Dynamical bar-mode instability of differentially rotating stars: Effects of equations of state and velocity profiles

As an extension of our previous work, we investigate the dynamical instability against nonaxisymmetric bar-mode deformations of differentially rotating stars in Newtonian gravity varying the equations of state and velocity profiles. We performed the numerical simulation and the followup linear stability analysis adopting polytropic equations of state with the polytropic indices n=1, 3/2, and 5/2 and with two types of angular velocity profiles (the so-called j-constant-like and Kepler-like laws). It is confirmed that rotating stars of a high degree of differential rotation are dynamically unstable against the bar-mode deformation, even for the ratio of the kinetic energy to the gravitational potential energy $\beta$ of order 0.01. The criterion for onset of the bar-mode dynamical instability depends weakly on the polytropic index n and the angular velocity profile as long as the degree of differential rotation is high. Gravitational waves from the final nonaxisymmetric quasi-stationary states are calculated in the quadrupole formula. For proto-neutron stars of mass $1.4M_{\odot}$, radius $\sim 30$ km and $\beta \alt 0.1$, such gravitational waves have the frequency of $\sim$ 600--1,400 Hz, and the effective amplitude is larger than $10^{-22}$ at a distance of about 100 Mpc irrespective of n and the angular velocity profile.