3.5 Nonaxisymmetric Instabilities3 Oscillations and Stability3.3 Axisymmetric Perturbation

3.4 Nonaxisymmetric Perturbations

3.4.1 Nonrotating Limit

For a spherical star, it suffices to study the m =0 axisymmetric modes of pulsation, since the tex2html_wrap_inline2283 modes can be obtained by a rotation of the coordinate system.

Thorne, Campolattaro and Price, in a series of papers [98, 99, 100], initiated the computation of nonradial modes by formulating the problem in the Regge-Wheeler (RW) gauge [101] and numerically computing nonradial modes for a number of neutron star models. A variational method for obtaining eigenfrequencies and eigenfunctions has been constructed by Detweiler and Ipser [102]. Lindblom and Detweiler [103] explicitly reduced the system of equations to four first-order ordinary differential equations and obtained more accurate eigenfrequencies and damping times for a larger set of neutron star models. They later realized that their system of equations is sometimes singular inside the star and obtained an improved set of equations which is free of this singularity [104].

Chandrasekhar and Ferrari [91Jump To The Next Citation Point In The Article] express the nonradial pulsations in terms of a fifth-order system in a diagonal gauge, which is independent of fluid variables. They thus reformulate the problem in a way analogous to the scattering of gravitational waves off a black hole. Ipser and Price [105] show that, in the RW gauge, nonradial pulsations can be described by a system of two second-order equations, which can also be independent of fluid variables. In addition, they find that the diagonal gauge of Chandrasekhar and Ferrari has a remaining gauge freedom which, when removed, also leads to a fourth-order system of equations [106].

In order to locate purely outgoing-wave modes, one has to be able to distinguish the outgoing-wave part from the ingoing-wave part at infinity. In the Thorne et al. and Lindblom and Detweiler schemes, this is achieved using analytic approximations of the solution at infinity.

W -modes pose a more challenging numerical problem because they are strongly damped, and the techniques used for f and p modes fail to distinguish the outgoing-wave part. However, Andersson, Kokkotas and Schutz [107] successfully combine a redefinition of variables with a complex-coordinate integration method, obtaining highly accurate complex frequencies for w modes. In this method, the ingoing and outgoing solutions are separated by numerically calculating their analytic continuations to a place in the complex-coordinate place, where they have comparable amplitudes. Since this approach is purely numerical, it could prove to be suitable for the computation of quasi-normal modes in rotating stars, where analytic solutions at infinity are not available.

The non-availability of asymptotic solutions at infinity in the case of rotating stars is one of the major difficulties for computing outgoing modes in rapidly rotating relativistic stars. A new development that may help to overcome this problem, at least to an acceptable approximation, is presented in [108] by Lindblom, Mendell and Ipser.

The authors obtain approximate near-zone boundary conditions for the outgoing modes that replace the outgoing-wave condition at infinity and that enable one to compute the eigenfrequencies with very satisfactory accuracy. First, the pulsation equations of polar modes in the Regge-Wheeler gauge are reformulated as a set of two second-order radial equations for two potentials - one corresponding to fluid perturbations and the other to the perturbations of the spacetime. The equation for the space-time perturbation reduces to a scalar wave equation at infinity and to Laplace's equation for zero-frequency solutions. From these, an approximate boundary condition for outgoing modes is constructed and imposed in the near zone of the star (in fact on its surface) instead of at infinity. For polytropic models, the near-zone boundary condition yields f -mode eigenfrequencies with real parts accurate to tex2html_wrap_inline2295 and imaginary parts with accuracy at the tex2html_wrap_inline2297 level, for the most relativistic stars. If the near zone boundary condition can be applied to the oscillations of rapidly rotating stars, the resulting frequencies and damping times should have comparable accuracy.

3.4.2 Slow Rotation Approximation

The slow rotation approximation has proven to be useful for obtaining a first estimate of the effect of rotation on the pulsations of relativistic stars. To lowest order in rotation, a polar l -mode of an initially nonrotating star couples to an axial tex2html_wrap_inline2275 mode in the presence of rotation. Conversely, an axial l -mode couples to a polar tex2html_wrap_inline2275 mode [91].

The equations of nonaxisymmetric perturbations in the slow-rotation limit and in the Regge-Wheeler gauge are derived by Kojima in [109, 110], where the complex frequencies tex2html_wrap_inline2307 for the l = m modes of various polytropes are computed. For counterrotating modes, both tex2html_wrap_inline2311 and tex2html_wrap_inline2313 decrease, tending to zero, as the rotation rate increases (when tex2html_wrap_inline2315 passes through zero, the star becomes unstable to the CFS-instability). Extrapolating tex2html_wrap_inline2311 and tex2html_wrap_inline2313 to higher rotation rates, Kojima finds a large discrepancy between the points where tex2html_wrap_inline2311 and tex2html_wrap_inline2313 go through zero. This shows that the slow rotation formalism cannot accurately determine the onset of the CFS-instability of polar modes in rapidly rotating neutron stars.

