I I Shapiro

R D Reasenberg

R D White

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Vol. 2, No. 5 GEOPHYSICAL RESEARCH LETTERS March 1975 
THE GRAVITY FIELD OF MARS 
R. D. Reasenberg, I. I. Shapiro, R. D. White 
Department of Earth and Planetary Sciences 
Massachusetts Institute of Technology 
Cambridge, Massachusetts 02139 
Abstract. A complete sixth-degree cribed below, to be uncertain by no more 
spherical-harmonic model of the gravita- than about 100 m for the region below 
tional potential of Mars has been produced 60 ø North Latitude. For higher latitudes 
from an analysis of Doppler data obtained the uncertainties increase to about 300m. 
from the radio tracking of the Mariner-9 This model represents a significant 
spacecraft for over 200 orbital revolu- improvement over those discussed previ- 
tions. The equipotential surface defined ously (Lorell et al., 1973). These ear- 
by this model is estimated to be accurate lier models exhibited unrealistically 
to within 100 m below 60 ø North Latitude large undulations in the northern hemi- 
and to within 300 m above. The dominant sphere, in part because the periapse of 
feature of this surface is a Tharsis-rela- Mariner 9 was located at about 23 • South 
ted bulge which rises nearly two kilome- Latitude. These artifacts were also due 
ters above the surrounding areas. Gravi- partly to the use of data sets with un- 
tational "lows" are found to correspond even longitude coverage and to the use of 
to the Hellas depression and the Valles a less effective filter. They persisted 
Marineris. even when fairly strong a priori con- 
straints were placed on the coefficients 
Introduction in the harmonic expansion. 
Our approach, which utilizes more uni- 
Knowledge of the gravitational poten- form longitude coverage, takes advantage 
tial of Mars provides strong constraints of both the short-period effects of the 
on models of its interior (Johnston et gravity field, manifested primarily 
al., 1974). Questions of isostasy'and through the tracking data obtained near 
depth of compensation, for example, can periapse passages, and the long-period 
be answered only with the combination of and secular effects which accumulate over 
topographic and gravitational information. many satellite orbital revolutions. Other 
Further, the accuracy of the Mariner-9 approaches that have been used recently 
and Viking tests of general relativity concentrated exclusively either on the 
that rely on ranging from Earth to a data obtained near periapse (Sjogren et 
spacecraft in orbit about Mars rests in al., 1975) or on the long-term and secu- 
part on the accuracy of the Mars gravity lar effects (Born, 1974). 
potential model used to determine the 
spacecraft's orbit. Method of Analysis 
Mariner 9 approached the surface of 
Mars closely enough to be sensitive to The data utilized in our analysis com- 
the higher-degree terms in the spherical- prised six disjoint 19-day spans, cover- 
harmonic expansion of Mars' gravitational ing the periods from 16 Nov 71 to 23 Dec 
potential. Experiences with artificial 71, and from 6 Jan 72 to 22 Mar 72. These 
satellites of the earth and with lunar data were the best available from the 
orbiters have demonstrated the difficul- points of view of continuity of tracking 
ties inherent in the determination of and, to a lesser extent, of signal-to- 
reliable coefficients in this expansion noise ratio. Each of these 19-day spans 
from the analysis of radio-tracking data corresponded to a "resonance period" in 
•from numerous spacecraft in diverse or- which the sub-periapse 'point of Mariner 
bits and a fortiori from'one spacecraft 9 completes a cycle in longitude. 
with a fixed periapse latitude. 
Nevertheless, the development by one Because of the presence of important ran- 
of us (RDR) of a limited-memory filter, dom, or quasi-random, accelerations of 
feasible economically for use in the ana- the spacecraft, due, for example, to im- 
lysis of the Mariner-9 data, has allowed balances in the gas jets used to control 
us ko obtain a reliable, complete, sixth- the orientation of the spacecraft (see 
degree model of this expansion of Mars' Figure 1 in Lore!• et al., 1973), it is 
gravitational potential. The heights of not fruitful'to process the data in the 
the corresponding equipotential near the usual "batch" mode least-squares analysis. 
ß surface of Mars are estimated, as des- Approximations to the Kalman-Bucy filter 
(Jazwinski, 1970) were therefore used- 
Copyright 1975 by theAmerican Geophysical Union. The si• resonance spans were divided 
89 
90 Reasenberg et al.' Gravity Field of Mars 
into three sets of two spans each, the procedure would produce a substantially 
first two sets being composed of alter- better estimate of the gravity field. 
nate members of the first four spans. To obtain our final model of the gray- 
Each set was analyzed separately and with ity field we used the simple and adequate 
two different filters. The first, a procedure of averaging arithmetically 
dual-span filter (DSF), consisted of the the means of the coefficients of the 6th- 
estimation, hy weighted least squares, degree models obtained with the'LMF. 
of the appropriate set of CoeffiCients Th•se means are shown in Table 1. Figure 
of the spherical-harmonic expansion 1 shows the equipotential surface cortes- 
(Appendix A), and of two spacecraft state Pønding to this mean model with J2 set 
vectors, one for each data span. The to zero. The DMEPS plots corresponding 
second, a limited-memory filter (LMF), to this 6th-degree model were examined 
involved the estimation of the same set in greater detail in order to estimate 
of coefficients but twice the number of the uncertainty in the EPS heights. The 
spacecraft state vectors, one for each local extrema below 60øN obtained from 
9.5 days of data. In all cases, para- COntours with 25 m separation, have 1) a 
meters describing such quantities as the mean absolute value of 95 m; 2) a mode 
mass of Mars, the direction of its rota- and median absolute value of 75 m; and 
tion axis, the location on the earth of 3) an rms of 110 m. From these statis- 
the tracking stations, and the radiation tics we estimate that the EPS shown in 
pressure exerted on the spacecraft, were 
fixed in accord with prior determinations. 
All solutions were of full rank and com- 
plete in the harmonic expansion to the 
particular degree chosen, except that 
the coefficients C21 and S21 were held 
at zero. No a priori constraint was 
used, either for the other coefficients 
or for the spacecraft state. 
The three models produced with a given 
filter for a given degree were intercom- 
pared through contour maps. For each 
model, a map (DMEPS) was formed by taking 
the difference of the heights of the 
equipotential surface (EPS) for that 
model and the mean of the corresponding 
heights for all three models. The con- 
tours, which had a separation of 25 m, 
were examined in each of two latitude 
regions: 70øS to 30øN and 70"S to 65"N. 
A study in these overlapping zones was 
Figure 1 is reliable to about 100 m below 
chosen in lieu of a more formal analysis _ 
because the EPS height errors do not have C41 - 0.052 
well understood statistics. The = 0 037 "greatest deviation" and the "typical S41 ' 
deviation" were recorded for each region C42 = -0.003 
of each DMEPS plot. These numbers howed S42 = -0.116 
that the LMF yielded significantly more _ 
consistent models, especially for those C43 - 0.064 
of higher degree. S = -0.021 
Intercomparisons of the LMF solutions 43 
for the spherical harmonic models of 6th, C44 TM -0.005 
7th, and 8th degree led to the conclusion S44 = -0.168 
that the highest resolution model that 
our analysis can support is of sixth C51 = 0.013 
degree. This model was found to be simi- S51 = 0.032 
lar to, and apparently slightly more re- 
liable than, the 7th-degree model. The 
two triads of 8th-degree models proved 
highly disparate. 
The post-fit rms residuals for the 
6th-degree LM• models were under 0.25 Hz. 
