NASA TECHNICAL TRANS L ATIO N Sl NASA U LJi.4 a', tr i Ins m o X ■< 5 AIRCRAFT NAVIGATION by S. S, Fedchin "Transport" Press Moscotv, 1966 LOAN COPY: RETURN TO AFWL (WLIL-2) KWTLAND AFB, N MEX NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • FEBRUARY 1969 TECH LrBRARY KAFB, NM 001=8=152 AIRCRAFT NAVIGATION By S. S. Fedchin Translation of: "Samoletovozhdeniye." "Transport" Press, Moscow, 1966 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - CFSTI price $3.00 TABLE OF CONTENTS ABSTRACT xl INTRODUCTION xlii CHAPTER ONE. COORDINATE SYSTEMS AND ELEMENTS OF AIRCRAFT NAVIGATION 1 1. Elements of Aircraft Movement in Space 1 2. Concepts of Stable and Unstable Flight Conditions 4 3. Form and Dimensions of the Earth 7 4. Elements Which Connect the Earth's Surface with Three-Dimensional Space 9 5. Charts, Maps, and Cartographic Projections 12 Distortions of Cartographic Projections 14 Elt-ipse of Dis tort-ions 14 Distortion of Lengths 15 Distortion of Directions 16 Distortion of Areas 17 Classification of Cartographic Projections 18 Division of Projections by the Nature of the Distortions 18 1. Isogonal or conformal projections 18 2. Equally spaced or equidistant projections 19 3. Equally large or equivalent projections 19 4. Arbitrary projections 20 Division of Projections According to the Method of Construction (According to the Appearance of the Normat Grid) 20 Cylindrical Projections 20 Normal (equivalent) cylindrical projection 20 Simple equally spaced cylindrical projection 22 Isogonal cylindrical projection 23 Isogonal oblique cylindrical projections 24 Isogonal transverse and cylindrical Gaussian projection 25 Conic Proj actions 27 Simple normal conic projection 28 Isogonal conic projection 29 Convergence angle of the meridians 30 Polyconic projections 31 International projection 32 Azimuthal (Perspective) Projections 34 Central polar (gnomonic projection) 36 Equally spaced azimuthal (central) projection 38 Stereographi c polar projection 38 Nomenclature of Maps 41 111 Maps Used for Aircraft Navigation 42 6. Measuring Directions and Distances on the Earth's Surface 45 rthodrome on the Earth's Surface 45 Orthodrome on Topographical Maps of Different Projections 55 Loxodrome on the Earth's Surface 60 General Recommendations for Measuring Directions and Distances 65 7. Special Coordinate Systems on the Earth's Surface.... 66 Orthodromic Coordinate System 67 Arbitrary (Oblique and Transverse) Spherical and Polar Coordinate Systems 71 Position Lines of an Aircraft on the Earth's Surface 73 Bipolar Azimuthal Coordinate System 74 Goniometric Range- F i nd i ng Coordinate System 77 Bipolar Range- F i nd i ng (Circular) Coordinate System 78 Lines of Equal Azimuths 80 Difference-Range-Finding (Hyperbol ic) Coordinate System 81 Ove ra 1 1 -Range-F i nd i ng (Elliptical) Coordinate System 85 8. Elements of Aircraft Navigation 88 Elements which determine Flight Direction 88 1. Assymetry of the Engine Thrust or Aircraft Drag (Fig. 1.59) 94 2. Allowable Lateral Banking of an Aircraft in Horizontal Flight 94 3. Coriolis Force 95 h. Two-dimensional Fluctuations in the Aircraft Course 95 5. Gliding During Changes in the Lateral Wind Speed Component at Flight Altitude 95 Elements Which Characterize the Flight Speed of an Aircraft 96 Navigational Speed Triangle 98 Elements Which Determine Flight Altitude 101 Calculating Flight Altitude in Determining Distances on the Earth's Surface 103 Elements of Aircraft Roll 107 1. Combination of Roll with a Straight Line 110 2. Combination of two rolls 110 3. Linear prediction of roll (LPR) Ill CHAPTER TWO. AIRCRAFT NAVIGATION USING MISCELLANEOUS DEVICES 113 1. Geotechnical Means of Aircraft Navigation 113 2. Course Instruments and Systems 114 IV Methods of Using the Magnetic Field of the Earth to Determine Direction 114 Variations and Oscillations in the Earth's Magnetic Field 119 Magnetic Compasses 121 Deviation of Magnetic Compasses and its Compensation 123 Equalizing the Magnetic Field of the Aircvaft.... 126 Deviation Formulas 128 Calculation of Approximate Deviation Coefficients 131 Change in Deviation of Magnetic Compasses as a Function of the Magnetic Latitude of the Locus of the Aircraft 133 Elimination of Deviation in the Magnetic Compasses 134 Gyroscopic Course Devices 141 Principle of Operation of Gyroscopic Instruments 142 Degree of Freedom of the Gyroscope 144 Direction of Precession of the Gyroscope Axis ... . 146 Apparent Rotation of Gyroscope Axis on the Earth ' s Surface 146 Gyroscopic Semicompass 149 Distance Gyromagnetic Compass 152 Gyroi nducti on Compass 158 Details of Deviation Operations on Distance Gyromagnetic and Gy ro i nduct i on Compasses 162 Methods of Using Course Devices for Purposes of Aircraft Navigation 165 Methods of Using Course Devices Under Conditions Included in the First Group 166 Methods of Using Course Devices Under Conditions of the Second Group 168 Methods of Using Course Devices Under the Conditions of the Third Group 172 3. Barometric Altimeters 175 Description of a Barometric Altimeter 180 Errors in Measuring Altitude with a Barometric Altimeter 183 4. Airspeed Indicators 186 Errors in Measuring Airspeed 193 Relationship Between Errors in Speed Indicators and Flight Altitude 196 5. Measurement of the Temperature of the Outside Air.... 199 6. Aviation Clocks 201 Special Requirements for Aviation Clocks 202 7. Navigational Sights 204 8. Automatic Navigation Instruments 210 9. Practical Methods of Aircraft Navigation Using Geotechnical Devices , 214 V Takeoff of the Aircraft at the Starting Point of the Route 215 Selecting the Course to be Followed for the Flight Route 218 Change in Navigational Elements During Flight 221 Measuring the Wind at Flight Altitude and Calculating Navigational Elements at Successive Stages 224 Calculation of the Path of the Aircraft and Monitoring Aircraft Navigation in Terms of Distances and Direction 227 Use of Automatic Navigational Devices for Calculating the Aircraft Path and Measuring the Wind Parameters 230 Details of Aircraft Navigation Using Geotechnical Methods in Various Flight Conditions 233 10. Calculating and Measuring Pilotage Instruments 234 Purpose of Calculating and Measuring Pilotage Instruments 234 Navigational Slide Rule NL-10M 235 CHAPTER THREE. AIRCRAFT NAVIGATION USING RADIO-ENGINEERING DEVICES 2 50 1. Principles of the Theory of Radionavi gational Instruments 250 Wave Polarization 251 Propagation of Electromagnetic Oscillations in Homogeneous Media 253 Principles of Superposition and Interference of Radio Waves 257 Principle Characteristics of Rad i onav i ga t i ona 1 Instruments 257 Operating Principles of Rad i onav i gat i ona 1 Instruments 258 2. Goniometric and Goniometri c-Rangef i nding Systems.... 259 Aircraft Navigation Using Groun-Based Radio Direction-Finders 263 Select-ion of the Course to be Followed and Control of Ftight Direot'ion 265 Path Control in Terms of Distance and Deter- mination of the Aircraft ' s Location 269 Determination of the Ground Speed, Drift Angle, and Wind 270 Automatic Aircraft Radio D i s tance- F i nders (Radiocompasses) 273 RadioGompass Deviation 279 Aircraft Navigation Using Radiocompasses on Board the Aircraft 283 Special Features of Using Radiocompasses on Board Aircraft at High Altitudes and Flight Speeds 292 VI Details of Using Radiooompasses in Making Maneuvers in the Vicinity of the Airport at Which a Landing is to be Made 295 Ultra-Shortwave Goniometric and Goniometric- Range Finding Systems 296 Details of Using Goniometric-Range Finding Systems at Different Flight Altitude^ 304 Fan-Shaped Goniometric Radio Beacons 306 3. Difference-Rangef indi ng (Hyperbolic) Navigational Systems 310 Operating Principles of Differential Range- finding Systems 312 Navigational Applications of Differential- Rangefinding Systems 317 Methods of Improving Differential RAngefinding Navigational Systems 318 4. Autonomous Radio-Navigational Instruments 320 Aircraft Navigational Radar 320 Indicators of Aircraft Navigational Radars 325 Nature of the Visibility of Landmarks on the Screen of an Aircraft Radar 327 Use of Aircraft Radar for Purposes of Air- craft Navigation and Avoidance of Dangerous Meteorological Phenomena 328 Autonomous Doppler Meters for Drift Angle and Ground Speed ; 339 Schematic Diagram of the Operation of a Meter with Continuous Radiation Regime 347 Use of Doppler Meters for Purposes of Aircraft Navigation 350 Preparation for Flight and Correction of Errors in Aircraft Navigation by Using Doppler Meters 357 5. Principles of Combining Navigational Instruments 366 CHAPTER FOUR. DEVICES AND METHODS FOR MAKING AN INSTRUMENT LANDING 370 SYSTEMS FOR MAKING AN INSTRUMENT LANDING 370 Simplified System for Making an Instrument Landing 374 Marker Devices 375 Low-Altitude Radio Altimeters 376 Gyrohorizon 378 Variometer 380 Angle of Slope for Aircraft Glide 380 Typical Maneuvers in Landing an Aircraft 381 Calculation of Landing Approach Parameters for a Simplified System 386 Calculation of Corrections for the Time for Beginning the Third Turn 387 Vll Cat cut at -Ion of the Covveotvon for the Time of Starting the Fourth Turn 388 Calculation of the Moment for Beginning Descent Along the Landing Course 389 Calculation of the Vertical Rate of Descent Along the Glide Path 390 Determination of the Lead Angle for the Landing Path 391 Landing the Aircraft on tiie Runway and Flight along a Given Trajectory with a Simplified Landing System 391 Course-Glide Landing Systems 394 Ground Control of Course-Glide Systems 396 Aircraft-Mounted Equipment for the Course- Glide Landing System 400 Location and Parameters for Regulating the Equipment for the Course-Glide Landing System 401 Landing an Aircraft with the Course-Glide System 403 Directional Properties of the Landing System Apparatus 406 Directional Devices for Landing Aircraft 408 Radar Landing Systems 410 Bringing an Aircraft In for a Landing with Landing Radar 415 CHAPTER FIVE. AVIATION ASTRONOMY 418 1. The Celestial Sphere 418 Special Points, Planes, and Circles in the Celestial Sphere 418 Systems of Coordinates 421 Apparent System of Coordinates 421 Equatorial System of Coordinates 422 Graphic Representation of the Celestial Sphere.... 424 2. Diurnal Motion of the Stars.... 426 Motion of the Stars at Different Latitudes 427 Rising and Setting, Never-Rising and Never-Setting Stars 428 Motion of Stars at the Terrestrial Poles 431 Motion of Stars at Middle Latitudes 432 Motion of Stars at the Equator 433 Culmination of Stars 433 Problems and Exercises 435 3. The Motion of the Sun 436 The Annual Motion of the Sun 436 Motion of the Sun Along the Ecliptic 437 Diurnal Motion of the Sun 439 The Motion of the Sun at the North Pole 439 Motion of the Sun between the North Pole and the Arctic Circle 439 Vlll Motion of the Sun above the Arctic Civcle 441 Motion of the Sun at Middle Latitudes 441 Motion of the Sun at the Tevrestviat Equator 442 4. Motion of the Moon 442 Intrinsic Motion of the Moon 442 Biveotion and Rate of the Moon ' s Motion 443 Phases of the Moon 443 Nature of the Motion of the Moon around the Earth 445 Location of the Moon Above the Horizon 445 5. Measurement of Time 446 Essence of Calculating Time 446 Sidereal Time 446 True Solar Time 447 Mean Solar Time 448 Local Civil Time 449 Greenwich Time 449 Zone Time 451 Standard Time 453 i^elation Between Greenwich, Local and Zone (Standard) Time 454 Measuring Angles in Time Units 455 Time Signals 457 Organization of Time Signals in Aviation 458 A Brief History of Time Reckoning 459 6. Use of Astronomical Devices 461 Astronomical Compasses 467 Astronomical Sextants 469 CHAPTER SIX. ACCURACY IN AIRCRAFT NAVIGATION 470 1. Accuracy in Measuring Navigational Elements and in Aircraft Navigation as a Whole 470 2. Methods of Evaluating the Accuracy of Aircraft Navigation 474 3. Linear and Two-Dimens i onal Problems of Probability Theory 478 4. Combination of Methods of Mathematical Analysis and Mathematical Statistics in Evaluating the Accuracy of Navigational Measurements 490 5. Influence of the Geometry of a Navigational System on the Accuracy of Determining Aircraft Coordinates 493 6. Evaluation of the Accuracy of Measuring a Navigational Parameter 497 7. Calculation of the l^lind with an Evaluation of the Accuracy of Aircraft Navigation 499 8. Consideration of the Polar Flattening of the Earth in the Determination of Directions and Distances on the Earth's surface 501 XX ll CHAPTER SEVEN. FLIGHT PREPARATION 507 1. Goals and Problems of Flight Preparation 507 2. Preparing Flight Charts and Marking the Route 508 3. Studying the Route and Calculating a Safe Flight Altitude 514 4. Special Preparation of Charts and Aids for Using Various Navigational Devices in Flight 517 5. Calculating the Distance and Duration of Flight 518 Calculating the Fuel Supply for Flight on Aircraft with Low-Altitude Piston Engines 518 Calculating the Fuel Supply for Flight in Air- craft with High-Altitude Piston Engines 521 Calculating the Fuel Supply for Flight on Aircraft with Gas Turbine Engines 521 Calculating the Greatest Distance of the Aircraft's Point of Closest Approach to a Reserve Airport 530 6. Pre-flight Preparation and Flight Calculation 532 CHAPTER EIGHT. GENERAL PROCEDURE FOR AIRCRAFT NAVIGATION 536 1. General Methods of Aircraft Navigation along Air Routes 536 2. Stages in Executing the Flight 538 Tal<e-Off and Climb 539 Executing a Flight Along a Route 540 Descent and Entrance to the Region of the Landing Airport by an Aircraft 542 Maneuvering in the Vicinity of the Airport and the Landing Approach 543 Supplement 1. Composite Chart of Topographical Maps 545 Supplement 2. Spherical Trigonometry Formulas 547 Supplement 3. Map of the Heavens 549 Supplement 4. Map of Time Zones 550 Supplement 5. Table of Greenwich Hour Angles of the Sun and Chart of Their Corrections for the Flight Date 551 Supplement 6. Table of Values of the Function $ (a; - a). 552 Supplement 7. Units often Encountered in Aircraft Navigation and Their Values 554 ABSTRACT : The theory and practice of aircraft 12_ navigation at the modern level of aviation tech- nology are summarized in this hook; the most im- portant practical problems of the utilization of general 3 radio-engineering, and astronomical means of aircraft navigation are set forth; the proce- dure of the pilot's preparation for flight, the means of calculating the distance and duration of a flight, and the carrying out of pve-landing maneuvering and landing of the aircraft under complex meteorological conditions during the day or at night are elucidated. The basic material of the book, sufficient for the practical mastery of the means and methods of aircraft navigation, is presented with the ap- plication of mathematics within the limits of a secondary school course. The problems which are necessary for a deeper study of the material are discussed in terms of principles of higher math- ematics . The book is intended for pilots and naviga- tors. It can be used as a textbook for students of civil aviation educational institutions . XI INTRODUCTION /3 Aircraft navigation or aerial navigation is a science which studies the theory and practical methods of the safe navigation of airplanes as well as other aircraft (helicopters, dirigibles, etc.) in the airspace above the Earth's surface. By the process of aircraft navigation, we mean the complex of activities of the aircraft crew and the ground traffic control, which are directed toward a constant knowledge of the aircraft's location and which ensure safe and accurate flight along a set course as well as arrival at the point of destination at a set altitude and at an established time. During the initial period of the development of aviation, air- craft did not have equipment for piloting when the natural horizon was not visible and for orientation when the ground was not visible, so that visual orientation was the basic method of aircraft naviga- tion. The position of the aircraft was determined by comparing vis- ible landmarks In the area over which the aircraft was flying, with their representation on a map. However, at this time the necessity for instrumental methods of aircraft navigation was already felt. The most simple devices for measuring airspeed, flight altitude, the aircraft's course, and several other flight parameters were Installed on aircraft. This period saw the appearance of the first navigator's calculating in- struments (wind-speed indicators and navigational slide rules). At the beginning of the 1920's, the first hydroscopic devices appeared on aircraft; they were turn and glide indicators which (in combination with indicators of airspeed and vertical velocity) in- dicators (variometers) made it possible to judge in a rather primi- tive way the position of the aircraft in space when the natural horizon was not visible. By means of these devices, the aircraft crews (after special training) were already able to carry out flights in the clouds and above the clouds. s and the beginning of the 30's, more re- developed: gyrohorizons and gyrosemi- ime reliably ensured pilotage of aircraft on llotage xne Deginning or xne ou's, moi loped: gyrohorizons and gyros liably ensured pilotage of all 'ound was not visible. Later, ces for automatic aircraft plJ /^ Xlll Achievements in the area of piloting aircraft when the Earth was not visible, as well as the growth by that time of the speed, altitude, and distance of aircraft flights, required the creation of means to ensure aircraft navigation independent of the visibility of terrestrial landmarks . During these years, zone radio beacons which allowed the air- craft's flight direction to be maintained along a narrowly directed radial line which coincides with the direction of the straight part of an aerial route began to appear. Ground radiogoniometers also appeared, by means of which direction was determined in an aircraft, as well as the position of the aircraft along two intersecting di- rections . Another aspect of the development of aircraft navigation at this time was astronomical orientation. To determine the location of an aircraft, various sextants were constructed and special com- putation tables and graphs of the movement of heavenly bodies were compiled for use with the sextants. In the mid-30's, devices ap- peared for determining the course of an aircraft according to the heavenly bodies . At the same time, optical sighting devices were used, by means of which (during visibility of the terrestrial landmarks) the ground- speed, flight direction and drift angle of the aircraft were meas- ured, all of which were later used for some time as constants for calculating the path of an aircraft according to flight time and direction . A very important stage in the development of means of aircraft navigation of the mid-30's was the appearance of aircraft radio- goniometers ( radiosemicompasses ) , a further modification of which were the automatic aircraft radiocompasses . Radiosemicompasses and radiocompasses were, for a period of more than 20 years, the basic means of aircraft navigation in aircraft with piston engines. During World War II and especially in the postwar years, radio- engineering systems of long and short-distance navigation of a dif- ferent kind as well as radio-navigation landing systems became wide- spread. Essentially, these were not autonomous means of aerial navigation but systems which included both ground-based facilities for the security of aircraft navigation and aircraft equipment. Radical changes in the area of means and methods of aircraft navigation occurred (and are occurring at the present time) in con- nection with the development of jet aviation technology. The sharply growing speed, altitude, and distance of flights have required automation of the most laborious processes of aircraft navigation. Magnetic course devices and non-automatic radio navi- /5 gational systems were of little use for ensuring the automation of aircraft navigation and the piloting of high-speed aircraft. There xiv J arose a necessity for developing highly stable gyroscopic compasses, autonomous speed and flight direction meters , and stricter consider- ation of the aircraft's flight ' dynamics to ensure the rapid and ac- curate solution of navigational problems by computers. The science of "aircraft navigation" grew and developed along with the development of aviation and navigation technology. The works of the outstanding Russian scientists and inventors, M. V. Lomonosov, N. Ye. Zhukovskiy, K. E. Tsiolkovskiy , and A. S. Popov were the basis of aircraft navigation theory. A large contribution to the science of aircraft navigation was made by the following Soviet navigators and scientists: B. V. Sterligov, S. A. Danilin, I. T. Spirin, G. S. Frenkel', A. V. Bely- akov, L. P. Sergeyev, R, V. Kunitskiy, G. 0. Fridlender, G. F. Molokanov, B, G. Rats, V. Yu . Polyak, et al . The successes achieved in the development of aircraft naviga- tion as a science made it possible, even in 1925-1929, to accomplish long flights by Soviet aircraft along the routes: Moscow-Peking (M. M. Gromov), Moscow-Tokyo and Moscow-New York (S. A. Shestakov). Further nonstop flights by Soviet aviators, organized from 1936-1939 (V, P. Chkalov, M. M. Gromov, and V. K. Kokkinaki) both over the territory of the Soviet Union and especially over the North Pole to the USA, were like a great school, in which the examinations were the successes achieved by Soviet scientists in the area of aircraft navigation. World War II was a verification of all the achievements in the theory and practice of aircraft navigation, especially in the field of long-distance aviation, with the carrying out of long-distance night flights. During this period, a rich store of experience was accumulated and further improvements in aircraft navigation methods were carried out. In the postwar period, the science of aircraft navigation under- went an especially vigorous development in connection with the ap- earance of high-speed jet aircraft, and also in connection with he great achievements of the radio and electronics industry. Long-distance flights of high-speed aircraft along aerial utes which include international and intercontinental flights, as 11 as flights to the Arctic and Antarctic, are becoming routine r civil aviation crews. At the present time, aircraft navigation science has been dis- guished as an Independent and orderly science in which the ievements of a number of the general and special branches of wledge are employed: physics, mathematics, geodesy, astronomy, Physics, aerodynamics, radio engineering, radio electronics, etc. XV Navigation technology is developing at a rapid pace; aircraft /_6_ and ground facilities for aircraft navigation are' continually being perfected and the professional training and navigational prepara- tion of flight and ground personnel has improved. All this has radically raised the reliability of aircraft navigation, its accu- racy, and its chief criterion, safety. Modern technical means of aircraft navigation are divided into four basic groups according to the principle of operation. 2. Radio-engineering means of aircraft navigation^ which are based on the operating principle of radio-electronic technology. These include goniometer radio-engineering systems (radio compasses with ground transmitting radio stations , ground radiogoniometers with aircraft receiving-transmitting radio stations, and radio bea- cons with aircraft receiving radio equipment), rangefinding systems, goniometer-rangef inding systems, ground and aircraft radar, Doppler meters and systems, radio altimeters, course-landing beam systems with their ground and aircraft equipment, etc. 3. Astronomical (radio astronomical) means of aircraft navi- gation^ which are based on the principle of measuring the motion parameters of heavenly bodies. These include aviation sextants, astrocompasses , astronomical orientators , etc. 4. Light engineering means of aircraft navigation^ which are based on the principle of using light energy radiation. These in- clude ground light beams, light and pulse-light equipment for take- off and landing strips as well as aircraft, enclosures for the light- ing equipment of the routes and airports (housings for ground in- stallations) , various pyrotechnic devices, etc. At the heart of a safe and accurate flight according to a set route, in the vicinity of the airport, or during take-off and land- ing, lies the principle of the overall usage of all the available technical means of aircraft navigation, both ground facilities and those aboard the aircraft. xvi ■■■■■■■I ■l^im IIUHHI NASA TT F-524 CHAPTER ONE COORDINATE SYSTEMS AND ELEMENTS OF AIRCRAFT NAVIGATION n* 1. Elements of Aircraft Movement in Space The fundamental problem of aircraft navigation in all stages of flight is maintaining a given trajectory of aircraft movement in altitude, direction and time by means of a complex utilization of navigational means and methods. A successful solution to these problems depends on constant and accurate information concerning the position of the craft relative to a given flight trajectory, the nature of the aircraft movement, and the actions of the crew. As a result of the curvature of the Earth's surface, any given flight trajectory of an aircraft is curvilinear. However, by taking into account the large radius of curvature of the Earth's surface, a small area can always be .delineated on it whose surface can be assumed to be plane (Fig. 1.1). Let us erect a perpendicular 0^1 from the center of the small area which we have chosen and continue it until it intersects the center of the Earth. Obviously, this will be a perpendicular line, which we can call the vevticat of the Zoaus. In the plane of the small area which we have chosen, let us draw a straight line through the point Oi and take it as the X axis ; then let us draw another straight line through the point O^ in the plane of the area, perpendicular to the first, and call it the Z axis. Thus, at point 0^ on the Earth's sur- face, we will obtain a rectangular system of space coordinates X, Y, Z. Fig. 1.1. Rectangular Coordinate System on the Earth's Surface. Numbers in the margin indicate pagination in the foreign text ^ The travel of an aircraft over the Earth's surface will in- volve both a shift in the point Oi (origin of the coordinates) and the rotation of the axes of the coordinate system around the center _/_8 of the Earth (point 0) . However, the system of coordinates which we have obtained can be used for determining the directions of the aircraft axes and the component flight speed vectors. Since the origin of this system is being continuously shifted, let us designate it as a gliding rea- tangulav system of eoord-inates . In this coordinate system, the following elements can be distin- guished : (a) Position of the longitu- dinal axis of the aircraft in the horizontal plane {aivovaft course) , (b) Position of the longitu- dinal axis of the aircraft in the vertical plane (angle of pitch of the aircraft ) , Fig. 1.2. Dip Angle of the Trajectory of Altitude Gain, (c) Position of the lateral axis of the aircraft in the ver- tical plane (lateral banking) , (d) Distance along the vertical from the Earth's surface (the area which we have chosen) to the aircraft (flight altitude) , (e) Vertical speed (altitude gain and loss), (f) Component flight speed along the X and Z axes or the vec- tor of groundspeed and its direction (groundspeed and flight angle), (g) Angular velocity of aircraft roll, (h) Component wind speed along the X and Z axes of the system, or the wind vector and its direction (wind speed and direction) . Usually the position of the craft on the Earth's surface is treated in surface-coordinate systems, the most widely used of which are the geographic system and the reference system whose major axis coincides with a given flight trajectory on the Earth's surface. The position of the aircraft in surface-coordinate systems is assumed to be the position of the origin of the gliding system. To analyze the elements of aircraft navigation, let us combine the X axis of the gliding-coordinate system with a given flight trajectory of the aircraft. In order to keep the aircraft in the rectilinear horizontal segment of this trajectory, the crew must maintain a flight condi- tion in which the aircraft will not be shifted along the vertical (altitude gain and loss), there will be no lateral deviation (to the right or left), i.e., the vertical velocity Vy and the lateral component of the velocity V^, will be equal to zero, and the longi- tudinal flight velocity V^ (along the X axis) will be as given. If the flight trajectory is inclined (segments of altitude gain and loss), the crew must hold this trajectory by maintaining the vertical and longitudinal flight velocities ( 7y and V^) t i.e., /J_ maintain a given dip angle of the trajectory 9 (Fig. 1.2). Obviously, at a constant dip angle of the flight trajectory, the latter will have a curvature in the vertical plane just as in horizontal flight. Therefore, if we neglect the curvature of the horizontal flight trajectory, we may assume X^-Xx ' (1.1) where Q is the dip angle of the flight trajectory; Xi, X2 are the coordinates of the initial and final points of the sloping segment of the trajectory; Hi, H2 represent a given altitude at the initial and final points. When the aircraft travels from the initial point Xi to the mov- ing point J, the flight altitude is changed by the value iiff=(X-Xi)tgg, (1.2) and the value of the moving flight altitude is H = ff^ + ^H^Hl + ^X—Xl)^gfl (1.3) or if we take Formula (1.1) into account, H^ff, + (X-X^) f-^' . (1.4) ■n.2 — ^1 Since the altitude during a sloping trajectory is a variable value, a given flight trajectory is maintained at a constant value of the vertical velocity Vy^V^tgQd r Vy^V,-^^^. (1.5) Checking of the position of the aircraft at given values of the varying flight altitude is carried out only at specific points on the sloping trajectory. Translator's note: tg = tan. 2. Concepts of Stable and Unstable Flight Conditions A navigational flight condition is determined by the motion parameters of an aircraft along a trajectory or by navigational elements of flight: course, speed, and altitude. The motion parameters of an aircraft are usually measured rel- ative to airspace. However, considering that the airspace also shifts , they are selected in such a way as to ensure retaining the given flight trajectory relative to the Earth's surface. Based on the nature of the trajectory and the conditions of aircraft navigation, four main flight conditions are distinguished: /.lO horizontal rectilinear flight, altitude gain, altitude loss, and roll. Horizontal rectilinear flight is characterized by two constant parameters: height and flight direction. Altitude gain and loss conditions each have two constant param- eters: flight direction, and vertical velocity or dip angle of the traj ectory . The condition of roll is always combined with one of the first three flight conditions , so that the flight direction becomes vari- able and can be replaced by a parameter which characterizes the curvature of the roll trajectory through the radius of roll or the angular velocity. A flight condition is stable if its parameters acquire constant values, and unstable if its parameters are variable. Flight practice shows that flight conditions , strictly speak- ing , are never fixed for any prolonged time, since there are always factors changing the aircraft's motion parameters. The main sign of a stable flight condition is the equality to zero of the first derivative of the given parameter with time d'^S or of the second derivative path with time — . For example, for the velocity parameter V = const, if = or r-=0. (dl) df dfl Analogously, for the flight direction parameter (.^) and the altitude parameter (ff) : dii dH .l;=const, If — T- = 0, //=const, if = 0. dt dt If forces arise during flight which change the aircraft's mo- tion parameters, the extreme values of the motion parameters (i.e., the points of the maxima and minima on the curve which characterizes the change of the given parameter with time) indicate equilibrium of these forces. A stable flight con on a given parameter exi the extreme points, sine derivative parameters ba at these points are equa while the disturbing for The disturbing fore maximum value at points i.e., when the second de rameters based on time a zero (Fig. 1.3). On a c structed for the velocit the points of a stable c designated by one line, while points of maximum distur are designated by two lines. Fig. 1.3. Graph of the Changes of a Navigational Parameter and Points with a Stable Flight Condition dition based sts only at e the first sed on time 1 to zero ces are absent es acquire a of inflection, rivative pa- re equal to urve con- y parameter, ondition are bing forces From aerodynamics, we know that in horizontal flight at a ve- locity significantly less than the speed of sound, the drag of an aircraft in a counterflow is /ll Qx = cj,s pV2 where a^ is the coefficient of drag of the aircraft, S is the cross- sectional area of the midship section, and p is the air density at flight altitude. It is obvious that the airspeed will be stable if the thrust of the engines (P) is equal to the drag of the aircraft P - Qx- With a disturbance of this equilibrium, there arises a disturb- ing force which changes the flight velocity. For example, with an increase in the thrust of the engines the disturbing force will be equal to : &P^P-—CjcS pV2 which causes an initial acceleration of the aircraft dt ~ m ' where m is the mass of the aircraft in kg. Later, with an increase in velocity, the drag of the aircraft will also increase. The value of this drag will approach the value of the thrust of the engines, i.e., the velocity very slowly ap- proaches a stable value logarithmically. Changes in airspeed which are analogous in nature arise during changes in the velocity of the headwind or the incident airflow at flight altitude. For example, with an increase in the velocity of the incident airflow, the airspeed diminishes. This provides a surplus of engine thrust. Subsequently, an increase in airspeed occurs logarithmically. If the lateral component of the wind speed changes pressure on the surface of the aircraft arises: a lateral Qz~CzSt pvl where Cg is the coefficient of lateral drag of the aircraft ; the cross-sectional area of the aircraft in the XI plane; V^ lateral velocity component equal to Ug . The initial lateral acceleration of the aircraft is: rfi^z Qz dt at is the Subsequently, the lateral velocity of the aircraft will log- arithmically approach the lateral component of the wind velocity, i.e., the flight condition will approach a condition which is stable in direction. Usually, during navigational calculations for each parameter, its mean value for a definite length of time is called a stable flight condition: mean velocity, mean vertical velocity, mean di- rection, etc. From the point of view of maintaining flight direction, air- craft roll is an unstable condition. If a given trajectory is curv- ilinear, the roll condition is also examined as stable or unstable. The entrance or exit of an aircraft from roll , as well as roll with variable banking, can serve as examples of unstable roll conditions. The rolling of an aircraft is considered to be coordinated if the longitudinal axis of the aircraft constantly coincides with the tangent to the trajectory of its movement, i.e., external or inter- nal aircraft glide is absent. This is achieved by tilting the rud- der of the aircraft for banking in a roll. During banking of an aircraft, its lift (J) is directed not along the vertical plane but along the axis of the aircraft, which is deflected from it (Fig. 1.4). /12 Rolling of an aircraft without descent or with stable vertical velocity is possible only when the vertical component of the lift (.Yl) is equal to the weight of the aircraft G. In this case, the horizontal (centrip- etal) component of the lift is: K«=GtgP. Fig. 1.4. Resolution of Forces During Rolling of an Air- craft . where craft is the banking angle of the air- Since we are examining a coordinate roll (without gliding of the aircraft), the centrifugal force in the roll mV2 R will be equal to the centripetal force , i.e., = Otgp, /? where m is the mass of the aircraft; and R is the radius of the co- ordinated roll. Transforming this equation, taking into account that m = g we will obtain formulas for determining both the radius and path of the aircraft with coordinated roll: R 1/2 fi-tgP ; ^'^ 2t:R. (1.6) Formulas (1.6) relate the radius of stable coordinated roll of the aircraft with the airspeed and also with banking in rolling, and they are used in calculations of the radius and path of the air- craft along a curvilinear flight trajectory. 3. Form and Dimensions of the Earth /13 In the practice of aircraft navigation, it is necessary first of all to deal with distances and directions on the Earth's surface which are the result of the mutual distribution of objects through which the flight path passes. The Earth's surface, its relief and mutual distribution of ob- jects can be most accurately expressed on a model of the Earth (a globe). However, a globe with a representation of the Earth's sur- face that satisfies the demands of aircraft navigation would be so large that its use in flight would be impossible. Therefore, dif- ferent means of representing the surface of the Earth, which is curved in all directions, on a plane (sheets of paper) are used. The Earth has a complex form called a geoid (without consider- ing the local relief, if we imagine that its entire surface is cov- ered with water at sea level). The surface of a geoid at any point is perpendicular to the direction of the action of gravity. A de- scription of a geoid by mathematical expressions is very complex, and if we consider the folds in the relief of the Earth's surface, then it is practically impossible to express its form mathematically, Therefore, in calculations the form of the Earth is taken as an e'Lt't'pso'id of revotuiion, the form closest to a geoid. Fig. 1.5. Great and Small Circles on the Earth's Surface. a) Semi- axis of the Earth and Great Circle; b) Small Circle. According to measurements made by Soviet scientists under the supervision of F. N. Krasovskiy, the major semiaxis of this ellip- soid (a), which coincides with the radius of the equator, is equal to 6,378,245 km. The minor semiaxis of the ellipsoid {b) , which coincides with the axis of the Earth's rotation, is equal to 6,356,863 km (Fig. 1.5, a). /14 The flattening of the Earth at the poles is — '^~* 1 ~ a ^ 298^3 ' These dimensions show that the Earth's ellipsoid of revolution is practically close to a sphere; to simplify the solution of the majority of problems in aircraft navigation, it is taken as a true sphere, equivalent in volume to the Earth's ellipsoid. The radius of such a sphere is equal to 6371 km. The maximum distortion of distances caused by the replacement of the Earth's ellipsoid by a sphere does not exceed 0.5%, and the distortion of directions is not more than 12 minutes of angle. In geodesy and cartography, the plotting of maps, as well as in other branches of science where more accurate calculations of distances and directions are necessary, the Earth's surface is taken as an etZipso'id of revolution. 4. Elements Which Connect the Earth's Surface with Three-Dimensional Space Taking the Earth as a true sphere , we will locate a perpendicu- lar (a resting pendulum) at any point above the Earth's surface. Then, disregarding the possible insignificant deviations caused by the varying relief, the irregularity of distribution of the densest masses in the Earth's crust, and the tangential accelerations con- nected with the Earth's rotation, it is possible to consider that the line of the perpendicular runs in the direction of the center of the Earth. The perpendicular line (see Fig. 1.5, a) joining the center of the Earth with the point of the observer's position, and continued in the direction of the celestial sphere (Y) , is called the geo- oentr-io veTtiaal of the locus. The plane on the Earth's surface, tangent to the sphere at the point of the observer and perpendicular to the true vertical of the locus, is called the plane of the true horizon. The direction and velocity of aircraft movement at every point on the Earth's surface are examined in the plane of the true hori- zon, while the altitude change is examined in the direction of the true vertical. If we cut the plane of this true horizon in any direction by another plane along the true vertical (through the center of the Earth), the line formed by the intersection of this plane with the Earth's surface forms a closed great circle, the mean radius of which will be equal to the radius of the Earth. The shortest distance between two points AB on the Earth's sur- face or part of the arc of a great circle is called the orthodrome (see Fig . 1.5, a ) . The mean radius of a great circle is assumed to be equal to 6371 km. The length of the circumference of such a radius is equal to 40,000 km. One degree of arc of a great circle is equal to 111.1 km, while one minute of arc is equal to 1,852 km. The length of a segment of the arc of a great circle at one minute of angle is called a nautical mile. /15 With an intersection of the Earth's sphere by a plane which does not pass through the center of the Earth, the line of inter- section of this plane with the Earth's surface forms a closed smatt a-VTote y the radius of which will always be less than the mean radius of the Earth. The small circles parallel to the plane of the equa- tor are called iparattets (see Fig. 1.5, b). M(X,!iZ} Fig. a Sp nate R fr gles dius and 1.6. heric s and om th : an -vect the d For the purposes of aircraft navigation, a coordinate system which unequivocally determines the position of an aircraft and objects on the Earth's surface is necessary. Obviously, a spherical coordinate system will be the most convenient (Fig. 1.6), A spherical coordinate sys- tem is distinguished from a rec- tangular system (Cartesian) by the fact that instead of deter- mining three distances to a point in the directions of the X, J, and Z axes, we determine the length of the radius-vector e center of the coordinate system to a point, and two an- gle A between the XY plane and the projection of the -ra- or (i?) to the plane XZ , and angle (j) between the XZ plane irection of the radius-vector (i?) . Relationship Between al System of Coordi- a Rectangular System. There is an obvious relation between spherical and rectangular- coordinate systems : X= Rcostf cos X; \ K=/?sin(f; I 2 = ;? cos if sin X. J (1.7) With a constant length of the radius-vector R, if angles X and ^ assume all possible values, the geometric location of the points of the end of the vector radius will be a sphere. To determine coordinates on the Earth's surface, there is no need to indicate the radius of the Earth (i?) each time. This coor- dinate is considered, once and for all, constant. /16 Thus, the spherical coordinate system is transformed into a two-dimensional surface system which is called a geographic system of coordinates. The plane of the equator and the plane of the prime (Greenwich) meridian are taken as the initial reference planes in a geographic coordinate system. The point coordinates on the Earth's surface bear the name "longitude of the locus" and "latitude of the locus" (Fig. 1.7). 10 The dihedral angle between the plane of the prime meridian and the plane of the meridian of a given point is called the longitude of the point (X). Determination of the longitude can be given in arc values : the length of the arc of the equator (or the paral- lel), expressed in degrees, be- tween the prime meridian and the meridian of a given point is called the longitude of the point. Fig. 1.7. Spherical Coordinate System on the Earth's Surface. Reading of the longitude is carried out from to 180° east of the rpime meridian {east long- itude ) and f rom to 180° west of the rpime meridian {west long- itude ) . In navigational calcu- lations , east longitude is taken as posi tive and is designated by a plus sign , while west longitude is nega tive and is designated by a minus sign However, in carry- ing out navi gational calculations , it is more convenient to carry- out a readin g of longitude in the easterl y direction from zero to 360° . The angle between the plane of the equator and the true verti- cal of a given point (or the length of the meridian arc, expressed in degrees, from the plane of the equator to the parallel of a given point) is called the latitude of the -point ( (j) ) . Since a set of true verticals at a constant latitude forms a cone with the vertex in the center of the Earth and an angle at the vertex equal to 90°-((), then in contrast to the dihedral angle between the planes of the m.eridlans , we shall call a similar angle in other spherical systems, the conia angle . Reading of the latitude is carried out from the plane of the equator to the north and south from to 90° {north and south lati- tude) . In navigational calculations, north latitude is considered positive and south, negative. A geographic coordinate system is a surface curvilinear system, /17 i.e. , the meridians of the coordinate grid on the Earth are not parallel. However, if we examine the meridians and parallels on any unit area of the Earth's surface, they turn out to be orthogonal (perpendicular in one plane). Two special points on the Earth's surface (the geographic poles) are an exception. A geographic coordinate system is used not only to determine the location of a point (object) on the Earth, but to determine direction from one point to another. 11 The angle included between the northern direction of the meri- dian which passes through a given point and the orthodrome direction to a point setting a course is called the heaving or azimuth. Read- ing of the angles of bearing or azimuth is done clockwise from to 360° . Since the meridians on the Earth's surface are generally not parallel, the value of the azimuth changes with a change in the mov- ing longitude along the line which joins the two points; the greater the latitude , the more it changes . Therefore , for the orthodrome direction together with an indication of the azimuth, it is neces- sary to mention from which meridian thia direction is measured. The change in azimuth with a change in the moving longitude does not make it possible to use magnetic compasses for moving along the orthodrome without introducing corresponding corrections, espe- cially when the two points are far apart. If the magnetic declination does not change, following a con- stant magnetic course will cause the meridians to intersect at iden- tical angles. The line which intersects the meridians at a constant angle is called the loxodrome . In order to proceed to a more detailed examination of the ele- ments of aircraft navigation and their measurement, it is necessary to become acquainted with the making of maps, their scales, and some features of cartographic projections. 5. Charts, Maps, and Cartographic Projections The representation of a small part of the Earth's surface on a plane is called a chart. Distortion as a result of the curvature of the Earth's surface is practically absent on a chart. The conventional representation of the Earth's surface in a plane is called a map. A map is a continuous representation of the surface of the Earth or a part of it without discontinuities and folds, made with a variable scale according to a definite rule. The sphericity of the Earth's surface does not allow it to be represented with com- plete accuracy on a plane surface. Therefore, there are many ways /18 of projecting the Earth's surface onto a plane which make it possi- ble to represent most accurately on the map only those parameters (elements) which are most necessary under the given conditions of application . Methods or laws of representing the Earth's surface on a plane are called cartographic projections. A common geometricat projection is the point of intersection of the line of sight (which passes through the eye of the observer 12 and the projected point) with the plane onto which the given point is projected. It is a special case of cartographic projection. A cartographic projection is set analytically as a function of geographical coordinates on the Earth (sphere) between the coordi- nates of a point on a plane. If we call one of the main directions on a map the X axis and the perpendicular to it the Z axis , then ^=^i('P; I) and 2 = ^2(9; ^); P = ^3(?; ^) an d 8 = Ft (<f,; X), where p and 6 are the main directions on maps of conic and azimuthal projections, and (J) and X are the geographical coordinates of a point on the Earth (sphere). The properties of the projections will depend on the properties of these functions (Fi, F2 , F3, and F14 ) , which must be continuous and well-defined, since the map is made without discontinuities so that a single point on the map corresponds to every point in the locat ion . Map Scales The map-making process is divided into two stages. a) The Earth is decreased to the definite dimensions of a globe . b) The globe is unrolled to form a plane. The extent of the overall decrease in the Earth's dimensions to the fixed dimensions of a globe is called a principal scale. A principal scale is always indicated on the edge of a map and makes it possible to judge the decrease of the length of a segment in transferring it from the Earth's surface to the globe. A principal scale is numerically equal to the ratio of the distance on the globe to the actual distance at a location: M: A. S, AS e. s . where M is the principal scale, ASg is a segment on the globe, and ^"^e . s . is a segment on the Earth's surface which corresponds to the segment on the globe. On maps, the principal scale is usually shown as a fraction (numerical scale) and by means of a special scale (linear scale). 13 The numeTicat soate is a fraction, the numerator of which is one, while the denominator shows how many such units of measurement fit into the location. For example, 1:1,000,000 means that if we take 1 cm on a map, then 1,000,000 cm at a location (i.e., 10 km) wil^ correspond to it. A t'lneav soate is a scale on a map in which a definite number of kilometers at a location correspond to special segments of the scale . However, a principal scale (numerical and linear) is insuffi- cient for accurately measuring distances on the entire field of a map. It is necessary to know the laws of distortion of distances and directions. The laws of change in the principal scale along the map field are determined by a special scale. A special scale is the ratio of an infinitely small segment in a given place on the map in a given direction, to an analogous seg- ment in a location (globe). At each point on the map, the special scale is different. It is either somewhat larger or somewhat small- er than the principal scale. Distortions of Cartographic Projections Ettiipse of Distortions Let us draw on a sphere (globe), an infinitely small circle with radius t\ let us also designate a rectangular coordinate system" on the sphere by x and z (Fig. 1.8, a). Then /■2 = ;e2 + z2. (1.8) Fig. Distortion of Scales on a Plane: (a) Scale on a Globe; (b) Scale on a Plane. 14 In the transfer of the coordinate system from the sphere (globe) to the plane, the direction of the coordinate • axes is dis- torted (Fig. 1.8, b). Having designated the special scales on a plane (map) by m in the direction X and n in the direction s, we obtain: *i = mxi m 'iwhile*= '=~ m n /20 Substituting the latter in (1.8), and then dividing both sides of the equation by p^ , we obtain \mr ) [ nr )~ (1.9) From mathematics , it is known that this is the formula of an ellipse with conjugate diameters; therefore: a) Any infinitely small circle on the surface of the Earth's sphere in any projection is represented by an infinitely small el- lipse . b) On the surface of the Earth's sphere (globe), it is pos- sible to choose two mutually perpendicular directions which will be transferred to a map without any distortions. These directions are called principal directions. Knowing the special scales (m and n) in the principal direc- tions, it is always possible to construct an ellipse of distortions which will make it possible to judge the nature of the distortions of the projection as a whole. In the majority of projections, the directions along the meridians and parallels are taken as the prin- cipal directions. Distortion of Lengths If an infinitely small circle on the Earth is represented by an ellipse (Fig. 1.9, b) with its transfer to a plane, the distor- tion of the special scale in any direction {hS^) can be expressed as follows: ,y /21 AS.= OM (1.10) 15 but from the circle in Figure 1.9, a: jf=slnor, wh i 1 ei^ = cosar, AS, = ym2sin2a + n2cos2o , then (1.11) Fig. 1.9. Distortion in a Plane: (a) Length on a Globe; (b) Length on a Plane . Z Z, Fig. 1.10. Distortion of Directions on a Map. (a) Direction on a Globe; (b) Direction on a Map i.e., knowing the special scales for the principal directions, we can always judge the value of the distortion of the special scale in any direction (and therefore, the distortion of the length of the segment as a whole). Distovt-ion of D-ireat'ions Let us take the radius r = 1 (Fig. 1.10) of an infinitely small circle on the Earth; then tg«=,-^, while tgp=-^. (1.12) 16 Dividing Equations (1.12) into one another, we obtain: tgP= tga. m (1.13) Obviously, knowing the special scales for the principal direc- tions , it is always possible to find an angle g on a map for an an- gle a in a location, and vice versa. Distortion of Areas The distortion of areas AP can be determined by a comparison or division of the area of the ellipse (.Sq±) by the area of a circle (5(2i); see Figure 1.11: /22 AP=- ' e 1 nab ab 'Cl 7cr2 r2 ' (1.14) but if we take the radius of the circle on the Earth as equal to 1, then ^P = ab or 5 if we express a and b by special scales for the principal direc- tions, we obtain: '^P="tn,, (1.15) Fig. 1.11. Distortion of Areas on a Map. (a) Area on a Globe; (b) Area on a Map. The distortion of areas is equal to the product of the special scales for the principal directions . Hence , we see that if we know the special scales for the prin- cipal directions , we can give the complete characteristics of any map projection.. 17 Classification of Cartographic Projections There are many cartographic projections. They can be divided according to two basic characteristics: (a) according to the nature of the distortions, and (b) according to the means of construction 'or the appearance of the normal grid. By normal grid we mean the coordinate system on a globe which is most simply represented on a map. Obviously, this is a system of meridians and parallels . Division of Projections by the Nature of the Distortions The choice of cartographic projections depends on the problems for whose solution they are intended. According to the nature of the elements which have the least distortion on a map, cartographic projections are divided into the following groups: 1. Isogonal or conformal projections These projections must satisfy the requirement of equality of angles and similarity of figures ( conf ormability ) within the limits of unit areas of the Earth's surface, i.e., so that in projecting a surface of a globe onto a plane (map)., the angles and similar fig- ures do not change. X I b) 11 Fig. 1.12. Conf ormability of Figures on Maps. (a) Preserving the Conformability of a Unit Area; (b) Destroying the Conf ormability of a Long Strip . According to the stipulation, the angle on a map must be equal to the angle at the location: L & = La, but from (1.13) it is ob- vious that in this case m = n . Therefore, the equation of special scales for principal direc- tions is a condition for isogonality. On large parts of the surface, within the limits of which it is impossible to disregard the change in scale, the conf ormability (and therefore the isogonality) are not preserved. Figure 1.12 gives an example of preserving the conf ormability of a unit area and destroying the conf ormability of a long strip. The unit area (Fig. 1.12, a) is transferred to the map on a definite scale without distortions. The long strip (Fig. 1.12, b) can be divided into a number of unit areas , each of which will be transferred to the map on a somewhat changed scale. Since the scales mx and ns are increased proportionally in the direction of the strip, each of the small areas is represented on the map with the conforma- bility being preserved, only on a different scale. By equating the lateral limits of the small areas , we do not obtain a conf ormal figure, i.e., the similarity of small figures in isogonal projec- tions is preserved, while the similarity of large figures (large lakes, seas, etc.) is destroyed. 2. Equally spaced or equidistant projections /2H- The equivalence to unity of the special scales for a principal direction (m = 1 or n = 1) is a necessary condition of this group of projections. .) b) o L] CJ -E^ .•^t Fig. 1.13. Conf ormabi Spaced Pro (a) Appear ure in a L Appearance on a Map . Distortion o lity in Equal j ect ions : ance of a Fig ocation ; ( b ) of the Figur f This means that the map scale will be preserved in one of the principal directions. Therefore, when using such a map we can measure the distance in one of the directions by means of a scale. The nature of the distortion of conf ormability in these projections is shown in Fig. 1.13. Here m = const, while n is a function of Z. 3. Equally large or equivalent projections This group of projections must satisfy the condition of equivalence of areas, i.e., the product of the special scales for the principal direc- tions must equal unity (mn = 1); there- fore, the relation between the special scales for the principal directions will be inversely proportional: 1 m> = These projections do not have an equivalence of angles and a similarity of figures. 19 ti 4. Arbitrary projections Projections of this group do not satisfy any of the conditions mentioned above. However, they are also used when comparatively small portions of the Earth's surface are projected onto a plane where the distortions of the angles and the scales for the principal directions and along the entire map field are insignificant and the similarity of figures and areas which satisfy the needs of their practical application is preserved. This group of projections in- cludes a basic flight map on a scale of 1:1,000,000, which is con- structed according to a special law and which has been accepted by international agreement. For the purposes of aircraft navigation, the most necessary conditions are (obviously) isogonality and equal scale of the maps. Equally large and equally spaced projections of maps are used in aircraft navigation only as survey maps for special applications. They include maps of hour zones, magnetic declinations, composite /2 5 diagrams of topographical map sheets, climatological and meteorolo- gical maps, etc. B-ivis-ion of ProQections Aoaording to the Method of Construation (Aaaording to the Appearance of the Normat Grid) Depending on the method of construction, cartographic projec- tions are divided into several groups, the bases of which are the following : (a) group of cylindrical projections; (b) group of conic projections and their variants, polyconic pro j ections ; (c) group of azimuthal projections; (d) group of special projections. Each of these projections is divided in turn into the following categories: normal ^ if the Earth's axis concides with the axis of the figure onto which the Earth's surface is projected; transverse ^ if the Earth's axis forms an angle of 90° with the axis of the fig- ure, and oblique J if the axis of the Earth does not coincide with the axis of the figure and intersects it at an angle which is not equal to 90° . Cylindrical Projections Normal (equivalent) cylindrical projection All cylindrical projections are formed by means of the imagin- ary transfer of the Earth's surface (globe) to a tangential or in- tersecting cylinder, with subsequent unrolling. In Figure 1 . 14- , a simple normal cylindrical projection is given, i.e., a projection of the Earth on a tangential cylinder, 20 the axis of which coincides with the axis 'of the Earth (globe), while the height of the cylinder is proportional to the length of the axis • © e Fig. 1.14. Normal (Equivalent) Cylindrical Projection ^ j3zr~~~^ — r* '" - o- ^' ___— --'^ Fig. 1.15. Simple Equally Spaced Cylindrical Projection In this projection, the meridians are compressed while the parallels are extended to a degree which increases with latitude. The projection includes a category of equally large and equivalent projections, since it satisfies the condition of an equivalence of areas . Its equation can be written in the following form: (1.16) where X represents the coordinates of a point along the meridian; Z represents the coordinates of a point along the equator; and R is the Earth's radius. Let us determine what the special scales for the directions are equal to in this projection: /26 21 m = dSmap rdf d5,globe R'if dS^map RdX R cos yrfy Rdf Rd\ ■■ COS f ; ''•^■qlobe '"''^ Rcosdl cosf = secf , (1.17) (1.18) where m is a partial scale along a meridian; n is a partial scale along a parallel; dS^-^^ is an increase in distance on the map; "-^•^elobe ^^ ^" increase in distance on the globe. The product of the special scales is ' =1 or m = -^, whi le n = — mn ==cos ^1 COS<f Therefore, the given projection is equal. Since m ^ n; m ^ 1 and n 7^ 1 in the principal directions (meridians and parallels) it is not isogonal and not equally spaced. Only in the equatorial band, in the limits from to ±5° along its latitude, is it practically possible to consider it isogonal and equally spaced. Simple equally spaced cylindrical projection If we take the height of a cylinder to be proportional not to the length of the Earth's axis, but to the length of a meridian, and instead of simply projecting we unfold the meridians to the cylinder walls, as shown in Fig. 1.15, then a simple, equally spaced cylindrical projection is obtained. It is regarded as normal since the axis of the globe coincides with the axis of the cylinder. In this projection, the meridians will be transformed to their full size during their transfer from the globe's surface to a map (i.e., m = 1), and the equator also will be transformed to full size (at the equator, n = 1), while the parallels will be extended just as in a normal (equivalent) projection. The magnitude of the effect increases with latitude. The coordinate grid of the map of this projection has the ap- pearance of a uniform rectangular ruling. Its equations have the form : x==Rr, Z^R\. The special scales are equal to: /27 along the meridian along the parallel dS map dS globe 1 COSip Rdtf Rdf = 1; = sec y. (1.19) (1.20) 22 Since m = 1, the projection is equally spaced along the meri- dians and also along the equator. Since m ^ n and mn ^ 1, the pro- jection is not isogonal and not equally large, except for the equa- torial band in the limits from to ±5° along the latitude, where it is practically possible to consider it Isogonal and equally large. Maps in normal (equivalent) and simple, equally distant cylin- drical projections are used in aviation only as references: maps of hour zones, maps of natural light, etc. Isogonal cylindrical projection An isogonal cylindrical projection (Mercator projection) is the most valuable of all the cylindrical projections for navigation. It is obtained from a simple, equally spaced cylindrical projection by artificially extending the scale along the latitude (lengthening the meridians), proportional to the change in scale along the longi- tude. The coordinate grid of the map of this projection is shown in Figure 1.16. The reason for its use is the fact that the angles measured on /28 the map are equal to the corresponding angles at the location, i.e. , m - n = sec(i). c„' W" 20''J0' 4 0' SO' SO' 70' SO' SO' Wd' IW' 120' "" I I I 1 1 1 — -T — 1 1 1 A~, .gg- 50' 40' 30° 20° 10° W 20' 30' MO' SO' 60' SO' fO' 30' 70' 10° W 20' 30' 40' 50' Let us write an equa- tion of this map projection along a meridian (J-coor- dinate) for which we can find m: dS map ax ''^ globe ^'^f where dS is an increase of distance along the meridian on the map; and Rd(^ is an increase in distance along the meridian at the loca- tion. We must have m = sec<p, We shall then equate the right-hand sides of these equations : 10' 20' 30° «r SO' eo° jo' so' so' loo'm' bo^"' Fig. 1.16, Coordinate Grid of an Iso- gonal Cylindrical Projection. -^^='sec*>hencedA-=-^^, (l.21) Rdf ^' cos<p 23 I After integrating (1.21), we will obtain the J-coordinate along the meridian: /29 A" = /? In tg («-.f). (1.22) while the Z-coordinate along the parallel is determined by the sim- ple equation : ^'=^^- (1.22a) Since m= n, the projection is isogonal but not equally spaced (m 7^ 1 and n / 1) and not equally large imn ^ 1). The basic advantage of maps in an isogonal cylindrical projec- tion is the simplicity of their use with magnetic compasses for moving from one point on the Earth to another, since the loxodrome in this projection has the appearance of a straight line. Therefore, the isogonal cylindrical projection has been used widely, primarily in marine navigation during the compilation of naval maps. The change in scale with latitude is a disadvantage of normal cylindrical projections. Here, in normal (equivalent) and simple, equally spaced cylindrical projections, the map scale is not identi- cal in the principal directions (north-south and east-west), so that the distance between two points in directions not parallel to the lines of the grating can be determined only by calculation. In an isogonal cylindrical projection, the map scale along the latitude is also variable, but at any point on the map it is identi- cal in the principal directions. This makes it possible to measure distances by means of compasses, for which a scale (varying with the latitude) is drawn on the western and eastern edges of the map. Means for measuring distances on maps with such a projection are indicated in manuals for marine navigation. Isogonal oblique cylindrical projections The basis for creating maps in an isogonal cylindrical projec- tion is a property of the Mercator projection: its isogonality. Such projections are used in the preparation of special flight maps on scales of 1:1,000,000, 1:2,000,000, and l:^, 000, 000 which are used in civil aviation. The tangential (Fig. 1.17) or intersecting (Fig. 1.18) cylinder is situated at such an angle to the axis of the globe that the tan- gent of the cylinder's surface to the globe or the intersection runs along the flight path. Usually the strip along the tangent does not extend more than 500-600 km to either side of the route (or the mid- dle line of the route, if it has discontinuities), while on the in- tersecting cone it does not extend more than 1000-1^+00 km to either side of the given middle line of the routes. 24 In practice, such flight maps are isogonal, equally spaced, and equally large; however, since the cylinder is in contact with the globe along the arc of a great circle or cuts the globe compar- atively close to the arc of a great, circle, the orthodrome on these maps will in practice be represented by a straight line. The distortions of lengths on flight maps of oblique tangential projections do not exceed 0.5%; for intersecting projections they do not exceed 0.8%-1.2%. /30 Fig. 1.17. Isogonal Oblique (Tangential) Cylindrical Pro j ect ion . Fig. 1.18. Isogonal Oblique (Intersecting) Cylindrical Pro j ection . Isogonal transverse and cylindrical Gaussian projection The axis of the cylinder in Gaussian projections is perpendicu- lar to the axis of rotation of the Earth (globe). The construction of maps with this projection is similar to the construction of maps with oblique cylindrical projections. For example, a flight map on a scale of 1:1,000,000 for Leningrad-Kiev has been compiled on such a projection. However, on the whole, isogonal transverse cyl- indrical Gaussian projection is used for compiling maps on a large scale, where the special principles of construction are used. A spheroid (Earth's ellipsoid) is taken as the figure from which the Earth's surface is projected, while the tangential cylin- der on which the Earth's surface is projected has an elliptical base according to the form of the Earth's ellipsoid. The entire Earth's surface is divided by meridians into zones, each of which has a latitude of 6° and is projected onto its own cylinder which is tangential to the Earth's surface along the mid- dle meridian of the given zone. Thus, in order to project the whole surface of the Earth, it is necessary to turn the elliptical cylinder mentally around the 25 axis of the Earth's ellipsoid through 6° at a time, In Figure 1.19, a, the projection of only one zone for 6° longitude is shown, while in Figure 1.19, b, the unrolling of a semicyllnder after its rota- tion around the Earth's axis in order to project several zones is shown. With such a projection, all maps are constructed on the scales: 1:500,000, 1:200,000, 1:100,000, 1:50,000, and 1:25,000. The latter are essentially charts. /31 V- 0' 3' 3' 15' 21' 27° 33' 39' Fig. 1.19. Isogonal Transverse-Cylindrical Gaussian Projection. Each zone on maps with a scale of 1:200,000 and larger has its own special X and J(Z) rectangular coordinate system, which is called the Gaussian kilometer system. Meridians and parallels on maps of this projection are curved lines and do not coincide with the Gaussian system. The vertical lines of the rectangular Gaus- sian system are parallel to the central meridian of the zone and do not coincide with other meridians of the zone. The angle between the vertical line X of the Gaussian system and the line to the object (point) is called the d'ivect'tonal angle. In order to obtain the true or magnetic direction (angle), the angles of the convergence of the system with the true and magnetic meridians are indicated on the edge of the map. In addition, the vertical section of a map (frame) always runs in the direction of the true meridian. By means of the Gaussian system and figures in the frames of the maps, it is possible to determine the distance from the equator and from the central meridian of the zone to the object (point). Distortions of lengths on these maps are insignificant and do not exceed 0.14-% along the edges of the zone in the latitude which is equal to zero (l^lO m at 100 km). Maps on an isogonal transverse-cylindrical Gaussian projection are used both in aviation for a detailed orientation and location 26 of targets , and in many branches of the national economy for linking projects, equipment, and radio engineering facilities in a location, /32 for determining geodesic reference points, and for accurate geodesic calculations of distances and directions, etc. Conic Projections Conic projections are constructed by projecting the surface of the Earth's spheroid (globe) on a tangent or intersecting cone, with its subsequent unrolling to form a plane surface (Fig. 1.20, a). °' /\ N N J /- K. /^ ~~>v /,trr: 4~~] / ^ \_ _y b) Pn Fig. 1.20. Construction of Conic Projections: (a) Tangent (inter- secting) cone; (b) Unrolling of the Cone to Form a Plane. According to the positions of the axes of the globe and cone, conic projections can be normal, transverse, and oblique. However, in our publications normal projections are generally used when the axis of the cone coincides with the axis of the globe. In a normal conic projection, meridians are represented by straight lines, while parallels are represented by arcs of concen- tric circles (Fig. 1.20, b). From Figure 1.20, a, it is easy to see that the radius of a parallel of tangency (Pq) can be expressed by the Earth's radius: po = /? ctg <po. where R is the radius of the Earth (globe) and tj) q is the latitude of the parallel of tangency. form ; The equation of this projection is written in the following P = Po + ^ ("Po + t); Translator's note: ctg = cot. (1.23) 27 It where 6 and p are the principal directions in the polar coordinate system along the parallel and meridian, respectively, and a is the coefficient for the angle of convergence of the meridians. Simple normal conic projection A simple normal conic projection is constructed with the con- sideration that the meridians on the whole map and the parallel of tangency be transferred from a globe without distortions to their natural value (i.e., m = 1), while for the parallels of tangency ((fiQ ) w = n = 1. Such a projection forms the basis of the improved intersecting conic Kavrayskiy projection (Fig. 1.21, a). It is equally spaced, since m - 1, while on the intersecting parallels it is isogonal and equally large (Fig. 1.21, b). /33 ^^ Inte rsection p aral lels Fig. 1.21. Simple Normal Conic Projection, (b) Unfolding of the Cone to (a) Intersecting Cone; a Plane . Many aircraft maps with scales of 1:2,500,000, 1:2,000,000, and even 1:1,500,000, which are used in aircraft navigation for general orientation and the approximate determination of the posi- tion of an aircraft by means of radio engineering facilities (air- craft radio compasses, ground radiogoniometers, etc.), have been published . Their positive feature if the insignificant distortion of lengths in the strip ±5° from the intersecting parallels, which does not exceed 0.34% (340 m for 100 km). Their disadvantage is the distortion of directions , which increases with distance from the intersecting parallels. 28 Isogonal conic projection By analogy with the construction of an isogonal cylindrical Mercator projection, destroying the equal spacing, a simple normal conic projection is transformed into an isogonal projection by re- ducing (equating) the scale along the meridians to the scale along the parallels (m = n) . This is more valuable for use in aviation. Aircraft maps with a scale of 1:2,000,000 and survey maps on scales of 1:3,000,000, 1:4,000,000, and 1:5,000,000 are published with a normal isogonal conic projection for aviation. Maps with a scale of 1:2,000,000 in this projection, besides having the basic advantage of isogonality, also have distortions of length which are permissible in the practice of aircraft navigation. On an intersecting cone in a strip from 40° to 70° in latitude, the maximum length distortions do not exceed ±1.8 km for 100 km. /34 Fig. 1.22. Angle of Convergence of the Meridians of a Tangent Conic Projection: (a) Arc of a Parallel on a Globe; (b) Arc of a Parallel on a Map. The orthodrome on maps of an isogonal conic projection for dis- tances up to 1200 km appears as a practically straight line. This valuable quality is used during flights on civil aviation airlines of average length by using gyroscopic and astronomical compasses for following the orthodrome. At great distances, the orthodrome (as a result of a change in scale) is bent by a bulge tending to- ward a larger scale. The loxodrome is represented by an arc of a logarithmic spiral. 29 This creates dif f iculties • in aircraft navigation by means of magnetic compasses. In these instances, for distances up to 500-800 km in directions which intersect the meridians on a map, a straight line is constructed, while measurement of the flight angle is carried out along the central meridian of the route which is maintained in flight by means of a magnetic compass. It is also possible to construct (continue) the loxodrome along an angle measured in the middle of the straight line joining the /35 control (rotating) landmarks of the route. The disadvantage of all maps with conic projections is the pre- sence of an angle of convergence of the meridians from the parallels of tangency (parallels of intersection) to the pole. It is neces- sary to consider this angle when determining directions (flight angles) or the location of the aircraft by means of aircraft radio compasses. In addition, depending on the parallels of intersection or tangency, the angle of convergence of the meridians will be dif- ferent . Convergence angle of the meridians The principal scale of conic projections is taken along the meridians and parallels of tangency or intersection ( (j) q ) • There- fore, the arc MU is equal to the arc Mi^i (Fig. 1.22). It is known that on a globe (spheroid) (Fig. 1.22, a), the arc MN - rAA , where T is the radius of the parallel. On a map of a conic projection (Fig. 1.22, b) the arc M^^^ = PqAS; then rAX = poA8. (1.24) But T = R cos (j) and p q = -R cot cjiQ, and from the equation of a conic projection (1.23), A6 = aAX. Substituting the values of r, pq, and A6 in (1.24) and carrying out the necessary reductions, we obtain: 0= iinipo. (1.25) Obviously, on the equator the coefficient of convergence of the meridians a = 0, since sin 0° = 0; at the poles a = 1, since sin 90° = 1, and in the general case for central latitudes, 0<a<l. Knowing the coefficient a, it is not difficult to determine any angle of convergence of the meridians 6 along a parallel of tangency or intersection : * = ^^*' (1.26) where AX is the difference in longitude between the given meridians. At any other latitude, the coefficient a will be different from 30 the coefficient a at a latitude of tangency (intersection). There- fore, for approximate calculations in the practice of aircraft navi- gation during the determination of flight angles or location of the aircraft, the mean latitude of the route, part of the route, or the distance between the aircraft and the radio station, is taken as 6 = (A2-Xi) sincfi^^^ or 6 = (Xr.-Xa) sine mid where X2 ^''^'^ ^l s^e the longitudes of the final and initial points, Xp and Xg are the longitudes of the radio station and aircraft , re- spectively, and (|>niid is the middle latitude between the indicated points (places). In some cases , for approximate determinations of the location of an aircraft or the flight angles, the coefficient a is assumed constant for a given map of a conic projection. Thus, for example, for a map with a scale of 1:2,000,000 and a normal isogonal conic projection, it is possible to let a (w 0.8, which corresponds to the sine of the latitude of the middle parallel between the intersection parallels, where the map scale will be minimum. Polycom ic projections Polyconic (multiconic) projections are the greatest perfection of conic projections for the purpose of decreasing distortions of lengths and angles in projecting the Earth's surface onto a plane. The principle of construction of such projections is shown in Figure 1.23, a. The central meridian of the projections is a /36 Fig. 1.23. Polyconic Projection: (a) Intersecting Cones on the Globe; (b) Unrolling of Cones on a Plane. 31 straight line, while meridians in the form of curved lines are sit- uated to the west and east of it. The parallels are concentric circles with different centers, lying on the central meridian (Fig. 1.23, b). As a result of the increase in scale in proportion to the distance from the central meridian to the west and east, such projections are used only to represent the Earth's surface in coun- tires extended along a meridian. International projection In terms of the method of construction, an international pro- jection is related to a modified polyconic projection; in terms of the nature of the distortions , it is related to an arbitrary pro- j ect ion , E^ h° of to 64°; for alj the pr; as a Ti parall« central 1 (Fig. sheet c cipal £ of the map wi 6° of 1 icted ac of a gi i is giv interse = 1) an ' the sh In the of long 56 sheet igh 4°) range itude ; s is ; and a'. 1,000,000, which encompasses a range of latitudes from :s own law, which is general lal strip. On each sheet, outer parallels of the sheet globe by a cone along these leridlans , separated from the the west and east, where m = Etudes from 64° to 80° , each -om 80 to 88°, 24°. The prin- ^en along the outer parallels ; meridians which are distant /37 —7—2 =/ L^ ■/ ^6' -"' ■^^^zmsiiilr^rwmB'^''^ Fig. 1.24. International Projection: (a) Construction of the Sheet; (b) Breaks in the Splicing of Sheets. 32 from the central meridian of the sheet by 1+ and 8°, respectively. The regions of the poles are projected onto separate sheets in a central (polar) projection. The meridians in this projection are represented by straight lines which have an angle of convergence to the poles, similar to the conic projections, while the parallels are curved lines which are constructed according to a special mathematical law. The centers of the circle-parallels are situated on the central meridian of a given sample of sheets, while their radii are proportional to the cotangents of the intersection latitudes: Ri = ctg 9i: R2 = ctg 92 e t c •., According to studies by Limnitskiy, distortions of lengths on maps with a scale of 1:1,000,000 with such a projection, in the mid- dle latitudes does not exceed 0.076% (76 m in 100 km), while distor- tion of directions is 5'. The greatest distortions arise in the region of the equator: distortion of lengths up to 0.1'+%, angles up to 7' . Insignificant distortions make it possible to consider the map as a practically isogonal, equally spaced, and with equally large pro j ectlon . f maps with scales of d. In a range of latl- th a scale of 1:2,000,000 ude (nine sheets of a e of 1:4,000,000 occupies cale of a 1:2,000,000 map sheet and the meridians of the sheet by 6° to the ap with a scale of t by 8°50' to the north Istortions , and the meri- e central parallel and According to this principle, s heets 1 : 2 ,000 ,000 and 1:4,000,000 are constructe tudes from to 64° , the sheet of a map wi occupies 12° of latitude and 18° of longit mi llionth , 3 X 3) , while a map with a seal 24 and 36° , respectively . The principal s is given along the outer parallels of the wh ich are di stant from th e central median we St and eas t (Fig. 1.24, b ) , while on a m 1 : 4 ,000 ,000 the parallels which are distan an d 8°10' to the south are given wi thout d dians are 12 ° to the west and east from th th e central meridian , res pectively . /38 The distortion of lengths in the middle latitudes on maps with a scale of 1:2,000,000 reaches 0.5%, and the distortion of the an- gles is 30'; on 1:4,000,000 maps, distortion of lengths reaches 1.5%, that of angles, 1°30'. A disadvantage of maps in the international projection on all scales is the presence of discontinuities in the splicing of several sheets, as a result of the features of its construction. Sheets of maps of only one strip or one column are spliced without breaks. During splicing of nine sheets of maps on a scale of 1:1,000,000 (3 X 3), the discontinuities which arise are partially evened out by deformation of the paper, and the use of such a map does not re- sult in perceptible distortions of lengths and angles. Splicing of a large number of sheets is not recommended. 33 It is even impossible to splice a map with a scale of 1:2,000, 000 from four sheets (2 x-2) without a break. At a latitude of 60°, the discontinuity of the spliced sheets reaches 1.8 cm, i.e., 36 km (see Fig. 1.24, b). Therefore, it is possible to splice only one strip or one column of these maps. The orthodrome with a length up to 1200 km on, maps with a scale of 1:1,000,000 and 1:2,000,000 (within the limits of one sheet) appears in practice as a straight line, while the loxodrome is the arc of a logarithmic spiral. Usually, in directions which intersect the meridians, the loxodrome sections with a length up to 600 km are likewise constructed in the form of a straight line, while the flight angle is measured in the middle of a part of a route in order to lessen by a factor of 2 the error of the measured angle during flight with the use of a magnetic compass. During the determination of the position of an aircraft by means of radio compasses , a correction is allowed for the convergence of the meridians just as in maps of conic projections, with an ap- proximate formula ^ f a "mid where X-^ is the longitude of the radio station; Xg^ is the longitude of the aircraft; "^mid is the mean latitude between the radio station and aircraft, or the mean latitude of the sheet (sheets) if the ap- proximate position of the aircraft is unknown. In civil aviation, maps with a scale of 1:1,000,000 and 1:2, /39 000,000 on an international projection are used as flight maps, pri- ' marily on piston-engine aircraft and helicopters , and secondly on aircraft with gas-turbine engines. Maps with a scale of 1:4,000, 000 are used as aircraft maps for general orientation and approxi- mate determination of the location of an aircraft by means of radio- engineering facilities. Azimuthal (Perspective) Projections Azimuthal (perspective) projections are constructed according to the laws of a simple geometric perspective; therefore, they are often called perspective projections. According to the position of the plane of the figure, azimuthal projections are divided into normal or polar (Fig. 1.25, a), trans- verse or equatorial (Fig. 1.25, b), and oblique or horizontal (Fig. 1.25, c); depending on the position of the center of the projection relative to the plane of the figure, they can be of the following types (Fig. 1.26): a) Centval or gnomonioj when the center of the projection co- incides with the center of the Earth (globe): point A; 31+ b) Steviogvaphio y when the center of the projection is sepa- rated from the point of contact with the plane of the figure by a distance equal to the diameter of the Earth (globe): point B; c) Orthographic J when the center of the projection is infi- nitely separated from the plane of the figure: point Cj d) External J when the center of the projection is located above the plane of the figure: point D. Plane of the'figure Fig. 1.25. Azimuthal Projections: (a) Normal; (b) Transverse; ( c ) Oblique . PI ane the f Fig. 1.26. Position of the Centers of Projection in Azimuthal Projections. As is evident from Fig. 1.26, on such projections points M and N on the Earth's surface will be pro- jected at a different distance from the point of tangency of the plane of the figure with the Earth's sur- face . Meridians in azimuthal (polar) projections are represented by straight lines which converge to a pole at an angle equal to the dif- ference in longitude: 6 = AX. Parallels are represented by concentric circles, the radii of which depend on the center of the projection and the latitude of the position . In aviation, central polar and stereographic polar projections are generally used. /i+0 35 Central polar (gnomonic projection) The center of projection in this projection coincides with the center of the Earth (globe) at the point (Fig. 1.27, a). From Figure 1.27, it is possible to write the equation of this pro j action : p = /?ctg<i>. In order to have a complete idea of the projection, let us find the special scales (m, n) for the principal directions (meridians and parallels): dSmap _ -dp _ -d(Rctgf) where dp is the increase in the radius of the unrolling, i.e., a positive increase in latitude ( (j) ) corresponds to a negative increase /M-1 in the radius (p). Integrating the latter, we obtain: __±Rdf 1 Rd<f sec2 <f [.sec2,y or ft = Here , r = i? cos dS mcosec2<p; map priB /? ctg <fd\ ctgy 1 "^^globe '■'^^ RcosfdX cosiy sinip ' is the radius of the parallel, i.e., n = cosec <p.. (1.27) (1.28) a) Plane of the F i gu re Fig. 1.27. Central Polar (Gnomonic) Projection: (a) Position of the Plane of the Figure; (b) Appearance of the Projection. Translator's note: cosec = esc. 36 Therefore, the projection is not isogonal {m ^ n) , not equally- spaced (m 7^ 1 and n 7^ 1 ) , and not equally large {mn ^ 1). Although the projection is not isogonal, the orthodrome on it is represented by a straight line. This remarkable property is ex- plained by the fact that the plane of the circumference of a great circle (plane of the orthodrome) always passes through the center of the Earth, which in this case appears -^s the center of the pro- jection, while the intersection of the plane of a great circle with the plane of the figure is a straight line. Since the projection is not isogonal, the moving azimuth of the orthodrome on it, if it is not equal to 0, 180, 90, or 270°, does not correspond to the azimuth on the Earth's surface. Distortion of directions on the map will be equal: /42 . „ n cosec 9 c t o o ^ tgP = tga = ^ (g a != sin <p tg a, (.1.29) m cosec2()) t & > while it is possible to calculate the actual direction of the ortho- drome at the location analogously with the aid of the measured angle on the map : m ~ ~ir '^^~ cosec 9-tgp, (1.30) where 3 is the measured angle on the map of a given projection, a is the corresponding angle in a location, and cj) is the latitude of the final point of the orthodrome. The distortions of directions and distances on this projection are great. In this connection, it is impossible to use a protractor to measure the directions and a scale to measure the distances on the map without corresponding corrections. A central polar projection is used for constructing gnomonic systems, while the regularity in the distortion of directions is used for calculating the nomograms of the orthodrome direction. The gnomonic system and the nomogram of the orthodrome direc- tion can be used for the graphic (approximate) calculation of the length of the orthodrome , the coordinates of its intermediate points . and the direction. The loxodrome and other lines of position of the given projection are represented by complex curves. The property of orthodromicity of a central polar projection has been used for the publishing of oblique central projections which have been used at radiogoniometric points in civil aviation. The middle of the base (the middle of the orthodrome distance be- tween two radiogoniometers) was taken as the point of tangency of the plane of the figure of such maps. In this case, the coordinates 37 of the position of the aircraft are very easily defined as the in- tersection of two straight orthodrome bearings (lines) extended from the radiogoniometers . Maps of the differential rangefinding (hyperbolic) system of long-range navigation (DSLN-1) are made on such a projection, since the spherical hyperbola on the projection is also expressed by a hyperbola . Equally spaced azimuthal (central) projection This projection is constructed by calculating and transforming conventional meridians (radii) to full size, equal to the principal scale transferred from the globe. The projection is used only for the publication of special small-scale maps (1:40,000,000), which are used as reference maps for measuring distances from a central point on the map . Usually a large administrative or aviation center, from which it is necessary to know the shortest orthodrome distance in any di- rection to a given point on a map, is chosen as the point of tan- gency of the plane of the figure of the projection. In such a pro- jection, for example, a map is constructed with the point of tan- gency at Vnukovo Airport , with circles plotted at equal distances from the airport. The geographic meridians and parallels are repre- sented by complex curves. This does not allow the directions to be measured . Stereographic polar projection The center of projection in a stereographic polar projection /43 a) Plane of the Figure Fig. 1.28. Stereographic Polar Projection: (a) Position of the Point of Projection; (b) Appearance of the Projection. 38 iiiiimii 1 1 III is separated from the point of tangency of the plane of the figure by two radii of the globe at the point B (Fig. 1.28, a). Here the angle 6 = 90° - <() , while the angle 6 90-? An equation of the projection can be derived from equations of the elements shown in Figure 1.28. R = X; 6 The meridians in the projection are straight lines which di- verge radially from the pole (Fig. 1,28, b), and from the point of tangency of the plane of the paper at an angle equal to the differ- ence in longitude: 6 = AX. The parallels are concentric circles, whose radii are propor- tional to the tangent of the latitude. /44 The special scale along the parallel is determined by the equa- tion dS map dp 2Rfltg — ''^qlobe -'^''V Rd(9(y^9) '• Here (j) = 90° - 9 , while after integration 1 : sec • 26 cos2 • (1.30) The special scale along the parallel is determined by the equa- tion dS map prfX 2RiS~ ''•^globe rdX R cos <f but cos (j) = cos (90 - 6) = sin 6, so that e sine = sec2_. (1.31) 1 . e m — n— sec2 9 J 90-<p \ Y=sec2(-^). 39 II n iiiiiii II iiiiiiiniiiiiinii ■iin iniiiiii iiiiiiii I Mil Therefore, this projection is isogonal (m - n) , but not equally spaced {m ^ 1 and n ^ 1) or equally large imn ^ 1). On maps of a stereographic projection, a circle drawn on the globe is represented by a circle on the plane (map); however, the center of this circle does not coincide with the projection of the center of the circle on the globe. This makes the projection in- effective for use in rangefinding systems, since lines of equal length will be represented by eccentric circles. The maximum distortion of lengths at 70° latitude does not ex- ceed 3% (3 km in 100 km), whereas if the plane of the figure is in- tersected (for example, at 70° latitude), the distortion of the lengths at the poles does not exceed 3% (and at 60° latitude, 4%). The orthodrome on maps of a stereographic projection has an /45 insignificant bend toward the equator and is constructed in prac- tice as a straight line. The loxodrome is represented by a logarithmic spiral. It is possible to continue it (just as in conic projections) along the flight angle, which is measured in the middle of the part of the straight line joining the control (rotating) points of the flight path . In determining the position of an aircraft by means of an air- craft radio compass, a correction for the angle of convergence of the meridians is allowed according to the formula where X^ and A^ are the longitude of the radio station and the air- craft, respectively. On maps of a stereographic projection, in order to facilitate determining directions in the polar regions according to a sugges- tion by V. I. Akkuratov, a supplementary system of "arbitrary" mer- idians (Fig. 1.28, b) parallel to the Greenwich meridian (A = 0°) and perpendicular to it (A = 90°) is plotted. Then the true Green- wich flight angle will equal: TFAcr = TFAar ± A^ where TFA^r is an arbitrary flight angle measured from a direction perpendicular to the Greenwich meridian (A = 90°); A^ = 90° is the i+0 location of the route (part of the path) to the east of the Green- wich meridian; and A^ = 270° is the location of the route (part of the path) to the west of the Greenwich meridian. Nomenclature of Maps At the present time, a map with a scale of 1:1,000,000 (1 cm = 1 km) which is executed in an international projection is considered the basic topographical map of the world. As described above, each sheet of this map encompasses a territory within the limits of ^■° of latitude and 6° of longitude. This has made it possible to com- pile an international designation for the sheets of maps. For the purpose of quickly choosing a given sheet of a map, each of them bears a designation of its rank in a definite system. This designation is called international map nomenclature . Th le sheets are situated in rows along parallels which run f lator to a latitude of 8"+° . There are a total of 21 rows imisphere. Each row is designated by a letter in the Lati ;t: A, B, C, D, E, F, G, H, I, J, K, L, M, N, 0, P, Q, R, laps for latitudes greater than 84° are constructed in per ■e projections). Each sheet of a row has an ordinal number from 1 to 60. Count- ing of the sheets begins from the 180th meridian and proceeds from west to east. The map sheets referring to the prime (Greenwich) meridian from the east have the ordinal number 31. Thus, columns of map numbers are obtained. To choose the necessary map sheet, it is necessary to know the approximate coordinates of the point for whose region the sheet is selected . /46 69° E For example: the point coordinates latitude 50° N, longitude Let us divide the latitude of the point by 4, and we will ob- tain the necessary row of map sheets: 5 v '+ > 12. Therefore, the map sheet is located in the thirteenth row, which is designated by the letter M. Let us divide the longitude of the point by 6 , and we will ob- tain: 69 V 6 > 11. The ordinal number of the sheet will then be 30 + 12 = 42. For convenience in selecting map sheets, composite tables have been constructed. These tables are executed on small-scale maps with a straight, equally spaced cylindrical projection by ruling the indicated map every 4 degrees in latitude and every 6 degrees in longitude, with corresponding designations showing the rows and columns of ordinal numbers of the maps (Supplement 1). 41 In addition, on the face of each map sheet is a diagram showing how the given sheet fits' to the adjoining one (Fig-.- 1.29). The sheet on which this diagram is drawn fits in the middle and is shaded . Sheets of maps with larger scales have standard schemes qf arrangement with- in the limits of a sheet with a scale of 1:1,000 ,000. For example, a map sheet with a scale of 1:1,000,000 contains 4 map sheets with a scale of 1:500,000, which are designated by letters of the Russian alphabet: A, B , C , and D . By an analogous method, the division of a map sheet with a scale of 1:1,000, 000 into sheets with larger scales is car- ried out. Roman and Arabic numerals are used for their designation. Here the map nomenclature retains the designation of the sheets in the initial division, beginning with a scale of 1:1,000,000 and up (Fig. 1.30). The nomenclature of map sheets with small scales (1:2,000,000, 1:2,500,000, and 1:4,000,000) is not international and is established when they are printed in accordance with the regions for which they are published and in accordance with the dimensions of the map sheets. Fig. 1.29. Scheme for Splicing Map Sheets with an International Pro j ect ion . Maps Used for Aircraft Navigation ' /M-? Depending on the nature of the tasks to be fulfilled, it is pos- sible to divide maps into several groups according to their scales. 1) Maps with detailed orientation, with scales of 1:500,000 and up, are used in civil aviation during flights for special pur- poses (geomagnetic mapping and photography, chemical treatment of areas, searching for small objects in the execution of special tasks, "joining" of radio engineering projects in airport regions, compi- lation of diagrams for piercing clouds, and for other purposes). 2) Flight maps with scales of 1:2,000,000, 1:1,000,000, and 1:500,000 are used in civil aviation as basic flight maps. Crews of light aircraft and helicopters at comparatively low speeds use maps with scales of 1:1,000,000 and 1:500,000, while crews of high- /48 speed aircraft use maps with scales of 1:2,000,000 and 1:1,000,000. 3) Aircraft maps with scales of 1:4,000,000, 1:3,000,000, 1:2,500,000, and 1:2,000,000 are used in civil aviation for general orientation and plotting of position lines with the aid of radio- engineering and astronomical facilities. For these purposes, crews of light aircraft at low speeds and helicopters use maps with only the last two scales. 42 h) Special maps with scales of 1:40,000,000 and up (to 1:2,000,^000), with special emphasis on different purposes of appli- cation: lines of equal distance from definite points, a hyperbolic system, azimuths from radio-engineering installations, etc. are used, These include maps with reference materials of smaller scales: maps with time zones, magnetic declinations, composite tables of map sheets , etc . //-4/ t'ltOOOO 1-200000 PZOOOOO 1:500000 (N-it1-8) Fig. 1.30. Scheme for Dividing a Map Sheet with an International Pro j ect ion . Also, special flight maps with scales of 1:1,000,000 and 1:2,000,000 with plotted and marked flight routes are published for civil aviation. As a rule, they are compiled on oblique cylindrical or oblique conic projections, with the least distortions of angles and lengths along the route. The orthodrome on such maps is prac- tically a straight line. The contents of a map depend on its scale, the aerographlc fea- tures of the regions for which it is compiled, and the purpose of the map . 43 On maps of all scales , the following are drawn in some kind of detail : (a) relief; (b) hydrography (seas, rivers, lakes); (c) populated points; (d) network of railroads, highways, and country roads; (e) vegetation or ground cover (large forests, meadow, swamp, sand, desert, etc.); (f) isolines of magnetic declinations and magnetic anomalies. The legends of the indicated elements are usually executed on the maps at the lower edge of the sheet. On maps, a relief is expressed by three methods: 1) It is expressed by isolines of equal height on the surface of the relief (horizontals), i.e., lines formed at the intersection of a relief with horizontal planes which are situated one above the other, with height intervals depending on the scale of the map; the height of the horizontals above sea level is designated by numbers. 2) It is expressed by layered coloring; a special color desig- nated on a special (hypsometric) scale on the lower edge of the map is assigned to each interval of relief height. 3) It is expressed by brown shading, i.e., by special coloring with thickening of brown in the highest areas of the relief and the steepest slopes. This use of color gives a natural, volumetric idea /^9 of the nature of the relief. In addition to the above methods of representing relief on maps, marks of command heights (which exceed neighboring heights), with an indication of the height of these points above sea level, are shown. Hydrography is shown on maps by a blue color. Its detail de- pends on the scale and purpose of the map. Populated points , depending on the scale of the map and the areal dimensions of the points , are represented by contours or con- ventional symbols in accordance with the point's dimensions or its population . In lightly populated areas, all populated points are designated. On small-scale maps of densely populated areas, some of the points are omitted. The number of points drawn depends on the scale of the map and the population density of the area. The detail of the highway network depends on its density, the vegetative or ground cover, and the scale of the map and its purpose. Besides the above general contents of maps, specially prepared 414 flight maps represent a navigational situation, i.e., the arrange- ment of radio-engineering facilities for aircraft navigation, posi- tion lines of aircraft, and special markings for navigational meas- urements and calculations are shown. On some forms of specially prepared maps (map-diagrams), some of the elements of the general contents are omitted or simplified for the purpose of a more detailed and graphic representation of the navigational situation. 6. Measuring Directions and Distances on the Earth's Surface Orthodrome on the Earth's Surface In the practice of aircraft navigation at the present time, an orthodrome direction is the main and most widespread direction. In order to explain all the problems connected with measuring moving angles, distances, and coordinates in flight along an ortho- drome, let us examine an orthodrome on the Earth's surface (Fig. 1. 31) . An orthodrome, in general, lies at an angle to the Earth's equator and intersects it at two points, the distance between which (along the arc of the equator) is equal to 180°. Only the equator, which likewise appears as an orthodrome, is an exception. In Figure 1.31, a and b, line XqMi is the arc of the equator, line \qM is the orthodrome examined by us; points \q and Xq + 180 are the points of intersection of the orthodrome with the equator; Pj^AqP^ is the meridian of the point of intersection of the ortho- drome with the equator; P^MMiP^ is the meridian of the point M on /50 Fig. 1.31. Orthodrome on the Earth's Surface: (a) Position of the Orthodrome on a Sphere; (b) Relationship between Longitudes and Lat- itudes of Points on the Orthodrome. ^5 the orthodrome ; 90° - ag is the angle between the plane of the equa- tor and the plane of the orthodrome ; X is the londigude of the point M ; (j> is the latitude of the point M ; -B , +B are points on the ortho- drome of maximum latitude, which are called vertex points. Let us erect a normal to the plane of the equator at point M-^ (see Fig. 1.31, b) and extend it to an intersection with the vertical of point M on the or- thodrome (point M2 ) . It is obvious that the triangle M1M2 will be a right tri- angle. Here M1M2 will be the tangent line of the angle ({' • Let us drop from points M^ and M2 , perpendiculars to the aperture axis of the orthodrome with the equator XgO. One of them will lie in the plane of the equator, the second in the plane of the orthodrome ; both will converge at one point on the aperture axis (point N). Fig. 1.32. Determining Distance on an Ortho- drome . It is obvious that line MiN will be a line of the sine of angle X, while an- gle M1NM2 will be the aperture angle of the plane of the equator with the plane of the orthodrome (90° - mq). Here the triangle NM1M2 will also be a right triangle. Thus, for point M and for any point on the orthodrome, the following equation will be valid: tgy /51 tgOO'-Oo)- slnX or tg oo =i ■ sin X tg<P (1.32) Formula (1.32) is valid only for cases when the point Xq is the point of origin of the longitude. When the longitude of the point is not equal to zero, the longitude of the point Xq must be subtracted from the longitude of the point M(Xj^), i.e., the refer- ence system of longitudes must be reduced to this point. Then 'g<»o= sin ( K„ — Xq) tg9« (1.32a) In the future, for the sake of simplicity, we will consider the longitude of Xq equal to zero. It is possible to determine the moving azimuth according to the formula tgo = tgdosecXsectp, (1.33) 46 Considering that tg cxq = — r- , it is possible to reduce (1.33) to the form: ^ ^ tga=^tg\cosec<f or ctga= ctgXsirt?. J (1.33a) '1 Formulas (1.33) and (l.33a) are obtained by differentiation of (1.32). Since the ratio — r = tg an = const remains valid for every tg ^ ^ " ^ length of an orthodrome , it is obvious that the elementary differ- ence quotient sin X and tg <j) will also be constant for every length of an orthodrome and will equal: rfsinX := tg oo = const. dtg<f Therefore, it is possible to write (1.32) in the form: dtgf d<f df whence cosX d\ sec2? ' d<f ~*^'^ or dif cos X cos3 9 On the Earth's surface, the linear scale of longitude is equal /52 to the linear scale of latitude multiplied by the cosine of lati- tude. Therefore, the tangent of the moving azimuth of the ortho- drome will be expressed by the derivative ^^ , divided by the co- sine of the latitude: ^j^ "-^ tga== "^ = 'g°° COS If cos X C0S2 <j) sinX or, considering that tgao= ^„ > we arrive at (1.33a): tg a = tg X cosec ?. In the practice of aircraft navigation, it is usually neces- sary to deal v/ith two points on the Earth's surface. With the ex- ception of special cases, neither of them is on the equator. Formulas (1.32) and (1.33) can be used only in those cases when the point of intersection of an orthodrome with the equator (i.e., ^■7 the longitude of a point on the orthodrome , the width of which is equal to zero) is known. Fig. 1.33. Elements of a Spherical Triangle. (a) Triangle on a Sphere; (b) Relationship of Angles and Sides of a Spherical Triangle, Let us derive an equation which makes it possible to determine the coordinate X of a given point on an orthodrome on the basis of the coordinates of two known points on it. Let us assume that we have two arbitrary points on the Earth's surface with coordinates <i>iAi and <|)2A2. We will take the differ- ence of the longitudes of these points as AA(AA - X2 " ^i)- Then, according to (1.32a), sin (Xi -^ Aq) sin [(Xi — Xp) + ii\] Transforming the right-hand side of this equation, we obtain: /53 sin (X; — Xq) ^ sin (X, — Xq) cos AX + cos ((X{— Xp) sin AX Dividing both sides of the equation by sin (X^ ing by tg 4>2i "^ will have: tg 92 stif (X] — Xq) cos ax + cos (X; — Xq) sin AX tg?i ~ sln(X, — Xp) ~ = cos AX + ctg (Xj — Xp) sin AX, Xq) and multiply- from which ctg (Xi — Xp) = tg <P2 ctg ipi cosec AX — cfgAX. (1.34) Equation (1.34), which makes it possible to determine the longitude of the point of intersection of an orthodrome with the equator (Xg), is very important. Knowledge of this coordinate makes it possible to calculate easily all the remaining elements of the orthodrome . 48 Having substituted the value X in (LS^l-) for the value (Aj - Xq), as before, and substituting into (1.33a) the value ctg X from (1.34), we obtain the following equation for a point with the coor- dinates <i)i Xi : ctg o = tg ij!2 cos 9i cosec AX — ctg AX sin <pi. (13 5) Formula (1.35) is usually used for calculating the azimuth of an orthodrome at the initial point of the straight-line segment of the path vfhen there is no necessity for determining the remaining elements of the orthodrome. In general, it is better to solve (1.34) independently, and then find the solution by substituting X into (1.32) and (1.33). Simple transformations of (1.32) reduce to formulas which make it possible to determine the coordinates of intermediate points on an orthodrome: tg<p = slnXctgoo, (1.36) slnX=tg<i>tgai. (1.36a) Given the arbitrary value of a point coordinate on the ortho- drome <J) or X , it is possible to obtain the value of the second coor- dinate of this point on the basis of these formulas. The formulas from (1.32) through (1.36), given by us, make it possible to determine the initial and moving azimuths of the ortho- drome, and also the coordinates of its intermediate points. In order to determine the length of the orthodrome or distances along it (.S) let us derive equations which connect the coordinates of the points of the orthodrome with its length. In Figure 1.32, the triangles ONMi and ON1M2 are similar. The straight line ON is a line of the cosine of the arc X, while ON' is a line of the cosine of arc S. The hypotenuse of triangle ONMi is equal to the radius of the /54 Earth, vihile the hypotenuse of triangle ON1M2 is the line of the cosines of arc (j) . Therefore, cos 5 = cos X cos?. (1.37) Equation (1.37) makes it possible to determine the distance from the starting point of the orthodrome to any of its points with knovfn coordinates . If the initial point of the orthodrome and the coordinates of any two points along it are known, the distance (S) between the latter is determined as the difference between the distances to the initial point: 49 ^\fl = 52 — Si. If the coordinates of the starting point are not known, and the necessity for determining the other elements of the orthodrome (besides the distance between the two points) is lacking, then the indicated distance can be determined by the formula cos 5 = slntpi sin<f2+ costpiCos<p2Cos 4X. / -, 30') Formula (1.38) is not derived from simple geometric ratios. For its derivation, it is necessary to use the spherical triangle {P^Mg^M-^) (Fig. 1.33a). Let us join points ?vr^a and ^j^ by verticals with the center of the Earth 0. Let us draw tangents to the arcs P^^a ^"^ ^N^b 3"^ "the point Pjf up to the intersection with the indicated verticals at points M^i and M-^i (Fig. 1.33, b). We will obtain two plane tri- angles Pf^^^al^bl ^^'■^ O^al^bl with "the common side Mg^iM-^i . Obviously, ^ai^bi=<'^N^ai^^ + <'PN%i>'^-2PN^al^N^bi,cosMa^PN%^ At the same time, M^:^Mi^^ = (0M^^)^+(0M^^)-20M^^0Mi,^oos M^^ OM^i (1.39) Since^ai ^■'-'1 ^^ the common side of the triangle, the left- hand side of the first equation is equal to the right-hand side of the second. Taking the radius of the Earth as equal to 1, from the right triangles OP-^M^i and ^^N^bl "® f ind ; %^a, = tg;b; Pj^M,.^ ~ tg a; OM^^ = sec b; Ml, ^^ sec a; L A^^.P^Mi^jP; L M^pM^ -=p. Substituting the indicated values into (1.39), we obtain: tg2b+tg2a — 2tgbtgacosP = sec2fl + sec2b— 2secasecbcosp; sec2a= 1 + tg2fl; sec2b=14-tg2b Therefore, 2tgatgbcosP = 2 — 2?ecasecbcosp. (1.40) >« 1 ^ • T • •!_ j.i_ • J ^ /■ T ,.r^\ -L cos a cos b , . . Multiplying both sides of (l.UO) by we obtain: sin «sin bcosP = cosacosb — cosp ojP cosp = cos a cos b + sin a sin b cos P. (1.41) /5[ Formula (1.4-1) is the first basic formula of spherical trigo- nometry and is widely used in aircraft navigation with the use of astronomical facilities (the remaining formulas of spherical trigo- nometry are given in Supplement 2). 50 In our case , LP = AX; Lp = Sabi L b'=f= 90° — ijij; Lfl = 90° — ?,. i.e., (1.41) has the form: cos 5 = sin <f 1 sin 92 + cos <i>j cos 92 cos AX. When determining point coordinates of the orthodrome , there is the same necessity to solve the inverse problems according to the knovin orthocromic distance (S) . For this let us return to Figure 1.32, in which it is obvious that the line MN i is the line of the sine of the arc S, while line MM2 is equal to MN i cos ag. At the same time, MM is the line of the sines for arc (j) . Therefore, sin If = sin 5 cos do (1.4-2) Formula (1.42) makes it possible to determine the coordinate (j) along the traversed orthodrome distance from the initial point. The coordinate A in this case is determined according to (1.36a). slnX = tgytgoo. Thus, we have an analytical form of all the necessary trans- formations for determining the elements of the orthodrome on the Earth's surface. However, in the practice of aircraft navigation it is sometimes more convenient to apply other formulas which de- termine separate elements of the orthodrome. For example, if the coordinates of two points on the Earth's surface and the orthodrome distance between them S are known, the azimuth of the orthodrome (a) at the starting point can be deter- mined by the formula cos tpo sin AX / -, ,, ^ \ sina= ~ . (1.43) sins Formula (1.43) can be transformed to determine the distance between points at a known azimuth: cos 92 sin AX sin 5— — . M U3n1 It is obvious that both formulas are obtained from the equation sin 5 sin o = cos 92 sin AX, which in turn is derived by means of Figure 1.34, where line BB i is a perpendicular dropped from point B to the plane of the equa- /56 tor, and is a line of the sine of the latitude of this point, while 51 line B1A2 is a perpendicular dropped from point B^ to the plane of the meridian which passes through point A. Obviously, A^Bi = cos tp2 sin 4X, Let us erect another perpendicular to the plane of the equator at point A2I we will then obtain plane A1A2B1B perpendicular to the plane of the equator and the plane of the meridian of point A . Fig. 1.34. Determining Special Elements of an Orthodrome . Fig. 1.35. Determining the Ini- tial Azimuth or the Vertex of an Orthodrome . If we rotate the indicated plane around the" line BAi in such a way that it remains perpendicular to the plane of the meridian of point A up to the moment when line A2A1 becomes perpendicular to the vertical of point A, the distance BA^ will not change. In this case, straight line BAi will be the line of the sine of the arc AB , while its length will be determined by the formula ^,5 = A.,B '2^1 sin a from which it follows that sin S sin a = cos <(2 sin AX. By a similar method, the initial angle of the orthodrome or the latitude of the vertex point is determined if the azimuth of the orthodrome at any point on the Earth's surface is known. In Figure 1.35, arc Xq^b is the equator; arc \qMB is the ortho- drome; line OP-^ is the axis of the Earth; OPq is the axis of the orthodrome; M is a point on the Earth's surface; and B is the ver- tex point. 52 Let us erect a perpendicular P^O to the vertical of point M from the center of the Earth so that it is located in the plane of the meridian of point M. Let us also draw a plane parallel to the plane of the equator through the poles P^ , Pq, and Py . The angles /57 PqP^O and PyiP-^0 will be right angles, since lines PqPn and Pf^Pj^ lie in a plane perpendicular to P^O. Angle PqPmO is also a right angle, since the plane of triangle PqPj^O has a slope to the axis perpendicular to P-^0 and parallel to PqPj^ . It is obvious that angle P^OP-^ is equal to the latitude of the vertex point, while angle P]i[OP]j is equal to the latitude of point M. Therefore, OPf^^ OPi cos 98 = 0P„ cos <p„; OPu^OPo sin a, whence cos ?a == sin Oo = cos <?„ sin a. (1.1^4) Formula (1.44) is used to find the latitude of the vertex point, vjhich also appears as a complement of the initial angle of the or- thodrome up to 90°. The longitude of point M relative to- the start- ing point of the orthodrome in this case can be determined by (1.35a). Vfith a known azimuth of the orthodrome at any point on the Earth's surface, the coordinates of its starting point can be ob- tained directly according to (1.33a), from which it follows that tgX = sin9tga, (1.45) Then it is not difficult to determine the initial angle of the or- thodrome . In some cases, in order to calculate the elements of the ortho- drome, coordinates of the vertex point rather than of the starting point are used. In these cases, the functions of the otQ angle are replaced by inverse functions of the latitude of the vertex point which are equal to them, just as functions of longitude are, since these angles differ by 90° . For example, (1.36a) has the form: COS?lB=tg<pCtg<fBl vrhile (1.45) has the form: ctg Xg = sin <f tg a. To explain the procedure for determining all the elements of an orthodrome, let us examine (as an example) an orthodrome which passes through two points on the Earth's surface with these coor- dinates: Ml = latitude 60° N, longitude 30° E; % = latitude 80° N, longitude 40° E. 53 First we shall carry out the general solution of the problem of finding the elements of an orthodrome . For this we shall use (1.34). Substituting into this equation the functions of the coor- dinates of the points Mi and M2, we obtain: ctg (h — Xo\ = tg 80° ctg 60°cosec 10° — ctg 10°; ctg(X,-Xo) = -~^ -5/n = 13.228; X = (Xj - >^) = 4°18.'; Xo = 25°42'. The initial azimuth of an orthodrome, according to (1.32), will be : tg oo = sin X ctg <p. Let us determine it on the basis of the coordinates of point /5i Ml : tg Oo = sin 4°18' -ctg 60° = 0.574-0.075 =■ 0.0433; do = 2°29'. According to (1.33a), the moving azimuth of the orthodrome (a) for point Mi is equal to ctg a = ctg 4°18' -sin 60° = 13.228-0.866 = 14.455; = 5°. The distance from the starting point of the orthodrome to point Ml, according to (1.37), equals cos 5i =- cos 4°1 8' - cos 60° = 0. 9972 -0.5 = 0. 4986; 5, = 60°4', while the distance to point M2 , according to the same formula, is cos 52 = cos I4°18'-cos 80° = 0.969-0. 1736 = 0.1682; 52 = 80°17'. The distance between Mj and M2 is then defined as the differ- ence between the distances to the starting point: S = 52 — 5i = 80°17' — 60°04' = 20°13'. Coordinates of any intermediate point can be determined accord- ing to (1.36) or (1.36a) . For example, the longitude of point M2 according to its lati- sin X2 = tg 80° tg 2°29' = 5.671 -0.0433 = 0.246; X= (Xj — Xo) = 14°15'; Xj = 39°57'. 54 Thus, all the necessary elements of an orthodrome are easily determined . Let us now assume that we had to determine only the azimuth of the orthodrome at point M^ . For this we will use (1.35): 0.5-5,671 ctg a = '^ ^^gg -0.866-5.671 = 11.4225,' a = 5°. Knowing the azimuth of the orthodrome at one of its points makes' it possible to determine the distance to any point by using (l.H3a), or in our example: .