Can we prove that the Moon isn't hollow?
Does the Moon's mass and related gravitational field prove that it is not hollow? If not, are there any observations that indicate whether or not the Moon is hollow?
The gravitational field of the Moon would be unaffected by the radial distribution of mass within the Moon if the lunar mass were spherically symmetric (i.e., density of the moon only varied radially). For example, had the Moon been replaced with a point object of identical mass, the same gravitational field would occur at distances greater than the ~1700 km (1060 miles) lunar radius. This can be derived quite easily for a spherically symmetric moon by applying the integral form of Gauss's law. Spherical symmetry is an effective approximation for most planet-scale gravitational calculations.
However, this means that the large-scale gravitational field of a planet (or satellite such as the Moon) cannot be used to constrain the (radial) distribution of mass. In other words, the large-scale gravitational field of a planet does not convey any information about the (radial) distribution of mass. Nevertheless, if a hollow Moon is assumed, the lunar density would have to be un-physically large to generate the observed gravitational field. Furthermore, fine-scale variation (e.g., variation along the orbit of the Lunar Prospector orbiter) of the lunar gravitational field is consistent with geologic processes involving a crust, mantle, and core.
The only effective ways to determine the distribution of mass within a planetary body are:
1. Moment of inertia parameters. For the Moon, moment of inertia parameters have demonstrated that the core is ~1.4% of the total mass. One such parameter, the normalized polar moment of inertia, is 0.393+/-0.001, which is very close to that for a solid object with radially constant density (0.4; for comparison, Earth's value is 0.33). In other words, the moment of inertia parameters indicate that the core of the moon is both dense and small, with the rest of the moon consisting of material with nearly-constant density.
2. Seismic observations. Besides Earth, the Moon is the only planetary body with a seismic observation network in place. The lunar seismic data have helped constrain the thickness of the crust (~45 km (30 miles)) and mantle, as well as the core radius (~350 km (220 miles)).
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