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AIRCRAFT NAVIGATION
by S. S, Fedchin
"Transport" Press
Moscotv, 1966
LOAN COPY: RETURN TO
AFWL (WLIL2)
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 ThreeDimensional Space 9
5. Charts, Maps, and Cartographic Projections 12
Distortions of Cartographic Projections 14
Eltipse of Dis tortions 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
DifferenceRangeFinding (Hyperbol ic)
Coordinate System 81
Ove ra 1 1 RangeF 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. Twodimensional 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 NL10M 235
CHAPTER THREE. AIRCRAFT NAVIGATION USING RADIOENGINEERING
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 cRangef i nding Systems.... 259
Aircraft Navigation Using GrounBased Radio
DirectionFinders 263
Selection 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
UltraShortwave Goniometric and Goniometric
Range Finding Systems 296
Details of Using GoniometricRange Finding
Systems at Different Flight Altitude^ 304
FanShaped Goniometric Radio Beacons 306
3. DifferenceRangef 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 RadioNavigational 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
LowAltitude 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
CourseGlide Landing Systems 394
Ground Control of CourseGlide Systems 396
AircraftMounted Equipment for the Course
Glide Landing System 400
Location and Parameters for Regulating the
Equipment for the CourseGlide Landing
System 401
Landing an Aircraft with the CourseGlide
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, NeverRising and
NeverSetting 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 TwoDimens 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 LowAltitude Piston Engines 518
Calculating the Fuel Supply for Flight in Air
craft with HighAltitude 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. Preflight 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<eOff 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 radioengineering, 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 pvelanding
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 (windspeed 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 mid30'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 mid30'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 shortdistance navigation of a dif
ferent kind as well as radionavigation landing systems became wide
spread. Essentially, these were not autonomous means of aerial
navigation but systems which included both groundbased 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 nonautomatic radio navi /5
gational systems were of little use for ensuring the automation of
aircraft navigation and the piloting of highspeed 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 19251929, to accomplish
long flights by Soviet aircraft along the routes: MoscowPeking
(M. M. Gromov), MoscowTokyo and MoscowNew York (S. A. Shestakov).
Further nonstop flights by Soviet aviators, organized from
19361939 (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 longdistance aviation, with the carrying out of longdistance
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 highspeed jet aircraft, and also in connection with
he great achievements of the radio and electronics industry.
Longdistance flights of highspeed 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. Radioengineering means of aircraft navigation^ which are
based on the operating principle of radioelectronic technology.
These include goniometer radioengineering systems (radio compasses
with ground transmitting radio stations , ground radiogoniometers
with aircraft receivingtransmitting radio stations, and radio bea
cons with aircraft receiving radio equipment), rangefinding systems,
goniometerrangef inding systems, ground and aircraft radar, Doppler
meters and systems, radio altimeters, courselanding 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 pulselight 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 takeoff 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
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NASA TT F524
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 eoordinates .
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 surfacecoordinate 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 surfacecoordinate 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 glidingcoordinate 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=(XXi)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, + (XX^) 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 crosssectional 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
fitgP
; ^'^ 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 ThreeDimensional 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
oentrio 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
aVTote 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 radiusvector
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 radiusvector (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 radiusvector 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
twodimensional 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 welldefined, 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 mapmaking 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).
Distovtion of Direat'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.
Bivision 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 (Jcoor
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
righthand 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 Jcoordinate
along the meridian:
/29
A" = /? In tg
(«.f).
(1.22)
while the Zcoordinate 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 (northsouth and eastwest), 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 500600 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 10001^+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 LeningradKiev 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 TransverseCylindrical 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 transversecylindrical 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 500800 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 = (A2Xi) 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 circleparallels 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 x2) 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 pistonengine aircraft and helicopters , and secondly on
aircraft with gasturbine 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 /M1
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 9tgp,
(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
longrange navigation (DSLN1) 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 smallscale 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 smallscale 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 radioengineering 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
1200000
PZOOOOO
1:500000 (Nit18)
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 smallscale 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 radioengineering 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 (mapdiagrams), 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 righthand 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 straightline 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 righthand 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=14tg2b
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 (1.41) 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 (1.42)
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° = 0.5740.075 =■ 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° = 13.2280.866 = 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° = 0.9690. 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):
0.55,671
ctg a = '^ ^^gg 0.8665.671 = 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:
.„  0.17360.1736
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°"= 0.50.0872 =^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.M3), and then to the
latitude of the vertex point according to ( 1 . MM ) .
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 eastwest 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)
straightline element will increase and will equal:
ctgv
r, o = = cosec tp.
''>"• cos <f
As is evident, the straightline element, situated along the
parallel (in general, in a direction perpendicular to the axis of
the cylinder) acquires a curvature. The straightline element sit
uated in the direction of the axis of the cylinder does not acquire
a curvature. Therefore, if the straightline 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
straightline 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
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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 straightline 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 = /?82y?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 2,0003,000 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 U0° , 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^RUcoSYJ (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 eastwest 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 straightline segments
of a path of more than 12001500 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 8001000
km .
