The solution of the inverse gravity potential problem for cylindrical bodies

Summary. The inverse gravity potential problem consists in the determination of the form and the density of the body by its exterior gravity potential. We describe two similar classes of bodies for which this problem has a unique constructive solution. (1) The first class contains the cylindrical bodies with finite length, arbitrary form of section and ρ( R , ø, z) =ρ 1 ( z )ρ 2 ( R , ø) density distribution, where z is the cylindrical coordinate; R , ø are the polar coordinates in a section plane. This class is important for prospecting geophysics in that it allows us to determine in a unique and constructive way, the function ρ 1 ( R , ø), the length, form and orientation of the cylinder if we know the function ρ 1 ( z ) and the exterior potential. The classical moment problem of functions is the basis for the solution of this problem. (2) The analogous problem for the class of the spherical cylinders, or bodies bounded by arbitrary similar sections of two different concentric spheres and the radial lateral surface, appears when bodies of planetary size are studied. (An example of these bodies would be the Moon mascons.) The density distribution of these cylinders is ρ(τ, θ, ø) =ρ 1 (τ)ρ 2 (θ, ø) where τ, θ, ø are the spherical coordinates. The function ρ 1 (θ, ø), length and form of spherical sections can be uniquely determined by exterior potential if we know the function ρ 1 (τ). We propose a new constructive method for harmonic continuation of the gravity potential into the region containing the perturbing masses for the solution of the problem.