On the r-mode spectrum of relativistic stars in the low-frequency approximation

TL;DR

This paper investigates the spectrum of axial r-modes in relativistic neutron stars using the low-frequency approximation, revealing conditions under which physical or unphysical solutions exist and their excitation properties.

Contribution

It demonstrates the existence of discrete r-mode solutions in relativistic stars within the low-frequency approximation and clarifies when these solutions are physical or unphysical.

Findings

Physical r-modes can be excited by initial data.

Unphysical solutions have divergent velocity perturbations.

Higher multipole modes may not exist in some models.

Abstract

The axial modes for non-barotropic relativistic rotating neutron stars with uniform angular velocity are studied, using the slow-rotation formalism together with the low-frequency approximation, first investigated by Kojima. The time independent form of the equations leads to a singular eigenvalue problem, which admits a continuous spectrum. We show that for $l=2$, it is nevertheless also possible to find discrete mode solutions (the $r$-modes). However, under certain conditions related to the equation of state and the compactness of the stellar model, the eigenfrequency lies inside the continuous band and the associated velocity perturbation is divergent; hence these solutions have to be discarded as being unphysical. We corroborate our results by explicitly integrating the time dependent equations. For stellar models admitting a physical $r$-mode solution, it can indeed be excited by arbitrary initial data. For models admitting only an unphysical mode solution, the evolutions do not show any tendency to oscillate with the respective frequency. For higher values of $l$, it seems that in certain cases there are no mode solutions at all.