Until recently most studies treated the stellar oscillations
in a nearly Newtonian manner, thus practically ignoring the
dynamical properties of the spacetime [202,
198,
171,
199
,
200,
141,
147,
79,
146]. The spacetime was used as the medium upon which the
gravitational waves, produced by the oscillating star, propagate.
In this way all the families of modes known from Newtonian theory
were found for relativistic stars while in addition the damping
times due to gravitational radiation were calculated.
Inspired by a simple but instructive model [128], Kokkotas and Schutz showed the existence of a new family of
modes: the
w
-modes [130]. These are
spacetime modes
and their properties, although different, are closer to the
black hole QNMs than to the standard fluid stellar modes. The
main characteristics of the
w
-modes are high frequencies accompanied with very rapid damping.
Furthermore, these modes hardly excite any fluid motion. The
existence of these modes has been verified by subsequent
work [138,
21]; a part of the spectrum was found earlier by Kojima [119] and it has been shown that they exist also for odd parity
(axial) oscillations [125]. Moreover, sub-families of
w
-modes have been found for both the polar and axial oscillations
i.e. the
interface
modes found by Leins et al. [138] (see also [17]), and the
trapped
modes found by Chandrasekhar and Ferrari [60] (see also [125,
122,
17]). Recently, it has been proven that one can reveal all the
properties of the
w
-modes even if one ``freezes'' the fluid oscillations (Inverse
Cowling Approximation) [22]. In the rest of this section we shall describe the features of
both families of oscillation modes, fluid and spacetime, for the
case
.
] which verifies and extends earlier work [141]. In Table 2 we show characteristic frequencies and damping
times of various QNM modes for a typical neutron star.
are given.
is the first
p
-mode,
is the first
g
-mode [87],
stands for the first
curvature
mode and
for the slowest damped
interface
mode. For this stellar model there are no
trapped
modes.
This relation is approximately correct also for the
relativistic case [17] (see also the discussion in section
5.4). The
f
-mode eigenfunctions have
no nodes
inside the star, and they grow towards the surface. A typical
neutron star has an
f
-mode with a frequency of 1.5-3 kHz and the damping time
of this oscillation is less than a second (0.1-0.5 sec).
Detailed data for the frequencies and damping times (due to
gravitational radiation) of the
f
-mode for various equations of state can be found in [141,
19]. Estimates for the damping times due to viscosity can be
found in [69,
71].
-mode) and the damping times for the first few
p
-modes are of the order of a few seconds. Their frequencies and
damping times increase with the order of the mode. Detailed
data for the frequencies and damping times (due to
gravitational radiation) of the
-mode for various equations of state can be found in [19].
); if unstable to convection the
g
-modes are unstable (
); and if marginally stable to convection, the
g
-mode frequency vanishes. For typical neutron stars they have
frequencies smaller than a hundred Hz (the frequency decreases
with the order of the mode), and they usually damp out in time
much longer than a few days or even years. For an extensive
discussion about
g
-modes in relativistic stars refer to [87,
88]; and for a study of the instability of the
g
-modes of rotating stars to gravitational radiation reaction
refer to [134].
of the star, becomes
An inertial observer measures a frequency of
From (58
) and (59) it can be deduced that a counter-rotating (with respect to
the star, as defined in the co-rotating frame)
r
-mode appears as co-rotating with the star to a distant
inertial observer. Thus, all
r
-modes with
are generically unstable to the emission of gravitational
radiation, due to the Chandrasekhar-Friedman-Schutz (CFS)
mechanism [53,
95]. The instability is active as long as its growth-time is
shorter than the damping-time due to the viscosity of neutron
star matter. Its effect is to slow down, within a year, a
rapidly rotating neutron star to slow rotation rates and this
explains why only slowly rotating pulsars are associated with
supernova remnants [23,
142,
131]. This suggests that the
r
-mode instability might not allow millisecond pulsars to be
formed after an accretion induced collapse of a white
dwarf [23]. It seems that millisecond pulsars can only be formed by the
accretion induced spin-up of old, cold neutron stars. It is
also possible that the gravitational radiation emitted due to
this instability by a newly formed neutron star could be
detectable by the advanced versions of the gravitational wave
detectors presently under construction [161]. Recently, Andersson, Kokkotas and Stergioulas [24] have suggested that the
r
-instability might be responsible for stalling the neutron star
spin-up in strongly accreting Low Mass X-ray Binaries (LMXBs).
Additionally, they suggested that the gravitational waves from
the neutron stars, in such LMXBs, rotating at the instability
limit may well be detectable. This idea was also suggested by
Bildsten [44] and studied in detail by Levin [139].
of the
curvature
modes increases with decreasing compactness, and for a typical
neutron star the first curvature mode nearly coincides with the
fundamental black hole mode. The behavior of the
interface
modes changes slightly with the compactness. The similarity of
the axial and polar spectra is apparent.
]. They are the most important for astrophysical applications.
They are clearly related to the spacetime curvature and exist
for all relativistic stars. Their main characteristic is the
rapid damping of the oscillations. The damping rate increases
as the compactness of the star decreases: For nearly Newtonian
stars (e.g. white dwarfs) these modes have not been calculated
due to numerical instabilities in the various codes, but this
case is of marginal importance due to the very fast damping
that these modes will undergo. One of their main
characteristics is the absence of significant fluid motion
(this is a common feature for all families of
w
-modes). Numerical studies have indicated the existence of an
infinite number of modes; model problems suggest this
too [128,
40,
13]. For a typical neutron star the frequency of the first
w
-mode is around 5-12 kHz and increases with the order of
the mode. Meanwhile, the typical damping time is of the order
of a few tenths of a millisecond and decreases slowly with the
order of the mode.
) i.e. when the surface of the star is inside the peak of the
gravitational field's potential barrier [60,
125]. Practically, the first few curvature modes become trapped as
the star becomes more and more compact, and even the
f
-mode shows similar behavior [122,
17]. The trapped modes, as with all the spacetime modes, do not
induce any significant fluid motions and there are only a
finite number of them (usually less than seven or so). The
number of trapped modes increases as the potential well becomes
deeper, i.e. with increasing compactness of the star. Their
damping is quite slow since the gravitational waves have to
penetrate the potential barrier. Their frequencies can be of
the order of a few hundred Hz to a few kHz, while their damping
times can be of the order of a few tenths of a second. In
general no realistic equations of state are known that would
allow the formation of a sufficiently compact star for the
trapped modes to be relevant.] are extremely rapidly damped modes. It seems that there is
only a finite number of such modes (2-3 modes only) [17], and they are in some ways similar to the modes for acoustic
waves scattered off a hard sphere. They do not induce any
significant fluid motion and their frequencies can be from 2 to
15 kHz for typical neutron stars while their damping times are
of the order of less than a tenth of a millisecond.
|
Quasi-Normal Modes of Stars and Black Holes
Kostas D. Kokkotas and Bernd G. Schmidt http://www.livingreviews.org/lrr-1999-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
