systems, the errors in measuring or maintaining 
some of the navigational parameters will be random variables. 

Let us assume that the value of some navigational flight param- 
eter (on the basis of some especially precise control device) is 
known exactly. However, in carrying out a number of measurements 
by the usual means, we always obtain new values for the parameter 
which differ from its precise value . 

The precise value of a measured parameter will be called its 



474 



mathematical expectation. If a series of measurements is suffi- 
ciently great, then in all probability we will obtain many values 
for the measured parameter, with both positive and negative errors. 
Here the mean arithmetic value of all the measurements will ap- 
proach (depending on the increase in their number) the mathematical 
expectation of the measured value. Therefore, to raise the accuracy 
of aircraft navigation, in many cases measurements are carried out 
repeatedly and the arithmetic mean of the series of measurements is 
found . 

The arithmetic mean of a measured parameter cannot characterize /HS 2 
the probable accuracy of carrying out individual measurements. 
Therefore, probability theory includes a concept of mean square 
deviation from the precise value. 

Let us designate the precise value of a measured quantity by 
a, and its measured values by X^, where i = 1 , 2 , 3 . . . 



Let us call the value (a;. 



a) the measuvement ervov. 



The value obtained by extracting the square root from the sum 
of the squares of the errors divided by the number of measurements 
is considered the mean square error of measurement: 




l-n 

2] (-</ ~ a)i 



(6.3) 



According to (6.3), the mean square error of measurement is 
determined when the precise value of magnitude a is known. 

If the value of the measured magnitude is determined as an 
arithmetic mean from a series of observations, it is considered 
that one of the measured magnitudes coincides with or very closely 
approaches the arithmetic mean. The error of this measurement is 
considered to be zero, resulting in an increase in the sum in the 
numerator under the root of (6.3) equal to zero. Therefore, in 
order to avoid decreasing the value of the mean square error, 
especially with a short series of measurements, the denominator of 
(6.3) reduces to 1. Then this formula assumes the form: 




n— 1 



(6.4) 



The mean square error characterizes the accuracy of the meas- 
urements in a rather definite way. With the raising of each of the 
errors to a square, its sign always becomes positive. Therefore, 



475 



in determining mean square errors , only the absolute value of each 
plays a role . 



It is considered that the mean square error does not have a 



sign, 



If we examine only one of a series of measurements, with a 
probability equal to 1 (complete probability), it is possible to 
say that the magnitude being measured will undoubtedly have some 
value. However the probability that the magnitude being measured 
will have a strict and absolutely precise value is practically 
equal to zero, except in cases when it can assume only a. discrete 
value. Therefore, in determining the probability of an error of 
measurement it is not the precise value of the error, but the limits /^5: 
in which it must be found, which are given (for example, the proba- 
bility of error in the range from 500-600 m or from 2 to 2.5 km, 
etc . ) . 

All the measured navigational magnitudes are (to a certain 
degree) calibrated magnitudes, i.e., they have errors limited by 
certain boundaries. These boundaries depend on the allowances in 
the regulation of the measuring apparatus and on the maximum pos- 
sible distortions of the measured magnitudes as a result of the in- 
fluence of external factors (electromagnetic wave propagation, the 
physical composition of the airspace, variations in the Earth's 
magnetic field, etc.). 

Allowances in the regulation of measuring apparatus are known 
quantities. Century-old observations permit the determination of 
the limit of change in the parameters of the environment. There 
are ways of evaluating the maximum influence and other factors on 
the accuracy of measurements. Therefore, it is always possible to 
predetermine the maximum errors of some kind of measurements . 

The quantitative characteristics of the distribution of errors 
from their zero to maximum values, in the majority of cases, are 
subject to the normal law of random variable distribution. 

If in some cases the law of error distribution is not normal, 
it will be close in any case. 

Considering that devices of probability theory are used not 
in calculating measurement errors, but only in evaluating limits 
and the probability of possible measurement errors within these 
limits 5 it is considered permissible in all cases to use the 
normal law of distribution of random variables. 

The normal law of random variable distribution (Gauss formula) 
characterizes the probability density of a random variable, in our 
case of the measurement errors (x - a), depending on its value: 



476 



f(x — a) = 



(x-ay 



(6.5) 



21/2^ 



where ^(x - a) is the probability density of errors of a given 
magnitude, a is the mean square error of a series of measurements, 
e is a Napier number equal to 2.71828, and a is the precise value 
of the magnitude being measured. 

It is obvious that the probability of finding the result of 
measuring (x) in the range of values from a to x can be determined 
by integrating (6.5) over x : 



X 

'{x-a)'= ' r 

K2.. J 



-{x-ay 



dx. 



(6.6) 



The graph of the probability of random variables subordinate 
to the normal distribution law is shown in Figure 6.2. 



/454 



The curve on the graph shows the probability density of random 
variable deviations from zero to maximum positive and negative 
values. The left side of the graph corresponds to errors with a 
negative sign, the right to errors with a positive sign. 

Since the absolute probability of obtaining any value of the 
measured magnitude is equal to one, the probability that the value 
of the magnitude will be negative or positive is 0.5. 

