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 [91] 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
and imaginary parts with accuracy at the
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.

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
for the
*l*
=
*m*
modes of various polytropes are computed. For counterrotating
modes, both
and
decrease, tending to zero, as the rotation rate increases (when
passes through zero, the star becomes unstable to the
CFS-instability). Extrapolating
and
to higher rotation rates, Kojima finds a large discrepancy
between the points where
and
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 , 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 [139,
112].

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, , as the eigenvalue.

Cutler and Lindblom obtain a formula that yields
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
.

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
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.

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*,
and
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
for less relativistic stars (*M*
/
*R*
=0.05) and about
for stars with
*M*
/
*R*
=0.2, and the error decreases for higher
*l*
-modes. For the
and
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 [121] use the Cowling approximation to estimate the onset of the CFS instability in rapidly rotating relativistic stars.

Rotating Stars in Relativity
Nikolaos Stergioulas
http://www.livingreviews.org/lrr-1998-8
© Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |