on the moment of inertia factor of 0.361

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102, NO. El2, PAGES 28,673-28,685, DECEMBER 25, 1997

Mars without Tharsis

Maria T. Zuber 1

Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge

David E. Smith

Laboratory for Terrestrial Physics, NASA/Goddard Space Flight Center, Greenbelt, Maryland

Abstract. The significant power in the Martian gravitational field due to the Tharsis rise may mask or modify gravitational signatures that contain important information on Martian geophysical processes. In order to isolate gravity signals in regions where the field is significantly affected by Tharsis as well to investigate characteristics of the global field, we present a numerical technique to "remove" Tharsis from the Martian gravitational field. Our analysis will show that to first order Tharsis can be represented as a sixth-degree spherical harmonic zonal gravitational feature in a reference frame in which the axis of symmetry runs through the center of the province. We produced the "Mars without Tharsis" (MWT) field by subtracting the gravitational signature of Tharsis from the full field and rotating the spherical harmonics back to the original coordinate system, in which the z-axis is coincident with the planetary rotation axis. Our study yields limits on the moment of inertia factor of 0.361 <C/MR2<0.366 for the range of allowable non-hydrostatic contributions to the flattening. The analysis indicates that the formation of Tharsis may have resulted in re-orientation of the spin axis, though if the planet retained rotational fossil bulge due to a thick elastic lithosphere then changes in the global figure could have been inhibited. The zonal and tesseral components of the degree 2 field in the GMM-1 and rotated configurations indicates that Tharsis is --5 times more axisymmetric about its "pole" than Mars as a whole is about the rotation axis. With the removal of Tharsis' long wavelength gravitational power, regional structures become better defined in a spatial sense while maintaining their local dynamic range. The Olympus Mons volcanic shield displays clear evidence of a gravitational flexural moat. Our method for removing Tharsis may also be applied to analyze other global-scale geophysical features that may be considered axisymmetric to first order, such as mantle plumes, major volcanic constructs, or large impact basins.

1. IntrodUction

The Martian gravitational field contains significantly more power at long wavelengths than does Earth's [Phillips and Lambeck, 1980; Balmino et al., 1982]. The difference reflects the fact that topographic excursions on Mars are greater than on Earth, because Mars is a smaller planet with a smaller mass, Topographic variations are characterized by large gravity and geoid anomalies, of which the most prominent planet-scale feature is the Tharsis bulge. Tharsis, which is conspicuous in the global Martian geoid in Figure 1, is the dominant locus of volcanism and tectonism on Mars and rises up to 10 km above its surroundings [Carr, 1981]. The widespread and diverse distribution of tectonic features and the high topography represent evidence that Mars has undergone complex thermal and stress histories which have been influenced on both the

global and regional spatial scale by Tharsis [Banerdt et al., 1992]. The large amplitude, long wavelength gravity and geoid anomalies associated with the Tharsis rise must certainly

•Also at Laboratory for Terrestrial Physics, NASA/Goddard Space Flight Center, Greenbelt, Maryland.

Copyright 1997 by the American Geophysical Union.

Paper number 97JE02527. 0148-0227/97 / 97JE-02527509.00

influence both long and short wavelength gravitational signatures that are relevant to a range of geophysical processes. In order to isolate gravity signals in regions where the field is especially affected by Tharsis, as well as to investigate aspects of the global field that have been influenced by Tharsis, we have developed a numerical technique to "remove" Tharsis from the Martian gravity field. Our method exploits the ability to quantify an axisymmetric signal in an arbitrary coordinate system in a spherical harmonic gravity model.

Other workers have also recognized the utility of isolating the gravitational influence of Tharsis, and our approach has both similarities and some important differences compared to previous work. Reasenberg [1977], Kaula [1979] and Kaula et al. [1989] accounted for the contribution of Tharsis to the second degree (flattening term) of the gravity field by assuming that the non-hydrostatic contribution to the moment of inertia is axisymmetrically distributed along an axis in the direction of the province. In a study concerned with reorientation of the planetary rotation axis, Melosh [1980] removed the attraction of Tharsis by deleting point masses from a then-current gravity model [Sjogren et al., 1975]. In order to estimate the contribution of Tharsis to the Martian

flattening, Willemann and Turcotte [1982] employed an approach that is conceptually similar to that invoked here: they moved the center of Tharsis to Mars' rotation pole using the spherical harmonic addition theorem [Kaula, 1968].

28,673

28,674 ZUBER AND SMITH: MARS W1THOUT THARSIS

Figure 1. The GMM-1 geoid of Mars [Smith et al., 1993] to degree and order 50. The Tharsis province corresponds to the broad geoid high on the left side of the figure (-250øE longitude). The geoid assumes a 5% nonhydrostatic component to the flattening of Mars (J2), discussed in the text. This map and others that follow utilize a Hammer equal-area projection. Zero longitude runs through the center and is measured positive east. Contours are 0.1 km.

However, as with other studies of this genre, they treated only the degree two zonal term of the gravity field rather than higher degrees as we do, and they did not rotate the residual field back to the original coordinate system and analyze it.

2. The Martian Gravitational Field

2.1 Spherical Harmonic Representation

The basis of our study is Mars' present-day gravitational field. We utilized Goddard Mars Model-1 (GMM-1) [Smith et

al., 1993], a spherical harmonic model derived from S band Doppler tracking observations from the Mariner 9 and Viking 1 and 2 orbiters. This field is complete to degree and order 50 with a corresponding (half wavelength) spatial resolution of 200-300 km where the data permit. We note that there exists a more recent Mars gravitational field model [Konopliv and Sjogren, 1995] that we believe resolves short wavelength structure in the vicinity of Tharsis better than our GMM-1 model. However, GMM-1 and Mars50c are virtually identical out to degree and order 20, which corresponds to a much higher resolution than the scale of Tharsis. The low degrees and orders of the Martian gravitational field are sufficiently well known [Esposito et al., 1992] that our results do not depend on the specific choice of recent gravity model.

The gravitational potential, V M, at spacecraft altitude may be represented in spherical harmonic form as

GM GM VM(r,•p,/•) = • + ß

• • • Plm(sin½)(Clm cosm/• + Slm sinm/•) /=2m=0 (1)

where r is the radial distance from the center of mass of Mars to

the spacecraft, 0 and )• are the latitude and longitude of the spacecraft, a is the mean equatorial radius of Mars, GM is the product of the gravitational constant and mass of Mars, Plm are the normalized associated Legendre functions of degree l and order m, C•, n and S•, n are the Stokes coefficients which were estimated using the tracking observations (the overbars indicate normalized coefficients), and N is the maximum degree representing the size or resolution of the field. Because this analysis is mainly concerned with the global-scale signature of Tharsis, we focus (though not exclusively) on the geopotential or geoid representation of the field. Because of the necessity to reduce the Doppler tracking data in a mass- centered reference frame, our analysis utilizes an areocentric coordinate system with an east positive longitude convention.

2.2 Moments of Inertia and Deviations from

Hydrostatic Equilibrium

Tharsis is centered at a low latitude, which complicates the attempt to distinguish its gravitational signature from that due to the planetary oblateness or flattening associated with rotation. The gravity signal due to the planet's rotational bulge is the hydrostatic part of the field and surface and internal mass heterogeneities associated with Tharsis (and other features) represent deviations from the hydrostatic state. Models of planetary internal structure are sensitive to the extent to which the flattening deviates from hydrostatic equilibrium [Bills, 1990], i.e., the nonhydrostatic contribution to the gravity field. The planetary oblateness observed from satellite orbit perturbations comes directly from the unnormalized degree 2 Stokes coefficient in (1) and is

ZUBER AND SMITH: MARS WITHOUT THARSIS 28,675

defined J2=-C2o . This parameter may be decomposed such that J2=J2h-J2 where J2h is the hydrostatic part and J2 is the nonhydrostatic part of the oblateness.

The second degree Stokes coefficients from the gravity field are related to planetary internal structure via the principal moments of inertia. Consider A, B, and C to be the minimum,

intermediate, and maximum hydrostatic moments and AA, AB, and AC to be the nonhydrostatic components. The hydrostatic component of the oblateness is related to the moments according to [Kaula, 1979]

C-I(s+A) 2

J2h = (2) MR 2

where R is the mean planetary radius. Also of relevance is the equatorial ellipticity J22=(C22+822) 1/2 of the gravity field that describes the deviation of the potential shape of the equator from a circular crosssection and can be written

1

J22 = (3) MR 2

Note that because J22 is independent of latitude it is fully nonhydrostatic and, consequently, depends only on nonhydrostatic moments.

Interpretation of planet-scale properties of the gravity field often utilizes the moment of inertia factor, C/MR 2. This parameter provides information on the radial stratification of mass in the planetary interior, with a value of 0.4 corresponding to a uniform density sphere and smaller values indicating mass concentration toward the center of the planet. It can be advantageous to express C/MR 2 in terms of the mean hydrostatic moment I=(A+B+C)/3 [Darwin, 1899], which yields

C I 2

MR---'-5- = MR • + • J2h' (4)

Alternatively, C/MR 2 can be expressed

C 2 1- 2 4m--3J2h MR 2 - -• -• m + 3J2h (5)

where m=oo2R3/GM, which is the ratio of the planet's centrifugal acceleration to the gravitational acceleration at the equator for a rotation rate to. The other hydrostatic moments take the form

B C

MR • = MR • - J2h q- 2J22 (6) and

A C

MR 2 MR 2 J2h -- 2J22. (7)

A direct (nonmodel dependent)determination of C/MR 2 requires two observations: J2 and the precession rate of the planetary rotation axis. While for Mars J2 is well known [Balmino et al., 1982; Smith et al., 1993; Konopliv and Sjogren, 1995], large uncertainties in the observed precession rate [Sinclair, 1989] dictate that constraints on C/MR 2 [Esposito et al., 1992], which requires knowledge of the

nonhydrostatic contribution to J2, are driven by assumption- laden theoretical considerations [Binder and Davis, 1973; Lorell et al., 1973; Cook, 1977; Reasenberg, 1977; Kaula, 1979; Bills, 1989b; Bills, 1989a; Kaula and Asimow, 1991; Zhang, 1994; Yoder and Standish, 1997]. Theoretically possible values of C/MR 2 span the range from 0.368 to 0.345, corresponding to nonhydrostatic contributions (J2)to J2, of 0%-20%. This mathematically allowable range translates to an approximately 800-km uncertainty in the size of the Martian core [Bills, 1990; Longhi et al., 1992], though it has been argued on the basis of analyses of SNC meteorites [Ohtani and Kamaya, 1992; Sohl and Spohn, 1997] that the lower limit of C/MR 2 is geochemically implausible. The upper limit may more plausibly equal 0.366 on the basis of geometric considerations.