In [111], it is shown that, for slowly rotating stars, the coupling between polar and axial modes affects the frequency of pulsation only to second order in rotation, so that, in the slow rotation approximation, to tex2html_wrap_inline2325, the coupling can be neglected when computing frequencies.

The slow rotation approximation has also been used recently in the study of the r -mode instability [139Jump To The Next Citation Point In The Article, 112Jump To The Next Citation Point In The Article].

3.4.3 Post-Newtonian Approximation

A first step towards the solution of the perturbation equations in full relativity has been taken by Cutler and Lindblom [113, 114Jump To The Next Citation Point In The Article, 115Jump To The Next Citation Point In The Article]; they obtain frequencies for the l = m f -modes in rotating stars in the first post-Newtonian (1-PN) approximation. The perturbation equations are derived in the post-Newtonian formalism of Gunnarsen [116], i.e. the equations are separated into equations of consistent order in 1/ c .

Cutler and Lindblom show that, in this scheme, the perturbation of the 1-PN correction of the four-velocity of the fluid can be obtained analytically in terms of other variables, similarly to what is done for the perturbation in the four-velocity in the Newtonian Ipser-Managan scheme. The perturbation in the 1-PN corrections are obtained by solving an eigenvalue problem, which consists of three second order equations, with the 1-PN correction to the eigenfrequency of a mode, tex2html_wrap_inline2335, as the eigenvalue.

Cutler and Lindblom obtain a formula that yields tex2html_wrap_inline2335 if one knows the 1-PN stationary solution and the solution to the Newtonian perturbation equations. Thus, the frequency of a mode in the 1-PN approximation can be obtained without actually solving the 1-PN perturbation equations numerically. The 1-PN code was checked in the nonrotating limit, and it was found to reproduce the exact general relativistic frequencies for stars with M / R =0.2 obeying an N =1 polytropic EOS with an accuracy of tex2html_wrap_inline2343 .

Along a sequence of rotating stars, the frequency of a mode is commonly described by the ratio of the frequency of the mode in the comoving frame to the frequency of the mode in the nonrotating limit. For an N =1 polytrope and for M / R =0.2, this frequency ratio is reduced by as much as tex2html_wrap_inline2349 in the 1-PN approximation compared to its Newtonian counterpart (for the fundamental l = m modes) which is representative of the effect that general relativity has on the frequency of quasi-normal modes in rotating stars.

3.4.4 Cowling Approximation

In several situations, the frequency of pulsations in relativistic stars can be estimated even if one completely neglects the perturbation in the gravitational field, i.e. if one sets tex2html_wrap_inline2353 in the perturbation equations [117]. In this approximation, the pulsations are described only by the perturbation in the fluid variables, and the scheme works quite well for f, p and r -modes [118]. A different version of the Cowling approximation, in which tex2html_wrap_inline2361 is kept nonzero in the perturbation equations, works better for g -modes [119].

Yoshida and Kojima [120] examine the accuracy of the relativistic Cowling approximation in slowly rotating stars. The first-order correction to the frequency of a mode depends only on the eigenfrequency and eigenfunctions of the mode in the absence of rotation and on the angular velocity of the star. The eigenfrequencies of f, tex2html_wrap_inline2367 and tex2html_wrap_inline2369 modes for slowly rotating stars with M / R between 0.05 and 0.2 are computed (assuming polytropic EOSs with N =1 and N =1.5) and compared to their counterparts in the slow- rotation approximation.

For the l =2 f -mode, the relative error in the eigenfrequency because of the Cowling approximation is tex2html_wrap_inline2381 for less relativistic stars (M / R =0.05) and about tex2html_wrap_inline2385 for stars with M / R =0.2, and the error decreases for higher l -modes. For the tex2html_wrap_inline2367 and tex2html_wrap_inline2369 modes, the relative error is similar in magnitude. However, it is smaller for less relativistic stars. Also, for p -modes, the Cowling approximation becomes more accurate for increasing radial mode number.

As an application, Yoshida and Eriguchi [121Jump To The Next Citation Point In The Article] use the Cowling approximation to estimate the onset of the CFS instability in rapidly rotating relativistic stars.

3.5 Nonaxisymmetric Instabilities3 Oscillations and Stability3.3 Axisymmetric Perturbation

image Rotating Stars in Relativity
Nikolaos Stergioulas
© Max-Planck-Gesellschaft. ISSN 1433-8351
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