Let us note that on the abcissa of the graph there are two 
values of a random variable, Xi and X2- The area bounded by the 
segment a:ia;2, by the ordinates Pxi? ^x2 ' ^^'^ ^^ tY^e curve is the 
probability of finding the result of measurement in the limits be- 
tween X I and X2 • 



iP[y2-a}-'P(yro) 



(X-a)<0^^ 




h ^1 



— Ix-a)> 



Fig. 6.2 Graph of the Proba- 
bility of Random Variables Under 
the Normal Law of Distribution. 



With the convergence of 
points xi and X2 at one point, 
the probability of finding an 
error of measurement between 
these points will diminish and 
converge to zero. 

The analogous problem for 
determining probability can be 
solved on the basis of the 
right side of the graph for 
errors of measurement which 
have a positive sign. 



477 



* without stopping at the methods of solving an integral (6.6), 
let us indicate that the overall probability of finding positive 
and negative errors of measurements is 68.3% in the range from to 
a, 95% from to 2a, and 99.7% from to 3a. A table of values of 
the function $(a; - a) for ix ~ a) from to 5a is given in Supple- 
ment 6 . 

For example, if the mean square error of measuring the drift 
angle with a Doppler meter is equal to 15 ' , then with a probability 
of 95% it is possible to expect that the measurement error will not 
exceed 30', and with a practically complete probability (99.7%), 45'. 

The value of the mean square error of measuring a given kind 
of parameter permits evaluation of the accuracy of other parameters 
which have a functional dependence on the first. 

Example : The mean square error of the direction-finding of an 
aircraft by means of a ground direction finder a = 1° . Determine 
the limits of linear error in determining the lateral deviation of 
an aircraft from the line of a given path with a probability of 95% 
if the aircraft is located at a distance of 300 km from the direc- 
tion finder . 

Solution: with a probability of 95%, the angular error of a 
direction finder in measuring does not exceed 2°. 

Therefore, AZ(P = 95%) = 300 tg 2° fi^ 10 km. Solving the same /455 
problem for a practical probability of 100% (more precisely, 99.7%), 
AZ(P = 100%) = 300 tg 3° R. 15 km. 

Let us now assume that we must solve the reverse problem, i.e. 
determine the necessary accuracy of a direction finder which en- 
sures the given accuracy of measurements of lateral deviations. 

Example : ^^man^'^ ~ 100^) = 10 km. Determine the necessary 
accuracy of a direction finder for distances up to 300 km. 

Solution: 3a^ = arctg TqK ^ 2°. Therefore, a^ = 0.7°. 

3. Linear and Two-Dimensi onal Problems of Probability Theory 

The normal law of random variable distribution examined in the 
preceding paragraph includes the linear ( one -dimensional ) problem 
of probability theory for one parameter of measurement. 

In aircraft navigation, it is often necessary to deal with 
several measurement parameters. For example, in calculating the 
path of an aircraft with respect to direction by automatic navi- 
gational devices, on the basis of results of measuring the drift 
angle and groundspeed of the aircraft with a Doppler meter, the 
following errors exert an influence on the accuracy of calculating 
this parameter: errors in calculating the given flight angle; 

478 



errors in measuring the course, drift angle, and groundspeed; 
errors in the operation of an integrating device. 

Each of these factors separately will create the following 
error components in calculating the path with respect to direction; 



AZ^ = X sin A.^ = Wt sin Ai; 

AZ^ = Wt sin Af ; 

AZ„-^ U^isinAa; • 

AZ^ = AlWsin(^^-<;;3); 

AZ j. = WtS^l' 



If the indicated components had the same sign and had a maxi- 
mum value within the calibration limits of each of the parameters, 
the general error would be equal to the arithmetic sum of these 
components. However, according to the law of normal random variable 
distribution, even when measuring one parameter, the maximum error 
is encountered rather rarely. The probability that all the errors /45 6, 
will take on a maximum value, and even one sign, will be extra- 
ordinarily low. 

In spite of the fact that we must deal simultaneously with 
many measured parameters, the solution of the above example includes 
a linear problem of probability theory, since the random variables 
are summed along one axis of the chosen frame of reference of their 
coordinates , 

To solve similar problems, the concept of the dispersion of 
random variables a^ is introduced into probability theory. 

It is known that the law of random variable distribution, ob- 
tained by adding other random variables which are subject to the 
normal distribution law, is also a normal distribution law. Here, 
the scatter of an overall random variable is the sum of the scatters 
of the values being added. 



In our example of calculating the path of an aircraft by means 



of automatic navigational devices, the value 
the sum, 



is the scatter of 



Here , 



°z = '22^ + =2Z^ + rfz„ + =2Z + ,2Z^. 



(6.7) 



The mean square error of the measurement is equal to the square 
root of the scatter: a = /a-^ . Therefore, the mean square error of 
the total value will equal the square root of the sum of the scat- 
ters. For our example. 



479 



•j/"a7Z^ + o2Z^ +a2Za+G2Z. + a2Z« 



(6.8) 



The value a^ in (6.8) is a small second-order value: 

Therefore, it is necessary to disregard this value. 

Let us assume that the remaining values included in (6.8) have 
been mean square errors as follows : 



o, =20'; a =20'; o„ = 15'; o. =0,5%o£x. 



Since the first three values are small, their sinces can be 
replaced by angle values. Then, considering 1° equal to 0.017 by 
1.7% X, their value can be expressed in percent of the distance 
traversed : 

a^ = 0,56%X; a =0,56°/o^; 5^ = 0,42%;?; op =0,5°/oA-, 

where X = Wt . 

