A Generalized Family of Post-Newtonian Dedekind Ellipsoids

TL;DR

This paper derives a family of first-order Post-Newtonian Dedekind ellipsoids, extending previous work to include axially symmetric limits and removing singularities, thereby broadening understanding of rotating equilibrium figures.

Contribution

It introduces a generalized family of PN Dedekind ellipsoids that include axially symmetric limits and resolve previous singularities in the sequence.

Findings

A family of PN Dedekind ellipsoids is derived.

The sequence includes the Newtonian limit and a PN Maclaurin spheroid limit.

Singularities in earlier models are shown to be removable with parameter choices.

Abstract

We derive a family of Post-Newtonian (PN) Dedekind ellipsoids to first order. They describe non-axially symmetric, homogeneous, and rotating figures of equilibrium. The sequence of the Newtonian Dedekind ellipsoids allows for an axially symmetric limit in which a uniformly rotating Maclaurin spheroid is recovered. However, the approach taken by Chandrasekhar & Elbert (1974) to find the PN Dedekind ellipsoids excludes such a limit. In G\"urlebeck & Petroff (2010), we considered an extension to their work that permits a limit of 1 PN Maclaurin ellipsoids. Here we further detail the sequence and demonstrate that a choice of parameters exists with which the singularity formerly found in Chandrasekhar & Elbert (1974) along the sequence of PN Dedekind ellipsoids is removed.