Non-linear pulsations in differentially rotating neutron stars: mass-shedding-induced damping and splitting of the fundamental mode

We study small-amplitude, non-linear pulsations of uniformly and differentially rotating neutron stars employing a two-dimensional evolution code for general-relativistic hydrodynamics. Using Fourier transforms at several points inside the star, both the eigenfrequencies and two-dimensional eigenfunctions of pulsations are extracted. The centrifugal forces and the degree of differential rotation have significant effects on the mode eigenfunction. We find that near the mass-shedding limit, the pulsations are damped due to shocks forming at the surface of the star. This new damping mechanism may set a small saturation amplitude for modes that are unstable to the emission of gravitational waves. After correcting for the assumption of the Cowling approximation (used in our numerical code), we construct empirical relations that predict the range of gravitational-wave frequencies from quasi-periodic post-bounce oscillations in the core collapse of massive stars. We also find that the fundamental quasi-radial mode is split, at least in the Cowling approximation and mainly in differentially rotating stars, into two different sequences.