Generalized r-Modes of the Maclaurin Spheroids

TL;DR

This paper derives analytical solutions for a broader class of r-modes in Maclaurin spheroids, revealing many previously unstudied modes and their potential gravitational radiation-driven instabilities.

Contribution

It introduces a new set of generalized r-modes with real eigenfunctions, expanding the understanding of oscillations in rotating uniform density stars.

Findings

About 30% of examined modes are gravitational radiation unstable.

Classical r-modes have simple analytical eigenfunctions.

The study provides solutions to a hyperbolic eigenvalue problem.

Abstract

Analytical solutions are presented for a class of generalized r-modes of rigidly rotating uniform density stars---the Maclaurin spheroids---with arbitrary values of the angular velocity. Our analysis is based on the work of Bryan; however, we derive the solutions using slightly different coordinates that give purely real representations of the r-modes. The class of generalized r-modes is much larger than the previously studied `classical' r-modes. In particular, for each l and m we find l-m (or l-1 for the m=0 case) distinct r-modes. Many of these previously unstudied r-modes (about 30% of those examined) are subject to a secular instability driven by gravitational radiation. The eigenfunctions of the `classical' r-modes, the l=m+1 case here, are found to have particularly simple analytical representations. These r-modes provide an interesting mathematical example of solutions to a hyperbolic eigenvalue problem.