Our analysis assumes a nominal 5% nonhydrostatic component to the planetary flattening but considers the full range of possible nonhydrostatic contributions. Our reasons for the assumption are discussed below. All our calculations have applied a correction to the second degree harmonic C2o only. A small hydrostatic correction to C4o ('•2.7x10-6; [Sleep and Phillips, 1985]) could also be applied but has been ignored, since this correction contributes only 4% to the power at degree 4. Most of the power in the GMM-1 field at degree 4 is in the nonzonal coefficients.

2.3 Defining Tharsis

We assume that the Tharsis province can be represented, to first order, as an axisymmetric gravitational signal. While a simplification, this assertion has also been made in several previous analyses [Kaula, 1979; Willemann and Turcotte, 1982; Kaula et al., 1989] for, we believe, good reason. Specifically, by invoking axisymmetry, it becomes possible to simply define Tharsis in a particular coordinate system by low-degree spherical harmonic coefficients. If the z axis of the coordinate system is taken to correspond to the central axis of the feature of interest, then the gravitational signature can be quantitatively described by zonal coefficients (Ci,O, where m=0), which have no longitude dependence. The representation of Tharsis as an axisymmetric signal is equivalent to treating the feature as a set of masses distributed along the central axis of the province that collectively provide a close match to the observed gravity. In addition to providing a means of analysis of the global gravity field using straightforward mathematics, the assumption of axisymmetry also allows a means of quantifying the extent of deviation from axisymmetry. This possibility has not been exploited in previous studies but is a focus of the present analysis.

We determined the size and center of Tharsis by examining the GMM-1 geoid at progressively increasing degrees and orders. We eventually chose to characterize Tharsis by a sixth degree zonal field because higher degree fields introduced complexities in the geoid that did not appear to be associated with the primary Tharsis signature. In addition, for all models of degree 6 and lower the center of Tharsis remained fixed, while at higher degrees the center shifted due to the influence of regional-scale structure at increasingly shorter wavelengths. Examination of the sixth degree and order geoid, shown in Figure 2, yielded a center of Tharsis of -111.67øE longitude, 6.67øN latitude. In determining the center of

28,676 ZUBER AND SMITH: MARS WITHOUT THARSIS

Figure 2. Sixth degree zonal geoid field corresponding to Tharsis, as defined in this analysis. The pole of Tharsis is at longitude -111.67øE, latitude 6.67øN. This representation assumes a 5% nonhydrostatic component to Mars flattening, though the precise location of the center of the province is insensitive to this parameter. Contours are 0.1 km.

Tharsis, we assumed that there is a 5% nonhydrostatic component to the flattening of Mars, but the precise location that we have designated is not sensitive to the value of this parameter. In contrast to our definition, previous studies [Reasenberg, 1977; Kaula, 1979; Willemann and Turcotte, 1982; Kaula et al., 1989] represented Tharsis as a degree two feature and either explicitly or implicitly assumed that the province was centered on the Martian equator.

In practice, it was necessary to perform finite rotations of the spherical harmonic coefficients, and so well-established methods for infinitesimal rotations on a sphere that have been used extensively in studies of the motions of Earth's tectonic plates [Minster et al., 1973; DeMets et al., 1990] were not appropriate. The formalism that we utilized instead [Goldstein, 1984] is based on a decomposition of an arbitrary, finite rotation of a coordinate system D into five elementary matrix rotations of the form

2.4 Removing Tharsis

A schematic illustrating the method by which we removed Tharsis from GMM-1 is given in Figure 3. Specifically, we (1) rotated the nonhydrostatic component of the GMM-1 field to the center of Tharsis and recalculated the spherical harmonic coefficients of the gravitational field in the new coordinate system, with the z axis constrained to pass through the center of Tharsis; (2) set all the zonal harmonic coefficients of the gravitational potential up through degree six in the new coordinate system to zero; and (3) rotated the new field back to the original coordinate system and recalculated the spherical harmonic coefficients to yield the gravitational potential for Mars without Tharsis. The calculation conserves the mass of

the planet such that the total gravitational attraction of Tharsis is redistributed over the planet.

Because Tharsis is situated near the Martian equator, some of the degree 2 power in the gravitational field, representing the planetary flattening, is associated with the province [Phillips and Lambeck, 1980; Willemann and Turcotte, 1982]. In removing Tharsis, we varied the amount of nonhydrostatic C2o that was left in the field from 0% to 20%, which fully encompasses the range of proposed values [Bills, 1989a,b; Kaula et al., 1989].

D= ArB-I Ai•BAa (8)

where A•, A n, and A r represent polar axis rotations andB and B '• are 90 ø rotations about a fixed equatorial axis whose effect on the spherical harmonics is described by a set of precomputed weighting coefficients. We determined the weighting coefficients recursively using formulas based on contiguous Jacobi polynomials [Arfken, 1970]. We validated the approach for our application by rotating coefficients back to the original reference frame without removing a Tharsis signature and verifying their recoverability. Our results confirmed analysis by Goldstein [1984], who demonstrated recoverability of spherical harmonic coefficients to within 1 part in 109 of the original values.