Therefore , the mean square error in calculating the path with /457 
respect to direction is 

0^ = WtV0,5& + 0,56? + 0,422 + 0,52 = U^f VT^ = 1 ,02%X. 

Hence, it is possible to consider that the mean square error 
in calculating the path with respect to direction amounts to ~ 1% 
of the distance traversed. 

Let us assume that we have set ourselves the goal of maintaining 
an aircraft within the limits of an air route with a width of 20 km 
(up to 10 km from LGP ) with a probability of 95%. Here the mean 
square error in determining the initial coordinates of the aircraft 
equals 2 km. 

For a probability of 95%, the error in the initial formulation 
of the aircraft's coordinates must be taken as 4- km, while the 
accuracy of calculating the path with respect to direction must be 
taken as 2%. The maximum error in calculating the path with respect 
to distance must not exceed 



A^niav = 1/102 — 42 = 1/84 «9'k 



m 



'max; 
The value 9 km must amount to 2% of the distance covered, 



l^80 



J 



Therefore , the allowable length of the stage of the path between 
the control points (S) must be not more than 



0,02 



= 450kn> 



If we set ourselves the goal of maintaining an aircraft within 
the limits of a route with a probability of 99.7%, the accuracy of 
the initial display of coordinates and the calculation of the path 
of the aircraft would have to be taken. as 3a or a reading accuracy 
equal to 6 km and an accuracy for calculating the path equal to 
3% X. Then 



AZ 



max 



:y 102 — 62 = yc4 = 8 kr 



8,km=3^oX; X=--- 



0,03 



266 km 



If we take the limits of calibrating each parameter as 3a, 
then by adding the errors on the basis of the calibration rules we 
would obtain the value 



A2max=3'+ + ^ + 3.„ + 3.j.^ 



or, in our example, 

AZ^g^:- 1,7 +1,7 +1,5+ 1,2 = 6,1%, 

i.e., in the case when all errors have a maximum value and the same 
sign, the error of calculating the path can reach 6%. Since it 
reaches 2% with a probability of 95%, with a practically complete 
probability of 99.7% it reaches 3%. 

The probability that calculating the path will occur with 
errors within the range of an overall calibration of the system is 
expressed by in hundred mlllionths of a percent. Therefore, when 
there is no threat of disturbing the safety of a flight, it is not 
necessary for practical purposes to take the limits of overall 
calibration into consideration. 



/458 




Fig. 6.3. Diagram of the 
Occurrence of Errors in 
Determining the Position of 
an Aircraft: (a) with Multi- 
lateral Errors in Rearings ; 
(b) With Unilateral Errors. 



f+Sl 



In the majority of cases, there is sufficient error in calcu- 
lating the path to calculate with a probability of 95%, and only in 
especially responsible cases, with 99.7%. 

The majority of problems in determining the accuracy of air- 
craft navigation or measuring its separate parameters with the use 
of some method reduces directly to linear problems pf probability 
theory. The final result of solving all the problems of aircraft 
navigation must be one-dimensional, since the goals and requirements 
for accuracy of aircraft navigation with respect to distance, 
direction, and flight altitude are different. 

At the present time, there are no navigational systems which 
determine the position of an aircraft in three-dimensional space. 
Therefore, the necessity for solving volumetric problems in proba- 
bility theory is superfluous. However, a number of navigational 
systems such as a hyperbolic, two-pole goniometer, or goniometric 
rangefinder (if only the location of a ground beacon is known for 
the LGP of a given segment), permit the solution of a problem in 
determining an aircraft's coordinates in two-dimensional space. 

An evaluation of the accuracy of aircraft navigation along 
each of the axes of the coordinate system chosen for aircraft navi- 
gation, in this case, can be carried out only after solving a one- 
dimensional problem in probability theory. 

Let us assume that we have an oblique-angled surface system of 
aircraft position lines, each of which does not coincide with the 
given flight path (Fig. 6.3, a,b). 

The linear error in determining the first (ri) and second (^2) 
aircraft position lines depends on both the accuracy of measuring 
the navigational parameter and its gradient. 

The gvadient of a navigational pavameteT is the ratio of its /M-5^ 
increase to the movement of an aircraft in a direction perpendicular 
to the position lines of the operating region of the system 



g 



da 
dr 



(6.9) 



where g is the gradient of the navigational parameter, and a is the 
navigational parameter being measured. 

For example, if the navigational system is a goniometer, then 



s = ■ 



da 
'dr'. 



dA__ 
dr 



_1_ 
5 • 



where A is the azimuth of the aircraft and S is the distance from 



482 



the ground beacon to point PA. In this case 



dr == 



da 



■■ daS or r = ^AS. 



With the introduction of the concept of the gradient of a 
navigational parameter, all the existing coordinate systems reduce 
to a generalized system, i.e., the problems of determining the 
accuracy of navigational measurements are solved on the basis of a 
general scheme, independent of the geometry of application of the 
navigational device. 

In Fig. 6.3 a,b two possible cases of the appearance of errors 
in determining the position of an aircraft on the basis of the 
intersection of position lines are shown: 

(a) Errors in r^ and ^2 have different signs; in this case, 
the measured position of the aircraft lies in an acute angle between 
the actual position lines. This leads to larger errors of deter- 
mination . 

