If a proto-neutron star has a sufficiently high rotation rate
(so that,
e.g.
T
/
W
> 0.27 in the case of Maclaurin spheroids), it will be
subject to a dynamical instability driven by hydrodynamics and
gravity. Through the
l
=2 mode, the instability will deform the star into a bar shape.
This highly nonaxisymmetric configuration will emit strong
gravitational waves with frequencies in the kHz regime. The
development of the instability and the resulting waveform have
been computed numerically in the context of Newtonian gravity by
Houser
et al.
[155] and in full general relativity by Shibata
et al.
[274] (see Section
4.1.3).
At lower rotation rates, the star can become unstable to
secular nonaxisymmetric instabilities, driven by gravitational
radiation or viscosity. Gravitational radiation drives a
nonaxisymmetric instability when a mode that is retrograde in a
frame corotating with the star appears as prograde to a distant
inertial observer, via the Chandrasekhar-Friedman-Schutz (CFS)
mechanism [60,
118
]: A mode that is retrograde in the corotating frame has negative
angular momentum, because the perturbed star has less angular
momentum than the unperturbed one. If, for a distant observer,
the mode is prograde, it removes positive angular momentum from
the star, and thus the angular momentum of the mode becomes
increasingly negative.
The instability evolves on a secular timescale, during which the star loses angular momentum via the emitted gravitational waves. When the star rotates more slowly than a critical value, the mode becomes stable and the instability proceeds on the longer timescale of the next unstable mode, unless it is suppressed by viscosity.
Neglecting viscosity, the CFS instability is generic in
rotating stars for both polar and axial modes. For polar modes,
the instability occurs only above some critical angular velocity,
where the frequency of the mode goes through zero in the inertial
frame. The critical angular velocity is smaller for increasing
mode number
l
. Thus, there will always be a high enough mode number
l
for which a slowly rotating star will be unstable. Many of the
hybrid inertial modes (and in particular the relativistic
r
-mode) are generically unstable in all rotating stars, since the
mode has zero frequency in the inertial frame when the star is
nonrotating [6,
117
].
The shear and bulk viscosity of neutron star matter is able to
suppress the growth of the CFS instability except when the star
passes through a certain temperature window. In Newtonian
gravity, it appears that the polar mode CFS instability can occur
only in nascent neutron stars that rotate close to the
mass-shedding limit [160,
159
,
161
,
326
,
204
], but the computation of neutral
f
-modes in full relativity [293
,
296
] shows that relativity enhances the instability, allowing it to
occur in stars with smaller rotation rates than previously
thought.
Going further A numerical method for the analysis of the ergosphere instability in relativistic stars, which could be extended to nonaxisymmetric instabilities of fluid modes, is presented by Yoshida and Eriguchi in [327].
In the Newtonian limit, neutral modes have been determined for
several polytropic EOSs [156,
218,
158,
326]. The instability first sets in through
l
=
m
modes. Modes with larger
l
become unstable at lower rotation rates, but viscosity limits
the interesting ones to
. For an
N
=1 polytrope, the critical values of
T
/
W
for the
l
=3,4, and 5 modes are 0.079, 0.058, and 0.045, respectively, and
these values become smaller for softer polytropes. The
l
=
m
=2 ``bar'' mode has a critical
T
/
W
ratio of 0.14 that is almost independent of the polytropic
index. Since soft EOSs cannot produce models with high
T
/
W
values, the bar mode instability appears only for stiff
Newtonian polytropes of
[164
,
283
]. In addition, the viscosity-driven bar mode appears at the same
critical
T
/
W
ratio as the bar mode driven by gravitational radiation [162] (we will see later that this is no longer true in general
relativity).
The post-Newtonian computation of neutral modes by Cutler and
Lindblom [79,
199] has shown that general relativity tends to strengthen the CFS
instability. Compared to their Newtonian counterparts, critical
angular velocity ratios
(where
, and
,
are the mass and radius of the nonrotating star in the sequence)
are lowered by as much as 10% for stars obeying the
N
=1 polytropic EOS (for which the instability occurs only for
modes in the post-Newtonian approximation).
