Relativistic corrections calculated for one way Doppler system used in calculating orbital velocit

Gaposchkin, E. M.

Wright, J. P.

English

NASA Technical Reports Server

c 
GENERAL RELATIVITY 
IN SATELLITE ORBITS 
E. M. GAPOSCHKIN AND J. P. WRIGHT 
https://ntrs.nasa.gov/search.jsp?R=19680024525 2020-03-12T06:24:59+00:00Z
Research  in Space Science 
S A 0  Special Report  No. 283 
A MEASURABLE EFFECT O F  GENERAL RELATIVITY IN 
SATELLITE ORBITS 
E. M. Gaposchkin and James P. Wright 
. 
August 1, 1968 
Smithsonian Institution 
Astrophysical Observatory 
Cambridge, Massachusetts 021 38 
808-1 
k 
TABLE OF CONTENTS 
Page 
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ABSTRACT i i i  
A MEASURABLE EFFECT OF GENERAL RELATIVITY IN 
SATELLITEORBITS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  
1 
4 
.. 
11 
ABSTRACT 
The reduction of data for  satell i tes having a one-way Doppler sys tem 
should include the general  relativistic t e rms .  
correct ion te rm,  which should be used in obtaining the velocity of a satell i te 
f r o m  the observed frequency, that is periodic and proportional to the eccen- 
tricity. 
g rea te r  than 0.05. 
using single and double Doppler systems simultaneously f rom the s a m e  ear th  
location. 
There is a general  relativist ic 
The effect should be observable f o r  satell i tes with an eccentricity 
We should also be able to measu re  the effect direct ly  by 
c 
L'analyse des donndes dans 12 cas de satellites ayant un systime 
Doppler sens unique devrait comprandre 1es t13rmes ggnhraux de relati- 
vit6. I1 y a un terme &n&ral de correction, terme relativiste, qui 
devrait 6tre employ6 pour obtenir la vitesse d'un satcllita partir 
de la frdquence observe'e, c'est 
tionnel l'excentricitg. L'effet devrait stre observable pour des 
satsllites excentricitd plus gr3nde que 0,05. Nous devrions aussi 
pouvoir mesurer l'effet dir2ctement en employant en &me temps des sys- 
times Doppler simples et doubles A un &me endroit sur la terre. 
dire un terme p6riodique et propor- 
iii 
KOHCITEXT 
iv 
A MEASURABLE EFFECT O F  GENERAL RELATIVITY IN 
SATELLITE ORBITS 
E. M. Gaposchkin and James  P. Wright 
There  a r e  some close ea r th  satell i tes that c a r r y  R F  t ransmi t te r  instru-  
mentation. By use  of the Doppler effect, the radiation f r o m  this t ransmi t te r ,  
which is received a t  ground stations, is used to infer the line-of-sight velocity 
component. 
of TRANET satel l i tes  u se  the Doppler system. These Doppler data have been 
successfully used for  a lmost  a decade in orbit  analysis,  geodetic research ,  
and navigation. 
The satell i te navigation sys tem known as TRANET and the s e r i e s  
The transmitt ing osci l la tors  in the satell i tes s e e m  to have a stability of 
This has been established by laboratory experi-  bet ter  than 1 par t  in l o l o .  
ments  (Sykes, Smith, and Spencer, 1962) and by position analyses by 
Newton (1966) and E. M. Gaposchkin (unpublished). 
Since the osci l la tors  (which a r e  clocks) will run a t  different ra tes  
(Landau and Lifshitz, 1962; Clemence and Szebehely, 1967) in different 
potential fields, the calculated Doppler velocity attributed to the satell i te 
will be different depending upon whether one uses  Newtonian o r  relativist ic 
(Einsteinian) theory to reduce the data  of the one-way Doppler satell i tes,  
such a s  TRANET observations. 
way Doppler satell i tes such a s  the Goddard Range and Range Rate Doppler 
satel l i tes  (Kruger,  1965). 
Doppler data should differ by the relativist ic effect. 
But this effect will not be present for  two- 
In fact, simultaneous one-way and two-way 
This work was supported in par t  by grant  NSR 09-015-018 
Aeronautics and Space Administration. 
f r o m  the National 
1 
In the framework of Einstein 's  theory, the ratio of the observed frequency 
) to the standard frequency on the ear th  (v,) would be (for a symmetr ica l  ('obs 
ear th)  
V 
C E c r  
2 2  
C 
2 c r  
S 
where M is the mass of the earth, V the velocity of the ea r th ' s  rotation 
at the observer ,  r and V a r e  the geocentric distance and total velocity 
of the satell i te,  respectively, r is  the distance f rom the observer  to the sa te l -  
lite, G is the gravitational constant, and c is the speed of light. This expres-  
sion includes the special  relativistic t ransverse  Doppler effect. In practice,  
the nonsphericity of the ear th ' s  potential is included in the reduction, and the 
ra te  of the ear th  clock will depend on the potential a t  the station. 
E E 
S S 
We may rewri te  the above expression in t e r m s  of the orbital  parameters  
as 
cos f t const t e r m s  
V 
obs 'E 
obs 
V 
V 
where the quantities a, e, and f a re  the semimajor  axis ,  eccentricity,  and 
t rue  anomaly of the orbit, respectively. In computing effects of this type, 
these parameters  must  be calculated in the framework of a relativist ic theory 
o r  Newtonian theory although a, e ,  and f will be near ly  the s a m e  numerically. 
The resulting expression is separated into two par ts .  One is a constant 
t e r m  that is indistinguishable f rom a ze ro  se t  e r r o r  in the oscil lator frequency. 
The second t e r m  is multiplied by the eccentricity and has a period of 
1 revolution. 
2 
The amplitude of the periodic t e r m  for  a typical geodetic satell i te (Geos i ) ,  
6 - 1  with a = 8.07  X 10 
variation of 5 c m  sec  There  will be post-Newtonian c o r r e c -  
tions to the orbital  parameters  in  the reduction of the orbit, but, since these 
a r e  essentially constant t e rms ,  we can calculate the periodic effect using 
the Newtonian values of G, ME, a, e, and c. I t  is much m o r e  difficult to 
detect a constant effect than a periodic effect. 
m and e = 0.07, is = 2 . 5  c m  sec  in velocity o r  a total 
-1 fo r  one orbit. 
-1 
A stability of 1 par t  in lo1' in the oscil lator is equivalent t o  3 c m  sec  
and there  is abundant evidence that the Doppler data can be acquired to an 
accuracy of 1 to 2 cm sec  
ionospheric effects (Willman and Tucker, 1968; Tucker and Fannin, 1968; 
Berbert ,  1967; Applied Physics Laboratory, 1968). 
, 
-1 with a two-frequency system, which removes the 
I t  is quite c lear ,  then, that this periodic t e r m  should be included in the 
reduction of the data. 
With an  improved capability in orbit  calculation, i t  would be possible to  
use  this effect to establish the validity of relativity, i f  that is required.  To 
achieve this capability the gravity-field and atmospheric  reductions and the 
ionospheric and tropospheric effects will have to be improved, as well a s  
other effects that have the same  spectral  charac te r i s t ics  as the periodic te rm.  
It s eems ,  therefore,  that we can make a positive statement on the 
validity of the t e r m s  in the formulation of Einstein 's  theory that predict  
the gravitational redshift. The calculations show that the residuals a r e  
reduced consistently if  the general relativist ic effect is included; and, unless 
there  is an effect that gives e r r o r s  in the Doppler velocity of the same 
magnitude and direction, the calculations show that the relativist ic effect is 
real .  At this stage we cannot choose between the various theories using the 
effect, but it may be possible to do this in the near  future (Dicke, 1967). 
3 
REFERENCES 
BERBERT, J. 
1967. Intercomparison of GEOS-A observation sys tems.  NASA 
X-5 14 -67 - 3 1 5, July. 
APPLIED PHYSICS LABORATORY (STAFF) 
1968. An intercomparison of GRARR and TRANET data. Johns Hopkins 
Univ., Document TG 967, 177 pp. 
CLEMENCE, G. M. andSZEBEHELY, V. 
1967. Annual variations of an atomic clock. Astron. Journ. ,  vol. 72,  
pp. 1324-1326. 
DICKE, R. H. 
1967. Gravitational theory and observation. Phys. Today, vol. 20, 
pp. 55-70. 
KRUGER, B. 
1965. The Doppler equation in range and range rate measurement .  
NASA X-507-65-385, October, 25 pp. 
LANDAU, L. and LIFSHITZ, E. M. 
1962. The Classical  Theory of Fields ,  2nd ed. Addison-Wesley Publishing 
Co. ,  Inc. ,  Reading, Mass.  
NEWTON, R. R. 
1967. The Navy navigation satellite sys  tern. In Space Resea rch  VII, 
North-Holland Publ. Co. , Amsterdam,  pp. 735-763. 
SYKES, R. A. ,  SMITH, W. C. and SPENCER, W. J. 
1962. Per formance  of precision quartz  -crystal controlled frequency 
genera tors .  IRE Trans.  Ins t rum. ,  vol. 1, no. 11, pp. 243-247. 
TUCKER, A. J. and FANNIN, B. M. 
1968. Analysis of ionospheric contributions to the Doppler shift of CW 
signals f r o m  artif icial  earth satel l i tes .  Journ.  Geophys. Res . ,  
V O ~ .  73, pp. 4325-4334. 
WILLMAN, J. F. and TUCKER, A. J. 
1968. Accuracy of satellite Doppler data fo r  ionospheric study, naviga- 
tion and geodesy. Journ. Geophys. Res . ,  vol. 73, pp. 385-392. 
4 
BIOGRAPHICAL NOTES 
E. M. GAPOSCHKIN graduated in Electr ical  Engineering f r o m  Tufts 
He received a Degree of Numerical Analysis in 1959 University in 1957. 
f r o m  Cambridge University in England and is now working toward a doctorate 
in geophysics a t  Harvard University. 
Since joining the staff a t  Smithsonian in 1959, Mr.  Gaposchkin has held 
positions as programmer and Division Chief of the Computations Division, 
and is now a mathematician in the Research and Analysis Department. He 
has helped to develop the basic computer program used in a l l  analyses of 
satell i te motion. 
His main interests  include satellite geodesy and geophysics and applied 
mathematics.  
JAMES P. WRIGHT received the B.S. degree from the University of 
F lor ida  in 1956 and the Ph.D. degree in chemistry f r o m  the University of 
Chicago in 1961. 
Dr.  Wright joined the staff of the Smithsonian Astrophysical Observatory 
a s  an astrophysicist  in 1964. F r o m  1961 to 1963 he was a National Science 
Foundation - National Research  Council Associate a t  the Institute for  Space 
Studies, and in 1963 and 1964 he spent a year  at the Mathematics Research  
Center,  Madis on, Wisconsin. 
J 
Dr.  Wright's f ie lds  of investigation include general  relativist ic effects 
and their  observation; high-energy astrophysics;  and interpretation of X-ray 
and cosmic- ray  background. 
NO TIC E 
I 
b 
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A measurable effect of general relativity in satellite orbits

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