(b) Errors in r i and r2 have identical signs; the measured 
position of the aircraft lies in an obtuse angle between the position 
lines. The errors in determining the position of the aircraft in 
this case are close to the linear errors of one bearing. 

It is necessary to note that the probability of errors in ri 
and T2 with the same sign in the majority of cases is more than the 
probability of errors with different signs. For example, in taking 
bearings with a radio compass, the error component as a result of 
an error in measuring the aircraft course will be general for two 
measurements. If the angle between the bearings is sufficiently 
acute, the radio deviation will have either one sign or different 
signs, but a small value in any case. 

A similar relationship between measurement errors in probability 
theory is called correlation (p). 

The general error in determining the position of the aircraft 
in our case will be (Fig. 6.1+): 



/-2 = 



'1 



^2 I 



+ 



sin2w sin2m 



2rir2 cos 10 
sin2 CO 



]/ • /-f + /^ — 2/'i/-2 cos ci) 



(6.10) 



Formula (6.10) characterizes only the magnitude of error in 



/460 



483 



1 



determining the position of an aircraft on the basis of two position 
lines with known errors in measurement for each of them. However, 
it does not give us an idea of the nature of the distribution of 
the indicated errors around the point of the actual position of the 
aircraft (center of scatter). 

In contrast to a linear problem, where the probability of an 
overall error in several measurements is examined, in a two-dimen- 
sional problem it is- necessary to examine the products of the 
probabilities of these errors. 

For simplicity of argument, let us assume that we have a 
rectangular coordinate system (Fig. 6.5); let us set ourselves the 
goal of limiting the area within which the aircraft is located with 
a probability of 95%. Here, the mean square errors of measuring 
the two position lines will be considered identical. 

Let us examine a certain large number of measurements (e.g. , 
10,000) and let us see what will be the probability that the 
measured position of the aircraft will be in an exterior angle at 
a distance from the center of the area of scatter which exceeds 
the diagonal of a square constructed with errors 2a, and 202- 

Since the probability of an error in the first measurement 
exceeding 2a equals 5%, then 500 of 10,000 measurements must be be- 
yond the limits of a side of the indicated square. The remaining 
9,500 measurements lie in the range from zero to 2a, and there is 
no need to calculate them during common measurements with the 
second position line . 

It is obvious that of the 500 remaining measurements, where an 
aircraft will be located a distance more than 2a, from the first 
position line, the errors in the second bearing will exceed the 
value 2a2 only in 5% of the cases, and thus there will be only 25 
cases (or 0.25%) simultaneously exceeding the errors of the values /H61 
2a , and 2a2 . 





Fig. 6.^. Total Error in 
Determining the Position of 
an Aircraft . 



Fig. 6.5. Probability of the 
Simultaneous Yield of Errors Be- 
yond the Limits of the Given Values 



484 



The example examined shows clearly that the probability density 
of errors directed toward the indicated angle diminishes sharply. 
It is practically possible to consider that a large number of the 
common measurements of the first and second position lines lie in 
a circle, the radius of which equals 2ai = 202, while the limit of 
equal probability of deviations from the center of scattering will 
be a circle. In general, when the errors in the first bearing are 
not equal to the errors in the second bearing, this boundary has 
the shape of an ellipse. 

If we examine a number of cases in which an aircraft is within 
the limits of an ellipse with axes equal to 2a, it turns out to be 
significantly less than 95%, since even in rectangles constructed 
with sides equal to 2a there will be 95% of 95%, or 90.025%. 

However, this is of value only when the probability of an 
aircraft's entering a given area is examined. From the point of 
view of aircraft navigation, it is not the location of an aircraft 
in a given area, but the deviation from the given path trajectory 
and the retaining of flight distance with respect to time which 
play a role. Therefore, the results of adding the probabilities 
are again distributed according to direction. This again raises 
the probability of each of them to 95% (Fig. 6.6). 



In the figure, an ellips 
certain position relative to 
shown. In the above case, th 
coordinates of an aircraft wi 
by tangents, parallels, and p 
path (the orthodrome coordina 
of the tangents from point PA 
lation dependence between the 
with respect to distance and 
dimensions, and orientation o 
lation dependence plays a rol 
when errors of aircraft navig 
nitely become errors of aircr 



e of measurement errors , located in a 
the path line of an aircraft, is 
e possible errors in measuring the 
th a given probability are determined 
erpendiculars to the line of the given 
tes are kept in mind). The distance 
, as well as the maintenance of corre- 
errors in the maintenance of the path 
direction, will depend on the shape 
f the ellipse of errors. The corre- 
e only at turning points in the route, 
ation with respect to distance defi- 
aft navigation with respect to direc- 
tion, and vice versa. 




*aX 



-&X 



Fig. 6.6. Ellipse of Errors 
in the Aircraft Position. 



The physical sense of the 
above arguments becomes clear if 
we assume that the ellipse of er- 
rors constructed from mean square 
measurement errors can be arranged 
with the major axis both in the 
direction of the path line and 
perpendicular to it. Then the 
accuracy of aircraft navigation 
with respect to distance and 
direction can be determined by 
mean square values of the axis of 
the ellipse. Obviously the 



/U62 



U85 



orientation of the axis of the ellipse at an angle to the line of 
the given path occupies an intermediate position between those in 
the figures. The accuracy of maintenance of the path along the 
axes of the coordinates in this case is determined in the same way 
as for the case shown in (Fig. 6.7 a,b). However, a correlation 
dependence between these measurements will be seen. 