On short path segments (up to 12001500 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 300MOO 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 300100 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 radioengineering 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 straightline path segment
(Fig. 1.43) is taken as the main axis X. The line perpendicular to
the Zaxis 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 straightline 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 straightline 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 2030 km), it is possible to take
the spherical surface of the Earth within the area of the possible
deviations of the aircraft from the Jaxis 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 ( ij ) :
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^=XcpRcos
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 highspeed 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
takeoff landing zone at an alr
Zaxis, 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 300400 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.M7.
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 twodimensional 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 position 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 radioengineering and astronomic facilities of
aircraft navigation. They include the following:
1) A twopole 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) Llnes 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.
^) Differencerange 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) Overalt 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 hyperbotiaeZtiptiaat 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 straightline 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. 1.48), 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 Zcoordinat es not as linear but as angular measures,
i.e. , we take the Jcoordinate 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 Xocoordinate , 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 rangefinding 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 rangefinding coor
dinate system is small (on the order of 3001+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 rangefinding 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 ij 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 rangefinding system (Fig. 1.50), the distance to
two points on the Earth's surface with known coordinates is a meas
ured navigational parameter.
Fig. 1.49. Conversion of Polar
(GoniometerRangeFinding) Co
ordinates to Orthodromic Coor
dinates .
Fig. 1.50. rsipolar RangeFind
ing Coordinate System.
The indicated distance is usually determined according to the
time of passage of radio signals from the aircraft to the ground
radiorelay 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 doublevalued.
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 Xcoordinate
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
rangefinding 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 erenceRange F i nd i ng (Hyperbolic) Coordinate System
The circular rangefinding system of aircraft position lines
examined earlier is used with comparatively small distances from
the ground radioengineering equipment to the aircraft, since the
sending and radiorelaying 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 radioengineering 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 differenaerange
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 radioengineering 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. DifferenceRange
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 Acoordinate 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>"lJflK2)(K2C0SpKi)
cos X. = ±
K? — 2yiK2C0sp+y^
[/ (>'?2yiK2
cos ? + yl) [(X2 Ki — ^1 r2)'' — 8iii2 p kJ]
where
Kf2KiK2COsp+ 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 RangeF i nd i ng (Elliptical) Coordinate System
Hyperbolic navigational systems are the most easily implemented
technologically of all the rangefinding 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 semiaxis, 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 hyperbolicelliptical 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 semiaxis of the ellipse and a^ is the parame
ter a of the hyperbola.
Obvious advantages of the
hyperbolicelliptical 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. HyperbolicEllipti
cal Coordinate System.
(b) Orthogonality of the
/89
86
position lines appears at any point of the system. In a confocal
hyperbolicelliptical 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 hyperbolicelliptical
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 (quartzcrystal 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 hyperbolicelliptical 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 hyperbolicelliptical 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
coursereading 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=TCAm
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 (500600 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 Xj 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 takeoff 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 = OCA.
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. Twodimensional Fluctuations in the Aircraft Course
During twodimensional 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 windspeed 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 ^ wv'
r^tgWAlg 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 takeoff 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 9001000 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
3000100,016
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
radioengineering 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 righthand side of the equation, we obtain:
Considering that at distances up to 600700 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
^ K57,3 16757,3 „„^ ,
Let us determine the additional shift of the aircraft as a
result of wind during rolling in the first instance:
70 in/sec113°
0,9deg/sec
and in the second instance :
7067
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 200300 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 12.5
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
0TPR 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=26,5tg20° = 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 radioengineer
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 astroinertia.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 magnetio 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 largescale 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 largescale 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 1012', 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 socalled 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 distancemagnetic 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 (Hshaped 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 (Pj) 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 . Devtatton 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 Xaxis
of the aircraft; Q along the Yaxis, and E along the Zaxis.
(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 Zaxis 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 Jaxis , and bars
Qj h, k are located along the Zaxis; 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 Jcomponent 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
% (Icos^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 + BoSin1iC„ cos t + Dq sin 2'y + £o cos 2^
I + B0COS7 — Cosln7 + £>ocos27 — £oSln2j " (2.9)
Expression (2.9) is called the pointdeviation formula, and
the coefficients Aq^ Sqj Cqj Dq and Eq are the point coefficients
of deviation.
The pointdeviation 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
810°, 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 +
+ //sin4j + A'cos47...
(2.11)
Here the coefficients A^ B, C, D, E have a somewhat different value
than in the pointdeviation 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 2r + £ 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)
Eliminatton of Deviation in the Magnetic Cornpasses
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 23 km; for larger aircraft with a greater radius of turn
on the ground, it should be at least 56 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 20100 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 20100 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 "NS", 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 "EW", it is possible to reduce the
deviation to zero 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 45° 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+31+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^jO
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.t0_ 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 / m1
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
Blveotion 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 errorfree 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 righthand 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 lefthand ro
tation of the gyroscope, at the upper end of the axis. Analogously,
with a righthand 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
lefthand 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 / m6
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 halfway 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 currentcarrying 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 currentcarrying 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 threephase 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 righthand 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 KPK52 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 "LR" (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
68° .
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 DGMC7 (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) Twochannel 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 threewire circuit con
necting it to the gyroassembly 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 12°, 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 threephase motor,
whose stator is mounted on the internal frame of the gyro assem
bly and whose shortcircuited 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 14° per minute.
When it is necessary to carry out a rapid coordination, the
motor is switched to reduced reduction by means of the rapidcoord
ination button and a special relay. The rate of rotation of the
selsyn in this case is raised to 1515° 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 A1 and a portion of the transmitter
potentiometer laAj, 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 ■2aBi). 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
13 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
01, 02, 03. Currents which are symmetrical in phase also arise
in the windings of the slave selsyn O^li, 0i2x, 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 23° 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
OBOA
^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
OBOA.
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 twostage 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 515 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 straightline 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 23°).
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 socalled 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 lefthand side of
, marked "NS", 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
COiiuLLjuiia 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
straightline 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 straightline 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 Conditions 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 12001500 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 (crosssectional 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 12 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
12
= 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 leMariotte 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 GayLussac 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 leMariotte and GayLussac 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
1.03322410
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 socalled 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 grp 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^ . jw 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. Zeropoint 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 VD10 and VD20 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 socalled 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 = Ap11.
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 1511 = 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 .
70 +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 NL10 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 socalled 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. AirPressure 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 MOO 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(PtotalPst)
(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 US350. 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 Kl ' 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 leyevClapeyron 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 = /^^BrJptotl::&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. , (/C1)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^lK^') (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 US350 or US700), as well
as the combined indicator (Type CSI1200) 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 70 (10,02:
^true = ^instF ^o " d  0fi22mf'^^
'^ Vue= 'S ^inst'^T'^(^^^ + '^)~T 'g 288 2.628 lg( 10 ,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 4lU 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 NLIOM 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 staticpressure 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 = Pfl. 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 210000 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)= 0.00482100 = 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 . 818. 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
^
jT
^
d
%
,\
\
\
\
^
Z
y
^
^
\
\
^
\
^
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 remotecontrolled 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 23°. 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
K1
/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
20003.62
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), onehalf 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 elapsedtime 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 7075°, 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 4050° 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 = 45°, 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 3540°
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
M5°+CAL2
bis
while the drift angle of the aircraft will be equal to CAL2jj_g90° , 720'+
so that
US
CAL2135°
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
12
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 NI50B, is
widely used. We shall now discuss its design and the method of its
application .
The NI50B 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
NI50B 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 coursesetting 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 automatio 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. 2.40).
If the aircraft is now to fly with an orthodromic course y, /207
the airspeed must be divided into two components:
K^= Vcos(7i;);
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,
6v
 Vsinoi
Fig. 2.40.
Fig. 2.41.
Fig. 2.40. Distribution of the Airspeed Vector along the Coordi
nate Axes .
Fig. 2.41. SineCosine 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 sinecosine potentiometer (Fig. 2.41).
The sinecosine 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 pickup 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 sinecosine 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 pickup shoes
on the sinecosine 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 sinecosine 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 sinecosine 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
Jaxis, i.e., along the orthodrome , while a pointer marked "E" shows
the travel along the Zaxis, 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 straightline 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 highspeed 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
socalled 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 onefourth of the initial lead,
and if this is insufficient or too much, a correction is made which
is equal to oneeighth 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 11.5°, 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) = tla + 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
hjL
=^
\ dlstance(kiii)
A^yii[\Hiij('l\iiMWM il llIIM
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 lll l lllllllllllllilll 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 ii 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.==^^^iJM;
To~^l
A
Fig. 2.44. Scales on Navigational Slide Rule NLIOM,
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 5070 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 2.45, 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)
«•©
®
wi'
®
ill
no
Fig. 2.45. 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 = 5631
= 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°, ij = 38°, AW = 48°.
av
:52'
Solution
6
true
The true wind direction is
= 48 + 38 + 150.8 = 98° ,
and the magnetic wind direction is
= 48 + 38 + 15 0 .8 + 5 = 103°
Fig. 2.48. 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
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fli
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thi
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Ititu
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and t
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ith t
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e fl
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ligh
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hat
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ights
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are
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Never
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ig
he
the
earlier
e air
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 largearea 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 highspeed 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 straightline 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 straightline 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 shortrange 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 MO km, etc. For this purpose,
a special stencil is included in the set of navigational instru
ments for the NI50B 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) WindChange 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 10001500
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 when 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 NLIOM.
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 NLIOM (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 NLIOM: (a) of the Ground
Speed; (b) of the Distance Covered on the Basis of Ground
Speed and Time .
