Dynamical bar instability in rotating stars: Effect of general relativity

We study the dynamical stability against bar-mode deformation of rapidly an d differentially rotating stars in the first post-Newtonian approximation o f general relativity. We vary the compaction of the star M/R (where M is th e gravitational mass and R the equatorial circumferential radius) between 0 .01 and 0.05 to isolate the influence of relativistic gravitation on the in stability. For compactions in this moderate range, the critical value of be ta = T/W for the onset of the dynamical instability (where T is the rotatio nal kinetic energy and W the gravitational binding energy) slightly decreas es from similar to0.26 to similar to0.25 with increasing compaction for our choice of the differential rotational law. Combined with our earlier findi ngs based on simulations in full general relativity for stars with higher c ompaction, we conclude that relativistic gravitation enhances the dynamical bar-mode instability, i.e., the onset of instability occurs for smaller va lues of beta in relativistic gravity than in Newtonian gravity. We also fin d that once a triaxial structure forms after the bar-mode perturbation satu rates in dynamically unstable stars, the triaxial shape is maintained, at l east for several rotational periods. To check the reliability of our numeri cal integrations, we verify that the general relativistic Kelvin-Helmholtz circulation is well conserved, in addition to rest-mass energy, total mass energy, and linear and angular momentum. Conservation of circulation indica tes that our code is not seriously affected by numerical viscosity. We dete rmine the amplitude and frequency of the quasi-periodic gravitational waves emitted during the bar formation process using the quadrupole formula.