„ - 0.1736-0.1736 sin 5 = — =; 34'ifi- 0.0872 -".>5*oo. 5 = 20°13'. Using the azimuth of the orthodrome at one point, it is possi- ble to determine the latitude of the vertex point or the Initial azimuth of the orthodrome according to (1.414-). For our orthodrome, using point Mi, we obtain: sin Oo = cos 60° sin 5°"= 0.5-0.0872 =^Q.0436; Oo = 2°30'. After this, the intermediate points of an orthodrome are easily /59 determined . Thus, it is possible to determine the elements of an orthodrome beginning with the distance between two points according to (1.38), changing to the moving azimuth according to (l.M-3), and then to the latitude of the vertex point according to ( 1 . M-M- ) . Orthodrome on Topographical Maps of Different Projections Let us examine an /I A Sin If orthodrome on maps of a simple equally spaced cylindrical projection, which essen- tially represents a geographical coor- dinate system on a scale of angles. I— 4A- Fig. 1.36. Elementary Segment of an Orthodrome on a Map of a Cylindri- cal Projection. To explain this, however, let us draw on the Earth's surface an elemen- tary normal cone at some latitude; we shall examine it on the above projec- tion ( Fig . 1.36). As is already known, the radius of this cone when unrolled equals: po = ^ ctg <po 55 or, taking the radius of the Earth as 1, Po = ctg <Po- According to (1.20) the scale of the projection along the paral- lel is equal to: 1 n = ■ COS<j> In order to draw our unrolled cone on a cylindrical surface, it is necessary to straighten the cone first and then extend it. Obviously, the segments of the meridians remain straight lines dur- ing straightening of the cone, but they must be unrolled together with the surface elements to an angle equal to AX sin (j) . Let us now draw a straight line AB in the east-west direction on an elementary cone. During straightening of the cone, the indicated straight line will acquire a curvature, the radius of which will be equal to the radius of the unrolled cone (r), but curved in the opposite direc- tion. Therefore, During extension of our cone along a parallel to a scale nz , each of its elements (including elements of our straight line) will undergo an extension equal to ~ . Therefore, the radius of the /60 ^ ^ cos (j) straight-line element will increase and will equal: ctgv r, o = = cosec tp. ''>"• cos <f As is evident, the straight-line element, situated along the parallel (in general, in a direction perpendicular to the axis of the cylinder) acquires a curvature. The straight-line element sit- uated in the direction of the axis of the cylinder does not acquire a curvature. Therefore, if the straight-line element is situated at an angle to the axis of the cylinder, its radius of curvature will equal : Ctg<j> ''a,b. — ; = cosec 9 cosec a.. -^'^^ cos <f sin a ^ (1.46) In geometry, the curvature of a curve is considered to be a value inverse to the radius of the curvature. Therefore, the curv- ature of our element will equal: 1 =sin<iisina. '■a.b, (1.47) 56 An orthodrome on the Earth's surface does not have its own curvature of each element of an orthodrome on a map in a normal, equally spaced cylindrical projection will be expressed by (1.47), from which it is obvious that the maximum curvature of the ortho- drome will be observed at its vertex points, while the starting points, i.e. , the points of intersection of the orthodrome with the equator (Fig. 1.37), will appear as points of inflection. Thus , the orthodrome on a map of an equally spaced cylindrical projection has a form reminiscent of a sine curve. This curve is the graph of the ratio of the coor- dinates of the orthodrome with a known initial azimuth (a). Fig. 1.37. Graph of an Ortho- drome in a Cylindrical Pro- j ect ion . As a result of the nonisogon- ality of an equally spaced projec- tion, the slope of a tangent to the curve of the orthodrome does not reflect its directly moving azimuth, with the exception of the azimuth at starting points. The moving azimuth of an orthodrome along a curve can be deter- mined if we consider the relationship between the scales m and n at the investigated points. With equal scales, the dip angle of the tangent to the curve is determined by the formula tgct = n ^ - riQ sec (() or n j^ = n^ sec (J). There- fore , the actually moving azimuth of the orthodrome in an equally In our case, the scale , the actually moving a: spaced cylindrical projection is determined by the formula /61 dX tg a = cos <f. d<( It is obvious that in an isogonal normal cylindrical projec- tion, the orthodrome will also have a shape reminiscent of a sine curve. However, as a result of the extension of the scale along the latitude (n - riQ sec <)) ) , the amplitude of this curve will be increased. The more it is increased, the smaller the initial azi- muth of the orthodrome will be, and the greater the latitude of the vertex points . In contrast to an equally spaced projection, in this projection the dip angle of the tangent to the curve will correspond to the moving azimuth of the orthodrome at any point ,, since the scales along the longitude and latitude are equal to : m = n = sec (f. 57 Thus, the orthodrome in a cylindrical projection has the form of a curve which is convex in the direction of the increase in the scale of the projection. This feature of the orthodrome is common to all projections which have an increase in one direction. Let us cite a brief analysis of the bend of the orthodrome with a varying map scale, in accordance with the general case. Let us assume that we have a spherical trapezoid which is rep- resented on a map in the form of a rectangle (Fig. 1.38). The length of any parallel (^x^ °^ 'the trapezoid is equal to its length on a rectangle divided by the scale of its representation. The scale of representation of the meridians in any part of the rectangle is equal to one . During extension of the trapezoid into a rectangle, each straight-line element on its surface acquires a curvature. fx rfcp '. r^ Therefore, for an equally spaced cylindrical projection 1 d cos y ''x df = — sin <j>. Since — = 0, for a straight line passing at an angle to the ■^ A meridian we will have 1 — = — sin 9 sin a. r A minus sign shows that bending occurs in the direction of a decrease in the value of — or in the direction of an increase in n the scale . /62 Fig. 1.38. Conical Trapezoid Represented in the Form of a Rectangle . Fig. 1.39. Bending of an Or- thodrome in the Directions of Scale Increases. 58 Let us now assume that we have a projection, a change in the scale of which occurs in two principal directions [for example, simultaneously in the north and east (Fig. 1.39)]. It is obvious that a straight line AB , passing at an angle to the meridian with a change in the scales in two principal directions, will simultaneously undergo bending in opposite directions, i.e., its component curvatures will be subtracted: rfip sin a ■ dk For the general case , R dz sin a — dx Therefore, the orthodrome on maps constructed with tangential cylindrical projections will have convexity: a) at latitudes greater than the latitude of a parallel which is tangent to a geographic pole; b) at latitudes lower than the latitude of a parallel which is tangent to the equator (Fig. 1.40). ■ Con i c straight 1 "0 r thod reme ne Fig. Tange the s stant limit gle t be a maps recti able , both 1.40. Orthodrome on ntial Conic Projecti cale alo , while s of eac o the me wavy lin with a s on will while i parallel ng t the h ma ridi e ar cale be s n th s an he latit scale al p sheet . an withi ound a s of 1:1, o insign e places d the or a Map o on . ude in ong the There n the 1 traight 000 ,000 if icant where thodrom f a In intersecting conic projections, the ortho- drome has the same form as in tangential projections. Here , its point of inflec- tion is situated on the middle parallel between the parallels of inter- section . /63 thes Ion fore imit pri , de tha sepa e wi IS in je e maps r gitude h , an ort s of one ncipal d viatlons t they a rate map 11 have Of the o an in ction . emains as sma hodrom sheet irecti from re pra sheet breaks spec rtho tern As pra 11 c e dr on on . the ctic s ar ial drom atio is ctic hang awn this How prin ally e sp inte e on nal know ally es i at a map ever cipa unn lice rest maps pro- n , con- n the n an- will , on 1 di- otice- d. 59 As we have already shown, the orthodrome in a central polar projection is expressed by a straight line. However, its moving azimuth, with the exception of the directions 0, 90, 180, and 270° cannot be determined by simple measurements on a map, but demand the introduction of corrections according to (1.29) and (1.30). In a polar stereographic projection, the orthodrome is also a nearly straight line. However, to determine its azimuth, it is necessary to use general equations of an orthodrome on the Earth's surface . Loxod rome on the Earth's Surface The loxodrome direction at the present time is used only to determine the mean path angle of flight on short segments of a path by the use of magnetic compasses. 'With the use of magnetic com- passes, not a geographic but a magnetic loxodrome direction is used, This leads to a bending of the flight path which does not lend it- self to precise analytical descriptions. As we already know, a toxodrome is a line on the Earth's sur- face which joins two points and intersects the meridians at a con- stant angle. In general, a loxodrome is a spiral line which goes to the Earth's poles. As a result of this, it has curvature not only in a vertical plane, but in a horizontal plane as well. Meridians, the equator, and parallels which are also loxodrome lines, expressed in the first two cases by a great circle and in the last case hy a small circle on the Earth's surface, are the exception. The curvature of a loxodrome in a horizontal plane increases sharply with an approach to the Earth's poles. As a result, it is not used at all for flights in polar latitudes. Let us determine the curvature of a loxodrome, its extension. /61+ 60 a Map of a Conic and its deflection as compared to the orthodrome direction at a given latitude <() . The maximum curva- ture of a loxodrome at a given flight altitude will occur when the flight is in an easterly or westerly direction, and it will vanish in a flight to the north or south . Let us assume that a flight at altitude ((> occurs in an easterly direction. In this case, the angle of turn of the loxodrome from point A to point B will be equal to the angle of convergence of the meridians (6) between these points (Fig. 1.41). B = _(Xb — X^)sin9. Its length (5) from point .4 to B will be •S = (Xb — X^)cos<p, where A^ and Ag are the longitudes of points A and B and (f> is the mean latitude between points A and B. The radius of curvature of the loxodrome i^/^x^ can be determined as the ratio of the length of part (5) to the angle of turn (6). /65 If we take the radius of the Earth as 1, then Fig. 1.41. Loxodrome on Pro j ection . '■x=— =ctgv. (1.48) The part of the loxodrome which runs along the meridian does not have a horizontal curvature. Therefore, if the loxodrome passes at an angle to the meridian, the radius of its curvature at any point will equal: r = r-^ cosec a = ctg 9 cosec a. (1.49) Example : Determine the radius of curvature of a loxodrome passing at an angle of 30° to the meridian at a latitude of 45°. Sol ution r = y?3 ctg 45° cosec 30° = 2/?3 = 12742 km where i?3 is the radius of the Earth 61 The curvature of the loxodrome in a horizontal plane creates some lengthening of the straight-line parts of the path. The later- al deviations from the line of the orthodrome direction may turn out to be very significant here. In Figure 1.42, the straight line AB is the orthodrome; arc AB is the loxodrome ; 6 is the angle of turn of the lo:kodrome from point A to point B. The length of the straight line is t ^B = 2/?sln — - . while the length of the arc is AB = m. Lengthening of the path along the loxodrome (hS) is determined by the formula AS = /?8-2y?sin-|-. ^^^^^^ Example : Determine the lengthening of the path along the loxo- drome passing through points A and B on the Earth's surface, with the following coordinates: A: latitude 55° N, longitude 38° E; B: latitude 55° N, longitude 68° E. Since the latitude of the starting and end points is the same, the direction of the loxodrome coincides with the Earth's parallel at a latitude of 55° . The radius of curvature of the loxodrome will be: /• = r^ = /?3 ctg 55° = 6371 -0.7002 = 4461 km The angle of turn of the loxodrome is determined by the formula — 6 = (Xa — X,) sin % 6 = ~ 30° sin 55° = — 30°-0,8192 = 24.576°, Then sin ~ = sin 12°17' = 0.2127. Substituting the value of the radius of curvature and the an- /66 gle of turn of the loxodrome into (1.50), we obtain: 24^576:4461__2^^gj 2127 ='15.6 km •^ " 57,3 From this example, it is obvious that at middle latitudes, with flight paths up to 2,000-3,000 km long, the curvature of the loxodrome creates relatively small lengthenings of the path (in our 62 example, less than 1%); however, in approaching the polar latitudes, lengthening of the path will increase , together with a decrease in the radius of curvature of the loxodrome . Significant lengthenings of the path along the loxodrome occur at middle latitudes with very long distances between points on the Earth's surface. For example, at a latitude of U-0° , with a distance of 11,000 km between points, lengthening of the path along the loxo- drome can exceed 4,000 km, i.e., more than 30%. In Figure 1.42, it is obvious that with a constant radius of curvature of the loxodrome, its greatest discrepancy with respect to the orthodrome (deflection) will be observed at half the path between points A and B. Loxodrome Fig. 1.42. Radius of Curvature of a Loxo- drome . Here , &Z = R- ■ R cos -—- or iZ^RU-coSYJ- (1.51) In the example analyzed by us , AZ = 4461 (1—0,9771) = 102,6 km Thus, the discrepancy between the loxodrome line of the path and the orthodrome, even at comparatively small distances between points on the Earth's surface, will be very substantial. This is the basic cause of the limitation of the length of the loxodrome segments of the path. In the practice of aircraft navigation, since the loxodrome direction of flight is used only in limited path segments, the azi- muth of the orthodrome (a), measured on the central meridian between the starting and end points of the segment is taken as the loxodrome direction of the flight. This angle can also be determined on the basic of the approx- imate formula tgo = X, — X f2 — ?! COSl^ av (1.52) /67 The length of the loxodrome segment of the path (S) is deter- mined by the formula T2 — 91 cos o (1.53) 5 = ".2 •—A I sin a (1.54) 63 • Formulas (1.5M-) and (1.52) are approximate and have a simple geometrical interpretation. Formula (1.53) is derived analytically. Considering that the loxodrome intersects the meridians at a constant angle, the ratio remains constant: dS 1 d<f cos a from which 5= ! \d:f^^iZUt2^, COS a i ^ cos a In the majority of cases, in calculating the distance along the loxodrome, it is more advantageous to apply (1.53). However, with loxodrome directions close to 90 or 270°, the values ^2 ~ ^\ and cos a simultaneously approach zero. This leads to large arith- metic errors in calculation and ultimately to an ambiguity in the solution. In these cases, it is more advantageous to use (1.54), the errors in which will be negligibly small, since a small differ- ence in the latitudes between the points means that the mean cosine of the latitude becomes practically equal to the cosine of the mean latitude . Example : Determine the loxodrome direction and the distance between points A and B on the Earth's surface, the coordinates of which are: A: latitude 56° N, longitude 38° E; B : latitude 68° N, longitude 47° E. Solution: According to (1.38), let us find the direction of the loxodrome : ,g o = /*^~^ cos 62° = 0.35Z1; a = 19°24'. 68 •"" 56 Let us determine the loxodrome distance according to (1.53): 5 = 111,1-^^=^, = 1413 km cos 19°24' Loxodrome on Maps of Different Projections /6E A loxodrome has the appearance of a straight line only on maps of a normal isogonal cylindrical projection. Oh maps of normal isogonal conic and azimuthal projections, the loxodrome is a curved line intersecting the meridians at a con- stant angle a. Therefore, knowing the direction of the loxodrome in order to draw it on a map it is sufficient at the starting point 64 to plot this direction up to the intersection with the next meri- dian, where the indicated direction must be extended to the next meridian in line. Continuing our plotting to the final point, we will obtain a broken line very close to the loxodroifie . On maps with nonisogonal projections, the loxodrome will have a variable angle to the meridians, which depends on the ratio of the scales tgam=«go— T (1.55) where a is the angle of intersection of the loxodrome with the meri- dian at a location; a^ is the angle of intersection of the loxo- drome with the meridian on a map; n and m are the scales of a map at a given point along the principal directions east-west and north- south, respectively. For example, on maps with an equally spaced normal cylindrical projection, where H. = sec (j) , m tg "07= *g ° 5^<= ■?' i.e. , the loxodrome will have a curvature in the direction of a pole, whereas it has a natural curvature in the direction of the equator . General Recommendations for Measuring Directions and Distances Orthodrome directions and distances for straight-line segments of a path of more than 1200-1500 km in all cases must be determined by analytical means, independently of the scales and map projections used. With a length of the path segments of more than 2000 km, the intermediate points of the orthodrome must also be determined in such a way that the distance between them does not exceed 800-1000 km . On short path segments (up to 1200-1500 km), the methods of determining directions and distances depend on the scale and pro- jection of the maps, as well as on the means and methods of air- craft navigation used. For example, in using precise automatic navigational devices, it is always advantageous to use analytical forms to solve these problems. It is possible to carry out direct measurement of distances and directions on maps by means of a scale and protractor, with the length of the path segments being not more than 1500 km if these maps are executed on an international polyconic projection and have a scale of 1:1,000,000 or 1:2,000,000 (the latter within the limits of one (or, in extreme cases, two) adjoining sheets). /69 65 We must note that good results in measuring directions and dis- tances can be obtained on route maps constructed on oblique cylin- drical or oblique conic projections when the flight direction coin- cides with or is located close to the axis of the route map. How- ever, in directions at an angle to the axis of the route map, the results of measurements are significantly worse than on maps with an international polyconic projection. In using maps constructed with all other projections, only the analytical form of determining distances and directions, with calcu- lation of intermediate points along the orthodrome after every 200- 300 km of the path, must be applied. The loxodromic flight direction can be measured directly only on maps with an isogonal normal cylindrical projection. Here, seg- ments of distances up to 300-M-OO km on this projection can be meas- ured by means of a varying scale located on the edge of the map. On maps in other projections, generally speaking, there is no need to measure and plot the loxodrome line of the path in parts of mDre than 300-100 km. Since the loxodromic flight direction in short path segments is used as the mean orthodrome direction, it is considered equal to the orthodrome as indicated by the mean meridian between the start- ing and end points of the path segment. In view of the fact that in short segments of the path the lox-. odrome line does not show significant deviation from the orthodrome as a rule , it is not plotted on maps but is considered coincident with the direction of the orthodrome. 7. Special Coordinate Systems on the Earth's Surface In the practice of aircraft navigation, rectangular and geo- graphic coordinate systems are insufficient, and it is necessary to use at least three or four coordinate systems simultaneously. Actually, elements of aircraft movement are examined in a mov- /70 ing rectangular coordinate system. The center of a rectangular system moves in one of the surface coordinate systems which is con- nected with the given flight path, which in turn is determined in a geographic coordinate system. The indicated order of the connection of the coordinate systems is minimal. For some purposes, it is advantageous to examine air- craft movement relative to the airspace, i.e., a supplementary co- ordinate system whose center shifts in the moving rectangular system. With the use of gyroscopic devices as well as astronomical ones, it is necessary to use a universal (stellar) coordinate sys- tem. The use of radio-engineering navigational facilities is con- 66 nected with the use of a whole series of special types of surface coordinates by which the position of the aircraft on the Earth's surface is determined. Let us examine the most important surface coordinate systems used in aircraft navigation. Orthodromic Coordinate System .The orthodromic coordinate system for calculating the path of an aircraft is the one most widely used at the present time. In this system, the direction of the straight-line path segment (Fig. 1.43) is taken as the main axis X. The line perpendicular to the Z-axis and also situated in the plane of the horizon is the sec- ond axis , Z . Fig. 1.43. Orthodromic Coordinate System, In Figure 1.43, angles a^ and a2 are the directions of the first and second straight-line segments of the path, measured from the meridians of their starting points. Points Oj and O2 are the starting points of the segments, the coordinates of which are de- termined in the geographic coordinate system. The orthodrome dis- tances O1O2 and O^O-^ are the lengths of the straight-line segments; the angle TA-^ is the angle of turn of the orthodromic coordinate system at point O2 • Since an aircraft moving above the Earth's surface in a given direction has only small random deviations from the given flight path (as a rule, not more than 20-30 km), it is possible to take the spherical surface of the Earth within the area of the possible deviations of the aircraft from the J-axis of the orthodrome system as a cylindrical surface. Then the unrolling of the cylinder gives us a rectangular system XZ on a plane. Let us assume that an aircraft moves from point O^ at a small angle to the OxXj axis equal to ij; - a^, and covers a distance S. /71 67 The coordinates of the aircraft at point M^^ are determined by the equations: ^a=5cos(.l/-ai); | 2'^=5sln(<j;-ai). J (1. 56) Measuring the Z^ coordinate constitutes checking of the path of the aircraft according to distance, while measuring the Z^ coor- dinate constitutes checking of the path according to direction. Periodic measurement of the Z^ and Z^ coordinates make it pos- sible to determine all the basic elements of aircraft movement; for example : a) Direction of aircraft movement ( i|j ) : 2a2~^ai ^ = arctg- -— + '^. (1.57) where Xg^ , Z^^^ are coordinates of the aircraft at the first point, ^aa » ^a2 ^^® coordinates of the aircraft at the second point; b) Speed of aircraft movement along a given flight path {W) '^a2~-^a2 W= (1.58) where t is the flying time of the aircraft between points Ja^ sn d ^a2 c) Remaining flying time to point O2 Xrem where ^pgm ~ '^l'^2 " -^a • d) Necessary flight direction for arrival at point O2 : Z 4, = ai— arctg-^r: — . (1.60) rem Formulas (1.55) to (1.60) are entirely obvious and do not re- quire special derivations or proofs. To refine the coordinates of the aircraft in the orthodrome system, we can use correction points (CP), visual or radar land- /72 marks on the Earth's surface, locations of ground radio facilities, etc. (Fig. 1.44). Translator's note: arctg = cot"-'-. 68 If the correction point is observed from an aircraft at an angle to the given route, at a distance from the aircraft equal to i? , the coordinates of the aircraft will be determined by the formulas : Zop Fig. 1.44. Determining the Orthodromic Coordinates of an Aircraft from a Correc- tion Point . Xg^=Xcp-Rcos n (1.61) During flight over the correc- tion point, i.e., when this point is observed at an angle equal to 90° to the flight path, (1.61) is simplified and takes the form: X^=X cp cp- m(x d) ' rem' Fig. 1.45. Transfer of the Next Stage in a Course to an Orthodromic Coordinate System: (a) with the Aircraft Position on the Path of the Given Course; (b) with Deviation of the Aircraft from the Path of a Given Course. The simplicity of the geometrical transformations and the na- tural perception of the coordinates of the aircraft in an orthodrome system, both of the path covered by the aircraft and of the devia- tion allowed from the given path, make it the most acceptable coor- dinate system for a given flight path. In high-speed aircraft (as a result of a large turning radius), in order to emerge without deviation at the next stage of the or- thodrome path, it is necessary to consider the linear advance to the angle of turn {TA) , This transfer is connected with transform- ations of the coordinates of the aircraft from the orthodrome sys- tem of the preceding stage to the system of the following stage. In Figure 1.45, a, point M located on the flight path of the preceding stage of flight is the point of the beginning of turn for arrival at the flight path of the following stage. Obviously, the coordinates of this point in the system of the following stage will be equal to /73 69 Z2 A2 = Areincos'rA; \ = A* rem sin T A. ) (1.62) In general, when the coordinate Z at the beginning of the turn is not equal to zero, i.e., if the aircraft is not located strictly on the given flight path when beginning the turn, the transforma- tion of the coordinates must be carried out according to the fol- lowing formula (Fig. 1.45, b): X2 = Zsinyn — Xrem cosTA; Z2 = Zcosyn+A' rem. sin TA; I TA. ) (1.63) In the process of turning, the coordinates of the aircraft are measured in the system of the following part of the flight in which their calculation after turning is carried out. The orthodrome system examined by us is sometimes called the stage orthodromia coord'inate system. In some instances , a rectangular co- ordinate system is used for flight over an area, e.g., for maneuver- ing of an aircraft in the region of an airport and for special-.pur- pose flights, etc. In these cases, the direction of the meridian at the point of origin of the coor- dinates or some other direction [for example, the direction of the take-off -landing zone at an alr- Z-axis, and a rectangular coor- Cx,z,) / N {X^i '.) k(Y«) - / (x,z,-o) 1 z Fig. 1.46. Rectangular Coor- dinate System for Flight over an Area. port (Fig. 1.46)] is taken as the dinate system is constructed from this. The flight is carried out along the given coordinates of the points of the route [for example, along the coordinates of the be- ginning of each of the four turns in the rectangular maneuver of making an approach to land at an airport (X.^Z;^), (Zj^z)^ (^323), The limits of applicability of an areal rectangular coordinate system are limited by the effect of the sphericity of the Earth on the precision of measurements. In practice, without noticeable dis- tortions, such a system can be used within a radius of 300-400 km from the point of origin of the coordinates. It Is also applied with the use of navigational indicators In flight, when the orthodrome direction of part of the course is taken as the X axis . /74 70 Arbitrary (Oblique and Transverse) Spherical and Polar Coordinate Systems In the solution of navigational problems with a geographical coordinate system in polar regions, very significant errors arise. A special chapter is devoted to problems of accuracy in air- craft navigation. In the present section, for the purpose of illus- tration, only (1.36a) is examined. It is obvious that with the approach of the aircraft to a lati- tude equal to 90°, the tangent (j) will approach infinity. Therefore, small errors in measuring the latitude of the location of the air- craft will cause the errors in calculating the longitude to grow indefinitely. To avoid a loss of accuracy in solving navigational problems, especially by automatic navigational devices, random spherical co- ordinate systems are employed. ^ Arbitrary spherical systems dif- fer from a geographical system by the fact that the poles of these systems do not coincide with the geographic poles. Therefore, in these systems all the analytical transformations of distances and directions which are carried out for a geographic coordi- nate system are justified. For transferring from a geograph- ical coordinate system to an arbitrary spherical system, or vice versa, it is necessary to derive special equa- tions: let us examine Figure l.M-7. igure 1.47, a cross section rth's sphere is shown. Here n in such a way as to pass poles of the geographic sys- s, i.e., so as to appear as eographic and arbitrary sys- t such a plane exists with itrary spherical system. Fig. of S on t the thro tem the terns any 1.47. pheri ca he Eart plane o ugh the and the plane o simult distrib Trans 1 Coo f the cent arbi f the aneou ution format rdinat urf ace cross er of trary merid sly . of th 1 on es In F of the Ea section is chose the Earth and the coordinate system ian in both the g It is obvious tha e poles of an arb /75 Let us agree that a reading of the longitude both in the geo- graphical and arbitrary systems will run from the indicated plane of intersection. Lines AB and AiBi in Figure 1.47, and the lines parallel to them, appear as lines of intersection with the planes of the equator and the parallels in the geographical and arbitrary systems. Point P is the pole of the geographic coordinate system; ?! is the pole of the arbitrary system; angle 9 is a combination of the axes of the geographical and random systems. 71 Let us choose point M ( (j) i X i ) on the Earth's surface and pro- ject it onto the plane of the cross section (point Mi). It is ob- vious that OL will appear as the line of the sines of the latitude of point M in an arbitrary system, while LMi will appear as the line of the cosines of the longitude of point M of this system in the plane of its parallel, i.e., LMi = cos Xj cos ^j. The latitude of point M in the geographical system will equal: siny = 0/:cose — ijW,sine or sin <f = sin 91 cos 6 — cos Xj cos (jjj sin 6. (1.64) It is obvious that a perpendicular dropped from point M to the plane of intersection (point Mi) will be the line of the sine of the longitude in the arbitrary system in the plane of the parallel of this point; at the same time, the line of the sine of the longi- tude in the plane of the parallel of the point in the geographic coordinate system will be sin X = sin Xj cos <pi sec <f . from which it follows that MMi ■-= sin Xi cos 91 = sin X cos <p. Formulas (1.64) and (1.65) make it possible to determi coordinates of a point in a geographical coordinate system ing to its coordinates , known in the arbitrary system under condition that the plane coinciding with the axes of both s is taken as the initial meridian. After solving the proble cording to (1.64-) and (1.65), it is necessary to introduce rection into the X coordinate equal to the longitude of the pole in the geographic coordinate system. (1.65) ne the accord- the ystems ms ac- a cor- Pi Since the principles of construction of spherical and geograph- ic coordinate systems are identical, for the solution of the reverse task (transferring from the geographic system to the arbitrary one), it is sufficient to drop the subscripts in the functions of the co- ordinates of (1.64) and (1.65) whereever they occur and to add them where they are absent : sin fi = sin ip cos 6 — cos X cos 9 sin's; sin Xj = sin X cos 9 sec f j. Formulas (1.64) and (1.65) were given with a consideration of the flattening of the Earth at the poles, i.e., the Earth was taken as a sphere with a mean radius . /76 72 Position Lines of an Aircraft on the Earth's Surface Thus far, we have examined coordinate systems on the Earth's surface as systems which connect the position of an aircraft with the Earth's surface during its movement in a given direction. In aircraft navigation, it is often necessary to determine the elements of aircraft movement according to consecutive coordinates. It is obvious that means and methods for measuring the coordinates of an aircraft are necessary for this purpose. Usually the two-dimensional surface coordinates of an aircraft are determined separately according to two lines of the aircraft's position measured at different times or according to two lines measured simultaneously. In some cases, it is sufficient to deter- mine one line of the aircraft's position. The geometric locus of points of the probable location of an aircraft on the Earth's surface is called the posit-ion tine of an aircraft . Similar groups of aircraft position lines are called a family of position lines. For example, if the latitude of the location of an aircraft is determined by astronomic means based on the elevation of Polaris, the parallel on which the aircraft is located will be a position line of the second family. Let us assume that the longitude of an aircraft was determined simultaneously on the basis of the altitude of a star, the azimuth of which is equal to 90 or 270°. The longitude obtained by such a method is a position line of the second family. Direct measurement of the geographic coordinates of an aircraft is possible only by astronomic means, and not in all cases. In determining the location of an aircraft by optical or radio- metric means, the families of position lines generally do not coin- cide either with the grid of geographic coordinates or with the given flight direction. At the present time, there are several types of coordinate systems which are used as families of aircraft position lines in the application of radio-engineering and astronomic facilities of aircraft navigation. They include the following: 1) A two-pole azimuthal system^ in which the radial lines {bearings) diverging from two points on the Earth's surface with known coordinates are families of position lines. 2) Polar or azimuthal range- finding system^ in which the bear- ings from a point on the Earth's surface with known coordinates are /77 the first family of position lines of this system, and concentric 73 ■■■■■■■ ■■■■ I circles at equal distances from the indicated point are the second family . 3) L-lnes of equal azimuths (LEA), which are position lines relative to known points on the Earth's surface, at each of which the azimuth of a known point retains a constant value. ^) Difference-range finding (hyperbolia system) ^xn which each family of position lines is bipolar; a constant difference of dis- tances to the poles of the system is preserved on each position line . 5) Over-alt range finding (eltiptioal) system^ in which the family of position lines is bipolar; a constant sum of the distances to the poles of the system is preserved on the position lines. 6) aonfoaat hyperbotia-eZtiptiaat system^ in which the fam- ilies of position lines are ellipses and hyperbolas confocal with them . From the above list of coordinate systems, it is evident that each has arisen from the nature of the navigational values measured by the devices used. The indicated values are called navigational parameters . For^example, for a hyperbolic system the difference in dis- tances serves as a navigational parameter, and in an azimuthal sys- tem (or for lines of equal azimuths) the azimuth serves as a navi- gational parameter, etc. In evaluations of the accuracy of navigational measurements, considering that the intersecting segments of the position lines of any system can be assumed to be straight-line segments in the region of the location of the aircraft, the concept of a unified coordinate system is sometimes introduced for the purpose of study- ing the general properties of all the above systems , including the geographic and orthodrome systems. In studying these coordinate systems , it is necessary to con- nect each of them with the geographic system for locating the in- termediate points of the position lines, in order to plot them on a map. In addition, it is necessary to know the analytical form for determining the coordinates of an aircraft in a geographic or ortho- drome system on the basis of known parameters of navigational sys- tems without plotting position lines on the map, as is done in auto- matic navigational devices. Bipolar Azimuthal Coordinate System Bearings for an aircraft, i.e., orthodrome lines diverging from two points on the Earth's surface with known coordinates, are position lines in the azimuthal coordinate system. 74 Let us assume that we have two points Oi and O2 on the Earth's surface (Fig. 1.48). If the map being used has been executed on a projection having the properties of isogonality and orthodromicity , e.g., on an inter- /78 national projection, the indicated position lines on the map can be taken as straight lines originating at points 0^ and 02- However, satisfactory accuracy in determining the coordinates of an aircraft at the intersection of the bearings as straight lines on a map is preserved at comparatively small distances and only on maps with an international projection. In general, for the precise plotting of position lines on a map, let us consider the points 0^ and O2 as poles of an arbitrary spherical coordinate system. Let us consider the distances S from Fig. 1.48. Bipolar Azimuthal Coordinate System. these points to any point on the Earth's surface M as complements of the latitude of point M in these coordinate systems, up to 90°: 5i = OiAf = 90° — <pi; S2 = O2M = 90° — <(2. In this case, the coordinates of point M in the geographical system are determined according to (1.64) and (1.65). Taking the meridian of point 0^ as the prime meridian of the geographic system, Oj as the azimuth a for the longitude in the spherical system, and a value of 90° for Si as the latitude in this system, let us obtain (in the geographical coordinate system) sin <p = cos Si cos 6 — sin Oj sin Sj sin 6, where O = 90°-<fo,; sin X = sin Oi sin Si sec 9. Given the definite value Si and substituting different values for ai , e.g., greater than 1°, formulas let us use the given formu- las to find the coordinates of the points of intersection of the azimuthal lines with the circle of equal distance Si in the geograph- ic system. Given another value for Si and having carried out the same operations with a^, we will obtain the coordinates of the points of intersection of the azimuthal lines with a circle of equal distance having this radius. Continuing to increase Si to a full radius of operation of a 75 navigational device, let us obtain the coordinates of the interme- diate points of the azimuthal position lines, running from pole Oi in the geographical coordinate system, with the longitude changed to the value Xqi • Introducing a correction in the values of the /79 longitudes of the intermediate points for the indicated value Xq , let us obtain the longitudes of these points from the prime meridian of the geographic system. On a map of any projection, by joining the points obtained by lines running from point 0^, we will obtain position lines of the first family. In this way, it is possible to obtain the family of position lines from point O2 , taking it as the pole of the second arbitrary spherical coordinate system. Let us now determine the coordinates of point M in the geo- graphical system, based on known azimuths measured at points 0^ and O2, without recourse to the plotting of position lines. Let us first solve this problem in the spherical system of one of the poles of a navigational device, e.g., O2 (see Fig. 1.4-8), taking the azi- muth of point Oi as the prime meridian. According to (1.64) and (1.55), the coordinates of point M in this system are determined by the equations : sin ^1 =i sin 92 cos 8 — cos Xj cos <(2 sin 6, where is the angular distance of O1O2; sin Xj = sin X2 cos (p2 sec'^j , where A1X2 are respectively ai , a2 . From (1.65) it is evident that slnX; _ cos 92 8lnX2 cos 91 8inX2 ~ C0S91 °'' .slnXj "" cosipj ' (1.66) Substituting into (1.52), instead of cos cjjj its value accord- ing to (1.64), we obtain: sinXo — , y =tg<P2COs9 — cosX2Sln6 or tg <P2 = sin X2 cosec Xj sec 6 + cos X2 tg 8. (1.67) Since the azimuth of point M in the O2 system is considered known, we obtained both coordinates of point M in this system. For transferring to the geographical coordinate system, it is again possible to use (1.64) and (1.65), considering as angle 9 the value ^^2 > ^^"i ^s the prime meridian the longitude of the point O2 . 76 It is obvious that here it is necessary to introduce a correc- tion into the X2 coordinate for the value of the azimuth of point Oi from point O2 » i.e., the corrected value of X2 will equal: X2j,= X.2 + Oq^. (1.68) It is also obvious that after transforming the coordinates into /80 geographical ones, it is necessary to introduce a correction into coordinate X2 for the longitude of point O2 • >^ + ^0.' (1.59) Formulas (1.6'+) and (1.65) also make it possible to Implement a transfer from a spherical system with pole O2 "to the orthodrome system. This is necessary for determining the position of an air- craft relative to a given flight path. Actually, it is possible to consider the orthodrome system as a spherical system if we measure the X- and Z-coordinat es not as linear but as angular measures, i.e. , we take the J-coordinate as A and the Z as <^ . In this instance, it is advantageous to take the ^-coordinate of point O2 as the prime meridian and the value 90° - Z of this point as the angle G • The coordinates of point M in the ortho- drome system will then equal: sin A' = sin 92 cos 6 — cos X2 cos <P2 sin 6; ^1n Z = sin X2 cos lyj sec X. If the Xo-coordinate , not equal to X02 > is taken as the prime meridian, then after transforming the coordinates according to (1.64) and (1.65) a correction equal to X02 is introduced in the X[^ coordinate. Goniometric Range- F i nd i ng Coordinate System The goniometric range-finding system is the most convenient system for conversion to the geographic or orthodrome system. Since direction and distance are measured simultaneously in this system, for conversion to the geographical system it is suffi- cient to use (1.64) and (1.65), taking the value 90° angle and Xqi for the prime meridian, '01 for In this instance the value 90° - 5 is considered the latitude of the point M in the coordinate system with pole Oi, while the azi- muth of point M is considered as the longitude. In the geographical system, the coordinates of the point M will equal: sin <f = sin ipi cos 6 — cos Xj cos <pi sin 8; sin X = sin Xi cos fi sec tp. After transformation, it is necessary to introduce a correction 77 to the coordinate X for the longitude of point Oi : (1.70) The conversion to the orthodromic coordinate system is imple- mented in the same manner as was done in the bipolar azimuthal sys- tem after solving (1.67). If the radius of action of the boniometer range-finding coor- dinate system is small (on the order of 300-1+00 km), it is possible to disregard the sphericity of the Earth in converting to the or- thodrome system and the problem of transfer is considerably simpli- fied (Fig. 1.49). In Figure 1.49, it is evident that with known values of R and a in the goniometer range-finding system, the coordinates of point M in the orthodrome system can be determined according to the fol- lowing formulas : (1.71) (1.71a) /81 where i|j is the direction of the orthodrome segment of the path rela- tive to point Oi. Bipolar Range- F i nd i ng (Circular) Coordinate System In a bipolar range-finding system (Fig. 1.50), the distance to two points on the Earth's surface with known coordinates is a meas- ured navigational parameter. Fig. 1.4-9. Conversion of Polar (Goniometer-Range-Finding) Co- ordinates to Orthodromic Coor- dinates . Fig. 1.50. rsipolar Range-Find- ing Coordinate System. The indicated distance is usually determined according to the time of passage of radio signals from the aircraft to the ground radio-relay equipment and back to the aircraft. 78 In Figure 1.50, it is evident that the task of determining the coordinates of an aircraft in a circular system is double-valued. The point of intersection of the circles of equal distance to the poles Oi and O2 is considered the location of the aircraft. Since there are two such points for any pair of circles , additional signs are used for choosing the actual point, e.g.: a) Provisional aircraft position at the moment of measurement. b) Tendency toward a change in distance during flight in a definite direction. In Figure 1.50, it is obvious that in flying from north to south, the distances i?x and i?2 will decrease at point M and increase /82 at point Ml . On maps with different projections, circular position lines will have a different appearance. Usually they are plotted on maps on an oblique central and international projection on which the ap- proximate form of the circles is preserved. For plotting the indicated lines on a map with any projection, it is necessary to determine the coordinates of their intermediate points. This problem is solved in the same way as for bipolar azl- muthal systems , with the sole difference being that after determin- ing the coordinates of intermediate points , the latter are not joined by radial position lines, but by circular lines. In converting from a circular to a geographic or orthodrome system, it is necessary first to determine the coordinates of point M in the spherical system relative to one of the poles of the cir- cular system. Considering the line lO 2 as the initial meridian of this sys- tem, the latitude of point M in the system 1 according to (1.64) will be sin.yj = sin i(>2 cos 6 — cos A2 cos tp2 sin 8, where 9 is the angular distance between points 0^ and O2 ; <(> 1 > <t'2 are 90° - Ri and 90° - i?2 , respectively. Carrying out simple transformations, we obtain: cos Xo = sin 92 cos 6 — sin ^^ cos f2 sin (1.72) Formula (1.72) makes it possible to determine the X-coordinate in the O2 system. Since tl^e ^-coordinate in this system is deter- mined directly as 90° - i?2 » it is possible to consider the problem solved . 79 The conversion to the geographic or orthodrome system is im- plemented by the same means as in the azimuthal and goniometer range-finding systems. Lines of Equal Azimuths Lines of equal azimuths (LEA) are a family of aircraft position lines which converge at one point on the Earth's surface, on each of which the azimuth of the known point retains a constant value (Fig. 1.51). For finding the location of an aircraft , it is used along one line of equal azimuths of two families , as is done along two bear- ings in an azimuthal bipolar system. Lines of equal azimuths were widely used in the period when the radiocompass (aircraft radiogoniometer), measuring the distance from the aircraft to the ground radio station, was the most refined nav- igational facility. Along with lines of equal azimuths , a method of determining the coordinates of an aircraft by plotting bearings from a radio station to an aircraft (taking account of the convergence of the meridians between them) has become widespread. An advan''"age of the lines of equal azimuths , in comparison with bearings for an aircraft, is the fact that the solution of the prob- lem of determining an aircraft's coordinates is independent of its location, whereas in order to plot bearings it is necessary to know the approximate coordinates of the aircraft for calculating the con- vergence of the meridians . In examining lines of equal azimuths, there is no sense in de- riving an analytical form of transformations for converting to the /83 T 7 T Equator Fig. 1.51. Line of Equal Azi- muths (LEA) . Fig. 1.52. Determining the Coordinates of Intermediate Points of an LEA. 80 geographic or orthodromic coordinate system. Let us limit ourselves to an examination of the means of calculating intermediate points for plotting them on a map in order to make it possible to determine the coordinates of an aircraft according to the intersection of the lines of equal azimuths of two families on a map with any projection, In Figure 1.52, one of the lines of equal azimuths of a family converging at point M is shown. At this point, the orthodromes in- tersecting the equator at different angles oq, j coo* etc. converge. According to (1.32), it is possible to find the longitude Xg of the points of intersection of a family of orthodromes with the equator, given the values of the initial angles oq . According to (1.UM-), cos <p, = COS <p sin a. Or since cos sm a > cos w = sin Op sin a ' (1.73) Formula (1.73) makes it possible to determine the latitude of a point on any line of the family of orthodromes which converge at point Af , where the azimuth of point M is equal to the given value of a . /Qi\ The longitude of the indicated point can be determined accord- ing to (1.36a) by substituting into it the given initial angle of the orthodrome and the latitude obtained from (1.73). It is obvious that the longitude obtained will be measured from the starting points of the family of orthodromes. Therefore, to reduce it to the geographic system, it is necessary to introduce a correction for the longitude of the indicated initial points. Having solved this problem for every value of oq with given values of a, let us obtain the intermediate points of the family of lines of equal azimuths . The problem of determining the coordinates of intermediate points on lines of equal azimuths of the second family, whose plot- ting on a map yields a grid of intersecting aircraft position lines, is solved analogously. D i f f erence-Range- F i nd i ng (Hyperbolic) Coordinate System The circular range-finding system of aircraft position lines examined earlier is used with comparatively small distances from the ground radio-engineering equipment to the aircraft, since the sending and radio-relaying of radio signals to the aircraft over great distances involves technical difficulties . 81 The technical solution of the problem is greatly simplified if, instead of relaying aircraft radio signals, we send simultaneous radio signals from two ground radio-engineering installations, with their subsequent reception by the aircraft. However, in this instance it is advantageous to measure not the absolute distances from the ground installations to the aircraft, but only the difference .in distances to them. The system of position lines for the difference in the distances to two points on the Earth's surface is called the differenae-range- finding or hyperbolia system. The geometric locus of points, the difference in whose distances to two given points (foci) is a constant value equal to 2a (Fig. 1.53), is called a hyperbola. The distance along the focal axis from the point of intersec- tion of the focal and conjugate axes to the peak of the hyperbola is the value "a". It is possible to designate hyperbolic aircraft position lines on a map by doubling the value of "a" as an ordinal / number. The distance along the focal axis from the focus to the intersection with the conjugate axis is designated by the value "e". To determine the position of an aircraft, two families of hyperbolic position lines constructed from three points forming pairs of focal axes are usually used. At each of these points, ground radio-engineering installations for synchronous transmission of radio signals are established. In order to plot hyperbolic position lines on a map with any projection, the intermediate points of hyperbolas in the spherical system of one of its foci, e.g. Fi, are determined first. Here the direction F \F 2 is taken as the initial meridian of this system. Bearing in mind the fact that the latitude of any point in the Fi system equals 90° - Si and 90° - S^ in the ^2 system, the value S2 = ^i + 2a, and taking the distance F iF 2 as the angle 9, it is possible to write (1.6H) in the following form: cos (5i + 2a) = cos Si cos 2c — cos Xj sin 5i sin 2c, Hence , V >1 a u fO w B •H •H X bO m (0 6 Fig. 1.53. Difference-Range- Finding (Hyperbolic) Coordinate System . 1 cos Sy co s 2c — cos (Si + 2a) COS A| — .—__„ , ^^ ._ ^_ sin 5i sin 2c (1.74) 82 Given the definite values of S as circles of equal radius and changing the values of 2a, it is possible to determine the value of X of all the hyperbolas of the family at points of intersection with the indicated circles. For conversion to the geographical coordinate system, the intermediate points are recalculated according to (1.64) and (1.65), after which they are plotted on a map and joined by smooth lines. Hyperbolic coordinate systems are usually used in the applica- tion of radionavigational devices with a large effective radius. Therefore , the automatic conversion of the hyperbolic coordinates to geographical or orthodromic coordinates is advantageous. The problem indicated is solved comparatively easily when all three foci of the hyperbolic system are situated on one orthodrome line (Fig. 1.54). According to (1.74J, , _ cos ^1 COS 2c J — cos (5] + 2ai) sin Sisln2ci cog ^1 cos 2c; — cos (Si + 202) sin Si sin2c2 Expanding the value of the cosines of the sum of the angles and carrying out a reduction, we obtain: cos 2ci — cos 2fli + tg 5i 8ln 2gi _ cosgCj — co3:^g2 4- tg-^i sin 2^2 sin 2c, ~ 8in2c2 Multiplying both sides of the equation by sin 2ci'sin 2^2 and rearranging the terms, we obtain: /86 cos 2*1 sin 2c2 — cos 2c2 sin 2 c, — sin 2c2 cos 2ai + sJn 2ci cos 203 =f = tg 5i (sin 2ci sin 2^2 — sin 2c2 sin 2«,^ or „ _ _sln 2c2 (cos 2ci — cos 2a i ) — sin 2ci (cos 2C2 — cos 2fl2) sin 2ci sin2a2 — sin 2c2 sin 2ai (1.75) The task is simplified even more if the distances FFi and FF2 which are chosen are identical, i.e. 2ai = 2C2' In this case cos 2^2 — ,cos 2fli *^ '"" sln2«3 — sia2ai ' (1.75a) Formulas (1.75) are used for determining the coordinates of an 83 aircraft in a spherical system with the pole at point F, bearing in mind that ^ = 90° - Si. The A-coordinate with a known value of Si is easily determined on the basis of il.T^). For conversion to the geographic or ortho- drome system, the same formulas (1.6U) and (1.65) are used. The problem of conversion to the spherical (and consequently, to the geographic coordinate system) if the foci of the hyperbolic system are not located on one orthodrome line (Fig, 1.55), is much more complicated to solve. It is obvious that (1.74) can be reduced to the form: cosXi = cos Si cos 2ci — cos Si cos 20] + sin Si sin 2ai sin5i sin 2^1 Carrying out simple transformations, we obtain: cos Xj = ctg Si ctg 2ci — ctg 5i cos 2ai cosec 2cj + sin 2«i cosec 2ci or ctg 5, cos Xi — sin 2ai cosec 2ci ctg 2ci — cos 2«i cosec 2ci (1.76) In Fig. 1.55 it is evident that in the FF2 system Aj^ = Xi + 3; /87 therefore, the following equation is valid: cteS — ^°^ ^^' + ^^ ~ ^'" 2^2 cofe c 2C; ctg 2c2 — cos 2^2 cose^2c2 (1.76a) Designating the second terms of the numerators of (1.76a) by X and the denominators by I and reducing to a common denominator, we obtain : Zr, F 2c, Srf!a^ Fig. 1.54. Conversion of Hyper- bolic Coordinates to Spherical Coordinates (Special Case) Fig. 1.55. Conversion of Hyper- bolic Coordinates to Spherical Coordinates (General Case). 8 4 OP (cosX, — Ai)K2 = [C0S(X, +p)-JV2]K, Vq cos Xj — K2A'i = Kj cos Xj cos ? — Kj sin Xj sin p — K1X2. Rearranging the terms, replacing sin X, by /I - cos'^X , and squaring both sides of the equation we obtain: or cosX, (Vj — cospK,) — Jfir2 + A^r, = — Vl — C082X, slnpKi cos2Xi ( y| ;^ 2yi K2 cos p + J1) + 2cos X, (Aijy, + Xi K2) X X (K2- n cosP) + (ATay, - ;Vi}'2)2- sinapKl Thus, the coordinate Xj is determined by the solution of a quadratic equation (;y2>"l-JflK2)(K2-C0SpKi) cos X. = ± K? — 2yiK2C0sp+y^ [/ -(>'?-2yiK2 cos ? + yl) [(X2 Ki — ^1 r2)'' — 8iii2 p kJ] where Kf-2KiK2COsp+ yj Xj = sin 2ai cosec 2ci; X2 = sin 2^2 cosec 2c2; Yi = ctg2ci — Cos2aj cosec 2cj; ^2 = ctg 2c2 — cos 2a2 cosec 2c2 . There is no sense in substituting the indicated values Xi , ^2, ^1 5 and Y2 into (1.77), since this hampers its solution greatly. In practice, it is easier to determine the numerical values of these magnitudes first, on the basis of the known values of 2ai, 2a2, 2ci, and 222, and to substitute them into (1.77). Knowing the coordinate X makes it possible to determine easily the coordinate <)) 1 , e.g., according to (1.76), keeping in mind the fact that (j) 1 = 90° - ^i , and then to convert to the geographic or orthodrome system using (1.64) and (1.65). Overs 1 1 -Range-F i nd i ng (Elliptical) Coordinate System Hyperbolic navigational systems are the most easily implemented technologically of all the range-finding systems. However, from 85 the point of view of use in flight, they are geometrically disadvan- tageous . Both families of position lines are divergent, and at distances exceeding 2c from the center of the system are practically directed along the radii of this center. This leads to an increase in error in determining aircraft coordinates with an increase in the distance from the center . In addition, with an increase in distance, the angle of inter- section of the hyperbolic lines of the two families decreases. This also lowers the accuracy of determining the aircraft coordinates Combination of hyperbolic position lines with elliptical lines turns out to be more advantageous (Fig. 1.56). It is known that the geometrical place of points, the sum of whose distances to two given points (foci) is a constant magnitude equal to 2a, is called an ettipse , The distance along the major axis from its intersection with the minor axis to the top of the ellipse, i.e., its semi-axis, is considered to be the value "a", in this case. The distance from the intersection of the axes to the foci is considered to be the "e" value. If, in addition to the difference in distances to the foci, the distance to one of them is measured, it is easy to implement the hyperbolic-elliptical system of position lines. Actually, if one distance Si is known and the difference in the distances hS = 2a^ , obviously the distance to the second focus is 2ae = 2Si + 2ah where a^ is the major semi-axis of the ellipse and a-^ is the parame- ter a of the hyperbola. Obvious advantages of the hyperbolic-elliptical system include the following: (a) There is an absence of divergence in the second family of position lines. The elliptical position lines are closed, so that the accuracy of determining the aircraft's position along them does not decrease with an increase in distance . A \ ' \ ;' /> \ \ A j ^ ^ \' ^ I 1 Fig. 1.56. Hyperbolic-Ellipti- cal Coordinate System. (b) Orthogonality of the /89 86 position lines appears at any point of the system. In a confocal hyperbolic-elliptical system, the position lines intersect only at a right angle . (c) Two foci, instead of the three for a hyperbolic system, are sufficient for the construction of a confocal hyperbolic- elliptical system. This leads to a simplification of the transfor- mations during conversion to a geographic or orthodrome system. However, in spite of the advantages of a hyperbolic-elliptical system indicated above, a wide distribution was not obtained. This was 'connected with great technical difficulties in measuring distance to points on the Earth's surface at distances exceeding straight- line geometric visibility of the object from flight altitude. The above problem is solved by keeping on board the aircraft a reference frequency (quartz-crystal clock) which permits syn- chronization of the transmission of radio signals from the ground with reference signals on board the plane. It is therefore possible to determine the travel time of the signals. Hence 5 strict stabilization of the reference frequency on board the aircraft is the main task for the technical implementation of hyperbolic-elliptical systems. To plot elliptical lines on a map with any projection, the intermediate points are determined according to the same formulas as the family of hyperbolas. For example, in (1.74), considering the second distance in it (.S2) not as the sum Si + 2aj^ , but as the difference 2ag - Si , cos X.J = C OS 5} CO S 2c — cos (2a — Si) ain .?. cin 9/- ' sln5i sin 2c (1.78) where Xi is an angle with the vertex at point Fi major axis of the ellipse. measured from the Given the different values of ^i and determining the values of Xi for each of them with a constant value of 2a, we shall obtain intermediate points of an ellipse in a spherical system Fi. Changing the value of 2a and performing these operations with Si, we obtain intermediate points of the next elliptical position line, etc. Recalculation of intermediate points is implemented in the geographic system according to (1.64) and (1.65) as in other cases analyzed by us. In the hyperbolic-elliptical system, the conversion to the geographic or orthodromic coordinate system is very simple. In fact, the value of Si and the parameter 2a in this system 87 are measured. Therefore, (1.78) is useful for the problem of calculating the spherical coordinates of an aircraft along measured parameters and for a subsequent transfer to the geographic or orthodrome system. 8. Elements of Aircraft Navigation /90 Aircraft flights are carried out in airspace. The physical composition of airspace, as well as the speed and direction of its shift relative to the Earth's surface, exert a substantial influence on the trajectory of aircraft movement in a geographic or orthodromic coordinate system. Until recently, the direct measurement of the speed and direc- tion of aircraft movement relative to the Earth's surface was a problem. At the present time, this problem has been solved. How- ever, it is not advisable to install the complex and expensive equipment which measures the indicated parameters on all aircraft. In solving navigational problems, parameters of aircraft move- ment relative to the airspace are usually measured to the greatest extent possible, and then additional parameters of the movement of the aircraft which are connected with movemert in airspace are found . Summarizing the measured parameters of aircraft movement , the value and direction of the speed vector of the aircraft relative to the Earth's surface are found. The parameters of aircraft movement with which we must concern ourselves in carrying out aircraft navigation are called elements of aircraft navigation. Elements of aircraft navigation are divided into three groups which determine the direction, speed, and altitude of flight. Elements which determine Flight Direction The basic element which determines the direction of aircraft movement in airspace is called the aircraft course. The aircraft course (generally designated by y) is the angle between the direction of a meridian on the Earth's surface and the direction of the longitudinal axis of the aircraft in a horizontal plane . Usually it is considered that the airspeed vector of an air- craft in the plane of the horizon coincides with the direction of the longitudinal axis of the aircraft, although this is actually not entirely true. Therefore, an understanding of the course often coincides with an understanding of the direction of the flight air- speed vector . 88 Depending on the reference system chosen, the following special varieties of aircraft courses can be distinguished: (a) A tvue aouTse (TC) is measured from the northern end of a geographic meridian which passes through the point of intersection of the Earth's surface with the vertical of the aircraft. The lat- ter is usually called the position point of the aircraft (PA): (b) The orthodrome course (OC) is measured from the northern end of a geographic meridian of the starting point of a rectilinear /91 (orthodrome) segment of the path or from another conditionally chosen (reference) meridian along which the zero point of the course-reading scale is established. (c) The magnetic course (MC) is read from the northern end of the magnetic meridian which passes through point PA, In addition to these varieties of aircraft courses, there is another concept, the compass course (CC), i.e., a course based on the responses of a compass. In textbooks on aircraft navigation, the concept of compass course has included only magnetic compasses, but we have broadened this concept to include all methods of measuring an aircraft course with an Instrument . Aircraft courses are measured by three different methods, i.e., stabilization of the zero reading of the compass along the meridians: Magnetic course, by means of magnetic systems. True course, by means of astronomical systems. Reference course, by means of gyroscopic devices. All of these methods have instrumental errors or a deviation designated by A^,. Individual components of errors in the course devices are components of the deviation. Any of the three types of aircraft courses can be obtained from responses of a course device, allowing for its deviation, e.g., OC TC = MC -CC„rth+ ^o\ = CCastr + ^c > = CCmag + ^c J (1.79) In the general case , Y = CC + Ac As a correction for any measurement , the value A^ is considered positive when the compass underestimates the value of the measured magnitude, and negative when the compass readings are too high. 89 In the future, when we study the relationship among the three types of aircraft courses, we will consider that the value of each has been corrected for the deviation of the device. The interrelationship between magnetic and true flight courses is established with the least difficulty, since the^e courses are measured from the meridians which pass through point PA (Fig. 1.57a). In Fig. 1.57a the northern geographic meridian is designated by Pjj , the direction of the magnetic meridian by P^. Since the magnetic meridian is shifted to the left relative to the geographic meridian, the magnetic declination at the given point is negative. With a positive deviation, the magnetic merld- /92 ian is shifted to the right relative to the geographic meridian. From the figure, it is evident that MC=TC-Am TC=MC + A M- J (1.80) In the case of Af^ , the value is negative; therefore, the abso- lute value of the true course turns out to be less than the magnetic course . Converting from the true or magnetic course to the orthodrome course (or vice versa) is more complex (Fig, 1.57, b). Fig. 1,57. Interrelationship of Aircraft Courses; a: Magnetic and True; b: True and Orthodrome. The direction of the geographic meridian passing through point PA is designated by P^', the direction of the reference meridian ^r . m . is shown by a dotted line which intersects the geographic meridian at angle 6 . Therefore TC = OC- OC = TC + :;) (1.81) 90 The value 6 , the angle of convergence of the meridians , is considered positive when the direction of the geographic meridian at point PA is proportional to the reference meridian extended clockwise (to the right) and negative when the geographic meridian is shifted to the left. On small path segments (500-600 km), the angle of convergence of the meridians is approximately equal to: S =Ur.m.- ^PA) sin cp^^ (1.82) In general, for converting from the orthodrome course to a true course or vice versa, it is necessary to determine the longi- tude of the starting point of the orthodrome of each rectilinear /93 path segment, e.g., on the basis of (1,34), or, if the azimuth of the orthodrome is known at the starting point of the path segment, on the basis of (1.33a). The angle of convergence of the meridians between the starting point of the path section Mi and any moving point M on a section, according to (1.33a), will be equal to B = o — «! = arctg (tg >. cosec f) — arctg (tg X-j cosec (pi), (1.83) where A and Xi are measured from the starting point of the ortho- drome . If the longitude of any other point on the Earth's surface, e.g., the point of take-off of the aircraft, is taken as the refer- ence meridian, the angle of convergence of the meridians will be determined as the sum: 6 = Si + 62 + . . . 6 ^ where & i ; &2 ■ • • sltb the angles of convergence of the meridians between the starting and end points of the preceding path segments determined on the basis of (1.83); 6^ is the angle of convergence of the meridians from the starting point to the moving point of the last path segment (on the basis of the same formula). The angle of convergence of the meridians calculated in this way allows conversion from a true course to an orthodrome course and vice versa at any flight distance with any number of breaks in the path. For conversion from an orthodrome course to a magnetic course and vice versa, (1.80) and (1.81) are used, from which it follows that 91 OC =P1C +A« + 8; MC OC + 4« + 5; 1 -A„-8. ) (1.84) The sum of the magnitudes Aj^ + 6 is taken as the overall cor- rection for conversion from a magnetic to an orthodrome course and vice versa, and is designated by A.... Then (1.84) assumes the form : OC = MC + A; MC = OC-A. The dvlft angle is the second element determining the direction of aircraft movement . In an aircraft, the angle between the airspeed vector and the groundspeed vector in a horizontal plane is called the drift angte (Fig. 1.58). In general, the drift angle is designated by the Latin letter a. In those instances when special designations for courses are used in the solutions of navigational problems the drift angle is designated by the Russian letters for DA. In Fig. 1.58. OP^ is the direction of the meridian at point PA; OT is the direction of the airspeed vector and the longitudinal axis of the aircraft; OW is the direction of the groundspeed vector relative to the Earth's surface; u is the wind speed vector. /94 The drift angle of an aircraft is considered positive when the groundspeed vector (vector of aircraft movement relative to the Earth's surface) is further to the right of the longitudinal axis of the aircraft and negative if it is further to the left. The angle between the northern end of the meridian and the groundspeed vector or the vector of the speed of the aircraft rela- tive to the Earth's surface is called the flight angle (FA). The general designation for the flight angle is i|) . The flight angle, like the course of the aircraft, can be measured from the reference meridian, the geographic meridian, and a magnetic meridian passing through point PA. Special values of the flight angles have the following designations: Fig, 1.58. The Path Angle of Flight. (a) The orthodrome flight angle is OFA , with obligatory indication by a sub- script of the longitude of the reference meridian. For example, OFAi^q = 96°. 92 (b) The true flight angle is TFA. (c) The magnetic flight angle is MFA. In Figure 1.58, it is evident that the flight angle is generally or in special cases : OFA = OC TFA = TC MFA = MC + DA>) + DA|> + DA;) (1.85) The interrelationship between the special values of the flight angles and the method of conversion from one special value to another corresponds completely to the interrelationship between special values of aircraft courses: OFA = TFA + 6=MFA+ A; TFA = OFA - 5 = MFA + Am; MFA = OFA - A = TFP - A^ . In determining the direction of aircraft movement relative to the Earth's surface, it is sufficient to know the course of the air- craft as the angle between the direction of the meridian and the lateral axis of the aircraft, and the drift angle as the angle be- tween the lateral axis of the aircraft and the direction of its movement. These elements, together with elements of flight speed, make it possible to determine approximately the speed and direction of the wind at flight altitude . For precisely determining the wind at flight altitude, it is necessary to separate the part of the drift angle of an aircraft caused by the wind. It is obvious that to do this it is necessary /95 Fig. 1.59. Moment and Control Force with Assymetry of Engine Thrust . Fig. 1.60. Lateral Glide with Transverse Roll. 93 to determine the direGtion of the airspeed vector of the aircraft as the course and glide of dynamic origin, arising in flight. There are several causes of lateral glide in aircraft during flight. The basic causes are the following. 1. Assymetry of the Engine thrust or Aircraft Drag (Fig. 1.59) Let us assaime that with symmetrical drag, one of the engines has a somewhat greater thrust than the other. The difference in thrust AP will produce torque in the aircraft relative to the ver- tical axis, i.e., the course of the aircraft will change. For stabilizing the flight direction, a moment must be applied to the empennage of the aircraft which is equal in magnitude and opposite in direction to the moment of thrust, i.e. where AP is the assymetry of the thrust; F^ is the control force; Lj is the arm of thrust assymetry (from the axis of the engine to the axis of the aircraft); and L^ is the arm of control (from the center of the empennage area to the center of gravity of the air- craft ) /96 The lateral force which causes gliding of the aircraft will be: F^ = APLr (1.86) In an analogous manner, the force which causes gliding of an aircraft with assymetry of drag arises. In this instance, the moment of rotation of an aircraft causes excess drag on one wing of the aircraft. The distance from the lateral axis of the aircraft to the center of its application is the arm of this force. 2. Allowable Lateral Banking of an Aircraft in Hor i zon ta 1 Flight. With allowable lateral banking (Fig. 1.60), the horizontal component of the lift will appear: ^,= 018 p. where G is the weight of the aircraft and 6 is the angle of lateral banking . For example, with a flying weight of the aircraft of 75 t, the allowable banking in horizontal flight, equal to 1°, causes a 91+ lateral component of lift >« 1.3 t. 3 . Cor i o 1 i s Force During flight in the Earth's atmosphere, as a result of the diurnal rotation of the Earth's surface, a lateral Coriolis force acts on the aircraft : /\-, = 2<i>e BTm sin <(, (1.87) where (Og is the angular velocity of the Earth's rotation; W is the speed of the aircraft relative to the Earth's surface; m is the mass of the aircraft; and (j) is the latitude of the point PA. k. Two-dimensional Fluctuations in the Aircraft Course During two-dimensional rolls (without banking), an aircraft (as a result of inertia) tries to maintain the initial direction of movement. This causes lateral gliding of the aircraft V^, equal to; V^= V sin At, (1.88) where V is the speed of the aircraft relative to che airspace, and Ay is the magnitude of the change in the aircraft's course. The indicated lateral gliding of an aircraft gradually dies down as a result of acceleration caused by the lateral airflow over its surface . 5. Gliding During Changes in the Lateral Wind Speed Component at Flight Altitude /97 This type of gliding arises as a result of the inertia of the aircraft . First , lateral airflow over the aircraft or something similar (gliding in airspace) will appear, followed by a change in the direction of aircraft movement. From the above five examples of lateral aircraft gliding, constant lateral forces are the causes of the gliding In the first three cases, while abrupt beginning and gradual diminution of gliding are the causes in the last two cases. The magnitude of stable gliding with a constantly acting lateral force can be determined according to the formula or * - V c,Sp (1.89) 95 where Z is the operative lateral force; Cg is the coefficient of lateral drag of the aircraft; S is the area of the longitudinal section of an aircraft with a vertical plane; and p is the mass density of the air flight altitude. To calculate gliding in flight, it must be integrated and converted to angular glide (agi): •^V=^' (1.90) where V^ is the lateral component of the airspeed and V^ is the longitudinal component of longitudinal speed. The direction of the airspeed vector is determined by the formula Tv =T + «gJ (1.91) The drift angle of an aircraft, whose cause is the action of the wind at flight altitude, will be: « = 'f — Tk = 4' — Tf — flgl (1.92) As we have already said, determining the gliding of an air- craft in airspace is necessary only for precise measurements of wind speed and direction at flight altitude. For the purposes of aircraft navigation, there is no need to separate out the causes of lateral aircraft movement. Elements Which Characterize the Flight Speed of an Aircraft The flight speed of an aircraft is measured both relative to the airspace surrounding the aircraft and relative to the Earth's surface . Measuring the speed of aircraft movement relative to the air- space is significant both from the point of view of flight aero- dynamics (stability and control of the aircraft) and from the point ov view of aircraft navigation. It is known that the lift of a wing, the drag of an aircraft, and the stability and controllability of an aircraft depend on the square of the airspeed. For example , at flight speeds which are significantly less than the speed of sound, the drag of an aircraft is determined by the formula /98 96 2 where Q is the lateral drag of an aircraft, o^ i^ ''^^e drag coef- ficient, S is the maximum area of the lateral cross section of an aircraft, and p is the mass air density at flight altitude. The value- -^—^ — characterizes the aerodynamic pressure of the atmosphere on the surface of an aircraft , All the aerodynamic characteristics of an aircraft are deter- mined relative to this value. In determining the aerodynamic characteristics of an aircraft, the- aerodynamic pressure (and therefore the speed of flight) reduce to conditions in a standard atmosphere, i.e., to flight conditions near the Earth's surface, with an atmospheric pressure of 760 mm Hg and an ambient air temperature of 15° C. Therefore, speed indicators which measure airspeed on the basis of aerodynamic pressure are calibrated according to the parameters of a standard atmosphere. With an increase in flight altitude, air density decreases. To preserve aerodynamic pressure at flight alti'tude, it is necessary to increase flight airspeed, although responses of the airspeed indicator which measure airspeed on the basis of aerodynamic pres- sure remain constant . Flight airspeed which is measured on the basis of aerodynamic pressure and which influences the aerodynamics of the flight of the aircraft is called aerodynamia speed (^aer^* It is necessary to consider, however, that with an increase in flight speed, especially in approaching the speed of sound, aero- dynamic speed does not completely correspond to the aerodynamic characteristics of an aircraft which are determined under the condi- tions of a standard atmosphere and which are inherent in flight speed. This is becuase the factor of air compressibility begins to exert an influence. To ensure safe pilotage of the aircraft in these instances, a corresponding correction is introduced into the indications of aerodynamic speed. For the purposes of aircraft navigation, it is necessary to /99 know the actual speed of an aircraft in space. The actual airspeed, which we shall call simply airspeed (V) , can be obtained from aerodynamic speed by introducing corrections for the change in air density with flight altitude and temperature: V- V, 'aeti-^»^H + AV^, + AV^c^p. 97 where ^^er ^^ ^^^ aerodynamic speed; AV^ is the correction for speed as a result of flight altitudes; hV^ is the correction for speed as a result of air temperature; and A7g „„„ is the correction for speed as a result of air compressibility cmp Correction for flight altitude is of basic i,mportance . Cor- rection for air temperature is significantly smaller and is intro- duced only in those cases when the air temperature at flight alti- tude is significantly different from the temperature calculated for this altitude . At the present time, there are devices which indicate flight airspeed directly, taking altitude into account. Corrections must be introduced in the responses of these devices only for instru- mental errors of the devices and (in individual cases) for a dis- crepancy between the actual air temperature and the calculated temperature at a given altitude . In published textbooks on aircraft navigation, airspeed has been classified as "indicated" (measured on the basis of aerodynamic pressure) but true, and as "indicated , corrected for methodological and instrument errors." Since there are now devices which measure both these speeds, each of them is "indicated". In addition, increasing airspeeds have required the introduction of corrections in aerodynamic flight speed. This has caused a new classification of air speeds. The speed of aircraft movement relative to the Earth's surface * is called flight groundspeed (.W) . Flight groundspeed can be measured directly by means of Doppler or inertial systems , determined by sighting along a series of land- marks on the Earth's surface, and also calculated on the basis of flying time between two landmarks on the Earth's surface. In addition, groundspeed can be determined by adding the airspeed and wind vectors , if the wind speed and direction at flight altitude are known . Navigational Speed Triangle The interrelationship of the elements of flight direction and speed in the chosen frame of reference of aircraft courses is clearly illustrated by a navigational speed triangle. In Fig. 1,61 a navigational speed triangle is shown for a /I general case, i.e., independently of the meridian which is used as the basis for measuring an aircraft course. Straight lines OPjj and OiP^ in the figure show the direction of the meridian at point PA; V is the airspeed vector; W is the groundspeed vector; y is the course of the aircraft (C), a is the 98 drift angle (DA); \p is the flight angle (FA), 6 is the direction of the wind vector relative to the meridian for reading the air- craft course; 6^ is the flight angle of the wind (WA) read from the given line of the path; and 6y = a + 6^ is the course wind angle (CWA), read from the longitudinal axis of the aircraft. A speed triangle can be solved graphically by construction of vectors on paper or by a mechanical apparatus, using a special de- vice (a wind-speed indicator which is a combination of rules, dials, and hinges with movable and immovable joints). A speed triangle is solved analytically on the uasxs of a known sine theorem. From Figure 1.63, it is clear that in the given case the sine theorem will have the form: sing U sin 8,,, sinb- (1.93) From (1.93) the value of the drift angle and the flight ground- speed are easily determined on the basis of known values of the air- craft course, airspeed, and the speed and direction of the wind. Now, let us define the path angle of the wind 8^ = S-^/. (1.94) The drift angle of an aircraft according to (1.93) is deter- mined from the formula slna= — sill 8^ (1.95) The value of the flight groundspeed is then easily determined Vsln8, W= ■■ I , sln5,„ (1.96) Fig. 1.61. Navigational Speed Triangle These problems are especially simple to solve with slide rules having a combination of sine logarithms with a logarithm scale of linear values. In this case , combining the log- arithm of the sine of the wind angle with the logarithm of the airspeed, we obtain directly: /lOl' 99 Igsinfl — Iga = IgsinS^— Ig K= IgsinB^ — Ig'W or on scales of navigational rulers with the designations used with special values for aircraft courses. D A __W A DA+WA [ Igsln y vy igsp (1.97) To determine wind speeds and directions at flight altitude on the basis of known values of airspeed, groundspeed, and drift angle, let us use Figure 1.62. that From the figure, it follows 0D= V^ cos DA; OiD= V sin DA; DM = OM — OD= W—Vcos DA. Fig. 1.62. Determining the Angle and Speed of the Wind with Known Values of the Groundspeed and Drift Angle of the Aircraft . Therefore , OiD VsinDA tg W A = -^— = — DM W— KcosDA (1.98) The flight angle of the wind determined in this way permits the further solution of problems on the basis of the sine theorem [Equation (1.93)]. With small drift angles (practically up to 10°), cos DA !=a i, i.e., it is possible to consider in approximation that tgWA=- Ksln DA W -V (1.99) For solution on a slide rule, (1.99) is reduced to the form: tg WA sin DA V ^ w-v' r^tgWA-lg V=lgslnDA— lg(W^— K) Translator's note; Ig = log. 100 or on a slide rule. DA. WA W—V V Ig sin Ig tg Igsp After finding the flight angle of the wind, the value of the wind speed is determined on the basis of the sine theorem. Elements Which Determine Flight Altitude The flight altitude of an aircraft (H) is measured from a special initial level of the Earth's surface. The initial level for measuring flight altitude is chosen depending on the purposes for which it is measured. For example, in order to distribute the counter and incidental movements of aircraft in airspace (flight echelons), the initial level for measuring the altitude on each aircraft must be general. To ensure the safety of flights of individual aircraft at low alti- tudes, it is desirable that the flight altitude be measured from the surface of the relief over which the aircraft is flying. In making an approach to land at an airport, flight altitude is meas- ured from the level of the landing point. Usually 2 or 3 kinds of altitudes are measured at the same time. Therefore, it is necessary to classify them and to establish a relationship between them. At the present time, the following kinds of altitudes are dis- tinguished (Fig. 1.63): /102 Level!r=760 mm Hg Fig. 1.63. Interrelationship of Different Systems for Measuring Flight Altitude 101 (a) Absolute flight altitude (^abs^ ^^ measured from the mean level of the Baltic Sea in the same way as the height of a relief on the Earth's surface. (b) Relative flight altitude (^rel) is measured from the /103 level of the take-off or landing airport, (c) Tvue flight altitude "^tr" i^ measured from the surface of the relief over which the aircraft is flying. (d) Conventional barometria altitude "Hy^^" is measured from the conventional barometric level on the Earth's surface, where the atmospheric pressure is equal to 760 mm Hg . Absolute, relative, and true flight altitudes are determined by barometric altimeters with correction of their readings for instrumental and methodological errors . The latter can also be measured by radio altimeters and aircraft radar equipment or deter- mined by aircraft sighting devices. There is a relationship be- tween the three indicated altitudes which makes it possible to switch from one kind of altitude to another. Conventional barometric altitude is measured by barometric altimeters without considering methodological errors. Therefore, it has no direct connection with the first three kinds of altitudes, and at a high flight altitude it can be distinguished from the abso- lute altitude closest to it by 900-1000 m. The main advantage of a conventional barometric altitude is the convenience of using it for echeloning flights according to altitudes when the important thing is not the precise measuring of altitude but only the preservation of safe altitude intervals between neighboring echelons. The latter condition is satisfied, since if we permit two aircraft to meet in one region and at one altitude, the methodological corrections in these aircraft will be identical. Therefore , such a meeting cannot occur if aircraft maintain dif- ferent altitudes based on instruments. From Figure 1.63 it is evident that true flight altitude is distinguished from absolute flight altitude by the height of the relief over which the aircraft is flying, and from relative alti- tude by the height of the relief above the airport level from which relative altitude is measured: tr tr = H = H abs rel H AH r; ) (1.100) where H.^ is the altitude of the relief above sea level; AH^ height of the relief above the level of the airport. is the 102 Relative altitude is distinguished from true altitude by the height of the relief, while it is distinguished from absolute alti- tude by the height of the airport above sea level: ^rel ^rel = fftr + = •^abs ■ r; \ ^air •' (1.101) Finally, absolute flight altitude can be determined on the basis of the values of true or relative flight altitude: /lOit ^abs ^abs --H ■■Ht tr + ^r; I tr + ^aivf (1.102) Calculating Flight Altitude in Determining Distances on the Earth's Surface In measuring directions on the Earth's surface, flight altitude does not exert a direct influence on the value of the measured angles or on the accuracy of the measurements. Actually, by direction on the Earth's surface we mean direction of the line of intersection of the horizon plane with the plane of a great circle (orthodrome) which joins two points on the Earth's surface . Since the vertical at any of these points on the indicated line lies in the plane of a great circle, flight altitude does not exert an influence on the direction of the orthodrome and therefore on direction on the Earth's surface. In measuring distances on the Earth's surface, flight altitude can play an important role and can lead to large measurement errors if we do not allow for errors in flight altitude (Fig. 1.64). In the figure, straight lines OAi and OBi are verticals of the position of an aircraft at points A and B. Obviously the distance S between points ^1 and Si at flight altitude is greater than distance S between points A and B on the Earth ' s surface : Fig. 1.6U. Calculating Flight Altitude in Determining Distances . 103 Re+H (1.103) whence -(-a jLS or '45 = 5 — (1.104) where i?„ is the radius of the Earth (equal to 6371 km) and H is the flight altitude. Each kilometer of flight altitude lengthens the path between /105 points on the Earth's surface by a value expressed in percent: MOO 6371 ' i 0.016%. For example, at a flight altitude of 10 km, a distance on the Earth's surface equal to 3000 km lengthens to the value 3000-100,016 6371 = 4,8 KM, The indicated lengthening of the path of the aircraft does not exert a substantial influence on the time of the aircraft flight along the path. The influence of flight altitude on determination of the position of the aircraft by rangefinding and, especially, hyperbolic devices turns out to be more substantial. Let us assume that a rangefinding device is located at point A on the Earth's surface, while the aircraft is located at point Bi, at flight altitude. As is evident from Figure 1.66, the distance from the ground radio-engineering apparatus to the aircraft R along a straight line will equal AB\, while the distance along the Earth's surface S is equal to AB . Let us drop a perpendicular from point A on the Earth's surface to point D on the vertical OBi . Obviously, AB\ = Am + DB\, since ABx = R; D = /JySin DBi = Rq— /?gCOs S + H, AD = /JySin 5; l0t^ Then __^ /?=K«|sln2S + (/?e-/?e<:os 5 + //)». ^^ ^^^^ With STaall angular distances 5 (up. to &° along the arc of the ortho- drome), when B sin S ?« 5 , while cos S pa 1 , (1.105) takes the form: R = Vs^+f^- (1.106) Figure 1.66 can likewise be used for determining the maximum distance of geometrical visibility of objects on the ground from on board the aircraft, or of an aircraft from the Earth's surface. It is obvious that with maximum visibility, line ABi must be tangent to the Earth's surface, i.e., it is located in the plane of the horizon. In this case, angle OABi will be a right angle. Therefore , OA2 + AB\ = OBl /105 d2 %+{ffe^SS)2 = (RQ + Hyi. Expanding the right-hand side of the equation, we obtain: Considering that at distances up to 600-700 km, R^igS rs S, and disregarding the value H^ as negligibly small in comparison with 2RqH , we obtain the approximate formula S = V2Rjf. (1.107) Substituting in (1.107) the value of the radius of the Earth (6371 km) we obtain: 5= Vm42H =113 Vff. Bearing in mind that as a result of the refraction of light or radio waves in a vertical plane, the distance of geometrical visibility increases approximately by 8%, the practical result will be: 105 S . =172 Vh. VIS (1.108) Formula (1.108) determines the limits of applicability of (1.101+) or (1.106). Since we have agreed to consider cos 5 = 1 and R sin 5 = £■ up to 5 = 6° , which on the Earth's surface corresponds to 666 km, it is obvious that at flight altitudes up to 25 km it is always possible to use (1.106). It is necessary to use the precise formula (1.104) at distances of more than 700 km. This is possible at flight altitudes exceeding 25 km. Fig. 1.65 Calculating Flight Altitude in Determining the Path Length of Electromagnetic Wave Propagation Let us pause now to discuss the influence of flight altitude on the accuracy of measuring distances at very small ranges, i.e., in cases when long radio waves capable of traveling around the Earth's surface are used (Fig. 1.65). In the figure , ground radio engineering equipment is located at point A on the Earth's surface; the aircraft is at point B at altitude H. Line AB is the curve of propagation of a radio wave front . If we conditionally move the Earth's surface to the right by a value equal to H/2, then the line of radio wave propagation be- comes concentric with the Earth's surface and will have a radius of curvature i?i = R^ + H/2. Therefore, the increase in distance from point A to point B can be considered as a lengthening of the orthodrome at a flight altitude equal to H/2, i.e.. /.10 7 AS = Si — S = S-;^- 2/?e 106 Elements of Aircraft Roll It is known that the radius of aircraft roll in airspace at a given banking 3 equals: V2 *tgP If a flight is carried out rolling of an aircraft through ^ with a counter or incidental wind, an angle of 90° involves an increase or decrease in the mean radius of roll of an aircraft relative to the Earth's surface (Fig. 1.66), u,'-Z50 km/hrs J u,'0;y'eoo 'k'm/hT u, - +250^ m /h r Fig. 1.66. Deformation of the Roll Trajectory in the Presence of Wind. craft from the original flight will be a deviation opposite to In fact , for a change in the direction of the groundspeed vector of an aircraft by 90° , with a shift from the plane of incident wind to a lateral plane, it is necessary to execute a roll of an aircraft to the right or left through an angle of 90° + DA, and in changing from the plane of incident wind to a lateral wind to a lateral plane through an angle of 90° - DA. During rolling of an aircraft in airspace , in the first instance there will be a deviation of the air- direction; in the second case, there /lO i the original. Exampte . Let us examine the roll of an aircraft through 90°, with a flight airspeed of 600 km/h and with a counter and incident wind speed of 250 km/h (70m/sec). According to (1.6), the radius of the aircraft in airspace with banking of 15° will be 1672 9,81 »g 15» = 10 500 m The drift angle at the end of rolling through 90° will have the following value : 250 DA=arc,g- = 23''. 107 Therefore, in the first case it will be necessary to turn the air- craft through 113° , and in the second case through 67° . The angular velocity of roll at F = 600 km/h (167 m/sec) and R = 10,500 m will be ^ K-57,3 167-57,3 „„^ , Let us determine the additional shift of the aircraft as a result of wind during rolling in the first instance: 70 in/sec-113° 0,9deg/sec and in the second instance : 70-67 Obviously, during roll (in the first case through 113° and in the second case through 67°) the movement of the aircraft in direc- tion X will not be identical, since Bjc = R sin yP. Then the general path of an aircraft in direction X will equal In the first case , In the second case , Rjc = 10,5 sin 113° + 9 = 18,5 km /?^= 10,5 sin 67° — 5 = 4,5 km Let us now determine the lateral shift of the aircraft R^ during roll: In the first case. and in the second case /?« = /?+■/? sin 23° =14,5 km Rz = R — R sin 23' = 6 k m. For comparison, let us examine the roll of an aircraft through 90°, with a radius calculated not on the basis of airspeed, but on the basis of groundspeed: B7= V± Ux. 108 In the first case, the radius of rolling is /109 „ 2372 '^^ 9,81 tg 15° =21 km and in the second case /? = 972 9,81 tg 15' — =3,5 km Let us compile a table with the results obtained: Roll parameters X z K;(=+250km/h 10,5 10,5 10.5 21 10,5 18,5 10,5 14,5 «^=— 250 km/h 10,5 3,5 4,5 6 From the table , it is evident that the results of the calcu- lations carried out on the basis of the groundspeed are much closer to the actual results than calculations on the basis of airspeed. Calculations o basis of groundspee high as 200-300 km/ a new line of fligh ried out with an ac km. Some inaccurac only in the lateral ever, this is not o nif icance , since th deviation coincides line of flight . f roll on the d with winds as h, when entering t , will be car- curacy of 1-2.5 ies arise , but direction. How- f practical sig- e direction of with the new Fig. 1.67. Aircraft to Approach of an a Given Line with the Presence of an Ap proach Angle . With a decreas of roll, the trajec according to the gr closer to the actua aircraft roll. The future we will proceed from flight groundspeed in c e in the angle tory calculated oundspeed comes 1 trajectory of ref ore , in the alculating roll In aircraft navigation, including maneuvering before landing, it is necessary to solve three types of problems, taking into ac- count the roll trajectory. 109 1. Combination of Roll with a Straight Line Let us assume that an aircraft is approaching a given line of flight at a definite angle (Fig. 1.67). It is obvious that the angle of roll of the aircraft for fol- lowing along the given line is equal to the approach angle (a). Let us determine the distance (Z) from the given line on which it is necessary to begin the roll so that the roll trajectory will be joined with the given line. In Figure 1.67, it is evident that this distance is equal to: /llO or Z = R — R cos o Z = /?(l— coso). (1.109) Example . An aircraft approaches a given line of flight with a groundspeed of 900 km/h at a 25° angle. Determine the lateral distance from the line of flight at which it is necessary to begin a roll for a smooth approach to the line. Sol ution R = 2502 = 26,5 km 9,81 tg 15° Z = 26 ,5(1 — cos 25°) = 2, 46 km 2. Combination of two rolls If, during flight along a given flight line, a deviation from it occurs and it is necessary to approach the given line by the shortest trajectory, an approach maneuver is used which is a combi- nation of two rolls (Fig. 1.68). Fig. 1.68. Approach of an Aircraft to a Given Flight Line with a Paral- lel Flight Line. Since the value of Z in this case is considered known, while the radius of roll is determined on the basis of the groundspeed and the given banking in the roll, it is necessary to deter- mine the value of the angles ai = a^ of the combined rolls. It is obvious that in this case , in each of the two combined rolls , the aircraft approaches the flight path by a value Z/2; therefore. 110 -- = '?(1 — COSo), whence '^°^"='-^- (1.110) For example , let us say that an aircraft having a groundspeed of 900 km/h has deviated from a given flight path by 5 km; to make the approach, it is necessary to execute two combined rolls with banking of 15° to angles up to 25°. 3. Linear prediction of roll (LPR) Let us examine two solutions to problems, with a consideration of the roll trajectory of an aircraft which includes one rectilinear part of the path. Linear prediction of roll is calculated in instances of a /111 break in the flight path at turning points in the route (Fig. 1.69). In the figure , TPR is the turning point in the route and TA is the turn angle of the flight path equal to the roll angle of an aircraft ( RA ) . As is clear from Figure 1.69, the radius of roll of an air- craft, at its beginning and end, is directed perpendicular to the preceding and following orthodrome segments of the path. The lines 0-TPR form the bisector of the angle of roll. Thus, we have two identical rectangular triangles with vertex angles equal to RA/2. The linear prediction of roll (LPR) is the line of tangency of the roll angle, divided in half: LPR TPR Fig. 1.69. Linear Prediction of Roll of an Aircraft (LPR). Fig. 1.70. Linear Lag of Air- craft Roll (LLR). Ill i LPR=/?tg RA (1.111) Example: Determine LPR with a flight groundspeed of 900 km/h and an angle of turn to the new flight path of 40° for banking in a roll of 15° . Solution. R 2502- =26,5 km 9,81 tg 15° LPR=26,5-tg20° = 9,6 km Linear predictions with roll angles from to 150° are given in Table 1.1. km/hr 400 500 600 700 800 900 R. M 4600 7'^ 10.600 14 700 18500 23500 Prediction with roll angles fromO to 150° ," km" 15° 30° 45° 60° 75° 90° 0,6 1,2 2,0 1.0 2,0 3.1 1.4 2,8 4,2 1.9 3,8 6,0 2.4 4,8 7.7 3,1 6,2 9.7 2,7 3,5! 4,6 4,3 5,7| 7,3 6,l| 8,210,6 8,3 11,0 M, 4 10,7]l4,o'l8,5 13,5 18,023,5 I 105' 120' 135° 150= 'roll to 90° sec 6,0 8,011,0,15,0 9,7 12,8,18,027,5 13,8 18,3'25,5]40,0 19,0 25,035,0 52,0 24.032,0'42, 070,0 30,040, o'58,087,0 '65 82 100 116 132 148 /li: In some cases, the necessity for flight above the TPR with the flight angle of the following part of the path can arise (Fig. 1.70) e.g., in flights of different kinds for testing aircraft and ground navigational equipment. In these cases, instead of linear predic- tion, linear lag of roll (LLR) is calculated, while the roll is carried out in the direction opposite to the turn of the new flight path by the angle RA=360°— TA In Figure 1.72, it is clear that the LLR is a line of the tangents of the turn angle of the flight path divided in half, i.e., with the same turn angles , the formula for the LLR remains the same as for the linear prediction of roll: LLR =«tg- RA 112 J CHAPTER TWO AIRCRAFT NAVIGATION USING MISCELLANEOUS DEVICES 1. Geotechnical Means of Aircraft Navigation Geotechnical means of aircraft navigation constitute a por- tion of the navigational equipment of an aircraft which has an autonomous character and is used under all flight conditions , independently of the use of other special devices such as those employing radio engineering or astronomy, for example. Such devices include those which measure the aircraft course airspeed, and flight altitude, as well as devices for automatic solution of navigational problems. /113 high phy s the red dent is b engi emat the Geo ly d ical fiel rang on eing neer i cal syst technic i verse fields d of el e , etc . the phy carri e ing or basis em for ievices for aircraft navigation are based on --• — .--T-- jr-„ -.-T-^ iiao of natural geo- Aircraft navigation using only geotechnical devices can be carried out in cases when it is possible to check the navigation- al calculations (even periodically) by determining the locus of the aircraft by other means or visually. Historically speaking, the development of radio-engineer- ing and astronomical means for aircraft navigation has been directed toward a solution of only one problem, namely, the determination of the aircraft coordinates on the Earth's surface, which proved a necessary adjunct to the geotechnical means of aircraft navi- gation in flight under conditions when the ground was not visible. In recent years, there has been a development of the radio- engineering, astronomical, and astro-inertia.l systems for solving problems which are inherent in geotechnical devices for aircraft navigation, i.e., measurement of the aircraft course, airspeed, turn angle, altitude, etc. /114 113 2. Course Instruments and Systems Course instruments are intended for determining the position of the longitudinal axis of an aircraft in the plane of the hor- izon or (what amounts to the same thing) for measuring the course of the aircraft. It is necessary to know the aircraft course in order to determine both the flight direction and the position of the aircraft relative to orientation points on the ground. As we have mentioned above, there are several systems for calculating the aircraft course, and the selection of the system of calculation is governed both by the requirements of aircraft navigation and by the technical possibilities for equipping the aircraft with the corresponding instruments. sent time, there are no course instruments which sfy the requirements of aircraft control under all eref ore , aircraft usually are fitted with several e instruments operating on different principles rent systems of calculation; each of them is used tions which are most favorable for it. In some struments are combined into complexes , called course the operation of the individual instruments is closely makes it possible to exploit the positive qualities in actual operation. At th e pre completely sati conditions Th different cours and using diffe under the condi cases , these in systems , w here related . This of each of them Methods of Using the Magnetic Field of D i rect i on the Earth to Determine Directions on the Earth's surface can be measured most ac- curately by astronomical methods. However, this requires opti- cal visibility of the sky, complex and accurate apparatus, and tedious calculation. Directions on the Earth's surface can be determined more simply and in many cases quite reliably by using the magnetic field of the Earth. The magnetic field of the Earth (Fig. 2.1) is characterized by the following parameters at every point on its surface: (H); (Z) (a) Directionality of the horizontal component of the field (b) Directionality of the vertical component of the field (c) The direction of the plane in which the vectors H and Z lie relative to the geographic meridian at the given point. The plane in which the vectors H and Z are located is called /115 the plane of the magnetic meridian. The angle between the planes im- of the magnetic and geographic meridians is called the magneti.Q deatination and is represented by A^^ . The points on the Earth's surface at which the magnetic mer- idians intersect are called magnet-io -poles. Obviously, the hori- zontal component of the magnetic field is lacking at the magnetic poles, while the intensity of the vertical component reaches its maximum value . Fig. 2.1. Magnetic Field of the Earth. is the resultant vector of E The magnetic poles do not coincide with the ones . The coordinates o Magnetic Pole are 7'4-°N a those of the South Magne are 68°S, 143°E (as of 1 The device of a fre magnetic pointer mounted of the magnetic meridian to determine direction o surface. Therefore, at on the Earth's surface t be a reliable indication parameters which charact magnetic field of the Ea The total intensity netic field of the Earth and Z. Consequently, j2 _ H' of the Earth geographi c f the North nd 100°W; tic Pole 952) . ely rotating in the plane is used n the Earth's every point here will of the three erize the rth . of the m a,g (vector T) (2.1) The oersted (Oe) is the unit of measurement for the total intensity of the magnetic field, as well as the intensity of its components; in other words, it is the intensity of a field which interacts with a unit magnetic pole with a force of one dyne. The limits of change in the intensity of the components in the magnetic field of the Earth are the following: (a) Horizontal : from zero in the vicinity of the magnetic poles to a maximum at the magnetic equator (0.4 oersteds in the vicinity of Indonesia); (b) Vertiaal: from zero at the magnetic equator to 0.6 oer- steds in the vicinity of the magnetic poles. A smaller unit of intensity, the gamma (y)? is used for very precise magnetic measurements; it is equal to one hundred thou- sandth of an oersted. The angle which characterizes the inclination of the vector 115 of total intensity of the magnetic field of the Earth to the plane of the true horizon is called the magnetic ■Lnotination "6". /116 larctg^ (2.2) Charts of the magnetic fields are prepared for convenience in using the magnetic field of the Earth to determine directions on the Earth's surface. A chart of magnetic inclinations is extremely important for aircraft navigation. Lines joining points on the Earth's surface which have the same magnetic declina-^ion are called isogonios . They are printed directly on flight and large-scale geographic maps . To determine the true course, the magnetic declination deter- mined from the chart at the locus of the aircraft (with its sign, as a correction) is entered in the readings of the magnetic compass. Figure 2.2 shows a map of the World with the magnetic declina- tions entered on it; the isogenics are shown as they appear on the Earth's surface. The positive isogenics on the chart are marked by solid lines, while the negative ones are marked by dashed lines. All of the isogenics meet at the magnetic poles of the Earth, and the compass readings (and consequently the magnetic inclin- ation) change by 180° when passing through the magnetic pole. In addition, the isogenics also meet at the geographic poles, since the directions of the magnetic and geographic meridians are opposite between the magnetic and geographic poles, but coincide after passing through the pole, i.e., the declination changes by 180° . The map of the World showing the magnetic declinations has the isogenics only for the normal magnetic field of the Earth. In addition to this normal field, there is also an anomalous field, caused by the magnetization of the soil in the upper layers of the Earth. Regions and areas of changes in the declination in such regions are marked on large-scale charts. The reliability of operation of magnetic compasses and the magnitude of the errors in their readings depend on the intensity of the horizontal component of the magnetic field of the Earth. Errors in the readings of compasses, particularly when the air- craft is rolling, depend only on the intensity of the vertical comonent . The lines on the Earth's surface which connect points with the same intensity of the horizontal or vertical components of the magnetic field are called isodynamic lines. 115 ■■ IHIII III IIIIIIIHII !■ Figure 2.3 shows a map of the World with the isodynamic lines for the horizontal component of the Earth's magnetic field, while Figure 2.h shows those for the vertical component. /117 Fij 2. 2 World Chart of Magnetic Declinations. 117 /Ill Fig. 2.3. World Chart of Isodynamic Lines for the Horizontal Com- ponent of the Earth's Magnetic Field. 118 /119 Fig. 2.4. World Chart of Isodynamic Lines for the Vertical Compo- nent of the Earth's Magnetic Field. Only general (outline) charts of isodynamic lines are used /120 in aircraft navigation. These lines do not appear on flight charts. Lines on the Earth's surface which connect points with the same declination of the magnetic field are called {.soatines . Form- erly, outline maps of isoclines were used jointly with charts of isodynamic lines showing the total intensity of the magnetic field to determine the errors of magnetic compasses. At the present time, these charts are no longer used, since it is better to use the isodynamic lines of the horizontal and vertical components of the magnetic field. Variations and Oscillations the Earth's Magnetic Field There are several hypotheses regarding the origin of the mag- netic field of the Earth, but none of them has been adequately proven as of the present time. Possible factors in the formation of the magnetic field are the subsurface and ionospheric electrical currents, as well as the magnetic induction and magnetic hysteresis 119 of the soil, composing the structure of the Earth's sphere. Even if these factors are not primarily responsible for the formation of the magnetic field of the Earth, they are in any case important influences on its structure and stability. An analysis. of the isolines of the intensity of the components in the magnetic field of the Earth and the magnetic declinations reveals that their configuration is determined both by general laws of the distribution of magnetic forces in the field of a mag- netized sphere, as well as by local disturbances in the general structure of the field. Therefore, the stationary magnetic field of the Earth is assumed to consist of a sum of fields: (a) The field of the uniform magnetized sphere; (b) The continental field, related to the nonunif ormity of the relief and the structure of the internal layers of the Earth; (c) The anomalous field, related to the existence of depos- its of magnetic material in the upper layers of the Earth's core. As systematic observations of the structure of the magnetic field of the Earth have shown, it does not remain strictly sta- tionary but undergoes constant changes. Changes or variations in the magnetic field of the Earth have a diverse nature. Annual or constant changes in the magnetic field of the Earth are called seoulav variation. These variations constitute the difference between the average annual values for the elements of the Earth's magnetism. The causes for the annual variations are changes in the components of the stationary field with time, i.e., the magnetic moment of the Earth and the continental field. The annual variations in the declination at middle latitudes reached 10-12', and up to 40' at high latitudes; therefore, when using charts of magnetic declinations, or isogenics on flight charts, it is necessary to consider the period when they were made. If the chart of magnetic declinations is obsolete, changes must be /121 made when using it for the variation in the declination during the time which has passed since the chart was made. The desired correction is determined from special charts of the secular vari- ations of the magnetic field of the Earth. The isolines of equal secular variations in declination on a chart are called isopors . In addition to the slow systematic changes in the magnetic field of the Earth, there are also periodic and even chaotic changes which are related to the so-called internal field of the Earth, the main cause of which is ionospheric currents. These are esti- mated periodically or are disregarded entirely. 120 Magnetic Compasses The magnetic compass is the simplest course device; in most cases, it is sufficiently reliable though not sufficiently accu- rate . However, a simple magnetic compass with a freely turning mag- netic needle is not suitable for use on board an aircraft, since its readings would be inaccurate and unstable. Various kinds of interference would influence the operation of the compass during flight, including: (a) Movements of the aircraft relative to its axis; (b) Vibrations produced by the operation of the engines and by the movement of the aircraft through the air; (c) The effect of the magnetic field of the aircraft, which would cause deflections of the magnetic needle from the plane of the magnetic meridian, i.e., compass deviation. Obviously, a magnetic compass which is intended for use on an aircraft must have devices for compensating the interference mentioned above. The simplest form of an aircraft magnetic compass is the inte- grated compass, i.e., one in which the course transmitter (sensi- tive element) and the indicator are combined in a single housing. Of the large number of types of magnetic compasses which have been devised as of the present time, the one most used nowadays is the "KI" (an abbreviation for the historic name of the magnetic compasses which were devised in the past for fighter aircraft). Compasses using other systems are called distance-magnetic or gyro- magnetic compasses. Any integrated aviational magnetic compass consists of the following main parts (Fig. 2.5): The bowl or container 1 of the compass, filled with a damp- ing fluid to decrease the oscillations, usually liqroin; on the bottom of the bowl is a pivot support for the movable part of the compass, with a damping spring and pivot bearing made of agate; /122 The movable part of the compass, consisting of is a combination of a magnetic system (H-shaped magnet), a floa to reduce the weight of the card and reduce the friction on the bearing, a needle pivot, and a rotating scale for the readings, mounted on the magnetic system; card 2 , which float e Chamber 3, compensating for thermal expansion and contraction 121 of the damping fluid; the expansion chamber is located above the bowl and is connected to it by holes of very small diameter. This allows air bubbles to escape from the bowl into the chamber and permits the fluid to flow back and forth with expansion and con- traction. It also prevents it from splashing in the bowl as the airplane moves ; A device ^ for getting rid of deviations of the compass, which contains several bar magnets pressed into drums which rotate in mutu- ally perpendicular planes with the aid of screws. Rotation of the drums permits them to be set to a position where the magnetic field of the bar magnets compensates for the magnetic field of the aircraft acting on the compass card. Fig. 2.5. Combined Magnetic Compass: (a) Cross Section; (b) External View. The design of the magnetic compass described above reduces the effect of interference with its operation to a considerable degree and the compass readings are quite stable. Nevertheless, magnetic compasses (especially integrated ones) have a number of shortcomings which prevent the course from being calculated under certain conditions. The most important of these shortcomings are the following: (1) A limitation of the choice of mounting location for the compass aboard the aircraft; the integrated compass must be lo- cated in a place which is suitable for determining the course, and therefore close to other instruments and moving parts for con- trolling the aircraft, which produce a large and varying devia- tion of the compass; (2) The impossibility of using the compass when the aircraft /123 is turning. When the aircraft makes a turn, several factors act on the compass card to move it from its customary position: the pressure of the damping fluid on the card, the action of centri- fugal force on the southern, somewhat elongated portion of the card, as well as a change in the structure of the magnetic field of the aircraft while turning. The deviations of the compass card 122 from the plane of the magnetic meridian when the aircraft is turn- ing are particularly noticeable when the aircraft course crosses the northern and southern directions. These deviations are called the northern and southern turning errors . The instability of the structure of the magnetic field of the Earth at the locus of the aircraft and its changes with time are the major shortcomings of using magnetic compasses of all types Deviation of Magnetic Compasses and its Compensation The cause of magnetic compass deviation is the presence of parts on board the aircraft which are made of materials exhibiting magnetic properties . Some of these parts have a constant magnetic field. Parts of this kind are called hard magnetic iron. Another group of parts are magnitized under the effect of the magnetic field of the Earth and are called soft magnetic iron. According to Coulomb's law, the force (F) of the interaction c masses (m) is inversely proportional to the distance — f ■^\ of magnetic mciaae between them (p). ^ = -^- (2.3) Therefore, the deviation of the magnetic compass increases very sharply with the approach of its sensitive element to parts which have high magnetization. According to the principle of independence of the action of forces at a given point in the aircraft, it is possible to sum the magnetic fields coming from individual parts of the aircraft and to subject them to the equivalent effect of a single magnetized bar located at a certain point. However, if we take into account the diverse nature of the action of the hard and soft magnetized iron on different courses and during different motions of the air- craft, it is better to subject this field to the equivalent action of bars which have a constant and varying magnetization. Let us assume that the equivalent bar of hard magnetized iron is located horizontally and coincides with the direction of the longitudinal axis of the aircraft (Fig. 2.6). With a magnetic course of the aircraft equal to zero, the vector F of the field intensity of the bar coincides in direction with the horizontal component of the magnetic field of the Earth /12H H, which does not produce any deviation of the compass card from the plane of the magnetic meridian. _In the case of aircraft courses equal to 90 or 270°, the vec- tor F of the field intensity of the bar is located at right angles 123 to the vector H, producing maximum deviation of the card from the plane of the magnetic meridian. Hence , when the aircraft is turning around its vertical axis through 360°, the resultant vector (P-j-) of the hard magnetic iron and the magnetic field of the Earth will coincide at two points with the direction of the magnetic meridian, and will be at a maxi- mum distance from it at two other points . Devtat-ton of this kind -is aalled semiairoutar, i.e., it has zero value with every 180° rotation of the aircraft (Fig. 2.7). Finally, it cannot be expected that in the general case the equivalent bar of hard Hi " \ 1 I >" magnetic iron will coincide in direction with the longitudi- nal axis of the air- craft. However, this does not alter the nature of the semi- circular deviation, but only shifts the graph of deviation relative to the course scale of the aircraft Fig. 2 a Bar F 6. Deviation of Compass Card by f Hard Magnetic Iron. by an craft angle which is equal to that between the axis and the axis of the equivalent bar. of the air- Semicircular deviation of a magnetic compass can be compen- sated easily. To do this, it is sufficient to make a bar of hard magnetic iron and place it near the compass installation in such a way that its field is opposite to the direction of the field of the equivalent bar of hard magnetic iron. Let us now assume that there is no hard magnetic iron aboard the aircraft, but a field of soft magnetic iron is located hori- zontally and contributes to the action of the equivalent bar, coin- ciding in direction with the longitudinal axis of the aircraft. The essence of the effect of the soft magnetic iron on the compass readings consists in the fact that the bar, which is lo- cated in a certain position relative to the magnetic field of the Earth, is not magnetized in the direction of the field but along the length of the bar. /125 ula The magnetization of the bar can be expressed by the form- B = pHcos a , (2.1+) 124 where B is the magnetic induction, y is the magnetic permeabil- ity of the bar, H is the intensity of the magnetic field, and a is the angle between the direction of the intensity vector of the field and the direction of the bar. ■ -^ e, n <>* ^ i s — *■ / — \ \ ) V y MC 90 m 270 360 Fig. 2.7 Graph of Semicircular Deviation . On courses in this case, the of the equivalent cides with the di of the horizontal nent of the vecto sity of the magne of the Earth (a = the magnetic indu the bar is maximu no compas will be ation . and 180° , direction bar coin- rection compo- V of inten- tic field 0); although ction of m , there s devi- wil its on in 2 . 8 In changing course from to from to 270° , t induction of the decrease, but the between the vecto 1 increase. It is obvious that the deviation will th maximum at a course of ^5 or 315° and will reach zero courses of 90 and 270°. A similar change in deviation the flight sectors from 90 to 180° and from 180 to 270 ). the aircraft 90° or he magnetic b ar wi 11 angle rs H and B en reach once again will occur ° (Fig. It is clear in the figure that the deviation from the soft magnetic iron during one complete turn of the aircraft around the vertical axis passes through zero four times, i.e., it has a quar- ternary nature . The action of one magnetic bar of soft iron clearly illustrates the quarternary nature of the alternating magnetic field of the aircraft. In practice, however, with the exclusion of rare cases, the alternating magnetic field of the aircraft cannot amount to the effect of one bar of soft magnetic iron. In fact, if we take two bars of soft iron and locate them at /126 90° to one another, the resultant vector of induction of the bars will coincide with the bisectrix between them (Si = B 2) only in the case when the intensity vector of the magnetic field (Hi) coin- cides with the bisectrix of the angle between the bars (Fig. 2.9). In all other cases, the induction vector will approach the axis of the bar which is closer to the intensity vector of the magnetic field. If we consider the action of one bar, th'e vector of magnetic induction will change in value but will always coincide with the 125 axis of the bar. This essentially explains the existence on the aircraft of both semicircular and quarternary deviation as well as deviations of higher order. 5 y^K i y^-=x, f . s -^ % N * i vy ^2 MQ " 90 m 270 Fig. 2.8, Fig. 2.9 . Fig. 2.8. Graph of Quarternary Deviation from Soft Magnetic Iron, Fig. 2.9. Magnetic Induction of Crossed Bars of Soft Iron. In addition, if we disregard the deviation of higher order, the deviation from soft magnetic iron cannot be eliminated by using a suitable bar of soft iron, since it will also be magnetized like all other parts of the aircraft and will not lead to a reduction but rather to an increase of the deviation. Equatizing the Magnetic Field of the Aivovaft The cause of magnetic compass deviation on board an aircraft is generally a lack of coincidence between the resultant components of the magnetic field of the aircraft with the vector of intensity of the Earth's magnetic field. When the aircraft rotates around its axis, the alternating /127 magnetic field of the aircraft not only rotates along with it, but simultaneously changes in magnitude and sign. Therefore, in order to determine the magnitude and sign of the deviation for various aircraft courses, it is advisable to express its field components in the form of forces acting along the axes of the aircraft. Obviously, the magnitude of these forces (with the exception of the components made of hard magnetic iron) will vary with changes in the magnetic course of the aircraft {yy[). Depending on the nature and character of the action of the components of the magnetic field on the sensitive element of the compass, we can divide them into three groups: 126 (1) Components of the magnetic field of the Earth along the axes of the aircraft; their designations coincide with the desig- nations for the aircraft axes X, ¥, Z. The resultant vector of these components is Y. (2) The components of the magnetic field of the aircraft made of hard magnetic iron have the designations: P along the X-axis of the aircraft; Q along the Y-axis, and E along the Z-axis. (3) Components of soft magnetic iron of the aircraft. As follows from what has been said above, they cannot be viewed as a simple part of the resultant vector along the axes of the air- craft . For convenience in mathematical operations, these components lead to an equivalent effect of nine bars of soft magnetic iron, of which three bars coincide with each of the axes of the aircraft. This means that each of the three bars which coincide with a given axis of the aircraft is magnetized by a component of the magnetic field of the Earth which is located only along some one axis of the. aircraft . Equivalent bars a, bj e are located along the Z-axis of the aircraft; bar a is magnetized by the component of the magnetic field of the Earth X, bar h by component Y, and bar a by component Z. Equivalent bars d, e, f are located along the J-axis , and bars Qj h, k are located along the Z-axis; they are magnetized by the same components of the vector T. The contribution of the magnetic field of the soft iron in the aircraft to the equivalent effect of nine bars acquires phys- ical significance in_summing the magnetic induction of the compo- nents of the vector T, along the axes of the aircraft. For example, the J-component of the magnetic field of the Earth acts on bars Uj d, g, and the resultant induction from these three bars shows how the vector of the magnetic field from the soft mag- netic iron of the aircraft IX would be located if the components of the magnetic field of the Earth Y and Z were equal to zero. In other words , the equivalent bars are equivalent to the vec- tors of division of the magnetic induction from the components of the magnetic field of the Earth along the axes of the aircraft (Table 2.1). In summing the magnetic forces along the axes, we obtain the equations for the magnetic field of the aircraft: /128 ■?' = ?+ QJ^'dX+ fy 4- fZ; Z' = Z + R + ^+Ty+kZ: (2.5) 127 These fields will be used as a basis for deriving formulas for the deviation of magnetic compasses on an aircraft. TABLE 2.1 axis of Resultant forces the aircraft T ^ IX J mY nZ OX X P aX bY cZ oy Y Q dX eY fZ' oz Z R gx hY kZ The sum of the vectors X'^ Y' and Z' gives a total vector T' acting on the sensitive element of the compass. Deviation Formulas In the equations of the magnetic field of the aircraft, the constant terms are only the components of the field of the hard magnetic iron, P, Q, R. However, to calculate the deviation in hori- zontal fligh_t, we can consider that the magnetic induction Z from the vector T is constant along the vertical axis of the aircraft (terms cZ , fZ, kZ) . In addition, horizontal flight will not involve the third equation in (2.5), determining Z'. If we also consider that the sum of the vectors X and I con- stitutes the horizontal component of the magnetic field of the Earth H, X= H cos 7; i/=//slnT, the first two equations in (2.5) can be rewritten to read as fol- lows : A"' = // cos 7 -)- aH cos 7 — ft// sin 7 -h c^ + P; ] f^' = //sin7 + cf//cos7 — e//sln7+/Z+ v/, J (2.6) where y is the magnetic course of the aircraft. The vectors X'y I' are the components of the magnetic field along the longitudinal and transverse axes of the aircraft at the locus of the compass . The magnetic compass deviation (6) is expressed by the angle between the direction of the horizontal component of the magnetic field of the Earth H and the horizontal component of the total mag- /129 netic field on the aircraft H' (Fig. 2.10). 128 Obviously, tg6 is equal to the ratio of the projection of vec- tor R' in a direction perpendicular to the magnetic meridian fl", to its projection on the magnetic meridian H"' : H" X' sin f + r cos -1 ^ H" A^'cosf— y'sin^ (2.7) If we substitute into Equation (2.7) the values of X' and Y' from Equation (2.6), and also reduce similar terms, replacing the values sinycosY, sin^y and cos^y by their obvious homologues % sin^Y % (I-cos^y) and h. (1+cos^y)j we will have: -^ // + (cZ + />) sin Y + (/Z + (?)C0S7+^^ // sin 2t -t- d +b tgB = (2,8) H + a + e H+(cZ + P) cos Y — (/2 + (?) sinY + . a'— e d + b + -T— // cos 2y— — r— // sin 2y The terms in Equation (2.8), with a coefficient equal to unity, have a constant character, i.e., they are independent of the air- craft course at a given magnetic latitude. The terms which have the coefficients 2 sin y and 2 cos y have a quarternary character. The terms with coefficients sin y and cos y have a semicircular character . All of the forces designa of Equation (2.8) are directed meridian while those in the de ted by values lo at an angle of nominator coinci cated in the numerator 90° to the magnetic de with it . The force d H is independent of the aircraft course; it is proportional to the horizontal the Earth and is directed at a ian. This force is related to of the aircraft by the magneti a fun the 1 nate component of th n angle of 90° t the magnetizati c field of the E ction of the mag ocus of the aire this force by Aq The force cZ+P i longitudinal axis of the result of the Ion of the field from the P and the induction f e magnetic field of o the magnetic merid- on of the soft iron arth and varies as netic latitude of raft. We will desig- \E. s directed along the the aircraft; it is gitudinal component hard magnetic iron rom the vertical Fig. 2.10. Deviation of Magnetic Compass Aboard an Aircraft. 129 ■component of the magnetic field of the Earth, This force is desig- nated BqXH and changes with the magnetic latitude of the aircraft /130 location only in accordance with the first term. The projection of the force on the normal to the magnetic meridian is proportional to the sine of the magnetic course of the aircraft. The force fZ+Q is d nature and character of along the transverse axi jection on the normal to the cosine of the magnet The forces a - b an iron on the aircraft, ma The former is designated the double course of the dicular to the double co The force H+^^ H is in the denominator, a of the magnetic meridian esignated by Cq}^H, and is analogous in the its changes to the force BqXH, but is directed s of the aircraft. Consequently, its pro- the magnetic meridian is proportional to ic course of the aircraft. /J t J-, d are related to the soft magnetic gnetized by the magnetic field of the Earth. DqXH and coincides with the direction of aircraft; the latter is EqXH and is perpen- urse of the aircraft. s designated XH . In Equation (2.8), it nd therefore coincides with the direction If we substitute into Equation (2.8) these designations for the forces and divide the numerator and denominator by XH , the latter will give us ^^_ A + BoSin1-i-C„ cos t + Dq sin 2'y + £o cos 2^ I + B0COS7 — Cosln7 + £>ocos27 — £oSln2-j " (2.9) Expression (2.9) is called the point-deviation formula, and the coefficients Aq^ Sqj Cqj Dq and Eq are the point coefficients of deviation. The point-deviation formula is inconvenient to use, so it has been simplified for practical purposes. Since it is almost always necessary to select a place for mount- ing the compass on the aircraft where the deviation does not exceed 8-10°, we can let tg 6 = 6 . The denominator of Formula (2.9) can be expressed in the form of a binomial: [1 + (Bo cos f — Co sin 7 + Do cos 27 — Eo sin 2f)]-i = (1 + a)->. We know that with a < 1, the expansion of the binomial gives the converging series: (1 +a)-i = l— a + a2 — a3. .. For practical purposes, we can limit ourselves to the first 130 two terms of the series (l+a)-iwl-«, so that Equation (2.9) assumes the form: t = (Ao f BoSlnT+ Qcos^ + £)os1b27 + £oC08 27)(l — BqCOSt + + Co sin T — Do cos if + Eg sin z^). (2.10) Having carried out the multiplication of the multipliers, re- /131 duced the similar terms, and carried out simple trigonometric conver- sions. Equation (2.10) assumes the form: 8 = i4 + B sin 7 + Ccos 1 + Z) sin 2 7 + £ cop 27 + Z' sin 3t + O cos 3f + + //sin4-j + A'cos47... (2.11) Here the coefficients A^ B, C, D, E have a somewhat different value than in the point-deviation formula: Bl r2 ^0 A = A<i\ B = Bo + AoCo; D = Do+-T-+-r-+ A^o', C — Co — j4oBoi ^ = ^0 — BqCq — AolJo, The coefficients of deviation of higher orders, i.e., propor- tional to the sines and cosines Sy , ^-y » • • • , can be disregarded, since they are much smaller than any of the first five coefficients Then Formula (2.11) assumes the form: 8 = vl + B sin -f + C cos Tf + /> sin 2-r + £ cos 2^, (2.12) where A is the coefficient of constant deviation, 5j C are the coef- ficients of semicircular deviation, and D^E are the coefficients of quarternary deviation. Formula (2.12) is called the approximate formula of deviation, and its coefficients are the approximate deviation coefficients. However, it is completely satisfactory for practical applications, especially if we recall that other factors are acting on the compass which are very difficult to allow for. Calautation of Approximate Deviation Coefficients We will assume that we know the deviation of a magnetic com- pass at eight symmetrical points: 0, 15, 90, 135, 189, 225, 270 and 315° . According to Equation (2.12), the deviation at these points must have the values: 8o = yl + BsinO° + C cos 0° + Z> sin 0* + BcosO°. 131 Since sinO° = 0, cosO° - 1, then 6 q = A + C + E ; la = A-\-B sin 45° + Ccos 45° + Z> sin 90° + £ cos 90° or, if we consider the values sin90° = 1, cos90° = 0, 645 = i4 + B sm 45° + C cos 45° + />. Similarly, we can obtain a system of equations for the devi- ation of the eight points: /132 60 = A ■(- C + £; \i = A + B sin 45° + C cos 45° + D; V = A + B — £,•< hh = A "-f B sin 45° — C cos 45° — D; \^ = A~C + E; ha.b = A — B sin 45° — C cos 45° + D; S270 = A — B — E; 8.11s = -4 — B sin 45° + C cos 45° — D. (2.13) Summing Equation (2.13), we obtain: *0 + S16 + %0 + 6186 + 6,80 + *22S + ^70 + 63,6 = 8i4 or i=0 consequently , A = 8 To find the approximate deviation coefficient B, we multiply each of the Equations (2.13) by the coefficient at B, depending on the aircraft course. Then, keeping in mind the fact that sinH5° = cos^+S", the equations for 60 and &iqq become zero and the remain- ders assume the form: 8,5 sin 45° = A sin 45° + B sln2 45° + C sln2 45° + D sin 45°; 68o = A + B — B; Sigj sin 45° = A sin 45° + B sin2 45° — C sln2 45° — D sin 45°; —8225 Sin 45° = A sin 45° + B sin2 45° + C sln2 45° — D sin 45°; —8270 = A + B + B; — B315 sin 45° = A sin 45° + B sin2 45° — C sin2 45° + D sin 45°, In summing the six remaining equations, the sum of the terms containing coefficient A becomes zero, since three of them have a plus sign and the remaining three, symmetrical to the first, have a minus sign. The sum of the terms containing coefficient B is equal to 132 J / — 9 2S + 4S sin^ 45°, but since sin^ i+5° -^~5") ~ 'K^ this sum will be equal to 4S . The sum of the terms containing coefficient C , as well as the /133 sum of the terras containing coefficient D, is equal to zero. Cons equently , S8,-ri=4B <=0 or B = — -^B/sln^,. Similarly, we can find the formulas for determining the re- maining three coefficients: o C"= —^5/ COSY/; 8 £» = — ^6/cos27,; 8 ^=—^8,- COS 2a;,. i = (2.1M-) J Change -in Deviation of Magnetie Compasses as a Function of the Magnetic Latitude of the Locus of the Aircraft The deviation of a magnetic compass determined for a given point on the Earth's surface, does not remain fixed for other points, but changps depending on the magnetic latitude of the locus of the aircraft . uDviously, a change in deviation cannot take place as a result of changes in the magnetic induction of soft magnetic iron from the horizontal component of the magnetic field of the Earth. By the same token, the induction from the component producing the deviation will change in the same proportion with a change in the horizontal component of the magnetic field of the Earth, as the principal directional position of the compass card. Consequently, the compass deviation remains constant. The deviation from magnetic induction of the horizontal com- ponent of the field of the Earth has a constant and quarternary character : 133 MH=^H; £><^//=«-=i//; s^ff^t±lff. Hence, we reach the conclusion that the constant and quartern- ary deviation at various magnetic latitudes remains constant. Essentially, the change in the deviation with a change in the /134 magnetic latitude is the result of the influence of hard magnetic iron and partially as a result of induction with soft magnetic iron from the vertical component of the Earth's magnetic field. This takes place because the magnitude of the vectors Pj Q remains constant with a change in the directional vector H. Consequently, with an increase in the magnetic latitude, the semicircular deviation must increas e . In addition, with an increase in the magnetic latitude, the induction of the soft magnetic iron from the vertical component of the Earth's field increases with a simultaneous decrease in the directional force H. However, if we consider the predominant influ- ence on the aircraft produced by the hard magnetic iron, we can consider in approximation that the semicircular deviation is inversely proportional to the horizontal component of the magnetic field of the Earth. Bo = cZ +P Co = fZ+Q XH ' which gives the following for the approximate coefficients of de- viation B and C: ^2 = Si 777 •" Cj = C, -J- , "2 "2 (2.15) where B i 3 Ci, Hi are the approximate coefficients and the horizontal component of the Earth's field at the point where the deviation is measured; Sjj (^23 ^2 ^^^ "the same values at a point with a dif- ferent magnetic latitude . With known coefficients B + C, the semicircular deviation at a given point on the Earth's surface can be determined by the form- ula H\ H B = B -— sin T + C -7- cos 7. "2 Ho (2.16) Eliminat-ton of Deviation in the Magnetic Corn-passes Modern magnetic compasses are fitted with a device for com- pensating only semicircular deviation, resulting from hard magnetic iron . 134 In addition, by a suitable rotation of the compass housing in its mountings, we can compensate for the constant component of deviation along with the adjustment error of the compass. Elimination of quarternary deviation by magnetic means encoun- ters considerable technical difficulty. Therefore, if we keep in mind the relatively low value of the quarternary deviation relative to the semicircular deviation, as well as its constant value at various latitudes, we will not be able to get rid of the latter but will enter it on special graphs for compass correction. Modern remote control magnetic compasses have devices for me- chanical compensation of deviation of all orders. /135 The device for compensating semicircular deviation consists of a system of four cylinders mounted in pairs, with permanent magnets installed in them (Fig. 2.11). When the magnets are tilted (Fig. 2.11, c), the horizontal component of their field appears, and can be set so that it is equal but directed opposite to the magnetic field of the aircraft (hori- zontal component), located along its transverse axis. The maxi- mum effect of the small magnets will be observed when they are in the horizontal position (Fig. 2.11, b). 135 II The cylinders for compensating deviation at courses of 90 and 270° are mounted in the transverse axis of the aircraft in such a way that the small magnets can be used to compensate for the compo- nent of the magnetic field of the aircraft which is directed along its longitudinal axis . The rotation of the longitudinal and transverse cylinders is accomplished by means of special handles made of diamagnetic mater- ial . To determine and get rid of deviations, the aircraft is placed on a specially prepared stand, made of concrete (for heavy aircraft ) /136 but without a metal core. The stand must be of sufficient size so that aircraft of any kind can be rotated in a circle and the distance from the stand to other aircraft and metal structures is at least 200 m. The accuracy of the setting of the aircraft on a given course for determining and getting rid of deviation can be checked in one of the following two ways : 1. Direction finding of landmarks from on board the aircraft. In the center of the area where the aircraft is to turn, a magnetic direction finder or theodolite is mounted on a stand so that the indicating dial is located exactly in a horizontal position, and the zero reading on the dial coincides with the direction of the magnetic meridian. For this purpose, these instruments are fitted with a bubble level and orienting magnetic needle. Then two or three distinct and prominent landmarks on the hor- izon are selected (towers and chimneys are best for this purpose), and their magnetic bearings (MB) are determined with the aid of a sight, rotating on the dial used for determining the bearings. The landmarks should be located as far as possible from the area so that the shifting of the aircraft from its center during rotation will not produce any noticeable changes in the bearings of the landmarks. For light aircraft, this distance should be at least 2-3 km; for larger aircraft with a greater radius of turn on the ground, it should be at least 5-6 km. After determining and recording the magnetic bearings of the landmarks, the aircraft is mounted on the stand. The direction finder is placed in front of or behind the aircraft at a distance of 20-100 m, depending on the length of the aircraft, exactly along its longitudinal axis so that the forward and rear points on the axis of the aircraft will be projected on the sight, e.g., the centers of the nose and keel. Then the dial on the direction finder is set to the magnetic meridian, and the direction of the longitudinal axis of the aircraft is measured, and its initial course is set. 136 It is necessary to recall that the minimum distance for the direction finder from the aircraft is limited by the effect of the aircraft on the magnetic needle of the deviation direction finder, and the maximum distance is set by the length of the aircraft, since at a distance of more than 100 m, with an aircraft which is not very long, this method will be insufficiently precise. After the direction finder has been moved to the aircraft, the magnetic needle is fixed and set so that one of the selected landmarks (Fig. 2.12) appears at a course angle ( CA ) equal to CA MBL - Mc (2.17) where MBL equals the magnetic bearing of the landmark and MC is the initial magnetic course of the aircraft. If the above condition is satisfied, the zero point on the direction finder dial will coincide exactly with the longitudinal axis of the aircraft . /137 To set the aircraft on definite courses, a table of course angles for landmarks for each aircraft course is compiled. For example, if the deviation has been determined at eight points, but the selected landmarks have magnetic bearings of 115 and 328°, then the course angles for the courses which we require will have the values shown in Table 2.2. TABLE 2.2 MC Cal^(MBL= ,,^Q Cal (MBT-= -115^ 2 3280) CC Ak 115 32(5 358 +2 45 70 283 42 +3 90 25 238 91 —1 135 340 193- 133 +2 180 295 148 178 +2 225 250 103 224 + 1 270 205- 58 271 —1 315 160 13 313 +2 When using this table, the sight of the direction finder is set to a given course angle for a landmark and the aircraft is then turned until the axis of the sight lines up with the direction of the selected landmark. It is clear that the aircraft is then set precisely on the desired course. The second landmark is an extra one in case the first is ob- structed by some part of the aircraft such as the empennage or wing, 137 The method of setting an aircraft on course by the method de- scribed above for obtaining the course angles of landmarks is the most precise and reliable one, especially since a fixed area can be set up at an airport for correcting devia;tions and doing other work to set the bearings of landmarks and compiling tables of course angles for given air- craft courses . However, this method is not always practicable. In some cases, it may be impossible to select suitable landmarks, and in other cases the visibility may be inad- equate for them to be seen. In some aircraft, there may be diffi- culty in fastening the direction / finder on board the aircraft in clear field of vision for observ- Fig. 2.12. Determination of Aircraft Course by the Course Angle of a Landmark. a place where there would be ; ing the landmarks . 2 . D F rec This method is an or j.ii^^ ...^uwwv^ -- used in craft on courses of 0, ^5 , t i on finding of J -• _ cases when it is impossible to set the by the tai 1 air- aii aircraft from the nose «. ,.>.... when it is impossible to set the air- 90°, etc., by the method described above. In this case, the aircraft is set each time(e.g., according to the readings of the magnetic compass) to a given course. Then the direction finder is located along the extension of the longi- tudinal axis at a distance of 20-100 m from the aircraft, depend- ing on the type of the latter; the correcness of the setting of the aircraft on course is then determined as in the first case before mounting the direction finder on board the aircraft. It may be necessary to turn the aircraft for a secondary check. This method is less convenient than the first, since it is necessary to shift the direction finder for each course, set it exactly along the extension of the aircraft axis, adjust the zero on the dial along the magnetic meridian, and make the dial level, in addition to measuring the distance to the aircraft. Under unfav- orable conditions aboard the aircraft, this operation may have to be repeated after moving the aircraft. The advantage of this method is its independence of the existence of landmarks, meteorological visibility, and peculiarities of aircraft design. Semicircular deviation of magnetic compasses is corrected and eliminated at four basic points: 0, 180, 90 and 270°. It is clear from (2.13) that semicircular deviation at the and 180° points is equal in value, but opposite in sign, and ex- pressed by the maximum value of coefficient C. Deviation from 138 coefficient B is equal to zero on these courses. However, all of these courses are subject to the action of a constant deviation in the coefficient A and quarternary deviation E in addition to the semicircular deviation. This means that the values of the constant and quarternary deviation are equal in value and sign. Consequently, if the deviation on course 0° is set to zero by turning the cylinder of the deviation- correcting apparatus with the marking "N-S", the semicircular deviation will be compensated for and the constant and quarternary deviation will simultaneously be compensated for. It will change with the same sign to a course of 180°, where its value doubles. Therefore, after setting the aircraft to a course of 180°, it is necessary to set the deviation not to zero, but to half the rotation of that cylinder, and in the reverse direction. Hence, the semicircular deviation from coefficient C can be eliminated completely and precisely without disturbing the constant and quarternary deviations . Analogously, by turning the cylinder of the deviation- corre ct- ing apparatus with the marking "E-W", it is possible to reduce the deviation to zer-o for a 90° course and by half for a 270° course, which completely gets rid of the semicircular deviation from coeffi- cient B without disturbing the constant and quarternary deviations. /139 TABLE 23 MC Deviation Shown Up to 12 180 +4 +2 90 +7 270 -2 —1 -s — N7 _ L^J ^^.,..— .^ s y / ( ■A /- \^ / V y — - - — «5" 30 135 m 22S 270 JI5 *5 -5 Fig', 2.13. Graph of Deviation of a Magnetic Compass . 139 The operation with semicircular deviation is described in a special table (Table 2.3). Obviously, the remaining deviation at these points will be equal to +2° for courses of and 180° and -1° for courses of 90 and 270° . After getting rid of the semicircular deviation, the aircraft is set to courses at 4-5° intervals and the remaining deviation is measured. An example of the recording is shown in Table 2.2. After summing the remaining deviation for eight courses (Graph 5, Table 2.2) and dividing the sum by eight, we obtain the value of the constant deviation A 2+3-1+2+2+ 1 -1+2 A = -^— = + 1^25°. The bowl of the compass must be set in its mounting to this value. If we disregard the value of 0.25° produced by turning the bowl of the compass through 1°, the remaining deviation for the eight courses will have a value of +1, +2, -2, +1, +1, 0, -2, +1 so that the graph of the corrections can be compared with the read- ings of the compass (Fig. 2.13). If the aircraft is intended for use on flights at magnetic latitudes where there will only be small changes, this will mark the end of the work with deviation. In preparing for long distance flights , with considerable changes in the magnetic latitudes, the coefficients of the semicircular /I^j-O deviation B and C must also be found with determination of their changes with magnetic latitude. In this case, the coefficient B will be equal to: B = + 1 sin (P + 2 sin 45° — 2 sin 90° + I sin 135° + 1 sin 180° + + — 2 sin 270° + 1 sin 315° + 1.4 — 2 + 0.7 + + + 2 — 0.7 „-i- ^ 0.35, and coefficient C will be l + l,4_0^0,7+ l.t-0_ 4.0,7 C =■ ■ . = 0,00. 4 140 Gyroscopic Course Devices Regardless of the fact that measures have been employed for a long period of time which are directed toward increasing the accur- acy of readings and the stability of operation of integrated mag- netic compasses , their shortcomings have not been completely over- come . In addition, magnetic course devices are difficult to use in a flight along an orthodrome for long distances , due to the complex- ity of the calculation of the magnetic declination as it changes along the route. All of this has made it necessary to seek new ways of devis- ing course instruments and systems which will satisfy the require- ments of aircraft navigation at all stages and all conditions of flight. The first steps in this direction were made by the remote con- trol magnetic compasses, containing a magnetic transmitter (a sensi- tive element) located at any convenient point in the aircraft, whose readings were transmitted by means of special potentiome trie transmitters to dials mounted in the cockpit. This made it possible to mount the compass in the pilot's field of vision and ensure optimum conditions for operation of the compass from the standpoint of deviation. However, there were still considerable shortcomings in the operation of the compass, such as instability of the readings with movement of the aircraft and the impossibility of using it when the aircraft was turning. In addition, the reliability of operation of the compass de- creased, since the potentiometric connection with reliable contacts produced an additional delay in the turning of the sensor card to a significantly greater degree than was the case for the rotation of a freely moving card on its bearing in an integrated compass. The next steps in increasing the accuracy and reliability of / m-1 operation of course devices was made by the gyroscopic semicompasses and magnetic course sensors linked with gyroscopic dampers. This made it possible to use the course instruments while the aircraft was turning and to achieve stability of course readings under any flight conditions. Analysis of the induction course sensors, free of friction during turning of an aircraft , significantly increased the reliability of magnetic compasses. However, the greatest reliability and accuracy in course meas- urements for aircraft has been achieved by the building of complexes of course instruments (course systems), combining the operation of gyroscopic, magnetic, and astronomic sensors. The principle of these systems is a stable and prolonged maintenance of the system I'll M for estimating the course, with a gyroscopic assembly having peri- odic correction of the readings by means of a magnetic or astro- nomical sensor, or input of corrections manually as desired by the crew . Fvino'ipte of Operation of Gyroscopic Instruments The gyroscope is a massive balanced body, rotating around its axis of symmetry at a high angular velocity. Gyroscopes are usually made in a form such that they have rel- atively low weight and small size, yet have a maximum inertial moment which is reached relative to the basic mass of the gyroscope as far as possible from the center of rotation within the given dimen- sions of the gyroscope. Let us recall that the inertial moment J in mechanics is the product of the mass times the square of the distance to the axis of rotation: ' (2.18) where rj is the distance from the mass to the axis of rotation. For a' complete cylinder, which constitutes the basic mass of a gyroscope (Fig. 2. 14-), the inertial moment is ■(-i-^?) (2.19) The gyroscope has two interesting properties which are used in a number of devices for pilotage and navigation: (1) Axial stabitity 3 i.e., the ability to maintain the di- rection of its axis of rotation in space in the absence of moments of external forces tending to change this direction; (2) Axiat precession of rotation under the influence of mo- ments of external force, i.e., a slow rotation of the axis in a plane which is perpendicular to the applied force, with maintenance of the direction in the plane of the application of the force. The first property of the gyroscope is usually used for sta- bilizing the directions of the axes of the coordinates for deter- mining the required values, the banking of the aircraft, the angle of pitch, and the course. The second property is used to set the axis of the gyroscope in the desired position, e.g., to the vert- ical of the locus of the aircraft, to the plane of the true hori- zon, for compensation of the apparent rotation of the axis due to the diurnal rotation of the Earth, etc. In addition, the property of precession is sometimes employed in devices which integrate the /142 142 action of the forces with time navigation devices. in the construction of inertial To explain the principles of operation of gyroscopic devices, let us consider the physical significance of the two properties of a gyroscope mentioned above. — r, -X ^ — - For the assume that is located a the axis of we shall sel at some poin Let us of a force F has been til Fig. 2.14. Gyros cope Rotor . Obvious of the eleme does not change when it passes through motion of the element at points A, Ai ference remain parallel. The tangents at the point C and diametrically oppos equal to Acj) . sake of simplicity, we shall the mass of the gyroscope long the circumference around rotation (Fig. 2.15), and ect an element of this mass t on the circumference . assume that under the influence , the axis of the gyroscope ted to an angle Acji . ly , the direction of rotation nt of mass of the gyroscope points A and B, since the and B, Bi tangent to the circum- to the direction of motion ite to it are at an angle Consequently, at these points there arises a difference in the velocities A7 = 7 sin A(() . (2.20) The greater the angular velocity of rotation of the gyroscope and the radius of the ring, the greater will be the circumferential speed of the element of mass and the magnitude of the vector A7. Obviously, the reaction duce resistance to the vector and Ci , i.e. , the forces F„ and Fp, _ P ^ 1 of the mass of the gyroscope must pro- velocity change at the points C arise at these points, directed opp axis '■ ' ' _ p Pi ■" ' osite to vector A7 and producing the precession of the gyroscope s . It is easy to see that the inertial forces directed against the external force will be exactly equal to the latter, so that no rotation of the axis of the gyroscope in the plane of the action of the external force will be observed. /143 143 The precession rate of the gyroscope can be determined easily if we know the moment of inertia of the rotor and the moment of the applied external force. A change in the moment of inertia of the gyroscope with time will be proportional to the moment of the external force il whence dt dt i u • M_ (2.21) (2.22) where U equals the moment of the external force , J equals the mo- ment of inertia of the gyroscope, w is the angular velocity of gyro- scope rotation, and to ^ is the angular velocity of precession. By the change in the moment of inertia of the gyroscope, we mean here the change in the direction of the vector of iner- tia . At the same time, the rotation of the axis of the gyroscope through 180° produces an opposite motion of all points on the rotor, which amounts to a braking of the gyroscope from its initial angular velocity to zero, with a subsequent speeding up in the opposite direction to the same angular velocity. Fig. 2.15. Precession of a Gyroscope Axis. Begvee of Freedom of the Gyroscope By degrees of freedom in mechanics, we mean the directions of free motion of a body which is not limited by connections of any sort. For example, an object sliding along a given line (rail) has one degree of freedom; an object moving in any direction in a plane has two degrees of freedom, and an object which is moving in three dimensional space has three degrees of freedom. Besides the degrees of freedom of linear motion, there are /lUU also degrees of freedom of rotational motion of a body around its three axes . Hence, a completely free body has six degrees of freedom. 14M- The rotors of gyroscopes in navigational and pilotage instru- ments have supports which limit their linear motion in a certain direction relative to the axes of the aircraft, so that when we are talking about the degrees of freedom of a gyroscope we are rer f erring only to the degrees of rotational motion. A gyroscope is considered to be free if all three degrees of rotational motion are free (Fig. 2.16). The first degree of of a gyroscope is the ro of its rotor around the bearings A^Ai. If these are tightly fastened to of the machine, as is do example for the flywheel ery , the gyroscope will one degree of freedom. if these bearings can mo an axis perpendicular to ings BjSj), then there w two degrees of freedom. If bearings B,Bi ca have the freedom to move still another (third) ax pendicular to B ,B i (bear the gyroscope will have of freedom and its axis set readily to any direction in space. Fig. 2.16. Gyroscope with Three Degrees of Rotational Freedom . free dom tation axis in bearings the body ne for s in machin- have only However , ve around A,Ai (bear- ill be ,n also around is , per- ings C,Ci), three degrees can be As we can see from Figure 2.16. the degrees of freedom of the gyroscope are ensured by pairs of bearings and (with the exclusion of the first) rotating frames. A gyroscope usually has two rotating frames, internal and ex- ternal. In course gyroscopic instruments, the internal frame, to- gether with the rotor and the bearings of the gyroscope, serves to set the gyroscope axis in the plane of the true horizon. The same frame contains a sensitive element for correcting the gyro- scope axis for this plane. The internal frame of the gyroscope along with the rotor and sensitive element for correction are called the gyro assembly. The external frame ensures free motion of the axis of the gyro- scope in the plane of the horizon; from its position in the unit, we can get an idea of the direction of the gyroscope axis relative /145 to the axis of the aircraft, or vice versa, thus making it possible to determine the aircraft course. 145 B-lveotion of Fveaession of the Gyroscope Axis The direction of the^ precession of the gyroscope axis under the influence of the moment of external forces can be seen in Figure 2.15. For a rapid and error-free determination of the direction of the precession of the gyroscope axis, we use the concepts of "pole of the gyroscope" and "pole of the external force", and use the rule of the right-hand screw. For example, in observing the rotation of a gyroscope which is turning clockwise as viewed from the top (turning the screw in- ward), the pole of the gyroscope will be considered as being located at the lower end of its axis (Points P and Pi); with left-hand ro- tation of the gyroscope, at the upper end of the axis. Analogously, with a right-hand direction of the moment of external force, the pole of the moment is considered as being directed along the screw, in its rear portion as shown in our diagram (Point C^ ) . With a left-hand direction of the moment of external force, its pole is located in the front part of the picture (Point C). The precession of the gyroscope is always directed in such a manner that the pole of the gyroscope attempts to reach the pole of the external force by the shortest path. In our diagram, the lower end of the gyroscope axis will tilt backward, and the upper one forward, i.e., if we look at the draw- ing from left to right, the axis of the gyroscope will rotate clock- wis e . Apparent Rotation of Gyroscope Axis on the Earth *s Surface A freely moving gyroscope, with an ideally stabilized external and internal support and the lack of noticeable friction in the bearings , tends to keep the position of the axis of rotation of the rotor in space. On the Earth's surface, however, due to the diurnal rotation of the Earth and partially due to the curvilinearity of its motion around the Sun, there arises an apparent rotation of the gyroscope axis in the vertical and horizontal planes. The apparent rotation of the gyroscope due to the motion of the Earth around the Sun is expressed as a slight deviation of the rotation of the gyroscope axis from the apparent diurnal rotation of the Earth, as a result of the fact that the Earth makes a com- plete rotation around the Sun along its orbit in the course of a year. This conditional rotation amounts to a total of about 1/365 of the apparent rotation of the gyroscope due to the diurnal rota- tion of the Earth. Hence, this value will not be considered in future . 1^6 Let us consider the apparent rotation of the gyroscope axis at various points on the Earth's surface, which appears as a result of the rotation of the Earth around its axis. We will assume that / m-6 we have a freely mounted gyroscope, whose axis at the initial moment coincides with the vertical of the locus (Fig. 2,17, a). Obviously, if such a gyroscope is placed on a pole of the Earth, the axis of its rotation will coincide with the axis of rotation of the Earth and there will be no apparent rotation of the gyro- scope axis (position A in the diagram). If the gyroscope with a vertical axis is placed on some lat- itude (j) (position B in the diagram), its axis will be at an angle to the axis of rotation of the Earth, equal to 90°-(j). As we can see from the diagram, the apparent rotation of the gyroscope axis will describe a cone with an aperture angle at the vertex equal to 2 (90-(|)). In the case when the latitude of the locus is equal to zero (position C in the diagram), the aperture angle of the cone will be equal to 180°, i.e., it will turn in the plane of rotation. Now let us examine the case when the axis of the gyroscope at the initial moment is located horizontally at various points on the Earth's surface (Fig. 2.17, b) and coincides in direction with the meridian of the Earth. It is obvious that the axis of the gyroscope located on the pole (position A) will remain horizontal and will rotate in the plane of the horizon with the angular velocity of the Earth. The axis of a gyroscope located at some latitude (position B) will de- scribe a cone with an aperture angle equal to 2(j). The axis of the gyroscope located on the Equator will remain horizontal and will have no apparent diurnal rotation. It is important to note in this regard that if there is any kind of correcting force which acts constantly on the gyroscope axis in the plane of the true horizon, the angular velocity of the rotation of the gyroscope axis in the plane of the horizon will be equal to (Fig. 2.17, c): at the pole, the angular velocity of rotation of the Earth; at the Equator, zero; at any other point, (0 = fi sin (j) , (2.23) where fi is the angular velocity of the Earth's rotation and oi is the angular velocity of the apparent rotation of the gyroscope axis. From the examples which we have seen, it is clear that a freely moving gyroscope can be used to determine the position of the air- craft axis only in the following cases: (a) To determine the position of the vertical axis (banking. 147 II pitch) only at the poles; (b) To determine the direction of the longitudinal axis (course of the aircraft) only at the Equator. In order to render the gyroscope useful for determining the position of the aircraft axis at any other point on the Earth's surface, we used devices which compensate for the apparent rota- tion of the axis of the gyroscope due to the diurnal rotation of the Earth, as well as its own drift, which arises as a result of imperfect balance, friction in the bearings, etc. /147 Fig. 2.17. Apparent Rotation of a Gyroscope on the Earth's Surface: (a) With Vertical Axis; (b) With Horizontal Axis; (c) With Constant Correction of the Axis in the Horizontal Plane . To keep the axis of a gyroscope constantly in the vertical position, pilotage devices ( gyrohorizon , gyrovert i cal ) , or in the horizontal position in the case of course instruments, are usually fitted with pendulum devices which act as sensitive elements react- ing to any deviations which may arise. The signals from these devices are converted to air currents in pneumatic devices and to moments of special electric motors in electrical devices. 148 Electrolytic gravitational correction (Fig. 2.18) is most widely used at the present time. This device consists of a bubble level attached to the lower part of the gyro assembly. Unlike a conven- tional level, its chamber is filled with an electrically conductive liquid (electrolyte), while on the top of the spherical surface are mounted four current- carrying contacts. When the gyro assembly is in a vertical position (Fig. 2.18, a), the bubble level is located so that all four contacts are cov- ered half-way by electrolyte, so that the moment applied to the frame of the gyro assembly by the correcting motor is equal to zero, /ms Fig- Gr avi and a the 1 one p in or frame If for some reason the sembly varies from the verti the current-carrying contact not be uniformly covered by (Fig. 2,18, b), resulting in able distribution of current windings of a small motor an moment which is applied to t of the gyroscope in such a w the precession which is prod the gyro assembly to a given ical position. For course d which have a vertical extern horizontally located axis of the gyroscope, in order atter to the plane of the horizon, it is sufficient t air of current-carrying contacts with a gravitational der to regulate the moment of the forces acting on th 2.18. Electrolytic tational Correction gyro as- cal position, s will the fluid a suit- s to the d in a he axis ay that uced brings vert- e vi ces al frame to correct o have leve 1 , e external Obviously, for those devices which measure direction on the Earth's surface, in addition to devices for correcting the axis of the gyroscope in the plane of the true horizon, there must also be other devices which compensate for the apparent rotation of the axis of the gyroscope in the horizontal plane due to the diurnal rotation of the Earth. Gyroscopic Semicompass In principle of operation, the gyros emi compass (GSC) is a gyro- scope with three degrees of freedom and its axis of rotation located in the horizontal, a vertical external frame, and a fluid gravita- tional corrector, attached to the gyro assembly. The rotation of the gyroscope rotor is produced by alternating three-phase current, while the correction of the axis in the horizontal position is achieved by an electromagnetic moment applied to the external frame. The gyrocompass has has a very sensitive balance and low fric- tion in the axes of the supports, which ensures a low intrinsic shift of the gyroscope (called "dvift") . In addition, in order to compensate for this "drift", the gyroscope is fitted in the 149 horizontal plane with a special balancing potentiometer and motor, which apply a moment to the external frame of the gyroscope in the vertical plane . This same motor is used for compensating the apparent diurnal rotation of the axis of the gyroscope, and is therefore fitted with a special latitudinal potentiometer, which regulates the moment of the motor in such a way that the rate of precession of the gyro- scope axis is equal to and coincides in direction with the rate of rotation of the Earth's meridian in the plane of the true hori- zon at the given latitude . By comparing the formula for the precession of the gyroscope axis (2.22) and the formula for the angular velocity of rotation of the Earth's meridian (2.23), we can determine the moment which is inquired to be applied to the gyroscope axis to compensate for the diurnal rotation of the Earth /149 M = QJta sin (2 .21+) where M is the moment applied to the gyroscope axis, Q is the ang- ular rotational velocity of the Earth, J is the inertial moment of the rotor of the gyroscope in the plane of its rotation, to is the angular velocity of rotation of the rotor, and cji is the lati- tude of the aircraft's location. With a constant rate of rotation of the rotor of the gyroscope, all of the coefficients which enter into the right-hand side of (2.2i+), with . the exception of sin cj) , are constants . The latter must be regulated in flight, Therefore, the potentiometer which regulates the moment according to the latitude of the aircraft , as well as the balancing potentiometer, „ ,„ „ , „ , are mounted on the control panel Fig. 2.19. Control Panel j. ., ,„. „ ^^. ^ ,.^„ r- rs r. • °^ "the gyrocompass (Fig. 2.19). of KPK-52 Gyrosemicompass . s>y t- a / The external frame of the gyro- scope is fitted with a scale for estimating the gyroscopic course and a selsyn- transmitter for transmitting the course to the indi- cators . The indicating dial and the selsyn- transmitter are free to rotate along with the external frame and can also be set with the aid of a motor to any angle relative to the frame. The setting of the indicator dial to the zero position is accomplished manually by turning a special handle on the control panel marked "L-R" (left - right), see Figure 2.19. Hence, the gyrocompass is a sort of "keeper" for the course 150 calculation set by hand: the direction of the zero setting of the course on the GSC remains constant in the plane of the horizon, so that the gyrocompass is an orthodromic course device^ and is cap- able of guiding a flight along an orthodrome over any distance. The advantage of a gyrocompass is its independence of operation from the magnetic field of the Earth, and consequently the fixed accuracy and stability, in operation at any point on the Earth's surface, as well as the ease of determining the course without any kind of methodological corrections; this is particularly important for automatic navigational devices /150 'rors It is relatively easy to eliminate errors in the operation of the GSC, which arise in the form of "drift". For this purpose, the operation of the GSC is tested on the ground for a period of one to two hours with an attempt being made to use the rotation of the balancing potentiometer to set the minimum excursions of the needle with time from the true settings. If a considerable deviation of the needle from the correct readings of the gyroscope is noticed during flight, this can be corrected by shifting the latitude scale on the control panel rela- tive to the average latitude of the given path segment. This means that the degree by which the scale is shifted for each degree at the time that the drift occurs will be the following at various flight latitudes : Range of Latitudes Degrees - - 32 32 - - 42 42 - - 60 60 - - 70 70 - - 90 Magnitude of Scale Deviation, Degrees 4 5 6 10 20 The latitude on the scale must be increased if the tendency of the GSC is directed toward a reduction of the readings for the course with time, and it must be reduced if the course readings increase with time. It should be mentioned that all shifting mentioned above with regard to the gyros emicompass is in reference to northern latitudes. In southern latitudes, the latitudinal compensations for the apparent 151 rotation of the axis of the gyroscope must be reversed, since the rotation of the meridian takes place in the opposite direction rela- tive to the northern latitudes. In addition, the system for intro- ducing corrections to the movement of the needle of the gyrosemi- compass must also be shifted to the opposite direction. Shortcomings of the gyrosemicompass include the fact that it is necessary to set its readings manually at the beginning of a flight and to make corrections en route. During flight, especially in rough air, this involves a certain amount of difficulty, since it is impossible to separate the movement of the indicator needle due to course variations from those motions which are caused by setting the course manually, i.e., the value of the course to which /151 the GSC must be set becomes variable. In addition, the GSC is subject to Cardan errors during turns. The essence of the Cardan errors is the shift in the reading of the indicator dial during banking. When the aircraft is banking less than 8°, these errors do not have any practical significance, but they rapidly increase with the degree of banking and can reach 6-8° . The Cardan errors have a quaternary nature . They are equal to zero in banking in the plane of rotation of the rotor of the gyroscope and in the plane of the position of the axis of its rota- tion. Maximum errors arise when the gyroscope axis is then at an angle of 45° to the plane of the banking. Therefore, the axis of the gyroscope can assume any position relative to the axes of the aircraft, and also with respect to the zero point on the course indicator scale, and the graph of the bank- ing error is "floating", i.e., its maxima and minima can assume any position on the indicator dial while retaining the values and periodicity of the errors . These errors automatically disappear when the aircraft comes out of the turn; however, they do constitute certain shortcomings in the pilotage of an aircraft, i.e., they disturb the correct esti- mation of the moment when the aircraft begins to stop banking in making a turn. Distance Gyromagnetic Compass The distance gyromagnetic compass (DGMC) has significant ad- vantages over the integrated and distance magnetic compasses, since it is suitable for use when the aircraft is banking at a certain angle and completely damps the oscillations of the magnetic card in flight in a turbulent atmosphere . The gyromagnetic compass is a combination of magnetic and gyro- scopic course devices, in which the role of the course sensor is 152 played by the magnetic transmitter and the role of the stabilizer of the readings is played by the gyro assembly. Let us consider the combined system which is presently used for distance gyromagnetic compasses, e.g., the DGMC-7 (Fig. 2.20). The basic parts of the distance gyromagnetic compass are the magnetic sensor, the gyro assembly, and the main course corrector. In addition to these main parts, the compass must be fitted with a power supply (not shown in the diagram), as well as compen- sating and regulating devices: (a) Compensating mechanism (combined with the gyro assembly); (b) Rapid compensation button; (c) A mechanism for compensating the remaining deviation (com-/152 bined with the main course indicator); (d) Outputs for course repeaters and other indicators; (e) Two-channel amplifier. The magnetic transmitter of the compass has a card whose axis carries a dial for showing the course directly on the transmitter (it can be used to get rid of semicircular deviation), as well as the brushes for the wires leading to the potentiometer on the trans- mitter . The transmitter potentiometer has a three-wire circuit con- necting it to the gyro-assembly potentiometer, through which it receives alternating current from the power supply. The transmitter in the damping suspension is mounted in the aircraft at a location where there is a minimum influence on the cards of the magnetic and electromagnetic fields of the aircraft. Compensation Button' correctio release 3 r rrr:^_ magnetic' transmitter com pen sating mechanism f H I iampii ,gyro assembly fier U ■D, Fig. 2.20. Functional Diagram of Distance Gyromagnetic Compass (DGMC) . 153 The transmitter housing carries a device for correcting semi- circular deviation. If the semicircular deviation at the point where the magnetic sensor is mounted does not exceed 1-2°, the devia- tion device is not used, since in this case it would not improve but would rather detract from the operating conditions of the trans- mitter . The gyro assembly consists of the gyroscope with a horizon- tal axis and a Cardan support, which ensures three degrees of free- dom for the gyroscope rotation. The external frame of the gyro assembly rotates around the vertical axis . The gyroscope is set in motion by means of a three-phase motor, whose stator is mounted on the internal frame of the gyro assem- bly and whose short-circuited rotor is the rotor of the gyroscope. For correction of the gyroscope axis in the horizontal posi- tion, the lower part of the gyro assembly is fitted with a two- contact gravitational corrector, whose activating mechanism is a motor which produces a moment of force that is applied to the ex- ternal frame of the gyroscope and acts in the horizontal plane. /153 If for some reason the axis of the gyroscope varies from the plane of the true horizon, the contacts of the corrector will be covered nonuniformly by the shifting conducting fluid, thus result- ing in a distribution of currents passing through the corrector. This in turn transmits a signal for a correcting moment of force to be applied to the external frame. As a result of the preces- sion of the gyroscope axis, it is shifted to a horizontal position, The external frame of the gyro assembly carries a master selsyn for connecting to the principal indicator of the compass (the pilot's indicator, PI) and a three- conductor cord for connec- tion to the magnetic transmitter. The master selsyn and cable are connected closely together and can rotate together with the external frame of the gyro assem- bly. However, they can also rotate relative to the external frame by means of a special coordination mechanism. The coordination mechanism consists of a small motor with a reduction gear for the s low- coordination regime, in which the rate of rotation of the selsyn is 1-4° per minute. When it is necessary to carry out a rapid coordination, the motor is switched to reduced reduction by means of the rapid-coord- ination button and a special relay. The rate of rotation of the selsyn in this case is raised to 15-15° per second. The potentiometer of the gyro assembly is firmly fastened to the housing. 154 The coordination of the magnetic transmitter with a gyro assem- bly is accomplished as follows (Fig. 2.21). The alternating current passes through contacts A and B to reach the potentiometer of the gyro assembly and is picked up by pickups 1, 2, 3 mounted on the external frame of the gyro assem- bly, from which it passes to the pickups on the transmitter poten- tiometer, la, 2a, 3a. It is clear from the figure that if the position of the brushes of the current pickups on the transmitter A^jB]^ relative to the current leads of the potentiometer la, 2a, 3a differs from the posi- tion of thfe current pickups of potentiometer A, B relative to their current connections 1, 2, 3 by 90°, there will be a current in the pickups of the transmitter. At the same time, between the current connection A and the current pickup A^ in this case, there will be a portion of the poten- tiometer in the gyro assembly A-1 and a portion of the transmitter potentiometer la-Aj, represented as a sum of the four circumfer- ences . Such a length of winding of potentiometer will be placed between current connection A and current pickup B^ (segments A- 2 and ■2a-Bi). Consequently, a potential difference will develop between points A^ and B^. We can reach an analogous conclusion if we consider the path of the current from connection B to pickups A j^ and B^. If the position of the brushes of current differs from the position of connectors A and is not 90° (considering their sections), there will be a relationship to current in pickups rent is channe 1 and the the coo The pot in the with th begin t low spe an equi rents o pickups Ai and B^ B by an angle which the potentiometer / 15H Aj and Bj. This cur-..„ fed to the first of the amplifier, n to the motor of rdination mechanism, entiometer brushes gyro assembly, along e selsyn- transmitter , o rotate at a very ed until there is librium of the cur- n pickups Aj and Bi. Fig. 2.21. Potentiometric Trans- mitter of Position Signal. me Li'd less of the apparent rotation of the gyroscope tion of the Earth and the natural changes in t Th the mas bly con agree w the tra us, the position of ter of the gyro assem- stantly shifts to ith the position of nsmitter card, regard- axis due to the rota- he gyroscope axis. 155 Inasmuch as the agreement of the readings of the selsyns of the transmitter and gyro assembly takes place at an angular veloc- ity which does not exceed 4° per minute, the readings of the gyro assembly cannot show the influence of rapid changes in the posi- tion of the transmitter card, i.e., the mechanism for coordination is a damper which averages out the readings of the compass for an average position of the card. In order that no transmitter errors be transmitted to the gyro assembly when the aircraft is making a turn, the DGMC complex includes a correction switch which automatically shuts off the correction mechanism of the gyro assembly from the compass card when the air- craft is turning. Estimation of the readings of the aircraft's course during turns is made with a purely gyroscopic operation regime of the DGMC. Inasmuch as the apparent diurnal rotation of the gyroscope axis cannot exceed 1° in four minutes of turn, while the turning time of the aircraft at an angle up to 90° as a rule does not exceed 1-3 minutes, no great errors in the compass readings are produced during the turn and the gyromagnetic compass can be used success- fully for turning an aircraft at a desired angle. Agreement of the gyro assembly with the basic course indicator is accomplished by means of a master selsyn (Fig. 2.22). Winding AB rotates inside the housing of the master selsyn, allowing alternating current to flow in the windings of the selsyn 0-1, 0-2, 0-3. Currents which are symmetrical in phase also arise in the windings of the slave selsyn O^-li, 0i-2x, O^-Sj. Hence, the magnetic field of the resultant currents of the slave selsyn will be parallel to the magnetic field of the supply winding AB . Therefore, if winding AjB^ of the slave selsyn occupies a position which is perpendicular to the supply winding AB , the current in it will be equal to zero. /155 If the angle between windings AB and A^B^ differs from a right angle , there will be a current in winding A;^B]^; this current passes through the second chan- nel of the amplifier to a motor which turns winding A^Bi, with an indicator scale showing readings up to a position where Fig. 2.22. Master Selsyn for Trans- AB is perpendicular. mitting Position Signal. The potentiometric and selsyn systems, with amplification of currents and analysis of signals by means of small motors , give very precise agreement 156 of readings and transmit them with high mechanical moments and good damping. This permits us not only to obtain precise and stable readings with the compass, but also to apply an additional stress to the course indicators or the intermediate links. For example, they can be used to set the mechanical compensators for deviation, and to take readings from otber indicators or devices which use course signals. The device for mechanical compensation of the residual devi- ation consists of a circular curved strip with special bends, which operates by means of a lever and pinion to produce an additional turning of the needle on the scale for showing the magnetic course. The adjustment screws are mounted along the edge of the strip, usually at every 15°, thus making it possible to compensate for the resid- ual deviation practically down to zero. However, it is not recommended that residual deviation greater than 2-3° be compensated, if it is possible to get rid of it by a deviation device with a magnetic transmitter, for the following reas ons : (a) Not getting rid of, but compensating for, semicircular deviation leads to considerable changes in it, depending on the magnetic latitude of the locus of the aircraft. /156 (b) When the aircraft is turning and the magnetic correction is switched off while the compass is operating in a regime of gyro- scopic stabilization, the mechanical compensation for deviation (if it is shown on the indicator) causes errors in the course readings in the form of overshooting and lagging, equal to the value of the compensated deviation, thus making it more difficult to turn the aircraft at a given angle . In addition to the mechanical compensator for the residual deviation, the main indicator has a declination scale whose revolu- tion to the value of the magnetic declination of the locus of the aircraft converts the compass readings from magnetic to true. To link it with other devices, the main indicator has both a master and a slave selsyn, whose indications can be transmitted either with the aid of the activating motors or by a direct selsyn connection . In the case of direct selsyn connection, the windings of the selsyn in transmitter AB and the selsyn of the indicator AjBj are connected in parallel with the alternating current source. In this case, the winding A B of the slave selsyn attempts to set itself according to the regulation of the magnetic field, produced by wind:: ings 0;ilj, 0^21, 0x3i, i.e., it automatically assumes the position of the power winding AB of the master selsyn. The direct selsyn connection has a lower sensitivity for the 157 matching of the selsyns and a smaller working moment, so that there is a reduced accuracy of transmission. Hence, it is used for trans- missions where there are no particularly high demands made on accur- acy, e.g., for pilotage course repeaters connected to the main indi- cator. Gy ro i nduct i on Compass In the preceding paragraph, it was mentioned that the distance gyromagnetic compass has considerable advantages over the integrated compass. However, the magnetic transmitter of this compass has a serious shortcoming. The fact is, that the magnetic moment which moves the trans- mitter card to the plane of the magnetic meridian is itself very small, and while it is sufficient for turning the floating card, it is frequently insufficient for overcoming the friction of the brushes on the current pickups, especially in flight at high mag- netic latitudes. Therefore, this transmitter is unstable in oper- ation and frequently goes out of order. To overcome this shortcoming, new types of induction magnetic transmitters have been developed; in addition to having an increased threshold of sensitivity, they do not have the ability to move in the horizontal plane (in the azimuth); consequently there are no- errors due to splashing of the fluid over the sensitive element or obstruction; they are less sensitive to the influence of accelera- tions when the aircraft is yawing, and the size of the transmit- ter is smaller. /157 The operating principle of the induction- type sensitive ele- ment is the dependence of the value of the alternating magnetic induction of the core upon the presence of its constant component, exerted in the core by the horizontal component of the terrestrial magnetism . For example, if the core has a constant component of magnetic induction in the direction of the vector OA (Fig. 2.23, a), then in order to bring it up to complete saturation in this same direc- tion we will require an additional vector AB . The change in indue- OB-OA ^d b) tf h- _J OA + OB Fig. 2.23. Induction Saturation of the Core of the Sensitive Element: (a) Induction Vector Coincides with Saturation Vector; (b) Induction Vector and Saturation Vector are in Opposite Directions. 158 tion in this case is expressed by the difference between the vectors OB-OA. As we see from Figure 2,23, b, when the magnetic induction is brought up to full saturation, the change in induction in the opposite direction will be equal to the sum of the vectors OA + OB . The transmitter of an induction compass has three sensitive elements, each of which is made as follows: Two parallel magnetic cores ■ made of permalloy (a material with a high magnetic permea- bility and a very low value of magnetic hysteresis) have separate primary windings, connected in opposite phase, and a common secon- dary winding around both cores (Fig. 2, 2^1, a). Alternating current /15 ! flows through the primary windings of the cores . Obviously, if the constant component of the magnetic induc- tion of the cores from the horizontal component of the Earth's mag- netic field is zero, the vectors of its change with passage of an alternating current through the winding will be the same in both c'ores , but in opposite directions, and there will be no alternating current in the secondary winding. If the cores have a constant component of magnetic induction, the vector of the change in magnetic induction will be greater in one and smaller in the other; this will produce pulses of alter- nating current as shown in the graph in Figure 2.24, b. The mag- nitude of the current pulses will be proportional to twice the value of the constant component of the magnetic induction of the cores. The sensitive elements in the transmitter are arranged in the form of a triangle and their secondary windings form a sort of master selsyn (Fig. 2.25). The rotating winding of the slave selsyn is connected to the amplifier and mounted in a position perpendicular to the resultant vector of the electromagnetic field of the slave selsyn by means of an activating motor with reduction gearing. The primary winding of this transmitter is mounted in an inter- mediate element between the transmitter and the gyro assembly in a correction mechanism which has a device for mechan compensation of residual deviation and is used as a correction mechanism for the following system. Fig. 2.24. Sensitive Element of Induction Transmitter; ing; (b) Graph of Current. (a) Wind- 159 The induction transmitters for the course are reliable and stable in operation, but their accuracy of operation drops when the transmitter is tilted to a sufficiently greater degree than is the case for magnetic transmitters. At the same time, if the tilting of the transmitter takes place in the plane perpendicular to the magnetic meridian, the vertical component of the magnetic field of the Earth, projected on the plane of the sensitive element, forms a magnetic induction normal to the magnetic meridian; the banking deviation will then be determined by the formula tg & = ■jT sm ^ sm n (2.25) where i is the banking of the transmitter, 9 is the angle between the plane of the magnetic meridian and the banking plane of the transmitter, and Z^H are the vertical and horizontal components of the Earth's field, respectively. For example, with the ratio — = 3 and the angle 9 90' each banking radius of the transmitter will produce an error of approx- imately 3° in the operation of the compass. 2 The ratio 77 = 3 corresponds (e.g.) to the latitude of Moscow /159 and increases rapidly with an approach to the polar regions. There- fore, the banking errors in the induction transmitter can take on very significant values . In order to reduce the errors in the induction transmitter, its sensitive element is mounted on a float mounted in a Cardan support. The body of the transmitter is filled with fluid to reduce the pressure on the axis of the frame of the Cardan suspension (a mixture of ligroin and methylvlnylpyridine oil). The Cardan suspension ensures the horizontal position of the sensitive element during banking and pitching to within 17° . The induction transmitter, like the magnetic one, is mounted aboard the aircraft in a position such that it is exposed to the smallest magnetic field of the air- craft and one which is as constant as possible; a deviation mechanism is mounted on it to record the semi- Fig. 2.25. Diagram Showing Con- nection of Elements in Sensor of Gyroinduction Compass. 160 circular deviation of the transmitter. However, the curvilinear trajectory of flight (although the radius of curvature is very great), in addition to the accelera- tion produced by Coriolis forces, produces a constant tilting of the sensitive element of the transmitter, the deviation from which is transmitted to the main indicator and its repeaters. For example, at the latitude of Moscow and an airspeed of 800 km/hr, the tilting of the sensitive element of the transmitter due to the acceleration of the Coriolis forces will be equal to approx- imately 20', which undergoes deviation equal to 1° in a flight in the northerly and southerly' directions. The gyroscopic induction compass (with the exception of the induction transmitter) is built in a manner similar to that of the distance magnetic compass. Its principal components are the induction transmitter, the gyro assembly and the course indicator. In addition amplifiers rid of res^^„^^ ^ mechanism for rap from the main dition to the principal units, ttiere is a_Lso a. power , correction mechanism with a curved device for gett idual deviation, a connecting chamber, a button with d for rapid coordination, a correction switch, and repeaters ain course indicator. there is also a power supply, for getting /160 a The correction mechanism is the intermediate link between the induction transmitter and the gyro assembly. The connection between the induction transmitter and the correction mechanism is made with a selsyn, while the connection between the correction mechanism and the gyro assembly, the gyro assembly with the main indicator, and the main indicator with the repeaters is made by potentiom- eters . The main indicator also has a curved device for getting rid of errors in the distance transmission of the course indications from the gyro assembly to the indicator at the factory. The correction switch is a two-stage gyroscope which serves for automatically disconnecting the gyro assembly from the correction mechanism; this disconnects the circuit for azimuth correction from the induction transmitter and disconnects the correction of the horizontal position of the axis of the gyroscope rotor when the aircraft is making turns with an angular velocity greater than 36 deg/min . Disconnecting the induction transmitter during turns gets rid of the considerable errors which arise due to the influence of the vertical component of the Earth's magnetic field Z. In order to ensure that the gyroscope correction will not be disconnected in a turbulent atmosphere when the aircraft is bumping and yawing. 161 the correction switch has a delay mechanism which disconnects the correction only after 5-15 sec have elapsed following the moment when the aircraft reaches an angular velocity of 36 deg/min. The course repeaters are simple in design and consist of three- phase magnetoelectric lagometers whose accuracy for determining the course is lower than that of the main indicator. Despite the numerous advantages of distance gyromagnetic and gryoinduction compasses over integrated compasses, they do not com- pletely satisfy the requirements of aircraft navigation, partic- ularly with regard to automation of its processes , since the follow- ing shortcomings of compasses still persist: (a) The dependence of the accuracy with which the course is measured upon the magnetic latitude and the impossibility of using the instrument at high magnetic latitudes. (b) The difficulty of maintaining an orthodromic direction of flight, since the magnetic flight angles which are then obtained vary . (c) The magnetic loxodrome along which a flight can be car- ried out with a constant magnetic flight angle is a complex curve, since it depends on the intersection of meridians and magnetic declin- ations, which limit the length of the straight-line flight segments , /161 along which the flight angle can be assumed constant. (d) Regardless of all the measures which have been taken to get rid of and correct for deviations, as well as the consideration, of magnetic declinations, the accuracy of the measurements of the magnetic course still remain low(within the limits of 2-3°). The majority of these shortcomings can be overcome by using gyroscopic semi compasses with high accuracy, or course systems which make it possible to fly in a regime using highly sensitive gyro- semi compasses (the GSC regime). Details of Deviation Operations on Distance Gyromagnetic and Gy ro i n duct i on Compasses Deviation operations on distance compasses are carried out using the same method as for integrated compasses, with certain changes necessitated by features of the design and mounting of these compasses , In several types of aircraft, the semicircular deviation at the point where the transmitters are mounted can be very low. In these cases, the deviation devices must be removed from the trans- mitters and all forms of deviation are compensated for by a mechan- ical compensator on the main course indicator or on the correction mechanism . 16 2 The compensation for the residual deviation, using a mechan- ical compensator, is carried out on 2H courses: 0, 15, 30, ..., 345° , in which the aircraft is set to the desired courses , and a screw is turned (corresponding to the course of the aircraft) in order to bring the remaining deviation to zero. The graph of the remaining deviation on the main course indicator is not plotted. However, if differences in readings between the main indicator and its repeaters are noticed, it is necessary to plot a graph of the corrections for the readings on the repeaters. After each two intermediate settings of the aircraft on course (at the points 0, 45, 90, 135, 180, 225, 270 and 315°), it is neces- sary to mark the readings of the compass transmitter on the scale of the compass course on the main indicator (for induction trans- mitters, on the scale of the correction mechanism), and use this to determine the coefficients of semicircular deviation B and C. The form shown in Table 2.4 is recommended for convenience in deter- mining these coefficients. The coefficients are calculated according to the formulas: B = 2 8/ Sin MC C = 2 6/ COS MC (=0 where 6. is the compass deviation on individual courses TABLE 2.4. MC ° 8° sinMC 45 0.7 90 1 135 0.7 180 o' 225 -0.7 270 — 1 315 -0.7 RslnMC cos MC 1 0,7 -0.7 — 1 -0,7 0.7 /162 8 cos MC The calculated coefficients must be in the form of tables, attached to the instrument panel along with the main course indi- cator. In addition to the coefficients on the table, it is also necessary to show the place where the deviations were corrected or the horizontal component of the magnetic field of the Earth at the point where the correction was carried out. Since the semicircular deviation, as well as all its other 163 forms, can be made by a mechanical compensator at the magnetic lati- tude of the point where the correction was made. Formula (2.16) for calculating the deviation for other magnetic latitudes assumes the form ' = ^(^-0^'"^-'^t^-')"''- (2.26) Course Sys terns The most complete devices for measuring the course of an air- craft are the course systems. Course systems are combinations or complexes of various course transmitters mounted on the aircraft, with their readings displayed on general indicators . Such trans- mitters include the following: Magnetic induction (MC regime); Astronomical (AC regime); Gyroscopic (GSC regime). In principle, the course system consists of a combination of the design features of a gyroinduction compass, gyrosemicompass and astronomical course transmitter, whose operating principle will be discussed in the chapter devoted to astronomical means of air- craft navigation. The primary feature of the design of the gyroscopic portion of the course system is the presence of a third frame for the gyro- scope with a horizontal axis, coinciding with the longitudinal axis of the aircraft. The purpose of the third frame is to select the Cardan errors in the readings of the gyrosemicompass when the air- / 16 3 craft is turning. The use of this third frame completely excludes Cardan errors from the transverse rolling of the aircraft, since the second frame of the gyroscope (with a master selsyn) will always be in a vert- ical position. The setting of the second frame of the gyroscope in a vert- ical position is accomplished by means of an electrical circuit and a mechanical device for matching it with the so-called gyro- vertical, mounted on aircraft for pilotage purposes. The second feature of course systems is the use (as a rule) of two gyro assemblies, a main one and a standby, which improve the reliability of the system and ensure reciprocal control of the readings . Figure 2.26 shows the control panel and the indicator of the course system. The course system operates on the main indicator in a regime in which the switch for the operating regime is set at the top part of the panel (MC, AC, or GSC). 164 r When switching the course system MC or AC regimes, in order to correct sary to press the button for rapid co the readi the readi After cor returned The for manua the cours The switc the panel switch th potentiom for the r Northern covers at "main" an for the b the main from the GSC regime to the the readings, it is neces- rrelation in order to adjust ngs of the gyro assembly to ngs of these transmitters, relation, the switch is again to the GSC position. pushbutton course control serves 1 setting of the values for e system only in the GSC regime. h on the left-hand side of , marked "N-S", is used to e polarity of the latitudinal eter in order to compensate otation of the Earth in the or Southern Hemisphere. The the bottom of the panel, marked d "standby", cover adjustments alancing potentiometers of and standby gyro assemblies. Methods of Using Course Devices for Purposes of Aircraft Navi- gation The methods of using course equip solving powers of /16I+ ment Fig. 2.26. Control Panel of Course System, Panel of Co if flight, the remainin .nd a given route (air depend upon the resolving powers of the complex of course devices mounted on the aircraft, the presence of other equipment for purposes of aircraft navi- gation, and also on the distance, geograph- ic and meteorological conditions of flight. While the meteorological flight nditions along a given route (path) ange in the course of time and can COiiu-LLj-uiia cixuiig a g±ve change in the course of ^^...^ ^ ary depending on altitude and distance "'""■' ^"' a given type of aircraft „ conditions for _ route) remain constant In discussing the methods of using course devices in flight, the constant conditions listed above can be divided into three groups (1) The aircraft is equipped with an integrated or distance gyromagnetic (induction) compass. Flights are carried out over long or medium distances without significant changes in magnetic latitude. The equipment for constant measurement of the airspeed, drift angle, and automatic calculation of the path are lacking on the aircraft. 165 II (2) The aircraft is fitted with a distance gyromagnetic or gyroinduction compass and a gyrosemicompass or course system of average accuracy. Flights are made over long distances with consid- erable changes in magnetic latitude. There is no equipment for automatic measurement of the drift angle or airspeed, or calculat- ing the flight according to these parameters on board the aircraft. (3) The aircraft is fitted with a course system of high accur- acy, as well as devices for automatically measuring the drift angle, the airspeed, and calculating the path. Flights are made at any geographical latitude and for any distance. Methods of Using Course Devices Under Conditions Inclu- ded in the First Group Under the conditions in the first group, i.e., when flights are being made over short distances in aircraft which have simple navigation equipment, the following methods are used to prepare the calculated data and use the course devices in flight. In preparing for a flight, the route of the flight to be made is entered on a flight chart. If the flight chart is one which is in an international or diagonal cylindrical projection, the straight-line portions of the flight between the turning points along the route are plotted as straight lines by means of a ruler. When using charts which are plotted with an isogonal cylindrical projection (Mercator), the straight-line portions of a flight whi ch /165 is very long are plotted as a curved line on the basis of the inter- mediate points along the orthodrome , calculated by analytical means. If the indicated correction is more than 3° in the straight- line portion of the flight, this segment is divided into two, three or more parts and the flight path angle is determined for each. This is usually not done by simple division of a straight line into equal parts, but by selecting characteristic orientation points along the section of the route, the flight between which can be made at the constant flight path angle. If we consider the low accuracy of the indications of the 166 magnetic compasses in a relatively short length of flight segment for a flight with a given flight path angle, the latter are deter- mined not by analytical means, but by simple measurement of the direction of the segment on the chart by means of a protractor. Measurement of the loxodromic flight path angle can be made relative to the meridian which intersects the segment at a point which is closest to its center, considering the magnetic declin- ation of this point. However, to increase the accuracy of the meas- urements, it is recommended that it be done at two points, at the beginning and end of the segment, considering the average declin- ation of these points. Obviously, in the first case the magnetic flight angle of the segment will be MFA = a m Mn,' while in the second case MFA = 2 where a b ' are the azimuths of the orthodrome at the begin- ning, the middle, and end, respectively. An advantage of the second method is the double measurement of the angles and the averaging of the declinations, since the accur- acy of two measurements and the averaging of their result is always / 166 higher than the accuracy of a single measurement. For the first group of conditions, it is possible to have some simplified preparation for the course equipment of the aircraft for the flight. Since the flights are made with relatively low measurements of magnetic latitude, there is no need to determine the coefficients of semicircular deviation B and C or to consider their changes during the flight. If the deviation is compensated by a mechanical compensator, it is assumed to be zero during the flight. In considering the residual deviation, a value is assigned to it as shown on the graph. During the flight, the course of the aircraft is checked so that its value together with the drift angle of the aircraft will be equal to a given magnetic flight path angle of the flight seg- ment . MFA. MC -t- US MFA g' On the other hand, since the magnetic course of the aircraft is equal to the compass course, it is necessary to add the compass deviation : 167 MFA^ = CC + Ac + US = MFAg. Froblems 1. The direction of a flight segment measured along the aver- age meridian is equal to 48° ; the magnetic declination in the middle of the segment is +7°. Determine the given magnetic flight path angle of the segment. Answer: mfa = 41° . 2. The direction of a flight segment measured along the in- itial meridian is equal to 136°, 132° at the final meridian, with an initial magnetic declination of +7° and a final one of +5°. Deter- mine the MFA„. Answer: MFA = 128°. 3. The given magnetic path flight angle of a segment is equal to 84°, the drift angle was equal to -6°, the deviation of the mag- netic compass is +4°. Determine the required compass course for following the flight lines . Answer: cc = 86° . 4. The compass course of an aircraft is equal to 54°, the compass deviation is +3°, the drift angle is +6°. Determine the actual flight path angle. Answer: MFA^ = 63° . Methods of Using Course Vevioes Under Cond-itions of the Second Group When flights are being made over long distances using distance gyromagnetic and gyrosemi compass es or course systems, but without any automatic course calculation, the use of course instruments in flight and preparation of charts for a flight is accomplished by devices which are somewhat different from those which are recom- mended for the conditions of the first group. The most important of these devices is the plotting of the orthodromic course along the straight- line segments of the flight with a gyrosemicompass or a course system in the "GSC" regime, with periodic correction of the gyroscope course by means of a magnetic or astronomic transmitter. As a rule, in flights over long distances, the flight chart /167 is one with a scale of 1 : 2 , 000 , 000 on the international projection. If a straight line within the limits of one sheet of this map, with distances ip to 1200-1500 km, can be assumed with insignificant error to be an orthodrome, then when two or more sheets are combined 168 and the route does not run along a meridian or when sheets of this chart are used separately at great distances, the orthodrome must be located along points which are determined by calculation. When splicing two adjacent sheets along the meridian, the orthodrome has a significant break in it, and in this case (when it crosses the adjacent sheets) a straight line cannot be taken as the ortho- drome . On the charts of all other projections, except the central polar and special route maps in a diagonal, cylindrical projection, when the line of the tangent (cross-sectional strip) of the cylin- der coincides with the axis of the route, the orthodrome is calcu- lated analytically and plotted on the chart according to the calcu- lated intermediate points . The distances for the sections of the orthodrome are also determined by analytical means. The orthodromic flight path angles of the route segments under these conditions are measured or calculated analytically relative to the initial meridian of each flight segment. If the straight- line segments of the flight have a very short length, the flight path angles calculated from the initial meridians of the segments can be applied to the system relative to the selected reference meridian (Fig. 2.27) according to the following formula: OFA TFA + 6 = TFA + (A, ref-^init-' sxn ■•m 5 where 6 is the angle of convergence between the reference and initial meridians of the segment. Since the condition for the second group assumes flights over long distances with considerable changes in the magnetic latitudes, the preparation of the magnetic compasses must be made with a consid- eration of determination of the changes in the semicircular devi- ation during the flight. second ual de not pi sary t and C: ourse devices intended for flights under conditions of the group have devices for mechanical compensation of the resid- viation. Therefore, the graph of the deviation for them is otted. However, in getting rid of the deviation, it is neces- o determine and write down the coefficients of deviation B B i=0 C = 2 h cos 7/ 1 = It is then necessary to write down the intensity of the horizon- /16! tal component of the Earth's magnetic field at the point where the deviations were corrected. 169 To calculate the changes in the semicircular deviation during flight, the corrections for the magnetic course at different seg- ments of the route must be determined when preparing for a flight. They are determined for a number of points along the flight path, on the basis of the magnetic flight angles of the route at these points with a frequency such that the difference between two adja- cent corrections along a straight line path does not exceed 1° and after each turning point on the route. In fact, the changes in the semi- circular deviation at correspond- ing points along the route will differ only slightly from the calculated cor- rections, since the course which is followed will be prepared with a consid- eration of the drift angle of the air- craft. However, the errors which arise in this process will be small and can be disregarded. During the flight, the gyrosemi- compass or the course system is cor- rected for the magnetic or astronomical transmitter when flying along the refer- ence meridians or the turning points of the route (TPR). If the correc- tion is made on the basis of the mag- netic transmitter, then the main indicator will have the required correction entered on its dial. This correction is equal to the sum of the magnetic declination and the change in the semicircular deviation along the magnetic latitude . Fig. 2.27. Calculation of Flight Path Angles from Reference Meridian, For correction, the course system is switched to the "MC" re- gime and the button is pushed to match the readings . The system operates for a period of 1-2 min in the slow coordination regime and is then switched to the "GSC" regime . In this manner, the systems are corrected for the astronom- ical transmitter. Having determined the latter on the basis of the coordinates of a star and the locus of the aircraft, the system is switched to the "AC" regime, the coordination is carried out, and then switched back to the "GSC" regime . This means that at the turning points of the route, no corrections are required on the scale of the declinations. The correction of the gyrocompass is made in the same manner, except that the course is set on the gyrosemicompass not by com- paring the readings of the transmitters , but by manual setting on the basis of the readings of the magnetic or astronomical trans- mitters . After correction, the flight is carried out with an orthodromic 170 course up to the next turning point of the route or reference merid- ian . When it is necessary to make a correction for the orthodrom- /169 ic course between two reference meridians, the correction is set on the main indicator and is equal to: for the magnetic transmitter, A = A + (X ^-X,,^) sin <}) ; M ref MC m for the astronomical transmitter ^ = ^^ref-^MC^ ^^^ *>"• Then the readings are matched in the manner described above. Prob lems 1. The east longitude of the reference meridian is 40°, the north latitude of the reference point is 52°. The coordinates of the setting point of the route are: longitude 43°, latitude 54°. The true flight path angle of the segment at the starting point is 67°. Determine the orthodromic flight path angle calculated from the reference meridian. rse Answer: +3.5°. 3. The east longitude of the reference meridian is equal to 70°, the north latitude of the reference point is 58°. The air- craft is located at the point X - 76° , cj) = 60° ; the magnetic declin- ation of the location of the aircraft is equal to +11°, while the correction for the change in the semicircular deviation Ag^ = +2°. Determine the correction for the readings of the magnetic compass for correction of the orthodromic course. Answer : + 8' 171 Methods of Using Course Devices Under the Conditions of the Third Group The third group of conditions for using course devices refers to flights in aircraft which are fitted with precise course sys- tems, apparatus for automatic measurement of the airspeed of the aircraft, the drift angle, and automatic calculation of the flight path of the aircraft. The aircraft with corr located s conditions of the third group assume a prolonged autonomic navigation with no visibility of the ground or over water, section of the aircraft coordinates only at individual points located significant distances apart. This places particularly strict requirements on the accuracy of the plotting of the orthodrome on the charts, the determination of the flight path angles, and the retention of systems for calculating the aircraft course, since the course is a basis for the automatic calculation of the flight in terms of direction. From the theoretical standpoint, a more precise and conven- /170 ient form for using the course devices under conditions of the third group is the following: In preparing the flight charts for each orthodrome section of the flight between the turning points on the route, regardless of their length, we determine the conditional shift in the longitude (X ), i.e., the difference between the longitude calculated from the point where the given orthodrome intersects the Equator (Aq) and the geographical longitude (A): ^s ~ ^0 ~ ^ • Here, the orthodromic longitude Ag is determined for the start- ing point of each segment by the formula ctg \ = tg<P2 ctgVi cosec AX — ctg AA. After determining the change in the longitude, the longitude of any point along the route can be converted easily to the ortho- dromic system, thus making it possible to determine relatively easily all of the required elements of the orthodrome for these points: (a) The azimuth of the point of intersection of the orthodrome with the Equator (ag) tgao = Sin \ »g"Pi (b) The coordinates of intert ermediate points for plotting the orthodrome on the map: 172 tg<p( sin Xo, fg«o (c) The initial. drome intermediate, and final azimuths of the ortho- «ga/ = Sin <fi (d) The distance to any point along the orthodrome iS^) from the point of its intersection with the Equator cos S . cos A , cos (|)^ (e) The distance between any two points along the orthodrome as a difference in the distance from the point of intersection with the Equator 1-2 = So ~ S 1 In this case, the path angle of the first orthodromic flight /171 segment is considered to be equal to the azimuth of this segment relative to the meridian of the airport from which the aircraft took off. The path angles of all subsequent segments are obtained by combining the orthodromic flight angle (OFA) of the previous section with the turn angle ( TA ) of the line of flight at the turning points along the route (Fig. 2.28): OFA- 1 ' OFA„ = ai + TAi + TA2...TA„_i The turn angles along the line of flight are found as the dif- ferences of the azimuths of the orthodrome, intersecting at the turning points of the route, determined according to the formula tga,= tgX, 0/ sin fj Obviously, the latitude of the turning points will be common for the two orthodromes; for one it will be final, for the other 173 it will be initial. As far as the longitude is concerned, it is determined on the basis of the geographical longitude of the turning point of the route, considering the shift in longitude of the prev- ious and subsequent segments. When flying above a continent , the best method of correction for the orthodromic course under conditions of the third group is to introduce corrections into the course as a result of calcu- lations of the aircraft path. )TA OFAi=ab'+/IX 1 Fig. 2.28. System for Calcu- lating Path Angles by Combi- ning the Turn Angles along the Flight Path. For example, if the readings of the calculating devices on board the aircraft at both the initial and final points indicate that it is on the line of flight, but has undergone a lateral devi- ation AZ during the flight, then obvious ly tgAT = -, where Ay equals the error in the readings of the orthodromic course, and S is the length of the control section of the flight. The sign of the correction to the compass reading coincides with the sign of AZ. With positive values of AZ, (a shift from the line of the de- sired flight to the right), the readings of the compass will be reduced and the correction must be positive; in the case of devi- ation to the left, the correction must be made with a minus sign. of more when determination of the correctness /172 -^^^ "ath in terms of direction is must be made the d In flights over water, when de the calculation of the aircraft path in terms of direction is e difficult, the correction of the gyroscope course must be m by astronomical methods. This means that the difference between the orthodromic and true courses at any point will be equal to t difference between the orthodromic path angle of the segment and the running azimuth of the orthodrome at a given point; OC - TC = OFA - a. If the positive difference of the courses turns out to be greater (or if it is negative, turns out to be smaller) than the differ- ence between the path angles , the reading for the orthodromic course will be increased and it will be necessary to reduce it manually by the course detector. When the readings of the orthodromic course are low, it must be increased. 174 In this manner, but with reduced accuracy, the orthodromic course can be corrected magnetically: OC (MC+ ..) = OFA M For the conditions of the third group, the preparation of the magnetic compasses must be carried out according to the rules given above for the conditions of the second group. However, the use of magnetic transmitters for correction of orthodromic course during flight is limited to cases when the readings of the orthodromic course cannot be checked on the basis of the results of calcula- tions of the path or by means of astronomical course transmitters. The meteorological conditions of a planned flight, especially over long distances, call for careful preparation of all course equipment on the plane, since it may become necessary to use devices for measuring the courses which belong to all three groups of condi- tions . 3. Barometric Altimeters The principal method of measuring flight altitude for naviga- tional purposes is the barometric method. It is based on the meas- urement of the atmospheric pressure at the flight level of the air- craft . For special purposes, such as aerial photography or aerial geodesic studies, as well as for signaling dangerous approaches to the local relief when coming in for a landing under difficult meteorological conditions, electronic devices for measuring alti- tude are used, which are more accurate in principle than the baro- metric method. However, they are not widely employed for navigational purposes because they are used only for measuring the true flight altitude. On the basis of the barometric method of measuring alti- tude, it is the law of change of atmospheric pressure with increase in height which means that the calibration of the altimeter dial must be made on the basis of the conditions of the international / 17 3 standard atmosphere. The conditions of the standard atmosphere are as follows: (a) The pressure at sea level is equal to 750 mm Hg , or 1.0333 kg/cm^ . (b) The air temperature at sea level is +15° C with a lin- ear decrease for flight altitudes up to 11,000 m of 6.5° for each 1000 m of altitude. Beginning at 11,000 m, the air temperature is considered constant and equal to -56.5°. To understand the operating principle of the barometric al- timeter, let us recall the familiar equations from physics which 175 describe this state of gases and the conditions of their change. Thus, according to the Boy le-Mariotte law, with isothermal compression (i.e., fixed temperature), the pressure of a gas changes in inverse proportion to its volume so that the product of the volume times the pressure remains constant: pv = const , where p is the pressure of the gas and u is the volume of the gas at temperature t. According to the Gay-Lussac law, heating a gas by 1° C at con- stant pressure causes the gas to expand to 1/273.1 of the volume which it occupied at zero temperature: V — Vo — Vo 273.1 i. where Vq is the volume at zero temperature and the same pressure. By combining the Boy le-Mariotte and Gay-Lussac laws, we obtain the state equation of a gas: ^''^ 2^ (' + 273.1). This equation is known as the Clapeyron equation. The temper- ature (t +273.1° C) is called the absolute temperature iT) , i.e., calculated relative to absolute zero (-273.1° C) -'- , and the constant p y value of — 2— 2_ is called the qas constant. 273.1 A gram molecule of any gas (gram mole, or simply mole), i.e., the number of grams of a gas which is equal to its molecular weight, always occupies exactly the same volume (22.1+1 liters) at zero temper- ature and a pressure of 1 atm. The gas constant for one mole of gas is called the universal gas constant (i?): PoVo /IT R = 273. 1 * With P = 1 atm, V = 22.'+l liters. The Clapeyron equation for one mole of gas in this case as- sumes the form -'- This value is usually assumed to ba approximately 273° in cal- culations , 176 pv = RT . The numerical value of the universal gas constant is 1.033-22410 R = 273.1 ^84,8 kg/cm( degrees/mole ) In technical calculations, the weight of the gas is usually expressed in kilograms. Therefore, we do not use the universal gas constant but rather the characteristic gas constant M where M is the number of grams of gas per mole, or its molecular weight . Then pv = BT. The constant B for air is 29.27 m/degree. By using the gas constant, we can find the weight density of air (y) at a given pressure p and absolute temperature T. y BT' Let us define an area on the Earth's surface measuring 1 cm^ , and erect a vertical column on it which extends upward to the limits of the Earth's atmosphere (Fig. 2.29). Obviously, the drop in pressure with increased altitude to the distance i\H at a certain height will be equal to: 1 ^j. or 'p ~ BT (2.27) By using Equation (2.27) and the altitude temperature gradient , /175 we obtain the so-called barometric formula /'// = /'o^l-^'>/Ki; (2.28) where Tq is the temperature on the ground under standard conditions equal to 281° K, and tgp is the vertical temperature gradient. 177 Formula (2.28) is obtained from Formula (2.27), switching to infinitely small values: d£_ _dH p ~ BT (2.27a) Integrating (2.27a) and keeping in mind that ffj^ we obtain: 0~ gr B, or 'H H f rfp 1_ P dH p, s-^ . Ph 1 , In — = ^r— Ig Po Bt 'gr- gr 1 1— £-//) gr -^ETl Fig. 2.29. Column of Air on Earth's Surface. Solving Equation (2.28) for H, we obtain the standard metric formula for the troposphere: the hypso- 'grL VPo/ J (2 .29 ) Substituting into Formula (2.29) the numerical values of Tq, t gr-p and B, we obtain: // = 44 308 1 — \Po) (2.30) We can use Formula (2.30) to calculate the hypsometric tables which relate the flight altitude up to 11,000 m to the atmospheric pressure; these tables are used to adjust and correct altimeters. Under the conditions of a standard atmosphere, the air tem- perature at altitudes greater than 11,000 m is considered to be constant, so that the barometric formula for these altitudes can be written as follows : 1^^ I //—1 1000 Pn BT„ (2.31) We obtain Formula (2.31) by integrating Equation (2.27) for /176 11,000 m and consider T^ equal to Tui 178 III nil ■■■ ■ II mil Pa 11 or Pa In — = ■ Pn BTn. //— llflOO BTn Solving Equation (2.31) for H, we obtain the standard state formula of the hypsometric table (Table 2.5) for altitudes greater than 11,000 m. H=nOOQ + BTi ' Ph (2.32) H. M —500 500 1000 1500 2000 2500 3000 3500 4000 4 500 5000 5500 6000 6 500 7 000 7 500 8000 8 500 9 000 9 500 lOOOO TABLE 2.5. Ph. T„. °K ' a, m/sec H. M Ph> T„. °K a, m/sec 806 .^2 291.25 342.1 10 500 183.40 219,25 297.2 760,2 228.00 340.2 11000 169.60 216.50 295.0 716.0 284.75 338.3 I200O 144.87 216.50 295.0 674,1 281.50 336.4 13000 123.72 216.50 295.0 634.2 278.25 334.4 14000 105.67 216.50 295,0 596.2 275.00 332,5 15000 90.24 216.50 295.0 560.1 271.75 330.5 16000 77,07 216.50 295,0 525.8 268.50 328.5 17000 65.82 216.50 295.0 493,2 265.25 326.5 18000 56,21 216,50 295,0 462.2 262.00 324.5 19 000 48.01 216.50 295.0 432.9 258.75 322.5 20 000 41.00 216.50 295.0 405.1 255.50 320,5 21000 35.02 216.50 295.0 378.7 252,25 318,4 22000 29,90 216.50 295.0 353,8 249.00 316.3 23 000 25.54 216.50 295,0 330.2 245,75 314.3 24 000 21.81 216.50 295.0 307.8 242.50 312.2 25 000 18.63 216.50 295,0 286.8 239,25 310.1 26 000 15.91 216.50 295.0 266.9 236,00 308.0 27 000 13,59 216.50 295.0 248.1 232.75 305.9 28 000 11.60 216.50 295.0 230.5 229.50 303.7 29 000 9.91 216.50 295.0 213.8 226.25 301.6 30 000 8.46 216.50 295.0 198.2 223.00 299.4 Note: The table for adjusting and correcting the barometric altimeters is given in abbreviated form. The value a represents the speed of sound at flight altitude under standard conditions, given in the fourth column of the table. Substituting the value of B and T to the log ten (In N= 2.30259 IgN), this .Pn //=11000+ 14600 Ig^ Ph = 216.5°, and shifting formula assumes the form: (2.33) 179 i I Formulas (2.30) and (2.33), suitable for compiling hypsomet- /111 ric tables and calibrating altimeters, are not completely suitable for calculating the methodological errors in the altimeter, related to a failure of the actual air temperature at heights from zero to the flight altitude of the aircraft to agree with the conditions of the standard atmosphere. Since the accuracy of altitude measurement is affected by the air temperature not only at the flight altitude but at all inter- mediate layers from the one on the ground up to that at the flight altitude, it is better to use the formula which relates the flight altitude not to the temperature gradient, but to the average temper- ature of the column of air which we have selected, and to use this to calculate a hypsometric table for adjusting and correcting baro- metric altimeters. This formula has the form: ^=^W"5- (2.34) Formula (2.34) is obtained by integrating Equation (2.27a) at a constant average temperature: "h h dp 1 Sf^'SK^"' p> whence av p^ If we consider that ^v- = 273 + (^,= 273(1 +2^j]. and the value B = 29.27, by using the from natural logarithms to the log 10 form: coefficient Formula ( 2 for transition 34) assumes the ff =. 18 400 Il + -^]l„l2.. [ ^273.1) ^Ph This formula is known as the Laplace formula. Description of a Barometric Altimeter The sensitive element in the barometric altimeter is a cor- rugated manometric box has two _ one of which is a oj.cj.»= c j.=;.i.= i^ .- j-w the barometric altimeter is a cor- metric (aneroid) box 1 (Fig. 2.30), made of brass. The rigid points (on the top and bottom corrugated surfaces), is fixed or tightly fastened to the casing of the appar- the other is movable. :>ne or which is rixed or tightly itus , while the other is movable 180 In principle, the aneroid box can be either evacuated or filled /17 i with a gas . Usually, the space within the box is filled with a gas to a pressure such that when the box is heated, the thermal losses of its elastic properties will be roughly compensated by an increase in gas pressure within the box when it is heated. The casing of the altimeter is hermetically sealed and con- nected by a nipple to a sensor of the atmospheric (static) pres- sure . Fig.. 2.30. Schematic Diagram of Barometric Altimeter . When the aircraft is located at sea level, the aneroid box is compressed to the maximum degree, since the atmospheric pres- sure acting on it has a maximum value. With a gain in altitude, the atmospheric pressure in the cham- ber decreases and the aneroid box expands due to its elastic prop- erties, shifting its movable center (with bimetallic shaft 2) up- ward . As it moves, the center displaces rod 3, which in turn acts through a lever U to convey a rotary motion to shaft 5. Shaft 5 carries a toothed sector 6 with a counterweight, fitted with a cog wheel 7, which transmits the movement to the pointer through another gear. Thus, the motion of the center of the box is used to indicate the flight altitude on the scale of the instrument. In addition to the parts listed above, the kinematic portion of the instrument includes elements intended for regulating the 181 11 instrument and adjusting the backlash in the transmission mechan- ism . 1. Zero-point bimetallic compensator. This device is intended /179 for compensating the temperature changes in the elastic properties of the box for zero altitude. If the atmospheric pressure in the casing of the instrument is set to zero altitude, but the temper- ature of the box increases, the loss of elastic properties of the material in the box creates additional compression, causing the indicator needle to shift from the zero altitude reading. The bi- metallic strip bends as the temperature changes, due to different coefficients of linear expansion for the two materials of which it is made. By rotating the strip in its socket, it is possible to set it in a position such that the deflection in the direction of the shift of the center of the box will exactly correspond to the additional travel of this center, but in the opposite direc- tion. Then rod 3 remains in place and the indicator needle will not move from the zero position. 2. The regulating mechanism of the device. This consists of strip 4 and an adjustment screw. Turning the screw pushes the strip away from rod 5, changing the arm of the lever. This is used to regulate the angular veloc- ity of rotation of the shaft, i.e., the transmission ratio of the apparatus. The transmission ratio of the rotation of the shaft is set so that the readings of the needle correspond to the atmo- spheric pressure in the casing of the apparatus. 3. High temperature compensator. When the elastic properties of the box change due to the effect of temperature, this not only causes an additional compression at zero altitude but also changes the amount by which its center moves with a change in altitude. For compensation of this error, strip ^■ is of bimetallic construc- tion. When the instrument is heated, and the travel of the center of the box increases, the end of the strip bends away from the shaft, thus reducing the transmission ratio for rotating shaft 5 and compen- sating for the increase in sensitivity of the box. It is important not to confuse the instrumental temperature errors of the instrument, which are compensated by the zero and altitude bimetallic compensators, with the methodological temper- ature errors in the altimeter. The instrumental errors are related to the temperature in the casing of the instrument, acting on the properties of the mater- ial from which the sensitive element is made, and can be overcome by compensators . The methodological errors which are related to the nature of the changes in pressure with flight altitude can only be corrected by special formulas. The building of a compensator for methodo- 182 logical errors is impossible, since in the general case the temper- ature of the casing is not equal to the average air temperature from zero altitude up to the flight altitude of the aircraft. In order to increase the accuracy of the altitude readings , /180 altimeters are made with two pointers . This means that the aneroid box is made double, increasing the travel of the movable center by a factor of two. Between the toothed sector and the axis of the pointer, there are additional gears which increase the trans- mission ratio of the mechanism several times. The main pointer of the instrument makes several revolutions; the number of revolu- tions of the pointer is equal to the change in altitude in thou- sands of meters. In addition, there is a pressure scale 8 for setting the al- timeter readings relative to a desired level. The altimeter mechanism, along with the axis of the main pointer, is rotated within the housing by means of a rack and pinion 9, consist- ing of a driving gear 10 and a driven gear 11. Thus, the main pointer of the instrument can be set to any division on the scale. Simultaneously, by means of driving gear 10, the pressure scale 8 is set in motion, which can be used in conjunction with the main scale to determine the pressure at the level at which the flight altitude is calculated. In the VD-10 and VD-20 altimeters, a movable ring is mounted around the main scale; it is rotated by means of a rack and pinion and driving gear 10 at an angular velocity equal to the rate of turn of the mechanism. It is used for shifting a movable index along the circular scale of the instrument, and can be set to the barometric altitude of the airport where the landing is to be made. This serves the same purpose as the pressure scale. However, the latter can only be used over a range of pressures from 670 to 790 mm Hg , while the movable index can be set to any airport altitude. In cases when pressure scales are not sufficient for airports located at high altitudes, the pressure at the level of the air- port is not measured aboard the aircraft, but rather the baromet- ric altitude of the aircraft is used for setting the movable index of the altimeter. Errors in Measuring Altitude with a Barometric Altimeter The errors in measuring the flight altitude with barometric altimeters can be divided into instrumental and methodological errors: Instrumental errors. These are related to incorrect adjustment of the altimeter, friction (wear) in the transmission mechanism, as well as temperature effects on the material of the sensitive element. The errors from so-called hysteresis are particularly 183 important, i.e., the residual deformation of the sensitive box with changes in flight altitude of the aircraft over wide limits. In addition, instrumental errors include errors in sensing /181 the static pressure, related to dynamic flight of the aircraft. Methodological errors. In the barometric method of measur- ing altitude, these include errors in correspondence of the initial atmospheric pressure, the pressure along the flight route, and the average air temperature with the calculated data. Under flight conditions encountered in civil aircraft, method- ological errors in measuring altitude in approaching aircraft are extremely rare, so that these errors do not disturb the mutual posi- tion of the aircraft and are not taken into account. However, they do have significant value in determining the safe flight altitude above the relief, as well as in making special flights (for pur- poses of aerial photography, e.g.). In practice, the baric stage at low flight altitudes (the dif- ference in altitude which corresponds to a drop in pressure of 1 mm Hg) is considered roughly equal to 11 m. However, at flight altitudes of 20,000 m, the baric stage is equal to 155 m, i.e., 14 times greater than on the ground. The increase in the baric stage with flight altitude, as well as the errors in measuring static pressure due to aerodynamic proc- esses, complicate a precise measurement of the barometric altitude at great altitudes and high speeds . In a flight according to a table of corrections, it is rela- tively easy to compensate for instrumental errors in the apparatus, related only to its regulation. Consideration of all other instru- mental errors presents greater difficulty, so that all measures are usually taken to reduce them to a minimum by carefully prepar- ing the apparatus, selecting the point of calibration, and design- ing the static pressure sensor. Methodological errors in altimeters are estimated by deter- mining the true altitude of the aircraft above the relief for special purposes and in calculating safe flight altitudes above the relief. Changes in atmospheric pressure along the flight route, relative to sea level, are calculated in baric stages, so that the lowest flight altitude oscillates as follows: Aff = Ap-11. For example , if the pressure measured at sea level at the point where the altitude is measured differs from 760 mm Hg to 15 mm, the methodological error in measuring the altitude from the level of 760 mm will be 15-11 = 165 m. 184- Hence, if the corrected pressure is greater than 760 mm Hg , i.e., equal to 775 mm in our example, the readings of the altim- /182 eter will be reduced and the correction will have to carry a plus sign, while if the corrected pressure is lower than the calculated pressure, it will have a minus sign. Methodological errors in the altimeter, which arise due to a failure of the actual mean air temperature to coincide with the ational ^ w^ w„ -_ (3_ . _„ „„__... ---ig from the fact that the instrument indicates a flight altitude on th a rai±ure or rne acTuax mean air xemperaxure to coinciae wiT:n calculated temperature, are accounted for by means of a navig slide rule, a description of which is given below. Proceedin the fact that the instrument indicates a flight altitude on the basis of the calculated mean temperature of the air, and the cor altitude must be determined on the basis of the actual altitude, the equation reads as follows: rrected Lows H. _^ = BT In— i^ ; mst av . c . p Po H = BT In , corr av . a . p ' where T ^ is the average calculated temperature and T ^.^ ^ ^ ^ is the average actual temperature . Whence Therefore , H H inst T corr T av . a , av. c . 7-0 +r„ av . a . ■ Ig H corr = 'g — :^ — + ig 7^ (2.35) av . c . where Tq and Tjj are the temperatures on the ground and at flight altitude, respectively. By using Formula (2.35), we can calculate the scales of the navigational slide rule NL-10 for making corrections in the readings of altimeters for air temperature up to altitudes of 12,000 m. For altitudes above 12,000 m, the corrected altitude is found by the formula 'H H - 11,000 = -sf corr T H (H . j_ - 11,000) , mst where Tjj and Tjj are the actual and calculated temperatures at a c the altitude . 185 The navigational slide rule for these altitudes is also pro- vided with logarithmic scales according to the formula lg(H -11,000) = lgT„ + Ig corr ° H ° a H. ^-11,000 mst ' 216.5° 4. Airspeed Indicators (2.36) /183 The flight of an aircraft takes place in the medium of air, so that a simplest and easiest method from the technical standpoint for measuring airspeed would be to measure the aerodynamic pres- sure or so-called velocity head of the incident airflow. For purposes of aircraft navigation, it is better to measure the speed of the aircraft relative to the surface of the ground, since the air mass practically always has its own movement rela- tive to the latter. At the present time, there are radial and iner- tial methods of measuring the speed relative to the ground, but the measurement of airspeed does not lose its significance even in the presence of such equipment. The fact is that the stability and maneuverability of an air- craft depends on the airspeed. In addition, the operational regime of the motors on the aircraft and the fuel consumption depend on the airspeed. The operating principle of airspeed indicators is based on a measurement of the aerodynamic pressure of the incident airflow. The relationship between the rate of motion of a liquid or gas and its dynamic and static pressure was first established by the St. Petersburg Academician Daniel Bernoulli (1738), working with incompressible liquids or gases (Fig. 2.31). According to the principle of inseparability of flow, the prod- uct of the speed of an air current (7) multiplied by the cross sec- tional area of a tube iS) must be uniform everywhere within its cross section. Consequently, in a narrow part of the tube, the speed of the flow must be greater than in a wide section. In the general case, if the tube is not horizontal, a mass of gas m enters the tube during a time At which introduces an energy consisting of three components: the potential energy of the gas mgh the kinetic energy rnVJ and the work of influx into the tube 186 where g is the acceleration due to the Earth's gravity, h is the difference in the gas levels , and p is the gas pressure inside the tube. These components determine the energy of the gas flowing out / 18^■ of the tube. Therefore — — +:PvSxV^U + mghx ■■ mVi + P^z^t^f + "*S'^2- The product SVht is the volume of fluid flowing through the cross section of the tube in a time At. Therefore, dividing the mass into the volume gives us the density (p), which is &^ = Fig. 2.31. Flow in a Tube with Varying Cross Section . If the tube through which the current is flowing is horizon- tal, hi = h2, therefore P^? + Pi = PVl + P2. (2.37) i.e. , the sum of the dynamic and static pressures at any point in the tube remains constant, since the dynamic component is propor- tional to the gas density (fluid density) and the square of the speed of flow . For adiabatic compression, i.e., when the process takes place with compression of the gas (air) without exchange of heat energy with the surrounding medium, which almost always can be considered valid for high speed events, this equation takes the form: ^^4-^+^, + ^, = g. + ^ + ^,4-^,. (2 .38) where y is the unit weight (weight density) of the gas, V is the internal (thermal) energy of the gas, and E is the potential energy of the gas . 187 Therefore, a change in the rate of airflow during flight due to the flow being retarded is usually negligible; the component E can be considered constant and may be omitted from the equation. Then each of the remaining terms of the equation, if we multiply them by mg , will characterize the component energy included in a unit mass of gas flow: V'^/2g equals the kinetic energy of the flow (for a unit mass m7^/2), p/y is the energy of the pressure, and U is the thermal energy. For measurements of airspeed, we can use sensors which allow /185 us to separate the dynamic air pressure from the static pressure. Figure 2.32 shows the operation of an air pressure sensor (Pitot t ub e ) . In the cross section of the airflow, the speed Vi will cor- respond to the airspeed, and the pressure pi will correspond to the static pressure of the air at flight altitude . .^^ ^ /Z_ /' ^p. 3^p total st Fig. 2.32. Air-Pressure Sensor (Pilot Tube). (1) Static Pressure Pst; (2) Total Pressure Ptotal' Within the limits of the opening in the sensor for total pres- sure, the rate of flow will be equal to zero (the critical current or current of complete braking). Obviously, at this point the pressure p2 will correspond to the total pressure (the velocity head plus the static pressure), and Equation (2.38) acquires the following form for this case: 2^ 11 total ( 2. 39) Let us consider that for airspeeds up to M-OO km/hr the com- pression of the air can be disregarded, i.e., the values y and U are constants. Then Equation (2.39) assumes the form: PtotalPst Y (2.40) E where y^^ is the unit weight of gas at a given altitude. Since y^ = Pn? (where p„ is the mass density), the difference 188 between the total and static pressures (velocity head) will be equal to p7^ ^total ~ ^st whence V = V 2(Ptotal-Pst) (2.41) This drop causes movement of the top of the box, which can transmit its movement by means of a system of gears similar to the mechanism in a single -pointer altimeter, eventually moving a pointer on an axis to show the airspeed on a scale which is graduated in kilometers per hr. Formula (2.'+l) can be used to describe airspeed indicators for low speeds, such as the US-350. If we introduce the weight density to this formula in the form whence st (2.H2) It is clear from this formula that in order to determine the true airspeed, it is necessary to know not only the value of the velocity head, but also the atmospheric pressure and the temper- ature of the air at flight altitude . The airspeed, which is measured only on the basis of the veloc- ity head, is called the aerodynamic or indicated speed. In view of the fact that calibration of the speed indicator is made for flight conditions at sea level at standard temperature and air pres- sure, during flight under these conditions the indicated speed will be equal to the true airspeed. Under other conditions, however, the indicated speed must be converted to the true airspeed. At high altitudes and speeds, the difference between the air- 189 speed and the indicated speed becomes so significant that it becomes difficult to use the latter for navigational purposes. In addi- tion, for airspeeds above 400 km/hr , it becomes necessary to take the compression of the air into account as well. Therefore, for aircraft operating at high altitudes and speeds, a combined speed indicator "CSX" has been developed, which measures both the indi- cated and true airspeed. In terms of its design, this indicator differs from the usual speed indicators in that the speed is measured in two ways: (a) The first method consists of the conventional system for indicating speed and is used to measure the indicated airspeed (the large pointer on the dial); (b) The second system incorporates a special compensator for changes in air density with altitude by means of a system of gears /187 and is used to measure the airspeed. The compensator is an aneroid box, which changes the length of the arm of a control lever, increasing the latter's mechanical advantage when the atmospheric pressure (as well as the density of the air at flight altitude) is reduced, and vice versa. It should be mentioned that in the case of high speed aircraft, the sensor for total pressure is usually separated from the static pressure indicator, so that it is possible to select the most suit- able position for mounting them on the aircraft. This means that the role of the static pressure indicator is played by openings which are made on the lateral surface of the fuselage of the air- craft and are linked to the instrument itself by tubing. In addition to the details of design described above, the reg- ulation of the systems in the CSI are made by taking the compres- sion of the air into account when the flow is retarded in the detec- tor for total pressure. Therefore, compression of the air on braking will be accom- panied by heating, and therefore by an increase in its internal energy . The relationship between the internal energy of the gas , its pressure, and weight densities is expressed by the formula: U- 1 £_ (2.43) where K = -^ is the ratio of the specific heata iifthe gas when it is heated, with retention of constant pressure and constant volume. 190 For air, this coefficient is K = 1,4. By substituting the value U into Formula (2.39), we can change it to read as follows: YL,Pst,__L_ P^t__ ptota l 1 /'tota l 2^ fi ^- 1 ■ -ri T2 K-l ' 72 • 2^ 71 ^-1 72 ^-1 (2.44) After making some simple conversions, V^_ K /PtotAlpst, 2^ f<-l\ 72 7J taking Pst/^i out of the parentheses, we will have: £=^ ^fet/Ptotal7i._,y (2.45) 2*? A:-1 7i \Pst T2 / For the adiabatic process, there is an equation which is known /18i as the Mende leyev-Clapeyron equation: Pi _ P2 ,K K • Til T2 from which we obtain for our case 1 T2 V /'total '^total Substituting the value Y1/Y2 in Formula (2.45), we obtain 2^ ^-1 ■ 7^ LI Pst ) i Assuming that y^ = y^y , so that Pst/Yi = ^^H ' "® ^^'^ rewrite this equation in the form: "=i^^«^«P°^^'=^' +')'-.] and finally obtain the formula which can be used to calibrate the combined speed indicator by the airspeed in the channel for subsonic airspeeds : K = /^^BrJptot-l::&t^,p;i, '^"^ "-^ -"st / 1 (2.46) 191 The temperature at flight altitude (T^) is assumed to be stan- dard according to the flight altitude (or Pst^' i.e., up to 11,000 m, Tg = 288°-6.5°fl, and above 11,000 m Tg = 216.5° K (-56.5° C). To calibrate the airspeed indicator, it is necessary to know the pressure in its manometric box and in the housing of the appar- atus, corresponding to the pressure in the sensors of total and static pressure under the given flight conditions. Therefore, (2.46) is solved relative to the pressures and assumes the form: Ptotal'^st _r. , (/C-1)F2 W_ p ^ L "^ 2KgBT J st or, if we insert the numerical values of Kj g^ B, Ptotal"Pst r, , _ZL?-5 (2.47) st r V2 -13.5 L "*" 20607J '' — 1. (2 .47a) As we have already pointed out, (2.46) is valid for subsonic airspeeds. At speeds which exceed the speed of sound, the flow /189 of the particles differs from their flow at subsonic speed. We know that the rate at which sound travels (a) in air de- pends only on the temperature of the medium and is expressed by the formula a = yfUgBT. In other words, if ^ = 9.81 m/sec^, and the coefficient for air is equal to 1.4, while B - 29.27 m/degree , a = /4127-=20.3y'r m/sec. The ratio of the airspeed to the rate of propagation of sound in air is called the Maoh number: M = I. a If we replace Kg BT in Formula (2.46) by a^ , we will have the expression for M (Mach number) for subsonic airspeeds: = |/_^r(itQialZ^t_/x'_ll. (2.48) M 192 II ■■■■II ■ III ■■■ The latter formula indicates that in order to determine the Mach number, it is necessary to know only the velocity head and the static pressure at flight altitudes. There is no necessity to measure air temperature for this purpose. For subsonic airspeeds, the relationship between the total pressure, the static pressure, and the Mach number is expressed as follows : ^+1 1 M ^^ P total- -^st ^ (K+IW'' /_J_)^ ^-^ ,— _ 1. (2.1+9) In this formula, if we replace M^ by its value as obtained in Equation (2.48), we will obtain the formula for calibrating the airspeed indicator for supersonic airspeeds: AT+l 1 K ^ total- ^st l_2_J- U -1/ UgBT) ^ ■ — • — = I — 1 . ^st [l^l-K^-') (2.50) If we substitute the numerical values of K for air, equal to /19 1.4, in Equation (2.50), we can convert it to the simpler form: ^otal- ^st _ _J66,7K' . Errors in Measuring Airspeed Errors in measuring airspeed, like those involved in measuring flight altitude, can be divided into instrumental and methodolog- ical ones. Instrumental errors include those which are related to improper adjustment of the apparatus and instability of its oper- ation with changes in the temperature of the mechanism in the device. In addition, instrumental errors also include errors in sensing dynamic and especially static pressures with sensors which depend on the mounting location on the aircraft. Instrumental errors are corrected by correction charts, which are compiled when the apparatus is tested, taking into account the errors in indicating the static pressure for a given type of air- craft . Methodological errors include those involving failure of the actual air temperature at flight altitude to correspond with the 19 3 calculated temperature for combined indicators of speed, and with the temperature and pressure for other speed indicators. Strictly speaking, the methodological corrections which must be taken into account in converting the indicated speed to the air*- soeed, are not instrument errors, since the indicated speed has its own independent value. However, from the navigational stand- point, it is convenient to consider them methodological errors. In aircraft navigation, it is possible to use both the single pointer dial for indicated speed (Type US-350 or US-700), as well as the combined indicator (Type CSI-1200) and others, so that the methods of calculating the methodological errors can be viewed sepa- rately . It should be mentioned first of all that the dials of speed indicators are calibrated to take into account the compressibil- ity of the air for a true airspeed equal to the indicated speed. In fact, at high altitudes, the true airspeed is almost always much greater than the indicated speed, so that it is necessary to consider that there is an error in the difference between the com- pression of the air at the actual and calculated airspeeds: A7 = AF -A7 comp comp.a. comp.c. There are special, precise formulas for determining the cor- rections for LV comp for use with indicators of instrument speed at subsonic and supersonic airspeeds, and they are used to draw up a table of corrections (Fig. 2.33); we will limit ourselves to discussing only the simple approximate formula /191 AV ^-LpndW.^,_i\ comp- 12 1^100 ) \P„ ')• (2.51) where l^ind is the indicated airspeed and AF^^j^p for the indicated speed, is the correction After making the corrections in the indicated speed for the compression of the air, conversion of the latter into airspeeds is done on an navigational slide rule. 194 Ill II II since the dial of the indicated airspeed is calibrated by the formula V =11 / ^^^otai-/'H ) and the airspeed is ^true=l/ ^ ^^^"' then if we divide the second formula by the first we will obtain; f Ph To ^true=^instl/ -^-:^- (2.52) Let us substitute into Formula (2.28) the following values: tgp = 6.5 deg/km, Tq = 288° K, and B = 29.27. We will then obtain p„=Po(\— 0.0226/y)^-2^ , and if we let the value p^ be substituted into Formula (2.52), we will obtain: instF 7-0 (1-0,02: ^true = ^instF ^o " d - 0fi22mf'^^ '^ Vue= 'S ^inst'^T'^(^^^ + '^)~T 'g 288- 2.628 lg( 1-0 ,0226//). (2.53) According to Formula (2.53) we can convert the logarithmic scales of navigational slide rules for converting the indicated airspeed into the true airspeed. Calibration of the combined speed indicator on the basis of the true airspeed is performed by taking into account the compres- sibility of the air over the entire range of the scale. The methodo- logical error in the reading is related only to the differences between the actual air temperature and the calculated temperature at the flight altitude. Since the airspeed, as shown by a combined speed indicator /192 under standard temperature conditions , is expressed by the formula 195 / St •' and the corrected value for the airspeed at flight altitude in ac- cordance with the actual temperature is F corr flg7-//a[( ^total ^st 4-lU -l], ^st if we divide the second formula into the first, we will have 7 =7 1/^^^a corr CSI y ~f 'Ho or / 273 + f„ ^corr'^CSI K 288 -0.0065// iAst (2.54) After looking up the logarithm of the latter, we will obtain a formula which can be used to construct the logarithmic scale on the NL-IOM for a combined speed indicator: -^S^corr = lg^CSI+T'S^^^^+'^>-T'2^^^^-'''^^^inst) (2 .54a) Relationship Between Errors in Speed Indicators and Flight Al t i tude In describing the errors in barometric altimeters and airspeed indicators, instrumental errors of aerodynamic origin are found, which are related to errors in recording the static pressure by the air pressure sensors. Experience has shown that aerodynamic errors in the speed in- dicators due to incorrect recording of the dynamic pressure are negligibly small by comparison with the errors in incorrect re- cording of static pressure. This is explained by the fact that it it immensely easier to measure the pressure of a retarded airflow with a sensor that is aimed into the airflow, than it is to select a location on an aircraft for a static-pressure sensor, such that the latter will not be distorted by the airflow over the body of the aircraft. In connection with the fact that the static pressure from the sensor is transmitted simultaneously to the hermetic chambers of the speed and altitude indicators, there must be a mutual rela- 196 tionship between the errors in the measurement of altitude and /193 speed owing to errors in recording the pressure. At the same time, the velocity head according to which the dial of the speed indicator is calibrated is equal to p ^ -p ^ = P^ . (2.55) '^total '^st 2 Since the errors in measuring the velocity head are equal to the errors in measuring the static pressure , then An = P-fl. A(f2). (2.56) '^ St 2 Under standard conditions, Pq = 0.125 kg/sec^/m^. The static pressure is usually given in mm Hg . The specific gravity of mer- cury is 13.6, so that the pressure of 1 kg/cm^ would equal 10,000/ 13.6 = 735 mm Hg . On the other hand, since the parameter p has m'* in the denomi- nator, the pressure expressed by (2.56), relative to an area of 1 m^ , must be divided by 10,000 to determine the value for 1 cm^ , so that we finally obtain 735 125 V=+= •■ ,„' A ( V2) ==-0.0048 S(V2). St 2-10000 V / • \ ' Example: At an indicated speed of 396 km/hr (110 m/sec), at a flight altitude of 5000 m, the aerodynamic correction for the speed indicator is 36 km/hr, (lOm/sec). Find the aerodynamic error in the altimeter. Solution: Vst = 0.0048(1102— 1002)= 0.0048-2100 = 10.8 mm Hg . According to the hypsometric table, the baric stage at a flight altitude of 5000 m is equal to 18.5 mm Hg ; hence, the aero- dynamic component in the altimeter error is hH = - 10 . 8-18. 5 = - 200 m. Formula (2.56) is an approximate one, but it yields sufficiently accurate results up to an indicated airspeed of 400 km/hr. The altimeter error can be determined more precisely if we know the dynamic pressure and take into account the compression of the air at different instrument readings . Table 2.6 shows the velocity head at various indicated air- speeds, and can be used to determine the aerodynamic corrections of the altimeter. The third column in Table 2.6 shows the mano- 197 metric stage, i.e., the change in pressure with change in airspeed by 1 km/hr. If we multiply the aerodynamic correction of the speed indicator by the manometric stage and then use the hypsometric table, it will be easy to determine the aerodynamic correction for the altimeter for a given flight altitude. mst P -P total st TABLE 2.6 Y for 1 km/hr inst P -P total St /19it 50 0.89 100 3.57 150 8 200 14.3 250 22.37 300 32.4 350 44.27 400 58.25 450 74.23 500 92.35 550 112.7 600 135.7 0.054 700 188,3 0.089 800 252 0.126 900 322 0.162 1000 418 0.2 1100 522.8 0,24 1200 645.8 0,28 1300 787.2 0.32 1400 947,2 0.36 1500 1125.4 0,41 1600 1317.6 0,46 1700 1525,7 0.53 1800 1748.8 Ap km/hr 0.65 0,8 0,96 1,04 1,23 1,42 1,6 1.78 1,92 2.08 2.23 2,4 Ay comp no m wo so so 70 so 50 uo 30 20 m ^.-- 7— [^ f V r /; \ // / ^ < \^ - -■ - / ^ 4/'' ^ j^ / lM. "V t^ '^/^ <^ \ M' s/^ / ^ "Si {'>. \ \ rA i vh ^ V /J \ y \ \ \ iv /A V <V/' / V r^ ^ -\ '-h (/^ tv \ 1 V \ \ \ \— L- A ^ y t ^ 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 remote-controlled instrument, i.e., its sensitive element is mounted outside the cabin of the aircraft and i'.s exposed to the airflow, while the indicator is mounted on the instrument panel in the cockpit. At the present time, electric thermometers are used for meas- uring the outside air temperature, and their operating principle is based on the changes in electrical conductivity of materials depending on their temperature. A schematic diagram of such a thermometer is shown in Figure 2.3M- and consists of an electrical bridge made of resistors. If the arms of the bridge 1 and 1^, as well as 2 and 2^, have .the same resistance when connected in pairs, no supply voltage will flow through bridge AB and consequently through the temperature indi cator . One arm of the bridge (2^) is made of a material which has a high thermoelectric coefficient, and is mounted on the surface of the aircraft to be exposed to airflow . Depending on the temperature of arm 2]^, its resistance changes, thus affecting the amount of current which passes through bridge AB with the temperature indicator connected to it. Fig. 2.34. Schematic Diagram of Electric Thermometer. Thermometers of this kind, when used at low airspeeds, indi- cate the temperature with an accuracy of 2-3°. However, at high airspeeds, due to drag and adiabatic compression of the airflow on the forward section of the sensor, the latter is subjected to local heating that creates methodological errors in measuring temper- ature . For an exact determination of the methodological errors of this thermometer, we will require a sensor with complete braking of the airflow, as is the case in sensors used to measure the total pressure in airspeed indicators. If we keep in mind that y = p/BT, (2.'44) can be changed to 199 read as follows : where Tx is the temperature of the retarded flow. Therefore K-1 /196 T„= T„- 2KgB V2 (2.57) If we substitute in (2.57) the values K - 1.4 and B = 29.27, we will obtain ^T=^^t = VI 2000 • where 7 is the velocity, expressed in m/sec, a) b) ^-^^^^ m^^^^ ^^^^^^^^ -<<^^^^^^^^^^^^ ^^^^^^^^^:-^^^ Fig. 2.35. Sensors for Electric Thermometer for Measuring Outside Air Temperature. (a) TUE ; (b) TNV , Since the conversion coefficient for changing from m/sec to km/hr is 3.6, for a speed expressed in km/hr ^t■- V2 V2 2000-3.62 26 000 (2.57a) Practically speaking, it is highly unsuitable to use thermom- eters for measuring outside air temperature which have complete retardation of airflow, since in this case the sensor will not be exposed to the flow and this will result in a high thermal inertia of the thermometer, i.e., rapid changes in temperature during flight, which could take place at high flight speeds, would not be detected by the thermometer. For sensors which are exposed to the airflow, the coefficient of drag is within the limits of 0.5 to 0.85. The TUE and TNV ther- mometers in use at the present time have coefficients of drag which are nearly the same (approximately 0.7). The scale of corrections for the thermometer for measuring outside air temperature (TUE), located on the navigational slide rule, can be used with sufficient accuracy for the TNV thermometers as well. The sensor of the TUE thermometer is in the shape of a rod 200 with a winding on the surface, covered by a cylindrical housing (Fig. 2.35, a). When a flow of air passes through such a sensor, it is heated on one side . /197 The sensor of the TNV thermometer is made in the form of a de Laval nozzle. The sensitive element is located in the narrowest portion of the nozzle (Fig. 2.35, b) and the air flows symmetrically over it. Therefore, this sensor has less thermal inertia and gives more accurate readings in different flight regimes . 6. Aviation Clocks The measurement of time plays an extremely important role in aircraft navigation, since the calculation of the path of the air- craft on the basis of the component airspeed and time is involved in almost all navigational equations. This means that an increase in the airspeed places increased demands on the accuracy of the measurement of time. It is especially important to have an exact determination of the moments of passage over control checkpoints, i.e., in this case, the exact measurement not of elapsed time but of time segments between the moments when the aircraft is passing over landmarks . There are also factors which demand high accuracy in deter- mining the time and the exact operation of aviation clocks . For example, the coincidence of the flight plans of individual aircraft, communication with the tower, and especially in astronomical calcu- lations, where an error in calculating the elapsed time of 1 min could produce an error in determining the aircraft coordinates of 27 km. The operating principle of all existing devices for measuring time is their comparison with the time required for some standard event to occur. In this case, the standard event is the period of oscillation of the balance wheel of a clock (a circular pendu- lum). All of the remaining mechanism of the clock acts mainly as a mechanical counter of the number of oscillations of the pendu- lum . However, it exerts a considerable influence on the accuracy of operation of the clock; when the main spring of a clock is wound completely, the clock runs somewhat faster, and when the spring has run down the clock runs slower. The most important role in measuring time is played by the accuracy of adjustment of the actual period of oscillation of the pendulum. We know that the period of oscillation of a body around its axis (torsional oscillation) is related to the deformation of the body as determined by the formula '-'Yi-' 201 where T is the period of oscillation of the body around the axis, J is the moment of inertia of the body, and D is the modulus of torsion , The product of the modulus of torsion times the angle through which the body rotates ( (j) ) is the torsional moment: M = D^. The period of oscillation of a balance can be adjusted both /19 8 by changing its moment of inertia (for which purpose adjusting screws are located along its outer circumference), or by changing the mod- ulus of torsion. The moment of inertia of the balance wheel is changed by screw- ing the adjusting screws symmetrically in or out along the entire circumference, in order not to disturb the balance of the pendulum. This means that a portion of the mass is brought closer to or moved further away from the center of rotation of the balance. The modulus of torsion is adjusted by means of a hairspring; the balance wheel is adjusted by changing the free length of the hairspring, for which purpose a movable stop, which acts as a regu- lator, is mounted near the point where the hairspring is fastened. It should be mentioned that many factors affect the precis- ion with which a clock operates, but the most important ones are temperature and magnetic effects. Therefore, a number of measures are taken to exclude these factors. The balance wheel of an accurate clock is usually made of bi- metallic material and divided along the plane of the diameter. When the temperature falls and the flexibility of the hair- spring increases (the modulus D increases), one-half of the balance expands and its ends move further away from the center of rotation, thus compensating for the temperature error in the clock. The harmful effect of magnetic fields on the accuracy of clocks can usually be overcome by using diamagnetic parts in the balance wheel, hairspring and escapement, or else the entire clock mechan- ism is placed within a shielded housing made of iron alloy. Special Requirements for Aviation Clocks In addition to the general requirements for clock mechanisms (high accuracy, compensation for temperature and magnetic effects), aviation clocks have additional requirements placed upon them: (a) Protection against vibration and shock, so that the clocks on an aircraft must be mounted in special shock mountings . 202 (b) Ensuring reliable operation under conditions of low tem- perature; for this purpose, aviation clocks are usually fitted with electric heaters. (c) Reliability and accuracy of operation under various con- ditions. The hands, numerals, and principal scale divisions are made larger and covered with a luminous material to permit their use during night flights. (d) The possibility of measuring simultaneously several time parameters. This means that several dials are usually driven by the mechanism. Aviation clocks of the ACCH type (aviation clock- chronometer /199 with heater) are made to satisfy all the conditions listed above. The elapsed time is indicated on these clocks by a main dial with a central pointer. To calculate the total flight time or the flight time over individual stages, there is an additional scale in the upper part of the clock. The start of the clock hands is marked on this scale, while the time when they stop as well as the resetting to zero are accomplished by pushing a button on the left- hand side of the clock housing. This same button, when pulled out, is used to wind the main spring of the clock. Below the "flight time" scale, there is a pilot light which is used to signal the following by means of a special shutter: (a) Start of mechanism: red light. (b) Stop mechanism: the light is half red and half white. (c) Pause: white light. To measure short time events , the clock is fitted with a sweep hand (thin central pointer) and an additional scale at the bottom of the apparatus where the minutes are counted. The sweep hand is started, stopped and held by pressing a button on the right- hand side of the housing. In addition to the ACCH, the aviation chronometer 13 ChP is currently in use. It employs a potentiometri c circuit; the version fitted with indicators is the 20 ChP. This chronometer, especially intended for purposes of astronomical orientation, is operated by remote control and consists of three main indicators: (a) An elapsed-time indicator whose readings are always linked to the chronometer at the transmitter. (b) Two time indicators for measuring the altitude of lumin- aries; their readings are also connected to the chronometer at the transmitter, but at the moment of measurement of the altitude of 203 the luminary by means of a sextant, a stop signal is sent to one of them and the time of measurement is noted. After the reading is made, the minute hand is set to the elapsed time according to the readings of the first dial by pushing the button. Each time the button is pressed, the hand moves forward one minute. The sweep second hand lines up with the readings of the transmitter immediately after the indicator is switched on. These dials do not have any hour hands . The time in hours is determined by readings from a Type ACCH clock. 7. Navigational Sights At the present time, nav.i gational sights are used only for special purposes such as aerial photography. They are not used in passenger aircraft. There are several types of navigational sights, which differ /200 in their design. However, all are intended for measuring the course angles of landmarks (CAL) and their vertical angles (VA). The course angle of a landmark is the angle between the lon- gitudinal axis of the aircraft and the direction of the landmark. The vertioat angle is the angle between the vertical at the point where the aircraft is located and the direction of the landmark. The sight can be used to solve a great many navigational prob- lems related to determination of the locus of the aircraft and the parameters of its motion. Fig. 2.36. Determining the Value (a) of Aircraft Bera- ing; (b) of the Distance from a Landmark to the Aircraft Verti cal . 1. Determination of the locus of the aircraft in terms of the course and vertical angles of the landmark (Fig. 2.36). In this Ccse, the true bearing from the landmark to the aircraft is (Fig. 2.36, a) TEA = TC + CAL ± 180° , 204 while the distance from the landmark to the vertical of the air- craft (Fig. 2.36, b) is 5 = ff tg VA, where TBA is the true bearing from the landmark to the aircraft, TC is the true course of the aircraft, CAL is the course angle of the landmark, H is the flight altitude, and VA is the vertical angle of the landmark . Obviously, if the aircraft course is determined by a magnetic compass, in order to solve this problem we must also add to the readings of the compass the corrections for the deviation of the compass of the magnetic declination of the locus of the aircraft. TC CC + A + c M' The correction for the deviation of the meridians between land- marks and the locus of the aircraft in this case is not taken into account, since the measurement of the vertical angles can be made satisfactorily up to 70-75°, i.e., at distances which do not exceed /20 1 three to four times the flight altitude. In solving this problem, it is particularly important to know the true flight altitude above the level of the visible landmark, since errors in determining the distance will be proportional to the errors in measuring the flight altitude. Therefore, the readings of the altimeter must be subjected to corrections for the instru- mental and methodological errors and the elevation of the landmark above sea level must also be taken into account if measurements are not being made in a level location. 2. Determination of the location of an the bearings from two landmarks (Fig. 2.37). aircraft in terms of In this case. IPSi = TC + CALj ± 180° ; IPS2 = TC + CAL2 ± 180°. The position of the aircraft is determined by the intersec- tion of bearings IPSj^ and IPS2 on the map. If the direction finding is made over great distances, especially in the polar regions, the measurements of the bearings must include a correction for the dis- placement of the meridians. An advantage of this method is its independence of flight al- titude, and consequently, of the nature of the local relief. However, this method requires careful measurement of the course -■ngle of the second landmark, since the aircraft may move consid- erably away from the line of the first bearing during a prolonged measurement . 205 a i ng 3 to Determination of the drift angle of the aircraft accord- risual points. To determine the drift angle by this means, it is set 'at a course angle of 180° and a zero vertical angl Fig. 2.37. Determining the Po- sition Line of an Aircraft by Two of its Bearings. With an exact maintenance of the course, by the pilot, observing, the directions of visual points and turning the sight to keep it parallel to the course chart, the sight is set in the direction in which the aircraft is moving. The ■drift angle of the aircraft is then calculated on a special s cale . This method is used for low flight altitudes, i.e., with rapidly changing visual landmarks . using a urement aircraft in terms the pilo the cros in these the pilo vertical 20° at h visual p k. Determination of the drift angle of an aircraft by backsight. The essence of this method lies in the meas- /20 2 of the course angle at which visual points recede from the . After setting the sight, as in measuring the drift angle of the location of visual points (CAL = 180°, VA = 0), t waits until the characteristic visual point appears in s hairs of the sight at the position of the bubble level cross hairs. Then, keeping the aircraft strictly on course, t waits until the landmark leaves the cross hairs in the plane at an angle of 40-50° at average altitudes or 15- igh altitudes. Then, by turning the sight, he matches the oint with the course marking and calculates the drift angle. 5. Determination of the drift angle of an aircraft by sight- ing forward. In measuring the drift angle by sighting forward, the sight is set to the zero course angle and a visual point is selected on the course chart, which preferably lies at a vertical angle of 45 or 26.5°. In this case, with VA = 4-5°, the distance to the landmark will be equal to the altitude, while at VA = 26.5° it will equal half the altitude: 206 S^ = H tg VA. The drift angle of the aircraft is determined as the ratio of its initial distance to its final distance: tg US = Sz tgVA2 57 "" tgVAi At drift angles on the order of 10° , the tangent US can be replaced by its value, while the tangent VU2 can be replaced by the value of the lateral deviation (LD): US LD tgLDi or, with an initial value of VAj: VAi=26 . 5° , US = 2 LD. 45°, US = LD; with an initial All three of these methods described above for determining the 'drift angle are used in locations which have many landmarks, i.e., where it is easy to pick out a visual landmark at the desired visual angle . 6. Determination of the ground speed of the aircraft by means of a backsight. To determine the ground speed by this method. The sight is set on the course angle scale to 180° , and to zero on the vertical angle scale. The bubble in the level is set at the inter- section of the cross hairs. Having selected the characteristic point as it passes through /20 3 the intersection of the sight, the sweep second hand is started and the pilot waits until this point has moved to a vertical angle of 35-40° where H is the flight altitude and t is the time measured by the sweep second hand. 7. Determination of the drift angle and the ground speed of the aircraft from a landmarl< located to the side. This method is used in the case when it is desired to measure the drift angle and the ground speed and the pilot has only one landmark at his disposal 207 which is not located along the line of flight of the aircraft. Being careful to keep the aircraft strictly on course, he looks through the sight at the landmark and waits until its course angle is equal to 45 or 315° , depending on whether it is to the left or right of the flight path of the aircraft. At a course angle for the landmark of US or 315° , the vert- ical angle of the landmark is measured and the sweep second hand is started. Fig. 2. 38. Determining the Drift Angle and Ground Speed by a Landmark Located to the Side. Leaving the vertical angle i tion, the sight the motion of th its fixed positi chart . At the b CAL = 90° + US) , the landmark wil then will increa sequently, the 1 first move away the intersection and will then ag it. At the mome mark is at the i the cross hairs , hand is stopped angle of the Ian lated . setting of the n the same posi- is rotated to follow e landmark , noting on on the course eginning (up to the distance to 1 decrease, but se again. Con- andmark will at to one side from of the cross hairs ain begin to approach nt when the land- ntersection of the sweep second and the course dmark is calcu- If CALi = 45°, the bisectrix of the triangle OAB (Fig. 2.38) will be located at the course angle, which is equal to: CAL M-5°+CAL2 bis while the drift angle of the aircraft will be equal to CAL2jj_g-90° , 720'+ so that US CAL2-135° If CALi = 315° , CAL 315° + CAL; bis US = CAL^ . -270° bxs or US = ^^^p-^^5° 208 At points 1 and 2 the distance from the aircraft to the land- mark is equal to S. = So = H VA. Consequently, the distance between points 1 and 2 is deter- mined by the formula 1-2 OE7 ^ WA • CALi+CALp 2H tg VA sm S; ^ Clearly, the reason for the change in the course angle of the landmark from CAL i to CAL2 , was the shift of the aircraft from point to point Oi, so that Si_2 = OOi. Consequently, the ground speed is W = ^^^ . t The majority of navigational problems which we have discussed, which are solved by means of mechanical or optical sights, can be solved using the radio devices which are installed nowadays aboard modern turboprop and jet aircraft, which will be described in the next chapter. 209 8. Automatic Navigation Instruments In Section 2 of Chapter I, it was mentioned that in the general case, all the elements of a flight regime are not strictly fixed, with the exception of the extreme points of deviation from a given trajectory. Therefore, the crew of an aircraft must con- stantly deal with average values of measured navigational elements (average course, average speed, average wind, etc.). If all the elements which have been mentioned had a constant given value, the practical problems of aircraft navigation could be solved quite simply and the question of automating the processes of aircraft navigation would be superfluous . /205 The simplest device used for automating the computation of the aircraft path in terms of the changing values of navigational param- eters and times is the automatic navigational device, which has been devised on the basis of the general features of aircraft navigation. At the present time, the navigation indicator Type NI-50B, is widely used. We shall now discuss its design and the method of its application . The NI-50B navigation indicator is an automatic navigation device which calculates the path of the aircraft on the basis of signals from sensors for the course and airspeed, taking into account the measured wind speed during flight. In addition, the indicator can be used to determine the wind parameters at the flight altitude. Calculation of the path of the aircraft with the use of the NI-50B can be performed both on the basis of orthodromic systems of coordinates for s traigh t- line flight segments, as well as in a rectangular system of coordinates with any orientation of its axes . Without going into the details of the design of the instrument, let us examine its schematic diagram, purpose, and operating prin- ciples of the individual parts , as well as the ways in which the system as a whole can be employed. The navigation indicator consists of the following parts: auto- - matic speed indicator, control unit, automatic course-setting device, wind indicator, and device for calculating the aircraft coordinates (Fig. 2.39). 210 The automatic speed control consists of a device which converts the pressure from the sensors of total and static pressure Into elec- trical signals, corresponding In value to the airspeed of the air- craft, according to Formula (2.'+7a) P -P total st ^st ~['^ 2060 7 ) The automatic speed control has two horizontal manometrlc boxes. One of them (aneroid 1) is used to measure the static pressure, while the other Is used to measure the aerodynamic pressure 2 as the differ- ence between Ptotal ^^'^ Pst • /206 Both boxes are connected by means of linking mechanisms to po- tentiometers 3, which regulate the current ratio in the balancing circuit, according to the ratio of the dynamic pressure to the static pres s ure . It is clear from Formula (2.47, a) that the ratio of the dy- namic pressure to the static pressure is not linearly related to the airspeed of the aircraft. In order to develop electrical signals which are proportional to the airspeed, the control unit contains an automatic speed control mechanism. This mechanism consists of a magnetic signal amplifier M- , coming from the automatic speed con- trol, activating motor 5, and a potentiometer 6 with a special pro- file, which levels out the nonlinearlty of the signals from the auto- matic airspeed control. Thus, the turn angle of the axis of the potentiometer of the analyzing mechanism becomes proportional to the airspeed . automati c speed control distributor unit magnetic amplifier wind sensor coordinate calculator automatic course control Fig. 2.39. Schematic Diagram of Navigational Indicator. 211 By means of a second potentiometer, connected by its axis of rotation to the activating mechanism, sends out electrical signals which are proportional to the airspeed, in the form of a DC volt- age . The automat-io course control is intended to distribute the sig- nals which are proportional to the airspeed, along the axes of the coordinates for calculating the path. Let us assume that we must make a flight over a path segment with the orthodromic flight angle ij; (Fig. 2.4-0). If the aircraft is now to fly with an orthodromic course y, /207 the airspeed must be divided into two components: K^= Vcos(7-i;); Vt= K sin (1 — 4/). It is clear that if there is no wind at the flight altitude, these components of the airspeed must be multiplied by the flight time to give us the change in the aircraft coordinates during this time : A^= K^Af;. AZ= V^M. V /^ -Vcosd' ^Vcosa, 6-v - Vsinoi Fig. 2.40. Fig. 2.41. Fig. 2.40. Distribution of the Airspeed Vector along the Coordi- nate Axes . Fig. 2.41. Sine-Cosine Distributor. The division of the course signals by the axes of the coord- inates in the automatic course control is accomplished by means of a sine-cosine potentiometer (Fig. 2.41). The sine-cosine potentiometer consists of a circular winding with power supplied to it at two diametrically opposite points. 212 Two pairs of pickups slide along the coils; they are located at right angles to one another. Obviously, if we say that the the one in which one pair ( cos ine ) and the second (sine) will be locat then the maximum current will flow while that through the second pair pickups from zero to 90° , the curre drop from maximum to zero and that from zero to the maximum. However , the pickups will not take place ace laws , but proportionately to the an zero position of the pickups is coincides with the supply leads ed at an angle of 90° to them, through the first pair of pickups will be zero. By turning the nt in the cosine pickups will in the sine pickups will increase the change in the current in ording to the sine and cosine gle of rotation of the pickups. In order for the law of change of currents to approach the sine- cosine, the winding of the potentiometer is given a profile or is / 20 i fitted with special regulating shunt resistors. Rotation of the pick-up shoes of the potentiometer is involved in figuring the course of the aircraft which is arriving from a course system or other course instrument. In order to apply the components of the aii ing system for calculating the aircraft coordina uea , tne c±rcu±ax' tentiometer is made movable and can be mounted in of the airspeed to the receiv- :: ui i u X- u cix l; u X d L J- ii^ L 11 e d J- x' u X' d ± L t^ u u X' u X ii a t e s , thc circular finding of the potentiometer is made movablf -,•„„ -K,, „„-,„„ „4: -, p3q]<; and pinion, :_n -I- JT __n-..n_-^,- j^ ^ ^ pOSltion in the automatic winding or the potentiometer is made movable and cai any position by means of a rack and pinion, located course control, and a special scale for calculating The angle for studying the system of coordinates for calculat- ing the path relative to the meridian from which the aircraft course is measured is called the chavt angte. In the majority of cases, the chart angle is made equal to the orthodromic path angle of the path s egment . Hence, by applying to the winding of the sine-cosine potentiom- eter a voltage which is proportional to the airspeed, we obtain signals at the outputs of the potentiometer which are proportional to the component of the airspeed along the axes of the coordinates V^ and For a precise regulation of the navigational indicator as a whole, these signals are calibrated manually by means of a poten- tiometer (see Fig. 2.39, Position 8), located in the control unit. The wind sensor has a schematic similar to that found in the automatic course control, with the exception that the voltage which is proportional to the windspeed is analyzed directly at the sensor by means of a potentiometer (see Fig. 2.39, Position 9) and is set by manually turning knob "w" so that the setting of the pick-up shoes on the sine-cosine potentiometer agrees with the wind direction. Thus, we have three set parameters on the wind sensor: the 213 ^ wind speed (u), the wind direction (6), and the chart angle ( i(j ) . It is clear that the difference between angles 6 and i|^ gives the path angle of the wind. As a result, we obtain signals at the output of the sine-cosine potentiometer which are proportional to the component of the wind speed along the axes of the coordinates for calculating the path . The outputs of the sine-cosine potentiometers of the automatic course control and the wind sensor are connected in series, so that we obtain signals at their common outputs which are proportional as folllows Vjc + Ujc= Kcos (7 — ij/) + tt^^os AW Vz-\-Ut= 7 sin (y --'}') + a sin AW i.e., signals which make it possible to calculate the path of the aircraft with time, considering the manual setting of the wind value for the flight altitude. The oooTdvYiate oateulatoT consists of two integrating motors /209 that work on direct current (see Fig. 2.39, Position 10), whose speed of rotation strictly corresponds to the magnitude of the signals coming from the automatic course control and the wind sensor. The revolutions of the motors are summed by two counters, whose readings are shown on a scale which is graduated in kilometers of path cov- ered by the aircraft along the corresponding axes . A pointer marked "N" shows the path of the aircraft along the J-axis, i.e., along the orthodrome , while a pointer marked "E" shows the travel along the Z-axis, or the lateral deviation from the desired line of flight. The names of the pointers ("N" and "E") were given because at a chart angle equal to zero, the pointer "N" will show the path traveled by the aircraft in a northerly direction from the start- ing point while the pointer "E" shows travel in an easterly direc- tion . To set the pointers of the counter to zero (at the starting point of a route) or to the actual coordinates of the aircraft when correcting its coordinates, there is a special rack and pinion which is used to turn the "N" pointer when it is pushed inward and to turn the "E" pointer when it is pulled out. 9. Practical Methods of Aircraft Navigation Using Geotechnical Devices Flight experience shows that in addition to a knowledge of the devices for determining each of the elements of aircraft navigation, successful completion of a flight, means that it is necessary to 214 obtain and use the measured values, i.e., to master the devices used for aircraft navigation prior to automation. These devices do not depend on systems of measuring flight angles and aircraft courses, since they have limited fields of application. In addition, in describing them, it is necessary to recall that the readings of navigational devices contain all necessary corrections. Therefore, in the formulas which have been found to be necessary, we have used the common designations for navigational parameters. Under practical conditions of aircraft navigation, an impor- tant role is played by the pilots' calculating and measuring instru- ments. However, in many cases, instead of using these instruments, approximate calculations are performed mentally. Approximate mental extimates can be used to advantage in all cases when the problem can be solved more precisely by means of calculating instruments in order to avoid any chance gross errors. Methods of approximate (yet sufficiently accurate for practical purposes) estimation of navigational elements in flight without the use of calculating and measuring instruments are called pilots' vis- ual estimates. The rules for pilots' visual estimates will be given later on in the description of the suitable methods of aircraft naviga- tion. Takeoff of the Aircraft at the Starting Point of the Route /210 The starting point of the route (SPR) is the first control land- mark along the flight path from which the aircraft will travel along the route at a given path angle \p . The final point on the route (FPR) is the last control land- mark along the route, from which the maneuver to land the aircraft begins . Regardless of the fact that the path angle of the flight is usually reckoned from the airport from which the aircraft took off up to the SPR, as well as from the FPR to the airport where it is to land, these values have significance only for general orientation in the vicinity of the airports . In connection with the fact that the first turn of the aircraft after takeoff is made after the aircraft reaches a certain altitude (200 m, e.g. ) and that many factors influence takeoff conditions (such as atmospheric pressure, wind speed and direction, flying weight of the aircraft, etc.), an exact determination of the location of the beginning and end of a turn is usually difficult. Therefore, the path angle and the distance from the first turn to the SPR has a variable nature and cannot be determined exactly. Methods of bringing the aircraft to the initial point on the route differ somewhat from the general methods of aircraft navigation along the flight route . 215 The basic difference between the methods of aircraft navigation involved in bringing an aircraft to the SPR, and the aircraft navi- gation along the route, is that in the first case we do not have a strictly determined path angle for the flight and can reach the given point from any direction, i.e., in the given case the navi- gation is made in a polar system of coordinates. In the second case, we have a given line of flight, and the aircraft navigation takes place along a straight-line orthodromic system of coordinates. In Figure 2.42, a, we see that the flight path angle from the center of an airport in the direction along the SPR and the short- est line for the aircraft's path to the SPR after takeoff and gain- ing altitude until the first turn are at right angles. Since the line of flight is not constant when the aircraft reaches the SPR, the problem involves bringing the aircraft to a given point with the minimum number of changes in the course, or (in other words) along the shortest path. Practically speaking, visual control of an aircraft to bring it to the SPR is done as follows . With the proper selection of the course to the SPR, i.e., when the lead angle (LA) is equal in value to the drift angle of the air- craft, the landmark will be observed at a constant angle to the axis of the aircraft, CAL = const (Fig. 2.42, b). In this case, it is necessary to continue the flight along the previous course until the SPR is passed or (in high-speed aircraft) until there is a linear lead on the turn. If the drift angle turns out to be less than the lead which has been taken (Fig. 2.42, c), a slipping of the landmark will be observed from the direction of the longitudinal axis of the aircraft. In this cascj, the aircraft must be shifted in the direction of the landmark so that- its course angle turns out to be less than the initial one . The slipping of the landmark in the direction of the longitud- inal axis of the aircraft (Fig". 2.43, d) indicates that the lead which has been taken is less than the drift angle, and the aircraft must be turned away from the landmark so that its course angle is greater than the initial one. 216 Thus , the course to be followed by the aircraft is set visually when the SPR is located along a straight line. This problem is SPR Fig. 2.42. Lining Up an Aircraft with the SPR: (a) Path Angle (ip ) and Shortest Distance (S); (b) Aircraft Course Chosen Correctly; (c) Aircraft Course must be Increased; (d) Aircraft Course Must be Decreased. best solved when there is a navigation level on board, by using the/212 so-called method of half corrections. This method involves the fol- lowing: if the lead which has been taken turns out to be greater or less than the required one, it then changes in the required direc- tion by half of the initial lead which was taken. If this turns out to be insufficient, it is changed again by half of the initial value until the course angle becomes stable or the sign of the cor- rection must be changed to the opposite. Reverse correction is made by one-fourth of the initial lead, and if this is insufficient or too much, a correction is made which is equal to one-eighth of the initial lead. It is not usually neces- sary to break down the corrections more than eight times , since the value of the correction will then be no more than 1-1.5°, which is no longer of practical importance for visual aircraft navigation. In the absence of a sight aboard the aircraft, the course angles for the SPR are determined by visual observation; to solve this prob- lem, the pilot requires a certain degree of experience which is gained in the course of the training of flight cruise in actual flight or in special training devices, as well as in practice flights. 217 Selecting the Course to be Followed for the Flight Route The course to be followed by the aircraft along the flight route not only must be set so the aircraft passes over certain control landmarks in the proper order, but must also ensure that the flight takes place exactly according to the given line of flight- There are three principal methods of selecting the course to be followed: (a) When deviations occur from the line of a given path ( LGP ) during the flight, (b) At a landmark along the line, (c) In the direction of the landmark points. The most universal and widely used method is the first one. This method involves the following: after flying over a certain control point, the calculated course to be followed along the given line of flight is determined as follows y = ^ 'calc ' which the aircraft follows until the first characteristic point along the flight path. If, at the moment that it is flying over this point, the air- craft turns out to be on the given line of flight, the course is then considered to be sufficiently correct. If the aircraft has undergone some shift to the right when it passes over this point, the linear lateral deviation from the desired line of flight is determined and the required correction is found for the course of the aircraft: tg Ay = LLD where LLD is the linear lateral deviation and S^ is the distance /213 covered . Example : An aircraft has flown from a control landmark for a distance of 36 km and has deviated 3 km to the right of the desired path. Determine the required correction in the course (Fig. 2.43): Sol uti on. tgAf = 1 36 12 ' A7 = -5°. 218 To reach the desired line of flight, it is usually pecessary first of all to make a double course correction (in our case, 10°), and then (when the aircraft has covered a distance equal to the base of the measurement, or is traveling along the line of the desired path) the lead in the course is reduced by a factor of two, leav- ing a correction in the course which is equal to the set angle of drift. If the closest turning point in the route (CTR) is located at a distance which is smaller than the base of measurement, then in order to attain it, correction must be made in the course for the distance covered for the travel parallel to the line of the desired path and over the distance covered, in order to reach the desired path at the moment when the next control landmark is being passed . Let us say that in our example the distance to the next land- mark is still 30 km; the correction for the remaining distance will be equal to : tgA,rem=^ = ^ Aj^— 6°. V -^ •Sc "^ lXd" Fig. 2.43. Corre ct i ons Followed . Determination of in Course to be Since the correction for the distance covered was equal to - 5° , the total correction for the course in order to get the air- craft to the CTR must be equal to -11°. The problem is solved similarly when the aircraft has wandered to the left of the desired path, but with the difference that the correction in the course to be followed is positive in this case. In solving problems in determining the desired corrections in the course to be followed, we preferably use methods involving visual observation by the pilot without the use of any calculat- ing instruments or tables. In the opposite case, while the pilot is solving the problems, the aircraft will cover a considerable distance, thus complicating the realization of the desired solu- tions . /214 The first method of pilot's visual estimation in this case will be the visual estimation of the lateral drift from the line of flight. 219 charac verse of the tance with a altitu tude . Interm by vis ical a is app f an ai teristi is dete point from th vertic de , whi These ediate ual obs ngle is roximat rcraft i c point , rmined b is close e point al altit le at an angles a values o ervation roughly ely equa s tra the y the to 2 whi ch ude o angl re us f ver and equa 1 to veling distan verti 6.5°, is eq f 45° , e of 6 ually tical interp 1 to 5 1. 5 fl to the side of the above mentioned ce from it by flight along the tra- cal angle . When the vertical angle the aircraft is located at a dis- ual to half the flight altitude; the distance is equal to the flight 3.5° it is twice the flight alti- determined by visual observation, angles and distances are determined elation. For example, if the vert- 5° , then the distance to the point ight altitudes . This method, with sufficient training, gives a very high accuracy for determining the location of the aircraft relative to a given point along the route, and consequently, with respect to the line of flight (on the order of 0.1 H) at vertical angles up to 65°. At very large angles (grater than 65°) from the vert- ical of the aircraft, the errors in distance will be greater and this method cannot be used. The second method of visual estimation by the pilot which is used in solving this problem is the mental calculation of the required course corrections following linear lateral deviation (LLD) . For convenience in metal calculation, one radian is assumed to be 60° rather than 57.3, but this does not introduce any consid- erable errors (the maximum error in angles up to 20° does not exceed 1°). This allows the required correction to be made in the course in terms of the approximate ratio of the lateral deviation to the distance covered: .LD/5 Ay. deg LLD/5 A7, deer LLD/5 -^T. de& 1/60 1 1/12 5 1/6 10 1/40 1.5 1/10 6 1/5 12 1/30 2 1/8 7 1/4 15 1/20 3 1/7 8 1/3 20 1/15 4 These ratios are easy to remember if we know that in order to obtain their required correction it is adequate to divide the number 60 into the distance covered, when the lateral deviation is taken per unit of measurement. Obviously, if this method for course correction is employed and the aircraft does not reach the desired point along the line 220 of flight, so that there is still some lateral deviation, the lateral deviation and the distance from the point at which the course was / 215 last changed can be used to correct the course. Selection of the course to be followed according to a land- mark along the route can be used in the case when the flight takes place along a straight- line portion of a railway or highway and means that the crew must change the course of the aircraft so that it follows this linear landmark. After changing the course by an additional turning of the aircraft , the crew returns to the desired course and travels in the desired direction once again. The selection of the course to be followed on the basis of orientation landmarks is a variety of the latter method. In this case, the course is selected so that the closer of two selected landmarks along the line of flight constantly (up to the moment that the aircraft flies over it) remains in a line with the further landmark. After passing by the closer landmark, the aircraft follows the desired course or choses the next land- mark, located beyond the second one, and continues its flight along this line . Change in Navigational Elements During Flight The majority of navigational elements (course, altitude, speed) are determined in flight on the basis of indications of the corre- sponding instruments, with introduction of corrections for instru- mental and methodological errors. Automatic radio devices, based on the Doppler principle, make it possible to make measurements directly (during flight) of such elements as the drift angle and the ground speed. Other methods of aircraft navigation do not permit direct measurement of the latter two elements, so that in order to de- termine them it is necessary to use various pilotage techniques. In the absence of sights, the drift angle of the aircraft can be determined as follows. Let us suppose that we are traveling along a given route and that a control landmark on this route has been passed. After 15- 20 min of flying time, we select another landmark by which we test the correctness of the course which has been selected. If no lat- eral deviation of the aircraft occurs on this segment, it means that the aircraft course has been properly set, i.e., the drift angle is equal in value to the previous course, but has the oppo- site sign a = \ii - y , 221 where a is the drift angle of the aircraft, y is the aircraft course, and ^ is the given flight path angle. It is not always possible, however, to correctly set the course to be followed. If a lateral deviation of the aircraft from tjie line of the /216 desired path arises in our flight segment, the course to be fol- lowed will be incorrect and the actual flight angle will be "I'll) = tl-a + arctg AZ where AZ equals the deviation of the aircraft from the LGF , and S is the length of the segment over which the drift angle was meas- ured . The angle of deviation of the aircraft from the line of the desired flight path arctg AZ/S is considered to be negative if the aircraft deviates from it to the left, and positive if it devi- ates to the right. As in the method of selecting the course, this angle is determined by methods of visual estimation by the pilot. In the case of improper selection of the course to be fol- lowed, the" latter can be determined as the difference between the actual flight angle and the course being followed: " = Y* — T = +3 + arctg — — — f . It is much easier in flight to determine the ground speed of an aircraft: the same landmarks are used for this purpose as those used for determining the drift angle of the aircraft. To do this, it is sufficient to determine the times when the aircraft flies over the first and second landmarks, after which the ground speed is determined by the formula ^ = 1^ where S is the distance between the landmarks and t is the flying time between the landmarks . The division S/t is done as a rule on scales 1 and 2 of a navigational slide rule (Fig. 2.HM-), with the exception of those cases when the flying time is less than 60 min. For example, 6, 10, 12, 15, 20 and 30, or even 40 and 48 min are possible. In these cases, the groundspeed will be equal to 105, 6S , 5S , 45', 3S , 2S , 1.5S and 1.255', respectively, and is easily determined men- tally by multiplying the distance between the landmarks by one of the numbers given above. 222 h-jL =^ \ dlstance(kiii) A^|yii[\Hi|ij|('l\i|i|MWM il ll|IIM turn angles i n 'i M' iiii ii ii i i timiuii i i iiii i i ii i iii i ii i ii ij. i ijm iii Mt iii i i i ii i[ ii | ^ t t is ' aV-" A / / 40 BO 00 70 I (in in1n or sec) a e 7 a 9 [io] je inir xiSfl ■ -t, i'-f. ■- Y i "-" 'I 1 1 I I I.I I I I I ll l lll l ll|l l lllllll|llllllilll lll lll ll ! »l » l H^- T H. ^^ ttnedn nr ot- ntn) 06^ ^^ J* ^ 2' ^' 4' ^8' a- r a' r ^' ip- ac ao* . 4o- y\ eo* eo' to* _t^"9ents r ^p. ?;^ Ml"l | l ' "*"^ l' » ' «^'!«'AI' ' 'tMM I . ' Jm l «, ' | l ,1 I l,' i ' ; ' ,M , HAM , l, ilili l i M ilililll/ MMMM ,li' l\W>l l ll)Ml i '^yil)^'J , Vai ) i i U i,li i i LL ii |^^ ^ a 8 7 B 9 10 i turn radius altitude 6 40 00 BO 70 80 90 100 \ 100 300 300 400 BOO tOO 700 SOO BO0 1000 Vj^"* ! 1 p I ' I I i l I t ■ . . t I I I I i I I I r I i I i-i I I'umiuiirnimiiiiiiMiiiiiNiiiiimmiii iipiiiimiii i \ i I i I i r i i i t i u m n m m mniiu 6 30 \ \ ( &^ \ for heights above Uemoeratu 12,000 m At*"« ruE »' ? *^ ^ /•i'»*M"i!f**!)V;*'i*i'i*jrB'ji*jrii*«'«*4i'ir' 7 ^ • • 10 It la "''■inst'";: jneasuredaltjtjjd. cted aititudel and^speed |^Wi»co*a(i(t,'t»J| ^^^!Mihll:fM.==^^^i-JM; To~^l A Fig. 2.44. Scales on Navigational Slide Rule NL-IOM, to K) CO H To measure the ground speed as well as the drift angle, it is desirable to select distances between landmarks which are no less than 50-70 km apart. Over short distances, in order to avoid gross errors, it is necessary to determine and mark down very exactly the time that the aircraft passes over the control landmarks . Measuring the Wind at Flight Altitude and Calculating /218 Navigational Elements at Successive Stages The principal factor which complicates the processes of air- craft navigation at flight altitude is the wind. With availabil- ity of exact data regarding its direction and speed, all problems of aircraft navigation can be solved by a combination of general methods of aircraft navigation independently of the visibility of terrestrial landmarks. When the aircraft has on board only the most general devices for aircraft navigation, the problem of determining the wind at the flight altitude as well as the drift angle and the ground speed can be solved if terrestrial landmarks are visible. The wind at flight altitude does not remain constant but is constantly changing with time and especially with distance. In order to be able to prepare the navigational data for the next stage of flight, it is necessary to determine the wind at the very end of the preceding stage and even in this case, the data on the wind which are obtained are obsolete to a certain degree and are not completely satisfactory for the needs of calculating. Under the conditions when an aircraft is flying along an air route, there are three navigational parameters which basically determine the speed and direction of the wind at flight altitude: the airspeed (F), ground speed iW) , and the drift angle for a given course . The wind calculated on the basis of these parameters will not be reckoned from the meridian of the locus of the aircraft (LA) but from the line of flight of the aircraft. The calculation of the path angle of the wind (AW) is car- ried out on the navigational slide rule by means of a key (Fig. 2.45, a) . Example: W = 360 km/hr; V = 320 km/hr; drift angle = +8°. Determine the wind angle . Solution : (Fig. 2 .45 , b) . Answer: AW = 48° . If we know the wind, it is easy to determine its speed by means of a key which is marked on the rule (Fig. 2.46, a). For our 224 example, see Figure 2.4-5, b. Answer: 60 km/hr. The direction of the wind relative to the meridian of the locus of the aircraft (LA) is determined by the formula 6 = AW + ij^. If the flight is made with magnetic flight angles , the wind direction is obtained relative to the magnetic meridian of the LA. This direction is also used to calculate the navigational elements in the next stage of the flight. Information on the speed of the wind and its direction is trans- mitted from the aircraft to ground stations, also relative to the magnetic meridian of the LA, and is used for controlling the flight of the aircraft. The angle of the wind for the next stage of the flight is /219 AW = 6 - \p , where 6 is the wind direction and (p is the flight path angle of the next stage of the flight. a) ® Sin US ;^AW® ■ b) «•© ® w-i' ® ill no Fig. 2.4-5. Calculation of the Path Angle of the Wind on the Navigational Slide Rule: (a) Key for Determining the Wind Angle; (b) Determining the Wind Angle. The values for the ground speed and drift angle of the air- craft for the next stage of the flight are calculated on the navi- gational slide rule by means of a key (Fig. 2.47, a). Let us assume that the flight in the preceding stage was made with a MFA = 38°, in the next stage with an MFA = 56°, and with an airspeed of 320 km/hr. The data obtained on the wind at the preceding stage are AW = 48°, u = 60 km/hr. a) ©US pm b> ® iS' ©" ® 50 ilO Fig. 2.46. Calculation of the Wind Speed on the Navigational Slide Rule: (a) Key for Determining the Speed; (b) Deter- mination of the Speed. 225 The direction of the wind relative to the meridian of the LA is 6 = 48 + 38 = 86° , while the angle of the wind for the next stage of the flight is AW = 86 - 55 = 30° . a) b) © us: .aV US+AW ® 5.5' JO' 35,5' ® ^ w ® 60 izo no • Fig. 2.47. Calculation of the Drift Angle and Ground Speed on the Navigational Alide Rule: (a) Key for Determining the Drift Angle and Ground Speed; (b) Determination of the Drift Angle and Ground Speed. The value of the groundspeed and the drift angle for the next stage of the flight are also determined by means of the naviga- tional slide rule (Fig. 2.47, b), i.e., \1 - 370 km/hr ; US = +5.5°. The values of the drift angle can be used to determine the calculated course to be followed in the next stage of the flight. In our case , Y = 5i 5 . 5 52.5° . If the flight is made with orthodromic flight angles, then in order to calculate the navigational elements for the next stage of the flight it is unnecessary to convert the wind angle to its direction relative to the meridian of the LA. In this case, the wind angle for the next stage of the flight is determined as the difference between the wind angle of the preceding stage of the flight and the angle of turn in the route (Fig. 2.48): AW; AW- TA In our example, AWi = 48°, TA = 56-31 = 30° . = If and AW2 = 48- However, in order to transmit information regarding the wind to ground stations, it is necessary to determine the wind direc- tion relative to the meridian of the LA. Obviously, the true wind direction at the point LA is ■^true = AW + a. /220 226 where a is the azimuth of the orthodrome at the point LA; the maj netic direction of the wind is «M = AW + a - Aj^ Consequently, if the calculation of the orthodromic path angles is made from the reference meridian, then 6 = AW + (A -X ^)sin(() - A„. M LA ref av M Example: \q-^-70°, Xla=85°, 5°, i|j = 38°, AW = 48°. av :52' Solution 6 true The true wind direction is = 48 + 38 + 15-0.8 = 98° , and the magnetic wind direction is = 48 + 38 + 15 -0 .8 + 5 = 103° Fig. 2.4-8. Determination of the Wind Angle in a Successive Flight Stage. Calculation of the Path of the Aircraft and Monitoring Aircraft Navigation in Terms of Distance and Direction In the preceding paragraphs, we have discussed the methods of placing the aircraft on course, determining the navigational elements during flight, and calculating them for the following stages of the flight. Therefore, it becomes necessary to use continuous calcula- tion of the aircraft path in terms of time at certain periods, when it becomes necessary to check the aircraft path with respect to distance and direction. Calculation of the aircraft path is always done with prev- iously calculated parameters (the calculated course and ground speed. 227 calculated time). At the same time, all the values and moments of change in the aircraft course are determined, which make it possible to determine the calculated position of the aircraft by plotting and thus to determine the additional errors in aircraft navigation . Calculation of the path of the aircraft means that after the last identified landmark has been left behind, the crew aims the aircraft toward the next landmark during a certain period of time which is used to fix all the values of the actual course of the aircraft . If the proper landmark has not been sighted when the sched- uled time has elapsed, due to meteorological conditions, the calcu- lated time for flying over this landmark is determined, and the aircraft is set to the next phase of calculated flight on the basis of the previous values for direction and velocity of the wind. Thus, calculation of the path (flight on the basis of prev- iously determined data) can continue until the conditions for visual orientation improve. However, it is necessary to keep in mind that the accuracy of aircraft navigation then decreases contin- uously due to the accummulation of errors with time, as well as in connection with the obsolescence of the data on the wind, meas- ured prior to the last reliably sighted landmark. When the conditions for visual orientation improve, the crew takes measures to check the path of the aircraft in terms of dis- tance and direction. To check the path in terms of distance, linear landmarks are usually employed, which intersect the route of the flight at an angle close to 90°. Five to ten minutes before the calculated time for flying over these landmarks, depending on the flying time according to the previously calculated data and the speed of the aircraft, the pilot carefully begins to examine the landscape, looking for the landmark; at the moment that he flies over it, the approximate position of the aircraft is determined relative to distance and time . /222 When flying over a control landmark, the pilot also tries to determine the lateral deviation of the aircraft from the desired path on the basis of additional features of the landmark (curves in rivers, tributaries, road junctions, populated areas, forest outlines , etc . ) . Having determined the point of intersection of the landmark, the pilot projects it along the line of the desired path, fixing the position of the aircraft (in terms of distance at the moment that it flies over the landmark)and the direction. 228 At the present when the ground is n tional equipment. L time, aircraft us ot visible are fi ight planes (whic side of acute angle Fig. 2.49. Lead Tow Angle of the Travers mark . than the calculated craft can be aimed a linear landmark. ard the Acute e of a Land- time for flying p t a control landm ed tte h f occ Ion the ori the occ on in tio int by 2.4 Ian fie as t ark for Ion d with ly at 1 asional g dista condit entatio less , w ur , the the bas the cou n of an ersecti a linea 9) . In dmark m Id of V it . A which g di spec ow a ly r nee ions n ar hen fli is o rse acu on w r la thi us t i ew f ter line stanc ial r Ititu equir fligh for e poo such ght c fas and t te an ith t ndmar s cas appea s omew this s up e fl adio des ) ed t ts w visu r . case an b ligh he d gle he r k (F e, t r in hat , th with ights naviga- are o make hen al Never- s do e made t lead ire c- of oute 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 large-area landmark (lake-, sea), and also by making inquiries from the ground, etc. Use of Automatic Navigational Devices for Calculating the Aircraft Path and Measuring the Wind Parameters To a considerable degree, automatic navigational devices sim- plify the work of the pilot in calculating the path of the air- craft and in measuring the wind parameters at flight altitude. These devices are mounted on high-speed passenger aircraft which have complete radio navigational equipment, thus considerably increasing the effectiveness of their use. Such devices, which are based on the general methods of air- craft navigation, can be used in straight-line systems of coord- inates at any orientation of their axes . The direction of the axes of the coordinates is selected by the pilot depending on the conditions for which the system is being used. For example, for flying along a route, it is most advan- tageous to combine the axis of the system OX with the directions of the straight-line segments of the flight, i.e., to calculate the path in an orthodromic system of coordinates in stages. To carry out special operations in this region, e.g., at test sites for radio navigational systems for short-range operation, the axis OX is combined with the average meridian of the flight area (magnetic or true), depending on which system for calculating the flight angles is being used to make the flight. In preparing to land and maneuvering in the vicinity of the airport, the axis OX coincides with the axis of the landing strip at the airport, etc. In all cases when an automatic navigational device is being used, a rectangular system of coordinates should be applied to the flight chart in the given region, parallel to the axes of the system OX and OZ . Parallel lines are drawn at 20 mm intervals, so that on charts with a scale of 1;1, 000, 000 this corresponds to 20 km, while on those with a scale of 1:2,000,000 it is M-O km, etc. For this purpose, a special stencil is included in the set of navigational instru- ments for the NI-50B indicator. In using an automatic navigational device with orthodromic coordinates in stages, no additional devices are needed other than the general navigational divisions of the chart. During flight, the apparatus is connected to a source of direct current, and the chart angle on the automatic course control is set 230 in accordance with the selected system for calculating the air- craft to coordinates. The windspeed and direction are set on the wind sensor on the basis of the results of measurements during the preceding flight segment. If the navigational indicator is used with an orthodromic system of coordinates in sections, the setting of the chart angle and wind is made at the end of the preceding stage of the flight before flying over a turning point in the route (TPR). On the coordinate calculator in this case, the pointer "N" is set to a value equal to the linear lead for the turn (LLT) and pointer "E" is set to zero . /224 At the moment when the aircraft emerges from the turn on the new course (Fig. 2.50), the alternating current is connected to the instru- ment and the indicator begins to calculate the flight path. At small turn angles in the line of flight (up to 30°), the turn trajectory of the aircraft is very close to TPR. In this case, the two pointers on the indicator should be set to zero, and the mechanism switched on when the TPR is passed as the air- craft is turning. At the beginning of the straight- line segment of flight, if possible, it is neces- sary to mark the established coordinates of the aircraft on the computer as the aircraft passes over a given landmark. Fig. 2.50. Transition to an Orthodromic System of Coor- dinates in a Successive Flight Stage. Constant knowledge of the aircraft coordinates facilitates both visual and radial orientation. However, aircraft coordinates obtained on the basis of a computer will not always correspond precisely with the actual coordinates, since the speed and direc- tion of the wind during flight change over the distance covered. The navigational indicator also makes it easier to determine the wind parameters at flight altitude. This is done as follows: At the end of a stage in the flight, the aircraft coordinates are recorded with the computer (Point B in Figure 2.51, a) and the 231 actual location of the aircraft is determined visually or by means of radionavigational devices (Point B^). These Points BBi deter- mine the vector of the change in the wind at flight altitude for the flight time of a given stage of flight. The problem of determining the wind vector in this case can be solved easily on a flight chart. To do this, a reverse line must be drawn from Point B and the length of the wind vector is set on the sensor (u -t) during the flying time from Point A to Point B (Point 0). Then the vector of OB will constitute the vector of the calculated wind (and OBj, the actual wind) at flight alti- tude . /225 Fig. 2.51. Measurement of the Wind by Means of a Navi- gational Indicator: (a) Wind-Change Vector; (b) Wind Vector, In order to obtain the value of the wind in km/hr, it is suf- ficient to divide the length of the vector OBi by the flying time between Points A and B, expressed in hours. The problem of measuring the wind can be simplified if we consider that the wind at the sensor is zero for the flight stage, i.e., we introduce the value of AW = 0, U = into the wind sensor. Then Point B will be the indication of the coordinates of the air- craft at the end of the flight stage, while Point Bj will repre- sent the actual coordinates (Fig. 2.51, b). Consequently, vector BB^ will be the wind vector for the flying time in this stage. The use of the navigational indicator in rectilinear coord- inates for flights in a given region is not different in principle from using it in orthodromic coordinates and stages. However, the important advantage of the orthodromic system of coordinates is then lost, i.e., the relationship of the coordinates to the checking of the path for distance and direction. Therefore, the position of the aircraft in this case can be determined only in terms of the coordinates of the network superimposed on the chart. The rectangular system of coordinates can be extended over a relatively small area (on the order of ^+00 x 400 km), since the effect of the sphericity of the Earth begins to show up in large areas . In conjunction with this, in the case of flights by a coord- inate system, it is not necessary to set a new chart angle for each 232 change in the line of flight and to describe the coordinates of the aircraft in a new system for calculation, which to a consid- erable degree compensates for the loss of those advantages which we have in the orthodromic system of coordinates in stages. Details of Aircraft Navigation Using Geotechnical Methods in Various Fligiit Conditions The conditions for aircraft navigation using geotechnical devices are determined primarily by the presence and nature of landmarks, as well as by their contrast relative to the surround- ing terrain. The best landmarks for visual aircraft navigation are lin- ear ones (large rivers, railways and highways, the shores of large bodies of water). Lakes, large and small populated areas, char- acteristic mountain peaks, etc., are also good landmarks, while grain elevators, water tanks, churches, industrial enterprises, etc., can be used for flights at low altitudes. For aircraft navigation in an area which is poor in landmarks, we can use separate sighting points on the Earth's surface in the form of spots, individual trees, foam on the surface of the water, etc. Such points are not landmarks, since it is impossible to determine their location on a flight chart, but they can be used to measure the drift angle and the ground speed when there is a sight on board and also make it possible to increase the accur- acy of aircraft navigation during flight between control landmarks. /226 The visibility of all landmarks, with the exception of il- luminated populated areas, is considerably decreased at night, especially when the Moon is not out. Therefore, populated areas are the principal landmarks at night; their appearance at night can differ from their appearance in the day. An important factor which determines the conditions for air- craft navigation is the stability of operation of magnetic com- passes. Conditions of aircraft navigation without the use of gyro- scopic compasses are unfavorable in the polar regions, as well as low altitudes in the vicinity of the magnetic anomalies. The flight altitude also has a significant influence on the aircraft navigation conditions. In clear weather, optimum conditions 233 for visual orientation exist at heights on the order of 1000-1500 m , since at this altitude the angular velocity at which the land- marks go by is small, all of their details can be seen clearly, and the field of view of the crew covers a very large area, which /227 is important in comparing the charts with the landscape. However, these altitudes can only be used wh-en there is a small amount of clouds along the flight route. In cloudy weather, flights are made at lower altitudes, as low as the relief of the terrain will allow. At low altitudes, the conditions for visual orientation are worse, since the angular velocity with which the landmarks go by increases and the area which the crew of the aircraft can scan is reduced. An increase in the flight altitude (above 1.5 km) in clear weather have a small influence on the conditions of visual orien- tation, but at great heights the visual visibility of landmarks (depending on weather conditions) is much worse than at low and medium altitudes . The selection of scales and chart projections for making a flight depend primarily on the altitude and speed of the flight. At low altitudes, it is best to use charts with a large scale of 1:500,000 or 1:1,000,000. At high altitudes and high speeds, it is best to use charts with scales of 1:2,000,000 and 1 : i+ ,00 , 000 . For flights along routes which are very long, charts are used which are made up of projections showing the properties of ortho- dromicity (the orthodrome on the chart has a shape close to a straight line), i.e., charts in the international or transverse cylindrical projection. For the polar regions, charts with tangent stereo- graphic projection are used. 10. Calculating and Measuring Pilotage Instruments Purpose of Calculating and Measuring Pilotage Instruments Pilotage calculating and measuring instruments are intended for the following: (a) Measuring distances and directions on flight charts. (b) Calculating navigational elements both in preparing for flight and when completing it. (c) Calculating methodological errors in the readings of navigational instruments (the readings of the airspeed, altimeter, and outside air thermometer) . (d) Calculating the elements of aircraft maneuvering. 23H Measurement of distances on flight charts is made by means of a special navigational slide rule. A feature which distinguishes this slide rule from conventional slide rules is the presence of several scales for measuring distances on charts with different / 22 1 scales . Measurement of directions on flight charts is made by means of navigational protractors, made of transparent material. The protractors are simultaneously used as triangles, which make it possible to make certain constructions on flight charts and diagrams (laying out the traverses of landmarks, parallel shift of lines , et c . ) . Calculations of navigational elements, corrections to nav- igational devices, and elements of maneuvering are presently car- ried out with the aid of navigational logarithmic slide rules, the best modification of which is the navigational calculating slide rule NL-IOM. In addition, to calculate certain navigational elements, we can use special devices for setting up the speed triangle (wind- speed indicators). However, due to the improvements in navigational calculating slide rules, they have a very limited application. Thus, the operations described above involving numbers can be applied to the summing of the segments of a scale on the ruler, which simplifies calculation to a considerable degree The scales of the navigational slide rule NL-IOM (see Fig. 2.44) are grouped so that one side is used for solving problems in determining navigational elements of flight as well as maneu- vering elements, while the other side is used for calculating the corrections for the readings of navigational instruments . In addition, the upper beveled edge of the ruler (Position 17) carries a scale divided into millimeters, which can be used to measure distances on the map. The scales on the ruler 1 and 2 are intended to determine the ground speed from a known distance covered in a given time, or from a given distance at a known ground speed and time. 235 Therefore so that W =: '^ and S = Wt. t IgW = \gS - Igt and IgS = IgW + Igt Scale 1 is the scale of logarithms of distances in kilometers /229 or flight speeds in km/hr; scale 2 is a scale of logarithms of flying time in min or sec up to the rectangular index marked 100 and beyond, in hrs or min. The principle of solving problems by determining the airspeed over a given distance at a given time is as follows : Let us say that an aircraft has covered a distance of 165 km in 12 min and that we must determine the ground speed in km/hr. We set the marking on the slider to the 165 position on the distance scale; by moving the adjustable scale 2, we set division 12 on it opposite the marking on the slider. We can then read off the distance covered by the aircraft in minutes of flight oppo- site the number 1 at the beginning of the scale: IgW km/min = Ig 165 - Ig 12 = Ig 13.8. However, since 1 hr is 60 min, the speed in km/hr would be equal to IgW km/h = Ig 13.8 + Ig 60 = Ig 825 or W = 825 km/h. By combining the first and second effects, we obtain IgW = Ig 165 - Ig 12 + Ig 50, i.e., in order to solve the problem, it is sufficient to set the number 12 on scale 2 opposite the number 165 on scale 1 and oppo- site number 60 on scale 2, which is marked with a triangular mark- ing, and then to calculate the ground speed from scale 1 (Fig. 2.52, a) . The problem is solved analogously if the flight time is meas- ured in sec. In our example, it will be 720 sec: IgW = Ig 165 - Ig 720 + Ig 3600 = Ig 825; W = 825 km/h. 236 The ground speed in this case is calculated from scale 1 oppo- site the number 3600 on scale 2 (the number of seconds in 1 hr), marked with a circular index. To determine the distance at a given groundspeed and flying time (5 = Wt) , the logarithms of these numbers are added: lg5' leW + let On the rule, the triangular or circular index on the movable scale 2 is set opposite the known ground speed on scale 1. The index marking on the slider is set opposite the given flying time on scale 2, after which the position of the indicator on scale 1 shows the distance covered in this time. /230 Example . W - 750 km/hr, t =1 hr and 36 min tance covered. Find the dis- solution. See Figure 2.52, b. Ig 5 = Ig 750 + Ig 1 h 36 min= 1200; Answer. S = 1200 km. a) ies S15 b> ® © 750 tzaa 12 ® ® 1 h 36 min Fig. 2.52. Calculation on the NL-IOM: (a) of the Ground Speed; (b) of the Distance Covered on the Basis of Ground Speed and Time . Let us apply the keys to NL-10 for solving problems in deter- mining the ground speed and distance covered on scales 1 and 2: (a) To determine the ground speed for a distance covered in a known time (fig. 2.53, a or 2 . 5 M- , a). (b) To determine the distance covered from the ground speed and time (Fig. 2.53, b or 2 . 5 1+ , b). a) O Skm b). © ® t sec "T 5 KM w 0- t man Fig. 2.53. Keys for Determining the Ground Speed on the NL-IOM, on the Basis of the Distance Covered and the Time. 237 I Movable scale 3 with the signs of the logarithms , which is the same (up to 5°) as scale 4 for the logarithms of the tangents and is also divided into scales 3 and 4, along with the fixed scale of distances or altitudes 5, which essentially repeats scale 1, are all intended for working with trigonometric functions. The majority of problems which are solved on these scales are based on the properties of a right triangle, so that the value of the sine of 90° and tangent 1+5° (scales 3 and H), whose loga- rithms are equal to zero, are marked on the rule by a triangular index . If the problem is solved from a known leg, e.g., determin- ing the error in the course on the basis of the distance covered and the linear lateral deviation (Fig. 2.55), we use scales U and 5 on the rule . Z tg^Y=-^0^ lgtgAY=5,IgZ-.lg^. The key to solving this problem is shown in Figure 2.56. In the case when the hypotenuse of the triangle is known, the problems are solved by using scales 3 and 5. For example, sup- pose we wish to determine the location of the aircraft in ortho- /231 dromic coordinates (-Sfg, Z^) on the basis of known coordinates of a landmark (^-[_, Zj), the distance and direction of which have been determined by means of a radar located on board the aircraft (Fig. 2.57) . It is clear from the figure that the orthodromic coordinates of the aircraft will be equal to Jg^ = X-L - i? cos e ; Zg = Z-|_ - fl sin e , where E is the distance to the landmark and 9 is the path bear- ing of the landmark (the angle between the given line of flight and the direction of the landmark). a) b) (D .V 5 ® ^ ^ ® h tv mm Fig. 2.54. Keys for Determining the Distance Covered on the Basis of the Ground Speed and Time, Using the NL-IOM. The difference in the coordinates of the landmark in aircraft are represented by X and Z, respectively, and are found on the logarithmic rule (Fig. 2.58, a, b), 238 In aircraft navigation, a number of problems are solved which are connected with the distances and directions (e.g.), the check- ing of a course in terms of the distance covered, determination of the position of the aircraft by using methods of visual and radar measurements, and many others. The essence of the solution of these problems is obvious from the examples given. © tgat ® X Fig. 2.55. Fig. 2.56 Fig. 2.55. Determination of the Course Error from the Change in the Lateral Coordinate. Fig. 2.56. Key on the NL-IOM for Determining the Aircraft Course Error . For cases when the angles measured are greater than right angles, the sine scale 3 is numbered backwards, so that sin 180-a - sin a, for example: Ig sin 135° = Ig sin 1+5°. Scales 3, 4 and 5 can be used to solve special problems of oblique-angled triangles, e.g., the solving of speed triangles. The key for solving this kind of problem is given on the right- hand side of the scale 3. /232 The theorem of signs, well known from trigonometry, deter- mines the relationship between the angles and lengths of the sides of oblique-angled triangles. In the case where the speed triangle is used (Fig. 2.59), this theorem has the form: sin US sin AW sin(AW+US) V W (2.58) Fig. 2.57. Determination of the Orthodromic Coor- dinates of the Aircraft . It is obvious that the relation- ship of Equation (2.58) is equiva- lent to the following: IgsinUS - lgu= lgsinAW-lg7 = lgsin(AW+US) - Igf/, which is expressed by the key on the navigational rule (see Fig. 2.60, a). 239 a) vi) Stnm-B) &X b) Stng Q) tl :SL Fig. 2.58. Keys for Determining the Aircraft Coordinates on the NL-IOM; (a) Z-Coordinates ; (b) Z-Coordinates . Example. MFAg = 35°, l^true = ^0° km/hr, 6 = 85°, u = 60 km/hr. Find the drift angle of the aircraft and the groundspeed. Solution. In our example, the wind angle is AW = MFAg = 85 35 = 50°. Having set the slider indicator to the division represent- ing 400 km/hr on scale 5, and also having lined up the 50° divi- sion on the logarithm sine scale 3 with the same slider indicator, we obtain the drift angle equal to 6.5°, and a ground speed of 440 km/hr (Fig. 2.60, b). This key for solving speed triangles is suitable for deter- mining speed and drift angle of an aircraft at known wind param- eters. However, it is not suitable for determining wind param- eters 'in measuring the drift angle -and the ground speed. This problem can be solved as follows. Let us say that on the basis of measurements, we know the airspeed of an aircraft, the ground speed and the drift angle, /233 and we want to find the speed and direction of the wind (w) at flight altitude (see Fig. 2.59). It is clear from the diagram that the running component of the wind at flight altitude is u„ = W - V cos US , (2.59) while the lateral component is V sin US = {W-V cos US) tg AW (2.60) If we consider that the drift angle of the aircraft rarely exceeds 15° , and the cosine of the angle of drift is practically always close to unity. Formula (2.60) can be written as follows: te AW = 7 sin US W - V ' However, since 240 then we have the ratio V sin US - ({/-7) tg AW, sin US _ tg AW \J - V ~ V which can be used as a key on the slide rule (Fig. 2.61, a). Fig. 2.59. Navigational Speed Triangle. Example . 520 km/hr; US angle . V . L w c w J. a, o. ree but with ^ key which = 450 km/hr; W = +10° . Find the wind Solution. The difference be- solution. The difference b tween the ground speed and airspeed (,W-V) is equal to 70 km/hr. If we set this value on scale 5 opposite 10° on scale ^■ (Fig. 2.61, c), we will find the wind angle to be equal to 48°. The wind speed is found with the aid of a key which is described in the sine theorem (Fig. 2.61, d). Answer. 105 km/hr. The fixed scale on the ruler 6, like scale 5 , is a scale of logarithms of linear values, but the scale is twice that used on first five scales _. - ms of linear values, but the scale is twice that used on t five scales. Therefore, when comparing any of the fir les to the fixed scale, a number is obtained on the latt garithm is equal to half the logarithm of the numbers of t five s cales . is as caL_ _ _ that used on f the first er the five s ca whose logax--L L 11111 xo c^u the first five scales. Example . In setting the marker of the slider to the number 400 on scale 5 or 1, this marker shows half the log of 400 on the sixth scale, which corresponds to the square root of 400 or 20. If the desired number is set on scale 6, we will obtain num- bers on scales 5 and 1 whose logarithms are equal to twice the loga- rithm of the given number, thus corresponding to that number raised to a power of two. The turn radius of the aircraft with a given banking angle (g), as we know, is determined from Formula (1.6). /234 i? = - g tgf 241 Therefore, the problem of determining the turn radius is solved by means of scales H, 5 and 5: Igi? = 21gF - Igg- - Ig tgg. Therefore, in solving this problem, it is necessary to have the logarithm of the square of the speed and to set it on scale 6. The logarithm of the tangent of the banking angle is calculated with the aid of scale ^■ . a) ©US ©" AW AW+US w b) (Dsy ® so 50- 56.5 m Ada Fig. 2.60. Calculation on the NL-IOM: (a) Key for Solv- ing the Navigational Speed Triangle; (b) Solution of Navigational Speed Triangle . If we consider that in order to determine the turning radius, the airspeed of the aircraft must be expressed in m/sec and not in km/hr, as we did on scale 6, and also that it is necessary to take into account the acceleration due to gravity g, we have a marking i? on scale 4 which corresponds to the logarithm of the number 5 = 0.00787, 3^62.9,81 i.e.. Formula (1.6) assumes the form: 0,00787^2 ^ = - tgP or lg;? = 21g K+ Ig 0, 00787 -igigp, which corresponds to the key for the navigational slide rule which was shown in Figure 2.62, and which is found at the beginning of the third scale of sines. The last scale on the slide rule NL-IOM is the scale la, which is intended to determine the turning time (tp) of the aircraft at a given angle ( UT ) at a known turning radius (i?) and flight speed (F). This scale is a scale of logarithms for the arc of the circum- ference, relative to the radius of turn of the aircraft. Obviously, the turning time of the aircraft at the given angle /235 will be IkR UT '" = ■ V 360 (2.61) 242 In this formula, the value 2tt/360 is a constant multiplder . In order not to have to calculate it each time, scale la is set to the value of the logarithm of this multiplier at the left-hand side . After dropping this multiplier. Formula (2.61) assumes the form : <p = RUT or ig'p = ig>? + igUT-ig v^. a) © 5inUS w-c c) © w- © 70 tgAW^ 4/' isa © tgUS tg AW V ® ,0- ',1' © m iSO Fig. 2.61. Calculation on the NL-IOM: (a,b): Keys for Determining the Wind Angle; (c,d): Determining the Angle and Speed of the Wind. which can be expressed on the rule scales by a key shown in Fig. 2.62, b . Example. R = M- . 5 km, V = HOO km/hr, UT = 90°. Find the turn- ing time of the aircraft . Solution. See Fig. 2.62, c. Answer. *_ = 6 4- sec. On the back of the rule are scales for making methodological corrections in the readings of navigational instruments (altimeters, airspeed indicators, outside-air thermometers). Adjustable scale 7, with a movable diamond-shaped index and the adjacent scales (fixed scale 8 and movable scale 9) are intended for making corrections in the readings of barometric altimeters in case the actual mean air temperature does not agree with the calculated temperature obatined when adjusting the apparatus. These corrections can be made with the formula Igff corr 1 H ^ ^ mst Ig 2 ^ T 243 According to this formula, the adjustable scale 7 is a scale of logarithms T q+T u/ 2 . For convenience in use, the logarithms Tq + Tjj/2 on the rule are marked tg+t. The arithmetic effects of converting temperatures from the centigrade scale to the absolute scale and their division in half are taken into consideration in the design of the scales in such a way that it is not necessary to make them each time during the flight. /236 a) B b) 40 V/Sff 90 (fa) Y-W I r UT © ip © V Si Fig. 2.62. Calculation on NL-IOM: (a) Key for Determin- ing Turn Radius. (b) Key for Determining Turn Time. (c) Determination of Turn Time. Fixed scale 8 (corrected altitude) is simply a scale of log- arithms of altitude, while the movable scale 9 (instrumental alti- tude) is a scale of logarithms of altitude, divided by the aver- age calculated temperature obtained for each altitude, i.e., H lg7 inst av . c , ^ mst IrT av . c . The key for solving problems by introducing methodological correc- tions to the readings of the altimeter are shown in Figure 2.63, a . Example. The flight altitude according to the instrument is ^inst = 6000 m; tjj = -35°. Find the flight altitude corrected for the methodological error. Solution. The actual temperature for a zero altitutde is de- termined from the temperature gradient equal to 6.5 deg/km: to = t^ + 6.5 km ■35 + 6.5-6 = +4° , so that ta+i O'^'^H •31° . If we set this temperature value on the slide rule (Fig. 2.63 b), we will obtain H corr 5.74 km, To introduce corrections in the readings of the altimeter at flight altitudes above 12 km, we use movable scale 10 with the adja- cent fixed diamond-shaped index, as well as the adjacent scales: fixed scale m for the corrected altitude and speed, and fixed 21+4 scale 15 for the instrumental altitude and speed. Corrections to the readings of the altimeter at flight alti- tudes above 12 km are made by Formula (2.36). Expressing the altitude in km, this formula can be written /237 as follows: lgiH^^^^-11) = IgT^ - Ig 216.5 + Ig(ff^^^^-ll). (2.62) av Adjustable scale 10 is a scale of logarithms ( Ig3'jy^^-lg216 . 5 ) . Scales 14- and 15 are scales of logarithms (H-ll km), so that they are simple, unique logarithmic scales -on which we can carry out multiplication and division of numbers, but with additional numbers which are shifted by 11 km to calculate altitude. a) b) Fig. 2,63. Calculation on NL-IOM: (a) Key for Introducing Methodological Correction in Altimeter Reading. (b) Deter- mination of Correction for Measured Flight Altitude. In accordance with Formula (2.52), the key for introducing corrections in the readings of the altimeter at flight altitudes above 12 km is shown in Figure 2.64 a. Example, ^inst ~ 1^ ^^'■' '^H = -50°. Find ^corr* Solution. See Figure 2.64, b. Answer: ^corr ~ 14,400 m. Note. Since the altitude of the tropopause at middle latitudes is not exactly at an altitude of 11 km, but can change within limits of 9-13 km, after solving the problem by means of the key shown in Figure 2.63, b, the flight altitude must be corrected for the additional correction AH - 900 + 20 {tQ+tj^) which is shown on the rule at the right-hand side below scale 14. a) b) ®(^ "corr® ® -so' M,« ® ^ ''■inst® ^ " ® Fig. 2.6tt. Calculation on NL-IOM: (a) Key for Intro- ducing Correction in Flight Altitudes above 12,000 m; (b) Determination of Correction for Flight Altitude above 12,000 m. 21+5 The methodological corrections due to the failure of agree- ment of the actual air temperature with a calculated value are made to calibrate the speed indicator (type "US") with the aid of Formula (2,53). In accordance with this formula, the scale IM- on the ruler for log7^jP^g and scale 15 for log V'j^jjg-i- are purely logarithmic scales of linear values. Adjustable scale 11 (temperature for speed) is a scale of logarithms ylg(273" + <//). while adjacent to it is fixed scale 12 (instrument altitude alti- /238 tude in km) with a scale of logarithms -— lg288 + 2,628 lg(l -J- 0,0226//). The key for introducing corrections in the readings of the speed indicator "US" is shown in Figure 2.65, a. Exam-pie. t^ = -30°, ^inst = '^ ^^' ^inst - ^^° km/hr. Find the airspeed. Solution: See Figure 2.65, b. Answer: 638 km/hr. ® t^ V,coTr ® -3S' SJ8 ® ® "'xn^X ''inst ® 45P QS) Fig. 2.65. Calculation on NL-IOM: (a) Key for Introducing Correction in Readings of Type "US" Speed Indicator; (b) Determination of Correction for Reading of Type "US" Speed Indicator . For speed indicators of type "CSI", the corrections given above are found by Formula (2.54 a). It is clear from this formula that fixed scale 11 and movable scale 15 (for ^inst^ ^"^ ■'-^ (for ^corr^ will be the same for the speed indicators of types "US" and "CSI". Instead of fixed scale 12, we can scale 13 on speed indicators of type "CSI", which is a scale of logarithms 1 =- lg(288 - 0.0065 H. ^) 2 ^ mst The key for introducing corrections in the readings of these indicators is shown in Figure 2.66, a. 21+6 Example. H = 10 km, tjj - -45°, Vqqj = 800 km/hr. Find the corrected airspeed. Solution: See Figure 2.66, b. Answer: 808 km/hr. Rule scale 16 is set up according to the formula 1/2 M^K 26000 and is used for introducing corrections into the readings of the thermometer for the outside air, type "TUE". This same scale can be used at subsonic airspeeds, and the error will not be greater than 1-2° for the type "TNV" . In practice, the front side of the navigational slide rule NL-IOM can be used to solve a number of other problems, the keys for whose solution are directly dependent on the nature of the prob- lem. One example of such a problem is the determination of the de- /239 viation angle of the meridians between two points on the Earth's surface . The angle of deviation of the meridians can be determined by Formula (1.82). Obviously, this problem can be solved on a ruler by means of a key shown in Figure 2.67, a. a) b) ® t„ Vco:^'g) @ -f SOS ® Fig. 2.66. Calculation on NL-IOM: (a) Key for Introducing Correction in Reading of Type "CSI" Speed Indicator; (b) Determination of Correction for Reading of Type "CSI" Speed Indicator . The scales on the back of the ruler can be used to solve some other problems. For example, movable scales 14 and 15 are the ones most suitable for multiplication and division of numbers. Scale 14 is marked off with the following values: AM (Ameri- can statute mile, equal to 1.63 km); NM (nautical mile, equal to 1.852 km), and foot (equals 32.8 cm). These markings are used for rapid conversion of measurements from one system to another. 247 a) h) i (A,-A,l (g) JIS S8S Fig. 2.67. Calculation on Nl-IOM: (a) Key for Deter- mining Angle of Deviation of Meridians; (b) Conver- sion of the Length of the Arc of the Orthodrome into Kilometers . Example . Convert the length of the arc of the orthodrome 5°16' to kilometers . Solution. 5°16' = 316 NM (nautical miles). Having set division 100 on scale 1^ on the navigational slide rule opposite 316 on scale 15 (Fig. 2.67, b), we obtain the answer (585 km) . On scales 14 and 15, by using the settings of scales 11 and 12, we can solve problems in determining the Mach number at a known airspeed and air temperature at a flight altitude, or determine the airspeed at a given Mach number and air temperature. Therefore, the speed of sound in air is found by the formula M = a = 20, 3 1/273° + %, V' true V^true <* 20,31/273° + % or /240 'g^ = 'gVue'220,3— |-lg(273°.+ %). ^^^^^^ Scales 14 and 15 are scales of log V, fixed scale 11 is the scale of 1/2 log (273°+*^), and fixed scale 12 is a scale of 2.628 log (1-0.02265), which is movable relative to scale 11 to the value l/21og288. H 1- @ J,25 277,5 M UM @ Fig. 2.68. Determination of Mach Number on NL-IOM, 248 In order to get log20.3 from the value 2.628 log (l-0.0226ff), it is important to replace H by a value of 3.25 km. Therefore, if we find an altitude of 3.25 km and set it on fixed scale 12, we will obtain the key for solving the problem with a certain M number (Fig. 2.68). Obviously, the value M = 1 corresponds to the airspeed (in km/hr) which is equal to the speed of sound. To determine the speed of sound in m/sec, it is necessary to set the value of 0.2775 (1/36) on scale 15. If we use the rectang- ular index with the marking of 1000 for M = 1, then division 0.2775 will correspond to the number 277.5. Note. In general, for converting zero altitude, correspond- ing to 1/2 log288, to the value of log20.3, it is necessary to shift it to the right to the value 2.51 km, and the functions of scales 14 and 15 in the key shown in Figure 2.100 will change places. Then Formula (2.63) will be valid. The fact that the numbers 2.51 (with a shift to the right) and 3.25 (with a shift to the left) are not equal is explained by the fact that zero altitude under standard conditions does not corre- spond to zero temperature but to +15°. Therefore, to make zero temperature match division H - 3.25, the scale must be moved by an amount such that it lines up with the marking H = -2.51 km. 249 CHAPTER THREE AIRCRAFT NAVIGATION USING RADIO-ENGINEERING DEVICES 1. Principles of the Theory of Radi onavi gational Instruments Geotechnical methods of aircraft navigation, although they /2'4l form the basis of the complex of navigational equipment on an air- craft, do not permit a complete solution of the problems of air- craft navigation when there are no terrestrial landmarks or when the latter are invisible. The principal reason for this is the variation of the wind at flight altitude, which means that the flight cannot be maintained for a significant period of time without checking the distance and direction of the path being followed. Astronomical means, however, are not always helpful in deter- mining the location of the aircraft, since the heavenly bodies are just as invisible as terrestrial landmarks when flying in clouds or between cloud layers. In addition, in order to determine the location of the aircraft, it is necessary to see at least two luminar- ies in the sky simultaneously, which is not always possible under normal flight conditions . sary to seek new methods of reliably anv Dhvsical and geograph- dence upon meteor- t of radio-engin- All radio-engineering devices for aircraft navigation use the properties of the propagation of electromagnetic waves in the Earth!s atmosphere to varying degree.s . We know that the phase velocity of the propagation of wave energy in dielectric media is Cl= ,r V^" 250 where c i is the rate of propagation of electromagnetic waves in the medium, a is the rate of propagation of electromagnetic waves in a vacuum, y is the magnetic permeability, and e is the dielec- tric constant. For a vacuum, p=e = 1. /242 In addition to the phase propagation rate of electromagnetic waves, there is also a group propagation rate of electromagnetic energy . In a vacuum, the phase and group propagation rates for elec- tromagnetic waves are the same in all cases. In dielectric media, especially in solids, liquids, and (to a much smaller degree) gases, the phase propagation rate depends on the frequency of the oscillatory process. This is explained by the inertia of the dielectric medium, i.e., the dielectric perme- ability of the medium depends on the oscillation frequency. The dependence of the phase propagation rate upon the oscil- lation frequency is called dispersion. If the waves propagate in an electromagnetic medium with different frequencies, their phase rate may not be the same. In this case, the total energy of the waves will be maximum at those points in space where the phases of the waves are closest to coincidence. In addition, there will be points where the total energy of all the waves will be equal to zero, i.e., where the positive phases of the waves will be bal- anced by the negative ones. The points with maximum total energy are called centers of wave energy. The rate at which the centers of wave energy move in space is the group rate of the waves . The group rate of propagation of electromagnetic waves in space '^g-r- c, — rfc. where Cgp is the group rate, oj is the average spectral frequency, and Ci is the average phase rate of the spectrum. It is clear from the formula that with positive dispersion, the group rate of the waves exceeds the phase rate of their prop- agation . Wave Polarization Figure 3.1 is a graphic representation of a propagating elec- tromagnetic wave in the horizontal plane as a function of the vertical open circuit. 251 In this case, the vector of the electrical field, and there- fore the displacement currents, will coincide with the direction of the dipole of the circuit (dipole open antenna). The plane of the vector of the magnetic field coincides with the horizontal plane. Obviously, the electromagnetic wave is a transverse wave, i.e. , /2H3 the amplitudes of the oscillations of the electric and magnetic fields are located at right angles to the direction of wave propa- gation . The direction of the plane of oscillation of the electric field is called the vector of wave polarization. In our sketch, we have electromagnetic waves with a vertical polarization vector. In receiving electromagnetic waves, it is important to be sure that the direction of the dipole of the receiving circuit coincides with the vector of wave polarization. In this case, the oscillations of the electric field and the axis of rotation of the magnetic field coincide with the direction of the dipole, and both of these factors will bring the electromagnetic force (and consequently the conduc- tivity currents) to the receiving antenna. isophasal circles Fig. 3.1. Propagation of an Electromagnetic Wave from a Verticle Dipole. to the transmitting antenna, ization vector of the waves. If the waves are vertically polarized and the receiving antenna is located in a horizontal position, no emf will be produced in the dipole . With the dipole in a horizontal position, the electromagnetic waves reaching the antenna will have a horizontal vector of polariza- tion. In this case, the receiv- ing antenna must be horizontal; in addition, the direction of the antenna in the horizontal plane must be perpendicular to the line . , it must coincide with the polar- The circles in Figure 3.1 join points in the horizontal plane which have identical phases for the electromagnetic waves. These circles are called isophasal. From the viewpoint of the receiving antenna, the isophasal circles (and the isophasal spheres in the propagation area) are the directions of the wave front. 252 Propagation of Electromagnetic Oscillations in Homogeneous Media In order to make use of the principles of design of various transmitting and receiving radio navigational instruments, it is necessary to become acquainted with the characteristics of the prop- agation of electromagnetic oscillations in inhomogeneous conduct- ing and nonconducting media. Electromagnetic wave processes in dielectrics constitute the /244 conversion of the potential energy of the electrically deformed medium to the kinetic energy of displacement currents and vice versa (the kinetic energy of the field into the potential deformation of the medium ) . ation is not )f dielectric materials, polari z,ci l j-uu j-s u^ because wave energy is prop ^+-nr^-n= A decrease For the majority of dielectric m related to absorption of wave energy, .. _ . . _ __ agated practically without losses in all directions. A decrease in the oscillation power with distance takes place due to the fact that the wave energy fills an increasingly large volume, which (as we know) is proportional to the cube of the radius of the sphere whi ch it f i lis . we know) is proportion which it fills Significant losses in wave energy can occur in solid dielec- trics with polar molecules. In this case, the polarization is not related to elastic deformation but to the motion of molecules, which causes a conversion of wave energy into heat. In conducting media, the electromagnetic waves carry alter- nating conductivity currents. This means that conductors always undergo absorption of wave energy and its conversion to heat. Thus, the propagation of wave energy in media, exhibiting both electronic and ionic conductivity, is practically possible to a slight depth which depends on the conductivity of the medium and the frequency of the oscillations. The higher the oonductivity of the medium and the greater the frequenoy of osoillatiorij the shatlower the depths to which the oscillations witt propagate . Since the propagation rate of electromagnetic waves depends on the dielectric and magnetic permeability of the medium, and the electronic or ionic conductivity of media can be assumed to be a very high (approaching infinity) dielectric permeability, the concept of optical density of media has been introduced. The minimal optical density (equal to one) is possessed by a vacuum (where the propagation rate of the waves is equal to c) . The optical density of all other dielectrics is greater than unity. In ideal conductors, the optical density is equal to infinity (the propagation rate of electromagnetic waves is equal to zero). In portions of a medium with varying optical density, electro- 253 magnetic oscillations change the direction of their propagation. The change in direction of propagation of electromagnetic waves on the surfaces of particles of the medium with different optical density is called refraation of rays. In addition, under certain conditions , there is reflection of waves from the surfaces of the sections. The coefficient of reflection depends on the difference between the optical densities of the media, the frequency of the oscillations, and the angle of incidence of the wave. When the path of a wave (propagation direction) runs from a /245 less dense medium to a more dense one, with a certain angle of inci- dence to the surface, there may be no separation of the reflected / wave. Such an angle is called the angle of total intevnal vefteo- tion of the denser medium. If the medium with the greater optical density is a conductor, irreversible absorption of wave energy may take place in it (conversion of wave energy into heat). With the gradual change in the optical density of the medium, there is a continuous refraction (bending) of the line of propaga- tion, called Tadiorefractton. The optical inhomogenei ty of a medium characterizes the prop- agation characteristics of waves of different frequencies in the Earth's atmosphere. All harmonic oscillations in a medium are characterized by an oscillation frequency ( o) ) and an amplitude oscillation (£") . If we say that the amplitude oscillation is the maximum value of the intensity of the electrical field, then at any fixed point the oscillation process will satisfy the expression: where E = Eq sin(a)t + (j) ) , is the initial phase of oscillation The derivative of the field intensity with time will charac- terize the magnitude of the displacement current dis dE -dt ■eE r\Msin(.b)t + (^ ) . while the second derivative will express the acceleration of the displacement current J^. = eEnixi^ cos ( wt + d) ) . dis ^ The distance between the two closest points in space which lie along the line of propagation of the wave front, in which the wave phase is identical, are called the wavelength (A), which is equal to Cj/u 254 Electromagnetic waves can be subdivided into four groups on the basis of their propagation characteristics in the Earth's atmo- sphere . 1. Long waves, from 30,000 to 3000 m (10-100 kHz). These waves have a surface type of propagation. Conducting media such as the Earth's surface and the upper ionized layers of the atmo- sphere have a deflecting effect upon them. 2. Medium waves, from 3000 to 200 m (100-1500 kH25) have a complex type of propagation. In the day, when the ionized layers of the atmosphere are lower, the type of propagation is superficial as in the case of long waves; at night, the medium waves have both a surface and spatial type of propagation. 3. Short waves J, from 200 to 10 spatial type of propagation. (1500-30,000 kHz) have /246 4. Ultra-short waves, less than 10 m, have a radial type of propagation. They can be reflected from conducting layers on the Earth's surface, but only under certain conditions can they be re- flected from the ionized layers of the atmosphere. Therefore, it is thesfe waves which are used within the limits of geometric visi- bility of objects. The resistance to these waves on the Earth's surface is insignificant. From the point of view of electrical conductivity and relief, the Earth's surface has a complex nature which depends on the time of year and weather conditions. The ionized layers of the atmo- sphere also have a varying nature. The ionized D layer, which is closest to the Earth's surface, is only observed in the daytime and depends on the time of year, time of day, and geographical latitude; it may appear at heights from 50-90 km. This layer has an effect on the propagation of long and medium waves. The critical frequency of the layer is . M- MHz (750 m). Waves with frequencies higher than the critical are not reflected from the layer. at a reta this to of m boun hour mum meas and in t Above heigh ins it layer .9 MHz edium dary w s , the ioniza uremen conseq he hor this is the E layer, whose ionization maximum is reached t of 120-130 km. This layer is the most stable one and s effect both day and night. The critical frequency of with maximumi llumination is 4.5 MHz; at night it drops this layer has a maximum effect on the propagation and intermediate waves (the short waves at the spectral ith the medium waves). During the evening and morning layer changes its parameters so that the surface of maxi- tion decreases. This leads to errors in radionavigation ts , since it reverses the vector of the wave polarization uently the direction of propagation of the wave front izontal plane. 255 The third ionized layer (.F) is the most unstable one both in terms of time of day as well as season of the year. Its average height is 270-300 km. During the daytime in summer, this layer divides into two parts (Fi and F 2) • In addition, the F layer shows some shifting in homogeneities, which make it difficult to predict the propagation conditions for electromagnetic waves. The F layer has an influence on the propagation of short waves. It should be mentioned that the medium and short waves are reflected both from the ionized layers of the atmosphere as well as from the Earth's surface, so that they may undergo multiple reflec- tion. All of this combines to give us the complex picture of the propagation of electromagnetic waves in the Earth's atmosphere, which must be taken into account in radionavigat ional measurements. The peculiarities of propagation of electromagnetic oscillations in a conducting feeder channel in receivers and transmitters include /2M-7 the following. Unlike constant and low-frequency alternating currents, high frequency currents propagate mainly along the surface of a conductor, since the reaction of the magnetic field within the conductor is greater than on its surface (the skin effect). This causes all high frequency conductors to be constructed with an eye toward increas- ing the surface, e.g., tubular and multiple-filament (stranded wire). However, these measures are insufficient for waves in the cen- timeter range. It is much better to use hollow conductors for these waves, called wave guides (Fig. 3.2), ■I R% Fig. 3.2. Propagation Electromagnetic Waves Along a Wave Guide. the surfaces of which i the vertically polarize walls of the box and be opposite wall also with reflection of the waves lations . In this case, along which the waves w any resistance. In a h propagation the vector — in the dire vector of p of of the wave gation dire are the s am type wave g s measured in whole d wave (striking th ing reflected from a whole number of will take place in the box-type wave ill propagate thems omogeneous medium, the rate of the field along of the wave polarization, ction of the perpendicular olarization, in the plane front, and in the propa- ction of the wave front e . Therefore , in a box- uide , the distance between numbers of half waves, e top and bottom internal them) will strike the half waves. Consequently, resonance with its oscil- guide will act as a channel elves practically without If the distance between the walls is equal to a whole, odd number which is one-quarter of the wavelength, then (as is easily 256 seen) the reflections from the walls of the wave guide will take place each time in opposite phase with the oscillations. In this case the wave guide will have infinite resistance, and the wave energy will not be propagated in it. In Figure 3.2, the vector of wave polarization must be turned 90° to accomplish this. Principles of Superposition and Interference of Radio Waves The principle of superposition is applied to wave processes, i.e. 5 each of the wave processes acts independently of other processes which are taking place in the medium or circuits. At the same time, the results of different processes can be summed by means of a simple superposition of oscillation vectors. If the vectors of two coherent (coinciding in frequency) processes such as the oscillations in the intensity of a field or displace- / 2 M-8 ment currents, are equal in amplitude and coincide in phase, the total amplitude of the oscillations will be doubled. Under these conditions, if the oscillations are in opposite phase, the total amplitude of the oscillations will be equal to zero and no method will suffice to detect the presence of the wave processes involved. Summing of the results of the processes in opposite phase is called wave -intevfeTenoe . The case in which the result of summing of the oscillations is equal to zero is called total -IntevfevenGe . The properties of interference of radio waves are widely em- ployed and r adionavigat ional devices both in receivers and trans- mitters, especially in measuring the direction of an object. Principle Characteristics of Rad i on a v i ga t i on a 1 Instruments The principle characteristics of transmitting radi onavigat i onal instruments are the following: (1) The radiated power, characterizing the operating range of the system . (2) Accuracy and stability of the frequency structure, as well as synchronization of special navigational signals. As far as the antenna arrays are concerned, which incorporate certain characteristics for radiation of signals, we will discuss them under the heading of "Principles of Operation of Concrete Naviga- tional Instruments". signals are combined and an intermediate frequency is produced which is equal to the difference between the frequencies given above. In such devices, further amplification of the signal is carried out with a constant, lower frequency, which makes it possible to use amplifier devices with very high coefficients of amplification, as well as to ensure a high selectivity of the receiver. Usually, receiving radionavigational instruments fulfill two functions: (a) reception and amplification of the signals from a transmitter (b) separation and indication of measured navigational parameters . The basic characteristics of receiver navigational instruments are the following: (1) Sensitivity of the receiver which characterizes the pos- sible receiving range for signals from a transmitter. (2) Selectivity of the reception; this parameter is usually obtained by narrowing the frequency band which the receiver will pass, which usually characterizes the freedom from noise of the re ceiver . /2H9 (3) The accuracy with which the navigational parameters are selected and recorded. Operating Principles of Radionavigational Instruments In accordance with the laws of propagation of electromagnetic waves in space, it is possible in principle to measure the follow- ing parameters of electromagnetic waves: amplitude, phase, fre- quency, and transmission time of the signal. According to the principle of technical operation, radionav- igational devices are divided into amplitude, phase, frequency and time devices . In addition, with a mutual exchange of radio signals between objects which have relative motion to one another, changes in the frequency characteristics of the signals occur which are known as the Doppler effect, which is used to build automatic airspeed indi- cators and devices for measuring the drift angle of an aircraft. Measurements of the parameters listed above for electromag- netic waves from the navigational standpoint make it possible to determine the following navigational elements: (a) The direction of the object, by means of goniometric sys- tems ; (b) The distance to an object, by means of rangefinding sys- tems ; 258 (c) The difference or sum of the distances to the object: hyperbolic or eliptical systems ; (d) Speed and direction of movement of the aircraft: auto- matic Doppler meters for ground speed and drift angle. For convenience of application, in many cases the navigational systems are compensated for measuring two navigational parameters simultaneously. For example, there are the goniometric-rangef inding systems, difference-rangef inding instruments, etc. The panoramic radar located on the ground and on the aircraft are goni ometri c-range finding devices with a single unit of navi- gational equipment. In studying methods of applying radionavigational systems , it is a good idea to classify them according to the principles by which the navigational parameters are measured. Therefore, the further subdivision of the material will be made on the basis of these principles. Radionavigational devices can also be subdivided into auto- matic and non- automati c . Non- automati c devices, when they consist of systems of ground control and apparatus aboard the aircraft, are called navigational systems. Automatic devices are called auto- matic nav-igat-ionat systems when the operation of several types of navigational devices is combined organically on board the aircraft. For example, the automatic Doppler system for aircraft navigation, which consists of a Doppler meter for the drift angle and the ground speed, course devices, and the automatic navigational instru- ments . /250 2. GONIOMETRIC AND GON I OMETRI C-RANGE FI NDI NG SYSTEMS The goniometric radionavigational systems are the simples ones from the standpoint of technical requirements, and are th fore those which are most widely employed at the present time. lest ere- Fig. 3.3. Reception of Electromagnetic Waves by a Frame Antenna . In the majority of these systems, the amplitude method of measurement is employed, based on the interference of electromagnetic waves. This principle serves as the basis of the operation of ground and aircraft-mounted radio direc- tion finders, which are also called radio compas ses . Let us imagine a frame-type receiving antenna, located in a field of outwardly directed radio waves (Fig. 3.3). If the frame antenna is located relative to the transmitter so that the direction^, of the 259 propagating waves will be perpendicular to the plane of the frame, the left and right vertical sides of the frame will be on the same isophasal circle. In this case, the high-frequency currents which are conducted in the sides of the frame will agree in phase and will consequently be directed toward one another. This gives complete interference of the oscillations of the currents in the frame, and there will no reception of signals from the transmitting station. If the frame is turned around the ver so that the direction of the plane of the direction of the transmitting station, the be at different isophasal circles, maximal device. Thus, the currents in the vertica undergo a certain phase shift which will g of the signals from the station. The maxi will be observed in the case when the dist is equal to half the wavelength. Then the ical sides of the frame will be in opposit tudes will be added. However, this requir of cases) cannot be fulfilled, since the a too unwieldy; therefore, we use that part obtained with a phase shift through a smal angle. In these cases, the receiving fram turns and a radio receiver with very high The vector diagram of the reception direct will have the form of a figure eight (Fig. tical axis through 90° , frame coincides with the sides of the frame will ly distant for the given 1 sides of the frame will ive maximum reception mum effect of the frame ance between its sides currents in the vert- e phase and their ampli- ement (in the majority ntenna device becomes of the effect which is 1 (frequently very small) /251 e is supplied with many sensitivity is employed, ionality of the frame 3.4) . The greatest accuracy in range finding is "obtained with min- imum reception, while at the maximum the change in amplitude of the received signal is obtained by turning the frame at a slight angle. Therefore, range finding by means of a frame is always done with minimum reception or audibility of the signal. )a Fig. 3.4. Fig. 3 . 5 Fig. 3.4. Diagram of Reception of a Frame Antenna. Fig. 3.5. Edcock-Type Antenna. 260 trom not nent f ram and base are for ment The agnet only of t e . I me cha d rad equiv verti ioned recei ic wa the V he po n add nical io ra alent cally abov ving frame antenna has the shortcoming that when elec- ves are being propagated through space, it picks up ertically polarized wave but also the horizontal compo- larization vector in the top and bottom sides of the ition, the frame antenna with its large dimensions rotation is inconvenient to use. Therefore, ground- ngefinding installations use special antennas which to a frame type in the characteristics of reception polarized waves, but are free of the shortcomings e; they are called Edcock antennas (Fig. 3.5). The picture shows one pair of Edcock dipoles with the coil of a goniometer between them. Obviously, in open dipoles, no inter- ference will be observed when they are located on one isophasal circle. However, the difference in potential at the ends of the goniometric coil will be equal to zero, since they are connected to symmetrical points on the dipole . If the dipoles are located on different isophasal circles, then the phase shift will disturb the potential equilibrium at the ends of the coil and a high-fre- quency current will pass through it. A similar pair of dipoles is mounted in the plane perpendic- ular to the first pair. The high-frequency current in the goniometer coils will depend /252 on the direction of the transmitting station relative to the crossed dipoles . In a goniometric instrument, in addition to the two fixed dipole coils mounted at an angle of 90°, there is a movable searching coil, connected to the input circuit of the receiver. If the searching coil is placed in the resultant field of the fixed coils, the reception will be maximum; when a coil is placed at an angle of 90° to the resultant field, reception will be min- imal . receiver Fig. 3.6. Inclusion of an Open Antenna for Solving Ambiguity of Reception. The horizontal wires connecting the antenna dipoles are located as close as possible to one another, so that the electromotive force con- ducted in them from the horizontal component vector of polarization will be in the same phase, and their total interference will appear at the inputs in the goniometer coils. Therefore, the antenna does not pick up component waves with horizontal polarization, thus considerably reducing the range finding error for waves in space . 261 The reception characteristics of the frame antenna (includ- ing the Edcock type) have two signs, i.e., we have two maxima and two minima of audibility, so that it can be used to determine the direction line on which the transmitting and receiving objects are located, but does not solve the problem of the sides of the mutual position of the objects (see Fig. 3.U). To solve the ambiguity of reception with radio rangefinding instruments, an open antenna with an externally directed (circular) reception characteristic is used in addition to the frame antenna (Fig. 3.6). The phase of the high-frequency current in the open antenna, depending on the reception direction, will coincide with the phase of one of the sides of the frame receiver and will be in opposite phase with the currents in the second side of the receiver. As a result, the current amplitudes of an open antenna will be added to one-half of the figure eight of the frame antenna and will inter- fere with the other half of the figure eight (Fig. 3.7, a). In combining the characteristics of the frame and open antennas, we obtain a total characteristic which has the form of a cardioid. If we connect the open antenna and turn the frame antenna through 90° clockwise, the maximum reception shown in Figure 3.7, b will shift to the upper part of the picture while the minimum will shift to the lower part (Fig, 3.7, c). This corresponds to one side of the minimum of the frame receiver being transferred to the maximum, /253 and the second remaining minimum. Fig. 3.7. Diagram of Directionality of a Frame Antenna Combined with an Open Antenna. Let us suppose that we have defined a line (bearing) on which the transmitting and receiving points are located at minimum aud- ibility. After connecting the open antenna and turning the goni- ometer coil through 90°, we can determine the direction of the trans- mitter. If the audibility of the signals increases sharply, the transmitter is located in the direction of the upper part. If it remains as before or changes slightly, the transmitter is located at the opposite side. 262 The principles described above for finding the direction of a transmitter are used in ground radio direction-finding installations. In this case, the transmitter is the radio on board the aircraft. Ground radio direction-finders in principle can operate at all wavelengths. The most widely used radio rangefinders operate on short and ultra-short waves . The position of the aircraft can be determined by means of the ground radio rangefinder in terms of the minimum audibility of the signal from the transmitter located on board. In addition, visual' indicators are mounted on the ultrashort wave (USW) range- finders, such as cathode-ray tubes. In this case, the frame of the direction-finder or the gon- iometer coil is set to rotating rapidly, and the scan of the cathode- ray tube is synchronized with it. The amplitude of the scan is related in magnitude to the amplitude of the received signals in such a way that at minimum reception the maximum amplitude of the scan is observed. Then, on a scale which is marked along the periphery of the tube face, we can determine the direction of the aircraft in terms of the position of the maximum deflection of the scan. With a relatively low density of air motion, the ground radio rangefinders are a sufficiently effective and precise method of aircraft navigation. An advantage of ground radio rangefinders is the lack of a need to mount special radio equipment on the air- craft. The radio rangefinders and receivers which are used for / 2 5 M- receiving signals from ground direction-finding points mainly have other purposes, and their use for navigational purposes is not related to the increased complexity and weight of the equipment on board. At the same time, however, the use of ground radio direction- finders has a number of serious shortcomings, which have led to a search to find new ways of radionavigational control of flight. The most important of these shortcomings are: (a) Lack of a visual indicator on board the aircraft to show its position, thus reducing the ease of aircraft navigation. (b) A small capacity for the ground installations; at the same time, the radio direction-finder can only operate with one aircraft, which is clearly inadequate when there are a great many flights . Aircraft Navigation Using Ground-Based Radio D i rec t i on- F i nde rs The use of ground-based radio direction-finders can be used to solve the following navigational problems: (a) Selection of the course to be followed and flight along 263 the straight-line segments of a route, at the beginning or end of which radio direction-finders are located. (b) Control of the aircraft path in terms of distance. (c) Determination of the aircraft location on the basis of bearings obtained from two ground-based radio direction-finders. (d) Determination of the ground speed of the aircraft, as well as the drift angle, direction and speed of the wind at flight altitude . Usually, the international "Shch"-code is used for determin- ing the bearings from on board the aircraft. The crew of the air- craft reports its position, gives the required code for its posi- tion, and presses the telegraph key of the transmitter for a period of 20 sec. In recent years, both state and local civil airlines have adopted USW direction finders, with visual indicators. They are oriented according to the magnetic meridian of the location of the USW direc- tion-finder, and (depending on the flight altitude) are used in a radius of 100-200 km as a form of trace direction-finder, report- ing on board the aircraft the "forward" (away) and "back" (return) magnetic bearings of the aircraft. If the crew of the aircraft requests the forward true bearing, the operator of the USW direc- tion-finder (supervisor) calculates the magnetic declination of the location of the distance finder and reports the forward true bearing to the aircraft . Distance finding by means of USW distance finders with a vis- ible indicator is used in the course of communication with an air- craft, i.e., with a depressed tangent of the connected USW trans- mitter on board the aircraft. /255 The operator of the ground radio direction-finder, after the required measurements, gives the call letters of the aircraft, the code expression for the bearing as requested by the aircraft or used for USW communication, and gives the magnitude of the bearing in degrees . The code expressions for the bearings in the international Shch code have the following meanings (Fig. 3.8): ShchDR: magnetic bearing from distance -finder to the aircraft, or forward bearing. ShchDM: magnetic bearing from the aircraft to the distance- finder (measured relative to the local meridian of the location of the distance-finder), or reverse bearing. ShchTE: true bearing from the distance-finder to the aircraft, or the forward true bearing. 264 ShchGE: azimuth of the aircraft at trol distance-finding station. a distance from the con- Fig. 3.8 Bearings hGE hTE and S^ Code Expressions for 1 the Shch-code. ShchTF: location of aircraft (coordinates or Due to the small eff radius of the USW rangefi they are not grouped into tance-f inding nets like 1 or medium-wave stations, operate independently, an not give the location of aircraft . Seteot-ton of the Cours be Followed and Control o D-treation the link) . G ctive nders , dis- ong but d do the e to f Flight The selection of the course to be followed and flight along a s tr aight- line path segment are accomplished by means of periodic inquiries and determinations of the forward or reverse bearings of the aircraft (ShchDR or ShchDM). If the radio distance-finder is located at the starting point of a flight segment (flight from the distance-finder), then the ShchDR bearings are requested. When the aircraft is passing pre- cisely over the ground radio distance detector and follows a constant course for a certain period of time, the first bearing of the air- craft after passing over the dis tance - finder can be used to deter- mine the drift angle (Fig. 3.9). Usually in this case the ShchDR bearing will be equal to MFA , so that US = ShchDR MC . If the ShchDR does not correspond to the given flight path angle for the path segment, then the aircraft is put on the desired line of flight after determining the drift angle and the course to be followed is set so that the total of the course and the drift angle of the aircraft will equal the given path angle. /256 It should be kept in mind that in the general case ShchDR is not equal to MFAg , since the former is the orthodromic bearing meas- ured at the starting point of the segment and the MFA is the loxodroraic path angle measured relative to the mean magnetic meridian: ShchDR MFA where Aw^^^ is the magnetic declination at the midpoint of the seg- ment , A Ml is the magnetic declination at the location of the radio distance-finder, and 6^^ is the deviation of the meridians between 265 the initial and middle points on the path segment ^ fc _^^^ \ VJ- -"Time of outward -— 1'^S \^ bearing _ Fig. 3.9 Fig . 3 . 10 . Determination of the Drift Angle After Flying Over a Radio Distance-Finding Station. Fig. 3.9. f Fig. 3.10. Path Segment Between Two Radio Distance-Finding Stations In principle, orthodromic control of the path for a loxodromic flight is inconsistent, because in practice the course to be fol- lowed in a loxodromic system of path angles is selected so that flight takes place along the orthodrome. In order to maintain the given flight direction over the path segment with sufficient accuracy, it is necessary to note that at each bearing (ShchDR or ShchDM) the aircraft will be located on an orthodromic line of the given path and will therefore maintain this bearing . Let us explain this by a concrete example. We will assume that we must make a flight from a point A(A - 105°, Af/j = -1°) to a point B(A = 115°, Aj^ = -7°) and return (Fig. 3.10). The magnetic flight angle of the segment is 95 or 275°, while the average latitude of the segment is 52°. Obviously, for flight in an easterly direction from the dis- tance-finder, located at point A ShchDR - MFA+A,, -A„ -6 = 95-1++1-4 = 88°. M M 1 av av ^ For flight in a westerly direction, the ShchDM from this dis- tance-finder must be equal to 268°. For a flight in an easterly direction, the initial course of /257 the aircraft must be set not on the basis of MFA = 95°, but from ShchDR = 88°. In the opposite case, the aircraft slowly begins to deviate from the line of the desired bearing at an angle of 7°. 266 Analogously, for the point B (ShchDR = 282°, ShchDM = 102°), the initial course must be set 7° greater than one would conclude on the basis of the MFA. Of course, it is impossible to make flights with a constant MFA at distances at which the magnetic direction of the flight changes by 14° , This example is given only to illustrate the geometry of the process. It would be more accurate to divide this segment into four parts 150 km long with the following flight angles: MFA^ = 90°, MFA2 - 93°, MFA3 = 97°, and MFA4 = 100°. In the first two segments, we must use a distance finder which is located at Point A (ShchDR = 88°), while for the latter we must use the distance finder- at Point B (ShchDM = 102°). This division of the flight segment into parts for the case of a flight according to a ground distance finder is an approxima- tion of the initial MFA to the ShchDR of the initial distance finder, while the latter is approaching the ShchDM of the range finder located at the terminus of the flight. In the orthodromic system of calculating flight angles, the distance between the OFA and bearings ShchDR and ShchDM from one of the distance finders will be constant in value and will depend only on the meridian selected for calculating the path angles. In the special case when the reference meridian coincides with the meridian where the range finder is located, OFA will differ from ShchDR only in the magnitude of magnetic declination for the loca- tion of the distance finder: OFA - ShchDR + A, , M fore , in an or1 1 uc j.ci wi-c , J.11 an orthodromic system of calculating flight angles, the course to be followed by the aircraft changes more rarely and to a much lesser degree than in a loxodromic system, but all elements of aircraft navigation, including the speed and wind direction, are determined more accurately. are determined more There the c to a mucn xesser aegree xnan 1 of aircraft navigation, includ mined more accurately For selecting a course and maintaining the flight direction of an aircraft in terms of ground radio distance- finders , the method of half corrections is used, which consists of the following: Let us say that at a point position of the aircraft on the line of a given path, the latter is on course with a certain anti- cipation of drift. After a certain period of time, on the basis of the bearing obtained from the distance finder, it is found that the aircraft is shifting from the line of flight toward the direction of the wind vector. This indicates that the correction in the course which has been taken is insufficient. Therefore, it is necessary to return the aircraft at an angle of 10-15° to the given line of flight, and the previously employed lead in the course to be followed is doub led . 267 If in this case the aircraft begins to shift from the line of flight toward the side opposite the wind vector, then after the second aiming of the aircraft along the given line of flight, it is necessary to make a correction in the course which is halfway between the latter and the former. If the deviation takes place along the direction of the wind vector, the correction in the course must be increased. In addition, if the deviation of the aircraft from the line / 2ZQ of desired flight takes place, the difference between the latter and the former corrections is divided and added to the course with a positive or negative sign, depending on the direction of the air- craft deviation. The placing of the aircraft on the desired line of flight by selecting the course with all the deviations mentioned is oblig- atory only in a flight from the distance-finder along a forward bearing (ShchDR). In a flight toward a radio distance-finder along a reverse bearing (ShchDM), the aircraft must follow the line of the desired path only in the case when it is going beyond the limits of the established trace. With small deviations (by distances from the radio distance-finder of up to 200 km within the limits of 1- 2°), it is sufficient to select the course to be followed by the same method of half corrections relative to the last ShchDM (reverse bearing), without going to the desired line of flight each time. The method of half corrections is the general one used for flight toward the radio dis tance- finder and away from it. However, in practical use, there are considerable differences between flight toward the distance-finder and away from it: (1) In a flight from the radio distance-finder, the drift angle can be measured at the beginning of the segment, while in a flight toward the distance-finder it can be determined only after selecting the course to be followed with a stable ShchDM. (2) In a flight from a radio distance-finder, the course to be followed by the aircraft must change in the direction opposite the change of the bearing: ShchDR increases, and the course must also decrease, and vice versa. In a flight toward a radio distance- finder, the change in the course must take place in the direction of the change in bearing: ShchDM increases, the course must be increased, and vice versa. (3) As we have already pointed out, a flight away from a radio distance-finder in all cases must be made strictly along the given bearing, while in the flight toward a radio distance-finder (within certain limits) it is permissible to select the flight to be fol- lowed according to the last stable bearing. 268 Path Controt in Terms of Distance and Determinat'lon of the Aircraft ' s Location For the purposes of controlling the path in terms of distance, as well as determining the location of the aircraft, we can use the true bearings from the ground radio distance-finder to the air- craft (SchTE) . For checking a flight in terms of distance, we usually select the control landmarks along the flight route and determine their precalculated bearings from the radio distance-finder located to the side of the aircraft route (Fig. 3.11). Three to five minutes before the aircraft reaches the control landmark, a series of "forward true" bearings are requested (ShchTE). When the bearing of the aircraft becomes equal to the calculated / 2 59 one, the passage of the control landmark is noted. By using long- and medium-wave radio direction-finders, the location of the aircraft is determined from bearings of two or three mutually related ground radio direction-finders, one of which is the command station. ftrue ,ShchTE=140° Upon request from the crew of an aircraft, with regard to the azimuth and distance from the command distance-finding sta- tion (ShchGE), the aircraft measures its distance simultaneously from two (three) distance measuring stations, while auxiliary distance finders report the measured bear- ing to the command distance station. Fig. 3.11. Previously Calcu- lated Bearing of a Landmark. The operator of the command radio dis tance - finding station uses a special plotting board to determine the true bearings of the aircraft with the aid of movable rulers with their centers of rotation at the points where the radio dis tance -finding stations are located; having measured the distance to the aircraft (the points of intersection of the bearings), the operator transmits to the crew of the aircraft its position (the true direction and distance from the command radio dis tance- finding station). If the crew of the aircraft desires to obtain data regarding the location of the aircraft in different forms (e.g., geograph- ical coordinates or relationship to some landmark), they must ask for the ShchTF bearing from the command radio distance-finding sta- tion . 269 Determination of the Ground Speed, Drift Angle, and Wind The ground speed of an aircraft can be found by using ground radio distance-finders as well as other non-automatic radionaviga- tional devices during flight on the basis of the distance covered by the aircraft between two successive indications of its position (LA): W = - . The successive landmarks for the LA are the points at which the aircraft passes over previously calculated bearings along the route or locations for the aircraft marked on a map which were obtained from the command distance-finders upon request of bearings ShchGE or ShchTF. The drift angle can be determined in three ways with the aid of ground radio distance-finders: (1) The difference between the "forward" bearing (ShchDR) /260 and the course of the aircraft after passing over the radio dis- tance-finding station: US = ShchDR - MC; (2) By the difference between the path angle of the flight and the course of the aircraft after selecting a stable "forward" bearing (ShchDR) or "reverse" ShchDM: a = ^ - y , where a is the drift angle of the aircraft, ^i is the path angle of the flight, and y is the course of the aircraft. (3) On the basis of the path angle and the mean course of the aircraft between successive indications of the PA {A and B): secona ana rnira mernoas give exacT resuxTs on±y in of the path segment, i.e., when crossing the meridi; to which the path angle of the segment is measured, ning and end of the segment, the errors are maximum, In the orthodromic system of calculating path angles and courses, the accuracy of determining the drift angle is approximately the same for all three methods . The speed and direction of the wind at flight altitude is 270 determined with the aid of ground radio distance-finders in two ways : (1) According to the ground speea of the aircraft, the air- speed, and the drift angle. To solve this problem, we can use a key oi the navigational slide rule for determining the wind angle (Fig, 3.12, a) and for determining the wind speed (Fig. 3.12, b). of t firs bear aire lati the on t the the on t time ampl (2) he a t iQ ing raft on i calm he c aire calm he b ove e : By i re r cati is m is s ma pos hart raft poi asis r a the aft on o arke f lyi de ( itio wit in nt a of give differ on the f the a d on th ng from accordi n of th h simul terms o nd the the She n path ence flig ircr e fl the ng t e ai tane f th s eco hGE segm between h t chart. aft on th ight char first lo o the ave rcraft is ous reque e ShchGE nd pos it i b earing , ent . Let the ac This e basi t . Du cation rage c deter St of or Sh c on of is the us CO tual met s of ring , th ours mine the hTF. the win ns id and calm hod means the Shch the time e calm pa e , airspe d, and al second po The vec aircraft , d vector er the fo coord i nates that the GE or ShchTF that the th calcu- ed and time ) , so entered sition of tor between determined for the flight llowing ex- After 2 4- min of flight between two successive locations, the wind vector is equal to 14-0° in direction and 30 km in magnitude. If we divide the modulus of the wind vector by the flight time /261 in hours (0.4), we will get the wind speed u 30 :0 .4 = 75 km/h . The first method of determining the wind is the one most widely employed. However, on large passenger aircraft with automatic navi- gational indicators on board (e.g., NI-50), by means of which auto- matic quiet calculation of the aircraft path can be carried out, the second method is the most suitable and precise. When this is done, it is no longer necessary to plot the wind vector on the flight chart . a) ® a.nUS ^^ tgAwi) b) ® Sin us ® Sin AW resting point 6i Xaw 77-^LA Fig. 3.12. ■' Fig. 3.13. Fig. 3.12. Keys for Determining the (a) Wind Angle and (b) Wind Speed on the NL-IOM. Fig. 3.13. Determination of the Wind by the Difference in the Coor- dinates of the Calm Point and the Location of the Aircraft. 271 It is clear from Figure 3.13, that AZ ^ AZ tgAW = -T-rr ; Ut = — : — T-rr * hX smAW where AZ = Z^ . - Z ; LA rp LA rp If we know the distance of the orthodromic coordinates of the location of the aircraft and the calm point, this problem is easily solved on the navigational slide rule using the following key: For determing AW (Fig. 3.1M-, a), and for determining ut (Fig. 3 . 14, b ) ) b) ® tg AW y ® s.nAV^ ^ (T) &i AX /iz ut Fig. 3.14-. Determination of (a) Wind Angle and (b) Wind Speed on the NL-IOM. Example: MFA = 110°; A^^ = -7°; AZ = H-0 km; AZ = 20 km; t - 15 min. Find the direction and wind speed at flight altitude. Solution: AW = 26° (Fig. 3.15, a) ut - 45 km (Fig. 3.15, b). 45 KM 45 km . „„ , u = — — :- = —— - = 180 «^/ hr lomin 0,25 t^p To determine the wind direction relative to the meridian of /262 the aircraft's location, the calculation of the given path angle of flight should be aipplied to that meridian, and then the wind angle should be added. a) b) Q) 20 «" (J) 10 «i Fig. 3.15. Determination of (a) Wind Angle and (b) Wind Speed on the NL-IOM. In a flight with magnetic path angles, we will have approx- imately 272 or m our case 6„ = MFA + AW M 6„ = 110 + 26 = 136° M "5 = fi.+A.. = 135-7 = 129° M M Automatic Aircraft Radio D i s tance- F i nders ( Rad i ocompasses ) Automatic aircraft radio direction-finders ( radiocompasses ) are very widely employed. Aircraft with piston engines use them as a reliable, operative, and highly precise method of aircraft navigation. Large passenger aircraft with jet engines, for a number of reasons, cannot make such effective use of radiocompasses, but they continue to use them successfully along with other more pre- cise means of aircraft navigation. The accuracy of distance-finding for ground radio stations with the aid of radiocompasses is somewhat lower than the accur- acy of distance finding for aircraft with ground radio distance- reasons : with the aid of radiocompasses is somewhat lower than 1 acy of distance finding for aircraft with ground radio finders, which can be explained by the following three an- (1) Stationary radio dis tance -finders can have special s which are equivalent to frame-type antennas but are fi-cc f the frame ; on air- tennas which are equ..- » a j.= aj l. .. ^ j.j.a.,--_ -^ ^ - f the effect related to the horizontal sides „^ ^..^ ^^^... ---c^ ■' ' ' ' -"""-■i-T^ *-„ ,-„„j-^TT „,,_u antennas due to their craft , it is no unwieldiness t possible to install su ch (2) The bearing of an aircraft is measured with the aid of ground radio dis tance -finder s directly from the direction of the magnetic or true meridian, passing through the radio distance-finder at a fixed setting of the antenna system relative to the vertical; /263 in distance-finding with ground radio stations by radiocompasses located on board aircraft, the error in the bearing includes the errors in measuring the aircraft course; in addition, the accur- acy of distance measurement is reduced due to the longitudinal and transverse banking of the aircraft. (3) Errors in distance-finding due to the effect on the prop- agation of electromagnetic waves over the relief of the surround- ing medium to a certain degree is taken into account in measuring the distance of aircraft with the aid of ground radio distance-finders (by means of preliminary test flights and the recording of a curve of radio deviation). 273 The considerable difference in flight conditions does not permit us to solve this problem for radio compasses located on board an aircraft. On the average (with a probability of 95%), the errors in locating aircraft with ground radio distance-finders in flat country is 1-2°, and 3-5° in the mountains. The errors in meas- uring the distances with the aid of radiocompasses in flat areas is 3-5°, and can reach 10-15° in mountainous areas, especially at low flight altitudes . Accordingly, the practical operating range of a ground radio distance-finder with satisfactory results of tracking is 300-400 km (except for those which work on USW, where the operating range is determined by the s traigh t- line geometric visibility). A satisfactory accuracy in determining the bearings of radio stations with the aid of on-board radiocompasses is obtained at distances up to 180-200 km. Nevertheless, radiocompasses have found increasingly broad application for purposes of aircraft navigation, and are more popular than the ground radio distance-finders due to their considerable autonomousness and the ease with which they can be employed. For purposes of increasing operativeness , as well as forming a reserve and ensuring reliable operation of radiocompasses, two sets of them are used in most aircraft. The basic control system for an on-board radio compass con- sists of the following: (a) Frame antenna with mechanical device for rotating it and a mechanism for compensating radio deviation. (b) Open antenna, (c) Superheterodyne receiver with a device for commutation of the phase of the frame antenna and an electrical device for turning the frame antenna (tracking system). (d) Indicator of course angles of radio stations. (e) A shield for the remote control of the radiocompass . 21^ which is equivalent to a drop in the wavelength of the received signal and therefore an increase in the phase shift between the sides of the frame. To increase the magnetic and dielectric per- meability of the medium, the effect of the frame will increase. low frequency generator Fig. 3.16. Diagram of Amplitude Modulator at the Output of the Radiocompass Receiver. In contrast to the amplitude ground radio direction-finders which we have discussed thus far and which belong to the "E" type (carrier-wave amplitude), radiocompasses presently use the method of amplitude modulation of the received signals (direction-finder type "M" ) . The essence of the method is that • r e cept i on of the signals takes place simultaneously with an open and a frame antenna, with the phase of the frame antenna being constantly switched by the low-frequency generator. This means that an amplitude-modulated signal is obtained at the input of the receiver. A simplified diagram of the amplitude modulator at the input of the receiver is shown in Figure 3.16. The control grids of L^ and L2 receive a negative voltage u„q , so that when the low-frequency generator is turned off, these tubes will be closed and the signals from the frame antenna will not be passed. When the low-frequency generator is turned on, tubes Lj and L2 open alternately, and the signal from the frame antenna reaches the input of the receiver in phases which are separated by 180°, and when these are combined with the signals from the open antenna, they undergo amplitude modulation. /265 275 r Fig, 3.17. Zero, Positive and Negative Modulation. Obviously, depending on the direction of the radio station (Fig. 3.17), with a fixed radiocompass frame, the amplitude modu- lation can be positive (Position 1), zero (Position 2), and nega- tive (Position 3). The tracking system at the output of the receiver is designed so that the frame of the radiocompass rotates in the direction which will produce a zero modulation of the signal, A diagram of the output section of the receiver is shown in Fig- _ ure 3.18. ^W^ \ I yi "" ^ reference voltage on the anodes of tubes L^ and L2 , formed previously in the positive half-periods of the rectangular pulse, is supplied to the switching circuit of the frame antenna at the input of the receiver. If the input signal is modulated by the frame signal, the average anode current of one of the tubes will be greater than that of the other. This produces a disturb- ance of the balance of the bridge circuit in the magnetic amplifier, made of permalloy cores , and a current passes through the rotor winding of a small motor. The stator winding of the motor is con- stantly supplied with a voltage which is shifted 90° in phase by capaciitor C, from a low-frequency generator which supplies the bridge circuit. The motor will continue to rotate until the direction of the radio station is no longer perpendicular to the frame of the radio- compass, and the modulation of the signal of the open antenna by the frame becomes zero. In the case when the frame antenna is turned toward the radio station in the opposite plane, the phase of the frame changes by 180°, In this case, in the presence of modulation, the rotation of the frame will take place not in the direction of reduction, but initially in the direction of increase of modulation, thus causing the frame to turn through 180°, In this manner, the readings from the radiocompass are all given the same sign, A block diagram of the radiocompass is shown in Figure 3.19. The controls for the radiocompass are mounted on a special control panel. Usually, the radiocompass has three operating regimes (besides the "off" position), so that a selector switch is mounted on the panel. 276 I. Tuning. In this regime, only the open antenna of the radio /266 compass is connected. A special vernier on the control panel is used to tune the device to the frequency of the ground radio sta- tion, either by ear or by a visual tuning indicator. When tuning by ear, reception takes place in the "telegraph" regime, i.e., the second heterodyne of the receiver is turned on to convert the inter- mediate frequency of the receiver to sound. In the telegraph regime, the call letters of the radio station are also heard, if the station is transmitting on a non-modulated frequency. ^\r- r^if I reference voltage JUirL o H o Hi U a 0) bD >> O a 3 cr 0) u o Fig. 3.18. Diagram of Output Section of Radiocompass Receiver . I I . Compass regime. In this regime, both the open and frame antennas of the radiocompass are connected. In this case, the track- ing system of the receiver turns the frame antenna depending on the direction of the radio station and the direction of the radio station is shown on an indicator (course angle or bearing). III. Frame regime. In this regime, only the frame antenna of the radiocompass is connected, and the bearing of the radio sta- tion can be determined with minimum audibility of its signals in the telegraph regime. The rotation of the frame is carried out by means of a special pushbutton switch on the control panel with the label "left-right". The reading of the bearing in this case has two signs . Recent models of the ARK-11 automatic radiocompass do not diffe in their principle of operation from the operating principle describ above for the ARK-5 radiocompass, but they have several design featu and advantages : /267 r ed res 277 fa (a) Complete electrical remote control. (b) Possibility of setting the apparatus to nine previously selected channels (frequencies) in the range from 120 to 1340 kHz and switching from one receiver channel to another by means of an automatic pushbutton switch, located on the control' panel . There is also a provision for smooth manual setting over the entire oper- ating range of the radiocompass (with the tenth button depressed). (c) Increased noise stability of the receiver. (d) Possibility of operation in combination with a non-con- trolled antenna of open type with a low aerodynamic resistance and a low cperating altitude (on the order of 20 cm). ,frame> M deviation | compensator NK input device selsyn trans- mitter rearing {indicator receiver output device control panel 1 Fig. 3.19. Functional Diagram of Radiocompass. The control panel of the ARK-11 differs in design from that of the ARK-5. In addition to the "off" position, there are four operating regimes. The first three regimes are the same as described above. The fourth regime "Compass II" is a spare and is used in the case of intense electrostatic noises when the usual distance- finding methods become unstable. In the "Compass II" regime, instead of the open antenna, a second frame antenna is used, mounted on a common frame-antenna block, perpendicular to the basic frame and forming a unit with the basic frame. The reference signal in this case reaches the input of the receiver not from the open but from the additional frame antenna, which is less sensitive to noise. However, the additional frame antenna which has the same properties as the main antenna, changes the phase of the reference signal by a further 180° when it is turned through 180°, so that both positions of zero reception of the main frame antenna will be positions of stable equilibrium, and consequently it is possible to have an error in determining the course angle of the radio station of 180° . 278 The control panel of the ARK-11 has a toggle switch for nar- /268 row and wide frequency bandpass: "wide-narrow". In the "narrow" position, the extraneous noises in the earphones are reduced and the desired radio station can be heard more clearly. Other control units on the ARK-11 panel (subrange switch, knobs for coarse and fine setting, toggle switches and buttons) have the same markings as in the ARK-5. Eadiooompass Deviation Conditions for directional reception of electromagnetic waves on an aircraft are not favorable and depend on the direction of propagation of the wave front in both the horizontal and vertical planes . If the reception of signals from a ground radio station is being made at considerable distances which exceed 5-6 times the flight altitude, the vertical component of the vector of propagation of the wave front has less of an effect on the reception conditions. In this case, we can use a compensated curve of radio deviation, which is a function only of the course angles of the radio station. The reason for the radio deviation is a reflection of elec- tromagnetic waves from the surface of the aircraft or their re- reflection from individual parts of the aircraft. Since the radio compass frame is mounted in the plane of symmetry of the aircraft X-Z, the deviation at course angles zero and 180° is close to zero. The transverse plane of the aircraft Y-Z is also close to the plane of symmetry, so that the deviation at course angles 90 and 270° is not great and passes through zero at course angles close to it. The maximum asymmetry of the aircraft takes place relative to the directions 45, 135, 225 and 315°. Therefore, the radio devi- ation at these course angles reaches a maximum. Hence, the curve of radio deviation has a quarternary appear- ance (Fig. 3.20) with extreme values AP = + 12 to 25° depending on the type of aircraft . Radio deviation is compensated by a mechanical compensator located on the axis of rotation of the frame antenna. The compen- sator has a control strip which produces an additional revolution of the axis of the master selsyn by means of a special transmis- sion. The required shape is given to the control strip by means of 24 compensating screws to set the readings for the radiocompass at 15° intervals on the scale from zero to 360°. Before the first determination of radio deviation, the com- pensator is usually neutralized, i.e., each of the screws is unscrewed 279 to such a position that the control strip has a shape with the correct curvature and the additional r-"^ation of the axis of the /269 master selsyn is equal to zero at all course angles. To determine radio deviation, a ground radio station is se- lected (preferably at a distance of 50-100 km from the airport) and the true bearing is measured as accurately as possible on a large-scale chart (usually 1:500,000), and then the magnetic bearing of this radio station (MBR) is determined. CAR Fig. 3.20. Graph of Radio Deviation. By means of a deviation distance finder, magnetic bearings of one or two separate landmarks (MBL) are measured from the center of the area where the radio deviation will be plotted, in the way which was described in Chapter II, with a description of their devi- ation of the magnetic compasses. If the area for the deviation operations at the aerodrome is constant, the MBL will be known earlier. The aircraft is then rolled out on the runway. The deviation distance finder is installed in the aircraft in a line 0-180° exactly along its longitudinal axis, and the course angle of the landmark (CAL) is calculated to get rid of installation errors in the radio- compass. The corresponding CAR = 0: CAL MBL MBR, In the deviation distance-finder, the level line is set to the cal- culated CAL, and the aircraft is turned until the sight line of the distance-finder coincides with the direction of the selected landmark. In this case, the longitudinal axis of the aircraft will be lined up exactly with the radio station (CAR = 0), and the mag- netic course of the aircraft will be equal to the magnetic bearing of the radio station as measured on the chart (MBR). Turning on the radio compass and setting it to the desired radio station, the reading of the radiocompass is taken ( RRC ) . If RRC is not equal to zero, we will have the installation error of the frame: 280 est CAR RRC Then, without turning off the radiocompass , it is necessary to loosen the fastening screws which hold the frame to the fuse- lage and then (by turning the base of the frame) adjust it until the indicator points to /270 RRC = CAR = , after which the frame is re-fastened to the fuselage. The remaining installation error, if RRC is not equal to zero after the frame has been fastened down, can be compensated for imme- diately either by the navigator or the pilot by turning the body of the selsyn relative to the indicator scale. After compensating for the installation error, the radio de- viation is determined successively at 24 RRC ' s at 15° intervals. To do this, it is necessary to set the sight line of the deviation distance-finder along the longitudinal axis of the aircraft to 0- 180°, loosen the dial of the deviation distance-finder and move it so that the line of sight 0-180° passes through the selected landmark, and then fasten the scale of the distance- finder once again. In this case CAR = and MBR = 0. The deviation dis tance - finder mounted in this manner makes it possible to calculate the course angles of the radio station (CAR) on the scale dial by turning the aircraft to any angle. Consequently, if we turn the aircraft according to the indi- cations of the radiocompass to a RRC = 15°, and then to 30, 4-5, 60°, etc., successively (setting the sight system of the deviation dis tance - finder each time to a selected landmark), we can calcu- late the CAR immediately from the scale on the dial. Thus, in each reading of the radiocompass, we determine the actual course angle of the radio station and can write the radio deviation as follows: A = CAR - RRC. r Compensation of radio deviation is performed after it has been determined. To do this, the graph of radio deviation is plotted and the extreme values of the graph are divided into three equal parts to avoid sharp bends in the strip, after which two intermed- iate graphs of radio deviation are plotted. The compensator is then removed from the axis of the frame; by turning the proper screws, compensation is made for the radio 281 deviation in terms of the first intermediate graph, calculating the correction made in the selected portion of the radio compass by means of a special pointer on the compensator. Then the devi- ation is compensated by the second intermediate graph, and finally by the curve of radio deviation Compe compensation for radio deviation by all three graphs is per- formed in an order such that after each introduction of a positive correction there is a correction of equal magnitude but negative, i.e., with a mirror image of the course angles. Usually, the order of compensation is selected as follows: 0, 15, 345, 30, 330, 45, 315, 60, 300, 75, 285, 90, 270, 10 5, 255, 120, 240, 135, 225, 150, 210 , 165, 195 and 180° . After compensation for radio deviation, the compensator is mounted on the mechanism of the frame; the aircraft is turned and the deviation distance-finder is used to check the correctness of the operations which have been carried out. If any errors in compen- sation are discovered, the radio deviation is compensated once again by an additional turning of the screws corresponding to the readings of the radiocompass . In addition to the method described above for correcting radio deviations on the ground, there are others. For example: (a) Determination of the magnetic course of an aircraft by distance-finding at the tail (nose), as described in Chapter II, and the calculation of course angles of the radio station on the basis of it. (b) Range finding of a radio station which is visible from the airport (e.g., a distant power radio station). In aircraft where the frame antenna of the radio compass is mounted below the fuselage, determination of radio deviation on the ground is impractical, since the reflection of electromagnetic waves from the surface of the ground causes a distortion of the electromagnetic field. In these aircraft, the radio deviation is determined in flight. /271 going away from it To save time, the flight can be carried out over a 24-angle route, i.e., practically along a course which crosses the straight- 282 line flight for 20-30 sec for each recording of the readings of the radio compass and course. However, in this case, it is neces- sary to determine the location of the aircraft at each point being measured and to enter it on a chart so that when the data is analyzed it will be possible to determine the bearing of the radio station from the point at which the reading was taken. In fact, the course angle of the radio station (CAR) at the moment when the recordings are made is determined by the formula CAR = TBR TK, and the radio deviation of the radio compass is determined as the difference : A = CAR r RRC Compensation for radio deviation is made after the aircraft lands in the same way as after determining it on the ground, but without checking the accuracy of the work which has been carried out, since this would require repetition of the flight. Aircraft Navigation Using Radiocompasses on Board the Aircraft / 212 Radiocompass es on board the aircraft make it possible to solve the same navigational problems as ground radio distance-finders. (a) Path control in terms of direction and selection of the course to be followed during flight toward the radio station and away from it . (b) Measurement of the drift angle after flying over the radio station . (c) Checking the path for distance by measuring the distance to a radio station located to the side. (d) Determination of the position of the aircraft by obtain- ing bearings from two radio stations. (e) Determining the drift angle and the groundspeed from suc- cessive positions of the aircraft, as well as the wind parameters at flight altitude. The solution of these problems by means of a radiocompass mounted on board the aircraft is very similar in principle of solution to the ground radio distance-finders, especially if the indicator for the course angles of the radio compass is combined with the course indicator of the aircraft and thus shows the reading for the bear- ing (Fig. 3.21). The figure shows the course indicator for the navigator. 283 combined with the bearing indicators of two radio compasses (USh- M). The course of the aircraft is measured on the inner, movable scale of this indicator (relative to a triangular mark on the outer scale), while the course angles of the radio stations are indicated on the outer fixed scale according to the position of the point- ers of the radiocompasses . On the inner, movable scale, opposite the arrows, it is possible to calculate the bearings of the radio stations, while the other ends of the pointers can be used to show the bearings of the aircraft . However, this method is practical only for use with one radio- compass, since the total correction is then shown on the scale of deviations, and is effective only for one radio station M where 6 is the difference between the meridian of the aircraft loca- tion and the meridian of the radio station, and Aj^^ is the magnetic declination of the location of the aircraft. Obviously, in the general case this correction will be dif- ferent for different radio stations . The necessity to make corrections for the deviation of the /273 meridians is one of the principal shortcomings of radiocompasses. This shortcoming to a certain degree can be reduced by using an orthodromic system foi' estimating the path angles and courses of the aircraft. In this case, the need to introduce corrections is no longer applicable, if the radio station is located on the refer- ence meridian for computing the path angles, and in any case the correction remains constant if this condition is not observed. The use of combined indicators considerably simplifies the operations related to the use of radio compasses mounted on board aircraft. Therefore, the methods of using them for navigational purposes must be viewed as non-recorded indicators of course angles of radio stations, assuming that in the combined indicators, the addition and subtraction of the angles according to those same rules is carried out automatically. It is clear from Figure 3.22 that the magnetic bearing of the radio station (MBR) and the true bearing of the radio station (TBR) are added from the course of the aircraft in corresponding systems of calculation and the course angle of the radio station: 281+ Fig. 3.21. Combined Indicator for Course and Course Angles of a Radio Station. TBR = TC + CAR; /27^ MBR = MC + CAR. Similarly, in the orthodromic system of calculating courses OBR = OC + CAR. where OBR is the orthodromic bearing of the radio station and OC is the orthodromic course. In determi- ing the location of the aircraft, the true bear- ings of the aircraft (TBA) are plotted on the flight chart from gi'ound radio stations. TBA = TBR + 6 + 180° , where 6 is the angle of convergence of the meridians. In calculating the true bearing of the aircraft , ±80° are added if the TBA has a numerical value less than 180° and subtracted when the value of the bearing exceeds 180° . Consequently, in calculating courses from the true meridian of the aircraft's location, where 285 TBA = TC + CAR + 6 + 180° , 6 = (A -X )sln<j) r , s . a ^av and in calculating the course from the magnetic meridian TBA = MC + A., + CAR + 6 + 18*0° . M — It is assumed that the LA is determined with the aid of mag- netic compass deviation. In the orthodromic system of calculating courses, TBA = OC + CAR +6 + 180°, r . m . r . s . — where ^-p .m.v . s . ^^ '^^^ angle of convergence of the meridians (ref- erence and radio station) which is equal to ( ^r . s . ~ ^r . m . ) ^ i'^'i'av • Let us consider the means of solving problems by means of radio- compasses located on board aircraft, taking the rules mentioned above into account. Obviously, a flight along a given line of flight from a radio station or to a radio station must take place with a constant true bearing of the aircraft. In /275 radio station. a flight from a raaio station, this bearing is equal to the azi- he orthodrome relative Fig. 3.22. Station Bearings of Radio and Aircraft nis Dearing is equal to rne azi- luth of the orthodrome relative o the meridian of the radio sta- tion. However, if the flight takes place in a direction toward the radio station, then the true bearing of the aircraft differs from the azimuth of the ortho- drome of the meridian of the radio station by a value equal to 180° . 3ath in terms of d rd a radio station and away from a r - "^ ^' ■ •■ ■ bearing of wa parison value of the actual true For example, in calculating the course from the magnetic merid- ian of the aircraft's location, we must satisfy the equation: 286 'init MC + A„ + M 6 + CAR; o J- = MC + ref M + 6 + CAR + 180° , where ainit i^ "the true bearing of the aircraft and a^ef is the true bearing of the radio station. In using a combined indicator for the bearing, the total cor- rection A = Af^ + 6 can be entered on the scale of declination. Then it is necessary to satisfy the condition that a^n^-j- = TBA and that apef = TBA +_ 180° . The true bearing of the aircraft is calculated from the other end of the indicator pointer, i.e., in flight toward a radio station, the direct reading of the pointer on the radiocompass must be equal to the final azimuth of the orthodrome , while in flight away from a radio station the other end of the pointer must show a reading equal to the initial azimuth of the orthodrome. Inasmuch as the total correction (A) over the length of the path segment will change constantly, it is necessary to determine this correction in each measurement in order to take into consider- ation other indicators or entering the declinations of the combined indicator on the scale. This poses considerable difficulty in using the radiocompass in flight. The problem is simplified considerably in an orthodromic sys- tem of calculations for the aircraft course. In this case TBA OBA + 6 r . m . r . s . where OBA is the orthodromic bearing of the aircraft. The correction 6-p m r s . -^ ^ constant for all s tr aight -line path segments; after setting it on the scale of deviations, the TBA can be read off immediately on the indicator over the entire straight- line path segment. Note. If a radio station is located at the starting point of the route (SPR) with a reference meridian, the flight can be made directly along the orthodromic bearing of the aircraft with a zero correction on the declination scale. Correction for devi- ation of meridians is only valuable when the radio station is located to the side of the flight route or the meridian of the radio sta- tion does not coincide with the reference meridian. As in the case of using a ground radio direction-finder in selecting the course to be followed along straight-line segments of a path by means of a radiocompass, it is necessary to be guided by the following rules : /276 287 (a) In a flight from the radio station, with a drift of the aircraft to the right, the bearing is increased and the course to be followed must be reduced; with a decrease in the bearing, the course must be increased. (b) In the case of a flight toward the radio station, the course must be increased when the bearing increases and decreased if the aircraft bearing decreases . In a flight along a course determined by a radio direction finder, if the course to be followed is selected on the basis of stable bearings ShchDR or ShchDM, the course is considered to have been selected by using the radiocompass on board if the course angle of the radio station remains constant. For example, in a flight toward a radio station at a constant course of the aircraft, an increase in the course angle of the radio station corresponds to a drift of the aircraft to the left (the TBA increases). In order for the aircraft to follow a constant bearing, the course of the aircraft must be increased. When the lead in the course is equal in value to the drift angle of the air- craft, the course angle of the radio station will remain constant and equal to the drift angle both in value and in sign. In selecting a course, the same method of half correc tion is us ed . Example . It is necessary to make a flight toward a radio sta- tion with a orthodromic path angle of 82° . Let us assume that the drift of the aircraft will be to the left within limits of approx- imately 10°. Select a course to be followed by using the method of half correction. In this case, after flying over the turning point in the route, it is necessary to assume an orthodromic course of 92°. The course angle the of radio station will then be equal to 350°, which is equivalent to a numerical value of -10°. In a flight with a course of 92°, if the course angle of the radio station is increased (i.e., if we acquire the values of 351, 352, 353° in succes sion), it is necessary to place the aircraft on the line of flight and to take a lead of 15° (course equals 97°, CAR equals 345° ) . Let us assume that this lead turns out to be too great; then the CAR begins to decrease, taking on values of 344, 343, and 342° Then, after a second placing of the aircraft on the path, it is necessary to take an intermediate lead in the course of 12-13° (CAR equals 348-347°). If the CAR is to be stable, it is necessary to ensure that the orthodromic bearing of the radio station is equal to the orthodromic path angle and to continue the flight with the selected course. 288 The course to followed in a flight away from a radio station is selected in the same manner, the only difference being that when the CAR increases, it should not be reduced but increased further. In using other indicators, the selection of the course is also accomplished by means of stable path angles of radio stations . How^ ever, in order to control the path of the aircraft in terms of direc- tion, it is necessary to determine on each occasion the bearing of the aircraft or the radio station by summing the course and course angles of the radio station, taking into account the deviation of the meridians of the radio station and aircraft and the magnetic declination at the point where the aircraft is located. It should be mentioned once again that in a flight toward a radio station, the selected stable course angle of the radio sta- tion is always equal to the drift ^ngle of the aircraft , regard- less of whether the aircraft is located on the line of the desired flight or as a slight deviation from it. For example, a stable TAR = 350° corresponds to a drift angle of -10° . In flight away from the radio station, the drift angle is always equal to a stable CAR minus 180° . 1211 The drift angle of the aircraft can be measured directly after flying over the radio station. At the same time, after flying over the radio station with any constant course, the course angles of the radio station will be stable, so that US CAR 180° However, in the majority of cases the drift angle is deter- mined during flight away from a radio station as a difference between the magnetic (true or orthodromic) bearing of the aircraft and the magnetic (true or orthodromic) course of the aircraft: US MBA MC , US = TBA TC , or US = OBA OC , where OBA is the orthodromic bearing of the aircraft and the OC is the orthodromic course of the aircraft. In this case, we can simultaneously determine the side to which the aircraft deviates (left or right) by comparing the given path angle and the determined range of the aircraft in the system of 289 coordinates being used;' the aircraft acquires the given line of flight according to the calculated course angle of the radio station, while on the line of flight corrections are made to the course which are e'qual to the average, angle of drift. The monitoring of the aircraft path in terms of distance by means of the radiocompass is accomplished with previously calcu- lated bearings of the lateral radio station In approaching a control landmark, the readings for the course and the course angle of the radio station are observed. At the moment when the sum of the aircraft course and the course angle of the radio station become equal to the previously calculated bear- ing (in combined indicators, the bearings of radio stations become equal to the previously calculated value), the moment for flying /27 8 over the landmark is determined. With the aid of a radiocompass located on board, it is also possible to determine the location of the aircraft on the basis of true bearings from two radio stations. However, the accuracy of determining the aircraft location by this method, involving consid- erable difficulty in the process, is insufficiently high. There- fore, the method is not widely employed in aircraft navigation, being used only for determining approximate aircraft coordinates in finding lost landmarks . The essence of the method is the following: when two radio- compasses are on board, one is set to the frequencies of two ground radio stations, located no more than 180-200 km from the aircraft. It is desirable when doing this to ensure that the bearings of these radio stations cross at an angle close to 90°. If the indicators of the r adiocompasses do not agree, the course ^of the aircraft, the course angles of the two radio stations, and the distance-finding time must all be described simultaneously for a given moment of time, aircraft are determined; TBAx TBA2 Then the approximate true bearings of the = MC + A[^ + CARi +_ 180° ; = MC + Af^ + CAR2 +_ 180° . 290 I nil I I ■■ I III iiiHiiin The bearings which have been obtained are plotted on the flight chart from the meridians of selected radio stations by means of a protractor and scale rule. Having thus determined the approximate position of the air- craft, we can find its true bearing by introducing the precise value of the magnetic declination and making corrections to the deviation angles of the meridians of the radio station and the aircraft loca- tion. These corrected bearings are again plotted on the chart to give a precise position of the aircraft at the moment of direction finding . If only one radiocompass is mounted on board the aircraft, it is necessary to consider its path when determining the location of the aircraft for the time between the moments of direction finding, and this is done as follows (Fig. 3.23). After determining the average bearings of the aircraft, the latter are plotted on a chart, and then the flight path of the air- craft is obtained from the point of location of the first radio station for the time between the measurements of the course angles of the radio station in a direction which coincides with the course of the aircraft. A line is drawn through the point which is obtained, parallel to the first bearing up to the intersection with the line of the second bearing, defining the position of the aircraft at the moment of direction finding for the second radio station. In addition, the true bearings of the aircraft are found in the same way as in the case of two radiocompas ses . The lab or ious nes s of the process of determining the position of the aircraft is considerably relieved if the flight is made with orthodromic courses, but the indicators of the radio compasses must match. In this case, the angles of convergence of the meridians of the radio stations with the reference meridian for calculating the course are determined beforehand. /279 lans At t for he time of measurement, the the first and then the seco t'^t Fig, 3.23 the Posit the Beari stions Diagram for Locating ion of an Aircraft from ngs of Two Radio Sta- angle of deviation of the merid- nd radio station are entered on the scale of deviations of the indicator in succession. They are calculated from the readings of the opposite ends of the pointers of the radio compasses and are designated as TBSi and TBS2. The bear- ings obtained are final and no corrections are required. As we have already ment when using radiocompasses mo on board aircraft, the drift led , mgle o: IS 291 determined from the stable course angles of the radio stations or measured after flying over a radio station. The ground speed of the aircraft is determined by checking the flight in terms of distance by means of radio stations located to the side of the course or from the moments when the aircraft passes over radio stations. The latter method is not accurate, especially when flying at high altitudes, due to the error in the readings of radiocompasses when flying over radio stations. The determination of the wind at flight altitude is accomp- lished on the basis of the ground speed of the aircraft, the air- speed, and the drift angle by the same methods as for ground radio distance-finders. The method of determining the wind by using the successive positions of the aircraft, obtained by distance measure- ment from two radio stations , usually is not used due to the inad- equate precision of the determination of the aircraft location. Special Features of Using Radioaompas ses on Board Aircraft at High Altitudes and Flight Speeds High altitudes and flight speeds cause deterioration of the conditions for using radiocompasses aboard aircraft for purposes of aircraft navigation. The use of radiocompasses and the observation of all rules for retaining accuracy of distance finding is a laborious process, so that the increase in flight speed, calling for operativeness of navigational calculations , creates difficulties in using radio- compasses on board the aircraft. This shortcoming can be largely overcome by using combinations /2 80 of bearing indicators, especially in the orthodromic system of calcu- lating aircraft courses . In addition, another shortcoming of aircraft radiocompasses which operate on medium and short waves, due to the increased speed of flight, is the effect of electrostatic noise on their operation. At high airspeeds, especially in clouds and in precipitation, a considerable electrification of the aircraft surfaces occurs. Static electricity, emitted at pointed portions of the aircraft (including open antennas) creates noise and radio interference in the frequency range at which radiocompasses operate. Despite the measures which are taken to prevent the charges from flowing by using special discharge devices, as well as shielding the open antennas of the radiocompasses , this shortcoming can be overcome only partially and manifests itself in very difficult flight conditions. High flight altitudes have an effect mainly on the accuracy of operation of radio compasses and especially on the accuracy of 292 b b, A _L c £,, I c l\ / Fig. 3.24. Operation of an Open Antenna When Fly- ing Past a Radio Station, determining the moment when the air- craft flies over a radio station. The decrease in the accuracy of operation takes place due to the change in the nature of radio deviation at different angles of deviation of the propagation vector of radio waves. The latter changes within wide limits when the aircraft approaches the loca- tion of a radio station. A diagram of the appearance of errors in determining the moment when the aircraft flies over a ground radio station is shown in Figure 3.24 where there is a picture of the electrical field radiated by an open vertical antenna on a ground radio station. At large distances from the radio station, the electromagnetic wave is vertically polarized. However, there is a space near the radio station and above it where the polarization shifts to the horizontal , then back to the vertical but in opposite phase. Let us assume that an open antenna of the radiocompass is tilted backward (position a. Fig. 3.24) and the aircraft is approaching the rad_io station at a high flight altitude in the direction of ve ctor V . Obviously, at position a the antenna will have zero reception. The reception of the antenna will then increase, but in a phase which is opposite to the reception up to the point a. This leads / 2 81 to a rotation of the radiocompass frame by 180° until the aircraft passes a radio station. Then, after passing the station, the phases of both the frame and open antennas change almost simultaneously ( at the point a;^ ) . Thus, the change in the readings of the radio compass by 180° takes place until the moment when the aircraft passes over the radio station (at point a) and only the oscillation of the needle will be observed from then on. Fig. 3.25. Equivalent of an Open Antenna on Board an Air- craft . 293 s ■ When the antenna is tilted forward (position c), the oscil- lations of the radiocompass needle will begin at point c, while the passage by the radio station with rotation of the needle through 180° will be noted at point c^, i.e., there will be a delay in markinj the passage. For a strictly vertical antenna (position b),'the movement of the needle through 180° can take place prematurely. Then the pointer can make a reverse turn and again show the passage by the radio station at point bi- It should be mentioned that an equivalent to the open antenna of the radiocompass in terms of its inclination in the vertical plane is the resultant combining the upper or lower points of the antenna with the electrical center of the aircraft, constituting its grounding or counterweight (Fig. 3.25). In this figure, point 1 is the top of the open antenna, point 2 is the receiver and point 3 is the electrical center of the air- craft . Obviously, straight line 1-3 is the equivalent of the inclin- ation of the open antenna, which is forward in this case. Thus, the setting of the open antenna above the fuselage in its forward section causes a delay in the reading of the moment when the radio station is passed. Mounting of the antenna in the same position with respect to the center but below the fuselage leads to a preliminary reading of the moment when the aircraft passes the radio station. The oppo- site picture is observed when the antenna is mounted behind the electrical center of the aircraft. is above or below the elec- ase an advance or delay these deviations depend Iso be a double The best place to mount the antenna is a trical center of the aircraft, but in this case an advanc in the readings is observed; in some cases, these deviati on the height and speed of flight, but there can also be reading involving both an advance and delayed indication. This system for the creation of errors in measuring a flight only approximately reflects the reasons for these errors. In prac- tice, they will depend both on the angle of pitch on the aircraft and on the accuracy with which the passage of the aircraft over the radio station is determined. For example, if the aircraft is passing a radio station to the side, then obviously there will not be an indication of pass- age with movement of the needle through 180°, but a deterioration in the passage over the radio station, i.e., errors in determin- ing the passage of the traverse of the radio station will be prac- tically non-existent. /282 294 Usually, the exact passage of an aircraft over a radio sta- tion occurs only in special and exceptional conditions. Therefore, in practice there is always a consideration of the effect of passage with the effect of error, which does not make it possible to con- sider the magnitude of the delay advance in marking the passage. Depending on the type of aircraft and the flight conditions , these errors can occur within limits equal to 1-3 flight altitudes, excluding the case of exact determination. However, exact deter- mination can occur at distances which exceed the flight altitude of the aircraft, beyond the limits of a zone with horizontal polar- ization, i.e., at very considerable deviations of the aircraft from the given line of flight. Details of Using Radioaompasses in Making Maneuvers in the Vicinity of the Airport at Which a Landing is to be Made The maneuver of approaching for a landing usually begins at a relatively low flight altitude (1200-4000 m) with a gradual reduc- tion of the airspeed. Therefore, the effects related to height and flight speed in this case are considerably reduced, Of course, if there is a tendency to drift in the aircraft course, the corresponding corrections must be made in the readings of the course angle of the radio station which are equal in magni- tude and sign to the lead which has been taken. The effectiveness of using radioaompasses in the vicinity of airports where landings are to made is increased also by the fact that the flight is made at short distances from ground radio sta- tions, which gives relatively small linear errors in determining the position of the aircraft in view of the errors already committed in measuring the course angles of the radio station. In addition, it is no longer necessary to calculate the magnetic declination and the deviation angles of the meridians . The accuracy of air- craft navigation in the vicinity of the aerodrome, using radiocom- passes, is considered to be quite satisfactory in all stages of the maneuver with the following exceptions: (a) Determination of the starting point of the maneuver by /2 83 295 flying past the power radio station, if the maneuver is beginning at a high altitude. (b) On a landing strip, where it is necessary to have very high accuracy of flight along a given trajectory for bringing the aircraft in for a landing. U 1 t ra- Shortwave Goniometric and Gon i omet r i c- Range Finding Sys terns As we have mentioned, radiocompasses have significant advan- tages over ground radio distance-finders with respect to uninter- rupted visual information on board the aircraft regarding its posi- tion. This means that they have been very widely employed and are installed in practically all types of aircraft as a rule in a double set. In addition, there are a number of important shortcomings for radiocompasses mounted on board aircraft, which reduce the accur- acy and feasibility of aircraft navigation. In addition to the errors caused by the effect of the local relief, which affect all systems for short-range navigation, radio- compasses have the following shortcomings: (a) Unfavorable conditions for directional reception of elec- tromagnetic waves on board the aircraft (radio deviation of an unstable nature ) ; (b) An increase in the errors in distance finding due to inac- curate measurements of the aircraft course; (c) The necessity to consider the deviation of the meridians and magnetic declinations when using magnetic compasses to deter- mine bearings ; (d) The effect of static noise in the range of received radio frequencies at high airspeeds; (e) The effect of flight altitude on the accuracy of meas- uring the range and determining the moment of flying over the radio stations . In addition to these shortcomings, radiocompasses are subject to a general disadvantage of goniometric systems: the need to plot bearings on the chart from two ground points to determine the loca- tion of the aircraft. Therefore it seems natural to try to build devices for short- range radionavigation which would have the advantages of the radio- compasses mounted on aircraft but would not have the shortcomings from which they suffer. Such devices are the goniometric and goniometric-rangef ind- ing systems which operate on ultra-shortwaves . 296 A common feature of these systems is the directional radiation of electromagnetic waves by ground instruments and their directional reception on board the aircraft. This feature, together with the range of waves employed, gives three very important advantages for navigational systems : (1) It frees the system from radio deviations on board; (2) The bearing of the aircraft becomes independent of the /284 aircraft course, magnetic declination, and deviation of meridians; (3) It sharply increases the freedom of the system from static and atmospheric interference . There are several types of goniometric directional radio bea- cons and receiving devices to carry aboard aircraft, which operate on ultra-short waves. Figure 3.26 shows the schematic diagram of a radio beacon with a rotating directional antenna. The generator of low frequencies produces a frequency which is synchronized with the rotation of the directional antenna. On the axis of rotation of the antenna is a special disk, which generates the reference signal related to the position of the rotating antenna. The reference signal passes through the modulator and transmitter to reach the open antenna of the radio beacon. I trans- mitter call signal modu- lator' modu 'lation suppres' SOT- I voltage indicator for refer lence phas M low frequency generator r= disc— generator of reference signals Fig. 3.26. Schematic Diagram of a Radio Beacon With Rotating Directional Antenna. 297 From the transmitter, the signal reaches the rotating antenna through a modulation suppressor, so that the amplitude of the signal radiated by the antenna in any given direction depends only on the position of the antenna relative to this, direction. Consequently, the signal from a directional antenna is modulated by a low fre- quency whose phase relative to the relative signal i's shifted through an angle equal to the azimuth of the aircraft. Thus, two signals reach the aircraft on the carrier frequency in addition to the call-letter signals: (1) Reference signal for beginning the reading. /285 (2) The signal from the directional antenna, whose amplitude maximum coincides with the moment when this antenna crosses the line to the aircraft . The receiver on the aircraft has three channels (Fig. 3.27): (1) Channel for picking up the call letters from the beacon ( earphones ) ; (2) Reference- channel ; (3) Azimuthal voltage channel. I receiver- earphone channel JO reference voltage channel azimuthal voltage channel phase discrim- inator zero indicator Fig. 3.27. Apparatus for Goniometric System Aboard an Aircraft. The indicator mechanism is usually based on measurement of the phase ratio of the reference and azimuthal signals with low frequency by the compensation method, i.e., the phase of the refer- ence signal is changed by an automatic phase shifter so that it 298 coincides with the phase of the azimuthal signal. In this case, the signal on the phase discriminator will be equal to zero while the bearing indicator on the aircraft will act as the pointer of the phase shifter. For flight along a given bearing, the phase shifter is set to a given position, so that the signal on the phase discriminator (and therefore on the zero indicator device) will be equal to zero only in the case when the aircraft is located exactly along the line of the desired bearing. The goniometric system for short-range navigation, operating on ultrasonic waves, in the case when the characteristic of the rotating directional antenna has a sharply pronounced maximum (which usually is achieved by using reflex reflectors), can be built on the principle of time ratios rather than phase ratios. In this case, the reference signal, when the rotating direc- tional antenna passes through zero reading, has a pulsed character, and the equipment on board must include a generator of a reference frequency as well as special delay devices to determine the bearing of the aircraft in time between the moments when the reference and azimuthal signals are received. /286 The geometry of navigational applications of USW beacons with directional radiation it exactly the same as the use of ground radio distancef inders . All the problems of aircraft navigation such as selection of the course to be followed, monitoring of the path for distance and direction, determination of the location of the air- craft from two beacons, measurement of the drift angle and ground speed, measurement of the wind at flight altitude, etc., are solved in exactly the same way as for ground radio distance-finders. In flight away from a radio beacon, the rules for flight along the ShchDR bearing are observed, while in flight toward a radio beacon it is the rules for bearing ShchDM which are followed. If the flight is made using a zero-indicator instrument, the method of selecting the course in flight from the beacon and toward the beacon with a corresponding switch in the mode of operation of the receiver leads us to only one type: the pointer of the zero indicator shows the direction of deviation of the aircraft from the LGF . The main difference between using USW beacons and ground radio distance-finders is only that in order to determine the location of the aircraft from two bearings using a radio direction-finder, the plotting of the bearings is done by the operator of the com- mand distance finding station, while in the case of USW beacons it is done by the crew of the aircraft. Nevertheless, goniometric USW beacons have a much wider range of application than ground radio-distance-finders and aircraft radio- 299 compasses, thanks to the constant indication of bearings on board the aircraft . The instrumental accuracy of goniometric USW navigational sys- tems is higher than for ground radio distance-finders . The prac- tical accuracy under average conditions of application is also somewhat higher or equal to the accuracy of distance-finders. However, in using radio distance-finders, it is possible to consider to a certain extent the influence of the local relief on the radius of appli- cation, which cannot be done for USW beacons. In this respect, the USW beacons have less favorable operating conditions than ground radio distance-measuring stations. The operating range of a USW system S is limited by the limits of direct geometric visibility from the ground beacon to the air- craft with an insignificant increase caused by radio refraction. It is determined by the approximate formula S = 122 /H. However, if there are some obstacles along the path of the propagation of the radio waves (e.g., mountain peaks), they will appear insurmountable for USW. From the standpoint of navigational applications, it is very /2 8 7 advantageous to combine the operation of a goniometric USW system with range finders. Rangefinding USW navigational systems are usually of impulse type (Fig. 3.28). The aii'craft transmitter sends out impulses of ultrashort waves, which reach the receiver aboard the aircraft at the same time as a reference signal. V re- ceiv- er* transH mittei: trans-l mitter^ re'- c elv- er" ^ indicator Fig. 3.28 System . Diagram of Long-Range Navigational 300 The ground receiver receives pulses of wave energy emitted by the aircraft, amplifies them and sends them out again through a transmitter into the ether, to be received by the aircraft. The range indicator aboard the aircraft has a generator of standard frequencies, a frequency-divider circuit, and a delay line for the reference pulse to measure the time required for the signal to pass from the aircraft to the ground beacon and back to the air- craft . While the signal is traveling, the duration of the delay in the reference pulse prior to its combination with the received signal determines the distance to the ground beacon, which is usually used as a visual indicator of the azimuth of the aircraft (the direct- reading instrument for distance and azimuth, DRIDA). The combination of azimuth and distance readings makes it very easy to solve the problems of aircraft navigation, especially if the beacon is mounted at the starting or end point of a straight- line flight segment. In the latter case, the crew of the aircraft has a constant supply of direct data regarding the position of the aircraft relative to the line of flight in terms of direction and distance . When the ground beacon is located to the side of the path to be covered by the aircraft, the problem of determining the aircraft coordinates is solved analytically, or very simple calculating devices are used to convert the polar system of coordinates for the posi- tion of the aircraft into the orthodromic system. One type of such device is the computer which is installed for zero indication of the position of the aircraft on the line of the path to be traveled during flight in a given direction. Let us assume that we have a straight-line path segment from point A to point B (Fig. 3.29). /288 If we are given the path angle of the segment (^), measured relative to the meridian for calculating the bearings (magnetic or true meridian of the location of the ground beacon), and we know the azimuth of the end point of the segment (-^fin^ ^s well as the difference from it to the beacon (^fin^» "the shortest distance from the beacon along the line of flight (i?s) (disregarding the spher- icity of the Earth), can be determined by the formula R - R^. sin (i|) s f m f m or for any point lying on the line of flight R = R . sin (.xp- A .) . SI ^ In other words, the given line of flight is the geometric locus 301 of points for which i? .sin( i|;-i4 . ) = const = i? _ . sLnX^-A . ) Fig. 3.29. Diagram Showing Operation of Computer for Zero Indication of Path Line. Thus, having set the path angle of the segment ( i|/ ) on the calculator, the distance from the beacon to the end point of the segment (i?fin) ^^^ "the azimuth of the end point (^fin)^ we can f.ind the navigational parameter i?g which corresponds to the position of the aircraft exactly on the line of flight. If it turns out in the course of the flight that ifg is greater than the given value, then in the example shown in Figure 3.29 the aircraft deviates from the LGF to the left, and the pointer of the zero indicator also moves to the left. With a deviation from the LGF to the right, the arrow of the zero indicator will move to the right . When a flight is made in a direction which is opposite to that shown in Figure 3 . UO , the value sin (^A^) and consequently i?g , will have a negative sign, so that when the aircraft moves to the left of the line of flight the pointer of the zero indicator will also move to the left regardless of the fact that the absolute distance i?g in this case decreases, and vice versa. The calculating device for zero indication of the given line of flight is very simple and makes it possible to solve only one problem, i.e., to select the course to be followed by the aircraft for a flight along a given line of flight, in a manner similar to that for a flight from a radio beacon or along the ShchDR bearings, using the method of half correction. It is better to solve the problem or to use computers to solve it using the computation of numerical values of orthodromic coord- inates of the aircraft, working on the basis of the indications from the DRIDA: /28' 302 Za^Z^ + RsiniA-iiJ. In this case, the angle of shift of the aircraft relative to the line of flight is determined very simply as the ratio of the change in the coordinate Z to the distance covered between the points of two measurements (-^cov^- ■^cov Example . The distance covered between measurement points is equal to 60 km. The coordinate Z varies from zero to +4- km. Find the required correction in the course for traveling parallel to the line of flight. Solution A4< = arctg-— - = 4°, We will assume that it is necessary to travel along this line of flight for another 60 km so that the correction in the course must be -8° , but at the moment when the coordinate Z becomes zero (and if this takes place as we have calculated at 60 km) the course will have to increased by 4°. If the juncture with this course takes place earlier or later, then it is necessary once again to determine the angle of shift of the aircraft (A(|;) and to move the aircraft to the right by this angle. For example, if we have an initial shift from the desired path of 4 km, the aircraft will reach the line of flight in 80 km, so th at Aij; = arctg 80 In the orthodromic system, the problem of checking the path for distance and determining the ground speed is solved simply For lack of a calculator, the problems in finding the angle of shift of the aircraft and checking the path for distance and direction, can be solved analytically by means of a navigational / 29 slide rule. In addition, these same problems can be solved by plot- ting on the chart the indications of the azimuth and distance of the aircraft as obtained from the beacon. On a flight chart which has an indication of the given line of flight, two points based on the bearings and distances from a ground beacon are plotted every 15-20 min. On the basis of the 303 positions of these points relative to the line of flight, we can determine their orthodromic coordinates X and Z. It is then easy to solve the problems in determining the angle of shift of the air- craft (AiJ;), and also the drift angle and the required angle for turning the aircraft , the ground speed as well as the wind param- eters at flight altitude. Details of Using Goniometr-ia-Range Finding Systems at Differ- ent Flight Altitudes A special feature of ultrashort waves is their ability to be reflected from the interfaces of media with different optical densi- ties 5 and especially from conducting media in a more sharply pro- nounced form than is the case for waves of shorter frequencies . In addition, at short wavelengths, the interference which arises with combination of oscillations shows up more rarely than in the case of long waves, since the small difference in the path of the coherent waves in the case of short wavelengths gives a considerable shift in their phase. Let us say that the antenna of a ground transmitter of a gon- iometric or range- finding system is mounted ax a certain altitude above the surface of the ground (point A in Fig. 3,30). The electromagnetic wave at the receiver point B will be prop- agated along two paths: Fig. 3.30. Diagram Forma- tion of Lobes of Maximum Radiation . (a) Along the straight line AB (b) Along the broken line ACB with reflection at point C off the Earth's surface. It is clear in the diagram that straight line AjB is equal to the broken line ACB, since the angle of incidence of the wave is equal to the angle of reflection. Let us draw line AAp in such a way that triangle ABA2 is an isosceles triangle. Obviously, line AjA2 will represent the path difference of the rays in the straight and reflected waves. The reflection of radio waves involves a phase shift in the /291 wave which depends on the optical properties of the reflecting med- ium. A purely mirror reflection changes the phase of a wave by 180°. With a small differene in optical densities of the media. 304 when the propagation of the reflected wave takes place along a curve with a dip in the reflecting medium, the phase shift can take place differently. Let us say that upon reflection, the phase of a wave remains fixed. Then the resultant of the direct and reflected signals at the receiving point B will have a maximum when the path differ- ence of the beams has a value which is an even whole multiple of the half wave : AS = 2*-—; ii. = 0, 2, 4, . . .2n and a minimum if k is an odd multiple of the half-wave: x=l,3,5. . .(2rt— 1). Thus, there will be an interference pattern for the propaga- tion of radio waves in the vertical plane with maxima and minima of directionality of the radiation characteristic (Fig. 3.31). A change in the phase of the wave with reflection from the Earth's surface causes corresponding changes in the distribution of the maxima and minima of the characteristic of directionality, but the total structure of the interference pattern will be similar to that shown in the diagram. ^■»"^«~r53rr;rr Fig. 3.31. Multilobe Radi- ation Characteristic of Electromagnetic Waves. The interference pattern of shading in the directions of radi- of radio waves by objects on the Earth's surface, as well as the altitude at which the antenna is mounted above the Earth's surface, introduce considerable corrections in the possible range of recep- tion of ultrashort waves. The operating range of a system, expressed by the approximate formula S = 122/H, is maximum at a sufficient power of the trans- mitter and sensitivity of the receiver, if the aircraft is located in the lobe of the maximum of directionality. However, at certain heights and distances, there can be "dips" in audiblity, when the aircraft passes through regions of radio shadow or interference minima. In addition, special features using USW goniometric-range finding devices at high flight altitudes arc related to their range- finding sections. Rangefinding instruments can be used to measure not only the /292 horizontal but also the sloping distance from- the aircraft to its radio beacon (Fig. 3.32). Therefore, s,~ Sjf cos e or 305 n In the special case when the aircraft is passing above the radio beacon 5jj--=0; 5h = //. Let us suppose that an aircraft is flying along a given route with R^ = 10 km, at an altitude which is also equal to 10 km with the use of a type i?£^sin( ip-j4£ ) = const calculating device. With ^-A = 90°, distance R must be equal to 10 km, i.e., the aircraft must deviate from the given course and pass over the radio beacon. The height errors in goniome tric-rangef inding devices have some important shortcomings in their use in the shortrange applications and especially in maneuverings in the vicinity of an airport. Fig. 3.32. Sloping and Horizontal Distance to Radio Beacon. Consideration of altitude errors is very important due to the rapidity with which the aircraft passes over the beacon, when the errors in measuring the distance change so rapidly that it becomes impossible to enter corrections without using special calculating devi ces . Therefore, the use of goniometric-range finding instruments for navigational measurements usually limits the distance from the beacon to 3-^■ flight altitudes, i.e., it defines an effective zone around the beacon with this radius. For example, at a flight altitude of 12 km, the radius of the inoperative zone thus defined must be equal to approximately 50 km . Fan-Shaped Gon i ome t r i c Radio Beacons The possibilities of aircraft radio compasses are increased considerably by using fan-shaped goniometric beacons (Fig. 3.33). The picture shows the schematic diagram of a radio beacon. The two outermost antennas are set to some wavelength and the power for them is in opposite phase. The total characteristic of the three antennas gives the multilobe picture of radiation as seen in Figure 3.34-. The number of lobes depends on the ratio of the length of the base line between the end antennas to the wavelength, and their direction depends on the ratio of the phases in the outer and inner antennas of the radio beacon. /293 306 ■^■11 ■■■mill With a change in the phase of the middle antenna by 180° , the positions of the lobes shift to their mirror images (the solid and dotted lobes in Fig. 3.3'+), while the points where the dotted and solid lobes intersect become axes of equal signals . n phase shifter for t QpO t transmitter main phasej— ' shifter Fig. 3.33 Fig. 3.34. Fig.' 3.33. Fan-Shaped Radio Beacon. Fig. 3.34. Radiation Characteristic of a Fan-Shaped Radio Beacon. During the periods between commutations, if we transmit short and long signals in the forms of dots and dashes in an overlapping pattern, signals of only one type will be heard within the edges of the solid lobes (e.g., long signals), while within the limits of the dotted lobes, only short signals will be heard. In zones of equal signals (near the axes of intersection of the lobes), one will hear a continuous tone. If we then smoothly change the phase ratio in the end antennas, the lobes will begin to rotate, e.g. , to the right, and the phase ratios will change in the reverse direc- tion: each of the solid lobes will change places with the dotted lobe to the right of it, and each dotted lobe will change place with the solid lobe to the right of it. Let us assume that an aircraft is located at Point B (see Fig. 3.34), i.e., within the limits of a dotted lobe, near the right- hand limit of the solid lobe, with each operating cycle of the beacon beginning after a pause in radiation. In this case, at the beginning of a cycle and after the pause, several fading dots will be heard, then a continuous signal, and finally a long series of dashes. If the aircraft is located in the middle of the lobe, the series of dots will be equal in length to the series of dashes. At a point/294 which is close to the right-hand limit of the dotted lobe, the series of dots will be longer than the series of dashes. 307 A similar picture for the audibility would be obtained when the aircraft is located within the limits of the solid lobe with the sole difference being that at the beginning of the cycle the dashes would be heard, and the dots would be heard only after the continuous tone . Thus of an air to plot t on a char location signals i bearing also to them by as thin is a mult of signal , to obtain the bearing craft, it is sufficient he orthodromic lines t according to the of the axes of equal n order to obtain the f the aircraft, and ivide the angles between rthodromes as well ines in a ratio which ■ iple of the number s in the cycle . Fig. 3. Lines f Basis o Beacons 35 . or a f Fa 3° . (Fig, Th 3 IS n . 35) With a sector width between the axes of equal signals of 15° and 60 signals per cycle, each signal will correspond to 15 min of angle . If the sector between the axes of equal signals is then divided into five parts , the thin orthodromic lines will diverge at angles of arrow sector will contain 12 signals of the same type Grid of Position n Aircraft on the n-Shaped Radio For example, if an aircraft is located at the sector of points on the first thin line to the right of the axis of equal signals, then 12 dots will be heard which will fade into a continuous tone, after which there will be 48 dashes. On the second line there will be 24 dots and 36 dashes, 36 dots and 24 dashes on the third, etc. At the limit of the sector (the axis of equal signals), a total of 60 dots and 60 dashes will be heard. Note. Practically speaking, if we consider that part of the signals (dots and dashes) are mixed with the continuous tone, the number of audible signals will be less than 60, so that after counting them the number of audible signals should be taken subtracted from 60 , then divided in half and added to the number of signals of both types that were heard. If the aircraft is located between the thin orthodromic lines /29 5 plotted on the chart, then the line of the bearing of the aircraft can easily be found by interpolation of the distance between the plotted lines . Fan-shaped beacons make it possible to determine very accurately 308 the position lines of an aircraft. To do this, with the aid of a radiocompass or by generally calculating the path of the aircraft, it is necessary to determine the approximate position of the air- craft with an error which is no greater than the width of one sector, Then, having listened to the operating cycle of the beacon with the radiocompass, or with the coherent receiver, we can determine the position of the aircraft in the sector. A similar method is used to determine the second line of posi- tion of the aircraft, using the second fan-shaped beacon, whose family of position lines intersects the lines of the first beacon. In order not to take into account the' shift of the aircraft during the time between the taking of bearings from the two beacons, it is desirable to listen to the operating cycles of the two beacons simultaneously using two members of the crew who are using two radio- compasses or one radiocompass and the coherent radio receiver. The accuracy of distance finding with the aid of fan-shaped beacons during the daytime is no worse than 0.1-0.3°. Under the most unfavorable conditions for distance measurement (in twilight when working with the space wave, or at the boundary for the use of surface waves), the errors can reach 3 and sometimes 5°. In a further zone of distance measurement, and also the short-range zone, with operation on a surface wave, the errors do not exceed .5-1° . The operating range of a fan-shaped beacon during the daytime reaches 1350 km on dry land and 1750 km above the sea. At night above dry land, this figure is 740 km and above the sea, 950 km. Unlike the radio beacons with non-directional and omnidirec- tional operation, which are mounted as a rule at the turning points of air routes , flight along the bearing line of a fan-shaped beacon is only a very rare case. Therefore, the principal method of air- craft navigation using fan-shaped beacons is determining all navi- gational elements including the wind parameters at flight altitude by successive measurements of the LA. This method is the most suitable one for fan-shaped beacons because the location of the aircraft can be determined in this manner with a sufficiently high accuracy. It is often desirable to carry out aircraft navigation during /296 309 V flight using fan-shaped beacons with conventional distance-finding from radio stations. For example, in a flight toward a radio sta- tion or away from a radio station, it is desirable to use bearings from fan-shaped beacons for checking the path for distance and deter- mining the ground speed. 3. DIFFERENCE-RANGEFINDING (HYPERBOLIC). NAVIGATIONAL SYSTEMS The azimuth lines of position are divergent because as the range of operation of a system increases, increasingly high require- ments are imposed on the measurement accuracy, while beyond the limits of direct geometric visibility it is very difficult to retain directionality of transmission or reception due to the effect of local relief and especially the ionized layers of the atmosphere. The situation is somewhat better as far as the circular p.osi- tion lines are concerned. Circular lines do not diverge, so that the requirement for accuracy in determining them remains constant at all distances . In addition, the linear error in determining the position of the aircraft in a . goniometric system is proportional to the sines of the angles of the propagation errors: ^Z = SslnAA. In range finding systems, these errors are proportional to the cosines of the angles of the propagation errors: A5 = 5(l — cosA^). At small angles, on the order of 6° , the cosine is practically equal to unity. Therefore, the errors in determining the distance are usually many times less than the errors in the azimuthal shift (Fig. 3.36). We can see from the figure that the linear error in determin- ing the direction CC^ - -SsinAA, and the linear error in distance is A5 = ABC - AC n S'Cl-cosAA). However, the technical achievement in measuring distance over long distances is much more complex than that in measurement of the azimuth . 310 I nil IB II ■HIIIW I IB I 11 I ■ As we saw in the case of USW systems, distance is determined by retranslation of signals from on board the aircraft by a ground beacon and their reception back on board the aircraft. This method, which is relatively easily accomplished at short distances, turns out to be unsatisfactory over long distances for use on medium and long waves . /297 Fig. 3.36. Errors in Measuring Bearing and Range with Reflec- tion of Electromagnetic Waves from Obstacles: A: Location of Ground Radio Beacons; B: Point of Mirror Reflection of Radio Waves; C: Location of the Aircraft (Actual); Ci: Measured Position of the Air- craft ; AA : Angular Error in Propagation . time with the signals from the quency , we can determine the di The best method of measuring long distances at the present time is the maintenance of a calibration frequency on board the aircraft. The generator for the calibration frequency is set at the frequency of a ground transmitter and retains a given frequency for long periods of time by means of special stabilizing elements. By means of these special timing devices, the calibra- tion frequency can be converted to a lower frequency which is synchronous with the signals of the ground beacon. If we take the signals from the ground stations and compare them in generator of the calibration fre- stance to these radio stations. However, this method has not been widely employed due to the complexity involved in keeping a highly stable reference frequency on board the aircraft, although it offers considerable promise in future . It is simpler to solve the problem of determining the posi- tion line of the aircraft on the basis of the distance between the distances to two ground radio stations. In this case, there is no necessity for a strict synchronization of the operation of the ground installations with those on board. Only the transmission of signals from the ground stations must be synchronized. The air- craft generator for the calibration frequency in this particular case acts only as a central measuring gauge to determine the time intervals between the moments of reception of the signals from the two radio stations ix-i-ons . lization of the operation of the apparatus at ground IS can be achieved incomparably more easily than syn chronization of a ground apparatus with one aboard an aircraft, since the distance between ground stations remains constant, thu allowing us to use a synchronizing device for two or three stati together. In addition, ground installations are not limited by 311 size and weight restrictions, not to mention the apparatus on board. The methods of measuring the difference in distance to ground /29 8 radio stations can involve either time (pulse) systems or phase systems. Each of these methods has its own advantages and disad- vantages . An advantage of the phase methods is the higher instrumental accuracy of the measurements, but in this case the result of meas- uring is obtained ambiguously, i.e., there may be several isophasal paths simultaneously with different distances to the ground radio stations, which differ in magnitude and are multiples of the length of the measured wave. On each of these paths, the result of meas- urement is the same and must be used as a measure for determining the pathway along which the aircraft is traveling. The pulse methods of measuring distance have somewhat less instrumental accuracy, but their results are more definite. Of course , it should be mentioned that for long-range navi- gational systems, the instrumental accuracy of measurement which can be attained at the present time both by the pulse and phase methods is sufficiently high so that their errors are many times less than other systematic errors which are related to conditions of propagation of electromagnetic energy. Since the errors in oper- ation of the systems under propagation conditions of radio waves are practically the same for both pulsed and phase systems, the advantage of phase methods of measurement may be restricted only to short distances from ground radio stations (in the short-range zone of effectiveness). Operating Principles of Differential Rangefinding Systems Differential rangefinding systems of aircraft navigation consist of two pairs of synchronously operating ground radio stations and a receiving-indicating apparatus aboard an aircraft. For purpose of reducing the amount of ground equipment for the system, one of the transmitting radio stations (the master) is made common for two pairs so that the system can include three ground radio sta- tions . The operation of the two slave stations is synchronized with the master station by synchronizing signals sent out by the master station . Let us begin by examining the geometry of the operation of one pair of ground radio stations (Fig. 3.37). Two ground radio stations are located at points F^ and F2. The line connecting points Fj and F2 will be considered as the focal line of the base, while the points Fi and F2 are the foci of the system . 312 Let us assume that at point M there is an aircraft which is receiving signals from radio stations F^ and F2. At the beginning, the aircraft will receive a signal from the first radio station and then from the second. The difference in the distances from the aircraft to these radio stations is determined by the differ- ence in time between the arrival of the signals in the pulse system or by the difference in modulation of the phases in the waves re- ceived from the two radio stations in the phase system. /299 1 \\\ I4"J ^ w \ Hr C ^ % ill /■f. Fig. 3.37. Hyperbolic Sys- tem of Position Lines. We know that the 1 is the geometrical locu the difference in whose to two given points is value, is called a hype given points, to which are measured, are calle of the hyperbola. Cons knowing the difference tances to the two radio we can always plot the line of the position wh craft is located. ine which s of points , distance a constant rbola. The the distances d the foa-i equent ly , of the dis- stations , hyperboli c ere the air- The hyperbolic line with a difference in distances equal to zero becomes a straight line perpendicular to the focal axis and dividing the distance between the foci of the system in half (see Fig. 3.37). This line is called the imaginary axis of the hyper- bola. The distance along the focal axis of the family of hyperbolas from the foci to the imaginary axis is called the parameter c. It is obvious that the difference in distances from the foci of the hyperbola to any point along its branches is equal to twice the distance along the focal axis from the imaginary axis to the peak of the hyperbola. This distance is called parameter a. Accord- ingly, the difference in distances from any point to the foci of the hyperbola is always equal to 2a. The maximum density of hyperbolic lines of position is found along the focal axis between the foci of the system, where the dis- tance between the peaks of the hyperbola is equal to the differ- ence in parameters a. The magnitude of the value 2a is measured by navigational param- eters of the system so that the accuracy in determining the lines of position of the aircraft depends on the accuracy with which this parameter is measured. Consequently, an error in determining the position line of the aircraft on the focal axis is equal to the error in measuring parameter 2a, divided in half. As we see from Figure 3.37, the family of hyperbolas is divided 313 1 by a family of position lines. At distances from the center of the system which exceed 2e, the hyperbolas practically become straight lines, whose direction coincides with the direction of the radii extended from the center of the system. Thus, the hyperbolic system is converted into a goniometric one. However, the density of the lines of position, in this case will not be equal along the circumference, as is the case in purely goniometric systems . At a given distance from the center of the system, the maximum density of position lines will be found at the imaginary axis of the hyperbola, gradually decreasing along the circumference as they approach the focal axis. The density of posi- tion lines at a distance greater than a on the focal axis becomes so small that the system becomes unsuitable for determining the location of the aircraft. /300 Fig. 3.38. Effective Area of Hyperbolic Navigational System. The master station of the second hyperbolic pair can be lo- cated along the extension of the focal axis of the first pair. In this case, the angle of fracture of the base (3) is equal to zero. If the master station of the second pair is not located on the focal axis of the first pair, there is a definite fracture of the base (Fig. 3.38). The angle of fracture of the base creates a more favorable condition for intersection of the position line in that region of application of the system toward which it is directed, since the angle of intersection of the hyperbolas in this case approaches a right angle and therefore the accuracy in determining the locus of the aircraft is increased when two position lines intersect. However, this involves a decrease in the quality of the condi- 314 The complex of equipment for a hyperbolic navigational sys- /301 tem aboard an aircraft usually consists of the following: a non- directional receiving antenna, a matching block for the antenna with d. receiving device, a receiver, and an indicator. The matching block serves to produce parameters of the receiv- ing antennas when signals are received from ground radio stations. Signals received by the antenna are transmitted to the indi- cator for measurement of the navigational parameter. The indicator has a generator for a calibration frequency, which produces stan- dard signals for purposes of measurement, and a number of frequency dividers which are required for forming electronic markings on the reading scales, as well as repetition frequencies for the scan on a cathode ray tube, synchronized with the transmissions of signals from ground radio stations. The signals which are received pass to the scan of the cath- ode ray tube, where the operator controls their size according to the amplification of the receiver. The synchronization of the scan on the screen is then regulated with the frequency of the received pulses so that the latter remain fixed on the screen. The operator then mixes a reference (selecting) signal from the generator with the signal from the master station, which is achieved by the intermittent introduction of small distortions in the generator for the calibration frequency, so that the pulses of the signals begin to move across the screen. The motion of the pulses stops when the signal of the master station coincides with the reference signal of the generator (usually a rectangular base at the beginning of the scan). To measure the time difference between the arrival of the signals and the signal from the slave station, a selecting pulse is given which is related to the delay in scanning of the reference signal, after which the indicator is switched to the reference regime and the reading is taken on the electronic scale. In some types of receiver indicators, the recording of the reading is made on a dial with two or three scales (for different scanning rates), for example, beginning with thousands of micro- seconds, then hundreds and finally tens, with interpolation up to units of microseconds. This provides increased accuracy of readings due to the many-fold increase in the scale of the indicator. 315 In systems with automatic tracking of the signals from ground stations , the time intervals between the moments of arrival of the signals are calculated on mechanical counting dials, whose rota- tion is related to the delay mechanisms for the selecting pulse. The reference signal from the generator is then reinforced, together with the signal from the master station, by an automatic frequency adjustment of the calibration generator. Thus, there is an automatic tracking of the signals from the radio station and a constant numerical indication of the output navigational parameter of the system, and the difference in dis- tances from the aircraft to the ground radio stations is expessed / 30 2 in microseconds of radio wave propagation. In phase systems, by means of distributing elements in the calibration generator, its phase is matched with the phase of the signals from the master radio station, after which a phasometer is used to measure the phase difference between the calibration generator and the slave radio station, and the position line of the aircraft is determined from this difference. As we have already pointed out, if the difference in distances to the radio stations includes several periods of the modulating frequency of the ground stations, the determination will be ambig- uous . The solution of the ambiguity of this estimate can be accom- plished by several methods . (1) An initial setting of the coordinates of the aircraft with automatic tracking of the radio station signals. In this case, using known coordinates of the aircraft (e.g., on the basis of the visual determination of the aircraft location), the indicator is set by hand to show the isophasal line on which the aircraft is located. If constant tracking of the radio station signals is then carried out, completely reliable readings of the position line will be obtained. A shortcoming of this system is the necessity to relate the aircraft to the local terrain on the basis of the initial reading of the hyperbolic coordinates. In addition, during flight, there may be readings of other isophasal lines, due to interference, which can be determined and corrected only by a repeated relation of the aircraft to the local terrain by means of other methods . (2) By modulation of the carrier frequency of the ground radio stations at very low frequencies (with long modulating waves , consid- erably increasing the possible difference in distances from the aircraft to the radio stations). In this case, at a low frequency phase, the rough position of the isophasal line of a carrier fre- quency or the frequency of the second modulation with a small, long period can be determined. 316 (3) By using several carrier frequencies for the ground radio stations, the isophasal line can be considered to be determined if it is simultaneously on the isophasal lines for all frequencies at which the measure ment is carried out (usually three frequen- cies, since two will be inadequate in some cases). On adjacent isophasal lines, for each frequency used, the isophasal lines of other frequencies will not coincide with the readings of the phaso- meter . Navigational Applications of D i f f e ren t i a 1 - Rangef i nd i ng Systems Dif f erential-rangef inding navigational systems , like fan-type beacons, are intended primarily for determining the locus of the aircraft on two position lines. Therefore, the principal method of aircraft navigation using these systems is the determination of the navigational elements on the basis of a series of determin- ations of the LA. /303 By recording and plotting on a chart a series of points for the locus of the aircraft, recording the time at which they were passed, and using a scale ruler and protractor to measure the dis- tances between them on the chart, as well as the distance from the first recording of the LA to the second, it is easyi to determine the speed and flight angle of the aircraft. 4' = "1 ,2! s, '1,2 where a ^ _ 2 is the azimuth of the second recording of the Lh from the first and Si 2 is the distance between the recordings of the LA. The drift angle of the aircraft is determined as the distance between the actual flight path angle and the average course of the aircraft over the segment between two successive recordings of the LA: a = ^li - Yav With a known groundspeed and drift angle, taking the airspeed into account as well as the course to be followed, the wind param- eters at flight altitude can be determined with the aid of a naviga- tional slide rule. In special conditions, when the flight direction coincides with one of the branches of the hyperbolic flight lines , the flight can be made along the latter. To do this, it is sufficient to main- tain a constant reading for the calculator of hyperbolic coordinates of one pair. The family of position lines for the second pair in this case is used to monitor the path for distance. 317 {^ M Monitoring the path for distance by means of the readings of one of the counters can be used in the case when the aircraft navi- gation in terms of direction is carried out using two devices, e.g. the USW bearing of a goniometric system or a fan-type beacon. To increase the feasibility of using dif f erential-rangef ind- ing systems, the hyperbolic coordinates can be converted to ortho- dromic or geographical ones (see Chapter I, Section 7). In some hyperbolic systems of aircraft navigation, e.g., that of Decca and Dectra (England), simplified methods of automatic plot- ting of the aircraft course on a special chart use the movement of a pen in mutually perpendicular directions. For this purpose, special charts are made on which the hyperbolic lines of the first and second family are laid out at right angles. Naturally this results in distortion of the contours of the terrain on the chart, as well as the scale and geographic grid, and the line of flight of the aircraft is also bent. /304 Such a method of recording has a number of shortcomings (e.g., in relation to the calculation of orthodromic coordinates for the air craft), but it is very easy to achieve from the technical stand- point and its shortcomings are considerably reduced if the path of the aircraft has markings for distance. Methods of Improving Differential Rangefinding Navigational Systems The design of hyperbolic systems contains elements whose im- provement leads to a conversion of the system to a hyperbolic-range- finding or hyperbolic-elliptical system. Such elements include the standard frequency generators aboard the aircraft. When these generators operate in a highly stable regime, the reference signals from these generators can be kept so precise that it becomes possible to measure distances to one of the ground radio stations. To do this, it is sufficient to combine the phases of the frequencies of the generator aboard the aircraft and the ground radio station with an initial distance setting (e.g., the takeoff point of the aircraft). Further changes in distance can be determined by the deviation of the phases of these frequencies or by the deviation of the pulse signals, if the system is oper- ating in a pulse regime. Measurement of distance in connection with one pair of hyperbolic position lines makes it possible to considerably improve the accur- acy with which the locus of the aircraft is determined over long distances, and one pair of g"?ound radio stations will suffice for measurements. However, the conditions for measurement between the foci of the system near the focal axis will remain unfavorable (Fig. 3. 39) . 318 It is more advantageous in this case to use the hyperbolic network of position lines (see Chapter I, Section 7). However, since we know the difference between the distances to the two radio stations as well as the distance to one of them, it is easy to determine the sum of the distances to these radio stations, e.g., if so that ^2 > Si and Aa- = ivj — Sj, S2 = Si + AS and Si + S2 = 2Si + M>, Similarly, for the case when S2 < Si , S, + S2 = 2Si ■ AS. Therefore, in order to obtain the number of the hyperbola, it is sufficient to use the difference in distances, while to obtain the number of the ellipse, we must double the distance to one of the radio stations and add the difference in distances with the / 30 5 corresponding sign. One great advantage of the hyperb oil c- e llipti cal network is the orthogonality of the intersection of the position lines at any point in the field which is involved. On individual sheets of the chart, the hyperbolic-elliptical network has the appearance of a nearly rectangular grid with noticeable curvature of the position lines only in the vicinity of the foci of the system. erbolic-range- ly a hyper- em , a sys tem e accuracy nates of the ed over long d several fold, range of appli- is also consid- h the use of nd radio sta- When using a hyp finding (and especial bolic-elliptical syst of position lines) th with which the coordi aircraft are determin distances is increase so that the practical cation of the system erably increased, wit only one pair of grou tions . Fig. 3.39. Combination of Hyperbolic and Rangefinging Navigational Systems. It should be mentioned, however, that a serious obstacle to the devel- opment of systems of long-range navigation for use on high-speed aircraft is the low noise sta bility 319 of operation of these systems, since only very long waves can be used for navigation over long distances . 4. AUTONOMOUS RADIO-NAVIGATIONAL INSTRUMENTS In recent years, there has been a considerable increase in the use of radio navigational instruments which are housed completely aboard the aircraft and operate without the need for ground facil- ities . Such instruments are called autonomous radio-navigatvonat instruments or, if their operation is combined with some other naviga- tional equipment aboard the aircraft, autonomous navigational sys- tems. These include aircraft navigational radar, Doppler systems for aircraft navigation, and radio altimeters. All autonomous radio-navigational instruments operate on ultra- short waves, since they have a very high (practically complete) free- dom from interference during operation (not counting artificial interference) . Doppler meters for measuring the ground speed and drift angle of the aircraft measure the motion parameters of the aircraft directly relative to the Earth's surface, which clearly differentiates them from all existing forms of navigational equipment, especially with regard to problems of automation of aircraft navigation and pilot- /306 age of aircraft. Aircraft Navigational Radar Aircraft navigational radar is a very flexible and effective method of aircraft navigation during flight over land or sea close to coastal regions . In terms of the geometry of their use, aircraft radar devices can be included among the goniometri c-rangef inding systems. However, in comparison to the goniometric-rangef inding navigational systems, they have a number of tactical advantages: (1) The high saturation of ground landmarks makes it possible to select the most suitable ones for measurement in navigation. (2) The lack of errors in determining the bearings of land- marks from the radio deviation of both the aircraft itself and the local relief, something which affects all non-autonomous naviga- tional systems . (3) The possibility of visualizing ground landmarks with the purposes of determining ground speed and drift angle to a better degree than with optical methods . ( M- ) The possibility of identifying dangerous meteorological conditions in flight (thunderstorms, powerful cumulus and cumulonim- bus clouds ) . 320 (5) The high accuracy and ease of the measurements using only one operational frequency. At the same time, the navigational use of aircraft radar has several shortcomings: (a) The bearing of the aircraft can be used only as a basis for measuring the aircraft course, thus lowering the accuracy of distance findings. (b) A certain amount of experience is needed for correct recog- nition of ground landmarks and the possibility of errors in deter- mining a landmark, since they are not labelled. The operating principle of radar is based on the ability of electromagnetic waves at high frequencies to be reflected from objects located along their propagation path (from the interface between media with different optical densities). To obtain a panoramic image of the terrain, a rotating or scan- ning antenna is used to cover a certain sector, so that its posi- tion must be synchronized with the position of the scanning beam on the screen of a cathode ray tube. In addition to synchronizing the direction of the antenna with the scanning direction of the beam, it is also necessary to ensure that the beginning of the scan is synchronized with the moment when the USW pulses are omitted from the antenna transmitter Thus, the radar screen shows the following: (a) The direction of the object on the basis of the antenna position at the moment of emission and reception of the signal. (b) The distance to the object on the basis of the time re- 321 quired for the signal to travel between the moment when it is emit- ted to the moment when it is received. (c) The nature of the object, on the basis of the brightness of the scanning beam at the point where the reflected wave is received, indicacor -\ receiver- transmitter modulator-l^ont^^ panel J ][ antenna antenna mechanism ' VWrv/N/- generator of standard freaue'ncy and dividers Fig. 3.40. Diagram of Aircraft Radar. The radar screen has a long afterglow so that when the antenna has made a complete revolution, the screen still shows a trace of all the irradiated objects on the Earth's surface which are located in the field scanned by the radar. The main section of the radar, controlling the operation of the entire system, is the standard-frequency generator with fre- quency dividers for forming distance markings and a signal-trans- mission frequency synchronized with the saw-tooth scanning image on the screen (Fig. 3 . M-0 ) . The signals from the standard-frequency generator reach the /308 modulator, where they are converted to high-voltage rectangular oscillations of a special length. The high-voltage pulses from the modulator pass to the transmitter magnetron, where high-fre- quency groups are generated according to the pulse length. The high frequency reaches the antenna through a wave guide and is radiated into space. At the same time, in synchronization with the pulses of high voltage which are sent to the transmitter, the scanning generator, forms a saw-tooth voltage which controls the scanning beam on the screen. The scanning rate depends on the steepness of the slope for the saw-tooth waves. At a low scanning rate, a fine image scale is obtained as well as long-distance detection of objects. When the scanning rate is increased, the scale of the image decreases proportionately with the distance covered by the radius of the screen. The control of the scanning rate is achieved with the aid of a switch on the control panel. 322 The resolving power of the radar in terms of azimuths is a function of the sharpness of the directionality of the antenna beam, The azimuths of ground landmarks can be determined immediately by the position of the antenna (and therefore by the scanning line on the screen), and the antenna mechanism is fitted with a selsyn mechanism for tilting the indicator. The azimuth reading is made on a scale located along the periphery of the screen. To measure the distance to a landmark, pulses from the fre- quency divider are sent to the receiver (and therefore to the scan- ning beam). These pulses increase the brightness of the beam at certain distances from the center of the screen, forming circular distance markings. When using the radar on different scales the distance mark- ings are shifted to different distance intervals. For example, with a scale of 10 km for the radius of the screen, the markings are usually 2 km apart; when using scales from 10 to 100 km, the markings are 10 km apart; at a scale of 200 km, they are 20 or M-0 Vm ^ rt ;:! rti- _ Now let US follow the path of the high-frequency pulses from the transmitter to the object and back again, and see how they control the brightness of the scanning beam. The high-frequency pulse passes through the wave guide to the radiating horn of the antenna, after which it is shaped into the required directional diagram for radiation by means of a reflec- tor. Usually, the directionality of the antenna in the horizontal plane is made as sharp as possible. To do this, it is necessary for the phase of the beam when emerging from the antenna to remain constant over its entire perpendicular cross section (Fig. 3.41), i.e. , the reflection in this plane must have a shape such that the wave path from the horn radiator to the surface of the reflector and along its chord of emergence is uniform. The characteristic of directionality of the radiation in the vertical plane must be such that the illumination ot the terrain from the vertical of the aircraft is as uniform as possible over the entire effective radius of the radar. To do this, it is neces- sary to have the maximum amount of wave energy transmitted at small /309 323 angles to the plane of the horizon, i.e., over the maximum range, and to have the smallest amount of energy radiated along the vert- ical of the aircraft. Such a aharaateT'ist-ho -is catted the cosecant- square , i.e., the reflectors in the vert- ical plane are given a shape such that the amount of energy radiated into space is roughly proportional to the square of the cosecant of the angle of the plane of the horizon to the propagation direc- tion . In some types of radar, an acicular characteristic of directionality is employed, i.e., one which is sharpest in both the horizontal and vertical planes, combin- ing it with the cos esant-square in the vertical plane, e.g., by a scanning cycle. This is achieved by using specially shaped reflectors with a telescoping deflector or by sending energy to the antenna by different wave guides for the acicular and cosecant-square antenna characteristics of the radar. Fig. S.M-l. Radar An- tenna for Use Aboard Aircraft . Antennas with cosecant-square characteristics are used for circular-scan radar, mounted below the fuselage of the aircraft. Antennas with combined radiation are used for sector-scan radars and are mounted in the nose of the fuselage to cover only the area ahead of the aircraft. In this case, the radar screen is made with the center displaced so that the maximum area of the screen can be used . Usually, the tilting of the antenna in the vertical plane (and therefore the characteristics of directionality of the radiation) is adjusted manually by means of a special electrical device and a switch on the control panel of the radar. The transmitting antenna of the radar acts simultaneously as a receiving antenna, since the directional characteristics of the antenna are reversed, i.e., used both for emitting and receiving the wave energy . /310 In order for the pulses of wave energy emitted from the trans- mitter not to return immediately to the wave guide of the receiver, special arresters are used which block the wave energy from enter- ing the receiver at the moment when the transmitter is operating. The transmission frequency of the pulses of wave energy from the transmitter is set so that the time intervals between them are not shorter than those required for propagation of electromagnetic waves to the most distant object at a given operating range for the radar and for its return to the aircraft. When using the radar at large-scale settings, the decrease in the pulse duration is 32^■ accompanied by an increase in the transmission frequency, thus pre- serving the average power of the transmitter. Hence, the recep- tion of the reflected signals takes place in the time intervals between the pulses of wave energy emitted by the transmitter. The radar receivers have special vacuum devices (klystrons for generating high frequency) which play the same role as heter- odynes in conventional receivers. The signals received by the antenna are mixed with the fre- quency of the klystron; an intermediate frequency is produced which then goes on (after detection and amplifi cation ) to control the bright- ness of the scanning beam. In addition to the special features of the radar which we have discussed above, the receiver has additional circuits and control units. In particular, to allow the frequency of the klystron to be changed, there is an automatic frequency adjuster (AFA), etc. For improved contrast of the image on the screen, in addition to the devices for adjusting the overall amplification of the receiver, the operator of the radar can use a separate signal amplifier which operates at high and low levels. This makes it possible to dis- tinguish shaded or illuminated objects on the Earth's surface as desired. For example, to examine populated areas, the high-level signals are increased and the low-level signals are reduced (by decreasing the brightness of the background of the screen). To pick out rivers and lakes, the low-level signals are increased, thus impr-oving the visibility of shaded objects against a brighter general background. It should be mentioned that for the formation of high-frequency pulses by the transmitter, very high voltages must be produced in the modulator; this means that at high altitudes (i.e., at low atmo- spheric pressure), there may be flashovers in the wiring of these units. Therefore, these units (including the wave guides of the transmitter) are hermetically sealed and the required pressure is maintained in them by a special pump or by systems for pressurizing the aircraft cabin. /311 Indicators of Aircraft Navigational Radars The aircraft radar is an autonomous goniometr i c-range- find- ing and sighting device, so that its indicator must be made so that all required navigational measurements can be performed satisfac- torily with it. Circular indicators are the ones which are of greatest inter- est from the navigational standpoint (Fig. 3.42). The center of this indicator, marking the 'position of the air- craft against the panorama of the field of vision, coincides with 325 the center of the screen. Around the edge of this screen is a scale of bearings, which can be rotated manually; in the upper part of it is a course marking which shows the position of the longitud- inal axis of the aircraft. The scale of bearings is set to its own divisions by means of a "course" ■ rack and pinion device, for setting the course of the aircraft by the course markings, accord- ing to the readings of the course instruments . The sighting lines of the indicator are marked on the protec- tive glass of the screen, which can be rotated by means of "sight" rack and pinion. For convenience in sighting, three movable points for longitudinal sighting lines are provided, and one transverse sighting line is provided for indicating traverses when flying over landmarks . When the radar is operating, circular distance markings appear on the screen, and the deflection of the luminous course lines of the aircraft may also be included. In the lower part of the indicator unit, in addition to the "course" and "sight" adjustments, there are other controls: "scale illumination", "beam scan focus", "beam brightness adjustment", and some types of indi- cators also have a "vertical and hori- zontal centering of scan". Thus, the circular screen of the radar can be used to measure bear- ings precisely or determine the course angle of a landmark, its distance, as well as the provisional line of motion of the landmark for purposes of determining the drift angle /312 and the ground speed on the basis of the traverse of the flight over the landmark . Fig. 3.M-2. Indicator for Radar with Circular Screen Sector-type radar screens have somewhat fewer possibilities (Fig. 3.43) . Instead, the screen is fitted with a system of divergent lines for the course angle of the landmark (CAL). The determination of the bearings in this case is made by adding the course angle of the landmark to the course of the aircraft by the formula: 326 TBL = TC + CAL; TBA = TC + CAL +_ 180° + 6, It is very difficul possible on these indica the moment of flying ove of landmarks . Instead of visualiz of landmarks, the soluti problems on these indica often accomplished by a measurements of the LA o exception to this is con marks which move across the immediate vicinity o marking, and can be used the drift angle by the p their shifting, using th course angles and the ground speed when passing ove markings on the screen. Fig. 3.43. Indicator for Sector-Type Radar, t and not always tors to determine r the traverses ing the movement on of navigational tors is more succession of n a chart . An stituted by land- the screen in f the course to determine arallelism of e lines of the r the distance Nature of the Visibility of Landmarks on the Screen of an Aircraft Radar For purposes of aircraft navigation using aircraft radar, the following landmarks can be used: 1. Large populated areas and industrial enterprises. The visibility and outlines of these landmarks depend on the number and location of metal structures and coverings in the object. Popu- lated areas and industrial enterprises appear as bright spots on the screen, as a rule, with sharply bounded outlines. This means that the outlines of the landmarks coincide closely with their out- lines on a chart or as they are seen by visual observation, as groups of structures with non-metallic coverings show up much less clearly / 313 and are visible from shorter distances than metal structures and coverings . Populated areas show up most clearly with maximum amplifica- tion of the high-level signals and a minimum amplification of the low- level signals . 2. Rivers and lakes. During the summer, these landmarks are visible as dark areas and spots whose outlines match those of the landmarks against the a lighter background of the surrounding ter- rain. In the winter, when these bodies of water are covered by a smooth layer of ice, only the river valleys are seen, especially against forested areas . Ice packs on rivers can be seen in the form of bright spots against a darker background of snow covered banks. Rivers and lakes can be distinguished by amplifying the low-level signals to increase the brightness of the entire background 327 I of the screen. Then the dark objects will be observed as still darker areas against the light background. 3. Mountains. These landmarks appear on the radar screen in a form which is very close to their natural one, i.e., as they appear to visual observation. Mountains can be distinguished by a suitable selection of signal amplification at both high and low levels . h. Forested areas. Landmarks of this type can only be seen clearly in winter, against a general background of snow-covered surface, by amplifying the low-level signals; in summer, against a background of vegetation and cultivated areas, forests are seen very dimly and cannot be used as landmarks. 5. Highway and railway bridges. These landmarks show up espec- ially well against the background of large rivers. The railways themselves show up clearly only when there are embankments or steel structures for supporting catenaries for electrified railways. In summer, the development of powerful cumulus and cumulonim- bus clouds shows up very clearly on radar screens. Areas which are dangerous for flight (with a large-droplet structure, and there- fore with intense turbulence and high intensity of electrical fields) appear on the screen in the form of bright spots with diffuse edges. These storms can be distinguished very well with maximum ampli- fication of high-level signals and minimum amplification of low- level signals. Amplification of low-level signals reduces the con- trast of the images of these dangerous storms, but areas of radar shadows begin to appear, which are very clear on the screen and are characteristic signs of storm clouds. In observing terrestrial landmarks and clouds in which there is thunderstorm activity, it is necessary (besides adjusting the amplification level of the receiver) to choose the proper inclin- ation of the radar antenna. As a rule, landmarks which are located close to the aircraft are observed with an increased inclination of the antenna downward, while those further away (and storm clouds ) /31'4 are viewed with a slight inclination downward or with the antenna aimed upward, depending on the flight altitude and the viewing range. The inclination of the antenna can be selected to provide the optimum clarity of the images of the landmarks on the screen. Use of Aircraft Radar for Purposes of Aircraft Naviga- tion and Avoidance of Dangerous Meteorotogiaat Phenomena Aircraft radar can be used to solve all problems of aircraft navigation, beginning with the recognition of landmarks over which the aircraft is flying and ending with measurement of all basic elements of aircraft navigation. 328 For recognition of terrestrial landmarks, it is desirable to use operating scales of the radar which coincide with the scales of flight charts . With an indicator screen radius of 55 mm, an image scale of 1:1,000,000 produces a range of 55 km on the screen. This oper- ating scale for the radar is most suitable when using maps with a scale of 10 km to 1 cm. Hence, when using charts with a scale of 1:2,000,000, one must use a radar scale of 100 km; 110 km is possible, if the design of the radar permits The harpest distinction of radar landmarks is obtained by proper selection of contrast in the image by using var- using the proper selection of contrast in the image by using var- thighandlowlevels,adjust- o the proper angle, and settin ious amplifications of the signals a_ ---„.. _ ing the inclination of the antenna to the proper angle, and setting the beam brightness on the screen. The location of the aircraft can be determined very accurately in terms of the bearing and direction from a point landmark. Point landmarks in this case can be the centers of populated areas, charac- teristic features of the shores of rivers and lakes, individual mountain peaks, etc. In using sector-type radars, the bearing of the aircraft is obtained by adding the aircraft course and the course angle of the landmark, as is done when using aircraft radio compasses with non- integrated indicators. As in the case when USW rangefinding systems are used, the measurement of distances with an aircraft radar means that the radar/315 measures not the horizontal but the oblique distance (OD) of the landmark. Therefore, when measuring distances to landmarks, which are less than five times the flight altitude (ff), the measurement must include a correction AR , which always has a negative sign: A/? = — (VOD2 —H2~ R); ;?=OD_A/?, where OD is the oblique distance , is the horizontal distance. H is the flight altitude , and R 329 If the oblique dista altitude (the correction the oblique distance), th zero. This is also refle a dark spot appears in th sharp limit for the begin of image formation is sep a distance which is equal scale. This spot is call uring the true altitude o nee to the landmark is equal to the flight for the flight altitude becomes equal to e horizontal distance will be equal to cted in the panorama of the image , when e middle of the indicator screen with a ning of image formation. The beginning arated from the center o£ the screen by to the flight altitude on the scanning ed the attimetrat and is used for meas- f flight above the local relief. TABLE 3.1, Oblique distance KM O 10 15 20 25 30 35 40 45 50 Flight f altitude . km 3 i 4 I 5 I 6 I 7 I 8 I 9 I 10 I 11 I 12 0^0 0,0 0.0 0,0 0,0 0,0 0,0 0.0 0,0 0,0 0,5 0.0 0,0 0.0 0,0 0.0 0,0 0.0 0.0 0.0 1.0 0,5 0.0 0.0 0.0 0,0 0.0 0,0 0.0 0,0 2,0 1.0 0.& 0,0 0,0 0,0 0,0 0,0 0.0 0^ rre ctior LSI, KM V 5.0 1,5 2,0 3,0 4.0 6,0 10,0 • — I.O •1.5 2.0 2,5 3^0 4,0 5,0 6»0 0.5 1,0 1.5 2.0 2,5 3,0 3.5 4,0 0,0 0.0 0,5 1,0 1.5 2,0 2.5 3.0 0.0 0.0 0.0 0.5 1,0 1,5 2.0 2,5 0,0 0,0 0.0 0.0 0,5 1,0 1,5 2,0 0,0 0,0 0,0 0,0 0.0 0,5 1,0 1,5 0.0 0,0 0,0 0.0 0,0 0,0 0,5 1.0 0,0 0,0 0,*0 0,0 0,0 0,0 0,0 0,0 For making corrections in the measured distances for the flight altitude, we can use Table 3.1. The location of the aircraft can be determined by means of aircraft radar and directly in stages of orthodromic coordinates. To do this, the scale of bearings on the indicator must be set not to the course of the aircraft, but to the lead angle (LA) on the / 316 course of the aircraft relative to a given orthodromic path angle of flight or drift angle. The sighting device can then be used to determine the path bearing of the landmark (PEL). For example, with LA = y-^ = -10°, the bearing scale must be set to 350° opposite the course marking; with a course angle of 40°, the path bearing of the landmark (PEL) will be equal to 30°; with a negative drift angle, and therefore a positive lead angle, such as 10°, e.g., the bearing scale must be set to 10° opposite the course marking. Knowing the path bearing and the distance to a landmark (i?) , we can very simply determine the orthodromic coordinates of the aircraft : 330 X = X - i?cosPBL = X - i?sin(90°-PBL) ; Z = Z - /?sinPBL. These formulas are different from (1.71) and (1.71a) only in the sign of the second terms on the right-hand side . This is explained by the fact that when we are using goniometric-rangef inding systems , the direction is reckoned from a ground beacon to the aircraft, while in this case it is reckoned from the aircraft to a ground landmark . Example . The radar landmark has orthodromic coordinates Jj^ = 250 km; Zj_ = 80 km and is observed with a path bearing of ^0° as a distance of 125 km. Find the coordinates of the aircraft X and Z. Sol uti on : A'= 250 — 125-sin 50° = 250 — 96 = 156 km. Z = 80— 125-cos40'' = 80 — 82 = — 2k^. Thus we have found that the aircraft is located at a distance of 156 km from the last PBL, 2 km to the left of the LGF , without resorting to a plotting of the bearings on the flight chart. In solving this problem, it is very convenient to use the cal- (S) 82 SB '^^ culating navigational slide rule. (D y '0' ^ To do this, the triangular Fig. 3.1+4. Determination of index on scale 4- is set to the Orthodromic Coordinates of distance of the landmark along an Aircraft on the NL-IOM. scale 5. The slider indicator is then set on scale 3 to the mark- ing which corresponds to 90° -PBL and PBL, while the values R sin (90° -PBL) and R sin PBL are set on scale 5 (Fig. 3.44). After this, there remains only the calculation of these values from the coordinates of the landmark and the determination of the aircraft coordinates. The problem is considerably simplified when the path bearing of the landmark is equal to 90° (flight over the traverse of the landmark). Then X = X^; Z = Z^ - R. It should be mentioned that the determination of the aircraft / 317 coordinates when flying over the traverse of a landmark is advan- tageous, since in this case the errors in measuring the path bear- ing of the landmark have absolutely no effect on the accuracy of determination of the lateral deviation of the aircraft from the 331 line of flight. This is very useful for monitoring the path in terms of direction and correcting the course of the aircraft by using autonomous Doppler measurements of the ground speed and drift angle . This method of determining the orthodromic coordinates of an aircraft is also suitable for use with sector-type radars. In this case, the path bearing of the landmark is determined by the formula PEL CAL + LA This problem can then be solved in the same way as for circ- ular-screen radars. However, on sector-type screens as a rule, it is not possible to determine the markings of the traverse of flights over landmarks. Therefore, for an accurate control of the path, taking into account the reduced accuracy of direction finding, due to the lack of sighting lines, it is necessary to image the landmarks at course angles which are as large as possible. The ground speed of mined most easily with th measurements of the LA, ( dromic coordinates, when the aircraft on a chart, scribed above. However speed on the basis of sue aircraft, is insufficient angle of an aircraft. Th ment to be made along a g the drift angle quite fre of successive measurement a large base for measurem the aircraft and drift angle can be deter- e aid of aircraft radar by using successive locus of the aircraft), especially in ortho- it is not necessary to plot the locus of The essence of this method has been de- the method used for measuring the ground cessive measurements of the locus of the ly practical for measurements of the drift e fact is that for an accurate measure- iven path, it is necessary to determine quently and rapidly, so that the method s of the locus of the aircraft requires ents . In some cases, it is advisable to use other methods for deter- mining the ground speed (e.g., if visual points lie in the field of vision of the radar which do not allow the position of the air- craft to be determined) since they do not appear on the chart. How- ever, they are suitable for determining the drift angle and the ground speed by visual methods. There are several methods of determining the drift angle and the ground speed by visual means. Let us discuss several of them which are most often employed: 1. Measurement of the drift angle of an aircraft on the basis of the secondary Doppler effect. The directionality of the charac- teristic of radiation from an aircraft radar in the horizontal plane is made as narrow as possible. The narrower the beam for the prop- ■ -■ - - -^ - "■ ^ netic waves, the better the resolving power ^.•--, j.-._^_^_-__ / ^icular to the radius s made as narrow as possible. ihe narrower tne beam ror tne prop- .gation of electromagnetic waves, the better the resolving power if the radar in a tangential direction (perpendicular to the radius if the scan). However, in order to produce a very narrow charac- /31i :eristic of radiation, we must use an antenna reflector on the radar 'hich has very large dimensions. Therefore, the practical width which has of 332 The widening of the characteristic of directionality within these limits is undesirable in principle for surveying the terrain, but can be used very advantageously for measuring the drift angle by the so-called secondary Doppler effect. The essence of this method is the following. Let us say that we have stopped the rotation of the radar an- tenna at a certain angle to the direction of the aircraft's motion (Fig. 3.45). Fig. 3.45. Creation of the Secondary Doppler Effect. In the picture, we can see the reflection of the electromag- netic waves from the elementary area S which we have selected. The high frequency reflected from the Earth's surface, when re- ceived aboard the aircraft, will not be equal to the frequency radi- ated by the radar, but will have a certain positive or negative fre- quency shift which is called the Doppler effect. Let us also note that the Doppler effect is proportional to the cosine of the angle between the direction of the aircraft's motion and the direction of the wave propagation (i.e., g-a). Angle 3 here represents the course angle of the antenna position of the radar, while the angle a represents the drift angle of the aircraft. For the sake of simplicity, let us consider the Doppler effect only for two extreme limits of the beam with a common character- istic of radiation directionality, the left-hand beam is marked L and the right-hand beam R in our diagram. The solution of the characteristic will be represented by the angle 6 , so that where /p is the Doppler frequency. Thus, we see that the Doppler effect on the left-hand edge of the beam is greater than on the right-hand side, so that the frequency received by the antenna from the left-hand side of the beam will be somewhat higher than that from the right. The frequencies of the left (L) and right (R) boundaries of the beam will be combined in the receiver and produce an intermediate 333 frequency as follows /319 /. ^D^ f^^. which will amount to amplitude modulation of the received signal. Now let us say that the direction of the antenna coincides with the direction in which the aircraft is moving, i.e., 3=cc . Then the Doppler frequencies of the left and right sides of the beam will be uniform in value and proportional according to the cosines 6/2: /pL~ + cos y;/d^ cos Y-. and the amplitude modulation from the edges of the beam will be abs ent . In actuality, there will be a very low-frequency amplitude modulation owing to the difference in the Doppler frequencies of the edges of the beam relative to the effect of the center of the beam (the bisectrix of the radiation characteristic), but due to the very small difference between the cosines of the angles, the beat frequency will be very low (expressed in Hertz), while the visual effect of the beat is maximum. With circular rotation of the antenna, the beating of the fre- quencies is not noticeable to the eye, since each of the luminous points is rapidly crossed by the scanning beam and appears on this screen as an individual point with subsequent afterglow. A slight impression remains of the secondary Doppler effect in a fixed antenna, when its direction differs considerably from the direction in which the aircraft is moving, since the flicker- ing of the points in this case takes place at high frequencies and is blurred by the afterglow on the screen. If the direction of the antenna slowly approaches the direc- tion in which the aircraft is moving, the luminous points all begin to flash at a reduced frequency and increased amplitude. A slow but bright flashing of the luminous points on the screen indicates a coincidence of the direction of the antenna with the direction in which the aircraft is moving. The drift angle of the aircraft is determined by the position of the scanning lines on the screen with naximum secondary Doppler effect. Measurement is performed best of all with a large-scale oper- ation of the radar (20 km for the screen radius), using a scanning delay of 20 km. It is then necessary to make a corresponding ampli- fication in the receiver for the common amplification channel, in both the high and low signal levels, as well as the corresponding inclination of the antenna. 334 One advantage of the method of determining the drift angle of the aircraft according to the secondary Doppler effect is its high accuracy. With a little experience in selecting the receiver /320 amplification and the angle for tilting the antenna, measurements can be made literally within several seconds. Several types of sector-type radars have a special operating regime and an additional indicator for measuring the drift angle according to the secondary Doppler effect. 2. Measurement of the drift angle and ground speed by sighting points near the course. If a clearly visible point is located near the line of flight of the aircraft on the radar screen, the ground speed and the drift angle of the aircraft can be measured by the movement of this point. To a void gross errors in T1+ riiio to altitude Fig. 3.^6. ± \^ a.v\_/_i.vj. ^j.v.'Ous c;xi\_/j.o measurement due to altitude errors, the sighting of the points must be made at distances from 60 to 30 km. At the moment when the point being observed crosses the 60 km marking, th« The drift angle is calculated directly from the bearing scale with negative drift angles being calculated as added to 360°. To determine the ground speed, the correction for flight altitude for a distance of 30 km is added to the length of the base, taking into account the correction for a distance of 60 km as equal to zero. Thus, at flight altitudes of 8-10 km, the length of the base turns out to be equal to: At a height of 8 km, 30.5 km; at a height of 9 km, 31 km; at a height of 10 km, 31.5 km. The ground speed is determined by means of a navigational slide rule (Fig. 3,4-7, a). Let us say that at a flight altitude of 10 km the time required to fly along the base between the 60 and 30 km markings is 2 min and 15 sec (Fig. 3.47, b). The ground speed in this case is 840 km/hr . 335 This method can be used with sufficient accuracy for measuring the drift angle of the aircraft. The accuracy of determination of the ground speed is obtained with a low and therefore very small / 321 measurement base. Thus, e.g., at an airspeed of 900 km/hr, the error in measuring the flight time on the baseCwhich amounts to M- sec) produces an error in measuring the ground speed of about 30 km/hr. In addition, at large drift angles, when the vector of the motion of the target point does not agree with the radius of the screen, errors arise in determining the measurement base from the distance markings on the screen. © w © t sec © 1 — - u © ® S1,S -I — 2 min 1 5 sec g4ff Fig. 3.47 Fig. 3.47. Determination of Ground Speed of a Point Near the Course Indicator on a Radar Screen. Fig. 3.48. Determination of Drift Angle and Ground Speed by Means of a Right Triangle . Fig. 3.4i 3. Determination of the drift angle of the aircraft and the ground speed by means of a right triangle. This method is more convenient and precise in comparison to the sighting of the motion of a landmark near the course. In addition, the use of the right triangle method makes it possible to select more freely the land- marks on the radar screen in order to track them. The bearing scale of the radar is set to zero opposite the course marking, after which the course angle of the landmark is measured with the sight, its distance on the circular markings is observed, and the timer is switched on. Leaving the sighting instru- ment in a fixed position, the operator observes the motion of the landmark across the screen. At the moment when it crosses the per- pendicular line on the sight (Fig. 3.48), the timer is switched off, the distance to the landmark is determined by the circular markings, and the flight time along the base is calculated. Corrections are then made in the first and second measurements of the oblique distance for the flight altitude; angle a between the position of the sighting line and the direction of the move- ment of the landmark is then determined as follows: tgo = R2_ and the length of the measurement base is : 336 coso sin (90 — «) This problem is easily solved on a navigational slide rule /322 (Fig. 3.49). a ®- T ?- b I J"-" f- Fig. 3.H9. Keys for Determining the (a) Acute Angle of the Triangle and (b) Measurement Base on the NL-IOM. The drift angle is determined as the difference between the first course angle of the landmark and the angle a (Fig. 3.49, a), while the ground speed is determined as the length of the base rela- tive to the time required to cover the distance (Fig. 3.49, b). Example . At a flight altitude of 10 km, the course angle of a landmark was initially equal to 8° at OD^ = 60 km. The oblique distance at the moment when the landmark crosses the transverse line in the sight was 23 km. The flight time along the base was 5 min and 35 sec. Find the drift angle of the aircraft and the ground speed. Solution. The correction for the flight altitude for the distance will be considered as equal to zero. The correction f the second distance (OD = 23 km, H = 10 km) is equal to 3 km, s that the horizontal distance is HD2 = 20 km. On the navigational slide rule, we find the angle a = 18.5° (Fig. 3.50, a) and the length of the measurement base is S = 60 km (Fig. 3.50, b ) . a ^ ^^ ¥ b -^— f^ ^ first or so (D 80 60 (£) SO (?) M 830 -t 1 — ® 4 min 3 5 sec Fig. 3.50. Determination of (a) the Acute Angle of a Tri- angle, (b) the Base and (c) the Ground Speed on the NL-IOM. Therefore US = 8 - 18. 5° = -10 . 5° . 337 The ground speed (W) is therefore equal to 830 km/hr (Fig. 3.50 , c) . k. Determination of the ground speed and drift angle of an aircraft by double distance finding using a sighting point with equal oblique distances. This method is the most precise of the methods which we have discussed which use sighting of landmarks. However, it calls for the maximum time for measurement and calculation. When a highly visible point shows up in the forward part of the screen, the crew waits until it reaches one of the circular distance markings (Fig. 3.51). At the moment when this point crosses the distance marking, the timer is switched on and the course angle of this point is measured. The crew then waits until this point moves across the screen and crosses the same circular distance mark- ing at the rear of the screen. At the moment when it crosses it, the timer is switched off and the course angle of the point is meas- ured once again. Since in this case HDi = HD2 , the line of motion of the point (from A to B) is perpendicular to the bisectrix between CALi and CAL25 i.e., if the point moves to the right of the course line of the aircraft, the drift line of the aircraft is determined by -the formula /323 US CAL]+CAL2 2 90° , and if the point moves to the left of the course line or US = CALi+CAL? 270< To determine the ground speed, a correction for flight alti- tude is made in the oblique distance at points A and B and the length of the measurement base is determined by the formula ^„ . CALy-CAL S = 2i?sin ^-r L If HDi = HD2 exceeds five times the flight altitude, the cor- rection for altitude is considered to be zero. Example. H = 10 km, HDi = HD2 = 60 km; CALi = 32°; CAL2 = 152°; the flight time along the base is 8 min and 15 sec. Find the drift angle and the ground speed. 338 Solution US 32° + 152° 1 -90° = + 2°; 152 32 5 = 2-60 sin = 120 sin 60°; By using a navigational slide rule, we can solve the latter equation and find the ground speed (Fig. 3.52). 5 <=^ 105 KMi IT" 765 km/hr. CAL ® ^P © W5 7S'S is- Fig. 3.51. ® W5 m Q) 8 min 1 5 sec Fig. 3.52. Fig. 3.52. Determination of (a) the Measurement Base and (b) the Ground Speed on the NL-IOM. Fig. 3.51. Determination of the Ground Speed and Drift Angle by Double Dis- tance Finding of a Landmark at Equal Oblique Distances. We should mention that in solving problems in determining the / 324 drift angle of an aircraft by the four methods enumerated above, the bearing scale of the radar may be set to the aircraft course rather than zero, e.g. , according to the orthodrome . Then the course angles in all the formulas will be replaced by the bearings of the landmarks, and the result of the solution will not be the drift angle but the actual flight angle of the aircraft. Autonomous Doppler Meters for Drift Angle and Ground Speed Autonomous meters for ground speed and drift angle of an air- craft, based on the Doppler effect, offer broad perspectives for automation of the processes of aircraft navigation and pilotage of aircraft. ^ 8 Fig. 3.53. Diagram of Formation of Doppler Fre- quency With a Moving Object. 339 eters of an aircraft Continuous measurement of the motion parame makes it possible to use simple integrating devi automatic calculation of the aircraft path in time. In addition, a constant knowledge of these parameters makes it possible to regu- late them in such a way that the aircraft follows a given flight trajectory with a minimum number of deviations. All other radio devices for aircraft navigation make it possible to determine only the locus of the aircraft. The motion parameters of an aircraft can be determined only discretely for individual path segments 5 using the navigational devices described above. As we pointed out at the beginning of Chapter One, the flight regimes of an aircraft are almost never stable, with the exception of the end points of curves along separate parameters. A strictly stable flight regime for all parameters simultaneously is never encountered. Therefore, automatic or semi-automatic calculation of the path on the basis of motion parameters measured over individ- ual segments is a very approximate and unreliable method. The operating principle of Doppler meters is the following. Let us say that we have a moving source of electromagnetic oscillations at a high frequency A and a fixed object B which reflects these oscillations (Fig. 3.53). If the source A remains fixed relative to object B, then after a period of time which is required for the electromagnetic waves to travel from point A to point B, electromagnetic oscillations will be set up in the latter at the same frequency as those emit- ted by the source. When the source of oscillations moves toward point B, each successive cycle of oscillations is emitted somewhat closer to this point; its propagation time to reach point B is somewhat less than in the preceding cycle, so that the moments at which the oscillation! arrive at point B can be compared. Let us call the wavelength of the source X, and the propaga- tion rate of electromagnetic waves a. With a fixed source, the frequency of the oscillations (/) both at the point of emission and at the point of reflection of the waves will be equal to With a movable source, the number of oscillations reaching point B per unit time is increased by the number of wavelengths contained in the distance covered by the aircraft in that same unit time , i.e., c-yw c w /325 340 the The increase in the frequency W/X, produced by the motion of source, is called the Doppler frequency (f-Q). Similarly, the oscillation frequency at the point of reflec- tion will decrease if the source recedes from the reflection point for the electromagnetic waves. Doppler meters work on the same principle of signal transmis- sion as aircraft radars, i.e., frequencies are received that have been emitted by aircraft sources after their reflection from the Earth's surface. Therefore, a double Doppler frequency is received which arises along the path of electromagnetic waves, from the air- craft to the reflecting surface and along the reverse route from the reflecting surface back to the approaching or receding aircraft. There are three ways of separating the Doppler frequency in receiving signals aboard an aircraft: (1) The internal coherence of the signals, when the received frequency is combined within the receiver with a frequency radi- ated by the source, as a result of which there is a beating of the double Doppler frequency; (2) External coherence, when the receiving antenna picks up signals which have been reflected from the ground as well as sig- nals radiated by the transmitting antenna through the external med- ium ; (3) Autocoherence of the signals; in this case, the frequen- /3 26 cies of signals reflected from the Earth's surface in the forward and backward radiation of the re cei ving- transmi tt ing antenna are combined in the receiver without the frequency radiated by the antenna. Since the oscillation frequency is increased by 2 f-Q relative to the preceding beam, and decreased by the same value for the follow- ing beam, the beat frequency will be equal to four times the Dop- pler frequency. If we agree to call the Doppler frequency the beat frequency separated in the receiver as a result of superposition of the signals, then for the cases of internal and external coherence we will have /t 2W_ X and for the case of autocoherence we will have: „ _ W •^ D X ■ In principle , Doppler meters with internal and external coher- ence can be made with a single-beam antenna, but with autocoher- ence a minimum of two beams is required. In practice, as we will see later on, it is convenient to use antennas with three or four beams. Recently, the most widely employed type is the Doppler meter with four-beam antennas . 341 Fig. 3. 54. Projection of the Ground-Speed Vector on the Direction of the Radiation of Electromagnetic Waves. Since the characteristics of directionality of the antennas of Doppler meters in the general case do not coincide with the vector of the ground speed of the aircraft, it is necessary to consider the' actual Doppler frequencies separated in the receivers Usually, the slope of the antenna beams of the meter is selected so that the areas of their reflection from the Earth's surface are not too far from the aircraft, i.e., the power of the transmitter is used most effectively. The slope angle of the beam rela- tive to the horizontal plane is called the angle 6 (Fig. 3.54). Obviously, when the beam is inclined relative to the plane of the horizon, the Doppler frequency will be proportional not to the modulus of the ground speed vector, but to its projection in the direction of the antenna beam. For example, for a meter with internal coherence , •^D - "T =°^ ®- On the other hand, the ground speed vector of the aircraft can be divided into two vector components: oir= oWi + Wi r. The vector WiW is directed perpendicular to the antenna beam, and therefore the Doppler effect is not produced. The vector ./327 OWi = OWcos is effective In addition to the fact that the antenna beam is set at a cer- tain angle to the vertical plane, the antenna beam is usually directed at a certain angle to the longitudinal axis of the aircraft in the horizontal plane. For example, with a four-beam antenna, the longi- tudinal axis of the aircraft is the bisectrix of the angles betweeh the directions of the forward and rear beams of the antenna (Fig. 3.55). The angle between the longitudinal axis of the aircraft and the direction of the antenna beam in the horizontal plane is called the angle 3. Hence, in receivers with internal and external coherence separated Doppler frequency the 342 ■ Hini II 11 ^D = 2r cos 8 cos (P — «)t Fig. 3.55. Diagram of the Positions of the Beams from an Antenna on a Doppler Met er . verse rolling of the aircraft where a is the drift angle of the aircraft . In the special case where the drift angle of the aircraft is ab- sent, the Doppler frequency for each antenna beam will be the same 1W f =— — cosBcosp. Three- and four-beam antennas are desirable because they make it possible to compensate automatically for errors in measurements which arise with longitudinal and trans- At the same time, in cases when single beam or two-beam an- tennas are used, they must be placed on gyros tabili zing devices. In the opposite case, longitudinal or transverse rolling of the aircraft will change the slope angle of the antenna 9, thus leading to a change in the Doppler frequency. In the case of a four-beam antenna, the longitudinal or trans- verse rolling of the aircraft produces a change in the slope angle of one pair of beams in a positive direction and changes the oppo- site pair in the negative direction by the same magnitude. If angles 6 are then located on an approximately linear section of the cosine /32i curve, the frequency shift of the opposite antenna beams will be opposite in sign but approximately the same in magnitude, which can also be used for compensating roll errors in the system (Fig. 3.56, a) . ''')CX'\ ■f. Fig. 3.56. Shifts in the Doppler Frequency with Tilting of the System: (a) Change in the Cosines of the Angles; (b) Frequency Shift. 343 For example, in the case of receivers with autocoherence , when the Doppler frequency increases in the front right-hand beam and decreases by the same magnitude in the rear left-hand beam, the beat frequency of one pair will simply be retained. In systems with internal and external coherence, turning of the antenna leads to doubling of the frequency spectrum of the oppo- site beams (Fig. 3.56, b). With a horizontal position of the longitudinal and transverse axes of the aircraft , the Doppler frequency in the forward and oppo- site rear beams will be the same. Tilting the system shifts the frequency spectrum of one of the beams forward, and that of the opposite beam backward to the same extent. However, if we add up these frequencies with time and divide them by the measurement time, the average frequency will turn out to be equal to the frequency of the horizontal position of the axis of the aircraft. Doppler meters which are presently in use can be divided into four types, depending on the regime of emission and reception of signals : 1. Pulse meters. In transmitters, these meters produce high- frequency pulses in the same manner as is done in aircraft radars. Reception of reflected signals takes place in the intervals between pulse emission. In order to separate the Doppler frequency, auto- coherence by beam pairs is employed. A shortcoming of this method is the presence of "dead" alti- tudes, i.e., when the reflected signals arrive at the moment coin- /329 ciding with the emission of pulses, and not in the intervals between them. In addition, when flying over mountainous terrain, the distance to the Earth's surface according to opposite beams of the antenna may not be the same, thus leading to a failure of the arrival of reflected signals to coincide for these beams and producing a disturb- ance of their coherence . Another shortcoming of pulse meters is the need for high volt- ages to drive the magnetron in analyzing the high-frequency pulses, thus necessitating a hermetic sealing of the transmitter units and subjecting them to a certain pressure. 2. Meters using continuous radiation of high frequency. In this case, the high frequency is radiated continuously by the trans- mitter. Reception of signals is accomplished with a separate antenna, having a certain by-pass coefficient with the transmitting antenna. The reflected signals are combined in the receiving antenna with the frequency produced by the transmitting antenna, so that the beat frequency is separated out in an external coherence system. The advantages of this method are the independence of the re- 3 44 iiiii~i ■■■■■II I ception conditions for the signals of flight altitude and the local relief. In addition, devices of this type worked at relatively low powers in the receiver mechanism. A shortcoming of this method is the need to have separate an- tennas for transmission and reception of the signals. 3. Meters with continuously pulsed radiation. These meters employ a constant regime of generation and transmission with pulsed emission of a portion of the high frequency into the antennas by means of commutating devices. The reflected signals are combined with the frequency developed by the transmitter in the intervals between the moments when the high-frequency segments are emitted (internal coherence). This means that it becomes possible to use part of the advantages of continuous emission (operation at rela- tively low voltages in the transmitter circuits) and the pulse systems (reception and transmission with a single antenna). However, the shortcomings still remain which afflict pulsed meters, i.e., the presence of "dead" altitudes and the effect of the relief on reception conditions. In addition, there are also difficulties in using these meters at low flight altitudes, since at a very short signal path, the moments of transmission and recep- tion are practically impossible to separate. h. Meters with frequency modulation of signal transmission. Frequency modulation of signal transmission can be used either in a pulsed or continuous -puis ed regime of operation for the meter. If the tran frequency at low of reception of tainous regions signals, when th sion of the sign to the combinati signals will be emission, since greater than the this case , thee tainous regions . smission of high-frequency pulses at a constant altitudes makes it possible to superpose the moments signals on the moments of emission, while in moun- /330 there may be disruptions in the coherence of the ere is a change in the frequency of the transmis- als, and only a portion of them will contribute on with the radiation moments. The majority of received in the intervals between the moments of the duration of the intervals is made sufficiently duration of the pulses. To a certain degree, in ffect of disruption of coherence is reduced in moun- The best properties are exhibited by continuous -puis ed meters with frequency modulation of signal transmission, since in this case all positive qualities of the continuous and pulsed systems are employed. However, the shortcomings of the pulsed systems remain, including difficulty in making measurements at very low flight alti- tudes . Of the types of Doppler meters which we have discussed, the ones which are currently used most widely are the meters with con- tinuous radiation and continuous -puis ed meters with frequency modu- lation of signal transmission. 31+5 1 t \\/\/\/\7\ wwv/wv T A^\>^W\M' Fig. 3.57. Doppler-Meter Antenna: (a) Waveguide Lattice; (b) Diagram of Beam Formation. In the first types of Doppler meters, beginning with the single beam versions, reflector- type antennas were used with an isosceles directional characteristic. Recently, antennas of the "waveguide network" type have come into use. The principle of operation of these antennas is the follow- ing (Fig. 3.57, a). Imagine a rectangular lattice, made up of waveguides, to one corner of which an electromagnetic wave of high frequency is applied. On the upper walls of the transverse waveguide in this lat- tice, there are slots for emission of wave energy into space. The wave energy, propagated along a transverse waveguide, emerges through the slits with a certain shift in time from one slit to /331 the next, so that there is an interf f erence of the waves emerging from the slits, as is the case in spaced antennas (Fig. 3.57, b). The direction of the radiation maximum and the isophasal lines are located at an angle to the surface of the waveguide. Since the electromagnetic energy propagated along the longitud- inal wave guide reaches the next transverse waveguide also with a shift in time, a similar picture of interference with a tilting of the isophasal line also takes place along the waveguide lattice. As a result, an isophasal surface is formed above the waveguide lattice, having a slope in the direction of its diagonal toward the corner opposite the corner at which the electromagnetic waves enter the lattice. Consequently, one of the beams will be formed along the diagonal of the antenna. If wave energy is also transmitted from the diagonally oppo- site corner of the lattice, two oppositely directed antenna beams can be formed simultaneously. 31+6 Flat, multi-beam antennas, especially when fixed in position, are very useful, since they can be placed below the radio-trans- parent housing flush with the skin of the aircraft and do not produce any additional aerodynamic resistance during flight. Schemattc Diagram of the Operation of a Meter with Contin- uous Radiation Regime The high frequency processed by the transmitter passes through a commutation device to the transmitting antenna, where the beams for propagation of electromagnetic waves are formed in pairs. The commutating device is connected to the counter of the meter, to separate the frequencies of the first and second pairs of beams (Fig. 3.58). A portion of the wave energy radiated by the transmitter reaches the receiving antenna, where it is combined with the received signals reflected from the Earth's surface, so that the Doppler frequency of the given pair of beams can be separated. The separated Doppler frequency, after amplification, passes to the calculating device, at whose output is an indicator for the ground speed and drift angle of the aircraft. As we already know, for the four-beam antenna of a Doppler meter with internal coherence, the separated frequency by beam pairs will be equal to: (a) For the first pair. f 2W ^Di = -Y- (b) For the second pair, cosO cos (P + a); /332 / Do = - 2W cosOcos(P — fl). For a Doppler meter, the sign of the angle is not important, but its absolute value is. Therefore, we can simply assume that with a positive drift angle, the drift angle in the right-hand pair of beams will be calculated from the angle 3, while in the left- hand pair these angles will be added. With a negative drift angle, the calculation of the angles will be performed in the left-hand pair of beams, and combined in the right. Therefore, the last two formulas given above can be written in the form JD^ = Ji51_ cos e COS (p + a); f = -— — cos 6 cos (p — a), •I T\ - A 347 ■ where the sign shows that the formulas change places for the left and right-hand pairs of beams when the sign of the drift angle changes . signals f'true signa ls y computer indicator ^,9 nii ei* receiving antenna _i.'ai)tnmatic 2 |navigationa!„ * device '^ receiver IC commur- tation device- 3 [ counter X,7 trans mitter i" transmitting antenna Fig. 3.58. Meter . Functional Diagram of A Doppler Note, Since the pairs of beams are diagonal to the wave guide lattice, and each of them contains a left and right-hand beam relative to the longitudinal axis of the aircraft , the left or right-hand pair of beams here is referred to as a pair whose leading beam is directed to the left or right of the longitudinal axis of the air- craft . Obviously, in the case of a fixed antenna on the aircraft (Fig. 3.59), the first problem for the calculating device of the Doppler /333 meter is the determination of an angle at which fr f D2 (P + a) \^ cos(p — a) ; Since the angle g is a constant value, and the frequencies /q , f-Q2 s^s variable, the solution of a problem of this kind does not ^ present any significant difficulties. The desired angle a is the drift angle of the aircraft. The second problem for the computing device is the determin- ation of the ground speed (W) with a previously known drift angle 348 and Doppler frequency for a pair of beams: W = w ^D,* COS e COS (P + a) cos 6 cos (P — a) Fig. Doppl the R Beams 3.59. D er Frequ ight- an of an A ifferece in encies of d Left-Hand nt enna . tion of the aircraft motion / D / D' The calculated drift angle of the aircraft and the ground speed are transmitted to the visual indicator of these parameters and also to the automatic navigational device for inte- gration of the aircraft path in time. The problem of the calculating device of the Doppler meter is sig- nificantly simplified by mounting a movable antenna on the aircraft. In this case, the direction of the antenna is set so that the Doppler frequen- cies of both antennas /p and /pi will 1 2 be the same, i.e., the bisectrix of the beams will coincide with the direc- = ^D.- 2W cos 008 p. Then the drift angle of the aircraft is determined by the course angle of the antenna setting, and the ground speed is found by the formula : (a) With internal and external coherence W = ^D^ (b) With autocoherence W-- 2 cos 8 cos p 4 cos e cos P' This means that all coefficients entered into the formulas (with the exception of /p ) are constants while f-Q is a variable quantity. We should mention that during flight above the ocean, Doppler frequencies from pairs of beams in a Doppler meter are somewhat lower than above dry land at the same airspeeds. This is caused by peculiar features of the reflection of electromagnetic waves from the surface of the water. /33^■ In flight above dry land, if the conditions for diffuse reflec- tion of the waves are approximately the same over all areas in contact with the Earth's surface, and the maximum amplitude coincides with 349 the center of the beam at the maximum of the radiation character- istic, then the reflection conditions above a watery surface will depend to a considerable extent upon the angle of incidence of the beam. Therefore, the leading edge of the beam will have a sharper angle of incidence (and therefore a lower signal amp,litude), while the trailing edge of the beam will strike more obliquely and have somewhat greater amplitude. Consequently, the maximum amplitude of the signals shifts from the center to a region of lower Doppler frequency (see Fig. 3.54). To compensate for errors in the operation of the meter above water, the circuit is designed to include a calibration element which is switched on from the control panel by turning a switch from the "land" position to the "sea" position. Over a smooth watery surface (with a swell less than a scale value of one), the potential of the reflected signals becomes inad- equate to ensure operation of the meter, and the latter then is turned off by switching the automatic navigational device to memory operation . The channel of the Doppler frequency receiver is fitted with a filter intended to damp out all parasitic frequencies produced by other electronic devices mounted aboard the aircraft which cou'ld disturb reception of reflected signals from the Earth's surface. The filter must have a narrow passband within the region of Doppler frequencies of the received signals. If the frequency of a carefully adjusted filter differs con- siderably from the midpoint of the range of Doppler frequencies being employed, it begins to introduce errors in the measurement of the Doppler frequency, shifting it toward the point of fine tuning of the filter. Therefore, filters are used with automatic tuning for the frequency of the signals employed. Use of Doppter Meters for Purposes of Aircraft Navigation Doppler meters for ground speed and drift angle are very effec- tive in aircraft navigation. The following problems can be solved directly by using a Doppler meter: (a) Maintainance of a given direction of flight along an ortho- drome or loxodrome, automatically if desired. To do this, it is only necessary that the sum of the course (y) and drift (a) angles /335 of the aircraft be constantly equal to a given flight path angle (^) : ip = y + a; (b) The calculation of the path of the aircraft in terms of distance can be solved on the basis of the ground speed and time: S = ¥t. 350 In view of the above, as well as the relative simplicity of automating aircraft navigation on the basis of Doppler measurements, the latter are practically impossible to use without combining them with automatic navigational instruments. Automatic navigational instruments connected to Doppler meters calculate the aircraft path with time in an orthodromic or geographic system of coordinates. To calculate the path of the aircraft in an orthodromic sys- tem of coordinates, the navigational devices are connected to trans- mitters of the orthodromic course (a gyro assembly for the course system, operating in the GSC regime). The automatic system includes a transmitter of the flight angle or (as it is usually called) the given chart angle (GCA). The signals for the drift angle of the aircraft, obtained from the meter, and the course signals of the aircraft, obtained from the course system, are combined and their sum compared with a given path angle fed into the transmitter. If the sum of the course and the drift angle of the aircraft is equal to the given path angle of the flight (Jj = y + a, the ground speed is directed along the J-axis: W = W^i W^ = 0. If this equation is not satisfied, the vector of the ground speed is divided into two components: UTt'^ ITcos (t + « — "l*); Wt= Wsln{i + a-^). The vector components obtained for the ground speed along the axes of the coordinates are integrated over time and calculators are used to find the running values of the aircraft coordinates X and Z . /336 Calculation of the aircraft path and geographic coordinates can also be done directly on the basis of the signals from the Dop- pler meter and the course calculator. However, to do this it is necessary to have an exact knowledge of the true course of the air- 351 craft and to express the division of the ground-speed vector of the aircraft according to the formulas: dt dk dt sin {f + a) = W cos T To ensure operation of the gyroscopic transmitter in a regime of true course 5 in addition to the moment which compensates for the diurnal rotation of the Earth 0)^ = Q sin cf, it is necessary to add the moment which compensates for the change in the true course with time due to the eastern or western compo- nent of the ground-speed vector of the aircraft: '•r= Wsia if + a) sin y cosy = Wsln(,t + a)tg<f. However, calculation of the aircraft course by this system cannot be considered adequate for three reasons: (2) The constant dependence of the operation of the course system on the operation of the Doppler meter and a calculating device introduces inaccuracies into the aircraft navigational elements . For example, when the Earth is not visible, a flight can be made over dry land; however, if the aircraft then begins to travel over a smooth watery surface, the reflected Doppler signals will not only introduce errors into the accuracy with which the path is calcu- lated with time, but will also incorporate errors in the operation of the course system. (3) The errors which appear in the calculation of the air- craft course at the points of correction of its coordinates cannot be used directly for correction of the aircraft course, as can easily be done in an orthodromic system of coordinates. A more logical calculation of the geographic coordinates of the aircraft would involve the orthodromic system of aircraft naviga- xne aircrarT wouxa involve xne orxnoarouii c sysxeui or aircrai tion, based on a constant conversion of the orthodromic cour the aircraft to the true course on the basis of the running inates of the aircraft se of running coord- ^true = «''=»g^^+^* tgV siny/ ort 352 where Ay = Y - ^ ort ort ort The true course for the aircraft obtained in this manner can be used to calculate the geographic coordinates of an aircraft as was shown earlier; it can also be used for correcting the ortho- dromic course by astronomical means. The advantages of a method of this kind are the independence /337 of the true course from the ground speed and its automatic correction along with the correction of the aircraft coordinates. However, we should mention that the calculation of the air- craft course and geographic coordinates should really be replaced by a constant conversion of the running orthodromic coordinates into geographic ones, e.g. , by Formulas (1.64) and (1.65): sine geog s inX sind) cosG-cosA ^sin( ort ort = smX cosd) seed) geog ort ort geog In this case, the geographic coordinates will always agree strictly with the orthodromic ones, so that there will be output parameters from only one integrating device and automatic correc- tion in the second system with correction of coordinates in one of them . In general, the geographic coordinates are not of much inter- est as far as aircraft navigation is concerned. However, they are important for ensuring accurate operation of navigational trans- mitters (latitudinal correction of course systems, analysis of gyre For purposes of aircraft navigation, automatic navigational devices are much more dependable for calculating the path of the aircraft in orthodromic coordinates. In addition to the basic regime of operation by signals from a Doppler meter, automatic navigational devices as a rule have an operating regime with "memorized" navigational parameters. The regime for operating by "memory' one of the following two versions. can be incorporated in 1. By "memorizing" the last values of the ground speed and drift angle of the aircraft. In this version, in the case when 353 there is an interruption in the arrival of Doppler signals for some reason, (e.g., when there are no waves in a flight over water), the path can be calculated by "memory" for a period of 15-20 min, only under the condition that the flight direction and airspeed have been recorded. With a changing flight regime for the aircraft, calculation by "memory" leads to large errors, since the ground speed and drift angle change on a new course or with a change in other parameters. this vari potentiom eters : Then, if the path of the wi the wind of the fl redis trib is not di By "memorizing" wind parameters at flight altitude. In ety, the calculating device is provided with special "memory" eters, which constantly set the value of the wind param- Uji = Wcos (Tf + a — iji) — V cos (7 — ij<); Ug=W3ln(T + a — ^)~ Vsin(Y — 4;). the signals should not be received from the Doppler meter, of the aircraft can be calculated by comparing the vector nd speed along the axis of the system of coordinates with /33 i vector components added to it. If the given path angle ight then changes , the components of the wind vector are uted among the coordinate axes and the calculation regime s turbe d . However, in both the first and second methods of "memorizing" navigational parameters, no provision is made for an exact calcu- lation of the aircraft path during a long period of time, since the wind parameters change with distance . In these cases , the navi- gational mechanism is used for calculating the path of the aircraft on the basis of discrete data obtained by measuring the ground speed and drift angle, e.g., by means of a aircraft radar or some other device, as is done (e.g.) when using the navigational indicator NI-50B. In some types of navigational instruments, inertial or astro- inertial instruments are used as memory devices. Inertial navigational devices are gyros tabi li zed platforms on which accelerometers and special gyroscopes are mounted which integrate the accelerations of the aircraft with time along the axes of the reference system. In the case when the motion of the aircraft along one or two 354 axes takes place with acceleration, a moment is applied to the axes of the gyroscope which is proportional to these accelerations, so that precession of the gyroscope axes takes place, i.e., there is integration of accelerations with time. Since dt and 0'' where a^ and a^ are the accelerations along the corresponding axes, we can use the position of the gyroscope axes to get an idea of the components of the aircraft speed along the axes of the coord- inates . The components of the ground speed along the axes of the ref- erence system can be integrated in turn with time by means of a navigational instrument. In an operating Doppler meter, the position of the axes of the integrating gyroscopes can be corrected by signals from this meter. In the case when the Doppler information does not arrive, the inertial device can be used for a long period of time to retain "remembered" values of the components of the speed along the axes of the coordinates, correcting them for any accelerations that arise in the way of wind changes, as well as in changes in the flight regime . Aircraft navigation using Doppler meters and automatic nav- igational instruments becomes extremely simple and practical, but very careful preparations for flight and exact measurements of the coordinates of the aircraft at the correction points are required. An exact measurement of the aircraft course is extremely important in this regard. /339 On the other hand, the fact that the crew is constantly aware of the ground speed, the drift angle of the aircraft, and its coord- inates makes it possible to maintain a given flight trajectory for long periods of time according to the indications of the instru- ments. To do this, it is sufficient that the sum of the aircraft course and the drift angle be constantly equal to the given path angle, and that the Z-coordinate of the aircraft be equal to zero. It is particularly easy to solve problems in aircraft navigation if the readings of the aircraft course and the drift angle are obtained from the indicator in the form of a sum, i.e., as the actual path angle of the aircraft flight. It is then sufficient to pilot the aircraft so that with Z equal to zero, the flight angle will actually be equal to the given one. 355 In a case when the path angle of the flight is not maintained precisely and the Z-coordinate of the aircraft is not equal to zero, or, if the improper operation of a system has caused the aircraft to deviate from the given flight path as revealed by correction of its coordinates 5 the path angle of the flight is set so that the aircraft approaches the given line of flight at an angle of 3-5°. When the Z-coordinate decreases to zero, the path angle of the flight becomes equal to the given value. The aircraft can be placed on the given line of flight by using the autopilot. For this purpose there must be a calculating unit aboard the aircraft for relating the Doppler meter with the auto- matic navigational device and an autopilot which solves the simple prob lem : AZ+kh^ = Cj where AZ is the lateral deviation from the line of flight, Ai|; is the angle of approach to the line of flight, and k is the selected coupling factor. The aircraft is then steered so that a lead in the path angle of the flight is taken when the aircraft deviates to a certain degree from the given line of flight with a certain coefficient. Then, in the presence of lateral deviation, the aircraft will automat- ically move into the line of flight, decreasing its lead as it ap- proaches the latter. Certain difficulties in aircraft navigation when using Dop- pler meters with automatic navigational devices are encountered in converting the computer to calculate the path in orthodromic coordinates of the previous stage, at the turning points along the route. The methods of conversion to the new system of calculation of coordinates is shown in Chapter II, Section 9. However, when using Doppler meters, it is better to set the aircraft coordinates to the reference system of the previous stage before beginning the /3140 turn of the aircraft. For example, with Z^ = , Zi = -LLT. X2 = -LLT cos TA; Z2 = LLT sin TA . c c I b se a m djusts itself according to the path angle of the ent, and it can be used to calculate the aircraft the reference system of this segment. coordinates in 356 The transition of the aircraft to the next orthodromic seg- ment of the path is accomplished by the indications of the second calculator, after which the first calculator is cleared and set for the next path segment. As we have already pointed out, in the case of double calcu- lators, their readings are mutually related, i.e., they are con- verted according to the formulas : X2 - X^cosTA-Z isin TA ; Z2 - XisinTA+ZjcosTA . Therefore, in correcting the coordinates of the aircraft on one of these computers, a correction is automatically made in the air- craft coordinate in the reference system of the next stage. Thus, at each turning point along the route, the aircraft makes a turn in a previously prepared and corrected system of coordinates for the next stage of flight, thus completely getting rid of any undesirable features of the transition which might occur if only one calculator were used. Preparation for Flight and Correction of Errors in Aircraft Navigation by Using DoppZer Meters Aircraft navigation using Doppler meters for measuring the ground speed and drift angle of an aircraft can be done very simply and rapidly. However, the required accuracy for aircraft navigation when using these devices can only be achieved with very careful preparation for flight, as well as careful correction for errors in aircraft navigation which arise during flight. When using Doppler meters, there may be errors in measuring the following elements in aircraft navigation due to errors in the transmitters : (a) Measurement of the aircraft course; (b) Measurement of the drift angle and ground speed; (c) In the programming of the given path angle and the dis- /3^1 tance of the flight stages; (d) In the integration of the aircraft flight along the axes of the coordinates by the automatic navigational device. The accuracy with which the aircraft course is measured is of extreme importance for aircraft navigation when using Doppler meters and is closely related to the proper programming of path angles for each flight stage. This is explained by the very high 357 requirements for accuracy in determining path angles in preparing for flight. Preparation for flight using Doppler meters must be carried out properly according to the third group of conditions in Chapter Two, Section 2. For each flight segment, all parameters of the orthodrome must be determined, beginning with X dis dis '0 1- 1 5 'g ^1 = tg <f2 ctg ?i cosec AX — cfg AX. It is then necessary to determine the original azimuth of the orthodrome ag by the formula sInXoi '8"" — ::; — ' tgTl and then the coordinates of the intermediate points on the ortho- drome for plotting them on the chart: slnXn tgy/ = ^. tg«o The distance between the turning points along the route along the orthodrome can be determined by the formula cos Si = cos Xq cos <fi. If we introduce into this formula the coordinates of the initial and final points of the flight segment, and (when necessary) any intermediate points, we can find the distance to those points from the starting point of the orthodrome. The distances between the points are determined by calculating the distances from the start- ing point of the orthodrome to them. For programming the flight path angle we determine the azi- muths of the orthodrome at the beginning and end of each segment according to the formula tgo, = — — S-. sin ft If it is proposed that we use astronomical methods for cor- recting the aircraft course in flight (e.g., in flight over water or terrain which has no identifying landmarks), the course correc- tion points are marked and the azimuths of the orthodromes at the correction points are determined by this formula. /3 42 Reference points are selected for correcting the coordinates of the aircraft during flight. Usually these are landmarks which show up clearly on radar or places where goniometric-rangef inding installations are located. Then the orthodromic coordinates of 358 these points are determined, and the ground goniometric-rangef inding instruments are used to determine the azimuths of the orthodromic segments of the path on which these devices will be used, relative to the meridians on which the ground beacons are located. The given path angle for the first flight segment is considered equal to the azimuth of the orthodrome at the starting point of this segment. The path angles of all subsequent path segments are considered to be equal to the sum of the path angle of the prev- ious segment plus the angle of turn in the path at the turning point on the route . Doppler meters have relatively low errors in measuring the drift angle of an aircraft, so that they can be compensated for in the total by the errors in aircraft course. In general, besides the errors in measuring the drift angle, depending on the operating regime of the meter, the height and speed of flight, which have a more or less constant character, there are errors which have a fluctuating nature (oscillations in the meter readings from the average value). The principal reason for fluc- tuations is the varying conditions of reflection of electromagnetic waves from the Earth's surface. When some point is encountered which reflects electromagnetic waves well, in the ellipse of reflection from the Earth's surface, the maximum of the amplitude of Doppler frequency is first displaced forward (for a rear beam, backward); then, as the point passes through the ellipse of reflection, the maximum of the amplitude shifts toward the average Doppler frequency and then backward, into a region of lower frequencies. Thus, there is first a positive "firing" of the Doppler fre- quency, then a leveling off, and finally a negative "firing". For the rear beam, the "firings" of frequencies take place in reverse order . The periods of fluctuating oscillations are short and depend on the time required for the reflecting points to pass through the ellipse of reflection. Practically speaking, they are located within the limits of 3-6 sec, so that they can be smoothed out to a consid- erable degree by selecting the proper rate of analysis for the read- ings of the drift angle and ground speed. As far as the calculations of the aircraft path for distance and direction are concerned, the fluctuating oscillations do not have any noticeable effect on it, since after 3-5 min of flight the integral value of the positive fluctuations becomes equal to the integral value of the negative fluctuations . The process of navigational exploitation of autonomous Dop- pler systems for aircraft navigation can be employed for adjust- ing the system itself, i.e., in correcting the aircraft coordinates /343 359 manually or automatically it is possible to determine and compen- sate simultaneously the systematic errors in the operation of the system as a whole. In fact, if the aircraft (at the starting point of a flight segment) is located precisely on the desired flight line, but its Z-coordinate is equal to zero, and we keep the coordinate Z equal to zero during all subsequent stages of the flight, the aircraft will have to remain on this line constantly. If this is not the case, an error will crop up in the calculation of the aircraft path with respect to direction, i.e., a certain angle will develop between the given and actual flight path angles of the aircraft. It is most likely that under the conditions of precise deter- mination and setting of a given path angle on the transmitter, an error in calculation will arise as a result of improper measure- ment of the aircraft course, since gyroscopic devices can show drift in their readings with time. Therefore, the total correction which is required for proper calculation of the path should most logically be made in the readings of the course instrument. If a portion of the errors in calculating is not related to the operation of the course instrument, then their contribution to the course errors will not make the accuracy of aircraft naviga- tion any worse . The latter statement is valid for a complex of instruments which permit calculation of the path in terms of direction, but it is theoretically not completely valid for instruments intended for distance finding of landmarks for the purpose of making correc- tions in the aircraft coordinates . Nevertheless, if we consider that the total error in measuring the drift angle and calculating the paths in terms of direction with an automatic apparatus is no more than 0.2 to 0.3° as a rule, we must recognize that correction of the aircraft course by the results of calculating the path is much more accurate than correct- ing it by any other me thods, including astronomical ones. During flight, the actual aircraft coordinates are determined by the distances B and the path bearings of the landmarks (PBL), selected for this purpose by the formulas: (a) In the measurement of aircraft radars X = Xy-E cos PBL ; Z = Zy-R sin PBL. (b) In the measurement of goniometric-rangef inding systems: X=X„ + /?cos(/l — 4i„); Z = ^^ + /?8ln(i4 — 4<„), 360 where ^m and Zx, are the orthodromic coordinates of a ground bea- 'M con . Obviously, the formulas for the aircraft radar and the gon- iome tric-rangef inding systems are invariable. The difference in the signs of the second terms on the right-hand sides is explained by the fact that the bearing of a landmark is obtained with the aid of an aircraft radar but the bearing of an aircraft relative to a ground beacon is obtained with the aid of a goniometric-range- finding system. /3^^■ At the moment when the distance and path bearing of a land- mark or aircraft are determined from a ground beacon , the indicator readings for the aircraft coordinates are recorded. After deter- mining the actual coordinates of the aircraft by means of a naviga- tional slide rule 5 they are compared with the coordinates on the indicator recorded at the moment of distance finding, and the errors in calculating the coordinates are found: act calc"^ AZ = Z ^ - Z ^ , act calc-" where '^^Q-t- and ^aQ-t are the coordinates of the aircraft on the basis of the measurement results and -^calc and Zl^alc s^r-e the coordinates of the aircraft according to the readings on the calculator. The corresponding corrections are then entered in the read- ings of the running orthodromic coordinates of the aircraft on the calculator . The characteristic feature of the solution of these problems is the lack of a need to fix the time of measurement of the air- craft coordinates and the introduction of corrections in the readings of the calculators when they change, as is necessary when using all other radi onavi gat i onal instruments. This feature is completely characteristic for Doppler systems. The relationship to time here is maintained only with selection of regimes of speed for reaching checkpoints at a given time. In measuring the aircraft coordinates and all other elements of air- craft navigation, the time need not be taken into account. Let us examine further the methods of getting rid of syste- matic errors in calculating the aircraft path and primarily the measurements of the aircraft course with the use of Doppler meters. For a precise determination of the errors in measuring the aircraft course, we need to determine the actual coordinates of the aircraft at at least two successive points, with the measure- ment base on the order of 200-300 km. 361 At the first point, the actual coordinates of the aircraft are determined and the readings of the calculator are corrected. At the second point, the actual coordinates of the aircraft are determined once again and the error in calculation is found, which has been accumulated during the flight time along the base from the first to the second measurement point. If we consider the error in the readings of the calculator at the first point to be equal to zero (since they have been cor- rected), the error in measuring the course is determined by the formula Af = arctg •^1,2 where AZ2 is the error in calculating the aircraft coordinates at the second point, and Xi 2 i^ "'^^e length of the measurement base between points 1 and 2. /31+5 This problem is easily solved on a navigational slide rule (Fig. 3.60). By using a Doppler meter, it is possible to find not only the errors in measuring the aircraft course, but also the nature of their accumulation with time. (t> . y ® a 2, v-z Fig. 3 .50 Fig. 3.61 Fig. 3.50. Determination of the Error in Measuring the Course on the NL-IOM. Fig. 3.61. Determination of Gyroscope Deviation on the Second Measurement Base. a © JS ISi ®"- b ® l(,S (sg'iff ) _3_ /70 22S Fig. 3.62. Use of the NL-IOM to Determine (a) Degree of Deviation of the Gyroscope and (b) Orthodromic Coordinates of the Aircraft from Ground Radio Beacons . 362 As we already know, the deviation of a gyroscope with time can be compensated by a suitable shift of the latitude on the compen- sator for the diurnal rotation of the Earth. If the deviation of the gyroscope is significant (2-3 deg/hr), it can be determined by changes in the errors in calculating the path on two adjacent bases, preferably of the same length (Fig. 3.61). With considerable deviations of the gyroscope axis, the path of the aircraft turns out to be curvilinear if the Z-coordinate recorded on the calculator is equal to zero, and the final error in measuring the course will be greater than this average error which appears on the first base at the initial value (Ayd) divided in half. Therefore, after introducing the corrections in the readings of the course instrument, an error remains in the measurement of the course which is equal to this value. If the measurement is repeated on the adjacent base, of approx- imately the same length, the error found in the measurement of the course will consist of two values: (a) The error in the initial setting, equal to ky^/2; (b) The average error due to the deviation of the gyroscope on the second base, also equal to hy^/2. Thus, the error which is found will constitute the magnitude /3H6 of the gyroscope deviation during the flight time along the second base . In order to determine the magnitude of gyroscope deviation per hour of flight, it is sufficient to divide the error which has been found into the flight time on the second base. Example. The flight time of an aircraft on the first and sec- ond bases is 20 min each. On the first base, an error in measuring the aircraft course was found and compensated for. However, on the second base the error in measuring the course turned out to be equal to 1°. Find the magnitude of gyroscope drift per hour of flight. Solution: Ay. 1° 0.33 hr 3 degrees/hr If the deviations of the gyroscope are small (0.5-1°), they cannot be found by measurement from a second base. However, in this case, it is not necessary to shift the latitudinal potentiometer in making compensation. It is sufficient to correct the readings of the course periodically (along with the correction of the air- craft coordinates) by the results of the measurements on one base. 363 When necessary, a Doppler meter can be used to determine the small-scale variations in the gyroscope (0.5-2 deg/hr). To do this, both the aircraft coordinates and error in measuring the course are determined on the first base. On subsequent bases, only the aircraft coordinates are deter- mined and corrected. On the last base, the error in the aircraft course is again determined. The error which is found will consti- tute the deviation of the gyroscope from the moment of the end of the first to the end of the last base. At the same time, the remaining error at the end of the first base is equal to t\y^/2, while the error found at the end of the last base, (e.g.) the fourth, is equal to: Ay, Ay + Ay-, + AYj + d2 d3 Ay. i.e. , if the last base is equal to the first, the error which will be found by measuring the course will be equal to the deviation of the gyroscope in the second, third, and fourth bases. Thus, the flight time will be sufficient for showing up even small-scale deviations of the gyroscope per hour of flight. Since the navigational use of Doppler meters does not pose any difficulties, but the detection of errors in calculating the path and measuring the aircraft course are much more difficult, we would like to conclude by providing several examples of how to determine these errors. 1. The last correction of aircraft coordinates was made at the point X = 156 km, where the error in the Z-coordinate was found to be zero . The flight then continued with maintainance of the Z- coord- inate on the computer equal to zero. At the point X = 330 km, accord- ing to the readings of the computer, the actual coordinates of the aircraft were determined on the basis of a radar landmark, having the coordinates X 1= 375 km, Z^ = 61 km. The polar coordinates /347 of the landmark were as follows: PEL = 54°, i? = 72 km. Find the errors in calculating the coordinates in measuring the aircraft course . Solution: following : by using a navigational slide rule, we find the R cos PEL = 42.5 km; H sin PEL = 58 km. Consequently, the actual coordinates of the aircraft are as follows: X = 375-42.5 = 332.5 km; Z = 61-58 = 3 km, while the errors in calculating the coordinates are: AX = +2.5° km, AZ = + 3 km. 364 The error in measuring the aircraft course is 3 A7 = arctg 332,5—156 = 0°58'. In this case, the aircraft deviated to the right from the given path, so that the readings of the course instrument were reduced an/ it was necessary to make a correction equal to +0°58' or approx- imately + 1° . 2. After correcting the coordinates in the aircraft course, considered in the first example, the aircraft traveled along a base equal to 180 km with a ground speed of 850 km/hr. A second check of the aircraft coordinates revealed that the error in measuring the course was +0°35'. The flight was made within latitudinal limits of 50-60°. Find the degree of deviation of the gyroscope per hour of flight and the required shift in the lati- tudinal compensator to get rid of it. Solution: The flight time of the aircraft along the base is equal to 13.5 min. The length of the second base is approximately equal to the first base, so that the deviation of the gyroscope along the second base is equal to the error found by measuring the cours e . The deviation of the gyroscope was found by means of a navi- gational slide rule (Fig. 3.62, a). Answer: the deviation of the gyroscope per hour of flight amounts to 154' or 2°3M-'. At latitudes of 50-60°, for each degree per hour of deviation in the gyroscope, it is necessary to shift the latitudinal compen- sator by 6°. In our example, the gyroscope deviated in the direc- tion of a reduction of the course indication so that the latitude on the compensator had to be set to the value: 6° x 2.6 = 15.6°. After the desired change in the setting of the latitudinal poten- tiometer is made, the deviation of the gyroscope should cease com- pletely . 3. For correcting the aircraft coordinates, a goniometric- rangefinding system is employed. The orthodromic coordinates of a ground radio beacon are: X„ = 187 km; Z„ = 142 km. The flight angle of an orthodrome segment, measured relative to the meridian of the point where the beacon is established, is equal to 64°. Find the orthodromic coordinates of the aircraft if its azimuth (A) is equal to 24° and R = 225 km. 365 1 . e Solution: ^-^^ = 320°; cos 320° = cos 40° = sin 50°; sin 320° = — sin 40°. By using a navigational slide rule, we find (Fig. 3.62, b), » R cos 320° = 170 km; /? sin 320° = — 145 K^. Consequently, the orthodromic coordinates of the aircraft are ^•=187 4- 170 = 357«*; Z= 142— 145 = — 3/«^. 5. PRINCIPLES OF COMBINING NAVIGATIONAL INSTRUMENTS /31+8 In Chapters Two and Three of the present work, we discussed the complexes of navigational instruments, which make it possible in one way or another to automate the processes of aircraft naviga- tion or measurement of individual navigational parameters. The first navigational complex is the course system. The basic principles of combining individual transmitters into a course system is the combination of the readings for purposes of automatic mutual correction (the MC , AC, GSC regimes), and also to combine the readings of individual instruments to improve the navigational values, constituting the sum of individual elements. For example: OBR = OC + CAR. The second complex is the navigational indicator NI-50B, in which there is a course transmitter, a transmitter of the airspeed, and a manually-set wind transmitter. The most complete of these complexes is the autonomous Doppler system of aircraft navigation, which works in conjunction with course transmitters and an automatic navigational device. Thus, the basic reasons for combining navigational instruments are the following: (a) Comparison of readings for purposes of mutual correction, (b) Combination of readings for purposes of automatic summation. Combination of individual transmitters into navigational sys- tems not only makes it possible to solve navigational problems auto- matically or semi- automati cally , but also makes it possible to realize their solution for automatic pilotage of an aircraft along the given trajectory. An example of such a realization is the automatic pilot- 366 age of an aircraft on the basis of signals from Doppler meters with automatic navigational Instruments. These complexes generally involve autonomous navig ational in- struments: transmitters for the course, airspeed. and drif t angle . The only exception is the aircraft radiocompass , w hose readings are combined with the reading s of course instruments t o obt ain hearings However, this is a result of a peculiar feature on the use of radio compasses (for obtaining the bearing it is necessary to add the course angle of the radio station to the aircraft course ) . The use of simple combinations of navigational systems sue h as ground ra- dars , radio distance-find ers , externally directed gonio metric and goniometric-rangefinding systems , fan-type beacons , and hyperbolic systems cannot be combine d satisfactorily. The first characteristic of combined navigational systems and aviational sextants is that they are intended only for determining discrete values of aircraft coordinates. Therefore, they can be used in navigational complexes as sources of information which dup- licate the results of automatic calculators of the aircraft path, i.e., only for purposes of correcting previously obtained navigational parameters . The second feature of these devices is that with a relatively high accuracy of coordinate measurement for the aircraft, they can- not be used to determine the first derivatives of these coordinates with time. Let us illustrate this with a concrete example. Let us say that some navigational instrument, taking its instru- mental errors into account (for electronic devices, considering the conditions for propagation of electromagnetic waves, and for astron- omical ones, the accelerations of the level of the aircraft) make it possible to determine the successive coordinates of an aircraft with an error which does not exceed 1 km, so that the error in measure- ment can change in value and sign. In this case, the error in determining the direction of the aircraft motion on the basis of two successive measurements may have a maximum value of AZ 2 With a measurement base of 30 km (approximately 2 min of flight in a jet aircraft), the angular error in the measurements can reach 367 _2_ = _!_«4°. 30 15 If the measurements are made more frequently, (e.g.) oftener than each minute of flight, the error in measuring the direction can reach 8° . With continuous measurement of the aircraft coordinates , the numerator in our example can retain its value , but the denominator will tend toward zero, i.e., the error in determining the direction of flight or (what amounts to the same thing) the first derivative of the Z-coordinate with time, will be equal to infinity. We can reach an analogous conclusion for the case of determin- / 350 ing the ground speed of an aircraft (the first derivative of the X- coordinate with time) by continuous measurement of it at a succession of points where the LA is measured. This example shows that communication and astronomical naviga- tional systems can only provide a rough pilotage of the aircraft along a given trajectory. With a very precise measurement of the aircraft coordinates along the route (with errors no greater than 200-300 m) and a very careful damping of the readings (averaging for time), automatic pilotage will take place with variations of the course within limits of 5-6°, i.e., 5-10 times greater than would be obtained by the results of measuring the drift angle by a Doppler meter. The only exception to this is the pilota-ge of an aircraft using strictly stabilized zones of landing beacons, where the errors in determining the deviations from a given trajectory are measured in several meters. Under these conditions, the pilotage of an air- craft can take place with variation of the course within limits of 1-2° with a very precise maintainance of the general direction of flight. etric-range- make the spherical conversions. It is somewhat simpler in this regard to use goniometric-r finding methods for short-range navigation and aircraft radars. Due to the limited radius of their operation, the polar coordin of these devices can be converted into orthodromic ones by solv simple equations for plane representations, using simple calcul ting devices with low accuracy. 368 Combined nav combined into nav 369 CHAPTER FOUR DEVICES AND METHODS FOR MAKING AN INSTRUMENT LANDING SYSTEMS FOR MAKING AN INSTRUMENT LANDING Landing an aircraft under conditions of limited ceiling and meteorological visibility in the layer of the atmosphere near the ground is the most complicated and difficult stage of the flight. Even under favorable meteorological conditions, a proper landing of the aircraft requires considerable attention and experience on the part of the crew. /351 @^^ Fig. 14.1. Setting the Aircraft Course for Lining Up with the Runway • Experience has shown that in order to land any kind of aircraft, it is neces- sary that it be located exactly on the landing path at a certain distance from the touchdown point (Fig. U . 1 ) . It is also necessary that the course followed by the aircraft be selected so that the vector of the ground speed is directed along the axis of the landing and take- off strip (LTS). However, it is not desirable to land an aircraft with a lead in the course being followed in order to compensate for the drift angle, since this causes considerable lateral stresses on the air- craft undercarriage when it begins to taxi along the runway. There- fore, the longitudinal axis must be lined up with the LTS immediately before landing, by making a flat turn without banking. Then (since the turn was flat) the aircraft will keep the desired direction of motion relative to the Earth's surface for a short period of time, showing lateral deviation relative to the air mass flowing over it. This shift gradually dies out, eventually turning into a drift angle relative to the new course of the aircraft. Therefore, the selection of the approach angle must be made several seconds (no more than 5 or 7) before landing the aircraft. It should be mentioned that the correct selection of an air- craft course while keeping it simultaneously on the given trajec- tory for landing poses considerable difficulties for the crew in preparing to land. /352 370 In cases when the aircraft is not lined up with the runway, it is necessary to carry out a maneuver which will bring it on to the axis of the LTS , and which involves a considerable loss of time and also a loss of distance along axis LTS (Fig. 4.2). fWTOTTI Let us say th along the LTS axis reached a point at deviation is equal at in a flight , the crew has which their lateral to Z. Fig. 4.2. S-Shaped Maneu- ver for Lining Up the Air- craft with the Runway . Obviously, in order to line up the aircraft with the LTS axis in the most economical fashion and without any remaining deviation rela- tive to the LTS axis, it is neces- sary to turn the aircraft toward the runway through a turn angle ( TA ) of a magnitude such that the lateral deviation of the aircraft from the LTS axis is reduced by a factor of two. Then the aircraft must be turned by the same amount in the opposite direction but through an angle such that the trajectory along which the aircraft is tra- veling when it emerges from the turn coincides with the LTS axis. In order to avoid loss of the selected direction of the ground speed vector while making the turns, i.e., shifting the aircraft after placing it on the landing course, the turns made by the air- craft must be coordinated as much as possible. It is obvious from Figure 4.2 that the magnitude of each of the two coordinated turns for bringing the aircraft on to the runway can be determined by the formula Z = 2(i?- cosTA) = 2i?( 1-cosTA) , wh ence cosTA Z 2i? where i? is the turning radius of the aircraft with a given bank- ing and airspeed during the turn. Obviously, while the aircraft is making the maneuver to land, it must travel through a path along axis LTS X 2i?sinTA. Example . When an aircraft is descending and is lined up with the runway on the desired course and with a horizontal ground speed of 280 km/hr, there is a lateral deviation from the LTS axis equal to 60 m. 371 Find the angles of the combined turns of the aircraft with /353 a given banking of 8° and the path of the aircraft along the descent path during the completion of the maneuver. Solution. The radius of the turns made by the aircraft are found by using a navigational slide rule (Fig. 4.3), which gives the answer 4-500 m. cosTA 60 4500 .9867 ; sinTA TA = 9°20 ' . 1625 ; X = 2 4500 . 1625 1463 m, ANSWER: R = 4500 m; TA = 9°20'; X - 1463 m, However, we must take into account the fact that the desired aircraft path in lining up with the runway must be chosen on the basis of the assumption that the turns are made with a constant banking angle, i.e., with a stable turn regime. At the same time, there is a delay in the maneuver produced by the reaction of the crew and mainly due to the inertia of the aircraft when entering and emerging from the turns . © (^ tsaa Fig. 4.3 280 Fig. 4.4 Fig. 4.3. Using the NL-IOM to Determine the Turning Radius of an Aircraft . Fig. 4.4. Landing Profile for a Jet Aircraft. As special tests have shown, the delay in the maneuver occurs primarily along the descent path and has practically no influence on the desired magnitude of the angles of the combined turns. This is explained by the fact that when the aircraft is entering and emerging from a bank at the beginning and end of the maneuver, the axis of the aircraft practically coincides with the axis of the LTS and the aircraft has practically no lateral velocity at these points . As far as the movement of the control surfaces when making the turns is concerned, the time required to move them is approx- imately two times less than the time required for the aircraft to enter and leave the turn, so that the lateral component of the air- 372 craft speed at turn angles up to 12° has a magnitude less than one- fifth of the longitudinal velocity. The delay time in the maneuver depends on the square of the horizontal velocity of the aircraft. At glide speeds of 280 km/hr, the delay time is equal to 4.5 sec of flight time on the average, or 350 m of the aircraft's flight along the LTS axis. This means that in our example, the required travel of the aircraft in lining up with the runway is equal to approximately 1800 m. At the same time that the course is b.eing selected which must be followed in order to make the landing, the crew must begin some distance away from the landing point to set up the desired descent /35M- trajectory in the vertical plane (Fig. H.4). In Figure 4 . M- , Point A is the point of transition from hor- izontal flight along the landing path to the descent regime of the aircraft . Point B is the point where the landing distance begins, which is also called the critical point for safe transition to making another pass. After this point has been passed, a second attempt at landing cannot be made, so that the aircraft must make a final selection of the aircraft course before this point is reached and the deviation of the aircraft from the given trajectory (upward and downward) must not exceed certain limits. Before this point is reached, a decision must be made either to make the landing or circle around the airport once again. After the starting point for the landing distance has been passed, the crew carefully observes the altitude. To do this, a leveling point C is selected along the approach to the airport (this is a conditional designation for the point where the descent tra- jectory of the aircraft crosses the Earth's surface), toward which the further descent of the aircraft is aimed. With proper descent and a constant pitch angle of the aircraft, this point is projected at a constant level on the cockpit window. If the approach is being made too rapidly, this point shifts upward on the glass, and if the aircraft is coming in too slowly it moves downward . Before reaching Point C (at an altitude of 8-15 m, depending on the type of aircraft) the aircraft levels off and then lands at Point D. The descent trajectory of the aircraft in the vertical plane is called the glide path. The aircraft is kept on a fixed glide path by selecting the proper angle of pitch for the aircraft and the correct amount of power to the engines. This process is much simpler in principle than the selection of the course to be fol- lowed by the aircraft, since it does not require maneuvering but 373 only the proper setting of the pitch angle and the levers which control the motors. However, it complicates landing as a whole because both processes must be carried out simultaneously while a given horizontal airspeed is being maintained. Unlike all other navigational devices, the systems used in making an instrument landing are intended specially for keeping the aircraft on a given descent trajectory before landing in the horizontal and vertical planes. The proper operation of these devices and the maneuverabil- ity of the aircraft determine the minimum permissible distance from the LTS at which the aircraft can be piloted by instruments or by instructions from the ground, with correction of any errors that may occur after changeover to visual flight. The more precisely the desired trajectory is maintained by instruments, the closer the transition to visual flight will lie to the landing point and the lower the altitude at that point. The limits within which an aircraft can be piloted by instru- /355 ments without the airport being visible and with no terrestrial landmarks in sight which could show approaches to the airport is called the weather minimum for landing the aircraft. At the present time, there are three principal types of sys- tems for making instrument landings: (a) A simplified landing system which involves lining up the aircraft with radio stations. (b) A course-glide landing system. (c) A radar landing sytem. A necessary complement to each of these systems is the sys- tem of landing lights at the airport. Simplified System for Making an Instrument Landing The complex of devices in the simplified system for making an instrument landing on the basis of information from two master radio stations includes the following: (1) Two master radio beacons, located on the LTS axis, whose standard designation is the short-range master station (SRMS), located 1000 m from the end of the LTS, and the long-range master station (LRMS), located 4000 ra from the end of the LTS. (2) Two USW marker beacons with a narrow vertical propaga- tion characteristic for electromagnetic waves, located on the same sites as the LRMS and SRMS. 371+ (3) The lighting of the approaches to the LTS and its out- line (4) The complex of aircraft radio navigational and pilotage- navigational equipment as a whole . This includes : (a) One or two radio compasses, (b) A marker receiver, (c) Course control of the aircraft, (d) A barometric altimeter, (e) A radio altimeter for low altitudes, (f) An airspeed indicator, (g) A gyrohorizon, (h) A vertical speed indicator (variometer). a very limited application j.wx j^^j-j^^^^^ ^^ its use is very simple from the standpoint of which must be taken into account. errors the methodological /356 In particular, we shall acquaint ourselves with the operating principles of the following pieces of equipment: marker devices, radio altimeters for low altitudes, the gyrohorizon and variometer. Marker Devices In order to make a landing with the simp very important to know (admittedly, at separa tance remaining until the end of the runway. landing with the simplified system, it is te points) the dis- As we know, aircraft radio compasses do not permit a precise nt when an aircraft flies over the control e to the special characteristics of the determination of the moment when an aircraft flies radio station; this is due to the special character __ ^_^„ „_ ^.._ operation of the open antenna aboard the aircraft. To solve this problem, marker beacons and aircraft marker receivers have been Marker radio beacons are transmitters with a directional trans- mission characteristic vertically upward, sometimes with a slight deviation toward the LTS so that the limit of the directional char- acteristic of the radiation is located to one side of the LTS and as close as possible to the vertical. In this case, an aircraft which is flying over the beacon towards the LTS will receive the signals from the marker transmitter at the moment when it is exactly above the beacon. For purposes of recognition, the transmission from the marker 375 beacon is not continuous but in the form of frequent short pulses (SRMS) or longer, less frequent signals (LRMS). These signals are heard aboard the aircraft for a period of 3-6 sec after it has flown over the vertical limit of the radiation characteristic and before it crosses the second, deflected limit of the characteristic. A still simpler device is the aircraft marker receiver. It is set to one frequency which is the same for all beacons . There- fore , it is very simple in design, has small dimensions, and requires no attention for use except to be switched on and off. When used in a complex together with course-glide devices, the marker receiver is turned on by a switch which is combined with the course-glide equipment, so that the crew does not have to inter- fere in its operation at all. In many cases, the marker receiver is combined with the switch for the radio compasses, the purpose being to ensure a low consumption of electrical energy, and allow stability and high reliability in the operation of this receiver. The marker receiver is connected to a light signal (a red light on the instrument panel in the cockpit marked "marker") and to a device which gives a simultaneous sound signal by means of a bell. Thus, when the aircraft flies over the marker, the lamp flashes and a series of short rings is heard. Low-Altitude Radio Altimeters /357 At the present time, low-altitude radio altimeters based on the principle of frequency modulation are the ones most widely em- ployed . 4. 5 A schematic diagram of such a radio altimeter is shown in Figure oscillation 'counter. indicator TTT^frmisT^f^f^^T^sx Fig. 4.5. Diagram of Low-Altitude Radioaltimeter , 376 The radio altimeter transmitter has a modulating device which produces a saw-tooth wave. For this purpose, we can use (e.g.) a variable membrane capacitor with mechanical oscillation of the membrane . The frequency of the signals reflected from the ground and picked up by the receiving antenna has the same saw-tooth character- istic, but is shifted in time by a value t , required for the electromagnetic waves to travel from the transmitting antenna to the ground and back again to the receiving antenna (Fig. 4.6). Fig. 4- . 6 . Frequency Char- acteristic of Radioaltimeter , It is clear from the figure that the frequency difference be- tween the emitted and received waves at any moment in time (with the exception of the segments between the extreme values of the frequency characteristic) will be strictly linear with respect to the flight altitude. For a complete retention of the linearity, these segments can be cut out by cutting off the receiving section with a n-shaped voltage at the end points of the emitted frequency. The emitted and received frequencies are combined in the bal- ancing detector, where a low frequency is formed which is propor- tional to the flight altitude. Following amplification, the low frequency is converted to /35 8 rectangular oscillations which are calibrated both in terms of ampli- tude and duration. Thus, the counting circuit will receive pulses which are of uniform magnitude, and whose number per unit time will depend on the flight altitude. The number of calibrated pulses is summed and fed in the form of a direct current to the indicator, whose pointer shows the alti- tude in meters. In the simplified landing system, the radio altimeter plays only an auxiliary role as an indicator of a dangerous approach to the ground, since its readings depend upon the nature of the relief and cannot be used for checking the rate of descent. To set up the descent trajectory of the aircraft, barometric altimeters are us ed . In more complete landing systems , the radio altimeter can be used to give a trajectory value as well, but only in the last stage of descent before landing above a given final area of safety adjoin- ing the LTS. Since those landing systems which ensure descent of the air- craft by instruments until the point where the landing distance 377 begins use the radio altimeter only to signal a dangerous approach to the ground, we can exclude them for convenience from the group of basic pilotage instruments located in the center of the field of vision of the pilot, and use audible signals. If an aircraft is making a descent and reaches the limit of permissible altitude above the ground, the audible signal warns the crew of the neces- sity to terminate descent. Gyrohortzon The artificial indicator of the position of the horizon rela- tive to the axis of the aircraft ( gyrohori zon ) is a common pilot- age instrument, intended for piloting the aircraft when the true horizon is not visible. However, it is very important in guiding the aircraft along a landing trajectory, where it is used for main- taining a desired landing trajectory. In principle, the design of the gyrohorizon is simpler than that of the gyrosemicompass , e.g., unlike the latter, the gyrohor- izon has a vertical axis of rotation for the gyroscope, and a grav- itational correction device suspended from the bottom of the gyro assembly. This serves to keep the gyroscope axis constantly vertical in the aircraft. The external frame of the gyrohorizon is located horizontally, while its axis of rotation coincides with the longitudinal axis of the aircraft. Therefore, we can immediately determine the exist- ence and magnitude of a lateral rolling of the aircraft by the posi- tion of the external frame relative to the axis of the aircraft. For this purpose, a silhouette of the aircraft has been pasted on the glass which covers the dial, and a horizontal strip which moves up and down imitates the position of the visible horizon. Figure 4.7 shows the schematic diagram of the gyrohorizon. Fig. 4- . 7 . Diagrams of Gyrohorizon: (a) Kinematics; (b) Indicator, 378 Gyro assembly 1, with a vertical axis of th and the gravitational correction device 2 mounte are suspended in the horizontal external frame 3 bly bearings 4. The carrier for the horizon lin to the casing of the gyro assembly and displaced relative to the horizontal axis of the gyro asse direction of the aircraft's flight from the forw instrument). The axis of the line is fastened a external frame, also along the flight direction Therefore, when reducing the angle of pitch of t strip of the gyrohorizon 7 moves upward, remaini horizontal axis of the gyro assembly. When the the horizon line moves downward as the true hori e gyroscope rotor d at the bottom, on the gyro assem- e 5 is fastened somewhat forward mbly (along the ard part of the t the front to the of the aircraft, he aircraft, the ng parallel to the pitch angle increases, zon does . During lateral rolling of the aircraft, the casing of the gyro- horizon (along with the silhouette of the aircraft) rotates rela- tive to the bearings of the external frame 9 in the direction in which the aircraft is rolling, which provides an indication of the rolling of the aircraft relative to the horizontal strip. For esti- mating and maintaining given longitudinal and lateral rolling of the aircraft, a scale is located between the outer frame and the horizon line and shows scale divisions for estimating the magnitude of the rolling in degrees. The gyrohorizon, fitted with the kinematic system described above, can be used within limited degrees of longitudinal and trans- verse rolling of the aircraft. Obviously, the rear bearing of the outer frame, i.e., the one located between the outer frame and the scale, must be mounted on a support in the unit. This support acts/360 as a pivot for the lever supporting the horizon line, e.g., when the aircraft rolls over on one wing. In the case of considerable changes in the pitching angle of the aircraft (e.g., in a Nestrov loop), a support will hold the lever for the strip in a notch on the outer frame. The projection of one of these supports limits the degree of freedom of the gyroscope, thus leading to a "dis-location" of its indications, and a very long period of time is required to readjust them by gravitational correction. To ensure "nondis location" of the operation of the gyrohor- izon, the gyroscopic section protrudes outside the housing of the instrument, i.e., constitutes a separate gyroscopic instrument, a gyrocompass without a limited degree of freedom. The readings of the gyrovertical are transmitted to the horizon indicator by means of master and slave selsyns. We should also note that gyrohorizons or gyroverticals are transmitters which indicate longitudinal and lateral rolling for the operation of autopilots, acting as transmitters of turn angles of the aircraft in the horizontal plane, in which gyroscopic semi- compasses are used. 379 Yaviometev A vavlometev is a device which measures the rate of vertical descent or climb of an aircraft. The operating principle of a variometer is based on the decel- eration of a current of air which equalizes the pressure inside the body of the unit with the external static pressure. This means that when vertical movement occurs, a pressure drop develops within the body of the unit and in the static tube (Fig. 4.8). capillary b: cxxr • pcco Fig. 4.i iome ter . from static pressure sensor Diagram of Var- The pressure from the static pressure intake passes directly into the manometric chamber of the instru- ment. Within the body of the instru- ment, this pressure passes through a capillary opening, i.e., with retar- dation. Therefore, when the aircraft gains altitude, the pressure in the unit will be somewhat higher (when the aircraft descends, somewhat lower) than inside the manometric chamber. This pressure drop is proportional to the vertical speed of the air- craft . To measure this drop, the variometer is fitted with a transmit- ter mechanism, similar in principle to the mechanism of the altim- eter or speed indicator. The indicator scale is graduated directly in terms of vertical speed, as expressed meters/sec. Angle of Slope for Aircraft Glide /361 The proper selection of an angle of slope for gliding is very important for all instrument landing systems, and especially for the simplified systems guided by master radio stations, both from the standpoint of making a safe landing and the meteorological minimum at which a landing can be made. When making an approach to land, it is very important that the flight altitude (^) correspond to the remaining distance (iS) to the point where the aircraft touches down: E = S tge rem where 9 is the glide angle. A simplified system of instrument landing makes it possible to determine the remaining distance to the landing point only when passing over the LRMS and SRMS. The point at which the aircraft begins to descend from the 380 altitude established for circling above the field is determined by calculating the time, and is therefore insufficiently exact. A descent between the LRMS and SRMS is also made by calculating the path of the aircraft with time, but this calculation takes only a short period of time and is performed after a certain point has been passed; it is therefore more accurate. According to the standards adopted in the USSR, the flight altitude for circling over an airport (for aircraft with gas tur- bine engines) has been set at 400 m; for piston-engine aircraft, it is 300 m. In both cases, however, the true flight altitude above the local terrain surrounding the airport must be no less than 200 m. This altitude reserve is retained even when coming straight in for a landing, until the beginning of descent in the designated gli de pattern . From the moment when descent begins in a gli aircraft passes over a certain marker (LRMS), the ab ove the relief is ] kept at a minumum of 150 m. th e LRMS, and before reaching the SRMS, the heigh ab ove the terrain is reduced from 150 to 50 m. D however , it is necessary to keep in mind the fact a possibl e premature loss of altitude , in case of St rong head wind. ¥• or this reason, it is conside fl ight altitude between the LRMS and the SRMS (fl ab ove the SRMS) must be at least 50 meters above in the vicinity, beg. inning at half the di stance b an d SRMS and extending to the point where the SRM de , and until the altitude reserve After flying over t of the aircraft uring this maneuver, that there may be an unexpected red that the minimum ight altitude the heighest point etween the LRMS S is locate d . These same altitude reserves are maintained even when using more complete landing systems, although in this case the given glide path for the aircraft is defined in space and the probability of a premature descent is sharply reduced. In this case, however, the basic method for checking the proper descent is the measurement of the barometric altitude when flying over the marker points, thus guaranteeing safety of flight in case the landing instruments aboard the aircraft or on the ground should malfunction. /362 In cases when the approaches to an airport are free of ob- structions, the angle of slope in the glide path is set equal to 2°40'. The flight altitude relative to the level of the airport in this case is set at 200 m above the LRMS and 60 m above the SRMS. Typical Maneuvers Landing an Aircraft Simplified systems for bringing an aircraft in for a landing are used at airports with a low traffic density, where the installa- tion of complex landing systems would not be justified. Conse- quently, it is difficult to know in advance whether these airports will have provision for radar control, to set up the approach and landing pattern on command from the ground. Hence, the approach for landing is made with the same devices which are used in landing 381 the aircraft along a straight line. For this reason, a successful accomplishment of the maneuver under these conditions will be assured if the starting point for the maneuver is one of the marker points of the system. 70sec.^""''>O M-1200 H = J900-t?00 Usually a LRMS is use this purpose, since at the ity of airports , it is the control facility at the ai There are then three possi ways to bring the aircraft the starting point for the neuver : (1) An approach of th craft to the LRMS, with a angle close to the landing (2) An approach to th with a path angle nearly p dicular to the landing cou (3) An approach of the aircraft to the LRMS, with a path H'2800 Fig. 4.9. Large and Small Rectangular Landing Patterns d for ma] or- mam rport . ble to ma- e air- path course e LRMS, erpen- rse . angle nearly the reverse of the landing course. In directing the aircraft toward the LRMS at path angle close to the landing course, the approach for landing can be made along a more or less straight- line course (Fig. 4.9). A large rectangular route is covered in this case, if the aircraft approaches the airport at a great altitude (for aircraft with gas turbine engines, this is 3900 to 4200 m), and an additional length of time is required for the aircraft to descend before land- ing. In this case, in making the approach to the LRMS, the aircraft makes a turn to the path angle for landing (in the following, the path angles will be referred to as magnetic), at which the aircraft descends to 2800 m (relative to the pressure at the level of the airport where it is landing). At an altitude of 2800 m, the double turn begins (first and second turns without a straight line between them) at 180° with a descent to 1200 m. Flight then continues with a magnetic path angle (MPA) opposite to the landing angle, with descent to the altitude set for circling over the airport. In aircraft with gas turbine engines , limits have been set for the horizontal airspeed with the undercarriage lowered. There- fore, in a flight with a MPA opposite to the landing angle, the flight altitude for circling the field is maintained for 5 to 6 km until the LRMS is passed, so that at the moment when it actually is passed, the speed of the aircraft in horizontal flight can be cut to the speed established for lowering the undercarriage. /363 382 After passing over the traverse of the LRMS, the flight continues opposite to the landing direction for 70 sec, prior to starting the third turn (usually at a flight altitude of 400 m , up to CAR = 120° to the right and up to CAR = 240° on the left straight-line paths). The undercarriage is lowered in this path segment . After a period of 70 sec flying time from the moment when the traverse of the LRMS is passed or until CAR-120 (240°) is reached, the third turn is made. Since the horizontal airspeed in the vicinity of the third turn is much less than in the vicinity of the doubling of the first and second turns, the radii of the third and fourth turns (with a banking angle of 15 to 17°) are then much less than the radius of the double turn. Therefore, between the third and fourth turns there is a period of straight-line flight which lasts 50 to 55 sec. This straight -line segment is used for preliminary lowering of the wing flaps before landing, and also acts as a "buffer", which compensates for errors in aircraft nav- igation in cases when the effect of a side wind in making the ma- neuver from the starting point until the end of the third turn has not been estimated sufficiently precisely. In these cases, the "buffer" line can be extended or shortened somewhat, but the last (fourth) turn must be always made on time. At airports where the nature of the local terrain or complex wind conditions render flight along a straight line at 400 m im- possible (for aircraft with gas turbine engines), but the estab- lished flight altitude is 600 or 900 m, the duration of the flight from the traverse of the LRMS to the beginning of the third turm is increased, so that after the aircraft emerges from the fourth /364 turn it is located below the glide path established for a given approach direction and has a segment of horizontal flight to the end of the glide path which is only 2D to 30 sec long. This time is needed to prepare the crew for landing and for extending the flaps fully . For example, if the flight altitude along a straight -line course is set at 500 m, and the slope angle of the glide path is 2°4' , the fourth turn must be executed no closer than 15 km from the end of the LTS, since the aircraft (at an altitude of 600 m) enters the glide path at a distance of 13 km from the end of the LTS, and 2 km are required for the horizontal flight segment before entering the glide path. Consequently, the start of the third turn under calm condi- tions, after passing the traverse of the LRMS, lasts 2 minutes and 30 seconds of flying time (at Y - 350 km/hr), with CAR approx- imately equal to 135° (225°). If the flight altitude along the straight-line path is set at 900 m, the flying time from the traverse of the LRMS to the beginning of the fourth turn is increased to 3 minutes and 30 seconds, so that it is advisable to increase 383 the glide path up to 4° for the purpose of shortening the time in- volved in making the descent. In cases when an aircraft is approaching an airport with a path angle close to the landing angle, at an altitude of 1500 m or less, the double first and second turns are made immediately after pas- sing the LRMS . The descent to circling altitude and reduction of speed to lower the undercarriage in this case are performed in the designated turn. Hence, the large rectangular flight pattern in converted to a small one, and the maneuvering time is shortened to about 4 . 5 min . If the aircraft approaches the airport at the altitude estab- lished for circling the field, the radii of all four turns are made approximately the same, so that in order to create the "buffer" line between the third and fourth turns, the first and second turns of the aircraft are executed in succession with a time interval between the end of the first and the beginning of the second turn which equals ^■0 sec. The fourth turn on the large and small rectangular patterns is made along the course angle. In aircraft with gas turbine engines , the CAR at the beginning of the fourth turn (when turning to the right) must be equal to 70° (and -290° when turning to the left). The landing approach for aircraft with piston engines' is made according to the small rectangular pattern, with different first and second turns, and the same time parameters between the first and second turns (40 sec), from the traverse of the LRMS to the beginning of the third turn (70 sec) (at a flight altitude of 300 m). The fourth turn for these aircraft begins at CAR = 75 or 285°. /365 However, due to the lower airspeed along the straight-line segments and the smaller turning radii, the linear dimensions of the maneuver for aircraft with piston engines are much less than for aircraft with gas turbine engines. In addition, due to the shorter time for each turn, the total time for executing the ma- neuver for aircraft with piston engines is shorter (for example) by 1 minute . Jfflsec, Fig. 4-. 10. Landing Maneuver When Approaching the LTS Axis at a 90° Angle. 384 When an aircraft is approaching an airport at an MPA which is perpendicular to the landing angle, the landing altitude for air- craft with gas turbine engines is usually set at 1200 m above the level of the airport (Fig. 4- . 10 ) . After passing the LRMS, the air- craft continues on a course which lasts for ^■0 sec until descent, and the second turn is also executed with loss of altitude. After completing the second turn, the flight lasts 30 sec until the beginning of the third turn, when the undercarriage is lowered. A similar maneuver is executed by piston-engine aircraft, with the sole difference that the flying time from the LRMS to the beginning of the second turn is set at no less than 1 min , since the airspeed of these aircraft in all stages of the landing approach until emergence from the fourth turn is roughly the same. If an aircraft approaches an airport with an MPA which is close to the reverse of the landing angle, the crew of a gas turbine aircraft travels along a small rectangular pattern with different sides for the first and second turns (Fig. M-.ll,a). In the case of aircraft with piston engines, the so-called standard turn is executed in this instance (Fig. 4.11,b) on the landing course. These maneuvers agree in terms of the magnitude and direction of the turns; in the former, however, there is a s t raight -line segment between the second and third turns for lowering the under- carriage, while there is a "buffer" line between the third and fourth turns. In addition, if the aircraft approaches the airport at the flight altitude for circling the field, and all the turns of the aircraft are made without loss of altitude (the radii of all the turns being the same), then between the first and second turns there will also be a period of flight along a straight line for a period of 40 sec. In the case of a standard turn, all four turns will be made in / 366 succession without there being any straight- line segments between turns . -J'i5',sec. CAi? O fT!X Fig. 4.11. Landing Maneuver with a Course Opposite to the Landing Course, (a) Along a Straight-Line Path; (b) Standard Turn. Analogous maneuvers for approaching to make a landing can also 385 be made by using more complete landing systems. However, air- ports that have such systems, as a rule, are also equipped with radar devices to monitor the aircraft maneuvering in the vicinity of the airport. Therefore, the beginning of the landing maneuver need not necessarily be made at the marker point on the LTS axis, thus making it possible to come in for a landing along the shortest path from any direction. A small or large rectangular pattern is usually used as the basis for setting up a landing approach along the shortest path. However, it is not generally completed, usually beginning at the point of tangency of the entrance into the maneuver to one of its turns . Calculation of Landing Approacli Parameters for a Simplified System In the preceding section, we discussed the typical maneuvers for landing an aircraft when approaching the airport from any di- rection. The execution of these maneuvers does not pose great difficulty for the crew of the aircraft, since the flight is made with a sufficient altitude reserve and sufficient speed, while the demands on the accuracy of making the maneuver are not very high. The main difficulty lies in flying along a given descent tra- jectory in the glide path, due to the very high demands on the maintenance of flight direction, altitude, and horizontal glide speed, depending on the remaining distance to the touchdown point. In the case of aircraft with gas turbine engines , there is the additional need to reduce the airspeed gradually as the airport is approached. In order to facilitate the task of descending along a given /367 trajectory to a certain degree, as well as to avoid serious errors in flight along the landing path, some preliminary calculations are made, of which the following is the most important. If the landing approach is made in a dead calm, the geometric dimensions of the maneuver (and consequently, the point where the descent begins along the landing path) are determined by simple relationships between the airspeed, time, turn radii, flight altitude in circling the field, and established steepness of the glide path . The calculated data for making a landing in a calm are usually plotted on special landing patterns, devised for each airport. Under actual conditions, however, it is necessary to take into ac- count the head wind and side wind components (for the landing course), which can have a very great effect on the making of a landing . 386 Calculation of Corrections for the Time for Beginning the Third Turn In preparing to land, especially with the aid of a simplified system, it is necessary to ensure that the aircraft emerges from the fourth turn onto the landing approach always at the same distance from the LTS. Obviously, in order to solve this problem, it is necessary to consider only the head-wind component for the landing course . In making an approach to land along a rectangular pattern, the last reliable point for determining the J-coordinate of the air- craft (the distance along the axis of the direction of the airport) is the traverse of the LRUS, while in a standard turn it is the passage over the LRMS with an MPA opposite to the landing angle. If we do not take the wind into account when coming in for a landing, the aircraft will enter the landing path at a distance from the LTS which exceeds the distance for calm conditions by the value hX = u t, where t is the flying time from the traverse of the LRMS to the emergence from the fourth turn, or from the moment when the air- craft passes over the LRMS until it emerges from the standard turn. Example : The flying time from the traverse of the LRMS to the emergence from the fourth turn in a calm is 4 min , divided into these stages : Traverse of the LRMS to beginning of third turn... 70 sec Third turn 50 sec Buffer line 50 sec Fourth turn 60 sec The speed of the head-wind component on the landing course is /36 i Ux = 15 m/sec. Find the value of AJ for emergence from the fourth turn. Solution: kX = 240 X 15 = 360C m. In order for the distance for emergence from the fourth turn to remain the same as in a calm, it is necessary to shorten the flying time from the traverse of the LRMS to the beginning of the third turn by a value At = 3600 V+u X 387 In our example, for an airspeed of 400 km/hr (110 m/sec) and a course opposite to the landing course. A/ = 3600 110+15 3600 125 = 29 sec Thus, a flight from the traverse of the LRMS to the start of the third turn would last 4-1 sec instead of 70 sec. By combining the formulas for obtaining the values for LX and Ai, we finally obtain the formula for determing the value ht : M = - tu. V + u, For our example. 240-15 ' „„ The problem for a standard turn is solved in the same way. In this case, the time t is the time from the passage over the LRMS in a course opposite to the landing course, to the end of the standard turn; the value of At is calculated from that for a calm in flying from the LRMS to the start of the standard turn. Calculation of the Correction for the Time of Starting the Fourth Turn The beginning of the fourth turn in coming in for a landing is usually determined from the course angle of the LRMS. For example, when executing a maneuver to the right: cigCAR=^.. where R is the radius of the turn made by the aircraft, and X is the distance along the LTS axis from the LRMS to the starting point of the fourth turn. Under the influence of a side wind, the fourth turn is begun earlier if the lateral component of the wind on the landing course is favorable between the third and fourth turns, later if this component is unfavorable. Obviously, if we take the wind into account: R-\-t-u^ ctgCAR= X /369 where t is the time of the fourth turn and Ug is the lateral com- ponent of the wind speed. For example, with a turning radius of 4500 m and X = 12.5 km. 388 r-AD 4500 etc C AF'= — ^ 12500 CAR =70°, If the lateral component of the wind appears on the "buffer" line and is favorable, with a speed of 10 m/sec, then for a turnin; time of 60 sec we will have: ^.„ 4500 + 60-10 ctg C AR= ^ 12500 CAR =68°, i.e., the turn must begin 2° earlier than under calm conditions. Cataulati-on of the Moment for Beginning Descent Along the Landing Course Under calm conditions, the distance at which the aircraft emerges from the fourth turn is determined by the flying time from the LRMS to the start of the third turn. For example, with 7 = 111 m/sec (M-00 km/hr), this distance will be : ^-70-111 wSk^ toLRMS The fourth turn will be completed at approximately this dis- tance if a correction for the effect of the wind is made in the time for starting the third turn. Consequently, with a standard location of the LRMS, the distance from the point where the air- craft comes out of the fourth turn to the touchdown point is 12 km. The distance for beginning the descent along the glide path is determined by the formula ^^ = //c,ctg6, where X is the distance for beginning the descent and E is the flight altitude for circling the field. ^ For example, at a circling altitude of B - 400 m and a slope angle for the glide path 9 - 2°40': X , = 400-clg2°40' = 8500 J<. a Thus , after coming out of the fourth turn at a distance of 12 /370 km from the LTS , the aircraft must follow the landing path without losing altitude for a period of time 389 Xr -Xd where tjj is the time of horizontal flight along the landing path. For example, if the horizontal airspeed after coming out of a turn is 360 km/hr (100 m/sec), and the head wind'is moving at 15 m/sec, the time for horizontal flight in our case will be 12000 — 8500 3500 'h = 100-15 =-ir='*' "^^ Under calm conditions, the time for horizontal flight in this case will be : 3500 '^ =1oo' = =^^'^^^ Practically speaking, the descent of the aircraft must begin 5 to 6 seconds before this time has actually elapsed, since a certain period of time is required to guide the aircraft into its landing regime. Calculation of the Vevtioal Rate of Descent Along the Glide Path The vertical rate of descent of an aircraft along the glide path is determined by the simple formula Ky=W^tgO = (K^_a^)tge. For example, with a mean horizontal rate of descent of 290 km/hr (80 m/sec), a head wind of 15 m/sec, and a slope angle in the glide path of 2°40': V-y = 65.tg2°40' = 3^/ sec The calculation of the vertical rate of descent is of partic- ular interest for piston-engine aircraft, whose horizontal glide is about 50 m/sec. Since the head wind can be as h5gh as 25 m/sec on landing, the vertical glide speed for these aircraft can change by a factor of 2, i.e., from 2.3 to 1.15 m/sec. In the case of aircraft with gas turbine engines, the ratio of the maximum rate of descent to the minimum rate, with the same steepness of glide, is 1.5. 390 Determination of the Lead Angle for the /371 Landing Path ~ A knowledge of the approximate value of the drift angle, and consequently the necessary lead angle for the landing path of an aircraft, considerably facilitates the choice of the course to be followed along a given descent trajectory. The value of the drift angle along the landing path can be determined by the approximate formula tg US= TT-^^^ . In flight along a given descent trajectory, however, the hor- izontal airspeed, altitude, and wind are variables, so that it is sufficient to use the following rule in finding the drift angle: (a) For aircraft with gas turbine engines, at glide speeds of 270-290 kra/hr, the lead angle is considered to be equal to 0.7° for each 1 m/sec of side wind. (b) For aircraft with piston engines, (glide speeds of 180-200 km/hr), the lead angle is considered to be 1° for each 1 m/sec of s i de wind . For example, with a side wind along the landing path of 8 m/sec, coming from the right, the lead angle will be: - for aircraft with gas turbine engines, 5.5° to the left; - for aircraft with piston engines, 8° to the left. The calculations given above for the time of starting the third turn, the course angle for beginning the fourth turn, the time for beginning the descent, the vertical rate of descent, and the lead angle for the landing path, must all be made by the crew of the aircraft before approaching the airport on the basis of landing- conditi on information. All calculations must be complete before the landing maneuver begins. Landing the Aircraft on the Runway and Flight along a Given Trajectory with a Simplified Landing System While making preparations for landing, the crew must prepare the course to be followed by the aircraft along all the straight- line segments of the approach pattern, with the exception of the line between the third and fourth turns, beginning with a calculation of the drift angle. The radiocompass must be set by the LRMS ; if there are two sets of radiocompasses , the second must be set by the SRMS. Along the line between the third and fourth turns , the course to be followed is always equal to the MPA of the "buffer" segment. 391 so that the start of the fourth turn will be determined by the CAR. The slight drift of the aircraft which occurs at this time, as we have seen, is compensated by redefining the time for starting the third turn. /372 When the course angle of the LRMS becomes equal to the cal- culated value, the fourth turn is executed with a banking angle of 15° before acquiring the calculated landing path. If all the calculated data are correct, the aircraft will come out of the turn precisely on the landing path with the desired course. At the moment when the aircraft emerges from the fourth turn, the timer is switched on to determine the time for beginning descent in the glide path. In the majority of cases, however, due to errors in the oper- ation of the radiocompass , improper maintenance of the course and air speed of the aircraft, errors in determining the side-wind com- ponent, and failure to bank at the proper angle when turning, the acquisition of the glide path by the aircraft is not accurate. The accuracy with which the aircraft acquires the landing path is determined by a comparison of the magnetic bearing of the LRMS with the MPA for landing. IF MC + CAR = MPA , but the com- bined reading of the radiocompass is MBR = MPA, , the aircraft will be exactly on the axis of the LTS. 'l' If MBR is greater than MPA-, , the aircraft will be to the left of the given landing path. With MBR smaller than MPA^, the aircraft will be to the right of the given landing path. The difference between MPAj^ and MBR is called the acquisition error a. Example: MPA^ = 68°, with a calculated drift angle of +3°; the aircraft emerged from the fourth turn with MC = 65°, the course angle for the turn over the LRMS was 358°; find the acquisition error . Sol ution a =68 — (65 + 358) = 5°, i.e., the acquisition error is 5° to the right. For lining up the aircraft with the landing path, the course followed by the aircraft is usually changed by doubling the acquisi- tion error. In our example, the course to be followed must be re- duced 10°, so that the CAR of the LRMS becomes 8°; the flight is continued at this course until the value of the course angle in- creases to the magnitude of the acquisition error, i.e., becomes 13°. 392 When the pointer of the radiocompass is on the 13° mark (on a combined indicator, a bearing of 68°), with a slight lead (no more than 1 to 2°), the aircraft makes another turn to the calculated landing path, and the CAR of the LRMS becomes equal to the calcu- lated drift angle of the aircraft (3° in the example). the As the aircraft continues to follow the landing path on tl" calculated course, the CAR will remain equal to the calculated drift angle if the course of the aircraft has been properly selected. If the CAR is increased, the aircraft will drift to the left of the LTS axis, and the path being followed will have to be increased for acquisition of the desired line of flight, and decreased later on, although it will remain somewhat greater than the calculated value (the CAR is then less than the calculated drift angle). If /373 the CAR is then to remain constant, the course to be followed must be selected properly. Similar operations in selecting a course are carried out when the aircraft deviates to the right of the desired line of flight. These operations will have the form of a mirror image of the oper- ations described above, i.e. , when the CAR is reduced, it is also necessary to reduce the course to be followed in acquiring the de- sired line of flight, then increase it somewhat, but still keep it below the calculated value. In the case when the course angle of the LRMS continues to change, after the first operation to correct the course by acquir- ing the line of the given course, the operations are repeated using the familiar method of half corrections. Thus, the readings of the radi ocompasses , beginning with the LRMS and then the SRMS, are used to maintain the given direction of the descent trajectory. When the aircraft is calculated to have reached the point for beginning its descent, it is shifted to a descent regime with a calculated rate of descent. The vertical rate of descent is main- tained by observing the variometer readings and those of the gyro- horizon, while maintaining the established regime of horizontal airspeed on the basis of the instrument-speed indicator. The gyrohorizon must be used to maintain the vertical rate of descent, because the readings of the variometer are less stable than those of the angle of pitch of the aircraft obtained with the aid of the gyrohorizon indicator. The readings of the variometer must be averaged over the time. In addition, the variometer has slight delays in the readings with a change in the angle of pitch of the aircraft. Therefore, the gyrohorizon is employed to select the angle of pitch for the aircraft at which the average readings of the variometer are equal 393 to the calculated values, and this angle is maintained by the readings on the gyrohorizon. If the horizontal airspeed is then increased or decreased rela- tive to the given value, it is regulated by changing the thrust of the engines and simultaneously changing the angle of pitch slightly to maintain the calculated rate of descent. A failure to maintain the calculated settiiq^ for the glide path, or errors in calculations, may cause the aircraft to pass over the LRMS earlier at the required altitude, so that the descent of the aircraft is terminated and the aircraft is once again placed in the regime of descent at the moment it passes over the LRMS. However, if the given altitude has not been attained when passing over the LRMS, the vertical rate of descent is increased at the stage of the flight between the LRMS and the SRMS . Similarly, the descent of the aircraft is terminated if it reaches the altitude set for passing over the SRMS before the sound of the SRMS is heard, marking the location of the latter. /374 The minimum weather for the ceiling when landing with a simpli- fied system, in the case of aircraft with piston engines, is not set any lower than the altitude for passing over the SRMS; in the case of aircraft with gas turbine engines, it is significantly higher. Therefore, the aircraft can be allowed to descend only in the case when the crew of the aircraft can see the lights of the approaches to the LTS and the end of the runway. Course-Glide Landing Systems The simplified system for landing an aircraft as described in the preceding section, using the master radio stations, has a number of important deficiencies: (a) The measurement accuracy of the aircraft bearing, using an aircraft radiocompass and course meter, is very low, so that it does not make it possible to land the aircraft (especially those with gas-turbine engines) with low weather minima. (b) The operation of radiocompasses during flight in clouds and precipitation is highly subject to atmospheric disturbances, thus complicating a landing with these devices as guides. (c) The simplified system requires constant checking of the position of the aircraft along a given descent trajectory in terms of direction only; the descent of the aircraft in a given glide path is accomplished by maintaining the vertical rate of descent of the aircraft and calculating the time, thus complicating the landing procedure and not ensuring safe descent under especially difficult conditions. 394 If we consider that the period of landing the aircraft with low ceiling and low meteorological visibility is the most difficult and dangerous stage of the flight, it is necessary to devise more complete systems of instrument landing. One such system is the course-glide landing system. The geometric essence of course-glide systems is the use of radio-engineering methods to define two mutually perpendicular planes in space (Fig. 4- . 12 ) : (,a) A vertical plane which intersects the Earth's surface along the LTS axis . (b) An inclined plane which represents the glide path of the aircraft . If the aircraft is in one of these two planes, the readings of the corresponding pointer on the indicator (direction or glide) must be equal to zero. When the aircraft moves out of one of these planes, the corresponding pointer shifts from zero. The shift of the pointer must be linear within certain limits (i.e., proportional to the deviation of the aircraft from the given plane). Obviously, the given trajectory for the descent of the air- craft is the line of intersection of these two planes. When the aircraft is on the given trajectory, both Indicator pointers must point to zero on the indicator. /375 Fig. 4.12. Radio-Signal Planes of a Course-Glide Landing System. For the best visual determination of the position of the air- craft relative to a given descent trajectory, the pointers on the Indicator are made in the form of strips, one horizontal for glide and one vertical for direction. The movement of the strips then occurs in a direction which is opposite to the deviation of the aircraft from a given trajectory (Fig. "4.13). 395 The center of the instrument, with a silhouette of an air- craft shown on the scale, shows the position of the aircraft rel- ative to the course plane and the glide plane. Thus, for example, in Fig. '^.IS the aircraft is located below the given glide path and to the left of the LTS axis . To set the aircraft on the de- sired trajectory, it must be turned in the direction of the planes, i.e., upward (to increase the angle of pitch) and to the right. The indicator for the direction and glide has the traditional name of "Landing System Apparatus", or LAS for short. Ground Control of Course-Gl-ide Systems The principal pieces of equipment in a course-glide landing system are two ground beacons which form the course zone and the glide zone marking the given trajectory for the descent of the aircraft . Both beacons operate on meter or centimeter wavelengths. The antennas of the beacons that use meter waves are crossed horizontal dipoles (horizontal frames) in course beacons and horizontal dipoles in glide beacons. Thus, the electromagnetic waves from the beacons are horizon- tally polarized, which to a certain degree reduces their effect on the directional characteristics of the antennas on the ground control facilities at the airport. /376 At* Aj A2 A A, A^ A^ ' to transmitting device Fig. tt.l3. Fig. i+.14. Fig. 1+.13. Indicator of Course-Glide Landing System. Fig. 4.14. Diagram of Location of Antennas of Course Radio Beacon. However, the Earth's surface plays a role in the formation of the course zone and the glide zone by these beacons. The course zone then becomes multilobed in the vertical plane, with the major lobe being the working lobe, which has a glide angle of the bisectrix which corresponds roughly to the slope angle of the glide plane of the aircraft. The Earth's surface is of still greater 396 importance for the formation of the glide zone, whose slope angle depends on the height of the antenna above the ground. The involvement of the Earth's surface in the formation of the beacon zones imposes limitations on the possibilities of the beacons in terms of ensuring the accuracy with which the aircraft can be landed. This is especially true for the glide zone, whose location can change with the state of the Earth's surface (wet or dry ground, grass cover, snow). The accuracy of the location of the course zone is subject to the influence of the local relief and equipment located within the limits of the directional charac- teristic of the antennas. The most important of these shortcomings can be overcome to a great extent by employing beacons which operate on the centimeter wavelength, using reflecting antennas to form very narrow directional characteristics . At the present time, however, these beacons have not been adopted sufficiently widely and are not used in enough locations. Therefore, we shall give a brief description of the course-glide systems only for the meter wavelengths. In addition to the beacons, which form the course and glide zones, the course-glide system for landing also includes marker devices, whose locations can coincide with those for the markers in a simplified landing system. In foreign practice, the first (long-range) marker is located / 377 7 km from the end of the LTS ; at a slope angle for the glide path of 2°30' and a circling altitude of 300 m, this marks the point at which the aircraft begins to descent in a glide. However, no significant advantages are gained by placing the marker at this spot, since the flight altitude of the aircraft when circling the field depends on the type of aircraft, while the slope angle for the glide path depends on the nature of the surrounding terrain. This means that the point for beginning the glide does not always coincide with the standard location of the aircraft (7 km). For purposes of checking for the correctness of the location of the glide zone, it is better to choose a marker located ^ km from the end of the LTS, since at this point the aircraft will al- ready have the selected rate of descent for following the glide path, and the altitude of its location will be determined more precisely . An inherent part of the course-glide landing system is also the lighting system for the approaches to the runways and along the edges of the runway itself. The Gourse beaaon is a transmitting device with an antenna system which consists (as a rule) of five or seven horizontal antennas (Fig. 4.14). 397 fl Antenna A has a radiation characteristic which is directed externally in the horizontal plane, and is powered by a transmitter operating without modulation on the meter wavelength. Antennas Ai and A2 receive amplitude-modulated frequencies from the transmitter, one at 90 Hz and the other at 150 .Hz. Antennas A^ and A^^ (as well as ^45 and A^, in some types of beacons) serve to regulate the directionality of the radiation characteristic, as well as the direction of the radio-signal zone of the entire system. The combined result of the electromagnetic oscillations of the entire antenna system forms the directional radiation characteristic of the electromagnetic waves in the horizontal plane; an example of this is shown in Fig. 4.15. Figure 4. 15. a, shows the shape of the radiation characteristic in the horizontal plane; the left side is tie one modulated' by the 150 Hz frequency, while the right side is modulated by the 90 Hz frequency. Along axis AB , where the radiation characteristics intersect the modulation frequencies of 150 and 90 Hz, the modulation depth of the carrier frequency by both low frequencies is the same (i.e., the difference in modulation depth is zero). When the aircraft moves to the left of axis AB , the depth of the modulation with the 90 Hz frequency increases and that with the 150 Hz frequency decreases. The picture is reversed when the air- craft moves to the right of the axis. The dotted lines in Fig. 4.15 show the projections of the radiation lobes of the electromagnetic waves in the vertical plane (as shown in Fig. 4.15,b) on the horizontal plane. Line AB is the common axis with a difference in modulation depths which is equal to zero for all lobes. However, the prin- cipal operating lobes are the first ones, located nearest to the ground . b) /378 Fig. 4.15. Ra,dtation Characteristics of Course Radio Beacon: (a) in the Horizontal Plane; (b) in the Verti- cal Plane . 398 The radiation characteristic o in such a way that the axis of the depth coincides exactly with the ax necessary that the difference in mo 3-4-° of the equal-signal axis incre deviation. With further deviation up to 10°), the difference in modul but not in proportion to the latera value or decrease, but without chan to the left or right of the radio-s f the course beacon is regulated zero difference in modulation is of the LTS. Hence, it is dulation depths within limits of ase linearly with the lateral from the LTS axis (within limits ation depths must also increase, 1 deviation; it can maintain its ging sign in the entire hemisphere ignal axis , The distance for possible reception of the beacon signals in the sector 10° from the equal-signal axis in the working lobe of the zone must be within the limits of 45 to 70 km. The gZ-ide beacon is also a transmitting device, operating in the meter wavelength, but at a frequency different from that of the cours e beacon . The antenna system of the glide beacon consists of only two antennas (an upper and a lower), mounted on a common mast. The upper antenna is double, as shown in Fig. 4. 16, a. Both the upper and lower antennas receive an amplitude -modula- ted frequency, but with different modulation frequencies (for example, 90 and 150 Hz). Each of the antennas, together with the ground forms an inde- pendent working lobe with its own modulation frequency (Fig. 4-.16,b). The points of intersection of the working lobes in the vertical plane also form a radio-signal axis AB with a zero difference in the modulation depth. Since the characteristic of the antenna directionality in the /379 horizontal plane is rather broad, the surface with a zero difference in modulation depth is conical, with AB as the generatrix. There- fore, the glide path can be an ideal straight line only in the case when the antenna system of the beacon is located at the point where the aircraft touches down on the runway. trans» H mining ^ y^ -y y Fig. Fig. 4. 17 Fig. 4.16. Glide Radio Beacon: (a) Diagram of Antenna Location; (b) Radiation Characteristic. Fig. 4.17. Hyperbolic Trajectory for Glide Plane. 399 Ill Mil I I II II II III I III I However, the glide beacon cannot be located on the LTS axis or even in the immediate vicinity of the LTS, since it would con- stitute a flight hazard. Therefore, the intersection of the cone with the zero difference in the modulation depth for the glide beacon of the equal-signal plane of a course beacon gives a hyper- bolic trajectory which does not touch the ground (Fig. 4.17). As the aircraft approaches along the landing path toward the traverse of the glide beacon, the glide path begins to "float" above the ground, moving upward after passing over the beacon. Since the location and shape of the directional characteristic of the two antennas of the glide beacon depends on the height of the antennas above the ground, the characteristic and the position of the line of their intersection in the vertical plane is regu- lated by the change in the height of the upper and lower antennas above the ground. As in the case of the course beacon, the increase in the difference of the modulation depth with deviation from the glide surface upward or downward must be linear with this deviation. However, the curvature of the curve of the change in the difference in modulation depth will not be symmetric in this case, as it is for the course beacon. A steeper curve for the change in the dif- ference of modulation depth is found above the glide surface, and a less steep curve is found below the surface. The operating range for a glide beacon in a sector of +_8° from the LTS axis must be at least 18 to 25 km. Ai-para ft -Mounted Equipment for the Course-Glide Landing System The following units make up the aircraft-mounted equipmeiit for the course-glide landing system: (a) Antenna and receiver for course-beacon signals. (b) Antenna and receiver for glide-beacon signals. (c) Control panel. (d) Landing-system apparatus (LSA) /380 The receivers of signals from the course and glide beacons contain essentially the same elements, with the exception of the AAC (automatic amplification control), which is not shown in the figure . '^ ^1 convertori-. IFA detec 1 tor - LFA (go Hz nfilter J_ heterodyne" 90Hz| recti ( fier 50Hz I fi lter . Fig. 4-. 18. Diagram of Aircraft-Mounted Glide Radio Beacon 400 The glide-beacon receiver uses the circuit for the reinforced AAC. The latter is not employed in course-beacon receivers, since it would add the microphone commands relayed via this beacon to the aircraft when the communications receivers are out of order. The signals from the course and glide beacons are picked up by the antennas and amplified by the HFA. The selection of the frequency channel is made on the basis of the first intermediate frequency by quartz returning of the heterodyne from the control panel. The signals are then amplified by the IFA and LFA channels, so that the signals pass through 90 Hz and 150 Hz filters to the rectifiers, then to the emergency blinker, and finally to the receiver ground. The indicator for the course or glide zone is connected in a bridge circuit between the rectifiers for the 90 and 150 Hz signals. If the signals do not reach the receiver or there is some malfunction in the receiver blocks somewhere ahead of the 90 and 150 Hz filters, the readings of the LSA indicator on that channel will be zero; if the equipment is operating properly, it means that the aircraft is located precisely in the corresponding zone. Therefore, the LSA system includes the emergency blinkers. When no current is flowing in the 90 and 150 Hz rectifiers, the current through the emergency blinker windings will not flow, and a signal indicating that the apparatus is malfunctioning will be displayed on -fhe indicator. The design circuit for the receivers of the signals from the / 381 glide and course beacons includes potentiometers for electrical balance of the LSA Indicators. Each of the rectifiers receives signals which do not pass through the 90 and 150 Hz filters. The in- dicator pointer should then point to zero. If the balance of the currents in the rectifier is upset, it causes the regulating poten- tiometer to rotate. The balancing potentiometer of the receiver for the signals from the glide beacon is usually mounted on the receiver housing, while the receiver for signals from the course beacon is mounted on the control panel. For smoothing the short-period oscillations of the course and glide indicator of the LAS, due to local disturbances in the radio-signal zone, the indicator circuit contains a special sealed unit damping capacitors in the circuit for turning on the apparatus. Location and Parameters for Regulating the Equipment for the Course-Glide Landing System The radio beacon for the course zone of the course-glide system for landing an aircraft is mounted at a distance of 600 to 1000 m from the end of the runway, along an extension of the axis of the LTS. 401 f The beacon for the glide zone is mounted to the side of the LTS (as a rule, to the left of the landing path), at a distance of r glide beacon fOOOn -^ 250- "Y^l} short range imarker -275m/ I50-200M course beacon BOO- 1000 m Fig. 4.19. Diagram of Location of Ground-Based Equipment for Course- Glide System. 150 to 200 m from its axis and 250-275 m from the end of the runway The axis of the zone of the course beacon coincides with the LTS axis. A controx point is chosen for measuring the parameters for regulating the system on the LTS axis . The control point is selected as a point where the antenna receiving signals from the glide beacon aboard the aircraft will be located at the moment when the aircraft touches down on the runway. It is considered that this point is located at an altitude of 6 m above the surface of the LTS, and is plotted from the location of the glide beacon, 75 m toward the end of the LTS (i.e., the dis- tance from the end of the runway to the control point is 180 to 200 m) . The slope angle for the glide path is calculated from a theo- retical plane located 6 m above the surface of the LTS. The ver- tex of the slope angle of the glide path is 1he control point (CP). The width of the zone of the course and glide beacons is reck- oned from the angles of deviation from the given descent trajec- tory, calculated respectively from the point where the course beacon is located and from the control point, within the limits of which the strips of the landing-system apparatus deviate from the zero position to the limits of the scale. Obviously, the angle of deviation of the LSA strip depends on the difference in the modulation depths in the beacon zones, as well as on the sensitivity of the receiver aboard the aircraft. Therefore, the angular width of the zones of the course and glide beacons is regulated by the sensitivity of the receivers mounted aboard the aircraft, which are used as standards. The standards for the width of the course-beacon zone are set as follows : (a) The angular width of half the zone must be located within 2 to 3° of the LTS axis. /382 402 (b) The linear width of half the zone at a distance of 1350 m from the control point (1150 m to the end of the runway) must be equal to 150 m. An expansion of the zone from the nominal value to 45 m and a narrowing to 30 m is considered permissible. The horizontal scale of the LSA (see Fig. 4.13) from the center to the scale stop has 6 divisions. The first division is the white circle on the silhouette of the aircraft, the second is the end of the vane, the third, fourth, and fifth are points on the horizontal axis of the scale, while the sixth is the scale stop. The vertical scale also has six divisions, of which the second division here is the first point on the vertical axis of the apparatus . Each division of the horizontal scale of the LSA corresponds to a deviation of the aircraft from the LSA axis (relative to the point where the course beacon is located) within limits of 20 25 m (+7, -5 m) from the LTS axis at a distance of 1350 control point . to 30' or m from the The angle width of the zone of the glide beacon is linked to the slope angle of the glide path, which is determined by the con- ditions of the formation of the zone. The width of the zone be- neath the glide path is then somewhat greater than above the glide path . The standards for regulating the glide zone are the following: (a) The position of the upper limit at an angle to the axis of the zone within the limits from 0.19 to 0.21 9, i.e., approxi- mately 1/5 of the slope angle for the glide path. (b) The location of the lower limit at an angle to the axis of the zone within limits from 0.29 to 0.31 6 (somewhat less than 1/3 of the slope angle for the glide path). Accordingly, one division of the vertical scale of the LSA in the upper part is equal to about 0.030 9, while in the lower part it is about 0.05 9, where 9 is the slope angle of the glide path. Landing an Aircraft with the Course-Glide System Setting up the maneuver for an aircraft approaching an airport to descend with the use of the course-glide system is performed according to the same rules as in the simplified system for landing an aircraft . The complement of equipment for the course-glide system for landing an aircraft is usually supplemented by one or two master radio stations with marker beacons, located in the system for simplified landing, which is used for setting up the maneuver for /383 403 bringing the aircraft in for a landing and to a certain degree reserves the course-glide system for cases of malfunction of the ground or airborne equipment, as well as during times when equip- ment is being repaired or adjusted. If in addition to the course-glide and master beacons, the air- port is equipped with radar for observing the aircraft, the maneuver for landing in minimum weather can be made along the shortest path for each landing direction and takeoff direction. By the same rules which govern the simplified landing system, preliminary calculations are carried out which ensure a simpler and more exact action of the crew in flight along a given descent tra- j e ctory . A portion of the preliminary calculations, such as (for example) the determination of the moment for starting the descent in a glide, cannot be done in this case if we keep in mind the fact that the given glide path is defined in space. The calculations of the drift angle of the aircraft and the vertical rate of descent along the landing path are of somewhat less importance in this case. When the maneuver for making a landing is made on command from the ground, the need for such calculations as the determination of the moment for making the third turn no longer exists. However, the moment for beginning the fourth turn must in all cases be determined by the crew of the aircraft, with the maximum accuracy poss ible . In setting up the maneuver for landing, the strips of the LSA can be located on any divisions of the scale and no attention need be paid to their readings; however, when approaching the fourth turn, both strips must be located on the scale stops. The strip for the course zone rests on the stop on the side opposite the direction of the maneuver, the strip for the glide zone rests on the stop at the top. The emergency blinkers must then be off. The strip for the course zone must move away from the scale stop during the fourth turn. The movement of the strip away from the stop is called deftection. When the fourth turn is made correctly, deflection of the strip for the course zone occurs at the moment when the turn angle is held until the aircraft acquires the calculated landing course (Fig. i<-.20,a). For aircraft with piston engines, this turn aisle is about 45°; for aircraft with turbojet or turboprop engines, it is about /38M- 30°. With a residual turn angle of 45° for aircraft with piston engines (30° for aircraft with gas turbine engines), if deflection of the course-zone strip does not occur, it means that the fourth turn is being made with a lead. 404 In this case, it is desirable to significantly reduce the banking angle during the turn or even to stop turning and follow the LTS axis at the residual turn angle until the LSA strip deflects b) LRMS bRMS Fig. 4.20. Acquisition of the Landing Path by an Aircraft Proper Turn; (b) With Turn Begun Late. (a) With When the course-zone strip deflects, the turn must be continued until the landing course is acquired. When the landing course is acquired, the course-zone strip must be located near the zero marking (center of the scale). In cases when the fourth turn is made with a delay (Fig. 4.20,b), the deflection of the LSA strip takes place earlier than 45 or 30° before acquisition of the landing course. In this case, the turn must last until the landing course and beyond, at a landing angle opposite to the LTS axis, depending on the magnitude of the transi- tion of the course strip through the center of the scale. For example, if the descaling occurs at the very beginning of the fourth turn, it is necessary to increase the banking angle in the turn up to 20°, and the aircraft will continue to turn to the opposite angle for landing (20° in aircraft with piston engines and 30° for aircraft with gas turbine engines). With less delay in turning, the opposite angle for approach can be within the limits of 5 to 20°. With reverse deflection of the course-zone strip, the aircraft makes a reverse turn onto the landing course, with a simultaneous flat turn onto the LTS axis. After the aircraft has acquired the /385 405 LTS axis, the flight continues for a time until deflection of the glide-zone strip takes place at a constant altitude. At the moment when the glide-zone strip moves away from the upper stop, the aircraft shifts to a descent regime with a smooth acquisition of the desired glide path downward. Dvreationat PToperties of the Landing System Apparatus The selection of the desired course and the vertical rate of descent are sources of considerable difficulty for the crew and require a certain degree of training. However, these difficulties do not arise from principles of piloting the aircraft along the LSA, but rather from the necessity of simultaneously observing several devices and instruments and selecting a flight regime in the vertical and horizontal planes simultaneously. Nevertheless, with a proper reaction of the crew to a change in the positions of the strips on the LAS, the landing maneuver should be successful in all cases and not very difficult. In piloting the aircraft by the LSA, two of its principal char- acteristics must be employed: (1) The indicating characterisli c , i.e., the indication of the position of the aircraft relative to a given descent trajectory. (2) The command characteristic, i.e., the ability to predeter- mine the actions of the crew ii selecting the flight regime. Inasmuch as the first property of the LSA is obvious, let us examine the second. The course and glide zones are rather narrow in space, suffi- ciently so that the limits of these zones can be considered parallel over short segments of the trajectory. Let us say that an aircraft at a given moment is located to the side of the LTS axis, and the ground speed vector of the air- craft does not coincide with the direction of this axis (Fig. 4.21). Obviously, the ground speed vector of the aircraft can be divided into two components: a longitudinal one W^ and a lateral one W^, The longitudinal component f/^ is not involved in the selection of the course to be followed. The principal role is played by the lateral or transverse component, W . z The component W „ determines the rate of motion of an LSA strip /386 along the horizontal scale of the apparatus. With the strip fixed at any scale division, the component W^ is equal to zero, which agrees precisely with the selected aircraft course, i.e., its path is practically parallel to the axis LTS. H06 The regulation of the LSA is set so that the change in the course of the aircraft (1,5 to 2°) makes the motion of the vertical iVx V .-_____V ^ /^ -^^ Fig. 4.21. Division of Ground Speed Vector into Longitudinal and Lateral Components along the Landing Path. strip LSA visible to the eye. For example, in aircraft with piston engines, a change in the course by 2° produces a lateral shift of the aircraft of 2 m/sec. This means that the most dangerous region of flight (1200 to 1500 m to the end of the runway), the LSA strip crosses each scale division in 11 to 12 sec, i.e., a sufficiently noticeable value equal to half a scale division after each 5 to 6 sec of flight. If a turn is made in the direction of the motion of the LSA strip by 2°, its motion can be halted at any division on the s cale . On this basis, the principle of selecting the course for the aircraft by the LSA must be the following: If the vertical strip is located at a significant distance from the center of the instrument (on the third or fourth division), it is then necessary that the rate of its shift to the center of the instrument be significant. To do this, it is sufficient to turn the aircraft in the direction in which the strip is moving, by 4 to 6° . As the strip approaches the center of the apparatus , its rate of motion must be arrested by turning the aircraft 1 to 2° in the direction shown by the arrow. At the moment when the strip reaches the center of the instrument, its motion is arrested by a final turn of the aircraft by 1 to 2°, and the aircraft will be set on the LTS axis, with the course already selected. This method requires a very precise flight of the aircraft along the axis of the course zone, with periodic changes in the course within the limits of 1 to 2°. An analogous method is employed to set the vertical rate of descent of the aircraft, with simultaneous acquisition of the desired glide plane and subsequent flight along it. Maintenance of the descent regime of the aircraft along a given trajectory by the readings of the LSA continues up to the moment when the aircraft emerges from the clouds and makes a 4-07 transition to visual flight, after which a tude is made and the aircraft touches down visual estimate of alti- /387 on the runway . Direat-ionat Devioes for Landing Airaraft Determination of the rate of shift of the strips calls for in- creased vigilance in observing each of them. In addition, local irregularities in the course zone and glide zone at individual points disturb the regularity of the process; this must be taken into consideration by the crew and carefully separated from the generally established tendency. All of this requires considerable caution and training on the part of the crew for making a descent along a given trajectory. Recently, special directional devices for piloting an aircraft in the course and glide zones have begun to be employed widely. Unlike the LSA, the directional properties of these devices are not expressed by the derivatives of the positions of the strips on the instrument with time, but directly by the positions of these strips . The most widely employed directional devices at the present time are those which are based on various laws of control, with an indication which is linked to the banking of the aircraft during a coordinated stable turn, or to the angle of pitch at a set rate of des cent . In pilotage of the aircraft in the horizontal plane, these laws represent a definite link between the course and the banking of the aircraft in a turn, with lateral deviation from the radio- signal plane of the course zone and the first derivative of this deviation with time. K^^t + ATp H KzZ + KvVz = 0, where Ay is the angle of approach to the landing path, 3 is the banking angle of the aircraft in the turn, Z is the lateral devi- ation of the aircraft from the zone axis, V^ is the rate of lateral shift of the aircraft, and K are the coefficients for the corres- ponding parameters. A similar law is employed for piloting an aircraft in the vertical plane: where V is the angle of pitch of the aircraft, Y is the deviation of the aircraft from the glide path in the vertical plane, and V, is the rate of vertical motion of the aircraft. y 408 Since the linear values Z and Y and their first derivatives cannot be measured directly in polar systems, their values are re- placed by angle values (a and AG) and their derivatives. The values a and Ae and their derivatives a and A6 are measured by the differences in modulation depths and their derivatives in the zones of the course and glide beacons. /388 Obviously, by selecting the proper banking angle and pitch angle for the aircraft, the aircraft can be positioned so that both strips on the directional indicator are located on zero. The coefficients for the converted position parameters for the aircraft axes are selected so that whatever deviations the aircraft may make from the given trajectory (if the indicator strips remain on zero), the aircraft will still travel along the given landing and glide path with a predetermined trajectory (whose course depends upon the coefficients selected). This means that the landing course and vertical speed must be selected simultaneously, since they are required for flying the aircraft along a given trajectory. Hence, instead of adjusting the rate of motion of the strips in accordance with their motion toward the center of the instrument as in a normal LSA, in directional instruments the crew need only bring the indicator strips to the center of the instrument by chang- ing the banking angle of the aircraft as well as its angle of pitch; this significantly facilitates the task of piloting an aircraft. To further reduce the work of the crew, directional instruments are usually combined with a gyrohorizon indicator. In this case, the entire attention of the pilot is concentrated practically on the readings of only one instrument. However, directional instru- ments based on the rules stated above have some important short- comings, which to a certain degree reduce the accuracy of piloting an aircraft relative to piloting by the indications of an LSA. The proper selection of coefficients for making a turn and the angle of pitch of the aircraft can be made only at a certain distance of the aircraft from the ground beacons. During measure- ment of the distance, the linear width of the course and glide zones changes, thus leading to a failure of the system regulation param- eters to agree with the dynamic flight trajectory of the aircraft. This shortcoming can be completely overcome if the system is regu- lated not only by the angular deviation of the aircraft from the radio-signal axis, but by calculation of the distance remaining to the ground radio beacons: r=£gtgAe. where L^ and L beacons . are the distances to the cours-e and glide radio 409 Control can then be effected in a rectangular system of ccor- / 389 dinates, and therefore with constant agreement of the regulation of the system with the dynamic trajectory of the aircraft's flight. In polar coordinates, shortcomings in the operation of the directional system can be eliminated by a special selection of converted signal coefficients (not proportional to the values of the signals in various sections of the trajectory) in accordance with the tactical characteristics of aircraft of various types. It should also be mentioned that in directional systems, the indication of the position of the aircraft relative to a given descent trajectory is lost. This means that on board the aircraft, in addition to the directional devices, there must still be a conventional LSA indicator, which is used as a standard to check the accuracy of pilotage according to the directional indicator. So-called paravlsual directional instruments are also beginning to be used nowadays; in principle, they represent a reinforcement of the directional properties of the LSA. In this case, the usual LSA indicators are located in the cen- ter of the field of the pilot's vision, while at the periphery of his vision there are imitators of the motion of the strips according to the first derivatives a and A9, which link the indication shown with the longitudinal and lateral rolling of the aircraft according to the laws of the design of directional instruments. Radar Landing Systems From the tactical standpoint, radar landing systems have no special advantages over course-glide systems; on the contrary, their use is less convenient, since there are no instruments a- board the aircraft for indicating the position of the aircraft and no commands for piloting it relative to a given descent trajectory. The accuracy with which an aircraft can be landed by means of radar landing systems is roughly equal to that of landing it with course-glide systems. Nevertheless, radar landing systems are widely employed, along with course-glide systems. The primary reason why radar landing systems have been employed so widely is the need for a constant check on aircraft making their landing approaches by course-glide systems, for the purposes of pointing out errors made by the crew and preventing the very danger- ous consequences of error. The second reason is the need to give the crew assistance in landing the aircraft if they should request it, if for some reason the course-glide system cannot be used. The same reasoning applies / 390 in retaining the course-glide system in case the ground control is not functioning. 1+10 I IIHIIIIIIIII I I I I The radar landing system consists of a complex of devices for observing the flight of approaching aircraft (radar screen, USW radio distance-finder) and those actually making a landing (landing radar). In addition, the system includes communication apparatus for transmitting information and necessary commands to the aircraft. The landing radar is the heart of the radar landing system, so we shall pause to examine the principles of its operation. Unlike ground radar installations with circular screens, the landing radars have a sector screen, i.e., there is no rotating directional characteristic of the antenna, but one which scans (oscillates) in a certain sector. Accordingly, the scanning line on the radar screen also oscillates. The landing radar has two antennas: (a) The course -s e ctor antenna, with a wide characteristic in the vertical plane and a narrow one in the horizontal. (b) The glide-sector antenna, with a wide characteristic in the horizontal plane and a narrow one in the vertical; the scanning of the characteristic of this antenna takes place in the vertical plane . The scanning of the directional characteristics of the landing- radar antenna can be achieved either by mechanical oscillation of the antenna reflector or by special devices which change the phase of the wave along the chord of the antenna reflector, thus causing the plane of the wave front to oscillate (so that all the wave- propagation characteristics also oscillate). The scanning sectors of the directional characteristics of the antenna are made narrow; (a) For a course sector of 15°: to either side of the LTS axis, (b) For a glide sector, 9° wide: +8° upward and -1° downward from the plane of the horizon. A peculiar feature of the landing radar is the special design of the scanning on the course and glide screens. Thus, instead of the circular distance marks on conventional circular radar screens, the distance marks on landing radars are straight lines, i.e. , the delay in the distance marks is made proportional not to R, but to i?/cosa, where a is the angle of deviation of the scanning line from the axis of the scanning sector. Hence, a rectangular system of coordinates is formed on the screen from the polar system of coordinates for the aircraft. In addition, the radar screen has a transverse scale three times larger than the distance scale for the course sector and five times larger than that for the glide sector. This means that there is a corresponding relationship between the increase in the scale indicating the position of the aircraft relative to the given trajectory for the same screen radius. itll A general view of the screens of the course and glide sectors is shown in Fig. 4.22. /391 glide path sector course sector Fig. 4.22 Fig. 4.23 Fig. 4,22. Landing-Radar Screen: (a) Glide Sector; (b) Course Se ctor . Fig. 4.23. Pattern on Course Screen of Landing Radar. The landing radar is mounted on the traverse of the center of the LTS , at a distance of 100 to 150 m to the side, so that the conditions for using it when landing at either end of the runway will be the same. In the immediate vicinity of the landing radar, there is a circular-scan radar for observing aircraft near and far from the airport . In setting up the landing maneuver, immediately before completin; the fourth turn, the short-range radar approach system is used, also called control-tower radar (CTR). Its screen can be used to show landing maneuvers for aircraft approaching from all directions. All turns of the aircraft are made on command from the flight supervisor, as are the course corrections on the straight -line segments between the turns, if the given flight directions are not maintained sufficiently accurately. Observation of an aircraft with the landing radar begins while it is making the fourth turn, using only the course sector s creen . In order to ensure that the aircraft lands precisely on a given descent trajectory, the required pattern is superposed on the landing radar screen. This pattern on the screen serves three purposes : (l) To show the given trajectory for the aircraft's descent. 412 (2) To provide auxiliary lines for giving commands to the crew of the aircraft. (3) To show the boundary lines for safe flight altitude and /392 the permissible zones for landing the aircraft. Since the landing radar is usually used for two directions of landing and takeoff, and can even be used for three or four is if other runways intersect, the patterns for the screens are printed on removable celluloid sheets which can be changed when shifting the landing radar to a new landing direction. The screen for the course sector of a landing radar (Fig. 4.23) usually shows the following: 1. The given landing path (axis of LTS), beginning at the end of the runway and extending to the limit of the screen. The follow- ing points are marked on this line: the beginning of descent along a set glide path and the locations of the LRMS and SRMS landing systems within the range of the master radio stations. The SRMS is usually fitted with a comer reflector, which produces a bright spot on the screen and is used in setting the radar for the given landing direction and as a control to check the accuracy of the setting of the radar after it is turned around. 2. The lines delimiting the zone of possible aircraft landings. These lines are defined on the basis of the assumption that the air- craft, being on a course close to that for landing, can be lined up with the LTS axis prior to the start of the landing distance only in the case when X>2;?sln UT, where yp = arccos ( 1 i'-w)' where X is the remaining distance to the start of the landing dis- tance, Z is the lateral deviation from the landing path, and B is the turning radius with a banking angle of 10° . The order in which these lines are plotted is the following: (a) Several points of deviation of the aircraft from the landing path are given (e.g., 30, 100, 200, 500, 1000, 2000, and 4000 m) and the required turn angles to correct these deviations are determined: cosUT=l--^; (b) The required course for lining up the aircraft with the LTS axis is determined: X=2/?sinUT. /393 413 To this path, we add the distance traveled by the aircraft (in 4 sec for piston-engine aircraft, 7 sec for gas turbine aircraft), required for receiving commands and carrying out the maneuver to line up the aircraft with the runway. (c) The path obtained forthe aircraft is measured from the start- ing point of the landing distance (as rule, from the SRMS), and we obtain the minimum attainable distances of the selected points for the lateral deviations of the aircraft. By connecting the points by a smooth curve, we obtain the limit of the possible landing zone of an aircraft, with permissible lateral deviations . In the course of landing an aircraft, if it shows up outside the indicated limits, the landing cannot be allowed and the command is given to make another pass at the field. The boundary lines are usually plotted for two typical glide speeds of aircraft : rith pis ton-engines 5 200 kro/hr; rith gas turbine engines, 2 80 km/hr. for those with for those The turn radius is calculated for a coordinated turn with a banking angle of 10°, with the lines for starting the turn plotted for making a landing at approach angles of 10 and 30°. If the aircraft has a significant deviation from the LTS axis after emerging from the fourth turn, we can in principle use any angle of approach to the LTS axis which makes it possible to line up the aircraft with the landing path before the landing distance is reached. However, as experience has shown, it is simplest to line up the aircraft with the landing path by using only two values for the approach angles: 10° if the deviation of the aircraft from the given line path is less than 500 m, and 30 m for deviations exceed- ing 500 m. Then the landing-radar screen can be bounded by a total of two auxiliary lines for beginning the turn onto the landing path . In this case, the distance from the line can be determined by the formula Z = /?(!_ cos UT^t LTS axis to the auxiliary However, experimental data show that there is an appreciable delay in the aircraft's acquiring the landing path, due to the time involved in transmitting commands and due to the reaction of the aircraft and crew in making the turn. Therefore, it is better to plot these lines on the basis of statistical data obtained from experience, as determined from a large number of aircraft landings. /394 4-14 According to these data, the turn to the landing course must begin : (a) For an approach angle of 10°, in aircraft with piston engines, 150 m from the LTS axis (5 mm on the screen scale); for aircraft with gas turbine engines, it is 250 m from the LTS axis (8 mm on the screen scale). (b) With an approach angle of 30°, these distances are 450 and 750 m, respectively (15 and 25 mm on the screen scale). The markings on the glide screen of the landing radar are shown in Fig. M- . 2 U . Fig. 4.24. Pattern on Glide Screen of Landing Radar. limits of these boundary lines In this case, the descent trajectory for the glide path is set at the airport . Above this glide path are two boundary lines for landing the aircraft; for aircraft with gas turbine engines it is 4°, and for aircraft with piston engines it is 5°. If the blip representing an aircraft appears above the boun- dary line designated for a given type of aircraft, the landing of the aircraft will be complicated. Therefore, when controlling the landing of an aircraft, it should not be allowed to go beyond the Below the established glide path, there are boundary lines for permissible descent of the aircraft below the glide path, i.e., the lines limiting the flight altitude above the local terrain: 200 m prior to beginning descent in a glide, 150 m before passing over the LRMS, and 50 m before passing over the SRMS . In addition, there may also be flight altitudes for circling the field, set at 300, 400 and 500m . These lines are used for aircraft coming in for a landing according to the CGS (course-glide system). In the case where the blip marking an aircraft intersects one of these lines, further descent of the aircraft is to be considered dangerous and the intervention of the flight supervisor operating the landing radar is required. Bringing an Aircraft In for a Landing with Landing Radar /395 The method of bringing an aircraft in for a landing with a landing radar is very simple and quite effective at the present time . 415 The setting up of the landing maneuver and the calculations of the elements of the descent is made by the same rules as in using the simplified or course-glide landing systems. The moment for starting the fourth turn is determined on the basis of the blip representing the aircraft on the flight super- visor's screen. No commands are given to the crew during the fourth turn . After the aircraft landing path must be fol landing-radar screen is course of the aircraft i craft need merely be lin is at an angle to the LT craft is not equal to th determine the desired co the angle of the blip is course error. For examp correction must be 3°. emerges from the fourth turn, the calculated lowed for 10 to 15 sec. If the blip on the parallel to the LTS axis, the calculated s equal to the landing course and the air- ed up with the landing path. If the blip S axis, the calculated course of the air- e landing course, but it is very easy to urse correction by visual inspection, since equal to three times the angle of the le , with a blip angle of 10°, the course Having thus determined the required correction in the course to be followed, the supervisor gives a command to the crew, telling them to acquire the desired landing path at an angle of 10 or 30°, thus setting the course to be followed. At the moment when the blip crosses the corresponding auxiliary line, a command is given to turn the aircraft onto the landing course, considering the correction given. In the majority of cases, when these two comm.ands are given, it is sufficient to line up the aircraft with the landing path on the desired course. If a tendency is observed during flight along the landing path for the aircraft to shift laterally, it can be corrected by commands for small changes in the aircraft course (by 2 or 3°), with indication each time of the course which must be followed . When the blip approaches the point where the aircraft is to begin its descent in a glide, a command is given to descend at a calculated vertical speed. If it then develops that the aircraft is deviating from the given glide path (either upward or downward), the flight supervisor corrects the vertical speed, giving new values for it and ensuring that the aircraft travels exactly along the given path. An advantage of the radar landing system is the relative simpli- city of the supervisor's task in directing the airi^raft to a landing and the uncomplicated actions of the crew in carrying out the super- visor's commands, with no previous training required. These advan- tages are also reinforced by the fact that the flight supervisor, /396 who constantly watches over several aircraft coming in for a landing 1+16 and gives them instructions, acquires a very great amount of exper- ience in the course of his work, a great deal more than that which the crew can acquire from the landings of their own aircraft alone. In addition, the supervisor, in the course of his work in guiding one aircraft after another to a safe landing, acquires a peculiar "feel" for estimating the navigational difficulties on a given day (selection of the required vertical speed and landing course on the basis of his experience with aircraft that have landed earlier). Therefore, in practice, the accuracy of landing an aircraft with a radar system is no worse than with a course-glide system. Nevertheless, the main shortcoming of the system (a lack of indi- cation for the crew as to the position of the aircraft on a given descent trajectory) creates a certain degree of inaccuracy in making the landing, and in this respect the radar landing system is inferior to the course-glide system. 417 CHAPTER FIVE AVIATION ASTRONOMY^ 1. The Celestial Sphere The shy appears to the observer as an immense hemisphere. The aeZestiaZ s'pher'e is an imaginary sphere of arbitrary radius,^ whose center is the eye of the observer (Fig. 5.1). /397 An observer on the Earth's surface can see only the half of the celestial sphere which is located above the horizon, since the other hemisphere is located below the horizon. If the Earth were transparent , an observer located at any point on its surface would see not one but two domes which together form the celestial sphere. Specfal Points, Planes, and Circles in the Celestial Sphere Zenith and nadir. If a line is plotted perpendicular to the location of the observer (through the center of the celestial sphere), it will intersect the imaginary limits of the celestial sphere at two points (see Fig. 5.1). The point which is located above the observer is the zenith (Z). The opposite point is the nadir (Z'). True horizon. If a plane is defined through the center of the celestial sphere and is perpendicular to the vertical line ZZ ' , we 7398 can call it the plane of the horizon. The plane of the horizon inter- sects the celestial sphere along the circumference of a great circle (the points NESW) which is called the true horizon. World axis. The imaginary line PP ' , around which the apparent rotation of the celestial sphere takes place , is called the World axis. It passes through the point of the observer, located at the center of the celestial sphere, and intersects the arbitrary limits ^ , ^ -, ^ • -, o T. of the celestial sphere at two Fig. 5.1. Celestial Sphere ^ ^This chapter was written by M.I. Gurevich 4-18 diametrically opposed points PP ' . The world axis is inclined to the horizon at an angle which depends on the latitude of the ob- server . Fig. 5.2. Vertical and Almucantar . Fig. Hour 5,3. Celestial Meridian, Circle, and Celestial Parallel, Celestial poles. The points where the Imaginary world axis intersects the arbitrary limit of the celestial sphere are called the ceZesti-al poles. Point P is called the superior (north) oetes- tiat pole, and the opposite point P' is called the -inferior (south) aetestiat poZe. Only the north celestial pole is visible in the Northern Hemisphere, and only the south celestial pole is visible in the Southern Hemisphere. Celestial equator. The plane which passes through the center of the celestial sphere and is perpendicular to the world axis is called the plane of the eetestial equator . The great circle QEQ'W, along which the plane of the celestial equator intersects the celes- tial sphere, is called the ceZest-iaZ equator. The celestial equator divides the celestial sphere into northern (QPQ') and southern (Q'P'Q) parts. The plane of the celestial equator is inclined to the plane of the true horizon at an angle which also depends on the latitude of the observer. Vertical . The great circle on the celestial sphere whose plane passes through the vertical line is called the vertical. Every vertical passes through the zenith Z and the nadir Z'. The plane of the vertical is perpendicular to the plane of the true horizon (Fig. 5.2). The vertical which passes through the east and west points (E and W, respectively) is called the primary vertioaZ . /399 LH9 The great circle ZMZ ' of the celestial sphere, which passes through the zenith of the observer and a certain star (Point M, Fig. 5.2), is called the vertical of that star. Almucantar. The small circle DMD ' on the celestial sphere, whose plane is parallel to the plane of the true horizon, is called the atmucantar . The almucantar which passes through a given star is called the atmucantar of that star. Hour circle. The great circle PMP ' of the celestial sphere, whose plane passes through the world axis , is called the circle of declination (Fig. 5.3). Since the world axis is perpendicular to the celestial equator, the plane of the hour circle is also per- pendicular to the equator. The hour circle which passes through a given star is the hour cirote of that star. Celestial meridian. The vertical PZP 'Z ' , which passes through the celestial poles, is called the celestial, meridian (since its plane coincides with the plane of the meridian of the observer). The celestial meridian divides the celestial sphere into the eastern and western hemispheres. The north point N and south point S. The celestial meridian crosses the true horizon at two points, called the north and south ■points . Meridian line. The plane of the celestial meridian crosses the plane of the true horizon to form the meridian tine. Obviously, the ends of the meridian line coincide with the north and south points. (N and S, respectively). This line is called the "noon line" in Russian because the shadows of vertical objects fall along this line at noon. The east point E and west point W. If we plot a straight line in the plane of the horizon perpendicular to the meridian line (see Fig. 5.3) and face north, the east point E will lie on the right at the point where the plane intersects the circumference of the true horizon, while the west point will be located on the left. As the figure shows , the east and west points are 90° distant from the north and south points. The same figure also shows that the east and west points (E and W, respectively) mark the points of intersection of the celestial equator with the true horizon. Celestial parallel. The small circle on the celestial sphere, /i+OO whose plane is parallel to the plane of the celestial equator, is called the cetestiat parattet (similar to the terrestrial parallels). 420 Diurnal circle of a star. The small circle on the celestial sphere, drawn through a star parallel to the celestial equator, is called the diurnal oiraZe of the star. Astronomical coordinates. As we know, in order to determine the location of any point on the Earth's surface, it is sufficient to know the two angular coordinates of this point, the latitude and longitude . In astronomy, the location of stars on the sphere is accom- plished by means of two angular systems of celestial coordinates: the apparent system of coordinates and the equatorial system of coordinate s . In each of these systems, the position of a point (star) on the celestial sphere is determined by two celestial coordinates. Let us examine the systems of celestial coordinates individually. Systems of Coordinates A'p^arent System of Coordinates The main circles relative to which coordinates are determined in this system (Fig. 5.4) are the true horizon and the meridian of the observer. The coordinates themselves are called the altitude of the star (h) and the azimuth of the star (A). Altitude of a star. The angle between the plane of the true horizon and a line from the center of the sphere to the star (angle M'OM, Fig. 5.4) is called the altitude of the star. The altitude of a star can also be measured by the arc of the vertical from the true horizon to the location of the given star (M'M). The altitude of the star is measured from to 90° (positive values toward the zenith from the ^ horizon, negative values from the horizon toward the nadir). Zenith distance. Instead of the star, we can also use the so- called zenith distance of the star as a coordinate , measured along the arc ZM . As we can see from Figure 5.4, the zenith distance is the arc from the zenith to the location of the given star. It is easy to set up a formula to express the relation- /401 ship between the altitude and the System zenith distance of a star, since the two add up to 90°: h + Z = 90°, Fig. 5.4. Horizontal of Coordinates. 421 h = 90° - Z, Z = 90° - h. Obviously, the value of the zenith dis- tance will be somewhere between and 180°. Azimuth. The second coordinate in the apparent system of co- ordinates is the azimuth of the star. The azimuth of a star is the spherical angle between the plane of the meridian of the observer and the plane of the circle of the vertical of the given star. The azimuth is calculated differently in different areas of astronomy: from the south point or from the north point toward the east and west. In aviation astronomy, the azimuth is always calcu- lated from the north point along the horizon in an easterly direc- tion (clockwise) from to 360°. We can therefore define the azimuth in aviation astronomy as the angle measured along the arc NSM ' of the true horizon from the north point through the east (the east point) to the vertical of the star (see Fig. 5.4), from to 360° . Hence, the first system of coordinates for celestial luminaries is called the apparent system. The coordinates of this system are the altitude of the star (h) and the azimuth of the star (A). The altitude and azimuth will suffice completely to determine the location of a star on the celestial sphere. For example, the star M, with h = 60° and A = 240°, is indicated on the sphere (see Fig. 5.4). Equatorial System of Coordinates The equatorial system of coordinates is the second system of coordinates which is used to determine the location of a star on the celestial sphere. The main circles relative to which calculations are made in this system are the celestial meridian and the celestial equator . The coordinates in this system are the declination of the star (5) and the hour angle of the star (t); see Figure 5.5. Declination of the star. The arc of the circle marking the distance from the equator to the location of the given star, or the angle between the plane of the equator end a line from the center of the sphere to the star, called the deelination of the star. ± the sphere to the star, called the aeoLinav Declination is measured by the arc of a circle which marks the istance from the equator to the location of the given star, from to ± 90°. If the star is located in the Northern Hemisphere, its eclination is considered positive, while if it is in the Southern emisphere, it is considered negative. lit; taLdX xo _Ltj'»;ciLt;u x declination is considered positive, w Hemisphere, it is considered negative It is clear from Figure 5.5 that if the star is located on the equator, its declination will be equal to zero, while the declination of the north celestial pole is + 90° and that of the south celestial 422 pole is -90°. Polar distance. Occasionally, instead of the declination, the polar distance is used as a coordinate, measured along the arc PM . /^Q^ The polar dvstanoe is the arc of the circle which marks the dis- tance from the north celestial pole to the location of the star. The relationship between the declination and the polar dis- tance is expressed by the formula or 6 + PAf = 90° PM = 90° — 8, B = 90° — PAf, i.e. , the declination and polar distance together add up to 90°. Therefore the point of the south pole has a polar distance equal to 180° . Hour angle of a star. The arc of the celestial equator Q'M' (Fig. 5.5) between the south point of the equator and the hour circle of a given star is called the houT angle of a star (t). In aviation astronomy, the hour angle is measured from the south point of the equator along the equator in the easterly and westerly directions from to 180° Tl 'he western hour angle is represented by the letter W, for example, t = 135° W; the eastern hour angle is represented by the ' E, for example, t = 60° E. In making calculations, the calculated from to 360° letter western hour a If the we stern A^^T / / / / / / / ( '/ 1 -^"^ \ 1 ^ . [' 1 K example, t = 60° E. In making calculations, the ngle must sometimes be calculated from to 360°. hour angle is found to be greater than 180°, it is related to 360°, but in this case the result is given as an eastern .,- — .^ hour angle. For example, t = 265° M Fig. 5,5. Equatorial System of Coordinates. the result is given as an hour angle. For example, t W or t = 360° - 265° = 95°E. Right ascension of a star. Instead of the hour angle , it is sometimes more convenient to use another coordinate, the right as- cension of the star (a). The right ascension of a star is the angle as measured along the equator from the point of the vernal equinox (y) to the hour circle of the given star (see Fig. 5.5). The point of the vernal equinox is the imaginary point of the 423 intersection of the ecliptic with the celestial equator, when the Sun passes from the Southern Hemisphere into the Northern Hemisphere, The opposite point on the ecliptic is called the point of the autumnal equinox {9.). In ancient Greece, the stars were used to reckon time. The /403 constellation Aries was located at the point of the vernal equinox, and was represented by the symbol (y)- Due to the precession of the Earth, Aries has now moved away from the point of the vernal equinox. This point has remained unmarked, though its name has been retained, and its position in the sky is determined by using some other star which is a fixed distance from the point of the vernal equinox. Right ascension is calculated from the point of the vernal equinox along the equator up to the hour circle of a given star in a clockwise direction (as seen from the north celestial pole), from to 360°. Like the hour angle of a star, the right ascension of a star can be reckoned in either degrees or hours, minutes, and seconds. This is because both of these coordinates (especially the hour angle) are closely related to the measurement of time. Thus , the equatorial system of coordinates can be used to determine the location of a star on the celestial sphere. If we know the declination and the hour angle or the right ascension, we can determine the location of a star on the sphere. For example, the star M, with 6 = +50°, t = 45°, is shown on the sphere (see Fig. 5.5). Graphic Representation of tiie Celestial Sphere In solving textbook problems in aviation astronomy, it is often necessary to sketch the celestial sphere and plot the stars on it according to their coordinates. Let us use a concrete example to study the order in which the celestial sphere is sketched. Example . 1. The latitude of the observer is (fi = 60°N, the altitude of the star h = 70°, and its azimuth A = 240°. Draw the celestial sphere and plot the position of the star on it . (Fig. 5 .6 ,a) . Solution. (1) Use a compass to draw the celestial meridian in the form of a circle of arbitrary radius. (2) Draw a vertical diameter (perpendicular line) and mark the zenith and nadir (Z and Z', respectively) at the points where it crosses the circumference. (3) Perpendicular to the vertical line, through the center of 4-24 the sphere, draw a large circle which will be the true horizon of the observer . (4-) Draw the world axis such that the angle it forms with the plane of the horizon will be equal to the latitude of the observer, i.e. , <j) = 60°N; mark the points where the world axis crosses the circumference (the north celestial pole P and the south celestial pole P ' ) . (5) At the points where the true horizon intersects the merid- ian of the observer, mark the north point N (close to the north celestial pole) and the south point S (close to the south celestial pole ) . (6) Perpendicular to the point of intersection of the celestial equator with the true horizon, mark the east point E (on the right, as viewed by someone facing north) and the west point W (on the left) . This completes the sketching of the celestial sphere. We have yet to plot the position of the star on the sphere on the basis of its coordinate data, as follows: (1) From the north point N, plot the azimuth of the star /HOM- (equal to 24-0°) along the circumference of the horizon, judging the angle by eye . (2) Through this point M', draw the circle ( semicir cumf erence ) of the vertical. (3) Along the circle of the vertical, from the plane of the horizon, plot the altitude of the star, equal to 70°, judging the distance by eye. S N Fig. 5.6. Examples of Graphic Construction of the Celestial Sphere; (a) At a Latitude of 60°; (b) At a Latitude of 50°. 425 The result of this construction will be the celestial sphere as seen by an observer at 60°N and the position of a star on the sphere according to its apparent coordinates . Example . 2. The observer is located at a latitude of 50°. Sketch the celestial sphere for this observer and plot on it the position of a star with the following equatorial coordinates: hour angle t = 130°, declination 5 = +40°. Solution. (1) Sketch the celestial sphere in the same order outlined in Example 1. (2) From the south point on the equator Q', proceeding along the circumference of the equator in a westerly direction, plot an hour angle t = 130° by eye (Fig. 5.6,b). (3) Through this point (M'), draw the hour circle (PM'P'). From the plane of the equator, along the hour circle, measure off the declination 5 = +40° and mark the location of the star on the sphere (point M). The result of this construction is the hour circle for an ob- server located at a latitude of (j) = 50°N; the star has been plotted on the sphere on the basis of its equatorial coordinates. Diurnal Motion of the Stars 7405 The reason for this apparent motion of the stars (or of the sky) is the diurnal rotation of the Earth on its axis from west to east . In order to facilitate a study of the diurnal rotation of the stars, we will assume for the sake of discussion that the Earth is fixed and the celestial sphere rotates on the world axis at the same rate that the Earth actually rotates on its axis, but in the opposite direction, from east to west (in other words, the way it actually looks to us). Since the entire celestial sphere rotates on the world axis, all the points (stars) located on the sphere 426 ,J will turn along with it, i.e., it is clear that each star describes a sort of circle around the world axis. Diurnal parallel of a star. All of the stars rotate together with the celestial sphere around the world axis. From this it is clear that every star, fixed permanently in the sky, describes a circle of some size in the course of 2^■ hours. The circle described by a star in 24 hours in the course of its movement around the world axis is called the diuvnat ciTote of the star. This circle is also called the aetestiaZ paraZtet . Since the entire celestial sphere rotates around the world axis, it is easy to see (and important to remember) that the di- urnal rotation of the heavenly bodies takes place parallel to the celestial equator, i.e., the diurnal parallel of the star (the path of the star around the world axis in 24 hours) is always located parallel to the celestial equator. The magnitude of the diurnal parallel of the star depends on the location of the star in the sky. Obviously, stars which are located closer to the celestial poles (and have higher declination values) have a small diurnal circle. The closer a star is located relative to the celestial equator (the smaller its declination), the larger its diurnal circle will be. The largest diurnal circle belongs to those stars which are located on the celestial equator, and whose declination is zero. Motion of the Stars at Different Latitudes If we observe the diurnal motion of the stars at different latitudes, we will see that the sky and stars turn relative to the observer's horizon at different angles. This phenomenon becomes understandable if we recall the location of the world axis relative to the horizon at different latitudes. The world axis is located relative to the horizon at an angle /406 which is equal to the latitude of the location. From this it fol- lows that the higher the latitude of a- location, the closer the celestial poles PP ' will be located to the zenith Z and the nadir Z', and the smaller the angle will be between the true horizon and the celestial equator. Conversely, the lower the latitude of the location, the further the celestial poles will be from the zenith and nadir, and the angle between the true horizon and the celestial equator will be larger. Figure 5. 7, a shows the angle between the true horizon and the celestial equator for an observer located at a middle latitude, e.g. 50° (angle 90° - <j) = 40°). Figure 5.7,b shows the angle between the true horizon and the celestial equator for an observer located on the Equator (angle 90° - (|) = 90°), while Figure 5.7,c shows the angle between the true horizon and the celestial equator for an 427 observer located at the North or South Pole. (The angle 90° - cf) = 0, the true horizon is parallel to the celestial equator, the zenith point Z coincides with the north celestial pole P, and the nadir /407 Z' coincides with the south celestial pole P'). It is clear in all three figures that the angle between the true horizon of the observer and the celestial equator is always equal to 90° minus the local latitude (90 - (|) ) . We can draw the following conclusion from the above: the slope of the diurnal parallel of stars relative to the true horizon of the observer depends on the latitude of the observer. The higher the latitude of the observer, the smaller the slope of the diurnal parallels of the stars relative to the horizon; the lower the latitude, the greater the slope. Rising and Setting, Never-Rising and Neve r- Set t i ng Stars If we know that the position of the celestial equator (and consequently the diurnal parallels of the stars) relative to the true horizon of the observer depends on the latitude of the observer, it will be clear why some stars at a certain latitude rise and set b) eU) / u K / o / — 1 ol / o TV fp\u o (1) VI" horizon] 1 cS \ c \ u \ p • iH >^ vj -o y Fig. 5.7. Angles Between the True Horizon and the Celestial Equator; (a) At a Latitude of 50°; (b) On the Equator; (c) At the Poles. ipir 428 at the horizon, others never set, and still others never rise. A star never sets if its declination is greater than 90° minus the latitude of the location, i.e., if 6 > 90° - A. For example, see Figure 5. 8, a. Given the latitude of the ob- server <l> = 60° , the declination of the star 6 - +45° . From this it is clear that 90° - (() = 90 - 60° - 30°. Since the declination 6 = +45°, i.e., greater than 90° - <j) , it is clear that the star can- not set below the horizon of the observer. In Figure 5. 8, a we have sketched the celestial sphere for an observer located at a latitude of 60°. We mark off the declination of the star along the meridian of the observer (i.e., the hour circle) so that 6 = +45°, and then lay out the diurnal circle (diurnal parallel) of the star parallel /^OS S /V Fig. 5.8. Examples of Never- Set ting Stars; (a) The Star Never Sets Below the Horizon; (b) The Star Touches the Horizon. aj ink* [ tf , ^^ <^\ ^x^/ /" /'^v / / / .-^ / ^\. p/ / / / \ / f^y / \ „ \Mi// lux //A \.^§^S^ ' n "\,^ / ^^--/_ S^A^><'/ ^Y / / /^j/f' \i/ y T< __,----''^^ 5 N 2' 3<^^ "^v^^ --'""3^^' '^ yf=sn\ /^-jiT/ 4 < / ^ )>^^C^ VJ^^^^^""^ ^ -''V^ y^ j ^^^\/ —-"^ ^-"^S r Fig. 5.9. Examples of Stars that Set; (a) Tne Star Rises and Sets; (b) The Star does not Rise. 429 to the celestial equator. As we can see from the figure, this circle is located above the horizon of the observer, and so a star which moves along this circle in the course of 2h hours will never set below the observer's horizon. The star touches the horizon, but does not go below it, in the case when its declination is equal to 90° minus the latitude of the observer, i.e., if 6 = 90- cf). Take Figure 5.8,b for example. The latitude of the observer (j) = 60°, the declination of the star 6 = +30°. From this it is clear that 90° - (f) = 90 - 60° = 30°. In accordance with what we have said, if 6 = 90° - (f , the star will touch the observer's horizon but will not set below it. In Figure 5.8,b we have sketched the celestial sphere for an observer located at a latitude of 60° . Along the meridian of the observer (i.e., the hour circle), we have plotted the declination of a star 6 = +30°, and have then drawn the diurnal parallel of this star parallel to the celestial equator. As we can see, the diurnal parallel of the star touches the observer's horizon, but does not cross it, i.e., a star moving along its diurnal parallel in the course of 2^- hours goes down to the horizon and then rises again in the course of its diurnal journey. A star rises and sets when its declination (in terms of abso- lute value) is less than 90° minus the latitude of the location, i.e., if 6 < 90° - (j) . Let us consider the following e observer is tj) = 30° , the declination Figure 5. 9, a we have sketched the ce located at a latitude of 30°. Along we have marked off the declination o drawn the diurnal parallel of the st q'). As we can see from the diagram course of 24 hours along its diurnal the horizon for a certain time (the parallel ) the rest xample: The latitude of the of the star is 6 = +4-0°. In lestial sphere for an observer the meridian (hour circle) f the star 6 = +4-0° and have ar parallel to the equator (q - , a star which moves in the parallel will be located below shaded part of the diurnal , and will be above the horizon of the time . A star never rises if its declina- tion is equal to or greater than 90° minus the latitude of the observer and has a sign which differs with latitude (the latitude Is positive and the Fig. 5.10 Division of the Celestial Sphere into Regions with Rising and Setting, Never-Setting and Never-Risinj Stars . /409 430 declination is negative, or vice versa), i.e., if - 6 >_ 90° or 6 = ()) - 90°. For example, the latitude of the observer i the declination of the star 6 = -30°. 6 0°N, In Figure 5.9,b we have sketched the celestial sphere for an observer at a latitude of (j) = 60°N. Along the meridian of the observer (hour circle) we have marked the declination of the star 6 = -30° (below the equator) and the diurnal parallel of the star parallel to the equator. As we can see from the diagram, a star which moves along its diurnal parallel will always be below the hori-zon and will never rise. This is completely understandable, since the declination of the star is negative. If the star had a negative declination still greater than 90 - (j) , its diurnal circle would be located still further below the horizon. Consequently, the entire celestial sphere of an observer lo- cated at a given latitude can be divided into three parts: (1) The portion of the celestial sphere with stars that never set . (2) The portion of the celestial sphere with setting and ris ing stars . (3) The portion of the celestial sphere with stars that never rise All three portions of the celestial sphere are shown in Figure 5,10- for an observer at a latitude of 60°N. The circumference is the plane of the celestial meridian, ZZ ' is the vertical line of the observer, PP ' is the world axis. The straight line BB ' is the section of the plane of the celestial meridian as cut by the diurnal circle of the star, touching the horizon of the observer but not /'+10 passing below it (a star whose 6 = 90° - (|) ) . This is the boundary of the region of stars that never set with that of the ones which rise and set for a given latitude of the observer. The straight line DD ' is the section of the plane of the celestial meridian as cut by the hour circle of the star in the Southern Hemisphere, touching the horizon but not going below it (a star whose declina- tion is - 6 = 90° - (j) ) . This is the boundary of the region of stars that never rise with that of the ones which rise and set for a given latitude of the observer. Motion of Stars at the Terrestrial Poles In order to get a better idea of the nature of the diurnal motion of stars at the terrestrial poles , let us construct a form of celestial sphere for an observer located at the North Pole. In this case, the altitude of the Pole above the horizon will be equal to the latitude of the observer. Since the observer is located at the North Pole, (j) = 90°N and consequently the altitude of the Pole above the horizon will be 90° (Fig. 5.11). t|31 nort cele the This will chan Nort the Hemi hori with The h eel stial celes mean move ge as hern hor iz spher zon 5 & < world estial pole t ial e s that paral the c Hemisp on , wh e ( and i.e . , 0° wil axis pole P' CO quato all lei t elest here ile t have all s 1 nev comci P coi incide r will stars , o the ial sp (havin he sta negat tars w er r is des wi ncides s with coinc depen horizo here r g posi rs whi ive de ith 6 e . th t wit the ids ding n an otat tive ch a clin > 0° he vert h the z nadir with th on the d their es . Al declin re loca ations ) will n ical line , i enith Z and Z' , while th e plane of t ir diurnal r altitudes w 1 stars loca ations ) will ted in the S will move b ever set and . e . , the the south e plane of he horizon . otation , ill not ted in the move above outhern elow the all those Motion of Stars at Middle Latitudes Let us examine the nature of the diurnal motion of the stars at middle latitudes, when < (f < 90°. (P'W Figure pearance of at a latitud to the incli axis , all st to the horiz celestial eq latitudes , a of stars ris the parallel which are lo these parall equator eith never set . 5.12 shows the ap- the celestial sphere e close to 45°. Due nation of the world ars move at an angle on (parallel to the uator). At middle considerable number e and set (between /^H s NK and DS) , Stars cated farther than els from the celestial er never rise or Fig. 5.11. Motion of the Stars at the Terrestial Poles region of never setting z stars p -0 Z region „f never risine stars Fig. 5.12. Motion of the Stars at Middle Latitudes . Fig. 5.13. Motion of the Stars at the Equator. 432 Motion of Stars at the Equator Since the latitude of an observer located on the Equator is equal to zero, it is clear that the world axis lies in the plane of the horizon and coincides with the meridian line on the plane of the horizon, while the terrestrial poles PP ' coincide with the north and south points N and S, respectively. Culmination of Stars The diurnal parallel of a star crosses the celestial meridian at two points (Fig. 5. 14, a). These points are called the culmination points. The moment of passage of a given star through the celestial meridian is called the moment of culmination, or it is said that the star culminates. The upper auZmination of a star is the moment when the star is at its greatest altitude above the horizon. The lower culmination of a star is the moment when the star is at its lowest altitude above the horizon. In the case of stars that set, the lower cul- I ^'L2 mination takes place below the horizon. Upper culmination of a star can take place on the southern portion of a meridian (between the south point and the zenith), and on the southern portion of the meridian (between the zenith and the north celestial pole), depending on the relationship between the a) ^^— lower ^ culmination'^ \. of the star ^, upper r»,cuimination of the star 'max' Fig. 5.14. Culmination of Stars on the Southern Section of the Meridian: (a) in the Apparent System of Coordinates; (b) in the Equatorial System. 433 latitude of the observer and the declination of the star. A star culminates on the southern part of the meridian (between the south point and the zenith) when the latitude of the observer is greater than the declination of the star, i.e., if ij) > 6 . A atar culminates on the northern part of th^ meridian (between the zenith and the north celestial pole) when the latitude of the observer is less than the declination of the star, i.e., if ^ < 6 . In Figure 5.14,b the celestial sphere has been sketched in simplified fashion, i.e., the circles of the horizon, equator, and parallels are not represented as circles but as diameters and chords. As we can see from the diagram, the latitude of the ob- server is greater than the declination of the star: NP > QM , i.e., (f) > 6 , so that the upper culmination of the star (point M') lies on the southern part of the meridian (between the zenith Z and the south point S ) . Let us determine the altitude of the star in this case . The altitude of the star (h) is the arc SM ' , but the arc SM ' = SQ ' + Q 'Z M'Z, and SQ ' = 90° ; Q'Z and M'Z By substituting these values, we will obtain h = 90° - (j) + (j; - ((f) - 6) or h = 90° - (|) + 5 . In Figure 5. 15, a the celestial sphere has also been sketched in a simplified form. Here the latitude of the observer ( NP ) is less than the declination of the star (Q'M'), i.e., <j) < 6 , so that the upper culmination of the star (point M' occurs on the northern part of the meridian (between the zenith and the point of the north celestial pole). Let us determine the altitude of the star in this /413 Fig. 5.15. Culmination of a Star on the Northern Section of the Meridian: (a) Coordinates of Upper Culmination; (b) Coordinates of Lower Culmination. 431+ case. The altitude of the star (h) is NM ' , but or I.e., NM' = 180° — M'Q' — Q'S ^=180° — S — (90 — 9), If the star does not set, we will sometimes be interested in its altitude at the moment of lower culmination. As we see from Figure 5.15,b, MQ = QN + NM , but MQ = 6 ; QN = 9 - (j) ; NM = h . By substituting the values of these arcs, we will obtain 6 = 90° - (|) + h, so that h = (() + 5 - 90°, i.e., the altitude of the star at the moment of lower culmination is equal to the latitude of the observer plus the declination of the star minus 90°. Problems and Exercises 1. What must be the declination of a star if at the latitude of Moscow ( ({) = 55°48') (a) it never sets, (b) it rises and sets, or (c) it never rises? Solution. (a) In order for a star never to set, we must have 6 > 90 - (|). If we substitute the value of the latitude of Moscow (55°48'), we will obtain 6 > 90 - 55°48', i.e., 6 must be greater than +34°12'. Consequently, all stars which have a declination greater than +34°12' will never set at the latitude of Moscow. Typical stars in this category are Capella, Alioth, Vega, Deneb , and Polaris. /^l^ (a) Stars rise and set, as we know, if the absolute value of their declination is less than 90° - (|> , i.e., 6 < 90° - ()> i . In our example, 6 < 34°12'. Stars in this category for the latitude of Moscow are Regulus , Arcturus , Altair, etc. (c) In order for a star never to rise, its declination must be equal to or greater than 90° - (fi and varies with the latitude of the observer, i.e., -6 >_ 90° - <|) . In our example, 6 must be equal to or greater than 3M-°12'. In addition, it must also be negative (inasmuch as we are talking about north latitude). (2) Calculate which of the following stars: Aldebaran, Alpherants, Capella, Sirius , Procyon, Arcturus) will never rise, rise and set, and never set at the latitude of Leningrad ( (j) = 59°59'N). 435 3. Calculate the altitude of the star Dubkhe at Moscow ( (j) = 55°U8') at the moment of upper culmination. 4. At what altitude does the star Sirius culminate (upper culmination) at Leningrad? 5. Show mathematically that all stars whose 6 > do not set at the Poles, while those which have 6 < never rise. 3. The Motion of the Sun The Annual Motion of the Sun The Sun participates in the diurnal motion along with all the other stars . The apparent diurnal motion of the Sun is also the result of the diurnal motion of the Earth in rotating on its axis. However, the Sun also has its own so-called intrinsic motion in the course of a year, called the annual motion of the Sun. The annual motion of the Sun is difficult to observe directly. However, if the stars were visible in the daytime, and we were to observe the mutual positions of the Sun and stars for a certain period of time, we would see that the mutual positions of these bodies would change in the course of time, while the mutual positions of the stars and constellations in the sky would not change. The direction of this intrinsic annual motion of the Sun is opposite to the diurnal motion of the stars, i.e., from west to east. The annual motion of the Sun is apparent (as is the diurnal motion), and occurs as the result of the annual rotation of the Earth around the Sun. As we did in describing the diurnal motion of the sky and stars, we will consider that the Sun is moving and the Earth stands still. Due to the existence of so-called annual motion of the Sun, the diurnal motion of the Sun has some unusual aspects, such as: /415 (c) The meridional altitude of the Sun changes constantly in the course of a year. 436 Ecliptic. In the course of its intrinsic motion, the center of the Sun moves along a great circle of a sphere called the ecZ-iptic (Fig, 5. 16, a). The plane of the ecliptic intersects the plane of the celestial equator at an angle of 23°27' at two points: at the point of the vernal equinox (y) and the point of the autumnal equinox (.0) . Tropic year. The Sun completes a journey around the ecliptic (through 360°) in 365.2422 mean days. The interval of time between successive passages of the center of the Sun through the point of the vernal equinox is called the tropic year. Sidereal year. In the course of its annual motion, the Sun makes a full rotation relative to the stars in a period of time somewhat longer than the tropic year (i.e., in 365.25635 days). This time interval, equal to the period of time required for the Earth to rotate around the Sun, is called the si-dereat year. After this interval, the Sun will have returned to its original position among the stars. Motion of the Sun Along the Ecti.pt'ia On cr OS se s This dat passes t equal to passes i increase June 22) called t is at it March 21, in the course of its annual motion, the Sun the celestial equator at the point of the vernal equinox, e is called the date of the vernal equinox. When the Sun hrough the point y, its declination and right ascension are zero. After March 21, the Sun continues its motion and nto the Northern Hemisphere, and its declination begins to (i.e., becomes positive). Thus, after three months (on the Sun is at the point K (see Fig. 5. 16, a), which is he point of the summer solstice . At this point, the Sun s highest position above the celestial equator. The OL'ISO' June 22 Mar 21 Fig. 5.16. Annual Motion of the Sun: (a) Motion along the Ecliptic; (b) Coordinates on the Dates of the Equinoxes and Solstices. 437 declination of the Sun at this ascension is 90° or 6 hours. F noon remains nearly constant , i which corresponds to the conste the date of the summer soZstiae and sets on this date will be a east and west points on the hor solstice , the Sun begins to app declination begins to decrease, sects the celestial equator at in the constellation Libra). point is +23027', and its right or several days , its altitude at .e., +23°27', so that the point K, nation Capricorn, has been named The points where the Sun rises t their maximum distances from the izon. After the date of the summer roach the celestial equator, its and by September 23 it again inter- the point of the vernal equinox (fi^. When the Sun passes through the point of the vernal equinox (fi_) , its declination becomes equal to zero, while its right ascension becomes 180° or 12 hours. TABLE 5.1 Date Vernal equinox Summer solstice Autumnal equinox Winter solstice Occurs on March 21 June 22 September 23 December 22 c oord Lnate s Declination (6) Right Ascension ( a ) 0° 0° + 23°27 ' 90° or 6 hours 0° 180° or 12 hours -23°27 ' 270° or 18 hours September 23 is called the date of the autumnal equinox. All of the events of the date of the vernal equinox are repeated on this date . After September 23, the Sun passes into the Southern Hemisphere and its declination becomes negative. On December 22, the Sun is at its lowest position relative to the celestial equator and is at /'^■17 the point of the winter solstice (the point L, in the constellation Leo). This date is called the date of the winter solstice . On the date of the winter solstice, the Sun has a declination of -23°27', while its right ascension is 270° or 18 hours. The points where the Sun rises and sets on this date are farthest south from the east and west points on the horizon. After December 22, the Sun begins its rise along the ecliptic, and on March 21 it has again risen to the point of the vernal equinox, where its declination and right ascension are once more equal to zero. Thus, we can draw up a special table for the annual motion of the Sun along the ecliptic, showing its coordinates (Table 5.1; Fig. 5 .16 ,b) . 438 In the course o fay ear, as it moves through the sky ( among the stars ) , the Sun passes through 12 constellations, calle d the signs of the zodiac. The y have received this name because the majority of them bear the names of animals (Aries, the Ram; Taurus , the Bull, etc . ) , and the word zodn in Greek means "animal" . As the Sun moves among the stars in the course of a year. it is in the foil owing positions: on the date of the vernal eq uinox (March 21) , in the constellation Pisces (the Fishes); on the date of th e summer solstice June 22, in the constellation Gemini (the Twins ) ; on the date of the autumnal equinox (September 23), in th e constellation Virg o (the Virgin ) , and on the date of the winter solstice ( December 22) , in the constell ation Sagittarius (the Archer) • Diurnal Motion of the Sun The Motion of the Sun at the North Vole During the other half of the year, when the Sun has a negative declination, it will be below the horizon as seen from the North Pole. Therefore, there are six months of day and six months of night at the terrestrial poles. North Pole on the date of the summer s The altitude of the Sun on that date i ation, i.e., 23°27'. At the South Pol mum altitude on the date of our winter urn altitude above the horizon at the summer solstice, i.e. , on June 22. t date is equal to its maximum declin- The Sun reaches its maxim on the date of the summer s ' ■" ' ' " te is equal to its maximum declm Pole, the Sun reaches its maxi- 3lstice, i.e. , December 22. Heavenly bodies do not set if their declination is equal to or more than 90° minus the latitude of the observer, i.e. if 6 >_ 90° - (j) . This situation also applies to the diurnal motion of the Sun. If, for example, the observer is standing at a latitude of 76°N (between the North Pole and the Arctic Circle), then according 439 to the condition set forth above the Sun will not set after the date when its declination is equal to or more than 90° - (f> , i.e. more than 90-76° = +x4° . This phenomenon (to cite a specific example) begins on April 26. After April 26 the Sun will rise higher and higher above the horizon. The Sun reaches a maximum altitude above the horizon on the date of the summer solstice. After June 22, the Sun will dip toward the horizon but will still not set. When its declination is again equal to 90° - (|) , the Sun will touch the horizon. After August 19, the Sun's declination will be less than 90° - (j) , i.e. 6 < 90° - (j) and for a fixed time it will appear to an observer located at this latitude as a rising and setting star. This phenomenon will continue until the declination of the Sun is not equal to or more than 90° - (j) , i.e. 6 >_ 90° - cj) and will have a sign opposite to that of the latitude (i.e. the latitude is positive and the declination is negative). For an observer located at 76°N this phenomenon begins on November 3. Beginning on November 3, for an observer located at a latitude of 76°, the Sun will not set, since -6 > 90° TABLE 5.2 /419 ter Latitude of - the position Sp ring Summer Autumn Win Beg. Duraticn Beg. Duration Beg . j Duration Beg. Duration in degrees date in days (fete in days date in days date 1 in days 68 4.1 143 27.V 53 19.VI1 144 lO.XII B5 70 17.1 120 17.V 72 28.VI1 121 26.XI 52 72 26.1 103 9.V 88 5.VIII 104 17.X1 70 74 3.11 88 2.V 102 12.V1I1 90 lO.XI 85 76 9.II 76 26.1V 115 19.V11I 76 3.X1 98 78 15.11 64 20. IV 127 25.V111 63 28.X 111 80 22.11 51 14.IV 139 31. VIII 52 22.x 123 82 27.11 41 9.1V 150 6.1X 41 17.X 133 84 4.111 31 4.1V 159 lO.lX 31 ll.X r44 86 9.1II 2! 30.111 169 15.1X 21 5.x 155 88 14.111 11 25.111 179 20.IX 10 30.1X 165 90 19.111 19.111 189 25.1X J 25. IX 176 ^The dates for nonsetting, nonrising or rising and setting of the Sun are given, taking into account the phenomenon of refraction. M-40 Thus , let us sum up the diurnal motion of the Sun in the course of a year for an observer located at a latitude of 76° (between the North Pole and the Arctic Circle). (1) From April 26 to August 19, i.e., for 115 days, the Sun does not set. This period of time is called the poZ-ar summer. (2) From August 19 to November 3, i.e. , for 76 days, the Sun will rise and set daily and the period of daylight will diminish each day. This period of time is called the potar autumn. (3) From November 3 to February 9, i.e., for 98 days, the Sun will not rise for the observer. This period is called the polar winter. ( M- ) From February 9 to April 26 i.e. , for 76 days the Sun will rise and set daily and the period of daylight will increase each day. This period of time is called the polar spring. The dates for the start of the seasons depend on the latitude of the observer. The dates given in our example are for 76°N. We have provided a table for the seasons as a function of the latitude of the observer (Table 5.2). Motion of the Sun above the Aratie Circle As we already know, the latitude of the Arctic Circle is equal to <|) = 66°33', or the complement of its latitude to 90° is 23°27'. Therefore, on the date of the summer solstice (June 22), 6 = /420 90° - (f) , and on the date of the winter solstice (December 22), 6 = (j) - 90° . On these dates at the Arctic Circle, the center of the Sun touches the horizon on June 22 at the north point at the moment of lower culmination (point N, Fig. 5.17), and on December 22 at the south point at the moment of upper culmination (point S). During the rest of the year, the Sun will rise and set daily. The period of daylight will increase daily from December 22 to June 22; after June 22, it will decrease. The Sun reaches its maximum altitude above the horizon on June 22. It will be: h = 90- 66°33' + 23°27' ^^&°S^'. Motion of the Sun at Middle Latitudes Knovjing the maximum declination of the Sun (equal to 23°27'), it is not difficult to calculate the latitudes of the observer on the Earth's surface where the Sun will be a rising and a setting heavenly body during the year. From the conditions for the rising and setting of heavenly M-41 ^ Motion of the Sun at the Tevvestviat Equator . In order t of the Sun at t have sketched s the Sun at the circles are div standable , sine server located to the equator, half by the hor the day and bel year the Sun ri length . o better und he Equator, chematically equator. As ided in two e we already at one of th Since the izon, the Su ow the horiz ses and sets erstand the nature of the diurnal motion let us analyze Figure 5.18. Here we the diurnal trajectories (circles) of 7421 we see from the drawing, all the diurnal by the horizon line. This is under- know that the true horizon of an ob- e poles is situated at an angle of 90° daily circles of the Sun are divided in n will be located above the horizon half on half the day, i.e. during the whole , and the days and nights are equal in Intrinsic Motion of the Moon The Moon, participating with all the heavenly bodies in the sky in the diurnal rotation of the celestial sphere, has its own intrinsic motion. Fig. 5.17. Diurnal Motion of the Sun Above the Arctic Circle . Fig. 5,18. Diurnal Motion of the Sun at the Equator. 442 If we observe the Moon for one night, we can easily see that it travels like the Sun in the sky (relative to the stars). The apparent motion of the Sun is the result of the Earth's motion around the Sun; the Moon actually moves around the Earth. Direotion and Rate of the Moon ' s Motion The Moon moves along the celestial sphere from west to east, i.e., in a direction opposite the diurnal motion of the celestial sphere . The great circle along which the Moon completes its motion around the Earth has the shape of an ellipse and is called the Moon's orbit. The Moon's orbit is intersected by the solar ecliptic at an angle of 5°08' (Fig. 5.19). The two diametrically opposed points at which the Moon's orbit is intersected by the solar ecliptic are called the nodes of the orhi-t. The Moon completes a full revolution along its orbit relative to the stars in 27.32 days. This time interval is called the sidereal (stellar) month. Thus, it is easy to calculate that during one day it moves 13.2°. Its hourly shift relative to the stars is approximately 0.5°. The motion of the Moon in its orbit may be studied in relation to the Sun (which has, as we know, its own motion). The period of revolution required for the Moon to return to a previous position in relation to the Sun is called the synodic /'+22 month. It is approximately 29.5 mean solar days. The Moon, revolving around the Earth, accompanies it in its motion around the Sun. The Moon is 356,000 km from the Earth at perigee (the closest point to the Earth) and i+07,100 km from the Earth at apogee (the most distant P point from the Earth). Phases of the Moon The Moon, like our Earth, is an opaque body which illuminates the Earth's surface with reflected sunlight. The illumination of the Earth's surface by the Moon, as we can see , is not always the same. At different times the Moon is visible in the form of a luminous disk, in the form of a luminous half-disk, or a cres- cent. There is a time when the Moon is entirely invisible. The Moon has various phases. The Fig. 5.19. Orbit of the Moon, t+43 periodically repeated change in the shape of the Moon is called the change ■in lunar phases. In order to explain the cause of lunar phases , let us look at Figure 5.20. At Point (in the orbit's center) is our Earth; at Points 1-8 we show the Moon in various positions in its revolution around the Earth, Below the figure we show the shape of the Moon in those eight positions for an observer located on the Earth's surface . When the Moon is in Position 1, its phase is new moon. An ob- server on the Earth's surface during this phase does not see the Moon, since its nonilluminated side is turned to the Earth and is located at an angular distance of not more than 5° from the Sun. In Position 2, the Moon is visible on the Earth in the form of a narrow crescent only during evening hours. When the Moon is in Position 3, its phase is called the first quarter and an observer sees it in the shape of half an illuminated disk. This phase is called the first quarter because at this time a quarter of the entire lunar surface is visible. During first quarter, the Moon is visible in the east at noon, in the south about 1800, and in the west at midnight. After first quarter, the illuminated part of the lunar disk begins to increase (Position 4); in Position 5, the entire lunar disk is illuminated. This is the half-moon phase. In this phase, the Moon is visible all night . After half moon, the illuminated part of the lunar disk begins/i4-23 to decrease from the right side of the disk (in Position 5), and in full moon last quarter Fig. 5.20. Phases of the Moon 44-4 Position 7 it reaches the phase of last quarter. In last quarter, the Moon is visible in the east at midnight , in the south about 0600, and in the west at noon. After last quarter, the illuminated part of the lunar disk diminishes even more and in Position 8 it is already visible in the form of a narrow crescent (on the left side of the disk); it then becomes invisible again, i.e., the phase of new moon again sets in. Nature of the Motion of the Moon around the Earth In view of the fact that the Moon, in its motion around the Earth, is sometimes closer to the Sun than the Earth is and some- times farther away, it receives acceleration from the Sun which is sometimes more and sometimes less than that of the Earth. As a result, the motion of the Moon around the Earth is complex, since the above factors not only change the shape and dimensions of the lunar orbit, but its position in space. The Moon, completing one revolution in its orbit, intersects the plane of the celestial equator twice; in the course of one sidereal month, its declination changes from a maximum positive value to a maximum negative one . The maximum declination of the Moon in a period of one month may be +28°27' and the minimum may be -28 27'. Location of the Moon Above the Horizon The location of the Moon above the horizon depends on the so- called waxing of the Moon (time elapsed after new moon) and the season . In the Northern Hemisphere, during the summer and the phase of full moon, the Moon is located comparatively low and for a short time above the horizon; in the winter, during the full moon, it is visible all night and rises rather high above the horizon. This depends on the declination of the Moon. In winter in the Northern Hemisphere, the full moons occur with a positive .declination, while the full moons in summer occur with negative declination. 445 5. Measurement of Time Essence of Calculating Time Measurement of time is the result of the presence of motion in universal space. Time and motion are synonymous. I'f motion ceased in Nature and in the Universe, there could be no discussion of time. It is entirely understandable that to measure time, some constant and uniform motion must be used. The rotation of the Earth on its axis (or, as a result of this, the apparent rotation of the celestial sphere around the world axis) could be such a motion. Special observations over a very long time interval have shown that the duration of the Earth's rotation around its axis has not changed even by a fraction of a second. This characterizes very well the constancy and uniformity of the Earth's rotation. In the practice of aviation, the following kinds of time must be studied and used: sidereal, true solar, mean solar, Greenwich, local, zone and standard time. S i derea 1 T ime 7425 Time measured on the basis of the apparent motion of heavenly bodies (stars) is oatted sidereat time. This time may be measured by the hour angle of some heavenly body with respect to the meridian of the observer. However, for convenience, it is advantageous to take the hour angle not of some star but of the point of the vernal equinox from which the right ascension is read. vernal equinox t y The western hour angle of the point of the is called sidereal time and is represented by the letter S (S = "ty) in Figure 5.21. Sidereal time is equal to the angle and the rij Fig. 5.2 the Poin Equinox . 1. Hour Angle of t of the Vernal star : Y sum of the hour ^ght ascension of a t . •Y Side rea 1 da val between two culminations of vernal equinox i day. The moment mination of the was taken as the real days . At t time is equal to days are divided hours , each hour sidereal minutes into 60 sidereal ys. The time inter- successive upper the point of the s called a sidereal of the upper cul- vernal equinox point beginning of side- his moment, sidereal 00:00:00. Sidereal into 24 sidereal is divided into 60 , and each minute seconds . 446 Sidereal time does not have a date and therefore in calculations when the sidereal time is more than 21+: 00, there is only a surplus of time above 24:00. Example. In calculations, there is a sidereal time of S = 29:20. Discarding 24-:00, we obtain a sidereal time of 5:20. The practical application of sidereal time. Sidereal time is not used in everyday life, but as the basis of time signals. In every astronomical observatory, there are special clocks which run according to sidereal time. In aviation, sidereal time must be used in observing stars for determining the position of the aircraft or the position line of the aircraft. In the aviation astronomical yearbook, the sidereal time is given for each date and each hour of Greenwich mean time. Therefore, the sidereal time for any moment at any point on Earth may be deter- mined by means of the aviation astronomical yearbook on the basis / 'j- 2 6 of Greenwich time. In everyday life, solar time rather than side- real time is used, since on the whole man's activity occurs in the daytime hours. True Solar Ti me True solar time t0 is the time measured on the basis of the diurnal motion of the true Sun. True solar time is measured by the western hour angle (t^^) of the center of the true Sun. True solar days are the time intervals between two successive upper culminations of the center of the true Sun. The moment of the upper culmination of the center of the true Sun is taken as the beginning of true solar days. At the moment of upper culmination, when the hour angle of the Sun is zero, the true solar time is 00:00:00. In proportion to the diurnal motion of the Sun, its hour angle increases; the true solar time also increases. At the moment of the lower culmination of the Sun, the true solar time is 12:00; when the center of the Sun is again in the position of upper culmination, the true solar time is 24:00. After this, new days begin. The duration of true solar days changes in the course of a year. This occurs because the true Sun moves during the year along the ecliptic, which is inclined to the celestial equator at an angle of 23°27' and is not a circle but an ellipse. For this reason, the daily shift of the Sun to the east is different on different days of the year. This shift is at a maximum near the solstices, 447 when the Sun moves parallel to the equator. On the other hand, near the equinoxes the shift to the east is smallest. The Sun lags behind the motion of the stars by either a very great or a very small value; the duration of true days changes all the time . In using true solar time in everyday life, it would almost be necessary (as a result of its nonunif ormity ) to regulate clocks every day, moving them ahead and back. This would be extremely inconvenient . In view of the inconvenience of calculating time on the basis of the true Sun, time in everyday life is calculated on the basis of the so-called mean Sun. The mean Sun is the imaginary or real Sun moving uniformly along the celestial equator in the same direction that the true Sun moves along the ecliptic, i.e., in a direction opposite to the diurnal motion of the celestial sphere . Mean Solar Time /427 Time calculated on the basis of the diurnal motion of the mean (imaginary) Sun is called the mean solar time (m). Mean solar time is measured by the western hour angle (t^) of the mean (imaginary) Sun. Mean solar days are the basic unit of mean solar time. Mean sotar days are the time intervals between two successive upper culminations of the mean Sun. They are divided into 24 mean hours, each hour is divided into 60 minutes and each minute into 60 seconds. The duration of mean solar days is constant. Mean civil time (m^). Since at the mom.ent of the upper cul- mination of the mean Sun its hour angle will be zero, the mean solar time at this moment (mean noon) will also be zero. In everyday life, with a 24-hour reckoning of time, this is very inconvenient since in this case the beginning of mean days comes at noon. Therefore, in everyday life, so-called civil time (which is different from mean solar time by 12 hrs . ) is used as a variety of mean solar time. Mean midnight, i.e. the mean time equal to the western hour angle of the mean Sun plus 12 hrs, is taken as the beginning of mean civil days : 448 where mc is the mean civil time, and m is the mean solar time. The plus sign is used when the mean time is less than 12 hrs and the minus sign is used if the mean time is more than 12 hrs. Example . Determine the mean civil time if the western hour angle of the mean Sun (mean time) is 6:00. Solution. mc = m + 12:00 = 6:00 + 12:00 = 18:00 Local Civil Time the h But t momen on th Earth In Fi plane Sun a on th point the h by th at th it is There mer id Side our he h t fo e Ea ' s s gure of t a e Ea on our e an 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 me , f a le ( e ave urf a it w e sh wing mom urf a th 's f th Mav s ex at t e sa e di true s heaven t) of nly bo ce var ill be ow a c Let ent : ce and surf a e mean and th pre s se he ang me phy f f eren olar ly b a he dy f ies lar e le s poi Radi rad ce . Sun e ho d by le A s ica t . and me ody or avenly rom the in valu ge and t ial sp us P^A ius P^B From t for th ur angl the an PnMav i 1 momen an solar time are measured by the point of the vernal equinox, body calculated at one physical meridians of various points e. For some points on the for others it will be small, here whose equator lies in the be the position of the mean is the meridian of one point / M-2 i is the meridian of a second he figure, it is apparent that e first point (ti) is expressed e for the second point (tz) gle BP^Mav- From this figure s greater than the angle BFj-iM^v- t the time at different Time calculated relative to the meridian of midnight of some point on the Earth's surface is called the toaaZ aiviZ time and is represented by T -[ . Local time may be sidereal or solar. It will be the same for all points lying on ^^^^ one meridian (i.e. having the same geographic longitude). G reenw i ch T i me Local civil time calculated from the Greenwich meridian is called Greenwiah time and is repre- sented by Tqj^ . In astronomical yearbooks, the times of some celestial phenomena are given as well as the astronomical values necessary for practical calculations on the basis of Green- wich t ime . The relation between local av im Fig. 5.22. Determining Local Time on the Basis of the Mean Sun . 4tt9 civil time and Greenwich time. With a knowledge of the Greenwich time and longitude of a place expressed in time units , it is simple to determine the local civil time and vice versa. The local civil time is equal to the Greenwich time plus or minus the longitude of the place, i.e. ^« = ^Qi:^W" Here the plus sign is used if the longitude of the place is east and the minus sign is used if the longitude is west. Exampte. 1. Greenwich time is 14:15. Find the local time for Moscow (Xj; = 2:28). Solution. Substituting the values for Tq^ and Ag in the formu- la Ti = Tq^ + XE, we obtain the local time T^ = 14:15 + 2:28 = 16.43. Example. 2. The Greenwich time is 10:42. Find the local time for Washington (Aw = 05:08). Solution. T^ = Tq-p - A^ = 10:42 - 5:08 = 5:34. When solving practical problems in astronomy, it is often /^29 necessary to change local time to Greenwich time. ^GR= ^« ± J. The plus sign is used if the longitude of the place is west (A;^) and minus if the longitude of the place is east (Ag), Example. 1. The local time in Ryazan (Ag = 2:39) is 1430. Determine the Greenwich time. Solution. Substituting the values for T-[_ and Ag into the formula Tq^ = T^ - Ag, we obtain Tq^ = 14:30 - 2:39 = 11:51. Example . 2. The local time in San Francisco (A^ = 8:09:44) is 1520. Determine the Greenwich time. Solution. Substituting the values for Tj_ and Ay into the formula Tg^ = Ti + Aw, we obtain Tq^, = 1520 + 8:09:44 = 23:09:44. Time difference on two meridians. It is easy to imagine that the difference in local time on any two meridians is equal to the difference in their longitudes. In Figure 5.23 the diameter EQ is the Greenwich meridian, the diameter BA is the meridian of a point on the Earth's surface, the 450 diameter DC is the meridian of a second point on the Earth's surface, Mg^ is the location of the mean Sun at a certain moment and the diameter KM^v is the circle of the Sun's declination. In Figure 5.23 it is evident that the hour angles of the mean Sun (tiQav and t zq ^v^ measured from both local meridians are dif- ferent from one another by a value equal to the difference in the longitudes of these meridians, since ti - t2 = X2 - ^i. Hence it follows that the local time on these meridians will differ by the difference in the longitudes of these two meridians. Example . Let the local time be Ti = 1204 at a point having an east longitude of Ag = 2:35. What is the local time at this moment at a point having an east longitude Xg = 4:35? Solution. Let us find the difference in the longitudes of these two points AX = 4:35 - 2:35 = 2:00. Let us find the local time for the point having an east longi- tude Xe = 4:35. Ti = 12:04 + 2:00 = 14:04. In solving such problems , it is important to remember that the /430 larger the east longitude of the point, the larger will be the local time at this point; the smaller the east longitude of the point, the smaller the local time . Zone Time We have already seen that each meridian of the Earth's surface has its own time. If we take Khabarovsk, whose longitude is 135°5' (9:00:20) its local time is 5:29:48 different from the local time of Moscow, which has a longitude of 37°38' (2:30:32). e(S) Fig. 5.23. Time Difference on Two Meridians Sine surface h it is too time in e when movi would be move the for every 1 hr for On the ot east to w have to b Therefore last cent Europe ha single ti This time meridians e each p as its o inconve veryday ng from ne cessar hour han degree every 15 her hand est , the e moved , s ince ury , the ve begun me in th is meas of the omt wn ( 1 nient life . west 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 so-called "Paris time", was introduced, in Italy "Rome time", and in England "Greenwich time", etc. The introduction of a single conventional time in these countries did not create great difficulties , since the local time of any meridian of these countries (in view of their small area relative to the meridian of the conventional time introduced) differed insig- nificantly (only several minutes in all). If we take England and France for example , their outermost populated points (to the east and west) are situated in a range of 7-8° from their respective meridians (Greenwich and Paris): the time difference amounts to a total of 30 min. If we take such a country as the USSR, we know that the difference in the longitudes of its eastern and western boundaries amounts to more than 10 hrs in time. However, in pre- Revolutionary Russia, a common Petersburg time (the local time of the Pulkovo Observatory meridian) was introduced only for railroads. This time was 00:28:58 behind Moscow local time (local time of the Moscow University Observatory meridian). The introduction of a common time in individual countries partially facilitated its calculation within each country, but it did not solve the problem on an international scale. The problem of calculating time was solved most successfully after the intro- duction of a zone time system. In some countries, this system was introduced at the end of the /^3j- 19th and beginning of the 20th century. In Russia, the zone time system was introduced only after the Revolution, on July 1, 1919 by a special decree of the Soviet Government. Essence of the zone time system. The entire Earth is divided into 24 hour zones. The outer meridians (boundaries) of each band are 15° of longitude (1 hr in time) apart from one another. The zones are numbered from west to east from the zero zone to the 23rd zone, inclusive. The zone included between the meridians 7°30'N and 7°30'E is taken as the zero zone, i.e. the initial zone. The Greenwich meridian, which has a longitude of 0°, is the mean meridian of this zone. Obviously, the first zone will be located between the meridians X = 7°30'E and X = 22°30'E and the mean meridian of this zone has a longitude of 15°; the second zone is located between the meridians X = 22°30'E and X = 37°30'E and the mean meridian of this zone has a longitude of 30°, etc. At all the points located within the limits of the same zone, the common time (time of the given zone) which represents the local time of the mean meridian of the given zone is taken. Such a con- ventional time is called the zone time (T^). In the zero zone, the time is calculated on the basis of the 452 local time of the zero (Greenwich) meridian. In the first zone, the time is calculated on the basis of the local time of the meridian having a longitude of 15°E. In the second zone, it is calculated on the basis of the local time of the meridian having a longitude of 30°E, etc. Since the mean meridians of two adjacent zones are 15° of longitude apart from one another, the difference between the zone time of adjacent zones is one hour. The number of each zone shown by how many hours the time in this zone is ahead of Greenwich time. For example, in the case of the time of the fourth zone, this means that its time is 4 hrs ahead of Greenwich time (Supplement 4). The introduction of zone time greatly facilitated the calcula- tion of time on an international scale, since the minute and second hands in all the zones indicate the same number of minutes and seconds. The hour hands must be moved a whole hour only when moving from one zone to another. When the boundaries of the hour zones were determined, the boundaries of states, regions and cities as well as natural boundaries (e.g. rivers, etc) were taken into ac- count. If the boundaries of the hour zones had been determined strictly according to the meridians, calculating the zone time (e.g. in Moscow, which is located on the boundary between the second and third zones) would have to be done on the basis of two zones: in the western part of the city on the basis of the second zone, and in the eastern part of the city on the basis of the third zone. I.e. the time difference in the two parts of the city would be one hour and in crossing the boundary the hour hands would have to be / 43 2 moved one hour. In view of this, the difference between the local time of the outer points of this zone may be somewhat more or less than 30 min relative to the zone time. Standard Time In the Soviet Union, on the basis clocks were moved ahead one hour beg" Since ^^ -' - -^ - - c bince this time, the entire USSR reckc called standard time. Thus, the zone ^^...v, .,^^ ^..^^^--^ ^^v,,^.^ ^ „^ , i.e. each zone lives not on the basis of its zone time but rather on the basis of the time of the adjacent eastern zone. For example, Moscow, which is located in the second zone, lives according to the time of the third zone . The time running 3 hrs ahead of Greenwich time (time of the 3rd zone) is called Moscow time. All the railroad, water, and air routes of communication in the Soviet Union operate according to Moscow time . 453 Relation between Greenwich, Local and Zone (Standard) Time When solving practical problems of aircraft navigation on the ground and in the air, it is very often necessary to convert from one form of time to another. These problems may be solved correctly only if 'the crew conscientiously masters the essence of calculating time. To facili- tate the work of the crew in solving such problems, there are formulas for converting time from one form to another. Converting Greenwich time to Mean time. Zone time (T^.) is equal to Greenwich time (Tq^) plus the number of the zone: In solving problems for the USSR, the number of the zone is used, taking into account the standard time, i.e. 1 hr . later. Example . What is the zone (standard) time in Novosibirsk when clocks in Greenwich show 12:00? Solution. On the basis of a map of hour zones or on the basis of a list of the most important populated points, let us find the number of the zone in which Novosibirsk is located. Taking into account that the standard hour in Novosibirsk , the clocks run ac- cording to the time of the 7th zone (6th zone + 1 hr ) , let us find the zone time on the basis of the formula r2= 7-j^g,+ Ar=12h+7h=19h Converting Zone Time to Greenwich Time. Greenwich time is /^33 equal to the zone (standard) time minus the number of the zone (taking into account the standard time): Example . What is the Greenwich time when the clocks show 17:00 in Krasnoyarsk? Solution. On the basis of a map of hour zones or on the basis of a list of populated points , let us find the number of the zone where Krasnoyarsk is located. Taking into account the standard time in Krasnoyarsk, the clocks run on the basis of the time of the 7th zone (6th zone + 1 hr ) . Let us find the Greenwich time. Tq^ = T^-N = 17:00 - 7:00 = 10 : 00 . 454 Converting Zone (standard time) time to local time. Local time is equal to the zone time minus the number of the zone plus or minus the longitude of the place (plus is used when the longitude is east, minus when the longitude is west): This formula assumes that the hour zones are counted from to 24 eastward from Greenwich. Example . What is the local time in Omsk when the zone (standard) time there is 16:00? Solution. Using a map of hour zones or a list of the most important populated points , let us find the number of the zone (taking into account the standard time) where Omsk is located and its longitude. Taking into account the standard time in Omsk, the clocks run according to the time of the 6th zone (5th zone + 1 hr); the longitude of Omsk A = 73°24'E (4:53:36). Let us find the local time T^ = T^ - N + Xg = 16:00 - 6:00 + 4:53:36 = 14 : 53 : 36 . Converting local time to zone time (standard time). Zone (standard) time is equal to the local time plus the number of the zone (taking into account the standard time) plus or minus the longitude of the place (plus is used when the longitude is west and minus when the longitude is east). Example . What is the zone (standard) time in Irkutsk when the local time there is 18:00? Solution. Using a list of the most important populated points, let us find the number of the zone (taking into account the standard time) and the longitude of Irkutsk. Taking into account the standard time, Irkutsk is located in the 8th zone (7th zone + 1 hr ) ; the longitude of Irkutsk A = 104° 18 'E (6 :57 :12) . Let us find the zone standard time T2; = T^ + N - Ag = 18:00 + 8:00 - 6:57:12 = 19:02:48. Measuring Angles in Time Units Since the values of hour angles and right ascensions are used for measuring time, it is often more convenient to express these 455 values in time units rather than in degrees . Also it is often necessary to express the longitude of a place in time units. To convert hour angles and right ascensions as well as longi- / ^3^■ tude from degrees to hours and back again, the following equations must be used : 24 hr = 360°; 1 hr = 15° or 1° = 4 min ; 1 min = 15 ' or 1 ' = 4 sec; 1 sec = 15" or 1" = 1/15 sec. These equations are based on the fact that the celestial sphere (or the Earth on its axis) makes a complete revolution in 24 hrs , which corresponds to 360°. To convert hour angles, right ascensions and the longitude of a place from degrees to hours, the following must be done: (1) Divide the number of degrees by 15 and obtain whole hours. (2) Multiply the remainder from dividing the degrees and ob- tain minutes of time. (3) Divide the number of minutes of arc by 15 to obtain whole minutes of time, which must be added to the minutes of time already obtained, and obtain the total number of minutes of time. (4) Multiply the remainder from dividing the minutes by 4 and obtain seconds of time . (5) Divide the seconds of arc by 15 and obtain an additional number of seconds. Add these seconds to the preceding ones and obtain a total number of seconds. (6) Discard the remainder of seconds of arc when it is less than 8; if it is greater than 8, consider it as 1 sec of time. Example, 1. Express the hour angle 163°57'35" in hours. Sol Ution. 150° = 10 : 00 13° = 0:52 45 ' = 0:03 12 ' = 0:00:48 00^30" = 0:00:02 Total: 10:55:50 To convert hour angles , right ascensions and longitude from hours to degrees , the following must be done : (1) Multiply the hours by 15 and obtain the radii. (2) Divide the minutes by 4 and separate out the whole degrees. 456 (3) Multiply the remainder of the minutes of time by 15 and obtain minutes of arc. (4-) Divide the seconds of time by 4 and separate out minutes of arc . (5) Multiply the remainder of the seconds of time by 15 and obtain seconds of arc. Example . 2. Express an hour angle of 11:27:15 in degrees. Solution. 11 hr = 155° 24 min = 6° 3 min = 45 ' 12 sec = 3 ' 3 sec = 45 ' Total : 171°48 '45" Time Signals / ^35 Accurate time in astronomical observatories is determined by means of astronomical observations of the culmination of heavenly bodies . Special transit instruments are used for this purpose. A transit is mounted in such a way that its main part (the terrestrial telescope) is always located in the plane of the meridian. Such a location of the terrestrial telescope permits the observation of a heavenly body only at the moment when it crosses the meridian, i.e. at the moment of culmination. Since the Sun is at the point of upper culmination (crosses the southern part of the meridian) at true noon, it can be observed only when the hour angle of the true Sun is zero. Therefore, when the Sun passes through the terrestrial telescope of the transit instrument, the moment of true noon on the given meridian will be recorded. Knowing the precise time of the moment of true noon and com- paring it with the actual indication of the clock at the moment of observation, it is possible to check the clock and determine its error. In the field of vision of a transit, vertical lines are drawn which permit more accurate determination of the moment that stars cross the meridian. In astronomical observatories, the accurate time is determined on the basis of observing the passage of stars is based on the fact that at the moment that any star is at the meridian (we already know), the sidereal time will be equal to the right ascension of the star. Thus, these observations make it possible to determine the exact sidereal time. The time obtained is set on special clocks, which run according to sidereal time. They differ from normal clocks in that (by the means of a special control) they run 3 min and 56 sec ahead of normal clocks per day. 457 Once the exact sidereal time is obtained, the mean solar time is calculated at astronomical observatories and set on mean solar astronomical clocks , The time obtained will be the local mean solar time on the meridian of the observatory. When necessary it is always possible to con,vert this time to zone and standard time. The necessary accuracy in determining the true time at astronomical observatories is computed in hundredths /U36 of a second, and therefore the process of determining accurate time is much more complex than we have described here. For accurately determining the moments of the passage of the stars through the meridian, the moments of the culmination of the stars are automatically recorded at astronomical observatories. With these methods, the determination of the true time is accurate to 2 or 3 hundredths of a second. At every observatory, there are accurate astronomical clocks which are manufactured on special order. Special care is required for these clocks, since the continuous changes in temperature and atmospheric pressure strongly affect the steadiness of the oscillation period of the clock balance wheels. Therefore, astronomical clocks are kept in a special room where a constant temperature is maintained, and they are placed under a hermetically sealed bell jar where a constant atmospheric pressure is maintained. Organisati-on of Time Signals in Aviation Time signals in aviation are organized to ensure accuracy of aircraft navigation. The basic task of the time signals in aviation is the systematic checking of clocks and the guarantee of knowing the accurate time at any time of day. The presence of accurate time is especially important when using astronomical means of air- craft navigation. This necessitates knowing the accurate time not only on the ground , but in flight . In order for the crew to know the correct time at any time of day there must be master clocks with a constant daily speed. They are installed in the cockpit, at the weather station, or in other special places. 458 Master clocks are used to determine the correction of other clocks in the periods between the transmissions of accurate time signals . Master clocks are checked and set according to the correct time on the basis of accurate time radio signals transmitted by broadcasting stations of the USSR. The correction of the clocks and its verification on the basis of signals is recorded in a special log. A Brief History of Time Reckoning. Sometimes we hear the terms "Old Style" and "New Style". Systems for measuring and calculating large time intervals are called aatendars . The basis for time reckoning is the tropical year, i.e. , the time interval between two successive passages of the Sun through the point of the vernal equinox. The length of the tropical year is approximately equal to 365 days, 5 hours, 48 minutes and 46 seconds, but it is very inconvenient to use the tropical year for time reckoning, since it does not contain a whole number of days. Thus, for example, if we take midnight on January 1 as the be- ginning of one year, the second year will begin not at midnight of January 1 but on January 1 at 5:48:46 AM, the third year at 11:37:32 AM, etc. We may conclude that each year the beginning of the new year would be shifted by 5:48:46. Old Style (Julian) Calendar. For Romans, the year was original- ly lunar and consisted of 12 lunar months. The length of the lunar year was 355 days, i.e., their year was 10 days shorter than the accepted year at the present time. With such a time reckoning, the beginning of the new year shifted rather quickly from one month to another. If, for example, we take a time interval of 10 years, then 100 days were accumulated during this period, i.e. , the begin- ning of the new year shifted by more than three months. To eliminate these inconveniences, approximately every three years the year was lengthened by one month, i.e. , this year had 13 rather than 12 months. The Roman dictator, Julius Caesar, introduced a calendar re- form in 46 B.C. This calendar was called the Julian calendar and we now call it the "Old Style". The essence of it was that the duration of one year was considered to be 365 days rather than 355 days. In addition, in February (which was then considered the last month) an extra day was included every fourth year, i.e. , that year had 366 days. The addition of the extra day once every 4 years nearly compensated for the difference accumulated in 4 years (about 6 hr per year) and thus the constancy of the date of the vernal /437 459 equinox (March 21) was preserved, until the present time. This principle has been retained The extra day as we know, is now added in February: instead of 28 days there are 29 days in all the years which are divisible by 4-, e.g. 1956, 1960, 1964, etc. These years have been called leap years up until the present time. New Style (Gregorian) Calendar. The length of the so-called tropical year is, as we know, 365 days, 5 hours, 48 minutes and 46 seconds. The Julian year (on the average) is equal to 365 days and 6 hours. Thus, the Julian year is 11 min and 14 sec longer than the tropical year. Although this difference is small, over a large interval of time it may also cause the beginning of the year to shift. It is not difficult to calculate that in order to shift the date of the vernal equinox one calendar date (one day), almost 128 years (24 hr or 1440 min, divided by 11 min 14 sec) are required. The gradually accumulating difference in the second half of the 16th century amounted to 10 days. As a result, the date of the vernal equinox came (according to the calendar) not on the 21st, but on the 11th of March. As a result of the shift of the beginning of spring from 21 to 11 March, the holiday of Easter (which must be close to spring) graduallly moved toward the summer. This greatly disturbed the clery, who did not want to depart from their rules. The Roman Pope Gregory XIII Introduced a new calendar reform in 1582 by decree. The essence of this reform, i.e., the transfer to a new style of calendar, was the following: after October 4, 15 he ordered everyone to consider the date to be October 15 , rather than October 5, i.e., he ordered the 10 days accumulated over 1200 years to be dropped so as to return the date of the vernal equinox to March 21. In order to avoid the accumulation of an error in future, it was decided that every 400 years those three days which differentiate the Julian year from the tropical year be dropped. To do this, it was decided that three leap years in every 400 years be considered regular years, i.e, not on the basis of 366 days but rather on the basis of 365 days. In order to remember this more easily, the centurial years in which the numbers of the century wer divisible by 4 were taken as leap years (for example, of the centur years 1600, 1700, 1800 and 1900, only the year 1600 remained a leap year. The others became regular years, since only 16 is divisible by 4). Of the centurial years, the next leap year will be the year 2000 . 82, /43S e ial As regards the other years (besides the centurial years), the calculation of the leap years remained the same as in the Julian calendar . The Gregorian calendar was gradually introduced in all civilized countries. In Tsarist Russia, the introduction of the new calendar 460 (New Style) met with great opposition. Only after the Great October Socialist Revolution on February 1, 1918 was the new style quickly introduced. Then the difference between the Old and New Styles has already 13 days. Neither the Old nor the New Style is absolutely accurate, but the Gregorian (New Style) calendar has less error (1 day in 3000 years). 6. Use of Astronomical Devices Astronomical means of aircraft navigation permit the deter- mination of flight direction and the position line of the aircraft on the basis of the stars. The advantage of astronomical devices is their autonomy. Their use in flight is not related to any ground equipment and their accuracy is not a function of flight distance. The use of astronomical devices is based on the principle of measuring the azimuths or the altitudes of heavenly bodies. The position of the aircraft is determined by astronomical instruments from the intersection of two astronomical position lines (APL) or (as they are still called) lines of equal attitude. The astronomicat position line (line of equal altitude) is the straightened arc of the circle of equal altitude whose center is the geographic position of the star. The geographic position of a star (GPS) is the point on the Earth's surface at which the given star is observed at the zenith, or the projection of the star onto the surface of the Earth. The coordinates of the geographic position of the star represent the equatorial coordinates of the given star, i.e. , the latitude is equal to the declination of the star ( (j) = 6) and the longitude is equal to the Greenwich hour angle of the star (A = "'^Gr-'' This is evident in Figure 5.24. Circle of equa Earth's surface (""" tude of from point M with ca rrom point M wi it will be --"'"' (^TO-f-3T1i^£3 -f- it will be called the circle distance from any point on th the zenith distance (radius). juat altitude. Let an observer at point A on the [Fig. 5.25) at any moment of time measure the alti M. On the celestial sphere, if we draw a circle 1 a spherical radius equal to the zenith distance. i the circle of equal zenith distances , since the J point on this circle to the star M is equal to ice ( radius ) . /439 The projection of the circle of equal zenith distances (each point on the circle) onto the Earth is called the circle of equal altitude (hh') and the center of this circle is the geographic position of the star M on the Earth (GPS). It is called the circle of equal altitude because at any point on this circle, the star M will have the same altitude, i.e. it is observed at an angle for the same altitude of the star. This may be proved rather simply if we recall the relation between the altitude of a star (h) and its zenith distance (Z): h = 90° - Z. But since the zenith distance for any observer located on the circle of equal altitude is the same 461 (R of the circle the same . Z) then the height (h) for all observers will be Knowing this principle, we may make the reverse conclusion: if an observer, by measuring, determines the altitude of a star at a certain moment of time, then by plotting the geographic position of the star (GPS) with a radius equal to the zenith distance of the star (Z = 90° - h), the circle of equal altitude may be drawn. Ob- viously, at the moment of measuring the altitude of the star the observer (aircraft) is located on the circumference of the circle of equal altitude. Therefore, the circle of equal altitude is the circle of the position of the aircraft. In practice, the altitude (h) and azimuth (A) of a star are determined on the basis of tables at a moment of time planned beforehand for the point being calculated whose coordinates approxi- mately coincide with the location of the aircraft at the given / '^■ '/t moment. On the map, a straight-line segment equal to Ah is drawn from the point being calculated in the direction of the azimuth of the star. The segment APL is drawn perpendicular to the straight lines through its end. At the moment for which the calculation of the APL of the point being calculated was made, the altitude of the star h is measured by means of a sextant. In general, the measured altitude does not coincide with the altitude of the star at the point being calcu- lated. The difference between these altitudes is equal to the difference between the radii of the circles of equal alti- tude of the calculated and actual points of the aircraft's posi- tion. Since all the circles of zenith distance' plane of the horizon Fig, 5 .24. of a Star. Geographic Position Fig. 5,25, Circle of Equal Altitude . 462 iriin I lifii innni II ■ -I iiBiiiiia II Hill equal altitude of the same star at the same moment of time are con- centric, the APL representing various altitudes of the star are parallel to one another. Astronomical position lines which represent high altitudes are situated closer to the geographic position of the star, and vice versa. Therefore, if the measured altitude (hji,) is greater than the calculated altitude (h^), this means that the aircraft at the moment of measuring was located not on the APL of the point being calculated but on the APL parallel to it moved in the direc- tion of the star by a value Ah = hm - h^. If the measured altitude is less than the calculated altitude, the APL is moved in a direction opposite the direction of the star. To construct the APL on a map, it is necessary to know: (a) The approximate coordinates of the aircraft's position (the point being calculated) ^, A. (b) The azimuth of the star (A) for the point being calculated: (c) The distance between the measured and calculated altitudes of the star ( Ah ) . Determining the astronomical position lines. Before beginning the measurements, the electrical supply and the sextant light are switched on, the averaging mechanism is wound up, and the repeaters of the chronometer are matched with the indicator of the current t ime . On the basis of a map of the stellar sky or tables of altitudes and azimuths, the azimuth and then the course angle of the star are determined roughly. By rotating the course-angle drum, the sextant is set to the course angle of the star. By rotating the altitude drum, we bring the star into the sex- tant's field of vision. The sextant is set on a level base; by rotating the course-angle drum, the star is lined up with the bubble level. By rotating the altitude drum, the star is made to coincide with the bubble level and the averaging mechanism is connected. Once the pilot has been notified beforehand about the beginning of measurement, and the star has been accurately superposed on the bubble level, the averaging mechanism is switched on. When the averager has completed its work, a reading is taken. On the basis of the measured altitudes of the stars, problems in determining the following are solved: (a) One astronomical position line on the basis of the Sun /l^l to check the path with respect to distance and direction. 463 (b) Two position lines on the basis of two navigational stars or on the basis of one navigational star and Polaris to determine the position of the aircraft To check the path with respect to distance by means of one APL, stars are used whose directions are similar to the direction of the line of the given path. To check the path with respect to direction, stars are used which are situated at right angles to the line of the given path. In determining the position of an aircraft on the basis of two APL's, stars must be chosen so that the directions to them differ by an angle close to 90°. In calculating the astronomical position lines, the following auxiliary tables are used: (a) detached pages from an aviation astronomical yearbook (AAY) for the flight date; (b) Tables of the altitudes and azimuths of the Sun, Moon and planets (TAA): (c) Tables of the altitudes and azimuths of the stars (TAAS). The astronomical position line is calculated on a form like the following: Calculating the APL Order of Date Name of Operation GPA, W the star 1 2 3 1 5 6 9 1+ •'Moscow TGr AtGr(ASGr) A 10 tl(Si) 3 * 7 6 13 FB 2 8 hm P 1^ S 15 -r 16 17 h 11 he 18 Ah 19 12 ^^KM A f64 Name of the star Calculation of the astronomical position line with respect to / ^■^2 the Sun, Moon or some planet is done in the order indicated in the left-hand column of the form: (1) The Moscow time of measuring the altitude of the star is recorded (Tmoscow). (2) The measured altitude of the star is recorded (h^). (3) and (4-). The latitude and longitude of the calculated point ( <t) and X) are recorded. (5) The Greenwich time of measuring is determined on the basis of the formula where (Ng; + 1) is the number of the hour zone plus the standard hour and is recorded on the form. (6) and (7) The declination of the star (6) and its hour angle (tQ-p) for the whole hour corresponding to the time of measurement are copied out of the AAY . (When using the Moon, the tgp i^ written for whole tens of minutes . ) (8) In measuring the altitude of the Moon, the parallax (P) is copied from the AAY. (9). On the basis of the interpolation table available in the TAA or the AAY, the correction (AtQr) for Tq^ in minutes and seconds is found and recorded. (10) The local hour angle of the star (ti) is determined by adding tgp, Atg-p and X. Increasing or decreasing X, it is neces- sary that t-| be expressed by a whole even number of degrees. If the western hour angle is more than 180°, its complement to 360° is taken. The value found for t]_ is considered the eastern hour angle and is recorded in the form. The value for the longitude of the calculated point X, written earlier on the form, is refined in accordance with the change introduced with the selection of tj. (11) and (12) From the TAA, the value for the altitude of the star at the calculated point (h^) is written, taking into account the correction for minutes of declination, and the azimuth A is recorded. If the hour angle is western, then the complement of its tabular value up to 360° is taken as the azimuth. (13) The path bearing (PB) of the star (path angle) is deter- mined on the basis of the formula PB = A - GPA and is written on 465 the form. (14-) (15) and (16) Corrections are written on the form from the pertinent tables: sextant (S), for the refraction ( -r ) and for the Earth's rotation (6), (17) The measured altitude of the star (h) is adjusted for corrections Nos . 8, 14, 15, 16. (18) The difference between the corrected value of the measured altitude (hjj,) and the altitude of the star at the calcu- lated point (hj,) is calculated on the basis of the formula Ah = h„ - h . (19) Another value for Ah is recalculated in kilometers. After the calculations, APL is plotted on the chart as shown above. The APL on the basis of stars is calculated in the same order as on the basis of the Sun, Moon and planets. Tables of the TAAS and AAY are used. In addition, instead of the Greenwich and local hour angles, the Greenwich and local sidereal time are determined. The sidereal Greenwich time (Sq^) is taken from the table entitled "Stars" in the AAY for the moment Tq^ . The local sidereal time is determined on the basis of the formula /41+3 where X is the longitude of the calculated point refined with this calculation so that S-]_ is equal to a whole number of degrees. When determining the position of the aircraft on the basis of the intersection of the APL from two navigational stars, the meas- uring and recording of the time of the readings are done successive- ly with the shortest possible time interval. When plotting on the chart, the first APL is shifted parallel to itself in the direction of the vector of the flight speed for the distance traversed in this interval of time . The correction for the movement of the aircraft between the moments of the first and second measurements is also determined by- means of a special table applied to the tables of the altitudes and azimuths of stars. The correction for the rotation of the Earth (a) is introduced in the altitude of the star measured first. It is not necessary to shift the first APL determined, taking into account the correction of 6 by the movement of the aircraft in this case. When determining the position of an aircraft on the basis of stars in the Northern Hemisphere, Polaris and one of the navigational 466 stars situated in a westerly or easterly direction are used. Polaris is approximately 1° from the north celestial pole and therefore its height above the horizon is always roughly equal to the latitude of the position. This simplifies the calculating and plotting of APL's. The accurate latitude of the position of an aircraft on the basis of Polaris is determined by simple addition: Its measured altitude h,^ ; the correction of the sextant S; the correction for refraction -r ; the correction for the Earth's rotation a and the correction for the altitude of Polaris A(|> . The correction A(f) is given in TAAS on the basis of the value of the local sidereal time S^. The altitude of Polaris is measured later than the altitude of the navigational star and therefore the parallel corresponding to the latitude found is shifted in the direction of the flight- speed vector for the segment of the path traversed during the time interval between the first and second measurements. /444 The correction due to the travel of the aircraft is also introduced directly in the calculated latitude by means of a table of corrections "D" in the TAAS. Astronomical Compasses Modern astronomical compasses are automatic devices for deter- mining the true course of the aircraft by the direction-finding of the Sun or other stars . Astronomical compasses of the type DAK-DB are used on aircraft. These astrocompas se s are mainly intended for: (a) Incidental determination of the true course on the basis of the Sun ; (b) Continuous measurement of the course in flight along the orthodrome on the basis of the Sun. Astrocompasses of the DAK-DB type can transmit the values of the true course to course system indicators, and they can also permit the true course to be determined on the basis of stars at night by means of a periscope sextant. Astrocompasses of DAK-DB type may be used in the range of latitudes from the North Pole to 10°S. Astrocompasses of a special type are intended for use in the Southern Hemisphere as well. They can operate when the Sun is not more than 70° above the horizon. Here the permissible error in determining the true course must not exceed ±2° . 467 An astrocompass automatically solves problems of determining the true course of an aircraft according to the equation: TC = A CA where A is the azimuth of the heavenly body and CA is the course angle of the heavenly body. The course angle of the Sun is determined automatically by means of a course-angle data transmitter (CAD). The photoelectric head is situated in a transparent case in the fuselage of the aircraft; by means of an electronic system, it is automatically oriented in the direction of the Sun and sup- plies an electrical signal representing the course angle (CA) to a computer device. The azimuth of the star is determined by a special computer whose basis is a spatial computer mechanism (spherant). When es- tablishing the equatorial coordinates on computers , the hour angle and declination of the star as well as the latitude and longitude of the position, the azimuth of the star, i.e. the horizontal co- ordinate, is given at the output in the form of electrical signals. The table for the Greenwich hour angles of the Sun is given in Supplement 5 . /t+45 star plane of Y\ the horizon v}> f/^/;^ s . — -D W — -0 ^ source Fig. 5.26. Optical Diagram of an Aviational Sextant. A signal representing the dif- ference in the azimuth and course angle, i.e. the value of the true course , is fed to the indicator of the astrocompass. When using the astrocompass to determine and retain the ortho- drome course , coordinates pertaining to the initial point of the ortho- drome path line are fed into the- astrocompass . During flight along the orthodrome , the course angles at the initial point of the route. To preserve a constant value of the true course relative to the reference meridian of the beginning of the path, a correction on the basis of the flight correction method is automatically fed in. This method entails the following: The axis of rotation of the head of the CAD is vertical at the be- ginning of the path. Later with movement of the aircraft along the 468 orthodrome , it slopes back toward the tail of the aircraft by an angle equal to the arc of the traversed part of the orthodrome at the same time remaining parallel to the original position. The automatic calculation of the angle proportional to the arc of the traversed segment of the orthodrome is performed by the flight corrector, with a manual setting of the airspeed of the aircraft. Astronomical Sextants Aviational astronomical sextants are intended for measuring the altitudes of stars to determine the astronomical position lines and the position of the aircraft, as well as for measuring the course angles of stars. At the present time, periscope sextants (PS) which are adapted for mounting on aircraft with hermetic fuselages are the most common variety. The optical system of the PS sextant (Fig. 5.26) includes a cubic prism 1 for sighting stars. The cubic prism turns in a verti- cal plane from to 85° , with a goniometer drum to indicate alti- tude of a star The sextant has a chronometer with two independent repeaters, the clock mechanism of the averager of the readings and the course- angle transmitting selsyn. 469 CHAPTER SIX ACCURACY IN AIRCRAFT NAVIGATION 1. Accuracy in Measuring Navigational Elements and in Aircraft Navigation as a Whole The process of aircraft navigation is directed toward a crew's maintaining given trajectories of aircraft movement with respect to direction, altitude, distance, and time. Since the coordinates of an aircraft and the parameters of its speed along the axes of coordinates of a chosen frame of reference are measured with definite errors, it is natural that a given trajectory of aircraft movement will likewise be maintained with some errors . By accuraoy of aircraft navigation is meant the limits within which the errors of any flight-trajectory parameter are included with a definite probability. In contrast to the accuracy of navigational devices, which characterizes (in the majority of cases) the errors in measuring one coordinate or two aircraft coordinates simultaneously, the accuracy of aircraft navigation depends on the conditions of imple- menting indicated measurements and, in some cases, on the dynamics of aircraft flight. Let us assume that an aircraft is moving in a field of constant wind or under conditions of calm. The direction of flight is main- tained on the basis of results of measuring the lateral deviation of the aircraft (Z) from the line of the given path at designated points ( Fig . 6.1). Points A and B in the figure correspond to the actual coordi- nates of the aircraft, while points Ai and Bi correspond to measured coordinates . It is obvious that on the basis of results of measurements (^1 and Si), the aircraft crew does not obtain an accurate notion concerning the direction of movement, i.e. there is an error in determining the actual angle of flight Ai|; . /4if7 470 In general, errors in measuring the Z-coordinate (and, there- fore, ijj ) will exert the same influence on the accuracy of aircraft navigation with respect to direction, independently of whether the actual trajectory of aircraft movement will coincide with the given trajectory or whether it is situated at some slight angle to it. /iJ^S 1 Uj ®— — — 6 = B Rt Fig. 6.1 Diagram of the Occurrence of Errors in Aircraft Navigation with Respect to Direction. However, for simplicity of argument, we will consider that on segment AB the actual path line of the aircraft accidentally turned out to correspond strictly with the given line. In this case, angle h^ and coordinate B\ will be magnitudes of misinformation for the crew which, in their graphic form, determine errors in the crew's actions in the flight segment BC . Actually, a crew located at point B precisely on the given path will assume that the aircraft is located at point Si . There- fore , for an approach to point C it will be obliged to make an ad- vance in the course: Ay = arctg ^ BC In addition, the crew will assume that an aircraft on segment AB did not travel parallel to the given path line, but at an angle Aijj , equal to A'^; — a ret J AA^ + BB AB Therefore , the total incorrect advance in the course ^ytotal = Ai2 = Aij + A-;. Therefore, if the distance BC is approximately equal to AB , the aircraft must go not to point C but to point Ci, situated the following distance from point C : CC2 = AAi + 2BBi, 471 where AAi = AZi; BBi = AZ2. Under actual flight conditions, it is difficult to expect that the wind in segment BC will be the same as in segment AB . There- fore, if a flight is made over BC by maintaining the condition se- lected in segment AB , the aircraft will not appear at point C2 , but at point C3 , displaced from point C2 by the value of the change in the wind vector in segment BC with respect to segment AB relative to the flying time BC . For aircraft navigation with respect to d_lrection, only the lateral component of the wind change vector AU ^ will have any sig- nificance. Thus, the general error in aircraft navigation with respect to direction in segment BC is: AZBc = 'i^i+2A^2+.A//^^ /^U9 (6.1) It is possible to come to an analogous conclusion by examining the accuracy of aircraft navigation with respect to distance if the condition of speed is chosen on the basis of results of measuring the J-coordinate at points A and B: Aj^Bc = AXi + 2AX2 + M-fj. (6.2) Formulas (6.1) and (6,2) determine the absolute errors in air- craft navigation with respection to direction and distance. In these formulas, only the third term on the right-hand side (hU^t and AU^t) is a value which depends on the length of the stage of the path and therefore, on flight time. Therefore, the absolute error grows smoothly with an increase in the length of the stage in the path between the control points , The ratio of the absolute error of a given parameter to the length of the stage in the path of the aircraft in which this error arises is called the relative error of aircraft navigation. There- fore, the relative error exerts an influence on the stability of the flight conditions of the aircraft. Let us illustrate this with a specific example. Let us assume that at the control stage in the path of an air- craft, with a length of 200 km, an error of aircraft navigation of 5 km in distance and 4 km in direction has accumulated. The relative error in aircraft navigation with respect to dis- tance and direction will be: \x X " 200 ~ AZ 4 40 ■=-2,5»/o; 200 50 472 The relative error with respect to direction characterizes the conditional errors of aircraft navigation: A(!; = arctgf AZ X In the following stage of flight of equal length (200 km) in order to balance the errors of aircraft navigation which were accumu- lated in the preceding stage, it is necessary: (a) to introduce a correction in the aircraft course equal to /U5 0| the error Aijj , in our case arctg 1/50 ~ 1° , and I (b) to change the airspeed, in our example by 2.5%. Let us assume now that the same error in aircraft navigation arose at a stage in the path about 50 km long. Then a;!' 5 X ~ 50., ' AZ 4 " X ~ 50 = 10%; 4°. In this case it would be necessary for us to change the air- craft course by 4° and the airspeed by 10% for every 50 km of the path, i.e. in modern aircraft, every 3-4 min of flight. Considering that the error in aircraft navigation Increases with respect to time only as a result of a change in the wind vector, it becomes entirely obvious that it is advantageous to choose control stages of flight which are very long, both from the point of view of the frequency of introducing corrections in the aircraft flight condition and in the values of the corrections being intro- duced . The necessary accuracy of aircraft navigation with respect to direction of the flight path is determined by thq set width of air routes and approach paths to airports, as well as national 473 boundaries . However, it is necessary to consider that at turning points on the paths, with significant turn angles for the route, the errors of aircraft navigation with respect to distance become errors with respect to direction, and vice versa. The accuracy of aircraft navigation during the approach of an aircraft landing on instruments acquires a special significance. The necessary length of the path of an aircraft's approach to a given trajectory, after changing to visual flight, depends on the magnitude of the aircraft's deviation from the given descent trajec- tory during an instrument approach for landing, and therefore on the weather conditions during which a landing can be made. /451 With automatic or semiautomatic approach to landing by air- craft up to low altitudes (for example, up to leveling off .or landing) the accuracy of aircraft navigation must be such that the landing of the aircraft in all cases will be ensured with the execution of safe deviation norms with respect to the landing position and direction of the aircraft vector in the path. 2. Methods of Evaluating the Accuracy of Aircraft Navigation In special books on the study of the accuracy of aircraft navi- gation with the application of navigational systems, the methods of probability theory (Laws of the distribution of random variables) are used . To evaluate the accuracy of aircraft navigation under practical conditions, it is sufficient to use only the basic conclusions of probability theory. Since the study of probability theory as a science is not the purpose of this textbook, in the majority of cases these conclusions will be given without proofs. In probability theory, variables which cannot be determined in advance by classical methods of mathematics, or are determined by methods so complex that they cannot be used for practical pur- poses, are considered to be random variables. In connection with problems of the accuracy of aircraft navi- gation or the accuracy of measuring aircraft coordinates by means of navigational systems, the errors in measuring or maintaining some of the navigational parameters will be random variables. Let us assume that the value of some navigational flight param- eter (on the basis of some especially precise control device) is known exactly. However, in carrying out a number of measurements by the usual means, we always obtain new values for the parameter which differ from its precise value . The precise value of a measured parameter will be called its 474 mathematical expectation. If a series of measurements is suffi- ciently great, then in all probability we will obtain many values for the measured parameter, with both positive and negative errors. Here the mean arithmetic value of all the measurements will ap- proach (depending on the increase in their number) the mathematical expectation of the measured value. Therefore, to raise the accuracy of aircraft navigation, in many cases measurements are carried out repeatedly and the arithmetic mean of the series of measurements is found . The arithmetic mean of a measured parameter cannot characterize /HS 2 the probable accuracy of carrying out individual measurements. Therefore, probability theory includes a concept of mean square deviation from the precise value. Let us designate the precise value of a measured quantity by a, and its measured values by X^, where i = 1 , 2 , 3 . . . Let us call the value (a;. a) the measuvement ervov. The value obtained by extracting the square root from the sum of the squares of the errors divided by the number of measurements is considered the mean square error of measurement: l-n 2] (-</ ~ a)i (6.3) According to (6.3), the mean square error of measurement is determined when the precise value of magnitude a is known. If the value of the measured magnitude is determined as an arithmetic mean from a series of observations, it is considered that one of the measured magnitudes coincides with or very closely approaches the arithmetic mean. The error of this measurement is considered to be zero, resulting in an increase in the sum in the numerator under the root of (6.3) equal to zero. Therefore, in order to avoid decreasing the value of the mean square error, especially with a short series of measurements, the denominator of (6.3) reduces to 1. Then this formula assumes the form: n— 1 (6.4) The mean square error characterizes the accuracy of the meas- urements in a rather definite way. With the raising of each of the errors to a square, its sign always becomes positive. Therefore, 475 in determining mean square errors , only the absolute value of each plays a role . It is considered that the mean square error does not have a sign, If we examine only one of a series of measurements, with a probability equal to 1 (complete probability), it is possible to say that the magnitude being measured will undoubtedly have some value. However the probability that the magnitude being measured will have a strict and absolutely precise value is practically equal to zero, except in cases when it can assume only a. discrete value. Therefore, in determining the probability of an error of measurement it is not the precise value of the error, but the limits /^5: in which it must be found, which are given (for example, the proba- bility of error in the range from 500-600 m or from 2 to 2.5 km, etc . ) . All the measured navigational magnitudes are (to a certain degree) calibrated magnitudes, i.e., they have errors limited by certain boundaries. These boundaries depend on the allowances in the regulation of the measuring apparatus and on the maximum pos- sible distortions of the measured magnitudes as a result of the in- fluence of external factors (electromagnetic wave propagation, the physical composition of the airspace, variations in the Earth's magnetic field, etc.). Allowances in the regulation of measuring apparatus are known quantities. Century-old observations permit the determination of the limit of change in the parameters of the environment. There are ways of evaluating the maximum influence and other factors on the accuracy of measurements. Therefore, it is always possible to predetermine the maximum errors of some kind of measurements . The quantitative characteristics of the distribution of errors from their zero to maximum values, in the majority of cases, are subject to the normal law of random variable distribution. If in some cases the law of error distribution is not normal, it will be close in any case. Considering that devices of probability theory are used not in calculating measurement errors, but only in evaluating limits and the probability of possible measurement errors within these limits 5 it is considered permissible in all cases to use the normal law of distribution of random variables. The normal law of random variable distribution (Gauss formula) characterizes the probability density of a random variable, in our case of the measurement errors (x - a), depending on its value: 476 f(x — a) = (x-ay (6.5) 21/2^ where ^(x - a) is the probability density of errors of a given magnitude, a is the mean square error of a series of measurements, e is a Napier number equal to 2.71828, and a is the precise value of the magnitude being measured. It is obvious that the probability of finding the result of measuring (x) in the range of values from a to x can be determined by integrating (6.5) over x : X '{x-a)'= ' r K2.. J -{x-ay dx. (6.6) The graph of the probability of random variables subordinate to the normal distribution law is shown in Figure 6.2. /454 The curve on the graph shows the probability density of random variable deviations from zero to maximum positive and negative values. The left side of the graph corresponds to errors with a negative sign, the right to errors with a positive sign. Since the absolute probability of obtaining any value of the measured magnitude is equal to one, the probability that the value of the magnitude will be negative or positive is 0.5. Let us note that on the abcissa of the graph there are two values of a random variable, Xi and X2- The area bounded by the segment a:ia;2, by the ordinates Pxi? ^x2 ' ^^'^ ^^ tY^e curve is the probability of finding the result of measurement in the limits be- tween X I and X2 • iP[y2-a}-'P(yro) (X-a)<0^^ h ^1 — Ix-a)> Fig. 6.2 Graph of the Proba- bility of Random Variables Under the Normal Law of Distribution. With the convergence of points xi and X2 at one point, the probability of finding an error of measurement between these points will diminish and converge to zero. The analogous problem for determining probability can be solved on the basis of the right side of the graph for errors of measurement which have a positive sign. 477 * without stopping at the methods of solving an integral (6.6), let us indicate that the overall probability of finding positive and negative errors of measurements is 68.3% in the range from to a, 95% from to 2a, and 99.7% from to 3a. A table of values of the function $(a; - a) for ix ~ a) from to 5a is given in Supple- ment 6 . For example, if the mean square error of measuring the drift angle with a Doppler meter is equal to 15 ' , then with a probability of 95% it is possible to expect that the measurement error will not exceed 30', and with a practically complete probability (99.7%), 45'. The value of the mean square error of measuring a given kind of parameter permits evaluation of the accuracy of other parameters which have a functional dependence on the first. Example : The mean square error of the direction-finding of an aircraft by means of a ground direction finder a = 1° . Determine the limits of linear error in determining the lateral deviation of an aircraft from the line of a given path with a probability of 95% if the aircraft is located at a distance of 300 km from the direc- tion finder . Solution: with a probability of 95%, the angular error of a direction finder in measuring does not exceed 2°. Therefore, AZ(P = 95%) = 300 tg 2° fi^ 10 km. Solving the same /455 problem for a practical probability of 100% (more precisely, 99.7%), AZ(P = 100%) = 300 tg 3° R. 15 km. Let us now assume that we must solve the reverse problem, i.e. determine the necessary accuracy of a direction finder which en- sures the given accuracy of measurements of lateral deviations. Example : ^^man^'^ ~ 100^) = 10 km. Determine the necessary accuracy of a direction finder for distances up to 300 km. Solution: 3a^ = arctg TqK ^ 2°. Therefore, a^ = 0.7°. 3. Linear and Two-Dimensi onal Problems of Probability Theory The normal law of random variable distribution examined in the preceding paragraph includes the linear ( one -dimensional ) problem of probability theory for one parameter of measurement. In aircraft navigation, it is often necessary to deal with several measurement parameters. For example, in calculating the path of an aircraft with respect to direction by automatic navi- gational devices, on the basis of results of measuring the drift angle and groundspeed of the aircraft with a Doppler meter, the following errors exert an influence on the accuracy of calculating this parameter: errors in calculating the given flight angle; 478 errors in measuring the course, drift angle, and groundspeed; errors in the operation of an integrating device. Each of these factors separately will create the following error components in calculating the path with respect to direction; AZ^ = X sin A.^ = Wt sin Ai; AZ^ = Wt sin Af ; AZ„-^ U^isinAa; • AZ^ = AlWsin(^^-<;;3); AZ j. = WtS^l' If the indicated components had the same sign and had a maxi- mum value within the calibration limits of each of the parameters, the general error would be equal to the arithmetic sum of these components. However, according to the law of normal random variable distribution, even when measuring one parameter, the maximum error is encountered rather rarely. The probability that all the errors /45 6, will take on a maximum value, and even one sign, will be extra- ordinarily low. In spite of the fact that we must deal simultaneously with many measured parameters, the solution of the above example includes a linear problem of probability theory, since the random variables are summed along one axis of the chosen frame of reference of their coordinates , To solve similar problems, the concept of the dispersion of random variables a^ is introduced into probability theory. It is known that the law of random variable distribution, ob- tained by adding other random variables which are subject to the normal distribution law, is also a normal distribution law. Here, the scatter of an overall random variable is the sum of the scatters of the values being added. In our example of calculating the path of an aircraft by means of automatic navigational devices, the value the sum, is the scatter of Here , °z = '22^ + =2Z^ + rfz„ + =2Z + ,2Z^. (6.7) The mean square error of the measurement is equal to the square root of the scatter: a = /a-^ . Therefore, the mean square error of the total value will equal the square root of the sum of the scat- ters. For our example. 479 •j/"a7Z^ + o2Z^ +a2Za+G2Z. + a2Z« (6.8) The value a^ in (6.8) is a small second-order value: Therefore, it is necessary to disregard this value. Let us assume that the remaining values included in (6.8) have been mean square errors as follows : o, =20'; a =20'; o„ = 15'; o. =0,5%o£x. Since the first three values are small, their sinces can be replaced by angle values. Then, considering 1° equal to 0.017 by 1.7% X, their value can be expressed in percent of the distance traversed : a^ = 0,56%X; a =0,56°/o^; 5^ = 0,42%;?; op =0,5°/oA-, where X = Wt . Therefore , the mean square error in calculating the path with /457 respect to direction is 0^ = WtV0,5& + 0,56? + 0,422 + 0,52 = U^f VT^ = 1 ,02%X. Hence, it is possible to consider that the mean square error in calculating the path with respect to direction amounts to ~ 1% of the distance traversed. Let us assume that we have set ourselves the goal of maintaining an aircraft within the limits of an air route with a width of 20 km (up to 10 km from LGP ) with a probability of 95%. Here the mean square error in determining the initial coordinates of the aircraft equals 2 km. For a probability of 95%, the error in the initial formulation of the aircraft's coordinates must be taken as 4- km, while the accuracy of calculating the path with respect to direction must be taken as 2%. The maximum error in calculating the path with respect to distance must not exceed A^niav = 1/102 — 42 = 1/84 «9'k m 'max; The value 9 km must amount to 2% of the distance covered, l^80 J Therefore , the allowable length of the stage of the path between the control points (S) must be not more than 0,02 = 450kn> If we set ourselves the goal of maintaining an aircraft within the limits of a route with a probability of 99.7%, the accuracy of the initial display of coordinates and the calculation of the path of the aircraft would have to be taken. as 3a or a reading accuracy equal to 6 km and an accuracy for calculating the path equal to 3% X. Then AZ max :y 102 — 62 = yc4 = 8 kr 8,km=3^oX; X=--- 0,03 266 km If we take the limits of calibrating each parameter as 3a, then by adding the errors on the basis of the calibration rules we would obtain the value A2max=3'+ + ^ + 3.„ + 3.j.^ or, in our example, AZ^g^:- 1,7 +1,7 +1,5+ 1,2 = 6,1%, i.e., in the case when all errors have a maximum value and the same sign, the error of calculating the path can reach 6%. Since it reaches 2% with a probability of 95%, with a practically complete probability of 99.7% it reaches 3%. The probability that calculating the path will occur with errors within the range of an overall calibration of the system is expressed by in hundred mlllionths of a percent. Therefore, when there is no threat of disturbing the safety of a flight, it is not necessary for practical purposes to take the limits of overall calibration into consideration. /458 Fig. 6.3. Diagram of the Occurrence of Errors in Determining the Position of an Aircraft: (a) with Multi- lateral Errors in Rearings ; (b) With Unilateral Errors. f+Sl In the majority of cases, there is sufficient error in calcu- lating the path to calculate with a probability of 95%, and only in especially responsible cases, with 99.7%. The majority of problems in determining the accuracy of air- craft navigation or measuring its separate parameters with the use of some method reduces directly to linear problems pf probability theory. The final result of solving all the problems of aircraft navigation must be one-dimensional, since the goals and requirements for accuracy of aircraft navigation with respect to distance, direction, and flight altitude are different. At the present time, there are no navigational systems which determine the position of an aircraft in three-dimensional space. Therefore, the necessity for solving volumetric problems in proba- bility theory is superfluous. However, a number of navigational systems such as a hyperbolic, two-pole goniometer, or goniometric rangefinder (if only the location of a ground beacon is known for the LGP of a given segment), permit the solution of a problem in determining an aircraft's coordinates in two-dimensional space. An evaluation of the accuracy of aircraft navigation along each of the axes of the coordinate system chosen for aircraft navi- gation, in this case, can be carried out only after solving a one- dimensional problem in probability theory. Let us assume that we have an oblique-angled surface system of aircraft position lines, each of which does not coincide with the given flight path (Fig. 6.3, a,b). The linear error in determining the first (ri) and second (^2) aircraft position lines depends on both the accuracy of measuring the navigational parameter and its gradient. The gvadient of a navigational pavameteT is the ratio of its /M-5^ increase to the movement of an aircraft in a direction perpendicular to the position lines of the operating region of the system g da dr (6.9) where g is the gradient of the navigational parameter, and a is the navigational parameter being measured. For example, if the navigational system is a goniometer, then s = ■ da 'dr'. dA__ dr _1_ 5 • where A is the azimuth of the aircraft and S is the distance from 482 the ground beacon to point PA. In this case dr == da ■■ daS or r = ^AS. With the introduction of the concept of the gradient of a navigational parameter, all the existing coordinate systems reduce to a generalized system, i.e., the problems of determining the accuracy of navigational measurements are solved on the basis of a general scheme, independent of the geometry of application of the navigational device. In Fig. 6.3 a,b two possible cases of the appearance of errors in determining the position of an aircraft on the basis of the intersection of position lines are shown: (a) Errors in r^ and ^2 have different signs; in this case, the measured position of the aircraft lies in an acute angle between the actual position lines. This leads to larger errors of deter- mination . (b) Errors in r i and r2 have identical signs; the measured position of the aircraft lies in an obtuse angle between the position lines. The errors in determining the position of the aircraft in this case are close to the linear errors of one bearing. It is necessary to note that the probability of errors in ri and T2 with the same sign in the majority of cases is more than the probability of errors with different signs. For example, in taking bearings with a radio compass, the error component as a result of an error in measuring the aircraft course will be general for two measurements. If the angle between the bearings is sufficiently acute, the radio deviation will have either one sign or different signs, but a small value in any case. A similar relationship between measurement errors in probability theory is called correlation (p). The general error in determining the position of the aircraft in our case will be (Fig. 6.1+): /-2 = '1 ^2 I + sin2w sin2m 2rir2 cos 10 sin2 CO ]/ • /-f + /^ — 2/'i/-2 cos ci) (6.10) Formula (6.10) characterizes only the magnitude of error in /460 483 1 determining the position of an aircraft on the basis of two position lines with known errors in measurement for each of them. However, it does not give us an idea of the nature of the distribution of the indicated errors around the point of the actual position of the aircraft (center of scatter). In contrast to a linear problem, where the probability of an overall error in several measurements is examined, in a two-dimen- sional problem it is- necessary to examine the products of the probabilities of these errors. For simplicity of argument, let us assume that we have a rectangular coordinate system (Fig. 6.5); let us set ourselves the goal of limiting the area within which the aircraft is located with a probability of 95%. Here, the mean square errors of measuring the two position lines will be considered identical. Let us examine a certain large number of measurements (e.g. , 10,000) and let us see what will be the probability that the measured position of the aircraft will be in an exterior angle at a distance from the center of the area of scatter which exceeds the diagonal of a square constructed with errors 2a, and 202- Since the probability of an error in the first measurement exceeding 2a equals 5%, then 500 of 10,000 measurements must be be- yond the limits of a side of the indicated square. The remaining 9,500 measurements lie in the range from zero to 2a, and there is no need to calculate them during common measurements with the second position line . It is