Combined solution for low degree longitude harmonics of gravity from twelve and twenty-four hour satellite

Wagner, C. A.

English

NASA Technical Reports Server

- 
’ X-643-68-131 
\ 
I ’  
I 
I 
I 
I 
COMBINED S O L U T I ~ N  FOR LOW 
I 
I DEGREE LONGITUDE HARMONICS 
OF GRAVITY FROM 12AND 24 i :% 
HOUR SATELLITES I \  
I t  ’ .  I , 
f f * 
GPO PRICE $ 
CFSTI PRICE(S) $ CARL A. WAGNER , I 
I 
G O  I -  4 Hard copy (HC) 2 
I 1 ,  
- 1  
Microfiche !M F) LS 
ff 653 July 65 
I 
d 
I 
I \ 
APRIL 1968 
, 
, 
https://ntrs.nasa.gov/search.jsp?R=19680012451 2020-03-24T00:12:12+00:00Z
X- 643- 68- 13 1 
COMBINED SOLUTION FOR LOW DEGREE LONGITUDE HARMONICS 
OF GRAVITY FROM 12 AND 24 HOUR SATELLITES 
Carl A. Wagner 
April  1968 
GODDARD SPACE FLIGHT CENTER 
Greenbelt, Maryland 
Combined Solution for  Low Degree Longitude Harmonics of 
Gravity from 12 and 24 Hour Satellites 
Car l  A. Wagner 
Special Projects Branch 
Goddard Space Flight Center 
Greenbelt, Maryland 
Abstract 
Over 600 near  stationary orbits of five high altitude communications satel- 
lites have been examined for the strong effects on them of low degree earth 
gravity harmonics. The accelerated dr i f t  of these 1 2  and 24 hour orbits, due to 
the earth 's  anomalous potential, has been measured at 50 longitude locations. 
From these accelerations, 5 low degree ear th  gravity harmonics which resonate 
with these orbits have been calculated (10 gravity coefficients). The best de- 
termined harmonic is felt to be H,,, which dominates the accelerated drift of 
the three 24 hour satellites (SYNCOMS 2, 3, and Early Bird) and the two 12 hour 
satellites (Molniya 1 and Cosmos 41). The constants found for  this harmonic 
a r e  (Normalized): 106C,, = 2.40 f .03, and 106S,,= -1.43 f .03. The other 
four harmonics in decreasing order  of their  estimated strength of absolute de- 
termination are: H31, H 4 4 ,  H,, and H31. Their normalized coefficients are 
found to be: 106C,, = 0.69 & 0.20, 106S3, = -0.53 & 0.20, 106C4, = 0.02, 
106S4,= 0.70, 106C3, = 0.16, 106S3, = 1.10, 106C3, = -0.42, and 106S,, =-1.58 
The measured drift accelerations themselves have been compared to accelera- 
tions calculated from a number of recently published satellite and combined 
satellite-surface gravity geoids. The "best'l geoid representing these accelera- 
tions is a 1967 S.A.O. determination of Walter  Kohnlein. 
1 
INTRODUCTION 
The refinement of knowledge about the gravitational field of the earth is 
still a subject of great interest  to geophysicists. The unique contribution of 
satellite geodesy in this effort is well recognized. That contribution is princi- 
pally to the accurate detection of large scale anomalies in  the field. 
The first major contributions in satellite geodesy have been towards the 
precise definition of the axisymmetric, or zonal, part of the potential, down to 
about the 10th degree harmonic. This definition has been principally due to the 
work of Kozai in Japan These ear ly  
efforts have been successful because of the ready availability of many different 
satellites whose orbits and orbit planes a r e  strongly rotated over long periods 
of time by the zonal components of the field. Because of the averaging effect of 
the rotation of the earth, no such easily observed effects, due to the non-zonal 
part of the potential, a re  obtained from satellites, except from those in so called 
resonant orbits. These a re  the orbits whose periods a r e  commensurate with 
the earth 's  rotation period. 
and King-Hele's group in England. 2'  
Yet, in  spite of the difficulties of extracting non-zonal geodetic information 
from observing small short period variations in the great majority of satellite 
orbits, much progress has already been made in this a r ea  too. It is generally 
recognized that the large scale geoid features due to non-zonal gravity anom- 
alies a r e  now fairly well established from independent and parallel studies of 
general satellite orbits at the Smithsonian, Johns Hopkins, the Naval Weap- 
ons Laboratory, Nevertheless, our knowledge of the de- 
tails of the non-zonal field i s  st i l l  much less  secure than that of the zonal field. 
