Phenomenological Relations for Axial Quasi-normal Modes of Neutron Stars with Realistic Equations of State

Here we investigate the axial w quasi-normal modes of neutron stars for several equations of state. In particular, we study the influence of the presence of hyperons in the core of the neutron stars. We have obtained that w-modes can be used to distinguish between neutron stars with hyperons and without hyperons for compact enough stars. We present phenomenological relations for the frequency and damping times with the compactness of the neutron star for wI and wII modes showing the differences of the stars with hyperons in the core. Also, we obtain a new phenomenological relation between the real part and the imaginary part of the frequency of the w quasi-normal modes, which can be used to estimate the central pressure of the neutron stars. We are able to construct explicitly the low compactness limit configuration for which the fundamental wII mode vanishes. Finally, we have studied the influence of changes in the core-crust transition pressure obtaining that it is very small. To obtain these results we have developed a new method based on the Exterior Complex Scaling technique with variable angle, appropriate for the treatment of the exterior part of the quasi-normal modes, which allow us to impose constringent enough conditions to generate pure outgoing quasi-normal modes. For the interior part we use a piece-wise polytrope approximation for several equations of state. A complete study of the junction conditions have been done.