The exterior gravitational potential of toroids

TL;DR

This paper develops a series expansion method to accurately compute the exterior gravitational potential of thin, axisymmetric toroidal shells, with high precision near their surface, applicable to both homogeneous and stratified bodies.

Contribution

It introduces a bivariate Taylor expansion approach for the Laplace equation to determine the exterior potential of thin tori, achieving high accuracy and convergence, and extends to magnetic potentials.

Findings

Series converges rapidly, especially near the surface.

Leading term matches the potential of a loop with same mass and radius.

High precision achieved with low-order truncations.

Abstract

We perform a bivariate Taylor expansion of the axisymmetric Green function in order to determine the exterior potential of a static thin toroidal shell having a circular section, as given by the Laplace equation. This expansion, performed at the centre of the section, consists in an infinite series in the powers of the minor-to-major radius ratio $e$ of the shell. It is appropriate for a solid, homogeneous torus, as well as for inhomogeneous bodies (the case of a core stratification is considered). We show that the leading term is identical to the potential of a loop having the same main radius and the same mass | this "similarity" is shown to hold in the ${\cal O}(e^2)$ order. The series converges very well, especially close to the surface of the toroid where the average relative precision is $\sim 10^{-3}$ for $e\! = \!0.1$ at order zero, and as low as a few $10^{-6}$ at second order. The Laplace equation is satisfied {\em exactly} in every order, so no extra density is induced by truncation. The gravitational acceleration, important in dynamical studies, is reproduced with the same accuracy. The technique also applies to the magnetic potential and field generated by azimuthal currents as met in terrestrial and astrophysical plasmas.