3. Interpretation of the MWT Field

3.1 General Characteristics

Figure 4 shows geoid and flee-air gravity versions of the gravitational field for Mars without Tharsis (MWT). For both representations of the MWT field, regional structures near Tharsis (discussed below) are more spatially localized and more


(a)

(b)

(c)

Figure 3. Two-dimensional schematic diagram illustrating the approach for removing Tharsis (shaded) from the Martian gravitational field: (a) define Tharsis as an axisymmetric gravitational signal, (b) rotate entire gravitational field so that pole of Tharsis is coincident with the rotation axis of Mars, ¸ remove low-degree (zonal) spherical harmonic coefficients corresponding to Tharsis from rotated field and then rotate the residual field back to the original coordinate system.

conspicuous than in GMM-1. Assuming 5% nonhydrostatic components of J2, the background signal in the Tharsis region of the MWT field is reduced by up to -1200 m in the geoid and nearly 300 mGals in gravity anomaly amplitude. As can be seen in Figures 1, 4, and 5, neither GMM-1 nor any of the MWT fields show visual gravitational evidence of the hemispheric dichotomy. This is consistent with our

topographic model that indicates a correlation of a broad-scale northern-southern hemispheric elevation difference with the center of mass/center of figure offset along Mars' polar axis [Smith and Zuber, 1996].

Elysium (-140øE longitude) is another spatially extensive locus of tectonism and volcanism in the northern hemisphere of Mars, though it is smaller than Tharsis. Assuming our definition of Tharsis, the geoid signature of the Elysium province is "contaminated" by the Tharsis contribution to the field. This is a consequence of the fact that the symmetric zonal coefficient has components on opposite hemispheres of the planet. Removal of Tharsis causes a reduction of --40 m (out of 600 m) in the maximum geoid in this region for a 5% nonhydrostatic J2. This illustrates that the long-wavelength zonal field that we have removed is not representative of Tharsis alone. We also computed the 5% MWT field including the small J4 hydrostatic correction as well as the correction to J2. The power increased at degree 4 by only 4.3% even though the change in J4 in MWT was significant as a percentage of its value. For these MWT fields the power at degree 4 is almost exclusively nonzonal.

Figure 5 shows the geoid representation of the Martian gravitational field with Tharsis removed assuming 0%, 10% and 20% nonhydrostatic components of J2 left in the gravitational field. In all cases the dynamic range of signal is significantly reduced from a planetwide value of about 2000 m in GMM-1 to -1500 m for a 20% nonhydrostatic MWT to about 400 m for a 0% nonhydrostatic MWT. The reduction is greatest at the lower nonhydrostatic values because Tharsis is more symmetric for these values (see next section), which suggests that Tharsis does not contribute significantly to the gravitational flattening of Mars. If it did, a larger value of nonhydrostatic J2 would fit the gravity model better. Note that the field for 20% nonhydrostatic J2 contains a significant concentration of signal in the equatorial plane that does not correlate with surface structure. We believe that this signal is associated with rotational flattening that has not been removed and interpret the pattern to indicate that such a large deviation from the hydrostatic state is inappropriate for Mars.

Table 1 lists the spherical harmonic coefficients through degree and order 6 for the MWT and GMM-1 fields. Figure 6 plots the RMS magnitudes of these coefficients for the 5% MWT and GMM-1 fields. This figure shows that significant gravitational energy has been removed only from degrees 2 through 4 despite modeling Tharsis out to degree 6. Figure 6 suggests that to account for the influence of Tharsis in the gravity field it is necessary to represent the province using at least four spherical harmonic degrees. Comparison of the two fields shows that Tharsis represents 75% of the gravitational potential energy at degree 3 and 50% at degree 4. The degree 3 term represents the asymmetry between the hemispheres, but the zonal term is the smallest of the coefficients because

Tharsis is near the equator, and it is apparently the longitudinal variation in gravity that is being removed. This is entirely consistent with our removal of the degree 3 zonal term in a coordinate system through the center of Tharsis.

3.2 Axial Asymmetry of Tharsis

As for several previous geophysical studies of Mars [Reasenberg, 1977; Kaula, 1979; Willemann and Turcotte, 1982; Kaula et al., 1989], our analysis assumes that the large-


(a)

(b)

Figure 4. The Mars without Tharsis gravitational field to degree and order 50 expressed in terms of (a) geoid in kilometers and (b) free-air gravity in mGals. Figure 4a corresponds to Figure 1 minus Figure 2. These fields assume a 5% nonhydrostatic component to J2. The contour interval is 0.1 km for the geoid and 100 mGals for the anomalies.

scale gravitational signature of Tharsis is axisymmetric about the center of the province. On the basis of visual inspection of the gravitational and topographic fields as well as the distribution of surface tectonics, it is clear that this is an

oversimplification of Tharsis' actual structure. However, we contend that axisymmetry is the most accurate first order representation of the province and, in addition, provides the least ad hoc quantitative description of its gravitational signature. In a critique of moment of inertia estimates, Bills [1989a,b] challenged the assertion of an axisymmetric Tharsis. To estimate the moment of inertia factor, he hypothesized the existence of unobserved mass sources and invoked assumptions about the statistical distributions of topography on the other terrestrial planets, none of which contain a surface structure that resembles Tharsis. While

mindful of the complexity of the Tharsis region, we note that possible mechanisms for the formation of the province include flexural and isostatic uplift [Banerdt et al., 1982; Sleep and Phillips, 1985; Tanaka et al., 1991; Banerdt et al., 1992],

volcanic construction [Solomon and Head, 1982], and mantle plumes [Harder and Christensen, 1996; Kiefer et al., 1996], all of which can be represented as fundamentally axisymmetric surface or subsurface loads.

