Differential rotation of the unstable nonlinear r-modes

TL;DR

This paper demonstrates that gravitational radiation-reaction uniquely determines the differential rotation profile in unstable nonlinear r-modes of rotating stars, removing gauge ambiguities present in non-radiative analyses.

Contribution

It shows that radiation-reaction forces eliminate gauge freedom, leading to a unique differential rotation law in unstable r-modes, with explicit formulas for Newtonian polytropic stars.

Findings

Radiation-reaction enforces a unique differential rotation profile.

Explicit expressions derived for slowly rotating polytropic models.

Gauge dependence in differential rotation is removed by gravitational radiation-reaction.

Abstract

At second order in perturbation theory, the $r$-modes of uniformly rotating stars include an axisymmetric part that can be identified with differential rotation of the background star. If one does not include radiation-reaction, the differential rotation is constant in time and has been computed by S\'a. It has a gauge dependence associated with the family of time-independent perturbations that add differential rotation to the unperturbed equilibrium star: For stars with a barotropic equation of state, one can add to the time-independent second-order solution arbitrary differential rotation that is stratified on cylinders (that is a function of distance $\varpi$ to the axis of rotation). We show here that the gravitational radiation-reaction force that drives the $r$-mode instability removes this gauge freedom: The expontially growing differential rotation of the unstable second-order $r$-mode is unique. We derive a general expression for this rotation law for Newtonian models and evaluate it explicitly for slowly rotating models with polytropic equations of state.