Let us apply the keys to NL10 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 NLIOM, 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^ = XL  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 NLIOM.
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 NLIOM 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 180a
 sin a, for example:
Ig sin 135° = Ig sin 1+5°.
Scales 3, 4 and 5 can be used to solve special problems of
obliqueangled 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 obliqueangled 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= lgsinAWlg7 =
lgsin(AW+US)  Igf/,
which is expressed by the key on
the navigational rule (see Fig. 2.60,
a).
239
a)
vi)
StnmB)
&X
b)
Stng
Q)
tl
:SL
Fig. 2.58. Keys for Determining the Aircraft Coordinates
on the NLIOM; (a) ZCoordinates ; (b) ZCoordinates .
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 = {WV 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 (,WV) 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 logaxL 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 NLIOM: (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 NLIOM 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 lefthand
side .
After dropping this multiplier. Formula (2.61) assumes the
form :
<p =
RUT
or
ig'p = ig>? + igUTig v^.
a)
© 5inUS
wc
c)
© w
©
70
tgAW^
4/'
isa
© tgUS
tg AW
V
® ,0
',1'
©
m
iSO
Fig. 2.61. Calculation on the NLIOM: (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, outsideair thermometers).
Adjustable scale 7, with a movable diamondshaped 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)
YW
I r
UT
©
ip
© V
Si
Fig. 2.62. Calculation on NLIOM: (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 + 6.56 = +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 diamondshaped 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 (Hll 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 NLIOM: (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 913 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 righthand side below scale 14.
a) b)
®(^ "corr® ® so' M,« ®
^ ''■inst® ^ " ®
Fig. 2.6tt. Calculation on NLIOM: (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^jjgi 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.
Exampie. 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 NLIOM: (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 12° for the type "TNV" .
In practice, the front side of the navigational slide rule
NLIOM 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 NLIOM: (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 NlIOM: (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 (10.02265), 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 NLIOM,
248
In order to get log20.3 from the value 2.628 log (l0.0226ff),
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 RADIOENGINEERING 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 radioengin
All radioengineering 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
'^gr
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 juu js 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 (10100 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 (1001500 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.
(150030,000 kHz) have
/246
4. Ultrashort 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 5090 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 120130 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 270300 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 /2M7
the following.
Unlike constant and lowfrequency 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 multiplefilament (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 boxtype 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 onequarter 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 M8
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 goniometricrangef inding
systems, differencerangef inding instruments, etc.
The panoramic radar located on the ground and on the aircraft
are goni ometri crange 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 navigationat 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 CRANGE 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 aircraftmounted radio direc
tion finders, which are also called radio
compas ses .
Let us imagine a frametype 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 highfrequency 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. EdcockType 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 highfre
quency current will pass through it.
A similar pair of dipoles is mounted in the plane perpendic
ular to the first pair.
The highfrequency 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 highfrequency 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 onehalf 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 directionfinding installations.
In this case, the transmitter is the radio on board the aircraft.
Ground radio directionfinders in principle can operate at
all wavelengths. The most widely used radio rangefinders operate
on short and ultrashort 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 cathoderay tubes.
In this case, the frame of the directionfinder 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 directionfinding 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 directionfinder can only operate with one
aircraft, which is clearly inadequate when there are a great many
flights .
Aircraft Navigation Using GroundBased Radio D i rec t i on F i nde rs
The use of groundbased radio directionfinders can be used
to solve the following navigational problems:
(a) Selection of the course to be followed and flight along
263
the straightline segments of a route, at the beginning or end of
which radio directionfinders 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 groundbased radio directionfinders.
(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
tionfinder, and (depending on the flight altitude) are used in
a radius of 100200 km as a form of trace directionfinder, 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
tionfinder (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 directionfinder, 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 distancefinder), or reverse bearing.
ShchTE: true bearing from the distancefinder to the aircraft,
or the forward true bearing.
264
ShchGE: azimuth of the aircraft at
trol distancefinding station.
a distance from the con
Fig. 3.8
Bearings
hGE
hTE and S^
Code Expressions for
1 the Shchcode.
ShchTF: location of
aircraft (coordinates or
Due to the small eff
radius of the USW rangefi
they are not grouped into
tancef inding nets like 1
or mediumwave stations,
operate independently, an
not give the location of
aircraft .
Seteotton of the Cours
be Followed and Control o
Dtreation
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 distancefinder is located at the starting point
of a flight segment (flight from the distancefinder), 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
distancefinder, 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 DistanceFinding Station.
Fig. 3.9.
f
Fig. 3.10. Path Segment Between Two Radio DistanceFinding 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
tancefinder, located at point A
ShchDR  MFA+A,, A„ 6 = 951++14 = 88°.
M M 1 av
av ^
For flight in a westerly direction, the ShchDM from this dis
tancefinder 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 wic , 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 1015° 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 distancefinder along a forward
bearing (ShchDR). In a flight toward a radio distancefinder 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 distancefinder 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 distancefinder and away from it:
(1) In a flight from the radio distancefinder, the drift
angle can be measured at the beginning of the segment, while in
a flight toward the distancefinder it can be determined only after
selecting the course to be followed with a stable ShchDM.
(2) In a flight from a radio distancefinder, 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
distancefinder in all cases must be made strictly along the given
bearing, while in the flight toward a radio distancefinder (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 distancefinder 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 distancefinder 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 mediumwave radio directionfinders, the
location of the aircraft is determined from bearings of two or three
mutually related ground radio directionfinders, 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 distancefinding 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 distancefinding 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 distancefinders as well as other nonautomatic 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 distancefinders upon request of bearings ShchGE
or ShchTF.
The drift angle can be determined in three ways with the aid
of ground radio distancefinders:
(1) The difference between the "forward" bearing (ShchDR) /260
and the course of the aircraft after passing over the radio dis
tancefinding 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 distancefinders 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 140° 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., NI50), 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 NLIOM.
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 = Trr ; Ut = — : — Trr
* 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 NLIOM.
Example: MFA = 110°; A^^ = 7°; AZ = H0 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 NLIOM.
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.. = 1357 = 129°
M M
Automatic Aircraft Radio D i s tance F i nders ( Rad i ocompasses )
Automatic aircraft radio directionfinders ( 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 distancefinding 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 frametype antennas but are ficc
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^ ■' ' ' ' """■iT^ *„ ,„„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 distancefinder
at a fixed setting of the antenna system relative to the vertical; /263
in distancefinding 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 distancefinding 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 distancefinders
(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 distancefinders in flat
country is 12°, and 35° in the mountains. The errors in meas
uring the distances with the aid of radiocompasses in flat areas
is 35°, and can reach 1015° in mountainous areas, especially at
low flight altitudes .
Accordingly, the practical operating range of a ground radio
distancefinder with satisfactory results of tracking is 300400
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 onboard radiocompasses is obtained at
distances up to 180200 km. Nevertheless, radiocompasses have found
increasingly broad application for purposes of aircraft navigation,
and are more popular than the ground radio distancefinders 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 onboard 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 directionfinders
which we have discussed thus far and which belong to the "E" type
(carrierwave amplitude), radiocompasses presently use the method
of amplitude modulation of the received signals (directionfinder
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
lowfrequency generator. This means that an amplitudemodulated
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 lowfrequency generator is turned off, these tubes
will be closed and the signals from the frame antenna will not be
passed.
When the lowfrequency 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 halfperiods 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 lowfrequency 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 nonmodulated 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 "leftright". The reading of the bearing in this case
has two signs .
Recent models of the ARK11 automatic radiocompass do not diffe
in their principle of operation from the operating principle describ
above for the ARK5 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 noncon
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 ARK11 differs in design from that
of the ARK5. 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 frameantenna
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 ARK11 has a toggle switch for nar /268
row and wide frequency bandpass: "widenarrow". 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 ARK11 panel (subrange switch, knobs
for coarse and fine setting, toggle switches and buttons) have the
same markings as in the ARK5.
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 56 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
XZ, the deviation at course angles zero and 180° is close to zero.
The transverse plane of the aircraft YZ 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 50100 km from the airport)
and the true bearing is measured as accurately as possible on a
largescale 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 0180° 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 distancefinder, the level line is set to the cal
culated CAL, and the aircraft is turned until the sight line of
the distancefinder 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 refastened 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
distancefinder along the longitudinal axis of the aircraft to 0
180°, loosen the dial of the deviation distancefinder and move
it so that the line of sight 0180° 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, 45,
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 distancefinder 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
distancefinding 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 24angle
route, i.e., practically along a course which crosses the straight
282
line flight for 2030 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 distancefinders.
(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 distancefinders, 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 nonrecorded 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 6p 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 directionfinder in
selecting the course to be followed along straightline 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 1213° (CAR
equals 348347°). 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 180200 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 distancefinding 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
distancefinders. 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 13 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 nonexistent.
/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 13 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 (12004000 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 distancefinders 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 shortrange 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 goniometricrangef ind
ing systems which operate on ultrashortwaves .
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 ultrashort 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 callletter 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 shortrange 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 distancefinders. 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 zeroindicator 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
distancefinders is only that in order to determine the location
of the aircraft from two bearings using a radio directionfinder,
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 radiodistancefinders 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 distancefinders . The prac
tical accuracy under average conditions of application is also somewhat
higher or equal to the accuracy of distancefinders. However, in
using radio distancefinders, 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 distancemeasuring 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:
transl
mitter^
re'
c elv
er"
^
indicator
Fig. 3.28
System .
Diagram of LongRange 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 frequencydivider 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 straightline 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^ + RsiniAiiJ.
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 1520 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 GoniometriaRange 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 halfwave:
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 goniometricrange
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( ipj4£ ) = 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 tricrangef 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 goniometricrange 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 .
FanShaped Gon i ome t r i c Radio Beacons
The possibilities of aircraft radio compasses are increased
considerably by using fanshaped 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. FanShaped Radio Beacon.
Fig. 3.34. Radiation Characteristic of a FanShaped 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 righthand 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
nShaped 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 .
Fanshaped 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 fanshaped 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 fanshaped
beacons during the daytime is no worse than 0.10.3°. 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 shortrange
zone, with operation on a surface wave, the errors do not exceed
.51° .
The operating range of a fanshaped 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 nondirectional and omnidirec
tional operation, which are mounted as a rule at the turning points
of air routes , flight along the bearing line of a fanshaped beacon
is only a very rare case. Therefore, the principal method of air
craft navigation using fanshaped 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 fanshaped 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 fanshaped beacons with conventional distancefinding
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 fanshaped beacons for checking the path for distance and deter
mining the ground speed.
3. DIFFERENCERANGEFINDING (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'ClcosAA).
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
ixions .
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 longrange 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 shortrange
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 receivingindicating 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 foai
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 manyfold 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 erentialrangef inding navigational systems , like fantype
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 fantype beacon.
To increase the feasibility of using dif f erentialrangef 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 hyperbolicrange
finding or hyperbolicelliptical 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 hyperbolicelliptical 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.
erbolicrange
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
bolicelliptical 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 longrange
navigation for use on highspeed
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 RADIONAVIGATIONAL 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 radionavigatvonat
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 radionavigational 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 crangef inding systems. However,
in comparison to the goniometricrangef 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 nonautonomous 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
modulatorl^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 standardfrequency generator with fre
quency dividers for forming distance markings and a signaltrans
mission frequency synchronized with the sawtooth scanning image
on the screen (Fig. 3 . M0 ) .
The signals from the standardfrequency generator reach the /308
modulator, where they are converted to highvoltage rectangular
oscillations of a special length. The highvoltage pulses from
the modulator pass to the transmitter magnetron, where highfre
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 sawtooth voltage which controls the scanning beam on the
screen. The scanning rate depends on the steepness of the slope
for the sawtooth waves. At a low scanning rate, a fine image scale
is obtained as well as longdistance 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 M0
Vm ^ rt ;:! rti _
Now let US follow the path of the highfrequency pulses from
the transmitter to the object and back again, and see how they control
the brightness of the scanning beam.
The highfrequency 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'istho 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 esantsquare 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 cosecantsquare antenna
characteristics of the radar.
Fig. S.Ml. Radar An
tenna for Use Aboard
Aircraft .
Antennas with cosecantsquare characteristics are used for
circularscan radar, mounted below the fuselage of the aircraft.
Antennas with combined radiation are used for sectorscan 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 largescale 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 highlevel
signals are increased and the lowlevel signals are reduced (by
decreasing the brightness of the background of the screen). To
pick out rivers and lakes, the lowlevel signals are increased,
thus improving the visibility of shaded objects against a brighter
general background.
It should be mentioned that for the formation of highfrequency
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 crange 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.M2. Indicator for
Radar with Circular Screen
Sectortype 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 SectorType 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 nonmetallic 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 highlevel 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
lowlevel 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 snowcovered
surface, by amplifying the lowlevel 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 largedroplet 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 highlevel signals and minimum amplification of low
level signals. Amplification of lowlevel 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 sectortype 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 righthand side . This is explained
by the fact that when we are using goniometricrangef 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 — 125sin 50° = 250 — 96 = 156 km.
Z = 80— 125cos40'' = 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 NLIOM. 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 sectortype 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
ularscreen radars. However, on sectortype 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 socalled 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., ga). 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 lefthand beam is marked
L and the righthand 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 lefthand edge
of the beam is greater than on the righthand side, so that the
frequency received by the antenna from the lefthand 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 lowfrequency 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 largescale 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 sectortype 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 810 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. 3,47, 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 NLIOM.
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
NLIOM.
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
^„ . CALyCAL
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 = 260 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 NLIOM.
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 semiautomatic 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.,
cyw c w
/325
340
the
The increase in the frequency W/X, produced by the motion of
source, is called the Doppler frequency (fQ).
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 fQ 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 singlebeam 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 fourbeam antennas .
341
Fig. 3. 54. Projection of the
GroundSpeed 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 fourbeam 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 fourbeam 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 twobeam 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 fourbeam 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 righthand beam and
decreases by the same magnitude in the rear lefthand 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 highfrequency 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 bypass 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 highfrequency 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 highfrequency 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. DopplerMeter 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, multibeam antennas, especially when fixed in position,
are very useful, since they can be placed below the radiotrans
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 fourbeam 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 righthand 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 lefthand
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 righthand 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 righthand beam relative
to the longitudinal axis of the aircraft , the left or righthand
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 ,
fQ2 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 LeftHand
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 fQ 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 Jaxis: 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 groundspeed 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 groundspeed 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) cosGcosA ^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 1520 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
NI50B.
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 Zcoordinate 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 Zcoordinate 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 35°.
When the Zcoordinate 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^cosTAZ 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 goniometricrangef inding
installations are located. Then the orthodromic coordinates of
358
these points are determined, and the ground goniometricrangef 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 36 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 35 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
Zcoordinate 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 = XyE cos PBL ;
Z = ZyR sin PBL.
(b) In the measurement of goniometricrangef 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 tricrangef inding systems are invariable. The difference in
the signs of the second terms on the righthand 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 goniometricrange
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 '^^Qt and ^aQt are the coordinates of the aircraft on the basis
of the measurement results and ^calc and Zl^alc s^re 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 200300 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,
vz
Fig. 3 .50
Fig. 3.61
Fig. 3.50. Determination of the Error in Measuring the Course on
the NLIOM.
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 NLIOM 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 (23 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 Zcoordinate
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 (0.51°), 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
smallscale variations in the gyroscope (0.52 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 smallscale
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 Zcoordinate 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 = 37542.5 = 332.5 km; Z = 6158 = 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 5060°. 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 5060°, 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 NI50B, in
which there is a course transmitter, a transmitter of the airspeed,
and a manuallyset 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 distancefind
ers , externally directed
gonio
metric and
goniometricrangefinding
systems , fantype 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 Zcoordinate 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
200300 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 56°, i.e., 510 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 pilotage 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
12° with a very precise maintainance of the general direction of
flight.
etricrange
make the spherical conversions.
It is somewhat simpler in this regard to use goniometricr
finding methods for shortrange 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. SShaped 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?( 1cosTA) ,
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 4500 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 NLIOM 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 815 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 courseglide 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 shortrange master station (SRMS), located
1000 m from the end of the LTS, and the longrange 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^^jj^^^^^ ^^
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 36 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 courseglide devices,
the marker receiver is turned on by a switch which is combined with
the courseglide 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.
LowAltitude Radio Altimeters
/357
At the present time, lowaltitude 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 LowAltitude Radioaltimeter ,
376
The radio altimeter transmitter has a modulating device which
produces a sawtooth 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 sawtooth 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 nshaped 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 "dislocation" 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 pistonengine 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
M1200
H = J900t?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
straightline 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 CAR120 (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 straightline 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
straightline 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 straightline
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 pistonengine 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 socalled 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 StraightLine 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 headwind component for the landing
course .
In making an approach to land along a rectangular pattern, the
last reliable point for determining the Jcoordinate 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 headwind 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 41 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.
24015 ' „„
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\tu^
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
rAD 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 + 6010
ctg C AR= ^
12500
CAR =68°,
i.e., the turn must begin 2° earlier than under calm conditions.
Cataulation 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 (M00 km/hr), this distance
will be :
^70111 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 , = 400clg2°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 = 10015 =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':
Vy = 65.tg2°40' = 3^/ sec
The calculation of the vertical rate of descent is of partic
ular interest for pistonengine 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
270290 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 180200
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 sidewind 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 instrumentspeed 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.
CourseGlide 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 gasturbine 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
courseglide landing system.
The geometric essence of courseglide systems is the use of
radioengineering 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. RadioSignal Planes of a CourseGlide
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 CourseGlide Systems
The principal pieces of equipment in a courseglide 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 CourseGlide 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 courseglide
systems only for the meter wavelengths.
In addition to the beacons, which form the course and glide
zones, the courseglide 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 (longrange) 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 courseglide 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 amplitudemodulated 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 radiosignal 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
34° of the equalsignal 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 radios
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 equalsignal axis in the working lobe of
the zone must be within the limits of 45 to 70 km.
The gZide 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 radiosignal 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 equalsignal 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.
Aipara ft Mounted Equipment for the CourseGlide
Landing System
The following units make up the aircraftmounted equipmeiit for
the courseglide landing system:
(a) Antenna and receiver for coursebeacon signals.
(b) Antenna and receiver for glidebeacon signals.
(c) Control panel.
(d) Landingsystem 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 AircraftMounted Glide Radio Beacon
400
The glidebeacon receiver uses the circuit for the reinforced
AAC. The latter is not employed in coursebeacon 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 shortperiod oscillations of the course and
glide indicator of the LAS, due to local disturbances in the
radiosignal 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 CourseGlide Landing System
The radio beacon for the course zone of the courseglide
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/ I50200M
course beacon
BOO 1000 m
Fig. 4.19. Diagram of Location of GroundBased Equipment for Course
Glide System.
150 to 200 m from its axis and 250275 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 landingsystem 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 coursebeacon zone are
set as follows :
(a) The angular width of half the zone must be located within
2 to 3° of the LTS axis.
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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 CourseGlide System
Setting up the maneuver for an aircraft approaching an airport
to descend with the use of the courseglide system is performed
according to the same rules as in the simplified system for landing
an aircraft .
The complement of equipment for the courseglide 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 courseglide 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 courseglide 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 coursezone 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 coursezone strip deflects, the turn must be continued
until the landing course is acquired. When the landing course
is acquired, the coursezone 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 coursezone 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
glidezone strip takes place at a constant altitude.
At the moment when the glidezone 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
407
transition to visual flight, after which a
tude is made and the aircraft touches down
visual estimate of alti /387
on the runway .
Direationat 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
radiosignal 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 course 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.
Socalled 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 courseglide 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
courseglide systems. Nevertheless, radar landing systems are
widely employed, along with courseglide 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 courseglide 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 courseglide system cannot be used. The same reasoning applies / 390
in retaining the courseglide 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 distancefinder) 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 glidesector 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. LandingRadar 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
circularscan radar for observing aircraft near and far from the
airport .
In setting up the landing maneuver, immediately before completin;
the fourth turn, the shortrange radar approach system is used, also
called controltower 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 pistonengine 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 tonengines 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 landingradar 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
414
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 (courseglide 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 courseglide 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
landingradar 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 courseglide 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 courseglide 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
418
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 ceZestial 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 ceZestiaZ 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
424
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 240°) 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, NeverRising 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 = +40°. In
lestial sphere for an observer
the meridian (hour circle)
f the star 6 = +40° 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, NeverSetting and NeverRisinj
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
horizon 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).
t31
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 socalled 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 socalled 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 ecZiptic
(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 sidereat 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 9076° = +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.
M40
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 poZar 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
M41
^
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 orhit.
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 halfdisk, 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 18 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 halfmoon phase. In this phase, the Moon is visible
all night .
After half moon, the illuminated part of the lunar disk begins/i423
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
444
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 socalled 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 24hour reckoning of time, this is
very inconvenient since in this case the beginning of mean days
comes at noon.
Therefore, in everyday life, socalled 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
Side
our
he h
t fo
e Ea
' s s
gure
of
t a
e Ea
on
our
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is m
obv
fore
ians
real ti
angle o
our ang
r one h
rth ' s s
urf ace
5 . 22 w
the dra
certain
rth ' s s
the Ear
angle o
gle APn
oment i
ious th
, at th
will b
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f a
le (
e ave
urf a
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e sh
wing
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urf a
th 's
f th
Mav
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at t
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true s
heaven
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nly bo
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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
ies
lar
e le s
poi
Radi
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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 / M2 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 BFjiM^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^ = Tqp  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
y "to
ds 4
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
ntro
erri
from
ipal
he E
) ti
use
r ex
ast ,
inuo
ahea
ude
itud
ng f
ds w
tant
e of
s of
duce
tori
the
obs
arth ' s
me ,
local
ample ,
it
usly
d
or
e .
rom
ould
ly
the
a
es .
erva
451
tories of these countries.
In France socalled "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 78° 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= 7j^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.
Organisation 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 socalled
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 "'"'
(^TOf3T1i^£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 straightline 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 courseangle 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 courseangle 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
lefthand 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
(tQp) 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, Atgp 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 directionfinding of
the Sun or other stars .
Astronomical compasses of the type DAKDB 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 DAKDB 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 DAKDB 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 courseangle 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 flighttrajectory 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 Zcoordinate (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 Jcoordinate at points A and B:
Aj^Bc = AXi + 2AX2 + Mfj.
(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 righthand 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 34 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:
ln
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 500600 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. Centuryold 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) =
(xay
(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
'{xa)'= ' r
K2.. J
{xay
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[y2a}'P(yro)
(Xa)<0^^
h ^1
— Ixa)>
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 directionfinding 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 TwoDimensi 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 secondorder 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 onedimensional, 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 threedimensional space.
Therefore, the necessity for solving volumetric problems in proba
bility theory is superfluous. However, a number of navigational
systems such as a hyperbolic, twopole 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 twodimensional 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 obliqueangled 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 /M5^
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 twodimen
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^,)24.;,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)p4(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 /M64
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
righthand side of the ellipse. The mathematical expectation of
the value of the Zcoordinate in this case will be found in the
middle of the chord of the ellipse, parallel to OZ and intersecting
the Jaxis 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.
fioients 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 Zcoordinate with errors for the
Zcoordinate 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 Xcoordinate with the errors of the
Zcoordinate, 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 twodimensional 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 righthand 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 /MSS
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:
491
oOOx "" ^'^°^ ^' '^'S 9i C0S2 Oo) aXo,;
01
ooo = (sin Xjjj cosec2 cpjx;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 05 units of the sixth sign, and (in the second case) from
010 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 2122 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 onedimensional problems of probability
theory for a generalized obliqueangled 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 .
Twopole goniometric, twopole circular and onepole 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
twopole circular system is solved analogously, with the sole dif
ference being that the vectorgradient 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 rangefinding 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 hyperbolicelliptical 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
bolicelliptical 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