The outer gravitational potential of an inhomogeneous torus with an elliptical cross-section

TL;DR

This paper presents an approximation method for calculating the outer gravitational potential of inhomogeneous elliptical tori in astrophysics, simplifying complex structures with high accuracy.

Contribution

The authors introduce a novel approximation of the outer potential of inhomogeneous elliptical tori using two thin rings, applicable to various geometries and density distributions.

Findings

Approximation with two rings has less than 1% error for homogeneous tori.

Outer potential weakly depends on density distribution law.

Model accurately predicts external force components.

Abstract

Toroidal structures are a common feature in a wide variety of astrophysical objects, including dusty tori in AGNs, rings in galaxies, protoplanetary disks, and others. The matter distribution in such structures is not homogeneous and can be flattened by self-gravity or become elongated in the vertical direction, as is the case with obscuring tori in AGNs. This led us to consider the more general case of the gravitational potential of an inhomogeneous torus with an elliptical cross-section. We begin by showing that the outer potential of a homogeneous elliptical torus can be effectively approximated with less than 1\% error by the potentials of two infinitely thin rings with a minor correction term. These two rings have masses each equal to half the total mass of the torus. The most notable feature is that each such infinitely thin ring is positioned at precisely the halfway point between the center and the focus of the elliptical cross-section, regardless of the torus' other parameters. The result, which holds for both oblate and prolate geometries, allows us to find a new expression to handle the outer potential of an inhomogeneous torus with an elliptical cross-section. The confocal density distribution is a special case. We have found that the outer potential of such a torus is only weakly dependent of the density distribution law. Consequently, even for the confocal inhomogeneous case, the outer potential is well represented by two infinitely thin rings. This approach can simplify problems of dynamics for the systems consisting of toroidal structures. We have also derived the expressions for the external force components exerted by a homogeneous torus with an elliptical cross-section, both for the exact form of the potential and for our approximation by two infinitely thin rings. Comparison of the two shows that our model fits the true trend of the force well.