In particular, while the discrimination of the zonal field has been accomplished 
(to a fa i r ly  high degree now) by dynamic satellite geodesy alone, the equivalent 
discrimination of the much more complex non-zonal field has required and will 
continue to require many different sources of accurate information and methods 
of analysis. 
and by Bill Kaula. 
Among these important supplementary sources of information a r e  the strong 
long term perturbations of the special resonant orbit satellites. Unfortunately, 
except for  the class of eccentric 24-hour satellites, these cannot tell us about 
the whole non-zonal field. But the limited parts they do measure,  they measure 
exceedingly well because of the amplification possible on a resonant orbit. 
Many of the recent non-zonal satellite and combined satellite-surface grav- 
ity geoids already incorporate some resonant orbit information. In this report, 
I will present acceleration data reduced from long term observations on five 
high altitude satellites in resonant orbits. This data is strongly relevant to the 
2 
solution for  the terms of low degree in the longitude (non-zonal) gravity field of 
the earth. 
OBSERVATION DATA AND CONDITION EQUATIONS 
In Table 1 is listed 50 reduced long te rm longitude accelerations of the five 
24 and 12-hour communications satellites 1963 - 31A (Syncom 2), 1964 - 47A 
(Syncom 3), 1965 - 28A (Early Bird), 1964 - 49D (Cosmos 41), and 1965 - 30A 
(Molniya 1). The longitude acceleration measured ti;) is essentially that of the 
nearly stationary ascending equator crossings of these commensurate orbit 
satellites. In those cases where the geographical drift  rates of the ground tracks 
were appreciable, the accelerations were derived by differentiating a smoothing 
function best f i t  to  drift rate data. Details of the theory governing the drift of 
these deeply resonant orbits, under the influence of the earth's longitude har- 
monics, are to be found in papers by Allan, 8 ,  
et. al., l2 and in Kaula's book. l3 The 24-hour data summarized in Table 1 
comes directly from a recent analysis by Wagner. l4 
data has  been derived from the gravity results of a recent study of the drift of 
11 the Russian communication satellites. 
Wagner, l1 Gedeon 
The 12-hour acceleration 
The definition of the earth's gravitational potential I use comes from 
Kaula's book: l3 
where ,u is the Gaussian gravity constant of the earty (3.9860 x lo5 km3/sec2),  
r is the distance to the satellite from the earth's center of mass,  ae is the mean 
equatorial radius of the earth, 4 is the geocentric latitude and A the geographic 
longitude of the satellite, (elsewhere A stands for  the satellites ascending equa- 
to r  crossing longitude) and the Pt,'s are associated LeGendre functions. The 
non-dimensional gravity constants Ct,, and S t ,  in this spherical harmonic series 
are unnormalized constants. From the above definition, Coo = 1, C,, = C , ,  
- S 11 = 0; and C 2 , ,  S,, 0, since the North Pole (the latitude reference axis) 
is very nearly a principal axis of inertia for the earth. 
- 
In reporting numerical values of gravity coefficients in this ser ies ,  it has 
become conventional to report normalized values C, S related to the unnormalized 
coefficients of (1) by: 
- -  
3 
where: 
for m # 0, o r  the longitude terms.  The normalized coefficients a re  better in- 
deces of the true magnitude of physical effects than the unnormalized ones. 
From the theory of resonant orbit perturbations ,lo the observations 
in Table 1 are related to the relevant longitude harmonics of the potential by 
(for the 24-hour data, zero eccentricity orbits): 
.. 
= la2 7 C4, [Fd, sinmh] f S t ,  [-F4, cosmh] 
4 - m  even 
r ad i  ans / s  i de r ea  1 day 
where: 
1 F,, = ( $ ) [ + ( l + c o s i ) 2 - - q  7 s i n 2 i c o s i ( l t c o s i )  
4 
and, (for the 12-hour data, orbits of any eccentricity): 
.. 
