Combining Eqs. (3.13) and (3.14) allows one to determine C/MR2 and A/MR2. C/MR is of particular interest since it indicates the amount of central condensation in a planet. A homogeneous spherical body has C/MR =0.4. As a body becomes more centrally condensed (i.e., forms a larger core), C/MR becomes smaller than 0.4. Mars' J2 varies seasonally due to the sublimation/condensation cycles of CO2 between the polar caps and atmosphere, but the average value from MGS Doppler tracking is

1.96 ± 0.69 X10-9 (Smith etal, 2001b). Comparison of Mars'

rotation rate between Viking lander and Mars Pathfinder data gives a precession rate of -7576 ± 35 milli-arcseconds per year, leading to a value of C/MR2 0.3662 0.0017

(Folkner et al, 1997). This indicates that Mars has a small core, but there is still debate as to whether it is solid (Smith et al., 2001b) or liquid (Yoder et al., 2003).

The primary contributor to J2 is the planet's flattening, producing the equatorial bulge. Since flattening primarily depends on how rapidly the planet is spinning, J2 can be related to the planet's angular rotation rate (x) through a dimensionless measure of the centrifugal potential (q):


Strictly speaking, Eq. (3.15) is only valid in planets where the interior density is constant. For more realistic planets, where density increases with depth, the coefficient before J2 becomes greater than 2. For bodies in hydrostatic equilibrium, J2 and q can be related to the flattening (f) by f (2) J2+(£) q- (3-16)

3.2.2 Gravity anomalies, isostacy, and crustal thickness

The equipotential surface corresponding to a spherical planet's mean radius (R) is called the geoid. Geophysicists define the martian geoid at a radius of 3396km from COM (Neumann et al., 2004). Variations in gravitational potential from the geoid are called gravity anomalies and represent uneven mass distribution on the surface or within the planet. Gravity anomalies are measured in gals, where 1 gal = 10-2 m s-2. Most gravity anomalies are very small and are measured in milligals (mgal). Smaller masses can be discerned as the order (l) and degree (m) increase in Eq. (3.7). Detailed trajectory analyses of the various Mars orbiter missions have led to gravity models up to order and degree 85 (Neumann et al., 2004).

Using Eq. (3.1), we find that the acceleration due to gravity (g0) at the geoid is g0 = GM (3-17)

At some height h above the geoid, this becomes g(h) = GM 2. (3.18)

Using a Taylor series expansion, Eq. (3.18) becomes

Thus, when a spacecraft is at some height h above the geoid, the gravitational acceleration that it measures is less than that measured at the geoid. The difference in gravity due to elevation above the geoid is called the free air anomaly and the term — (2GM/13)h is the free air correction, indicated by gFA.

The free air anomaly ignores any mass which exists between the geoid and the spacecraft position at h. Thus, if the spacecraft passes over a mountain, it might experience an additional gravitational acceleration due to the mountain's excess mass. In 1749 the French mathematician Pierre Bouguer applied Gauss' s law in a gravitational field to an infinite slab of density p and thickness h to obtain the Bouguer correction:

One final correction occurs for latitudinal variations in g due to rotation and the planet's oblate shape. This latitudinal correction is indicated by y0. The complete description of gravity is called the Bouguer anomaly and is given by


g(h) = ~R---1F~ h — 2nGph — C0 = g0 + gFA — gB — C0. (3-21)

Topography and the Bouguer anomaly are often correlated on Mars, providing insights into how topographic features are supported. If the free air anomaly (Eq. 3.19) is approximately zero, the feature is isostatically compensated. Isostacy is the balance between the weight of a crustal block and the buoyant force exerted on it. In Figure 3.4a, we see a crustal block of cross-sectional area A and density (p-Ap) resting in material of density p. The block straddles the reference surface, extending a height h above the surface and a depth d below it. The upward buoyancy force (FB) on the block is given by Archimedes' principle:

Fb must be balanced by the block's weight for the block's position to be stable:

Equation (3.23) reduces to the isostacy equation:

A zero free air anomaly (Eq. 3.19) or a negative Bouguer anomaly (Eq. 3.20) indicates complete isostatic compensation. There are two ways in which isostatic compensation can be achieved. Topography which has a constant density but where the depth of the below-surface root is greater than the height of the feature above the reference surface is compensated through Airy isostacy (Figure 3.4b). Alternately, Pratt isostacy (Figure 3.4c) assumes that the depth of compensation is the same for all topographic features but that the density varies, with higher features having lower density. Although Belleguic et al. (2005) find density variations across the




Figure 3.4 Surface mass distributions such as mountains extend below the surface to a depth which depends on the difference in density between the block and underlying material (p —Dp and p, respectively). (a) The height (h) and depth (d) of a block of material (with cross-sectional area A) relative to the surface is determined by the balance between the block's weight and the buoyant force (FB) from the underlying material. (b) Airy isostacy says that this balance occurs because mountain blocks have roots that extend a greater depth than the mountain's height above the surface. (c) Pratt isostacy has all blocks reaching the same depth, but the above-surface height varies because of variations in block density.

martian surface and suggest that Pratt isostacy might apply, most researchers argue that Airy isostacy is the major compensation mechanism operating on Mars (Zuber et al., 2000; McGovern et al., 2002).

Figure 3.5 shows a Bouguer anomaly map obtained from the MGS MOLA and Radio Science investigations (Neumann et al., 2004). Comparison with the MOLA-derived topography map of Mars (Figure 3.6) shows correlations between gravity and topographic features such as volcanoes and impact basins. Large impact craters and basins typically display a positive Bouguer anomaly because of post-impact uplift of the underlying Moho (the boundary between the crust and the mantle) and infilling by volcanic and sedimentary deposits. Isostatically compensated mountains such as Alba Patera and the Elysium volcanic region display highly negative Bouguer anomalies.

Crustal thickness variations can be modeled using the gravity data, topography, and assumptions about density variations in the crust and mantle (Zuber et al., 2000; Neumann et al., 2004). Crustal thickness varies with latitude, being thicker in the southern hemisphere than in the northern, but also varies with longitude (Figure 3.7). Neumann et al. (2004) report an average thickness of 32km for the northern plains crust versus 58 km for the southern highlands crust. Interesting, the transition in crustal thickness between the two hemispheres does not exactly correlate with the hemispheric dichotomy, arguing against an impact origin for the dichotomy (Zuber et al., 2000).

3.2.3 Topography

Resolution of the topographic variations on Mars has been greatly improved by the MOLA experimental results. The laser altimeter provided a vertical accuracy of ~1m relative to Mars' COM and the topographic grid has a resolution of 1 /64° in latitude and 1/32° in longitude (Smith et al., 2001a). Zero elevation corresponds to the geoid, the equipotential surface corresponding to a distance of 3396km from COM at the equator. The highest point on Mars is the Olympus Mons volcano summit at 21.287km and the lowest point is in the Hellas impact basin at - 8.180km. Figure 3.6 clearly shows the hemispheric dichotomy between the low-lying northern plains and the elevated southern highlands.

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