On a method of computing the gravitational fields of inhomogeneous bodies

Summary

A solution of the direct gravity problem for a finite body with variable density is given. The method is based on Green's formula and is applicable when a particular solution of Poisson's equation is known. The attraction due to the body is expressed by integrals over its surface

The exact solution of the direct gravity problem, as known from the theory of two-dimensional fields [1–3], is closely connected with the problem of the analytic continuation of the exterior field of the attracting mass system into its interior. In the first place, this is a problem of determining the singularities of the exterior field, their distribution within the system and their nature. This approach to the solution of the direct problem is also meaningful from the point of view of determining the characteristics of the attracting system and, therefore, also of solving the inverse problem. In the case of two-dimensional fields the methods of analytical continuation were widely developed in a series of well-known papers by V. N. Strakhov, and they are mainly based on the methods of the theory of the functions of the complex variable. These methods were also successfully applied by Tsirulskii and Golizdra [1, 2] in treating the homogeneous and inhomogeneous, two-dimensional direct problem by means of Cauchy's integrals. However, as regards three-dimensional fields a number of fundamental problems has not been solved in this respect.