In the above examples of a circle and ellipse of errors, we 
assumed that the aircraft position lines were situated at right 
angles to one another, although an angle to the given path line is 
possible, and that the correlation dependence between the measure- 
ments of the position lines is absent. 

Let us now present, without derivation, the formulas which can 
be used as a basis for determining the dimensions and orientation 
of the axes of an ellipse for a situation when the position lines 
are situated at an angle (not a right angle) with an independent 
accuracy of measurement for each of them (correlation coefficient 
equal to zero ) : 



a2 == c2 '^'■^°'' + K (°;.+<)-4.^ . .1 sin2 0. 



A2=c2 



•^sin-'u) 



'2sin2u) 
tg2a=- 



2 sin2 w 



<+:< /(=^.+o^,)2-4.;,o^^sin2. 



2sin2(o 



5? sin 2a) 
'1 



0, C0s2o: 
'I 



(6.11) 



(6.12) 



where a is the major semiaxls of the ellipse; h is the minor semi- 
axis of the ellipse; u is the angle between the position lines; a 
is the parameter of the ellipse chosen for a given probability; and 
a is the angle between the bisector of the angle of intersection of 
the position lines and the major axis of the ellipse (it is plotted 
in the direction of the position line with the smallest error). 

These formulas are used in the majority of cases, since the 
accuracy of determining position lines is usually independent or 
the correlation coefficient is unknown. 



Only in individual cases, e.g., in determining position lines 
with an aircraft radio compass, is the correlation coefficient 
determined rather simply, since part of the error of measuring the 
bearing (depending on the aircraft course) will be general. For 



example 



' course 



= 3°; 



= 2° 



The general mean square error in measuring a bearing will be : 



1+86 



iA 



= K^ 



+ ''coursei=1^13: 



;3,C'. 



In measuring two bearings on one aircraft course , it is neces- 
sary to expect that both errors will be shifted in one direction by 
the mean square error of measuring the aircraft course, in our case 
by 2°, which amounts to 0.55 of the total error. Therefore, the 
correlation coefficient p = 0.55. 



As shown above (see Fig. 6.3), the presence of a total compo- 
nent in the errors of measuring position lines raises the accuracy 
of determining the position of an aircraft. According to (6.10), 
since the third term under the radical in the numerator can be 
given both positive and negative values, for independent measure- 
ments it is possible to write 



/463 



v 



V^. + =?, 



(6.13) 



With dependent measurements, the third term under the radical 
must be multiplied by the correlation coefficient p and (6.13) takes 
the form : 



V 






(6.14) 



With a correlation coefficient equal to zero, or with angle 
u equal to 90°, (6.11+) is transformed into (6.13). 

In the presence of correlation, the dimensions and orientation 
of the axes of the ellipse are determined according to the formulas 



tg2a: 



a2=.c2 



2pG^^ a^^ sin 0) — 0^ sin 2u 



°r, + °?, t^OS^ « — Spo^^a^, COS < 



+ 



2sin2 (0 

V'(°?. + °r,-2f>ar.°r.COS0)p-4(l - p2) g^^ o^^ sin2 , 

2sin2u 



(6.15) 



(6.16) 



The length of the minor axis of an ellipse is also determined 
on the basis of (6.16), with replacement of the positive sign in 
front of the radical by a negative sign. 



487 



In probability theory, the probability of locating an object 
within the limits of the indicated ellipse of errors (given definite 
values of the magnitude e) is examined. 

Since we have agreed to examine the accuracy of aircraft navi- 
gation separately with respect to distance and direction, this prob- 
lem will not interest us. We are using the ellipse for an evalua- 
tion of the accuracy of aircraft navigation with respect to both 
distance and direction. 

Let us assume that we know the orientation of an aircraft's 
position lines and a given path line on a map, and have determined, 
on the basis of (6.11) and (6.12) or (6.15) and (6.16), the lengths 
of the axes of the ellipse 2a and 2Z> , as well as the orientation 
of the major axis of the ellipse with parameter c = 1 (mean square 
ellipse ) . 

In this case, the mean square error in the maintenance of the 
path with respect to distance and direction is determined by tangents 
to the ellipse at points Xq ^^^ ^Oj perpendicular to the path line, 
and at points X\ and Z^, parallel to it (Fig. 6.7), 



It is possible to show that the mean square errors in main- 
taining the path (with respect to distance a^ and direction a^ ) in 
this case are 



/ a'^ cos2 g r 



/■ 



a' sinSa 



a2+, A2ctg2a 



+ 



b^ sin2a 



a2 ctg2 a + A2 



/■ 



b^ cos2a 



a2 ctg2 a + 62 



(6.17) 



where a is the angle between the line of the given path and the 
major axis of the ellipse. 

In Fig. 6.7, it is obvious that in a general case, when the /M-64 
major axis of an ellipse does not coincide with the given path line 
or the line perpendicular to it, there is a correlation dependence 
between the errors in the maintenance of the path with respect to 
distance and direction. 

Actually, if we have a positive error in determining the X 
coordinate, the measured position of the aircraft is located in the 
right-hand side of the ellipse. The mathematical expectation of 
the value of the Z-coordinate in this case will be found in the 
middle of the chord of the ellipse, parallel to OZ and intersecting 
the J-axis at a point corresponding to AX. 

The diameter of the ellipse, dividing its chords (which are 



488 



perpendicular to some other diameter) in half, is called the 
oonQugate diametev . 

The direction of the conju- 
gate diameter is determined ac- 
cording to the formula 




•-jr 



Fig. 6.7. Correlation Depend- 
ence of the Errors in the Con- 
trol Path with Respect to Dis- 
tance and Direction. 

fi-oients of regression. 



tgP = 



62 



a (tg90 — a) 



(6.18) 



In probability theory, lines 
which determine the dependence 
between random variables are 
similar to the conjugate diameters 
of an ellipse of errors. In the 
case described by us, they are 
called regression tines, while 
the angular coefficients of these 
lines (tangents of the angles to 
the axes of the frame of refer- 
ence) are called angular aoef- 



In our case, the direction of the conjugate diameter connecting 
the errors of measurement of the Z-coordinate with errors for the 
Z-coordinate is determined on the basis of (6.18), where a is the 
rotation angle of the major axis of the ellipse relative to the X- 
axis . 

The direction of the second conjugate diameter, which connects 
the measurement errors of the X-coordinate with the errors of the 
Z-coordinate, is determined according to the formula 



tg h = 



a'i tg a 



(6 .19) 



Here, the angular coefficients of regression will be 



^^of,-^ = tg(a + Pi). 



(6.20) 



489 



4. Combination of Methods of Mathematical Analysis and /465 
Mathematical Statistics in Evaluating the Accuracy of 
Navigational Measurements 

In the preceding paragraphs , we examined methods of mathematical 
statistics (probability theory) used to evaluate the' accuracy of 
navigational measurements. As a group, these devices permit the 
solution of any two-dimensional or linear problem encountered in 
aircraft navigation. 

However, in examining the methods of probability theory, we 
assumed that the accuracy (in general, mean square error) of 
measurements of separate parameters was known. 

Actually, the accuracy of measurements of navigational parame- 
ters has a functional dependence on other physical or geometric 
values connected with the principles of measuring or determining a 
navigational parameter. 

Since this functional dependence is always known, the simplest 
(and presently most universal) method of determining the accuracy 
of a navigational parameter is that of the variation of independent 
variables included in the equations of formulas which determine a 
navigational parameter within the limits in which the indicated 
variations are encountered in the practice of aircraft navigation. 

For example , the basic equation for an orthodrome which deter- 
mines its shift in a geographic coordinate system has the form: 

ctgXoi = tg"f2Ctg¥i cosec AX — ctgAX. 

Let us determine the accuracy of solving the above equation, 
assuming that the measurement accuracy of each of the parameters 
included in the equation is known. 

The dependence of the accuracy of the solution of the equation 
on the accuracy of measuring the coordinates of <^ 2 i^ expressed by 
the equation: 

'^^ctgXpi _ aT (tg <p2 ctg cpi cosecAX — ctg AX) 
d<f2 ~ d<f2 ' -~ =ctgcficosecAXsec2cf,2. (5.21) 

The final result of solving (6.21) would have to be viewed as 
an arc tangent of the right-hand side. However, from the point of 
view of a mathematical solution, this would lead to a significant 
complication of the given problem. It is advisable to use the 
following method: 

rfctgXpi ^_rfctgXoi_ riXpi 

rftf2 rfXpi d'-f2 



1+90 



■ ■ ■IIIIHIH 



or ^^l£^=_cosec2Xo:. 



Therefore, jftp^ ^ _ ^g^i cosec AX sec2v-^ ^ 

rftf2 cosec2 Xfli (, b . ^ .i ; 



Thus, if the mean square error In measuring the <|) 2 coordinate 
equals a(l>2, it causes an error in determining Xqi: 

, ctgip] c osec AX sec2 <p2 

'^01,^ — \, '—l-;;^^ (6.23) 

The dependence of the accuracy in determing Agi on the accuracy /M-SS 
of coordinate (j) 1 can be obtained analogously: 



01. = =¥1 .....,>_. ; (6.24) 



tg tf2 cosec AX cosec2 (fj 
cosec2 X(,i 
For the parameter AA 

,, tg 92 ctg vi ctg AX cosec AX — cosecZ AX 

<'^nl.^ = "^^ ' ' ' ol ~ (.6.25; 

"'AX cosec2 Xoi 



The total error in solving the equation will be: 

aXo = ]/^o2Xoi^^+'a2Xo,^ + a2Xo,^^ . 

Since we have examined as an example the accuracy of solving 
the basic equation for an orthodrome , it is appropriate to examine 
the accuracy of solving all the special equations which determine 
its parameters: 

(a) Initial azimuth of an orthodrome: 

sinXn; . . 

tg ao = — = sin Xo; ctg ?,■; 

tgtfi 

Using this method of transition from the arc tangent of the angle 
to its value, as in the preceding example, we obtain: 



4-91 



oOOx "" ^'^°^ ^' '^'S 9i C0S2 Oo) aXo,-; 



01 



ooo = (sin Xjjj cosec2 cpj-x;os2 Oq) 99/; (6.26) 



"9i 



(b) The moving azimuth of an orthodrome 



tgar- 



sinifi 



ca,. =— (ctg <fi cosec <fi tg Xq/ cos2 a,) of/; 

' (6.27) 

oa- = (sec2 Xq,- cosec 9/ cos2 a,-) uXq/; 
''0/ 

"tt; = ya^ai^^ + a2a/x^^ ; 



(c) Coordinates of intermediate points 



sin Xq. 



iga.0 
"Vx = cos Xoi ctg Oo cos2 <paXo;; 

a^a^ = sin Xq/ cosec2 oq cos2 9000; 
"9 = V^^Ki '^''^'^''- '• 



(6.28) 



(d) Distance along the orthodrome from its source: 7^67 

cos 5/ = cos Xo; COS O;; 



'S,-^ = sin Xq; COS <p,- cosec 5,aXo/; 

'i 

aSi = -|/"o25 +025,. 