This was, of course, lower than the cor- 
responding DSF number because of the 
Table 1. Coefficients for Spherical- 
Harmonic Expansion of the Gravitational 
Potential of Mars* 
C21 -- 0.0 
S21 --- 0.0 
C22 = -0. 841 
S22 = 0. 494 
C31 = 0. 047 
S31 = 0. 261 
C32 = -0.163 
S32 = 0. 079 
C33 = 0.358 
S33 = 0. 249 
C52 = -0.015 
S52 = -0.012 
C53 = 0. 027 
S53 = -0. 029 
C54 = -0.132 
S54 = -0. 032 
C55 = -0. 051 
S55 = 0.036 
C61 = 0. 027 
S61 = -0.015 
C62 = 0. 058 
S = 0.012 
62 
C63 = 0. 018 
S63 = -0. 019 
C64 = -0. 026 
S64 = 0. 081 
C65 = 0. 006 
S65 = 0. 020 
C66 = 0. 023 
S66 = 0. 016 
J2 = 19.550 
J3 = 0.113 
J4 = -0. 573 
J5 = -0.452 
J6 = -0.665 
larger number of parameters adjusted in Each coefficient has been multiplied by 
the LMF. By dividing the data into suf- 104 ; see Appendix for relevant defini- 
ficiently short arcs it should be possi- tions. We note that for the model given 
ble to bring the residuals quite close by Lorell et al. (1973), the second 
to the nominal noise level of under 0.01 degree terms, normalized as above, are 
Hz. However, we do not expect that this J2 = 19.6, C22 = -0.7.9, and S22 = 0.53. 
Reasenberg et al.' Gravity Field of Mars 91 
NORTH 
180 ø 120 ø 60 ø 0 ø 300 ø 240 ø 180 ø 
:'•:'"•:-N..• •'• .•::. . - .... .. ':-• .... •': ...... . . '• ' • ( • • ..... '• - ß -.' ' ' (- ....... : . •-', ....:.•:.-.:-:•:•...r . : •-.- • .. •:• --.:... ............. ......,. ........... . . .. • • • .... ,..• ? .... , ...... •: .....,• ..... -:• 
-3• ....... •=.' ' •:' .- '. 
': 
", ,.':;," :.k' '"'" ..... " ' ' :"' "" .... ' :' 
180,: 120 ø •o 0 o 300 ø 240 ø 180 ø 
SOUTH 
•lgu•e 1. Co•ou• mag •o• • e•ulgo•e•l•l •u•oe o• ua•' g•a•l• •leld 
gored o• • montage o• g•o•og•g• o• •e •u•aoe, g•o•lded • •. Ua•u•k•, oou•e• 
o• •e U•l•ed ••e• geologlo•l •u•e•. •e oo•ou• oml• •e e•eo• o• •2 a•d •e 
•e•e•ed •o • •dlu• o• 339• km. Co•ou• •e •o• • 0.2 km 1•e•1•. • •e•o•o• 
g•o•eo•1o• 1• •o•. 
60øN. Of course, the rms of the extrema rise in the region of Tharsis, 105 ø West 
should be substantially larger than the Longitude. The other major elevated 
rms errors of the contour in Figure 1; in region of the equipotential surface, near 
addition, some advantage is presumably 285 ø West Longitude, and the depressed 
gained by averaging the three separate areas in between all seem to be manifes- 
sets of coefficients. However, the pos- tations of the Tharsis construct. The 
sibility of systematic effects, common to indentations of the contour lines that 
all three LMF solutions, despite each appear on the East side of Tharsis are 
being dependent on a totally different probably associated with the Valles Mari- 
data set, requires that we not make a neris (Sjogren et al., 1975). 
smaller estimate of the uncertainty. By Shallow structures in the EPS tend to 
a similar method we assign uncertainties be obscured by the Tharsis-related undu- 
of up to 300 m for the EPS heights above lations. By offsetting the center of the 
60øN. spheroid used as reference for the EPS, 
S'jogren et a1.(1975) have recently this obscuring effect can be reduced. (A 
developed a Mars gravity model with EPS possible geophysical basis for this ap- 
height uncertainties "ranging from 60 m proach will be discussed in a separate 
near the equator to 500 m at the poles." publication.) In such an "offset-con- 
We have compared this model to ours by tour" plot, the Hellas depression, as 
plotting contours of the difference EPS well as the Valles Marineris, appear as 
with 25 m separation. An examination of gravitational "lows". There seems to be 
the extrema of these contours below 60øN a pattern of undercompensation for major 
yields: 1) a mean absolute of 120 m; surface features (Christensen, 1975). 
2) mode and median absolutes of 100 m; Such undercompensation' is consistent with 
and 3) an rms of 135 m. For the region the assertion that the Martian litho- 
below 30øN the rms drops to 100 m. Above sphere must be thick (approximately 200 
60øN the difference contours are domina- km) and rigid (Carr, 1974), and thus able 
ted by a single feature that reaches 450 to support a substantial disequilibrium. 
m at the pole. We find this agreement 
highly satisfactory considering that the 
models were generated by different analy- 
sis methods and disjoint sets of data. 
Discussion 
The equipotential-surface of Mars 
(Figure 1) is clearly dominated by the 
Appendix 
The spherical-harmonic expansion of 
the gravitational potential of Mars is 
written as 
U = GM• • • n - r {1- Z J ( ) Pn(sinS) + n=2 n 
92 Reasenberg et al.: Gravity Field of Mars 
co n n 
a (sinS) ß Y' Y' (•) Pnm 
n=2 m=l 
[CnmCOS mk + SnmSin mk] }; r > a, 
where Pn and Pnm are, respectively, the 
Legendre and associated Legendre polyno- 
National Aeronautics and Space Adminis- 
tration, NGR 22-009-804. 
References 
Born, G. H., Mars Physical Parameters as 
Determined from Mariner 9 Observations 
of the Natural Satellites and Doppler 
Tracking, J. Geophys. Res. 78, 4837- 
mials; a = 3394 km is the value assumed 4844, 1974. -- 
for the--equatorial radius of Mars; M• = Carr, M. H., Tectonism and Volcanism of 
(3098700) -1 M• is the mass of Mars w•th the Tharsis Region of Mars, J. Geophys. 
M• the mass of the sun; G is the univer- Res. 79, 3943-3949, 1974. 
sal constant of gravitation; r, 8, and k Chr•-•en•n, E. J., Martian Topography 
are the usual Mars-centered coordinates: Derived from Occultation, Radar, Spec- 
radius, latitude, and longitude, referred 
to a pole of rotation defined by right 
ascension 317728 and declination 52?77 
(1950.0); and the Jn, Cnm, and Snm are 
the coefficients to be estimated with 
the normalization defined by 
1 2 -1 1 2 
œ P (x)dx = 2(2n + 1) ; œ P (x)dx= 4. 
-1 n -1 nm 
Acknowledgements. We thank M. E. Ash 
and G. L. Slater for their important con- 
tributions to the early phases of our 
research, J. Lorell, J. F. Jordan, and 
other members of the staff of the Jet 
Propulsion Laboratory for their indispen- 
sable cooperation, and the computation 
group at the Charles Stark Draper Labo- 
ratory, Inc., for their cooperation in 
the processing of the data. This re- 
search was supported by a grant from the 
tral and Optical Measurements, 
submitted to J. Geophys. Res., 1975. 
Jazwinski, A. H., Stochastic Processes 
and Filtering Theory, pp. 189-193, 
Academic Press, New York, 1970. 
Johnston, D. H., McGetchin, T. R. and 
Toksoz, M. N., The Thermal State and 
Internal Structure of Mars, J. Geophys. 
Res. 79, 3959-3971, 1974. 
Lorell, J., Born, G. H., Christensen, E. 
J., Esposito, P. B., Jordan, J. F., 
Laing, P. A., Sjogren, W. L., Wong, S. 
K., Reasenberg, R. D., Shapiro, I. I. 
and Slater, G. L., Gravity Field of 
Mars from Mariner 9 Tracking Data, 
Icarus 18, 304-316, 1973. 
Sjogren, W•--L., Lorell, J., Wong, L. and 
Downs, W., Mars Gravity Field Based 
on a Short-Arc Technique, J. Geophys. 
Res., 1975. 
(Received January 17, 1975; 
accepted February 6, 1975.) 


The Gravity Field of Mars

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The Gravity Field of Mars

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