Bar Mode Instability in Relativistic Rotating Stars
Rapidly spinning neutron stars, with angular velocity higher than a given threshold, may become secularly unstable to an l = m = 2 mode and deform into a bar configuration. This instability sets in only in the presence of a suitable dissipative mechanism, such as viscosity or gravitational wave emission, and in Newtonian theory its point of onset does not depend on the nature of dissipation. The first numerical investigations, however, seem to suggest that, in general relativity, the instability is weakened when driven by viscous dissipation and strengthened when driven by gravitational radiation.
After a review of the physical problem and of previous investigations, we schematically present an analytic treatment carried out in the framework of post-Newtonian (PN) gravitation. In this analysis, we model rotating stars by homogeneous, rigidly rotating, triaxial ellipsoids employing an energy variational principle to construct relativistic equilibrium sequences and locate the point of onset of the bar mode instability along each sequence. The spacetime metric is obtained by solving Einstein’s equations of general relativity in 3+1 ADM form, and we focus on the viscosity-driven instability.
We find that the value of the eccentricity, as well as related ratios like Ω2/(πϱ 0) and T/|W| (= rotational kinetic energy / gravitational potential energy), all increase at the onset of instability as the star becomes more relativistic. Since higher degrees of rotation are required to trigger a viscosity-driven bar mode instability as the star becomes more compact, the effect of general relativity is to weaken the instability, even to PN order.
KeywordsGravitational Radiation Gravitational Potential Energy Newtonian Limit Newtonian Theory Triaxial Ellipsoid
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