Uniformly Rotating Polytropic Rings in Newtonian Gravity

TL;DR

This paper introduces an iterative method to analyze uniformly rotating, self-gravitating rings in Newtonian gravity, deriving formulas and coefficients for various polytropic equations of state, and validating results against numerical solutions.

Contribution

It presents a novel iterative approach for modeling rotating rings without a central body, including analytic and numerical coefficients for different polytropic indices.

Findings

Derived a simple mass-pressure relation for general equations of state.

Calculated analytic coefficients for n=1 polytropes up to third order.

Validated the iterative solutions against high-precision numerical methods.

Abstract

An iterative method is presented for solving the problem of a uniformly rotating, self-gravitating ring without a central body in Newtonian gravity by expanding about the thin ring limit. Using this method, a simple formula relating mass to the integrated pressure is derived to the leading order for a general equation of state. For polytropes with the index n=1, analytic coefficients of the iterative approach are determined up to the third order. Analogous coefficients are computed numerically for other polytropes. Our solutions are compared with those generated by highly accurate numerical methods to test their accuracy.