On the shape, gravity field, and strength of the Moon

Abstract

The available astronomical data concerning the moon are inconsistent, probably because of errors or incorrect estimates of terms in the theory of the moon's motion or because of an error in the reduction of observations. The theory of a rotating gravity field which has a triaxial ellipsoid as an equipotential surface is developed here to the first order. This theory allows direct comparison of the dimensions of the ellipsoid with the components of the gravity field by means of the solution of a Dirichlet problem without any hypothesis on the density distribution. The moments of inertia are introduced and the density distribution is discussed. If the surface of the body is not equipotential, the results are still valid for one of its equipotential surfaces and the gravity field. The shape of an equipotential surface of the moon is discussed by means of this theory. It is also proved that the physical surface of the moon cannot be equipotential; the moon's moments of inertia are discussed in terms of density discontinuities. The origin of the bulges in the lunar gravity field is also discussed, and the strength for supporting them computed for various lunar models with or without a core. It is evident that a lunar orbiter would give long-awaited answers to these problems. Therefore a useful expression of the lunar potential and gravity fields for use in the study of orbits of satellites is obtained. Moreover, by means of this expression of the potential, a formal solution of the Hamilton-Jacobi equations is obtained. This solution is simpler than those already given and also has a closer relation to the model field. In this potential, in elliptic or spherical coordinates, the zero- and second-order zonal harmonics are retained, and the perturbations begin with the zonal harmonics of order 3 and with the tesseral harmonics which are smaller than the zonal harmonic of second order, according to the present data on the moments of inertia.