Outer gravitational potential of a homogeneous torus with an elliptical cross-section: I. Representation by two massive circles

TL;DR

This paper introduces a new, accurate method to compute the gravitational potential of a homogeneous elliptical torus by approximating it with two massive circles, applicable to both oblate and prolate shapes.

Contribution

It provides a novel expression for the gravitational potential of elliptical tori using two massive circles, simplifying calculations and applicable to different cross-sectional shapes.

Findings

Outer potential approximated by two massive circles with half the torus mass

Approximation is accurate for both oblate and prolate cross-sections

Error maps confirm robustness of the method

Abstract

This paper deals with the gravitational potential of a homogeneous torus with elliptical cross-section. We present a new expression for its gravitational potential which is valid in any point of the space, obtained by modeling the torus with a set of massive circles (infinitely thin rings). We found that the outer potential can be represented with good accuracy by the potential of two massive circles with masses which are half of the torus mass. These massive circles intercept the elliptical cross-section at two points along the major axis which are in opposite directions and at half of the distances to the foci of the cross-section. The same formula works for both cases: oblate and prolate cross-sections. For the case of the prolate cross-section of the torus the distances to massive circles are imaginary and conjugate ones but the values of the torus potential for this case are real. The obtained approximation is robust as the error maps show.