Exact gravitational potential of a homogeneous torus in toroidal coordinates and a surface integral approach to Poisson's equation

TL;DR

This paper derives exact gravitational potential solutions for a homogeneous torus using series of toroidal harmonics and introduces a novel surface integral method for solving Poisson's equation applicable to various geometries.

Contribution

It presents new exact solutions for a homogeneous torus's gravitational potential and introduces a general surface integral approach to Poisson's equation.

Findings

Derived rapidly converging series solutions for the potential

Developed a surface integral method applicable to all geometries

Reduced volume problems to surface integrals for potential calculation

Abstract

New exact solutions are derived for the gravitational potential inside and outside a homogeneous torus as rapidly converging series of toroidal harmonics. The approach consists of splitting the inter- nal potential into a known solution to Poisson's equation plus some solution to Laplace's equation. The full solutions are then obtained using two equivalent methods, applying differential boundary conditions at the surface, or evaluating a surface integral derived from Green's third identity. This surface integral may not have been published before and is general to all geometries and volume density distributions, reducing the problem for the gravitational potential of any object from a volume to a surface integral.