Instability of rotating stars to axial perturbations

Sharon Morsink, University of Wisconsin, Milwaukee

A great wealth of information about the internal structure of neutron stars can be obtained through the study of neutron star oscillations. Just as helioseismology has recently revealed important details of the sun's structure, it may be possible in the fu ture to detect gravitational waves caused by the oscillations of a relativistic star and obtain the star's mass and radius [1]. The possibility of making such exciting measurements underlines the importance of understanding the theoretical details o f the pulsations of neutron stars. Earlier this year Nils Andersson [2] made a surprising discovery while numerically investigating the subset of non-axisymmetric perfect fluid oscillations known as axial perturbations: all axial perturbation s with azimuthal angular dependence are unstable when the star rotates, for any value of the star's angular velocity. As a result, all rotating stars are unstable to small perturbations!

It has been known for some time that rotating stars are unstable to gravitational radiation reaction [3,4,5] via the Chandrasekhar-Friedman-Schutz (CFS) instability. It turns out that Andersson's result can be explained by the CFS mechanism, but the way that the instability sets in is different from the usual result which holds for polar perturbations. (Recall that the non-radial fluid velocity field created by a polar perturbation of a spherical star can be expressed as a gradient of a spherical harmonic, while for an axial perturbation it is a cross-product of a radial vector and a polar flow.) For a polar perturbation with fixed value of m, the perturbation is stable for small stellar angular velocity, , until a critical velocity, is reached. When , the perturbation's frequency vanishes as seen by inertial observers. For all angular velocities , the mode is unstable. Andersson's result is that axial modes are unstable for all .

The difference in critical velocities for the two types of perturbations is really not too surprising. For static stars, axial fluid perturbations are trivial [6] and their frequencies of oscillation must vanish [7]. This implies that the critical angular velocity is zero and that axial modes are unstable for any non-zero angular velocity. Indeed, Papaloizou and Pringle [8] have studied these modes for Newtonian stars (which they call r-modes) and the form of the frequency which t hey calculate conforms to the CFS instability criterion. However, the implied instability of the Newtonian r-modes went unnoticed until Andersson pointed it out [2]. A formal proof of the instability for the general relativistic analogue of t he r-modes in the slow rotation limit is presented in a paper by John Friedman and me [7].

Of what astrophysical significance is this new instability? If the instability's growth time is shorter than the time scale for viscosity to damp it out, the axial mode could be an important source of gravitational radiation. For a l=m=2 axial mode the instability's growth rate scales as (in geometrical units, where ) while the damping rate due to shear viscosity is independent of . An order of magnitude calculation (which agrees with preliminary numerical results [9]) shows that for a neutron star with a temperature of , the two time scales are equal when the rotational period is of the order of a millisecond. (Assuming a coefficient of shear viscosity which takes account of superfluid effects [10].) As the star rapidly cools, shear viscosity will increase and quickly damp out the instability. This leaves open some interesting questions for future research. When viscosity is included in a full relativistic computation, do axial or polar perturbations place the lower limit on the angular velocity of neutron stars born with high angular momentum? As the newborn star cools and spins down, is it possible for the star to be in a marginally unstable configuration for a long enough time so that an appreciable amount of gravitational radiation is emitted? We look forward to the resolution of these problems.


[1] N. Andersson and K.D. Kokkotas, gr-qc/9610035

[2] N. Andersson, gr-qc/9706075. [3] S. Chandrasekhar, Phys. Rev. Lett., 24, 611 (1970). [4] J.L. Friedman and B.F. Schutz, ApJ, bf 222, 281 (1978). [5] J.L. Friedman, Commun. Math. Phys., 62 247 (1978). [6] K.S. Thorne and A. Campolattaro, ApJ, 149 591 (1967). [7] J.L. Friedman and S.M. Morsink, gr-qc/9706073. [8] J. Papaloizou and J.E. Pringle, MNRAS, 182 423 (1978).

[9] N. Andersson, personal communication. [10] C. Cutler and L. Lindblom, ApJ, 314 234 (1987).

Jorge Pullin
Wed Sep 10 15:05:58 EDT 1997