"Sj = cos Xo; sin 9/ cosec 5;C9,-; (6.29) 



The above formulas (6.23) to (6.29) have the following practi- 
cal significance. 

Let us assume that in solving the basic and special equations 
of an orthodrome, we use trigonometric tables to five decimal 
places or a computer with 18 binary digit bits (which are also 

492 



equivalent to 5 decimal places). In the first case, the error in 
the value of each independent variable will have a magnitude of 
from 0-5 units of the sixth sign, and (in the second case) from 
0-10 units of the sixth sign. Substituting the values of the 
possible errors of each independent variable into these formulas, 
we will obtain the possible errors in the solution of the equations. 

Thus, it is possible to determine the necessary accuracy of the 
tables (number of signs) or the computers (number of orders) for 
obtaining a satisfactory result in solving equations within the 
given value limits of the independent variables. 

Calculations show that for geographic latitudes from to 80°, 
while solving equations for an orthodrome , it is necessary to use 
tables with 6 decimal points or computers with 21-22 binary digit 
bits . 

5. Influence of the Geometry of a Navigational System on the 
Accuracy of Determining Aircraft Coordinates 

The accuracy of determining the coordinates of an aircraft by 
means of navigational systems depends both on the accuracy of 
measuring a navigational parameter and on the geometry of the navi- 
gational system being used. 

Means of solving one-dimensional problems of probability 
theory for a generalized oblique-angled coordinate system were 
examined above. The azimuth coordinate system was given as an 
example for determining the gradient of a navigational parameter. 

Since it is necessary to know the value and direction of the 
gradient vector (g) of the navigational parameter to solve problems 
in a generalized coordinate system, only the reduction of different 
coordinate systems to a generalized system is examined in this 
sect ion . 

Two-pole goniometric, two-pole circular and one-pole range- 
finding are most simply reduced to a generalized coordinate system. 

As has already been indicated, for an azlmuthal system at 
distances on the order of up to 3,000 km, we can consider 



^=-T 



(6.30) 



where S is the distance from the aircraft to the ground radio beacon 



For greater distances, we must consider the convergence of the / '^& i 
position lines as a result of the sphericity of the Earth's surface, 
and (6.30) assumes the form: 



493 



1 

^~ RslnS ' (6.31) 



where E is the radius of the Earth and 

dr = dAR sin S, 

where dr is the increase in linear error; dA is the increase in 
azimuth . 

The directions of the position lines in this case can be deter- 
mined as the moving azimuths of the orthodromes at a given point M, 
which Intersect foci of the systems ^4 1 and A2. 

We must take point M as the starting point of both orthodromes; 
in this case , 

c'g '•OH. = tg fA, ctg <f„ cosec AXi — ctg AXj; 
ctg ^OM, = tg 9a, '^'g ¥m cosec AXj — ctg AXj; 

tg^'OM. tgV 

tgai = — ■; tga2=- 



sin<f„ sincp„ 

Si = arccos (cos X^^^ cos fu) — arc cos (cos Xq cos 9^ ); 

52 = arc cos (cos X^^^ cos f^) — arc cos (cos Xq cos t(^^) . 



The problem of finding azimuths of the position lines for a 
two-pole circular system is solved analogously, with the sole dif- 
ference being that the vector-gradient will not be directed per- 
pendicular to the azimuths of the orthodromes, but along these 
orthodromes; accordingly, the formulas for determining the azimuths 
of the position lines take the form: 



ctg ai = 

CtgOo — 



'g^OM. 

sin <?„ 

'g'-OM. 

sinifM 



Since the density of circular position lines 1/g does not de- 
pend on distance, 

^=1: dr = dR K Ar = A/?, 
where Ai? is the error in measuring distance. 
^94 



In goniometric range-finding systems, the task of finding the 
density and position of the azimuthal position lines is solved in 
the same way as for goniometric systems. In the case of circular 
position lines at point M, their direction will differ from the 
azimuthal lines by 90°. 

The axes of the ellipses of errors in this case will coincide 
with the position lines. Here, at short distances from the focus 
of the system (without taking into account the convergence of the 
azimuthal position lines), the minor axis of the ellipse coincides 
with the position line which is determined most accurately (usually 
the circular line ) . At great distances , we must also consider the 
convergence of the azimuthal position lines according to (6.31). 

The problem of conversion to the generalized coordinate system /469 
from the hyperbolic or hyperbolic-elliptical system is somewhat 
more complicated. Let us use (1.74) for this purpose: 



cosX] 



cos 5i c os 2c — cos {S^ — 2a) 
sin Si sin 2c 



Developing cos{S\ - 2a), we can present this formula in the 
form (Fig. 6.8a): 



cos X, ~ 



cos 5i COS 2c — cos 5i cos 2a — sin S^ sin 2a 
sin 5i sin 2c 




Fig. 6.8. Determining Hyperbolic Position Lines: (a) Direction; 

(b ) Distance . 

