Axisymmetric modes of rotating relativistic stars in the Cowling approximation

Axisymmetric pulsations of rotating neutron stars can be excited in several scenarios, such as core collapse, crust‐ and core‐quakes or binary mergers, and could become detectable in either gravitational waves or high‐energy radiation. Here, we present a comprehensive study of all low‐order axisymmetric modes of uniformly and rapidly rotating relativistic stars. Initial stationary configurations are appropriately perturbed and are numerically evolved using an axisymmetric, non‐linear relativistic hydrodynamics code, assuming time‐independence of the gravitational field (Cowling approximation). The simulations are performed using a high‐resolution shock‐capturing finite‐difference scheme accurate enough to maintain the initial rotation law for a large number of rotational periods, even for stars at the mass‐shedding limit. Through Fourier transforms of the time evolution of selected fluid variables, we compute the frequencies of quasi‐radial and non‐radial modes with spherical harmonic indices l=0 , 1, 2 and 3, for a sequence of rotating stars from the non‐rotating limit to the mass‐shedding limit. The frequencies of the axisymmetric modes are affected significantly by rotation only when the rotation rate exceeds about 50 per cent of the maximum allowed. As expected, at large rotation rates, apparent mode crossings between different modes appear. In addition to the above modes, several axisymmetric inertial modes are also excited in our numerical evolutions.