A surface integral approach to Poisson's equation and analytic expressions for the gravitational field of toroidal mass distributions

Abstract

A new boundary integral approach is proposed for evaluating the potential of an arbitrary given static mass/charge distribution with a definite boundary, reducing the problem from a volume to a surface integral. In this approach, the internal potential is split into a particular solution to Poisson's equation plus some solution to Laplace's equation. The surface integral solution is then derived from Green's identities and the boundary conditions. For rotational symmetry the surface integral may be reduced to a line integral. Using this approach, the gravitational potential of a homogeneous torus is expressed as a series of toroidal harmonics, and results are confirmed with the solution obtained via volume integration. Analytical results are also derived for a torus with a mass distribution concentrated around the plane of symmetry. The surface integral approach is tested numerically against the standard volume integral for a spheroid.