Uniformly Rotating Homogeneous Rings in Newtonian Gravity

TL;DR

This paper develops an analytical series expansion method to study uniformly rotating homogeneous rings in Newtonian gravity, providing high-order approximations and validating them against numerical results.

Contribution

It introduces a systematic analytical approach using series expansions around the thin ring limit, extending to the 20th order, for rotating rings without a central body.

Findings

Series expansions accurately approximate ring properties

Analytical results agree with numerical simulations

Method extends to high-order precision

Abstract

In this paper, we describe an analytical method for treating uniformly rotating homogeneous rings without a central body in Newtonian gravity. We employ series expansions about the thin ring limit and use the fact that in this limit the cross-section of the ring tends to a circle. The coefficients can in principle be determined up to an arbitrary order. Results are presented here to the 20th order and compared with numerical results.