The direction of the position lines at point M can be deter- 
mined after differentiating (1.74) on the basis of S : 



dXi cos Sj sin 2c (cos .S; cos 2c — sin Si cos 2a — sin Si sin 2a) 

dS^'^ 



sinZS'i sin2 2csinXi 
sin 5] sin 2c (sin Si cos 2e^ sin Si cos 2a — cos Sj sin 2a) 
sin2 5i sin2 2csinXi 



(6.32) 



495 



The hyperbolic position line intersects the azimuth line of 
point M , drawn from the focus Fi , at an angle 






(6.33) 



Let us determine the density of the hyperbolic position lines 
after differentiating (1.74) on the basis of the parameter and with 
a constant S (Fig. 6.8, b): 



fift] _ sin Si sin 2c (sin Si cos 2a — cos 5] sin 2a) 
da sin2Sisin2 2csinXi 



Then, 



dr 
da 



da 



-5 cos a. 



(6.31+) 
(5.35) 



For conversion a generalized coordinate system from a hyper- 
bolic-elliptical system, it is sufficient to solve the problem for 
hyperbolic position lines. 

The directions of the elliptical position lines are then easily 
determined as being normal to the hyperbolic ones. The density of 
the elliptical lines is constant for the whole area of the activity 
of the system, since these lines do not diverge. /470 

The reduction of special coordinate systems" of radio-engineer- 
ing devices to a generalized system permits the construction of 
ellipses of errors in determining the coordinates of an aircraft 
by means of these devices and the estimation of the maintenance of 
the aircraft path with respect to distance and direction. 

In some cases, the boundaries of the operational area of the 
system are designated, within the limits of which the dimensions of 
the axes of the ellipses of errors do not exceed the given values. 
The operational areas of the system can be constructed on the basis 
of other factors (for example, on the basis of the accepted accuracy 
of the maintenance of the path with respect to direction alone or 
with respect to distance alone. 

The examined means of evaluating the accuracy of the navi- 
gational measurements include, on the whole, radio-engineering 
devices. However, the accuracy of the calculationof an aircraft 
path by means of geotechnical means of aircraft navigation can be 
examined with an azimuthal range-finding system. The accuracy of 
determining aircraft coordinates by astronomical means can be 
examined by a circular system. 



496 



6. Evaluation of the Accuracy of Measuring a Navigational Parameter 

In the preceding paragraph, the effect of the geometry of a 
navigational system on the accuracy of determining aircraft coordi- 
nates (assuming that the accuracy of measuring a navigational param- 
eter is known) was examined. 

By measured navigational parameter of a system, we mean the 
value being measured at the output of a navigational device: azi- 
muth, course angle, distance, difference in distances, or sum of 
distances to objects on the ground. 

In addition to measured navigational parameters, there are 
parameters which are determined by calculating the path of an air- 
craft on the basis of its speed and time components; for example, 
the orthodrome or geographic coordinates of the aircraft. 

Let us examine briefly the reasons for errors and the methods 
of evaluating the accuracy of measurements or determinations of the 
indicated parameters. 

During visual aircraft navigation with the use of geotechnical 
devices and with the use of astronomical and nonaut onomous radio 
devices, the calculation of the aircraft path at each succeeding 
stage is carried out on the basis of the results of measuring the 
parameters of aircraft movement in the preceding stage . In this 
case, two factors will influence the accuracy of aircraft naviga- 
t ion : 



(a) The accuracy of juncture (determination of the location 
of the aircraft) at the beginning and end of the control stage of 
the path . 

(b) Wind variation at flight altitude from stage to stage. 

The accuracy of visual junctures depends on flight altitude 
and on methods of measuring both the vertical and course angles of 
reference points. 

Visual methods of aircraft navigation are usually used at low 
flight altitudes, in conjunction with closely spaced reference 
points. Therefore, the errors of juncture are very small, and do 
not have values comparable to the wind variation at flight altitude 




/471 



under different operating conditions. 

The accuracy of navigational junctures of an aircraft by means 
of navigational systems is evaluated by the means set forth in the 
preceding section, proceeding from the mean square error in measuring 
a navigational parameter, considering the geometry of the system in 
this area of application. 

The accuracy of measuring a navigational parameter by astronom- 
ical means (as a rule, the altitude of a star) is determined, in 
the first place, by the accuracy of the installation at the level 
from which the measurements are carried out . 

The bubble levels of aviation sextants, which are subject to 
constant acceleration under the effect of Coriolis forces , which 
can be taken into account in flight if the groundspeed is known, 
to a still greater degree, they can be subjected to varying acceler- 
ations by unstable flight conditions. 

Gyroscopic levels are also subject to Coriolis accelerations 
and also to long-period fluctuations in the shape of a precession 
cone, connected with the errors of balancing a gyroscope, where 
calculation is practically impossible. 

However, gyroscopic levels are free from the short-period 
interferences which are connected with disturbance of the aircraft's 
flight condition. 

If, as a result of the astronomical observations, we introduce 
corrections for the instrument error of the sextant, the refraction 
of the atmosphere and the Coriolis accelerations of the level, then 
the errors as a result of short-period fluctuations of the bubble 
level or long-period (gyroscopic) fluctuations will be p