Quote:
Originally Posted by Hornblower
I can imagine the possibility that irregular, lumpy concentrations of heavy material deep inside the Moon could displace the center of mass diagonally from the geometric center, rather than merely 2km along the geometric long axis of the surface. That could cause the best-fit prolate ellipsoid to be misaligned from the Earth.
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There
are two different shapes involved here--one is the actual surface shape, the other is the mass distribution, which is reflected in the shape of the gravity field. On the Earth, the gravity field equipotential surface (the geoid) has a relief of a few hundred meters after the twenty kilometer equatorial bulge is accounted for, whereas the surface obviously has a much greater relief (Mt. Everest is almost nine thousand meters above sea level).
The
moon shape is can be represented by spherical harmonics ("Degree 359 shape model of the Moon derived from the USGS Unified Lunar Control Network 2005 "), the coefficients of which are:
Code:
0 0 1737025.82132502 0.000000000000000
1 0 147.488719624524 0.000000000000000
1 1 -985.035952958352 -422.005429835170
2 0 -705.984306653320 0.000000000000000
2 1 -778.071552512909 -0.397858856690212
2 2 85.3781176271541 395.764151767781
3 0 63.7994229702025 0.000000000000000
3 1 568.005959384810 87.1612559253587
3 2 469.590091713719 153.342161993025
3 3 423.630343614396 -16.3014823621007
The degree n (n,0) coefficient is always zero (actually it doesn't exist), except for n=0. The (0,0) coefficient is the average radius, which the table reports as 1737 kilometers (about a half kilometer different from what appears at
Planet Scapes). The (1,n) coefficients represent a shift of the center of mass from the geometric center, and can be resolved into a single shift along an axis. The (2,0) coeffiicient represents the equatorial bulge, and the (2,2) coefficients the "pinching" of the equator. The (2,1) coefficients represent how much the degree two shapes deviate from axial symmetry. The (3,0) coefficient is the so-called "pear-shape" coefficient.
I'll see if I can dig up the lunar gravity field data.
ETA: The
same website has some coefficients (Spherical harmonic coefficients of the lunar potential field LP150Q (Konopliv et al., 2001))
Code:
4902.80107600000
2 0 -9.090109494810000E-005 0.000000000000000
2 1 -1.862736081840000E-009 -1.424538946100000E-009
2 2 3.463762742080000E-005 1.440635035400000E-008
3 0 -3.203071679590000E-006 0.000000000000000
3 1 2.634183586220000E-005 5.463078608820000E-006
3 2 1.418533167860000E-005 4.889139117950000E-006
3 3 1.228626450440000E-005 -1.782462707200000E-006
Hmm, these seem to be in a bit different scale. There are no (1,x) coefficients (they're identically zero), because when the center of mass is the center of coordinates, there is no shift.
ETA:
This article from 1969 (Science, Kaula) is online, and it shows that the 4902 figure is
GM, the product of the gravitational constant and the mass of the moon. The other figures are fairly close to the Michael
et al. figures in its Table 2 (the (2,1) coefficients are smaller in the above--the size of the (2,1) coefficients result in what is known in the Earth as the Chandler wobble).