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If the moon is a rigid body, why do the mass concentrations on the moon make orbits unstable? Doesn't a satellite just orbit the center of mass of it's parent body?

Satellites (e.g. GRACE) seem to be able to measure these things by perturbations in their orbit but I'm not sure how or why. What oversimplification am I making?

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It's often worth considering an extreme example. Suppose, for instance, that we put virtually all of the mass of the moon into four extremely dense "mascons" located near the surface and at the vertices of a square (whose center would, therefore, necessarily coincide with the center of the moon.) Now consider the net gravitational force on a satellite as it orbits at relatively low altitude in the plane of that square. You should be able to convince yourself that the orbit would consist of four relatively low curvature sections when the satellite is far from any of the mascons (and, therefore, subject to a relatively low gravitational force) and four relatively high curvature "corners" as it passes each mascon (and is, therefore, subject to a relatively large gravitational force.) If you are visualizing the resulting orbit correctly, you should see that the satellite might graze the surface in between mascons and reach maximum altitude as it passes each one. The orbital shape would reveal the location of the mascons.

Repeating the gedankenexperiment for different orbital parameters or for different distributions of such high density mascons should make the general principle clear.

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The replacement of a body, for gravitational purposes, by the same mass located at the center of mass, is based on the assumption that the body is spherically symmetric. This is usually a good approximation for large body, but not a perfect one.

The earth can be considered as spherical, then as a sphere with an added equatorial bulge, and then as a sphere with bulge with various lumps and bumps down to finer and finer detail...

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Because the moon, like many planetary bodies, is not homogenous in its composition and doesn't express an absolutely even gravitational pull from every side.

When you orbit a body, you are countering the gravitational pull of its entiriety. When you come over a spot with higher density and/or proximity, you are pulled more due to the fact that gravitational force increases based on proximity and mass.

The effects of this can be fairly easily measured with the proper equipment, but I am far from knowing the specifics of that side of things.

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