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

In this paper 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 that we have extended to the post-Newtonian case. This method, based on Landau's theory of second-order phase transitions, provides the critical value of the ellipsoid polar eccentricity e at the onset of instability, i.e., at the bifurcation point from the axisymmetric Maclaurin to the triaxial Jacobi ellipsoids, 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 that 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 qualitative agreement with previous investigations but with a less marked amount of such an increase. Then we have studied the dependence of this critical value on the configuration equation of state. Considering polytropic matter distributions, we find that the increase in the eccentricity at the onset of instability 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.