The bar-mode instability in differentially rotating neutron stars: Simulations in full general relativity

Abstract

We study the dynamical stability against bar-mode deformation of rapidly spinning neutron stars with differential rotation. We perform fully relativistic 3D simulations of compact stars with $M/R\ge 0.1$, where $M$ is the total gravitational mass and $R$ the equatorial circumferential radius. We adopt an adiabatic equation of state with adiabatic index $\Gamma =2$. As in Newtonian theory, we find that stars above a critical value of $\beta \equiv T/W$ (where $T$ is the rotational kinetic energy and $W$ the gravitational binding energy) are dynamically unstable to bar formation. For our adopted choices of stellar compaction and rotation profile, the critical value of $\beta ={\beta }_{dGR}$ is $\sim 0.24-0.25$, only slightly smaller than the well-known Newtonian value $\sim 0.27$ for incompressible Maclaurin spheroids. The critical value depends only very weakly on the degree of differential rotation for the moderate range we surveyed. All unstable stars form bars on a dynamical timescale. Models with sufficiently large $\beta$ subsequently form spiral arms and eject mass, driving the remnant to a dynamically stable state. Models with moderately large $\beta \gtrsim {\beta }_{dGR}$ do not develop spiral arms or eject mass but adjust to form dynamically stable ellipsoidal-like configurations. If the bar-mode instability is triggered in supernovae collapse or binary neutron star mergers, it could be a strong and observable source of gravitational waves. We determine characteristic wave amplitudes and frequencies.

17 pages, accepted for publication in APJ

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