Neutron star oscillations in pseudo-Newtonian gravity

We investigate the oscillations of neutron stars using a purely Newtonian approach and three other pseudo-Newtonian formulations. Our work is motivated by the fact that pseudo-Newtonian formulations are commonly used in core-collapse supernova (CCSN) simulations. We derive and solve numerically the radial and nonradial perturbation equations for neutron star oscillations using different combinations of modified Newtonian hydrodynamics equations and gravitational potentials. We pay special attention to the formulation proposed recently by Zha et al. [Phys. Rev. Lett. 125, 051102 (2020)] that implements the standard Case A effective potential in CCSN simulations with an additional lapse-function correction to the hydrodynamics equations. We find that this "Case A+lapse" formulation can typically approximate the frequency of the fundamental radial mode of a $1.4 M_\odot$ neutron star computed in general relativity to about a few tens of percent for our chosen EOS models. For the nonradial quadrupolar $f$ mode, which is expected to contribute strongly to the gravitational waves emitted from a protoneutron star, the Case A+lapse formulation performs much better and can approximate the $f$ mode frequency to within about a few percent even for the maximum-mass configuration in general relativity.Comment: 13 pages, 6 figures, accepted for publication in MNRA