Dynamical Bar Instability in Rotating Stars: Effect of General Relativity

We study the dynamical stability against bar-mode deformation of rapidly anddifferentially rotating stars in the first post-Newtonian approximation ofgeneral relativity. We vary the compaction of the star $M/R$ (where $M$ is thegravitational mass and $R$ the equatorial circumferential radius) between 0.01and 0.05 to isolate the influence of relativistic gravitation on theinstability. For compactions in this moderate range, the critical value of$\beta \equiv T/W$ for the onset of the dynamical instability (where $T$ is therotational kinetic energy and $W$ the gravitational binding energy) slightlydecreases from $\sim 0.26$ to $\sim 0.25$ with increasing compaction for ourchoice of the differential rotational law. Combined with our earlier findingsbased on simulations in full general relativity for stars with highercompaction, we conclude that relativistic gravitation {\em enhances} thedynamical bar-mode instability, i.e. the onset of instability sets in forsmaller values of $\beta$ in relativistic gravity than in Newtonian gravity. Wealso find that once a triaxial structure forms after the bar-mode perturbationsaturates in dynamically unstable stars, the triaxial shape is maintained, atleast for several rotational periods. To check the reliability of our numericalintegrations, we verify that the general relativistic Kelvin-Helmholtzcirculation is well-conserved, in addition to rest-mass energy, totalmass-energy, linear and angular momentum. Conservation of circulation indicatesthat our code is not seriously affected by numerical viscosity. We determinethe amplitude and frequency of the quasi-periodic gravitational waves emittedduring the bar formation process using the quadrupole formula.Comment: 10 pages with 10 gif figures, emulateapj5.sty. Accepted for publication in The Astrophysical Journa