Uniformly Rotating Homogeneous and Polytropic Rings in Newtonian Gravity

TL;DR

This paper introduces an analytical expansion method for modeling uniformly rotating, self-gravitating rings in Newtonian gravity, providing high-order solutions for homogeneous rings and polytropes, and validating results against numerical data.

Contribution

It develops an iterative analytical scheme for rotating rings, extending solutions to high order for homogeneous and certain polytropic rings, and offers a simple formula relating mass and pressure.

Findings

High-order analytical solutions for homogeneous rings

First-order solutions for polytropes with n=1

Validation of analytical results against numerical methods

Abstract

An analytical method is presented for treating the problem of a uniformly rotating, self-gravitating ring without a central body in Newtonian gravity. The method is based on an expansion about the thin ring limit, where the cross-section of the ring tends to a circle. The iterative scheme developed here is applied to homogeneous rings up to the 20th order and to polytropes with the index n=1 up to the third order. For other polytropic indices no analytic solutions are obtainable, but one can apply the method numerically. However, it is possible to derive a simple formula relating mass to the integrated pressure to leading order without specifying the equation of state. Our results are compared with those generated by highly accurate numerical methods to test their accuracy.