Post-Newtonian treatment of bar mode instability in rigidly rotating equilibrium configurations for polytropic stars

Abstract

We determine the onset point of secular instability for the nonaxisymmetric bar mode in rigidly rotating equilibrium configurations in the Post-Newtonian approximation, in order to apply it to neutron stars. The treatment is based on a precedent Newtonian analytic energy variational method which we have extended to the Post-Newtonian case. This method, based upon Landau's theory of second-order phase transitions, provides the critical value of the ellipsoid polar eccentricity e at the onset of viscosity-driven instability and it is valid for any equation of state. The extension of this method to Post-Newtonian fluid configurations has been accomplished by combining two earlier orthogonal works, specialized respectively to slow rotating configurations but with arbitrary density profile and to constant mass density but arbitrarily fast rotating ellipsoids. We also determine the explicit expressions for the density functionals which allow the generalization of the physical quantities involved in our treatment from the constant mass density to an arbitrary density profile form. We find that, considering homogeneous ellipsoids, the value of the critical eccentricity increases as the stars become more relativistic, in qualitatively agreement with previous investigations but with a less relevant amount of such an increase. Studying then the dependence of this critical value on the configuration equation of state, we find that for polytropic matter distributions the increase of the critical eccentricity with the star compactness is confirmed for softer equations of state (with respect to the incompressible case). The amount of this stabilizing effect is nearly independent of the polytropic index.