Gravitational potential of a homogeneous circular torus: new approach

TL;DR

This paper introduces a new integral expression for the gravitational potential of a homogeneous circular torus, providing approximate formulas for inner and outer regions and a method to ensure a continuous potential solution.

Contribution

A novel integral expression and a sewing method for the gravitational potential of a homogeneous circular torus are developed, improving approximation accuracy across the entire region.

Findings

Outer potential approximates that of an infinitely thin ring

Inner potential is quadratic in coordinates

Continuous potential solution achieved through sewing method

Abstract

The integral expression for gravitational potential of a homogeneous circular torus composed of infinitely thin rings is obtained. Approximate expressions for torus potential in the outer and inner regions are found. In the outer region a torus potential is shown to be approximately equal to that of an infinitely thin ring of the same mass; it is valid up to the surface of the torus. It is shown in a first approximation, that the inner potential of the torus (inside a torus body) is a quadratic function of coordinates. The method of sewing together the inner and outer potentials is proposed. This method provided a continuous approximate solution for the potential and its derivatives, working throughout the region.