h = la2 c qm 
2 , m  r e  levant 
- 
- cos m(h t %) 
- s i n m  ( A t  - y:m; 
-4 
ma (=).em 
- 
4 - m  even  
m 
8-rn even 
4 - m  odd- 
9 rad. / s id-day2.  (4) 
The (FG)t, and the yd, are the amplitudes and phase angles of composite 
harmonic vectors which depend on the inclination, eccentricity and argument of 
perigee of the resonant orbits." For the eighteen 12-hour observations, these 
orbit-harmonic functions a r e  listed in Table 1 for  the relevant harmonics (Htm)  
through degree five: H22y H S 2 ,  H 4 * ,  H 4 4 y  H S 2 ,  and Hs4 .  In equations (3) and (4), 
a is the semimajor axis of the satellite's orbit in earth radii, and i is its in- 
clination. The observation data in Table 1, along with the condition equations 
(3) and (4), is suitable for incorporation into existing solutions for the gravity 
field if  the data is weighted according to the given estimates of their standard 
deviations. These estimates a re  felt to be quite realistic. They take into ac- 
count many sources of small non-resonant gravity model e r ro r s  including sun 
and moon effects and atmospheric drag. In fact, I have solved (in a weighted 
least squares sense) for a set of ten low degree coefficients from this data using 
equations (3) and (4). The characteristics of this solution a r e  discussed in the 
next section. 
SOLUTION FOR TEN GRAVITY COEFFICIENTS 
Previous calculation has shown l5 that resonance effects of degree higher 
than four should have negligible influence on the 24-hour satellite. Similarly, 
considering the higher e r r o r s  for the observations on the Russian satellites, it 
appears that the rapid decline in the anomalous potential at 12-hour altitudes 
will result in small or negligible effects for  5th and higher degree harmonics. 
Detailed calculations show that of the two harmonics of 4th degree resonant on 
these satellites, H,, should at best be only barely discernible from the data, 
while H,, should have strong influence on the 12-hour observations. Murphy 
and Victor, l6 though, have measured the influence of the 5th degree harmonics 
H,, and H,, on the 12-hour satellites and it seems that though their  effects are 
small, at least H,, should be included in  future solutions from observations on 
these objects. 
.In Table 2, I list the results of the weighted least squares solution of equa- 
tions (3) and (4) according to  the data in Table 1. The solution is for the ten un- 
normalized resonant coefficients which appear to  be most sensitive to  this data: 
(C,S),, , (C,S)31, (C,S)32,  (C,S),, and (C,S),,. The internal quality of this solu- 
tion can be gauged by comparing the accelerations computed from it (as shown 
in Table 1 in the column labeled "Wagner 1968") with the actual observations. 
The residuals of the solution, shown in Table 1 also, are normalized with respect 
to the estimated standary deviation of the measured observations. These nor- 
malized residuals are also displayed against the data longitude in Figure 1. The 
normalized solution is listed in Table 3 ,  and displayed in Figure 2 for compari- 
son of individual coefficients with other recent combined-data geoids. 
For two cases I have computed drift accelerations for  these satellites as 
predicted (according to equations (3) and (4)) from the published geoids of 
Kohnlein-SA0 1967, and Kaula 1966. The results and residuals of these 
predictions a re  also listed in Table 1 and displayed in Figure 1. These com- 
parison computations included the effects of H,, on the 24-hour but not on the 
12-hour data. However, inclusion of these effects on the 12-hour satellites, as 
well as those of 5th and possibly higher degree, is not expected to alter the 
major results of this study. 
DISCUSSION OF THE SOLUTION 
The chief point to be made about the solution presented in the last section is 
that it agrees quite well overall with the recent satellite-surface gravity results 
for  the low degree field. On the other hand, it is also clear from the excess of 
large residuals, that the full field geoid investigations would benefit from an 
incorporation of the acceleration data presented here. These comparison resid- 
uals (Figure 1) show no glaring bias with respect  to longitude, except perhaps 
for the very well determined Early Bird ohscrvations7 but do average about 
twice the residuals of the resonant solution. The Kohnlein-1967 solution is 
marginally superior to that of Kaula. This is interesting because the Kohnlein 
solution does not incorporate any high altitude resonant orbit data while the Kaula 
solution does, albeit with small  weight. It also appears that the Bjerhammar- 
1967 solution (Table 3 and Figure 2) is of about the same quality as the Kohnlein 
and Kaula geoids with respect to this resonant data. The Bjerhammar field l7 
was constructed by combining, accumulating and solving normal equations rep- 
resenting satellite data from the work of Kozai, Anderle and Gaposhkin, and uses 
the same surface gravity data as in Kaula's studies. 
The concordance of the resonant solution with the recent comparison geoids 
appears best with regard to H,, and H,, and worst with regard to H,, and H,, 
(see Figure 2). It is interesting, and perhaps significant, that the strongest cor- 
relation coefficient in the resonant solution, 0.84, is between C,, and S,, (see 
Table 2). This attests to the relative uncertainty of the discrimination of these 
harmonics by this data. Other than this, all other correlations have absolute 
values less  than 0.60. Absolute bounds ( 1 ~ )  only on H,, and H,, a r e  found in 
Table 3. They a re  somewhat greater than those given by the statistics of the 
solution (Table 2). They appear justified by the resonably low correlations in 
the resonant solution and by the good agreement with the new, independently 
derived geoids. 
There is some evidence from Murphy and Victor's study l7 that the amp- 
litude of H,, for the resonant solution will be reduced when the effects of H,, on 
the 12-hour data a re  considered. If true, this would improve the agreement of 
this harmonic and the 12-hour residuals with respect to the recent more com- 
plex geoids. Future gravity data reductions of these high altitude resonant orbit 
observations will include the effects (either as known or as solved for) of all 
relevant harmonics through at least the 5th degree. 
CONCLUSIONS 
Data reduction of over 600 orbits of 12 and 24-hour satellites has established 
a set of drift accelerations which appear to be strong determinants for many 
components of low degree in the geopotential. Recent complex geoids, inde- 
pendently derived, a r e  quite compatible with most of this data. But inclusion of 
the best of the high altitude resonant orbit accelerations in the complex geoid 
solutions should result  in considerable improvement in the overall precision 
and accuracy of those solutions. 
7 
REFERENCES 
1. Y. Kozai, "The Earth's Gravitational Potential Derived from Satellite Mo- 
tion," Space Science Reviews, Vol. 5 ,  p. 818 - 879, Sept., 1966. 
2. D. G. King-Hele, G. E. Cook and D. W. Scott, I'Even Zonal Harmonics in  the 
Earth's Gravitational Potential - A Comparison of Recent Determinations ," 
Planetary and Space Science, Vol. 14, p. 49 - 52, Jan., 1966. 
3. D. G. King-Hele, G. E. Cook and D. .W. Scott, "Odd Zonal Harmonics in the 
Geopotential, Determined from Fourteen Well Distributed Satellite Orbits ," 
Planetary and Space Science, Vol. 15, p. 741 - 769, April, 1967. 
4. W. Kohnlein, "The Earth's Gravitational Field as Derived from a Combina- 
tion of Satellite Data with Gravity Anomalies,'' Preprint  of a Paper  Pre- 
sented at the 14th General Assembly of the I.U.G.G., Lucerne, Switzerland, 
Sept.-Oct. , 1967, Printed by the Smithsonian Astrophysical Observatory, 
Cambridge, Mass., 1967. 
5. W. H. Guier and R. R. Newton, "The Earth's Gravity Field as Deduced from 
the Tracking of Five Satellites," Journal of Geophysical Research, Vol. 70, 
#18, p. 4613 - 4626, Sept. 15, 1965. 
6. R. J. Anderle, "Geodetic Set NWL-5E-6 Based on Doppler Satellite Observa- 
tions," NWL Report 1978, U.S. Naval Weapons Laboratory, Dahlgren, Vir- 
ginia, 1965. 
7. W. M. Kaula, "Tests and Combination of Satellite Determinations of the 
Gravity Field with Gravimetry,'' Journal of Geophysical Research, Vol. 7 1, 
p. 5303 - 5314, NOV. 15, 1966. 
8. R. R. Allan, "Resonance Effects due t o  the Longitude Dependence of the 
Gravitational Field of a Rotating Primary," Planetary & Space Science, 
VO~.  15, p. 53 - 76, 1967. 
9. R. R. Allan, "Satellite Resonance with Longitude-Dependent Gravity-11, Ef- 
fects Involving the Eccentricity," Planetary and Space Science, Vol. 15, 
p. 1829 - 1845, 1967. 
10. C. A. Wagner, "The Use of Resonant Librating Orbits in Satellite Geodesy," 
NASA TN D-4081, Washington, D. C., August, 1967. 
8 
. 
11. C. A. Wagner, "Determination of Low Order Resonant Gravity Harmonics 
from the Drift of Two Russian 12-Hour Satellites," NASA - GSFC Document 
X-643-67-621, Greenbelt, Maryland., Dec., 1967: Also, Journal of Geo- 
physical Research, In Press, 1968. 
12. G. S. Gedeon, B. C. Douglas and M. T. Palmiter, "Resonance Effects on 
Eccentric Orbit Satellites ,'I Journal of the Astronautical Sciences, July-Aug., 
1967, p.. 147 - 157. 
13. W. M. Kaula, Theory of Satellite Geodesy, Blaisdell Publishing Company, 
Waltham, Mass., 1966. 
14. C. A. Wagner, "Resonant Gravity Harmonics from 3 1/2 Years of Tracking 
Data on Three Synchronous Satellites," NASA - GSFC Document X-643-67-535, 
Greenbelt, Maryland, Nov., 1967. 
15. C. A. Wagner, "The Earth's Longitude Gravity Field as Sensed by the Drift 
of Three Synchronous Satellites," NASA T N  D-3557, 1966. 
16. J. P. Murphy and E. L. Victor, "A Determination of the Second and Fourth 
Order  Sectorial Harmonics i n  the Geopotential from the Motion of 12-Hour 
Satellites," Planetary and Space Science, Vol. 16, p. 195 - 204, 1968. 
17. A. Bjerhammar, "On a Coalescent World Geodetic System,'' Special Report 
by The  Research Institute for Geodetic Science, 70 Prince Street, Alexan- 
dria,  Virginia, July, 1967. 
9 
10 
I -  
d 
T 
N 
k 
0 m 
d 
0 
I 
0 m 
.r( 
c, 
4 
rcl 
0 
m 
0 .r( 
+ m 
.r( 
c, 
cd x 
rn 
F: 
0 
cd 
k 
a, 
a, 
0 
3. 
4 
2 
E 
0 
Is 
a, 
9 
d 
d 
u*l-l 
0 0  
rc, 91 
l-l 
m 
m m w o  
w ml- l  0 1 9  
rl 
m 
mcil m 
ucvrl 
w 9 9  
0 0  
rl 
rl 
N X 
0 
~ r l m a 3 m m N m + t - o  
m ?  999 u?19 9 9 9  
I I  I I I I I l l - l  
l-l 
11 
cu 
0 
r! 
W 
0 
I 
9 
0 
0 
'9 
0 
0 
c? 
cu 
rl 
Y 
b 
0 
v? 
In 
0 
p: 
Q) 
0 
Q) 
0 
L'. 
1 
ua 
rl 
c: 
W 
rl 
I 
c? 
c.1 
cu Ttl 
CD 
W ua 
rl 
G 
?c 
4 
3 
CD 
0 
c? 
m 
0 
I 
9 
b 
0 
v? 
m 
0 
c? 
0 
I+ 
'9 
Q) 
0 
v? 
0 
0 
I 
v? 
Q) 
0 
(9 
rl 
I 
0 
dr 
rl 
c? 
c.1 
rl 
I 
c? 
Q, 
a c? 
5 
ii 
%I? 
i3s a w  
I I 
24 HOUR 
SATE LL I TE S 
(32  OBSERVATIONS) 
I 
I - 2  
-3 
- 4  
0 
Y 
i I 
- 
- I 
R M S  RESIDUAL (0) I.. 5 
I I I I I 
R M S  RESIDUAL (0 )  0.765 
4 r 3 
- 4  
0 40 8 0 1  
x .  
!ESIDUAL (0) l . u J 4  
I I I I 1 %  
) 160 -160  - 1 2 0  -80 -40 
I I j 
~ 
1 2  HOUR 
SATELL I TES 
(18 OBSERVATIONS ) 
- 
- 
4 
3 
2 
1 
0 
-1 
- 2  
- 3  
- 4  
LONGITUDE (DEGREES 1 
Figure 1. Observation Residuals for 12 and 24 Hour Satellite Accelerations From Recent Geoids. 
13 
0 
a x  
I 
I N 1 
I m 
4 
h 
W 
I 
2 
Y 
0 
u) t
C 
C 
u) 
t 
c 
9) 
U 
9) 
0 
>. 
> 
.- 
3 
- 
v 
.- 
.- 
Y 
U 
t .- 
e 
c3 
9) 
7 
3 
m c 
0 
-I 
73 al 
N 
0 
0 
.I- .- 
.- -
E 
Z 
~ 
2 
3 
m 
U 
.- 
14 


Combined solution for low degree longitude harmonics of gravity from 12 to 24 hour satellites

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server