Chandrasekhar and Ferrari [61] expressed the nonradial pulsation problem in terms of a fifth
order system in a diagonal gauge, which is formally independent
of fluid variables. Thus, they reformulate the problem in a way
analogous to the scattering of gravitational waves off a black
hole. Ipser and Price [163] show that in the RW gauge, nonradial pulsations can be
described by a system of two second order differential 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 [246].
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. This is typically 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. The first accurate numerical solutions were obtained by Kokkotas and Schutz [181], followed by Leins, Nollert, and Soffel [197]. Andersson, Kokkotas, and Schutz [15] 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 plane, 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 method that may help to overcome this problem, at least to an acceptable approximation, has been found by Lindblom, Mendell, and Ipser [206].
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 spacetime 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 0.01-0.1% and imaginary parts with accuracy at the 10-20% 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 are derived in a diagonal gauge by Chandrasekhar
and Ferrari [61], and in the Regge-Wheeler gauge by Kojima [173,
175], 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 [174], it is shown that, for slowly rotating stars, the coupling
between polar and axial modes affects the frequency of
f
- and
p
-modes only to second order in rotation, so that, in the slow
rotation approximation, to
, the coupling can be neglected when computing frequencies.
The linear perturbation equations in the slow rotation approximation have recently been derived in a new gauge by Ruoff, Stavridis, and Kokkotas [257]. Using the ADM formalism, a first order hyperbolic evolution system is obtained, which is suitable for numerical integration without further manipulations (as was required in the Regge-Wheeler gauge). In this gauge (which is related to a gauge introduced for nonrotating stars in [27]), the symmetry between the polar and axial equations becomes directly apparent.
The case of relativistic inertial modes is different, as these
modes have both axial and polar parts at order
, and the presence of continuous bands in the spectrum (at this
order in the rotation rate) has led to a series of detailed
investigations of the properties of these modes (see [180
] for a review). In a recent paper, Ruoff, Stavridis, and
Kokkotas [258
] finally show that the inclusion of both polar and axial parts
in the computation of relativistic
r
-modes, at order
, allows for discrete modes to be computed, in agreement with
post-Newtonian [214] and nonlinear, rapid-rotation [294
] calculations.
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; this is
similar to the perturbation in the three-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 3-8%.
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 12% 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 [331] 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 30% for weakly relativistic stars (M
/
R
=0.05) and about 15% for stars with
M
/
R
=0.2; the error decreases for higher
l
-modes. For the
and
modes the relative error is similar in magnitude but 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 [328,
329
] use the Cowling approximation to estimate the onset of the
f
-mode CFS instability in rapidly rotating relativistic stars and
to compute frequencies of
f
-modes for several realistic equations of state (see Figure
6).
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Rotating Stars in Relativity
Nikolaos Stergioulas http://www.livingreviews.org/lrr-2003-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei.mpg.de |