Differential rotation of the unstable nonlinear<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>r</mml:mi></mml:math>-modes

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 $\ensuremath{\varpi}$ 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 exponentially 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.

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