Analytic description of the r-mode instability in uniform density stars

TL;DR

This paper provides an analytic, second-order accurate description of the r-mode instability in uniform density neutron stars, including growth and damping timescales, with implications for gravitational wave emission and stellar evolution.

Contribution

It introduces a second-order analytic model for the r-mode instability in neutron stars, accounting for gravitational radiation and viscosity effects with minimal dependence on the equation of state.

Findings

Current-multipole radiation dominates the instability timescale.

Second-order results differ slightly from lowest order approximations.

Analytic formulas facilitate quick assessment of effects like viscosity and cooling.

Abstract

We present an analytic description of the $r$-mode instability in newly-born neutron stars, using the approximation of uniform density. Our computation is consistently accurate to second order in the angular velocity of the star. We obtain formulae for the growth-time of the instability due to gravitational-wave emission, for both current and mass multipole radiation and for the damping timescale, due to viscosity. The $l=m=2$ current-multipole radiation dominates the timescale of the instability. We estimate the deviation of the second order accurate results from the lowest order approximation and show that the uncertainty in the equation of state has only a small effect on the onset of the $r$-mode instability. The viscosity coefficients and the cooling process in newly-born neutron stars are, at present, uncertain and our analytic formaulae enables a quick check of such effects on the development of the instability.