Pulsations of relativistic stars

The rotational evolution of neutron stars (NS) can be affected by several instabilities. If hot protoneutron stars are rapidly rotating, they can undergo a dynamical bar-mode instability. When the NS has cooled to about 1e10K after its formation, it can be subject to the Chandrasekhar-Friedman-Schutz instability and it becomes an important source of GW. It was recently found that the l=m r-mode has the shortest growth time of the instability and it can transform a rapidly rotating newly-born NS to a Crab-like slowly-rotating pulsar within about a year after its formation. The numerical study of the r-mode instability is beyond present perturbative codes which compute the normal modes of the linearized pulsation equations. Instead, it requires non-linear effects, both from the gravitational field and the hydrodynamics, to be taken into account. Pulsations of rotating NS can be excited in several scenarios, such as core-collapse, crust and core-quakes and binary mergers and could become detectable either in GW or high-energy radiation.

In collaboration with Dr. Stergioulas (Thessaloniki) and others I have started a program for the study of non-linear evolutions of rotating relativistic stars. Contrary to previous perturbative approaches we use an axisymmetric code to integrate (using HRSC schemes) the non-linear relativistic hydrodynamic equations in the star's fixed background geometry. As initial data we use axisymmetric configurations, which are perturbed from their stationary equilibrium using quasi-normal mode eigenfunctions. The evolution can be followed over many hydrodynamical timescales which allows the study of the non-linear dynamics of radial and non-radial oscillations. This investigation is ultimately aimed at the study of the growth of the gravitational radiation driven non-axisymmetric instability in the fundamental (f) and rotational (r) modes of oscillation.

As shown in Font, Stergioulas and Kokkotas (2002) HRSC schemes -- like the Piecewise Parabolic Method -- are suitable for the evolution of perturbed NS and for the accurate identification (via Fourier transforms) of normal modes of oscillation. This was demonstrated for radial and quadrupolar pulsations in the non-rotating limit, where we found good agreement with frequencies obtained with a linear perturbation code. More recently in Font et al (2001) we have performed a comprehensive study of all low-order axisymmetric modes of uniformly and rapidly rotating relativistic stars. The frequencies of the axisymmetric modes are affected significantly by rotation only when the rotation rate exceeds about 50% of the maximum allowed. As expected, at large rotation rates, apparent mode crossings between different modes appear.

With the recent discovery of the r-mode instability, a new GW driven instability in rotating NS, there is also renewed momentum into studies of such effects. Our understanding of such instabilities has improved considerably in the last year or so. The r-modes are a set of axial fluid oscillations (in the limit of spherical stars) whose dynamics is governed by rotation. They resemble the Rossby waves existing in the Earth's oceans. The restoring force is the Coriolis inertial force, normal to the velocity. Therefore, the fluid flow resembles oscillatory circulation patterns. In Stergioulas and Font (2001) we presented the first study of nonlinear r-modes in isentropic, rapidly rotating relativistic stars, via 3D general relativistic hydrodynamical evolutions. For our study we used the CACTUS code. We found that (1) on dynamical timescales, there is no strong coupling of r-modes to other modes at amplitudes of order one -- unless nonlinear saturation occurs on longer timescales, the maximum r-mode amplitude is of order one. An absolute upper limit on the amplitude (relevant for the most rapidly rotating stars) is set by causality. (2) The spectrum of r-modes and inertial modes in isentropic stars is discrete. (3) In addition, a kinematical drift associated with r-modes appears to be present in the evolutions, but an unambiguous confirmation requires more precise initial data. A parameter space survey extending our initial investigation is currently underway.

For further information about these topics is worth a visit to Nick Stergioulas web site.

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