R-mode instability of slowly rotating nonisentropic relativistic stars

We investigate the properties of r-mode instability in slowly rotating relativistic polytropes. Inside the star slow rotation and the low frequency formalism that was mainly developed by Kojima are employed to study axial oscillations restored by the Coriolis force. At the stellar surface, in order to take into account the gravitational radiation reaction effect, we use a near-zone boundary condition instead of the boundary condition usually imposed for asymptotically flat spacetime. Because of the boundary condition, complex frequencies whose imaginary part represents a secular instability are obtained for discrete r-mode oscillations in some polytropic models. It is found that such discrete r-mode solutions can be obtained only for some restricted polytropic models. The basic properties of the solutions are similar to those obtained by imposing the boundary condition for asymptotically flat spacetime. Our results suggest that the existence of a continuous part of the spectrum cannot be avoided even when its frequency becomes complex due to the emission of gravitational radiation.