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

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 sta r. We obtain formulae for the growth-time of the instability due to gravita tional-wave emission, for both current and mass multipole radiation and for the damping timescale, due to viscosity. The l = m = 2 current-multipole r adiation dominates the timescale of the instability. We estimate the deviat ion of the second order accurate results from the fewest order approximatio n and show that the uncertainty in the equation of state has only a small e ffect on the onset of the r-mode instability. The viscosity coefficients an d the cooling process in newly-born neutron stars are, at present, uncertai n and our analytic formulae enable a quick check of such effects on the dev elopment of the instability.