Applying the turning point theorem provided by Sorkin [286], Friedman, Ipser, and Sorkin [115] show that in the case of rotating stars a secular axisymmetric
instability sets in when the mass becomes maximum along a
sequence of constant angular momentum. An equivalent criterion
(implied in [115]) is provided by Cook
et al.
[69]: The secular axisymmetric instability sets in when the angular
momentum becomes minimum along a sequence of constant rest mass.
The instability first develops on a secular timescale that is set
by the time required for viscosity to redistribute the star's
angular momentum. This timescale is long compared to the
dynamical timescale and comparable to the spin-up time following
a pulsar glitch. Eventually, the star encounters the onset of
dynamical instability and collapses to a black hole (see [274
] for recent numerical simulations). Thus, the onset of the
secular instability to axisymmetric perturbations separates
stable neutron stars from neutron stars that will collapse to a
black hole.
Goussard et al. [134] extend the stability criterion to hot proto-neutron stars with nonzero total entropy. In this case, the loss of stability is marked by the configuration with minimum angular momentum along a sequence of both constant rest mass and total entropy. In the nonrotating limit, Gondek et al. [127] compute frequencies and eigenfunctions of radial pulsations of hot proto-neutron stars and verify that the secular instability sets in at the maximum mass turning point, as is the case for cold neutron stars.
Axisymmetric (m
=0) pulsations in rotating relativistic stars could be excited in
a number of different astrophysical scenarios, such as during
core collapse, in star quakes induced by the secular spin-down of
a pulsar or during a large phase transition, or in the merger of
two relativistic stars in a binary system, among others. Due to
rotational couplings, the eigenfunction of any axisymmetric mode
will involve a sum of various spherical harmonics
, so that even the quasi-radial modes (with lowest order
l
=0 contribution) would, in principle, radiate gravitational
waves.
Quasi-radial modes in rotating relativistic stars have been
studied by Hartle and Friedman [145] and by Datta
et al.
[83] in the slow rotation approximation. Yoshida and Eriguchi [330] study quasi-radial modes of rapidly rotating stars in the
relativistic Cowling approximation and find that apparent
intersections between quasi-radial and other axisymmetric modes
can appear near the mass-shedding limit (see Figure
4). These apparent intersections are due to
avoided crossings
between mode sequences, which are also known to occur for
axisymmetric modes of rotating Newtonian stars. Along a
continuous sequence of computed mode frequencies an avoided
crossing occurs when another sequence is encountered. In the
region of the avoided crossing, the eigenfunctions of the two
modes become of mixed character. Away from the avoided crossing
and along the continuous sequences of computed mode frequencies,
the eigenfunctions are exchanged. However, each ``quasi-normal
mode'' is characterized by the shape of its eigenfunction and
thus, the sequences of computed frequencies that belong to
particular quasi-normal modes are discontinuous at avoided
crossings (see Figure
4
for more details). The discontinuities can be found in numerical
calculations, when quasi-normal mode sequences are well resolved
in the region of avoided crossings. Otherwise, quasi-normal mode
sequences will appear as intersecting.
Several axisymmetric modes have recently been computed for
rapidly rotating relativistic stars in the Cowling approximation,
using time evolutions of the nonlinear hydrodynamical
equations [104] (see [106
] for a description of the 2D numerical evolution scheme). As
in [330
], Font
et al.
[104
] find that apparent mode intersections are common for various
higher order axisymmetric modes (see Figure
5). Axisymmetric inertial modes also appear in the numerical
evolutions.
The first fully relativistic frequencies of quasi-radial modes
for rapidly rotating stars (without assuming the Cowling
approximation) have been obtained recently, again through
nonlinear time evolutions [105] (see Section
4.2).
Going further The stabilization, by an external gravitational field, of a relativistic star that is marginally stable to axisymmetric perturbations is discussed in [308].
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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 |