Rotational modes of non-isentropic stars and the gravitational radiation driven instability

We investigate the properties of r-mode and inertial mode of slowly rotating, non-isentropic, Newtonian stars, by taking account of the effects of the Coriolis force and the centrifugal force. The Coriolis force is the dominant restoring force for both r-mode and inertial mode, which are also called rotational mode in this paper. For the velocity field produced by the oscillation modes the r-mode has the dominant toroidal component over the spheroidal component, while the inertial mode has the comparable toroidal and spheroidal components. In non-isentropic stars the specific entropy of the fluid depends on the radial distance from the center, and the interior structure is in general divided into two kinds of layers of fluid stratification that is stable or unstable against convection. Because of the non-isentropic structure, low frequency oscillations of the star are affected by the buoyant force, which has no effects on oscillations of isentropic stars. In this paper we employ simple polytropic models with the polytropic index n = 1 as the background neutron star models for the modal analysis. For the non-isentropic models we consider only two cases, that is, the models with the stable fluid stratification in the whole interior and the models that are fully convective. For simplicity we call these two kinds of models “radiative” and “convective” models in this paper. For both cases, we assume the deviation of the models from isentropic structure is small. Examining the dissipation timescales due to the gravitational radiation and several viscous processes for the polytropic neutron star models, we find that the gravitational radiation driven instability of the nodeless r-modes associated with l = |m| remains strong even in the non-isentropic models, where l and m are the indices of the spherical harmonic function representing the angular dependence of the eigenfunction. Calculating the rotational modes of the radiative models as functions of the angular rotation frequency Ω, we find that the inertial modes are strongly modified by the buoyant force at small Ω, where the buoyant force as a dominant restoring force becomes comparable with or stronger than the Coriolis force. Because of this property we obtain the mode sequences in which the inertial modes at large Ω are identified as g-modes or the r-modes with l = |m| at small Ω. We also note that as Ω increases from Ω = 0 the retrograde g-modes become retrograde inertial modes, which are unstable against the gravitational radiation reaction.