By isolating the axisymmetric component of the gravity field associated with Tharsis, we aim to obtain an objective measure of the extent to which Tharsis deviates from

axisymmetry. A particularly straightforward assessment can be made by comparing the symmetric (C22) and asymmetric ([C222+S222] 1/2) parts of the second degree gravity field for the cases where the coordinate system z axis is coincident with the rotation pole (i.e., GMM-1)and with the pole of Tharsis. Table 2 summarizes this comparison for the assumption of 5% nonhydrostatic C2o, though the qualitative conclusions hold for other values. Note that when Tharsis is rotated to the pole, there is 3.27 times more "power" in the second degree zonal term than there is in the tesserals. For the normal GMM-1

field the ratio of the zonal to tesseral terms is only 0.63. So a 5% nonhydrostatic Mars viewed from the rotation pole is


(a)

(b)

(c)

Figure 5. Mars without Tharsis geoid to degree and order 50 assuming (a) 0%, (b) 10%, (c) 20% nonhydrostatic components of J2. The contour interval is 0.1 km.


Table 1. Spherical harmonic coefficients of GMM-1 [Smith et al., 1993] and MWT to Degree and Order 6

GMM- 1 MWT

I m C!m Stm Clm Stm

2 0 -4.3730e-05 6.1011e-06

or multiple mechanisms contribute to the long wavelength geoid and gravity signature of this region. As such, the observed departure from axisymmetry could be used as a constraint in models for the origin and evolution of the province.

3.3 Flattening/Moment of Inertia

2 1 3.7121e-09 6.5220e-09 6.6502e-06

2 2 -8.4177e-05 4.9608e-05 -2.0072e-05

3 0 -1.1887e-05 -3.9906e-06

3 1 3.8350e-06 2.5237e-05 -7.4147e-06

3 2 -1.5896e-05 8.4208e-06 -8.4302e-06

3 3 3.5357e-05 2.5121e-05 -1.8433e-06

4 0 5.1407e-06 -2.9099e-07

4 1 4.2679e-06 3.6871e-06 2.8749e-06

4 2 -1.1195e-06 -8.9548e-06 -7.1406e-06

4 3 6.4612e-06 -2.4462e-07 3.4032e-06

4 4 1.0599e-07 -1.2837e-05 -5.7057e-07

5 0 -1.7606e-06 -1.9689e-06

1.6734e-05

We next consider the range of allowable moment of inertia -1.0887e-05 factor for the gravity field without Tharsis as compared to that

for the full Martian gravity field. Figure 7 shows the variation of C/MR 2 as a function of the fraction of nonhydrostatic J2 for the two gravity models. When Tharsis is removed, the

-3.0757e-06 parameter J2 is reduced by approximately 5% for a fully

1.3756e-06 hydrostatic J2 to approximately 10% for J• that is 20% nonhydrostatic. The corresponding values of C/MR 2, plotted

7.7818e-06 in Figure 7, vary from 0.368 to 0.361. In Figure 7, MWT fields are shown for the cases where Tharsis is fully compensated and where there are equal contributions to the compensation from the planetary flattening and from Tharsis.

1.8130e-07 We have introduced the possibility of a variable compensation of Tharsis because it is a near-global feature of the planet and,

-3.2729e-06 as such, might be expected to be compensated to a similar extent as the flattening. Note that for the MWT field the

-1.6699e-06 moment of inertia factor is not nearly as sensitive to the deviation from the hydrostatic state as is the case for GMM-1, - 1.17 44e- 06 even considering uncertainty in the isostatic state of Tharsis. For comparison, Figure 7 also shows the values of the mean moment of inertia, I/MR 2, for the case of a fully compensated

5 1 6.2057e-07 2.0309e-06 7.9574e-07 2.4717e-06

5 2 -4.1596e-06 -1.2360e-06 -4.3680e-06 - 1.0393e-06

5 3 3.3581e-06 2.7677e-07 3.8441e-06 5.0332e-07

5 4 -4.6433e-06 -3.4267e-06 -4.6579e-06 -3.1737e-06

5 5 -4.3883e-06 3.8021e-06 -3.6433e-06 3.5550e-06

6 0 1.3318e-06 1.8934e-06

6 1 1.8198e-06 - 1.4533e-06 2.0480e-06 -8.7890e-07

6 2 8.3546e-07 1.6579e-06 1.4418e-06 1.0857e-06

6 3 8.3959e-07 2.8911e-07 1.3787e-06 5.4037e-07

6 4 9.8305e-07 2.6753e-06 1.0398e-06 1.6971e-06

6 5 1.6663e-06 1.6393e-06 2.1636e-06 1.4743e-06

6 6 2.8035e-06 8.0493e-07 1.8510e-06 -3.2937e-07

GMM-1 values assume Mars has a 5•o nonhydrostatic component to C2o. All terms above 6x6 are the same as GMM-1. Read -4.3730e-05 as -4.3730 x 10 -5.

considerably more irregular in longitude than in latitude, in contrast to when it is viewed from Tharsis. Comparing the two configurations, it is apparent that Tharsis is 5.2 times more symmetric than the planet as seen from the pole, which indicates that Tharsis is considerably more axisymmetric than Mars as a whole. We interpret the deviation of Tharsis from axisymmetry as an indication of the extent to which complex

10'

KI KI

N 10 '

Z

olo- o

10'

! ! !