In full general relativity, neutral modes have been determined
for polytropic EOSs of
by Stergioulas and Friedman [293
,
296
], using a new numerical scheme. The scheme completes the
Eulerian formalism developed by Ipser and Lindblom in the Cowling
approximation (where
was neglected) [161], by finding an appropriate gauge in which the time independent
perturbation equations can be solved numerically for
. The computation of neutral modes for polytropes of
N
=1.0, 1.5, and 2.0 shows that relativity significantly
strengthens the instability. For the
N
=1.0 polytrope, the critical angular velocity ratio
, where
is the angular velocity at the mass-shedding limit at same
central energy density, is reduced by as much as 15% for the most
relativistic configuration (see Figure
7). A surprising result (which was not found in computations that
used the post-Newtonian approximation) is that the
l
=
m
=2 bar mode is unstable even for relativistic polytropes of index
N
=1.0. The classical Newtonian result for the onset of the bar
mode instability (
) is replaced by
in general relativity. For relativistic stars, it is evident
that the onset of the gravitational-radiation-driven bar mode
does not coincide with the onset of the viscosity-driven bar
mode, which occurs at larger
T
/
W
[39]. The computation of the onset of the CFS instability in the
relativistic Cowling approximation by Yoshida and Eriguchi [328] agrees qualitatively with the conclusions in [293,
296
].
Morsink, Stergioulas, and Blattning [230] extend the method presented in [296] to a wide range of realistic equations of state (which usually
have a stiff high density region, corresponding to polytropes of
index
N
=0.5-0.7) and find that the
l
=
m
=2 bar mode becomes unstable for stars with gravitational mass as
low as
. For
neutron stars, the mode becomes unstable at 80-95% of the
maximum allowed rotation rate. For a wide range of equations of
state, the
l
=
m
=2
f
-mode becomes unstable at a ratio of rotational to gravitational
energies
for
stars and
for maximum mass stars. This is to be contrasted with the
Newtonian value of
. The empirical formula
where
is the maximum mass for a spherical star allowed by a given
equation of state, gives the critical value of
T
/
W
for the bar
f
-mode instability, with an accuracy of 4-6%, for a wide range of
realistic EOSs.
In newly-born neutron stars the CFS instability could develop
while the background equilibrium star is still differentially
rotating. In that case, the critical value of
T
/
W, required for the instability in the
f
-mode to set in, is larger than the corresponding value in the
case of uniform rotation [333] (Figure
8). The mass-shedding limit for differentially rotating stars also
appears at considerably larger
T
/
W
than the mass-shedding limit for uniform rotation. Thus, Yoshida
et al.
[333] suggest that differential rotation favours the instability,
since the ratio
decreases with increasing degree of differential rotation.
while the radial eigenfunction of the perturbation in the
velocity can be determined at order
[176]. According to Equation (55
),
r
-modes with
m
>0 are prograde (
) with respect to a distant observer but retrograde (
) in the comoving frame for all values of the angular velocity.
Thus,
r
-modes in relativistic stars are generically unstable to the
emission of gravitational waves via the CFS instability, as was
first discovered by Andersson [6] for the case of slowly rotating, relativistic stars. This
result was proved rigorously by Friedman and Morsink [117], who showed that the canonical energy of the modes is
negative.
Two independent computations in the Newtonian Cowling
approximation [208,
16] showed that the usual shear and bulk viscosity assumed to exist
for neutron star matter is not able to damp the
r
-mode instability, even in slowly rotating stars. In a
temperature window of
, the growth time of the
l
=
m
=2 mode becomes shorter than the shear or bulk viscosity damping
time at a critical rotation rate that is roughly one tenth the
maximum allowed angular velocity of uniformly rotating stars. The
gravitational radiation is dominated by the mass current
quadrupole term. These results suggested that a rapidly rotating
proto-neutron star will spin down to Crab-like rotation rates
within one year of its birth, because of the
r
-mode instability. