[1st Edition] 9781483224657

Table of contents :
Content:
Front MatterPage iii
Copyright pagePage iv
Contributors to Volume 4Page v
ForewordPages vii-ixFREDERICK I. ORDWAY III
Contents of Previous VolumesPages xiii-xiv
Doppler Effect of Artificial SatellitesPages 1-38J. MASS, E. VASSY
On the Possibilities of the Existence of Extraterrestrial IntelligencePages 39-110ROGER A. MACGOWAN
The Development of Multiple Staging in Military and Space Carrier VehiclesPages 111-137H.E. NYLANDER, F.W. HOPPER
Spacecraft Entry and Landing in Planetary AtmospheresPages 139-201MAURICE TUCKER
Development of Manned Artificial Satellites and Space Stations*Pages 203-317SIEGFRIED J. GERATHEWOHL
On the Utilization of Radioactive Elements as Energy Sources for Spacecraft PropulsionPages 319-399, 401-409, 411-415J.J. BARRÈ
Author IndexPages 417-421
Subject IndexPages 422-431

Citation preview

Advances in

Space Science

and Technology Edited by FREDERICK I. ORDWAY, III George C. Marshall Space Flight Center National Aeronautics and Space Administration Huntsville, Alabama

Editorial Advisory Board

Wernher von Braun Frederick C. Durant, III Eugen Sänger Leslie R. Shepherd George P. Sutton Etienne Vassy

VOLUME 4

ACADEMIC PRESS NEW YORK and LONDON 1962

COPYRIGHT ©

1962, BY ACADEMIC PRESS INC.

ALL RIGHTS RESERVED NO PART OF THI& BOOK MAY BE REPRODUCED IN ANY FORM BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS

ACADEMIC PRESS INC. Ill

F I F T H AVENUE

N E W YORK 3, N.

Y.

United Kingdom Edition Published by ACADEMIC PRESS INC. (LONDON)

LTD. BERKELEY SQUARE HOUSE, LONDON W. 1

Library of Congress Catalog Card Number 59-15760

PRINTED IN THE UNITED STATES OF AMERICA

Contributors to Volume 4 J. J. BARRÉ, Fabrications d'Armement du Ministère Français de la Défense Nationale {cadre de Réserve), Versailles (Seine-et-Oise), France SIEGFRIED J. GERATHEWOHL, Biotechnology Division, Ames Research Center, National Aeronautics and Space Administration, Mountain View, California F. W. HOPPER, The Martin Company, Denver, Colorado ROGER A. MACGOWAN, Army Ordnance Missile Support Agency, Redstone Arsenal, Alabama J. MASS, Laboratoire de Physique de VAtmosphère, Université de Paris, Paris, France H. E. NYLANDER, The Martin Company, Denver, Colorado MAURICE TUCKER, Lockheed Missiles and Space Company, Palo Alto, California E. VASSY, Laboratoire de Physique de VAtmosphère, Université de Paris, Paris, France

v

Foreword Progress in space science and technology has been rapid and often spectacular since the appearance of Volume 3 a year ago. In the field of space carrier vehicles, the Atlas Agena B has proven capable of handling a variety of difficult satellite and probe missions. Successful launchings of the giant Saturn Cl in October 1961 and again in April 1962 promise relatively early availability of an operational carrier capable of orbiting Earth with payloads weighing 20,000 lb. The much more powerful follow-on Saturn C5 will project satellites ten times as heavy into orbit, from where spaceships may be launched towards the Moon. In the area of artificial satellites equally important progress was recorded. Unmanned scientific and applications satellites* successfully performed their missions, adding greatly to our knowledge of the universe around us and, more than incidentally, of the planet Earth and its atmospheric envelope. First the Soviets and then the U.S. launched manned satellites and successfully recovered their cosmonauts and astronauts. These achievements encouraged the U.S. to accelerate development of the Dynasoar and Gemini advanced manned satellites and to embark upon the Apollo lunar spaceship program. Much activity was expended in attempting to define how Apollo should be sent to the Moon —along a direct flight trajectory from Earth's surface to Moon's surface powered by a gigantic Nova-class carrier, or by way of Earth or lunar orbital rendezvous and assembly, in which case smaller, but still huge, Saturn C5's would be required. Increased attention was given to the Moon and the nearer planets Venus and Mars. One Ranger probe by-passed the Moon and another impacted upon it during a series of tests aimed at hard landing seismometers on the surface. Missions for Mariner R Venus fly-by probes and larger, heavier Mariner B probes were clarified as the exploratory onslaught of Venus and Mars began. Against this background of progress the six chapters in this book examine many vital areas of basic and applied astronautics. The first chapter deals with one of the more practical aspects of artificial satellites, measurement of the Doppler effect. Knowledge of, and means of applying, the Doppler effect of Earth-circling satellites are reviewed. The * "Working" satellites designed primarily to provide information in a single applied field, e.g., meteorology, communications, ICBM detection, etc. vii

viii

FOREWORD

authors first explain the Doppler effect, then cover experimental devices and perturbations, and finally describe ways in which the effect can be utilized, e.g., for tracking and navigation. They write that of the schemes for using satellite transmissions for navigation, the "most promising . . . seem to be those based on measuring the Doppler effect." For advanced astronautical missions they show that the orbital parameters of a probe or spaceship moving around the Sun can be determined by the Doppler technique, yielding an important means of interplanetary navigation. The second chapter in this book deals with one of the most fascinating subjects known to science, and indeed, to humanity in general—the possibility of the existence of intelligent beings other than man. The prospect of finding life, however primitive, on other planets in the Solar System encourages biologists, astronomers, chemists, geologists, and geochemists to probe deeply into the origin of life on Earth and elsewhere. While they entertain little hope of finding advanced forms of life in the Solar System, many consider it very possible that intelligent beings exist elsewhere in the universe. There seem to be two ways to detect them: (1) send probes to other suns, search for planets, and then for intelligent beings on them; and (2) listen, from the Earth, for signals that may be produced by intelligent beings on extrasolar planets. If signals emanate from such planets around stars within roughly 50 light years from the Sun, they may be received by large radio telescopes and subsequently interpreted. In order to set the stage for the chapter on extraterrestrial intelligence, the author reviews the various planetary origin theories. He looks into the development of intelligence biological life on Earth, and reviews the rise of mechanical thinking automata. Considerable space is devoted to the thinking process, both biological and mechanical. One of the more interesting conclusions reached is that intelligent beings elsewhere in the universe may be mechanical, not biological. The early rocket and astronautical pioneers were quick to recognize the importance of the staging concept to the development of carrier vehicles capable of propelling spacecraft beyond the Earth's atmosphere. The authors of the chapter on multiple staging first review the historical development of multistage rockets and space carrier vehicles and then examine such subjects as the equations of rocket motion, performance analyses of tandem and parallel-staged vehicles, stage sizing, and staging techniques. The fourth chapter of this volume is concerned with problems of bringing spacecraft safely through planetary atmospheres and onto the surface. Since Mars and Venus, the two prime target planets in the

ix

FOREWORD

Solar System, are surrounded by atmospheres the importance of this subject is obvious. Following a short introduction, the entry equations are reviewed, attention being focused on aerodynamics and heating. The section on entry analyses covers aerodynamic loading and heating problems in various planetary atmospheres, atmospheric braking, and entry techniques (ballistic and controlled). Various configurations for entry vehicles are described. The use of parachute recovery techniques is assumed. The author concludes that problems of entering the atmospheres of Earth and Venus "as regards deceleration levels, guidance accuracy, and convective heating" are about equal. Convective heating problems for entry into the Jovian atmosphere appear to be unrealizable within the present state of space technology, whereas "no particular difficulties are anticipated for Mars entry." Manned artificial satellites and space stations are accorded a verycomplete survey by an acknowledged world authority on manned space flight. For some reason the space pioneers spent little time thinking about small, one-man space capsules with which we are familiar today; rather they conjured up an astounding stable of large space stations, many of which contained the conveniences we take for granted here on Earth. These early concepts are first surveyed, following which operational and other details are given of such modern craft as the X-15, Dynasoar, Mercury, and Vostok. New concepts and approaches to manned orbital flight are considered in Section IV of the chapter. The author believes that large space station technology will come about as the result of achieving: (1) advanced capsules and satellites with increased controllability, (2) multicrew orbital vehicles, and (3) manned space stations and winged return vehicles. In this section many modern space station proposals are reviewed. Structural design, materials, control, communications, human factors, power supplies, maintenance, and logistics are also covered. The final section deals with the all-important matter of cost, something no space planner can ignore. One of the many energy sources applicable to the propulsion of spacecraft are radioactive elements. Various radioactive elements are discussed and their constraints on propulsion examined. Then, in considerable detail, so-called radiothermal and radionic engines are described. The author concludes that while radioactive elements could not be used to power space carrier vehicles taking off from Earth, they do hold promise as energy sources for spacecraft propulsion in orbital transfer and for travel between the worlds of the Solar System. Huntsville, Alabama June, 1962

FREDERICK

I.

ORDWAY,

III

Contents of Previous Volumes Volume 1 Interplanetary Rocket Trajectories DEREK F. LAWDEN

Interplanetary Communications J. R. PIERCE and C. C. CUTLER

Power Supplies for Orbital and Space Vehicles JOHN H. H U T H

Manned Space Cabin Systems EUGENE B. KONECCI

Radiation and Man in Space HERMANN J. SCHAEFER

Nutrition in Space Flight ROBERT G. TISCHER

Appendix. A Decimal Classification System for Astronautics HEINZ HERMANN KOELLE

Volume 2 Experimental Physics Using Space Vehicles CHARLES P. SONETT

Tracking Artificial Satellites and Space Vehicles KARL G. HENIZE

Materials in Space FREDERICK L. BAGBY

Plasma Propulsion Devices MORTON CAMAC

Electrostatic Propulsion Systems for Space Vehicles ERNST STUHLINGER and ROBERT N. SEITZ

Attitude Control of Satellites and Space Vehicles ROBERT E. ROBERSON Xlll

xiv

CONTENTS OF PREVIOUS VOLUMES

Volume 3 The Role of Geology in Lunar Exploration JACK GREEN and

JACK R. VAN LOPIK

Venus as an Astronautical Objective PATRICK MOORE and

S. W.

GREENWOOD

Mars as an Astronautical Objective SEYMOUR L. H E S S

The Exploration of Mercury, the Asteroids, The Major Planets and their Satellite Systems, and Pluto RAY L. NEWBURN, JR.

Interplanetary Matter EDWARD MANRING

Structures of Carrier and Space Vehicles A. ALBERI and

C. ROSENKRANZ

Advanced Nuclear and Solar Propulsion Systems WILLIAM C. COOLEY

Human Factors: Aspects of Weightlessness PAUL A.

CAMPBELL

Doppler Effect of Artificial Satellites J. MASS* AND E. VASSY

Laboratoire de Physique de VAtmosphère Université de Paris I. Introduction A. Generalities on the Doppler Effect B. The Doppler Effect for a Transmitting Earth Satellite C. Straight Path Constant Velocity Approximation D. Circular Orbit Approximation, Stationary Earth E. General Case II. Experimental Devices A. Receivers B. Filtering C. Frequency Tracking D. Frequency Measuring E. Reduction of Doppler Data III. Perturbations A. Refraction B. Noise, Modulation and Polarization Rotation C. Tropospheric Effects D. Relativity Effects IV. Utilization of Results A. Tracking Applications B. Navigation Applications C. Scientific Applications References

1 2 3 4 5 9 10 11 12 13 16 17 19 19 24 27 27 28 28 31 33 34

I. Introduction When a radio transmitter is placed in an artificial satellite and its signal is received on Earth, the frequency registered at the receiver will seem to vary according to a well defined curve. This is due to the DopplerFizeau effect, more frequently called the Doppler effect [1]. It was originally found in acoustics where it is most easily observed by the change in pitch of the tone emitted by the horn of an automobile as it approaches the listener and moves away again. The tone is at first high, becoming very quickly low-pitched as the car passes by. The effect is due to the *On study leave from the Scientific Department, Israel Ministry of Defense. 1

2

J. MASS AND E. VASSY

finite velocity of propagation of the sound waves which arrive at the observer in a compressed or diluted form according to whether the car approaches or recedes from the observer. Virtually the same effect is found for electromagnetic waves (light, radio waves, etc.), except that there are certain differences due to the principles of relativity. The distinction between moving transmitter-stationary observer and stationary transmittermoving observer becomes meaningless without an absolute medium of wave transmission. Measurement of direction or distance by radio-means is more complicated and less accurate than measurement of frequency. The Doppler effect, when measured on satellites and space probes, consists of frequency measurements only, and is therefore very attractive for many purposes. It has been found, for instance, that in principle a Doppler-frequency curve (frequency function of time) of a single pass of a satellite should suffice to measure its orbital parameters [2, 3] ; and also that accurate tracking of objects within the Solar System using only the Doppler-effect was possible and within the present state of the art [4]. At present a program is under way (the United States Navy Transit satellite program) to use the emissions of several satellites to facilitate navigation on the surface of the Earth by accurate frequency measurements of their Doppler shift. It is assumed, of course, that the orbits of the navigational satellites are very accurately known. This procedure is the inverse of calculating the orbit parameters from the Doppler-shift and the known position of receiver; this procedure, furthermore, should be simpler to carry out because the number of unknowns is smaller (longitude and latitude only). The distortions of the Doppler curve introduced by ionospheric effects, such as refraction and change of phase velocity, create a disadvantage for tracking and navigation, but they do enable one to take some interesting measurements of the ionosphere such as total electron contents and heterogeneity of the ionized layers. It is the purpose of this chapter to review the important results so far published concerning the measurement and utilization of the Doppler shift from artificial satellites. A. Generalities on the Doppler Effect

An electromagnetic wave from a transmitter propagated through a vacuum moves at a velocity c, equal to the velocity of light. When the wave passes through a dispersive medium its velocity is changed. If the velocity of propagation of the wave crests is measured, one may find that it has become greater than the velocity of light. This greater velocity is

DOPPLER EFFECT OF ARTIFICIAL SATELLITES

3

called the phase velocity of the wave in the medium. When the medium contains free electrons and ions, its refractive-index n < 1 can be calculated, the phase velocity being vv = c/n to a good approximation. When a transmitter sends a continuous wave A cos cot, a receiver will receive the same wave with a delay due to the time of propagation B cos œ(t — At) Assuming the medium between transmitter and receiver to have a refractive index being a function of the space coordinates n(x, y, z), the wave will travel from transmitter to receiver through a curved trajectory due to refraction, and at phase velocities varying along its path. The total time delay for a wave crest will be

At= f * - - 1 f

nds J vv cJ the integral being taken along the path. When there is relative movement between the transmitter and the receiver, the delay At is a function of time which is noted as a change of frequency at the receiver, as calculated by the time derivative of the phase angle. The received frequency will be

-/-is/"*

4f=A " / - - f a / " *

(1)

where Δ/ is the frequency "shift" due to the Doppler effect. The frequency shift is proportional to the transmitted frequency / and to the modified relative speed measured by the rate of change in "phase path," i.e., by (d/dt) J nds. When n = 1 A

f=cit

(la)

where r is the distance between satellite and receiver. In the above formulas, it was assumed that the velocity dr/dt is much smaller than the velocity of light, so that relativity effects can be ignored (see Sec. H I D ) . B. The Doppler Effect for a Transmitting Earth Satellite

As it has been shown, the frequency shift produced by the Doppler effect is proportional to the relative velocity dr/dt between transmitter

4

J. MASS AND E. VASSY

and receiver, or, more accurately, the relative phase velocity, which is the relative velocity modified by the propagation medium. Calculation of the Doppler shift by receiving on Earth and using a transmitter placed in a satellite having a known orbit is a rather long and tedious process. The formula for distance between the observer and the satellite involves satellite coordinates (in a geocentric system) as function of time as well as observer coordinates (which include the Earth's rotation). When differentiating this distance with respect to time, one gets the relative velocity and hence the Doppler shift, neglecting propagation effects (refraction and change of phase velocity). These effects are more important for the lower frequencies and can be almost neglected at about 100 Mc. In order to introduce propagation effects in the calculated Doppler shift, a certain "standard" ionosphere must be assumed. Unfortunately, there exists no easy, general solution for the complete Doppler effect and one usually works with first approximations sufficient for the problem at hand. In the following section the Doppler effect is calculated for some simple assumptions and later on, as the various uses of the Doppler effect are considered, the necessary approximation will be made. C. Straight Path Constant Velocity Approximation

Let us assume the transmitter to travel on a straight line at constant velocity V and to pass at a distance r0 (see Fig. 1) from the observer at time t = 0. From Fig. 1 one obtains r·2 = r02 + (Vt)2

(2)

Differentiating this successively, one gets 2rf = 2VH

when from (4)

. _ VH_ VH 2 r Vro + VH2

(3)

,_Γ.(1_±,)

(4)

t = 0, then r = r0 and from (3) r = 0 r = V2/r0.

(5)

This means that the relative velocity is zero at the point of closest approach (and therefore the Doppler shift is also zero). At closest approach, the rate of change of Doppler shift, i.e., the "Doppler slope," is inversely

DOPPLER EFFECT OF ARTIFICIAL SATELLITES

5

F I G . 1. Geometry for calculating Doppler shift, linear approximation.

proportional to the miss-distance TO; the slope at closest approach is also maximum as can be easily verified by calculating When t —» ±oo, then r —> Vt, and r —» V; the complete curve of the Doppler shift in this special case is shown in Fig. 2a. D. Circular Orbit Approximation, Stationary Earth

If a satellite moves in a circular orbit it is assumed that its velocity V will be constant. Naming the angles and distances according to Fig. 3, we get the following relations (assuming that the Earth is stationary) 7 =

Vt

R + ti from triangle COS:

r2 = Ä2 + (R + h)* - 2R(R + h) cos ß 2

from triangle COS 0: r0 = Ä + (Ä + hy - 2R{R + h) cos a Subtracting (7) from (6) we obtain r2 = ro2 + 2R(R + h)(cos a - cos ß)

(6) (7) (8)

6

J. MASS AND E. VASSY

J1 Af=

fr-f

L/A'>maX f m

^

-4-v (a)

il a=o

a*o

+t

•

-— (b)

Fig. 2a and 2b

' \

DOPPLER EFFECT OF ARTIFICIAL SATELLITES

1i

7

Af

(c)

FIG. 2a. Doppler shift for straight trajectory, constant velocity approximation. (Δ/) π

ÎY1 C

7*0

FIG. 2b. Doppler shift for circular orbit, stationary Earth approximation. End points marked for zenithal pass (a = 0) and nonzenithal pass (a =^ 0). fV2

R

Δ/max = - V ^ C O S 2 « c

COS 2 ffhoi

tan ßhor

FIG. 2C. Doppler shift for far-away pass, elliptical orbit. Because OG0G is a right spherical triangle, we have cos ß = cos a - cos y subtituting this in (8) r2 = r02 + 2R(R + h) cos a(l - cos y) Differentiating with respect to y 2rdr = 2R(R + h) cos a sin y dy) dr dy

=

R(R + h) cos a sin 7 r

(9)

8

J. MASS AND E. VASSY So

FIG. 3. Geometry for circular nonzenithal orbit. SoS—satellite orbit; G0G—subsatellite ground track (orbit projected on Earth); 0—Observer's position; C—center of Earth; R—Earth's radius; h—orbit altitude; So—point of closest approach.

dy V and because -ττ- = p , , we get finally dr dr dy — τ~ΊΓ dt = dy dt

=

Trn c o s a VR

1 . Vt "rs l n cTTT R + h

(10)

According to (la), the Doppler shift is Δ/=

/T/E> 1 . Vt Jfdr - - -r. = — - VR cos a - sin n , , c dt c r R + h

(H)

In order to deduce the peculiarities of the Doppler shift curve in this case, we differentiate further. The slope of the Doppler curve is proportional to (12) When t = 0, sin Vt/(R + h) = 0 and therefore r = 0, i.e., at closest approach, the Doppler shift is zero. Substituting t = 0; f = 0 in f we get (13)

DOPPLER EFFECT OF ARTIFICIAL SATELLITES

9

It is seen that the minimum distance r0 can be directly found from the measured maximum slope. Equation (13) reduces to (5) on substituting R —> oo and a = 0. The slope of the Doppler curve for a circular pass is lower than for a straight line pass by a factor cos a R/(R + h). By differentiating further it can be shown that the value of r at t = 0 is maximum, i.e., that the slope of the Doppler curve is maximum at the time of closest approach (for the above assumptions). By substituting in (10) the geometrical conditions at the moment the satellite crosses the horizon, namely, cos ß = R/(R + Λ), r hor = R tan 0hOr one obtains, using (9) r

^ ~ ^

v

U4;

tan/3 hor

This expression, for zenithal circular passes (a = 0), reduces to /"hör = ^ F c O S ß h o r = Τ 7

R

,

(15)

It can be shown that for zenithal passes the value of f (and therefore also of Δ/) reaches a maximum at the horizon, that is f — 0. For nonzenithal passes (a ^ 0), the geometrical maximum is not reached at the horizon (f = 0 below horizon). See also Eq. (22). E. General Case

The general equation for the Doppler effect has to take into account the following factors which have so far been neglected. (1) The ellipticity of the orbit: this factor causes the orbital velocity to vary. The Doppler effect may have a maximum before or after horizon. The slope of the Doppler curve need not be maximum at the moment of closest approach, especially for far-away passes [5]. The Doppler curve will be asymmetric (see Fig. 2c). (2) The Earth's rotation: this motion causes an additional asymmetry in the Doppler curves; an effect which depends, of course, on the latitude of the observer and the orbit's inclination and tends to reduce the maxima of the Doppler-curves by up to about 5 per cent. (The maximum slope, however, may be reduced by 10 per cent.) For polar orbits, this influence is negligible. When exceptionally a satellite moves opposite to the Earth's rotation the effect is, of course, to increase the Doppler shifts. (3) Ionospheric effects: the ionized layers of the upper atmosphere cause refraction and change in phase velocity [see Eq. (1)], so that the measured Doppler effect will vary considerably from the one calculated for vacuum from geometrical and kinematical data alone. This variation is especially pronounced for the lower satellite frequencies of 20 Mc, and may

10

J. MASS AND E. VASSY

reach about 15 per cent of the vacuum Doppler effect. This fact and its use for measurements on the ionosphere will be treated in Sec. I l l below. An inhomogeneous or nonspherical ionosphere may introduce additional and considerable effects, especially under twilight conditions and on days of great solar activity. (4) Phase reversals, due to satellite rotation, and polarization rotation (Faraday effect) tend to split the Doppler frequency into nearby frequencies at a spacing of twice the rotation frequency. When using a circularly polarized antenna, the Doppler frequency is merely shifted by the rotation speed (see also Sec. I l l , B). All of the above factors assume a perfectly frequency-stable transmitter in the satellite and a similarly perfect receiver on the ground. A formula for the general Doppler effect in a vacuum (including items (1) and (2) above), when satellite coordinates are known as a function of time, is given, for instance, by Williams [6] c 1 — 7 Δ/ = f = - {rs — R[sin ls sin l0 + cos l8 cos Ζ0(λο — Xs)]}fs — rsÄ[sin k · cos k — cos k sin ls cos (λο — Xs)]?8 — raR cos k cos k sin (λ0 — λ*)(λο — K)

(16)

where r is given by r2 = (r s ) 2 + R2 — 2r8Ä[sin ls sin U + cos k cos k cos (λ0 — Xs)]

(17)

where r8) Zs, and Xs are the satellite coordinates in a geocentric spherical system with longitude X measured from the equinoctial line. Ä, l0, and X0 are the observer coordinates in the same system, and Xo is the rotation rate of the Earth. It is seen that when the satellite crosses the meridian of the observer then sin (Xo — Xs) = 0 and the effect of the Earth's rotation is zero. When the orbit parameters are known, the coordinates of the satellite have to be computed first, from which the Doppler effect can be calculated, as above. It is evident that a computer program is necessary. II. Experimental Devices

Before discussing the perturbations of the Doppler curve and the various proposals and experiments which have been made to utilize the Doppler effect, a summary will be given of the experimental arrangements used to measure it [7]. The apparatus used may vary from a simple commercial receiver with

DOPPLER EFFECT OF ARTIFICIAL SATELLITES

11

an audio-oscillator and oscilloscope, to a completely automatic station with data print-out on cards ready for computer use. The different equipment set-ups employed are almost as numerous as the experimenters working in the field. In order to present the different schemes in a coherent manner, the set-ups are compared stage by stage, starting from the receiver and ending with the recorder. A. Receivers

The Doppler shift is noted as a shift in frequency of the incoming signal. It can therefore be measured after R F amplification or after heterodyning to a lower frequency; that means after the I F or AF stages. Directly measuring the high frequency (RF stage) is usually impractical because sufficient amplification is difficult to attain; also, the relative frequency change is much too small at the high frequencies. This point may be clarified by noting that assuming the original signal at frequency /o to be Doppler shifted to / 0 + Δ/0, the relative frequency change is Δ/0//0 of the order of 1:105. When the incoming signal is heterodyned to a lower frequency / 2 by mixing with a constant frequency /i (that is / 2 = /o — /i), we get the same absolute Doppler shift on the lower frequency. We find that / 0 + Δ/0 — /i = / 2 + Δ/ and the relative frequency change is now (Δ/)// 2 . A measurement after the I F stage of the receiver is a better proposition for the two reasons cited above. Measurement after further heterodyning down of the signal is often done. In this case, one reads the frequency after AF amplification of the signal mixed with a beat-frequency oscillator, which is a fixed oscillator tunable around the IF center frequency. When measuring frequency after heterodyning, the stability of the local oscillator used for heterodyning will directly influence the Dopplershift readings. Special care is taken, therefore, to ensure their drift-free operation by using crystals with thermostatic control and power supply stabilization. Oscillators can be thereby easily stabilized to within 10~5 of their operating frequency for long-time drifts and to within 10~7 for short-time drifts. This will give a 10 cps short-time drift for a 100 Mc local oscillator, which practically means a 10 cps drift on the 108 Mc satellite channel during the time of one passage as compared to a peak-to-peak Doppler-shift of about 5000 cps. For more accurate Doppler measurements special Chronometrie local oscillator circuits are sometimes used to give an accuracy of 10~9 or about 0.1 cps on the 108 Mc band. Maximizing the signal-to-noise power ratio requires the use of small effective bandwidths around the received frequency. This requirement is also more easily met after heterodyning to lower frequencies. The total

12

J. MASS AND E. VASSY

bandwidth, however, must be large enough to include the total Doppler frequency-shift, possible drifts in transmitter and receiver (local oscillator) frequency, as well as expected frequency calibration errors. Further narrowing of the effective bandwidth is possible when the center-frequency of the filter is variable. This will be dealt with later on. B. Filtering

The signal picked off from one of the receiver stages (usually from the AF stage) will now undergo further filtering and frequency measuring. 1. MANY FIXED AUDIO FILTERS

In one method which is sometimes used, the received signal is supplied in parallel to several narrow-band audio filters regularly spaced in frequency. As the Doppler-effect shifts the frequency past these filters the summed response will reach a peak value every time the received frequency passes a center frequency of one of the filters. This method provides fairly accurate time measurements between fixed frequency points, but its equipment is rather cumbersome. Moreover, part of the information is lost. On the other hand it can easily be made completely automatic (Fig. 4c). A scheme called "Rayspan" [7] consists of rapidly switching over a band of narrow-band filters and recording the outputs on a facsimile-type recorder. This operation enables one to follow the Doppler frequency on the sidebands of a modulated signal, and also permits one to recognize interference and even to distinguish simultaneous passages of several satellites. 2. FIXED NARROW BAND (LOW-PASS) FILTER, MANY FIXED BEAT FREQUENCIES

A variant of the many filter method uses a single narrow-band low-pass filter while a beat frequency oscillator beats the received frequency down to near zero. When the heterodyned frequency passes zero, the received frequency equals the local-oscillator frequency. This is seen on the oscilloscope very clearly [8]. Time is measured accurately every time the received frequency passes certain preset values marked on the local oscillator; read-out is accomplished with an oscilloscope [9]. Using phototechniques, these times can be read to an accuracy of about 0.01 sec [8]. To use this accuracy one needs, of course, a comparable time standard. Variants of this method consist of generating the reference frequencies by a single oscillator. One variant consists of taking the powerline or other stable low frequency and generating its harmonics by squaring. The mixed signal is applied via a low-pass filter to an oscilloscope. Every time the

DOPPLER EFFECT OF ARTIFICIAL SATELLITES

13

received frequency is equal to one of the harmonics a signal will appear, giving a record of time between fixed frequency points. A well known similar scheme consists of creating a circular sweep on the oscilloscope at powerline frequency [10]. The signal superimposed on the vertical plates will now give a stationary pattern every time the received frequency is a multiple of the line frequency. Another way of generating the required harmonics by short pulses at a P R F (Pulse Repetition Frequency) of 100 cps has been proposed [7]. The harmonic coincidence method consists of chopping the received signal by an accurate 100 cps chopper. At the output the signal will increase every time the received frequency is a multiple of 100 cps [10]. 3. VARIABLE FILTER

The additional narrow-band filtering which is sought can be achieved by a variable audio-filter (Fig. 4a), manually tuned. In this case the operator simply adjusts the filter for maximum response. Tunable filters having a bandwidth of about 20 cps at 1000 cps are commercially available. 4. FIXED NARROW-BAND FILTER, VARIABLE OSCILLATOR

The methods discussed above all measure time intervals between fixed frequency points. This may present certain difficulties when the Doppler curve is not monotonie (see Sec. I, E) and some of the methods are confused by momentary loss of signal through fading. Continuous measurement of frequency using a single fixed filter is possible. A variable oscillator is used to beat the signal down to zero or to a certain fixed frequency. This process of beating down the signal requires that the oscillator track the Doppler-shifted satellite frequency; when tracking is ideal, that is, when the locally generated frequency is always equal to (or at constant difference from) the received frequency, then the local oscillator frequency can be measured instead of the received one, thus leading to the advantages of having the measured frequency free from noise and its power level high. The means of tracking with a local oscillator is described presently. C. Frequency Tracking 1. MANUAL TRACKING

In order to tune a variable oscillator to track a received frequency a method of comparison is needed. For very inacurrate tracking the ear can be used to compare the two frequencies. A better solution is visual tracking and many devices have been used to this effect.

14

J. MASS AND E. VASSY

Ψ receiver with BFO

audio

variable filter

Ψ

counter

oscilloscope variable filter

variable audio oscillator

9 0 phase shifter

NARROW-BAND AUDIO FILTERS

RECEIVER WITH BFO

FIG. 4(a) Simple assembly for measuring Doppler effect; (b) assembly for manual tracking of the received frequency with the help of a circular sweep ("Toothwheel" pattern on screen); (c) assembly for automatic recording of Doppler shift. Simultaneous passes as well as modulated carrier can be recorded.

Circular Sweep. Let a circular sweep (Fig. 4b) [7] be created on an oscilloscope by feeding sin 2irf\t and cos 2irfit to the vertical and horizontal plates, respectively and let the trace be intensity-modulated by the received frequency. When the received frequency / is equal to /i a half-moon shape

DOPPLER EFFECT OF ARTIFICIAL SATELLITES

15

will be seen. As the received frequency changes the half-moon will start to rotate in a sense depending on whether / > /i or / < /i. It is therefore easy for the operator to change the circular sweep frequency/i to keep the half-moon stationary. When the change in Doppler shift is relatively fast (on higher carrier frequencies) it is easier to work with a tooth-wheel pattern, that is when /i is a submultiple of /. In this case the pattern will not rotate so fast for a certain difference in frequency. This method has proved itself under very difficult conditions, when the signal is very noisy and severely fading. Good results were obtained for the first Sputniks which had a pulsed signal. The operator was required to correct his local oscillator rapidly during the short time the signal was on [11.] In the circular sweep method the human eye is very efficiently used as a filtering device, as all noise at frequencies which are more than one cycle away from the desired frequency will "race around" the circular sweep and will thereby be distributed equally around. The half-moon or toothwheel shape will appear clearly, even when faint, on the background of evenly distributed noise. A certain difficulty lies in the generation of a circular sweep of variable frequency. 2. LISSAJOUS PATTERNS

In this method the received frequency is connected to one pair of oscilloscope plates and the comparison frequency to the other. When the signal is clear enough this arrangement gives the well known Lissajous patterns providing the two signals are harmonically related (an ellipse when /i = /, a figure " 8 " when / = 2/i, etc.). The operator can tune his local oscillator by attempting to keep the Lissajous figures stable. The sign of the difference in frequencies is not so easily recognizable as in the circular sweep method. 3. AUTOMATIC TRACKING OR PHASE-LOCK

Results equivalent to the manual tuning of the local oscillator can be achieved by automatically tuning the oscillator to follow the received frequency by automatic phase comparison [12]. This is the system now generally used for accurate, routine Doppler measurements [5] [13]. It is extremely sensitive, the effective noise bandwidths being reduced to about 10-20 cps (Fig. 5). When / and /i are equal and exactly 90° out-of-phase the output of the phase comparator will be zero. When / changes slightly the relative phase will start to change. The phase comparator will now read a voltage which, when passed through a low-pass filter, will change the frequency /i. The tendency of the circuit is to keep /i equal to / and at a constant phase

16

J . MASS AND E .

VASSY

PHASE (COMPARATOR

LOW PASS FILTER

*f,

VOLTAGE CONTROLLED OSCILLATOR

CONTROL VOLTAGE

FIG. 5. Basic phase-lock loop.

relationship given by the voltage necessary for keeping the voltage-controlled oscillator at the required frequency. It is evident that the low-pass filter must be wide enough to allow the voltage-controlled oscillator to change fast enough to follow a changing / ; in practice this width will be about 20 cps. Not only the calculated Dopplerrate must be accommodated but also frequency scintillations created by ionospheric effects. When the signal fades to zero for a certain time, phase-lock is lost and must either be reestablished by manual tuning or by rather difficult automatic search techniques. D. Frequency Measuring

After the received signal has been amplified and filtered as far as possible, its frequency, or that of the comparison oscillator, must be accurately measured. The basic methods used are periodic counting, continuous average rate counting, and frequency discrimination. 1. PERIODIC COUNTING

An electronic counter shifts an electronic memory device by one digit every time a signal passes a certain threshold in a positive sense. When combined with an accurate time gate this will "count" the frequency of an incoming signal. Counting during one second one gets the frequency correct to ± 1 cps, depending on the phase of the signal when the time gate opens. For a tenfold increase in accuracy the counting must be continued for ten seconds, a period rather long for recording Doppler effect. Manual reading of counter frequency is practical to about once every

DOPPLER EFFECT OF ARTIFICIAL SATELLITES

17

4 sec. Automatic print-out is usually done once every 2 sec. The read-out frequency can be directly printed on cards for computer processing. 2. FREQUENCY M E T E R

The so-called frequency meter usually converts the measured frequency into square pulses and then measures the average rectified current. The averaging circuit introduces, of course, a certain lag; the output, however, is continuous and can be recorded automatically on a pen-type recorder [14]. 3. FREQUENCY DISCRIMINATOR

It is possible to use a circuit which is similar to an FM demodulator based on the phase relationship in tuned circuits, but this method has a rather low accuracy. E. Reduction of Doppler Data

For many applications the data required from the Doppler-curve are : (1) time of maximum slope, and (2) value of maximum slope. For close to zenithal passes they measure almost exactly the time of closest approach and indirectly the distance at closest-approach, as can be seen from Eq. (13). As the recorded Doppler curves usually show some irregularities some skill is required to deduce the required data. Several graphical and computational schemes have been proposed as well as some automatic reduction techniques. 1. GRAPHICAL METHODS

Four graphical methods are briefly discussed below. (1) Satellites with nearly circular orbits will give almost symmetrical Doppler curves, when one neglects the small influence of the Earth's rotation (even this is symmetric to a first approximation). This property of symmetry can be utilized to find the exact point of closest approach. Most simply [9], the Doppler curve is copied on transparent paper, and a point is sought on the curve around which the copied curve can be turned by 180° to cover the original curve again. This point is the point of closest approach, and the slope of the curve at this point should be maximum. (2) Another graphical approach requires the drawing of lines at various slopes in such a way that (see Fig. 6) ΑχΒι = BiCi)

A2B2 = B2C2

The points B will almost coincide, their average being the point of closest approach [8.]

18

J. MASS AND E. VASSY

F I G . 6. Illustration of graphical method for finding point of closest approach.

(3) A third approach consists of estimating from the curve its point of inflection (where Δ/ = 0) by assuming the recorded curve to approximate the analytical expression Δ/(ί) = at* + bt2 + ct + d Differentiating this twice one gets Δ/(ί) = &at + 2b Setting Af(t) = 0 one gets for the inflection point U = — b/3a and for the slope at that instant Afi(t) = —biU + C. Assuming A, B, C, and D to be the measured Δ/ at 4 equally spaced time points, th t2, k, and U as far apart as possible, and solving the set of 4 linear equations for the constants a, fo, c, and d) one gets U-

*2-ΓΔί

(B - A) + (B - O 3 ( 5

_

Q

+

(Z)

_

C)

Where At is the time spacing between th t2, h, and U. Quick graphical or arithmetical solution for ti is thus possible. Similar formulas are easily developed for Afti [15]. (4) Several authors have noted that the Doppler curve can be modified by change of variables to give a straight line, from which one can directly measure the velocity of the satellite (from the slope) [7] [8] [16] and the distance of closest approach (from the y intercept). This is the case when one plots (t/f)2 versus t2, as can easily be shown for the straight line approximation case by substituting (3) in (2). The point of closest approach (t = 0), however, must first be found by other methods. In criticism of these methods it should be noted that for elliptical orbits there is no symmetry, and neither are the circular passes accurately symmetric when one considers the effect of the Earth's rotation. The real point of closest approach is the one which gives zero Doppler effect (that is where r = 0). This point, found on the curve when the transmitting frequency is accurately known, can be distorted by refraction effects as well as by the finite velocity of the radio waves.

DOPPLER EFFECT OF ARTIFICIAL SATELLITES

19

2. AUTOMATIC DATA REDUCTION

Three automatic reduction methods are mentioned. (1) When the frequency is measured at fixed intervals it is, of course, not difficult to establish a computer program for finding the point of symmetry [8]. (2) A method has been suggested [10] by which the magnetic tape containing the Doppler frequency is picked off at two reading heads spaced some distance apart. The two picked-off frequencies are beaten against each other, and when the beat frequency is maximum the point of inflection has been reached. (3) When the frequency of the satellite transmitter is accurately known, an oscillator of equal frequency can be beaten with the incoming signal. The beat frequency is passed through a very narrow band filter centered at, say, 100 cps. This filter will register a signal twice during a passage. The midpoint between the two signals is the point of closest approach, and the distance between the signals measures the slope at closest approach [40].

III. Perturbations

For practical utilization of the Doppler curves in tracking and navigation it is required that they be as undistorted as possible by perturbing effects. On the other hand, it is measurement of just these perturbing effects, when the satellite's orbit is known, that gives interesting scientific information. In this section, an analysis of the main effects will be given, while the practical utilization and scientific results will be reserved for Sec. IV. A. Refraction 1. SPHERICAL IONOSPHERE

It is well known that the upper layers of the Earth's atmosphere are ionized, mainly by the Sun's radiation. The ionized part of the atmosphere, the so-called ionosphere, extends from about 80 km to about 1000 km and has a maximum in the so-called F layer at about 250 km (Fig. 7) [17]. An electromagnetic wave entering this partly conducting medium is refracted according to the refractive index μ which is dependent on the concentration of free electrons, on the direction and amplitude of the magnetic field, and on the frequency of shocks suffered by each electron. For most practical purposes at frequencies 20 Mc and above, μ is assumed to depend

20

J. MASS AND E . VASSY

9 0 0 km

800

'

700

"

600

'

500

'

400 300

'

200

'

100 ■

5·Ιθ"

l0-l0"el/m8

FIG. 7. Approximate typical electron distribution in the ionosphere. A. summer day; B. summer night; C. winter night. Winter day curve to the right of A.

on the electron concentration only, μ ~ 1 — [(40.25iVe)//2] where Ne is the number of electrons per cubic meter. Refraction is manifested by a bending of the ray path from a satellite away from Earth (Fig. 8), as well as an increase in the phase velocity above the velocity of light. Since the time differential of the ray-path gives the Doppler effect [see Eq. (1)], it can be seen that the combined effect of the ionosphere is rather complex. For certain simplifying assumptions concerning the shape of the ionosphere, several calculations are available for the effects of refraction as a function of maximum electron concentration at the F layer peak [15] [18] [19]. All results so far have been limited to the assumption that there are spherical layers in the ionosphere, i.e., thatiVe, the concentration of electrons, is a function of the altitude only, or, in still other words, that the horizontal gradient is zero. For this assumption, Weekes has derived an interesting general result [20], which is stated below. Using Bouguer's refraction law for spherical layers μϋ sin ψ = Ro sin i

(18)

DOPPLER EFFECT OF ARTIFICIAL SATELLITES

21

FIG. 8. Geometry for calculating Doppler effect with refraction.

When defining q = (Α/Α0)μ we get sin ψ = (l/q) sin i and from geometric considerations

The total geocentric angle between satellite and observer can therefore be expressed as a function of μ(Α) or q(R) by (19) Inversing the order of differentiating and integration we get, assuming the upper limit Ra to be variable (20) The measured Doppler effect is

as ds = dR/cos ψ from geometrical considerations. By carrying out the differentiation and substituting (20) it can be shown that (21)

22

J. MASS AND E. VASSY

When the satellite is in a circular orbit Rs = 0 and the formula reduces to a very simple law - Δ/ = R0èa sin i

(22)

This may also be written as follows - Δ/ = Α808μ8 sin i/>s

(22a)

As in a circular orbit 0S is constant; it follows that the Doppler effect measured is directly proportional to sin i. This is true, of course, also for the case of zero refraction (μ = 1) in which case the Doppler effect can be calculated by calculating sini from purely geometrical considerations. For actual cases, a model ionosphere, that is μ(Α) is assumed, and by way of iteration [19] or direct calculation [15] [18] the variation of the angle of incidence as a function of time i(t) is found. This gives the Doppler effect by the Weekes formula (22). Assuming a parabolic distribution of electron density around the altitude of maximum ionization [18] or an unsymmetric model [19], a maximum for sini which corresponds to a critical angle i smaller than 90° is obtained. The critical angle is about 55° for normal daytime ionization at a satellite frequency of 20 Mc and it is 83° for 40 Mc. For 108 Mc, the angle is greater than 90°, that is, signals from a satellite can be received at about 2° below the horizon. Occasional signals are received due to anomalous effects much farther away [21]. Table I gives the values of i max for different frequencies as well as the maximum deviation of the incident ray from the geometrical angle of incidence % (Fig. 8). The ionosphere model assumed differs for various authors, but has a maximum plasma frequency of 12.7 Mc. The assumed satellite altitude is 1000 km. Millman's calculations of the Doppler error due to refraction for a satellite at 300 km and ionosphere having a maximum T A B L E I. VARIATION OF W X WITH F R E Q U E N C Y ; MAXIMUM DEVIATION OF INCIDENT R A Y FROM GEOMETRICAL ANGLE OF INCIDENCE %Q.

(

Frequency

imaxe

5maxa

imaxb

sin t'max \ Maximum Doppler —;—— J . _ A„ sin to ) I error due to sin io / refraction c _ |-A/max(M g* 1 ) Ί imax in 3maxb ίΔ/πιβχίμ = 1) J Ä % cps LA/max(M h = (670km

20 Mc 40 Mc 108 Mc

~53° 83° —90°

~15° ~10° 0.9°

~55° 83° ~90°

0.6°

° Carru et al., réf. 19. Rawer, réf. 18. e Bournazel, réf. 15.

b

—

~0.80 ~0.99 ~1.0

~57°

— —■

-14% 1% 0.1%

—57 —8.6 —2.3

DOPPLER EFFECT OF ARTIFICIAL SATELLITEZ

23

plasma frequency of 10.0 Mc at this altitude, show it to be 68 cps at 108 Mc and 33 cps at 200 Mc. The great difference is due to the refractive index at satellite altitude. It is seen that the refraction effect on the Doppler shift is almost negligible for 108 Mc or higher frequencies. The absolute deviation due to refraction is approximately inversely proportional to frequency squared. When i = 0 (satellite overhead) the Doppler effect is proportional to qB, that is to μ8 the refraction index at satellite altitude as can be seen from Eq. (21). When the satellite orbit is accurately known (rs known), a convenient method of measuring electron density at satellite altitudes is thus available. An analogous method for measuring electron density from Doppler measurements with an ascending rocket has been used as well (see Sec. IV, C, 2). A formula equivalent to the one in Eq. (21) can be obtained directly from geometrical considerations at the satellite [22]. Considering Eq. (22a), it is noted that even for a circular orbit the actual Doppler effect is influenced by the refractive index at the satellite altitude, and by the electron distribution below the satellite as expressed by bending of the ray implicit in sin ^ s . The relative importance of these two components varies according to the angle of incidence i and also to the height of the satellite. At great altitudes μ8 is nearly 1 and the electron distribution below the satellite becomes more important. When the satellite orbits at low altitudes the electron density at satellite height (as implicit in μ8) becomes decisive. 2 . NONSPHERICAL IONOSPHERE

In the previous section the ionosphere was assumed to consist of constant spherical layers, that is ionization was a function of altituda only. In reality, the ionosphere is constantly changing as a function of time and place. One even speaks of ionized "clouds" moving at speeds of about 100 meters/sec. This tends to distort the Doppler curve even further. A horizontal gradient of electron concentration in the direction of the satellite orbit will distort the Doppler curve unsymmetrically. Such horizontal gradients will almost always be present when the satellite moves from high latitudes towards lower ones or vice versa, and in the morning and evening when the ionosphere shifts between nighttime and daytime conditions [17] [23]. The distortion for a horizontal gradient (/i2 — / 2 2 )/AL (where /i and / 2 are the critical plasma frequencies at two points in the ionosphere at distance AL, in the direction of satellite orbit) is given by Tischer [24] as approximately (in slightly different notation)

lh.fi*-frr 2cf AL

24

J. MASS AND E. VASSY

where ft, c, /, and V are as defined earlier and e is the error in frequency. This gives for /i = 8 Mc and f2 = 7 Mc a shift in the Doppler curve of 5 cps when the satellite is above the observer. Near sunrise or sunset the ionospheric layers are not spherical but rather inclined relative to the local horizon. This makes the assumptions of the Weekes formula invalid. The change in maximum Doppler frequency due to layer inclination will reach about 1 per cent for a 5° inclination at 20 Mc or about 13 cps. If, as sometimes assumed, the upper layers of the ionosphere tend to be inclined to follow the magnetic lines of force, then one should also measure a marked change in the Doppler frequency [43]. 3. HETEROGENEOUS IONOSPHERE

Very often the ionosphere is not calm, especially when so-called spread F or sporadic E conditions prevail. This cloudiness or turbulence in the distribution of electron contents will cause the received signal to arrive on a ray-path and continually change back and forth. It will also cause interference effects between rays arriving from different directions. This can be compared to viewing a moving point through an imperfect glass window. The over-all effect on the received signal is scintillation in amplitude as well as in Doppler frequency. These scintillations can reach important amplitudes as has been shown by Arendt and Hutchinson [25], of the order of 20 cps, usually on frequencies of 20 and 40 Mc, but often also on 108 Mc. Two curves are reproduced in Figs. 9 and 10 showing deviations from a smooth Doppler curve on 20 Mc and 108 Mc as recorded at ValJoyeux, France. It should be mentioned that part of the registered frequency scintillations may be due to transmitter instabilities [5] and partly also due to the passage of the satellite through local "clouds" of electrons (see Eq. 22a). A "scintillation" of 1-2 cps must also be allowed for the measuring system which in the case of the reproduced curves consisted of a tunable narrowband filter followed by a counter, counting during 1 sec at intervals of 5 sec. It is interesting to note in this connection that the equivalent frequency noise at 100 Mc for a system using first order refraction correction by the harmonic frequency method (see Sec. IV, B, 2) leaves only about 0.1-0.2 cps rms frequency noise, when a highly stable transmitter in the satellite is used (Transit 2A, 1960 Eta 1). B. Noise, Modulation and Polarization Rotation

When the received signal is weak and approaches the noise level in the filtered signal fed to the frequency-meter, then the frequency read will tend to be wrong. It depends on the characteristics of the frequency meter

FIG. 9. Doppler shift and ampUtude record a t / = 20 Mc, recorded at Val-Joyeux, France (48° 49' N, 2°01Έ). Note periodic amplitude changes in the left part caused by polarization rotation (Faraday effect). Also note frequency scintillations at ends. Doppler frequency is counted from arbitrary zero. T.C.A. means time of closest approach. fcO

>4-£_·

• I I 55 UT

μ.

#

/ ··

\

/

T.Ç.A.

\

\

,

V

/

1

12 0 0 UT

Δ (Ai)

1_* • " - ^ - ^ . - r - - - T l .

60 ETA 2 REV. 3078 25 JAN 1961 VAL-JOYEUX

-20

-30

cps/sec

♦ Δ(ΔΙ)

F I G . 10. Record of Doppler shift with arbitrary reference and Doppler slope showing frequency scintillations. Satellite frequency 108 Mc. Recorded at Val-Joyeux, France (48° 49' N, 2°01Έ).

-2500

-2000

1500

-1000

-500

500

1000

1500

cps 2000

2250

CO

< >

Ö

>

> GO

to

DOPPLER EFFECT OF ARTIFICIAL SATELLITES

27

employed whether the frequency read will be larger or smaller. A zero crossing counter will tend to read too high a frequency. Amplitude or frequency modulation of the received carrier should not influence a frequency-meter which counts zero crossings or crossings of a level not affected by the amplitude modulation (modulation index smaller than 1). Polarization fading due to the Faraday effect or due to the rotation of the satellite itself may produce amplitude zeros in the received signal. These zeros are frequent when linear transmitting and receiving antennas are used, especially, at the lower frequencies (below 100 Mc). At the time of zero the frequency meter will tend to read the noise frequency or, if set so that it will be unaffected by noise, will read zero. In both cases an error is introduced. When the transmitting antenna in the satellite is circularly polarized the received wave will generally be elliptically polarized with the major axis of the ellipse rotating due to satellite rotation and due to rotation produced by the magneto-ionic effect on passage through the ionosphere [26, 41]. The frequency received in this case will be either greater or smaller by the frequency of ellipse rotation, according to whether the ellipse rotates with or against the sense of rotation of the electric vector. This effect will cause an error of about 0.5 cps at 20 Mc at daytime due to Faraday rotations alone. It is approximately inversely proportional to / 2 . C. Tropospheric Effects

The refractive index in the lower layers of the atmosphere (the so-called troposphere) is almost independent of frequency up to 30,000 Mc and is a little greater than 1. It depends mainly on the humidity. This refraction will cause an additional error in the measured Doppler effect. The Doppler error will be 0 at zenith, about 2 cps at i = 40° for / = 108 Mc, about 6 cps at i = 65° and nearly zero again for i = 90°. These figures have been estimated from values of angular errors due to refraction in a wet troposphere given by Millman [27] and from the Weekes formula [Eq. (22)], assuming the satellite at h = 1000 km. The relative error due to tropospheric refraction should be almost constant, and therefore the absolute error in cycles per second proportional to frequency. D. Relativity Effects

Due to the finite speed of electromagnetic waves the Doppler curve will be slightly distorted in time. The time of closest approach accurately reduced from a Doppler curve would of course be the actual time of closest

28

J. MASS AND E. VASSY

approach plus the time needed for the wave to travel down from the satellite. For a satellite height of 1000 km this corresponds to 1/300 sec, during which time the satellite has actually moved away about 28 m. In fact, the whole Doppler curve should be shifted towards earlier time by about 3 m sec at the center and by about 10 m sec at the ends. The Lorentz transformation has to be applied in measuring time (or frequency) in one system for a phenemenon produced in another system moving at relative velocity V. The frequency received on an Earth-bound receiver / r from a transmitter in a satellite (and vice versa) of frequency / t will therefore be accurately (in vacuum) [1]

1 + z-

/ r = ft

I

fi

where VR = V cos 0, and 0 is the angle between the relative velocity vector and the direction of the receiver as measured in the coordinate system of the transmitter. For V/c -

te

J

\

S

v) 20°

ô