!

Mars without Tharsis • --•--GMM-1; 5% non-hydrostatic Mars

....

5 10 15 20 25

DEGREE, L

Figure 6. RMS magnitude of spherical harmonic coefficients of the Mars without Tharsis gravitational field compared to that for GMM-1. Both fields assume a 5% nonhydrostatic component to J2. The figure shows the amount of gravitational potential energy that is associated with the removal of Tharsis.


Table 2. Degree 2 Spherical Harmonic Coefficients for Nonrotated and Rotated Gravity Fields

GMM-1 (5% Nonhydrostatic) Not Rotated

A change in pole position, information needed to

determination of C/MR 2.

if detected, would provide the derive a nonmodel-dependent

I rn C S RMS

2 0 -4.37x10 -5

4. Polar Wander

2 2 -8.417x10 -5 4.96x10 -5 6.9x10 -4

C2/(C222-1-S222) 1/2-- 0.63 GMM-I (5% Nonhydrostatic) Tharsis Rotated to Rotation Pole

2 0 1.443x10 -4

2 2 -6.379x10 a

C2/(C222.j.•, 2hl/2_ 3.27 •-'22 ! --

2.518x10 -6 4.4x10 '•

Tharsis; this quantity does not show a range significantly different from that of C/MR 2.

Reasenberg [1977], Kaula [1979], and Kaula et al. [1989] also estimated the moment of inertia assuming axisymmetry of Tharsis and found a most likely value of approximately 0.365. Those analyses differed from ours in that they assumed that Tharsis is situated on the Martian equator and they did not (at least explicitly) specify the deviation of J2 from a hydrostatic state. Nonetheless, our range of permissible values bounds their estimate. Bills' [1989a] lower limit of C/MR 2of 0.345 is based on the assumption that Mars' intermediate principal moment is midway between the greatest and least principal moments. More recently, a reanalysis of Viking range data to estimate Mars' observed precession rate [Yoder and Standish, 1997], which made assumptions about the planet's polar cap- atmosphere mass exchange, finds a value of C/MR 2 of 0.355+0.015. Geochemical studies based on SNC meteorites

have end-member values of 0.357-0.366 [Sohl and Spohn, 1997].

Tharsis is an ancient feature [Tanaka et al., 1992]' however, it is believed to have formed subsequent to planetary differentiation [Schubert et al., 1992], at which time Mars' fundamen. tal radial density distribution (i.e., presence of core and mantle) was established. While the formation of Tharsis

undoubtedly affected the density structure of at least the Martian upper mantle, it would not have necessarily significantly changed the moment of inertia factor. If Mars' present-day long-wavelength gravity field is fundamentally similar to that early in Mars' history, then the moment of inertia factor implied by the MWT field may have relevance to the value before Tharsis formed. Indeed, it is possible that the presence of Tharsis has obfuscated attempts to constrain C/MR 2 .

Doppler tracking of the future Mars Global Surveyor orbiter [Tyler et al., 1992] and Pathfinder [Edwards et al., 1992' Folkher et al., 1997] and Mars Volatiles and Climate Surveyor (D.A. Paige, personal communication, 1996) lander spacecraft are expected to provide accurate estimates of the position of Mars' rotation pole. These results will be compared to the pole determined from Viking observations [Davies et al., 1992; Standish et al., 1995] that were collected two decades earlier and will allow a direct estimate of Mars' precession rate.

Various studies have proposed that there has been significant polar wander earlier in Mars' history [Murray and Malin, 1973; McAdoo and Burns, 1975; Schultz and Lutz,

1988], an assertion that appears to be inconsistent with observed tectonic patterns and predicted magnitudes of lithospheric stresses [Melosh, 1980; Willemann, 1984; Grimm and Solomon, 1986]. Previous stress models of a reoriented Mars assumed either an oblate spherical shell or a thin shell on which is emplaced an axisymmetric load corresponding to Tharsis. As with the elastic shell models, i t is not possible to quantitatively test specifically proposed polar wander paths. However, it is feasible to investigate whether global-scale reorientation could have been associated with the formation of Tharsis.

Reorientation of the Martian lithosphere relative to the rotation axis could occur as a result of the change in the moment of inertia tensor associated with the removal of the

excess gravitational load corresponding to Tharsis. Goldreich and Toomre [1969] showed that the rotation axis corresponds

0.38

0.37

0.36

0.35

ß ß n•n ßnnn '

ß ß • • nmll ß ''' ' :;.' ' nnlnn nnllnn Kaula et '-'"N;•.• ....

