Nonradial oscillations of neutron stars and emitted gravitational waves: Computing strongly damped normal modes

In this paper, we compute eigenfrequencies of strongly damped normal modes arising from the coupling of the nonradial oscillations of a neutron star to the oscillations of the space-time metric, so-called “w-modes”, by integrating all involved differential equations in the complex plane. Regarding the interior of the star, we use the so-called “complex-plane strategy”. Specifically, we integrate the differential equations of the nonradial fluid oscillations of a general-relativistic polytropic model, simulating the star, along a straight-line contour placed parallel to the real axis and at small imaginary distance from it, thus avoiding a singularity at the stellar center. Regarding the exterior of the star, we use a method proposed by Andersson, Kokkotas and Schutz, following a slightly different terminating procedure. Specifically, (i) we integrate the equations along a straight-line contour lying parallel to the so-called “anti-Stokes lines”, on which the exponential divergence of the solution is drastically suppressed, so that the outgoing and ingoing waves become comparable; and (ii) we carry out one final integration up to a “common reference point”, thus comparing all results at this point. We verify the reliability and accuracy of the method by comparing our numerical results to corresponding ones appearing in the bibliography.