_ ai:---[J-989] '••"•...• ...... .---. Yoder & G &T i

_ Standish [1997] •. _ :--- Bills [1989b] '

[ [ [ , I [ .[ ] • I [ • [ [ I .[ [ r

0'340 0.05 0.1 0.15 0.2 Fraction Non-hydrostatic J2

MWT

Figure 7. Comparison of moment of inertia factors for GMM-1 and .MWT gravitational fields. The solid line shows C/MR 2 as a function of the percent nonhydrostatic component of the Martian flattening, J2. The dashed line shows C/MR 2 for the MWT model for the case where Tharsis is assumed to be

fully compensated, and the closely spaced dotted line shows C/MR 2 for the MWT field for the case where there are equal contributions to the compensation from the planetary flattening and Tharsis. Other published values of C/MR 2 are shown for comparison. The dotted line with the greater spacing shows !/MR 2 for the MWT field. Also shown is the maximum value of C/MR 2 required prevent planetary reorientation according Goldreich and Toomre [ 1969] (equation (9•)). The various lines are discussed in the text.


to the axis of maximum inertia for both the nonhydrostatic AC and the total polar moment C+AC. The Goldreich and Toomre formalism places an upper bound on the value of C/MR 2 that can occur without reorientation of the rotation axis of the

planet. This occurs for the case where zM-0 and

AC >_ AB = 4MR 2 J22 (9)

or

AC/MR 2 _> 0.252 x 10 -3 (10)

for the present Mars gravity field [Sleep and Phillips, 1985; Kaula et al., 1989]. The corresponding maximum value of C/MR 2 = 0.36631. Note that the condition for planetary reorientation depends on the equatorial ellipticity as well as the planetary oblateness.

The reorientation of Mars due to Tharsis was previously estimated by Melosh [1980], under the assumption that the second degree harmonic on Mars is essentially fully compensated (as on Earth [Goldreich and Toomre, 1969]) and does not contribute to the stability of the orientation of the lithosphere. That study predicted a maximum planetary reorientation of -25 ø. A subsequent study by Willemann [1984] included the effect of the incomplete compensation of the degree 2 term on the stability of the orientation and, accordingly, bounded the range of reorientation to 3 ø- 9 ø. That estimate is based on removal of terms higher than degree 2 and thus considered the deviation from compensation of shorter wavelength loads. But note that all studies of Mars' spin axis reorientation make either direct or implicit assumptions about the long-wavelength compensation state.

As applied to Earth, Goldreich and Toomre [1969] argued that the timescale for adjustment of the oblate rotational figure was much less than that of changes in nonhydrostatic density variation. This would be expected because an adjustment in shape would require a single shift, while that due to density heterogeneities may require complex motions associated with the internal thermo-mechanical structure. Kaula et al. [1989] asserted that this argument should also hold for Mars. Goldreich and Toomre's condition for stability of the rotation axis due to global-scale mass re-distribution as given by equation (9) is plotted as a dashed/dotted line in Figure 7. Note that for theoretically plausible values of the nonhydrostatic component of the Martian flattening, J2, the value of C/MR 2 exceeds the Goldreich and Toomre limit. Consequently, the planetary figure for the models without Tharsis is prone to reorientation of the spin axis. However, it must be remembered that the Goldreich and Toomre formulation was

developed for Earth, which lacks a strong elastic lithosphere. On Mars, which likely has a cold shallow interior [Solomon

and Head, 1990], at least in the present day, the lithosphere could conceivably have retained a remnant fossil bulge that could have prevented the shell from being easily reorientated by mass variations (N. Sleep, personal communication, 1997). Flexure at the second harmonic, which could have relaxed quite slowly, must ultimately be taken into account to determine the possible reorientation of the global figure. This will require an improved understanding of Mars' thermomechanical structure over time.

5. Regional Structures

A number of studies have attempted to constrain the effective elastic thickness (T e) of the Martian lithosphere via analysis of gravity or gravity/topography observations of individual structural features, mostly in the Tharsis and Elysium provinces [Comer et al., 1985; Hall et al., 1986; Janle and Erkul, 1991; Anderson and Grimm, 1995; Turtle and Melosh, 1995]. The parameter Te can be related to the near- surface thermal gradient [McNutt and Menard, 1982; McNutt, 1984] and has implications for Martian thermal history and thermal and compositional heterogeneity [Solomon and Head, 1990]. Our analysis has some relevance to regional gravity studies.

Both the MWT geoid and gravity anomaly maps show improved definition of major geological structures, such as the Tharsis montes, Alba Patera, and Valles Marineris. In all cases

the peak amplitudes of the gravity signals are decreased by the removal of the Tharsis signal, which is generally manifest as a long-wavelength slope, but the local dynamic range of the geological structures is virtually unchanged. For example, the peak gravity anomaly associated with Valles Marineris decreases by 30%, and that for Alba Patera decreases by 25%, but both maintain their dynamic range of short-wavelength signal. Obviously, in regional studies, care should be taken to isolate gravity (and topography) signals in the appropriate wavelength range to assure the absence of "contamination" from Tharsis' significant long wavelength power.

Figure 8 shows the geoid in the vicinity of Mars' largest shield volcano, Olympus Mons, before and after the removal of Tharsis. Olympus Mons is located on the western slope of Tharsis. Removing Tharsis' gravitational signature reveals a distinct gravitational "moat" surrounding the volcano that has been interpreted as evidence of lithosphere flexure in response to the volcanic load [Zuber et al., 1994]. Isolation of the Olympus Mons anomaly should facilitate calculation of Te from the predicted gravity associated with flexure of the elastic lithosphere. Previous analysis based on stress considerations [Comer and Solomon, 1981; Comer et al., 1985; Janle and Jannsen, 1986], which utilized the lack of observed fractures concentric to the volcano as the principal model constraint, found the elastic thickness in this region to be at least 150 km. In contrast, more recent analyses that matched the amplitudes and widths of the volcanic construct and flexural moat for the MWT model instead found values of the order of

50 km [Zuber et al., 1994; Kiefer et al, 1995], which are comparable to flexural lithospheric thicknesses of the other Tharsis montes--Ascraeus, Pavonis and Arsia Montes [Comer et al., 1985]. }towever, the latter models did not yield good fits to both the amplitude and width of the observed gravity. This may be in part a consequence of the a priori constraint applied in the spherical harmonic models. The Kaula constraint [Kaula, 1966] provides stability in spherical harmonic solutions derived from data sets with non-uniform

coverage or density, as are currently present for Mars, but causes power at short wavelengths to be underestimated because of the suppression of viable signal in some areas. The Mars50c gravity model of Konopliv and Sjogren [1995] selectively relaxes this constraint in the Tharsis region and shows greater power for Tharsis features as compared to the


(a)

(b)

40 ø

30 ø

20 ø

10 ø

0 o -160 ø

Figure 8. Geoid in the vicinity of the Olympus Mens volcano from (a) GMM-1 and (b) MWT, both assuming a 5% nonhydrostatic component to J2. This volcanic shield, which is the largest in the solar system, is located on the western flank of Tharsis. Removing the gravitational signature associated with Tharsis eliminates the regional (westward) slope and elucidates an apparent flexural moat around the volcano.

GMM-1 model. Nonetheless, neither gravity field yields satisfactorily good fits for elastic lithosphere thickness to the Olympus Mens gravity data to enable significant confidence in the estimate of Te. Analysis of global spherical harmonic solutions (with Tharsis removed) to be produced with the uniform tracking coverage from Mars Global Surveyor will hopefully help resolve this issue.

6. Summary

We quantitatively represent the gravitational signature of Mars' Tharsis province in terms of low-degree zonal spherical harmonic coefficients. We show that removal of the Tharsis

signal from the Martian gravitational field via a set of matrix rotations yields a residual field that can be used to address certain questions relevant to the global and regional geophysics of Mars. For example, consideration of the dynamic range of signal for various nonhydrostatic contributions to the J2 term of the gravity field shows that Tharsis does not appear to contribute significantly to the planetary flattening. Results also show that if Tharsis formed subsequent to the global differentiation event that produced the approximate present-day planetary radial density stratification, then Mars' moment of inertia factor is expected to fall in the range 0.361<C/MR2<0.366. The analysis indicates that the formation of Tharsis may have resulted in large-scale polar wander, though if the planet had a sufficiently thick elastic lithosphere at the time Tharsis formed such that the planet retained a rotational fossil bulge, then changes in planetary figure would be inhibited. Understanding of the relaxation of the degree 2 flexural harmonic, which will require improved knowledge of the planet's thermemechanical structure over time, should be factored into future efforts to

estimate the effect of Tharsis on the planetary orientation. A comparison of the zonal and tesseral components of the degree 2 field in the GMM-1 and rotated gravity fields indicates that Tharsis is more than 5 times more axisymmetric about its pole than Mars as a whole is about its rotation axis. Without the

removal of Tharsis, the geoid in the Tharsis region decreases in amplitude by up to a kilometer. Regional structures are better defined in a spatial sense and show decreases in peak amplitude while maintaining dynamic range. The Olympus Mens volcanic shield displays clear evidence of a gravitational flexural moat.

Anew Mars gravitational field of globally uniform spatial resolution will be derived from X band Doppler tracking of the Mars Global Surveyor spacecraft [Tyler et al., 1992], which will be in a polar orbit with an altitude of approximately 400 km. This gravity field, in combination with an even higher resolution topography field that will also be obtained in the mission [Zuber et al., 1992], will enable detailed analysis of the geophysics of Mars using many techniques, including the one presented here. An analysis using this approach that addresses the effect of Tharsis topography on global atmospheric circulation models is under way [Zuber et al., 1996]. Our method may also have value if applied to other planetary geophysical observations, such as magnetics, that contain global-scale signatures which may be considered axisymmetric to first order. Isolation of structures such as mantle plumes, major volcanic constructs, or large impact basins should be attainable given adequate data distribution and resolution.

Acknowledgments. We are grateful to Norman Sleep and Walter Kiefer for insightful reviews, John Robbins for programming support, and Gregory Neumann for assistance in producing the maps. We also acknowledge insightful discussions with Bruce Bills, David Rubincam, and Bruce Jakosky. This work was supported by the NASA Planetary Geology and Geophysics Program and the NASA Mars Global Surveyor Project.


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(Received March 31, 1997; revised September 3, 1997; accepted September 8, 1997.)

on the moment of inertia factor of 0.361

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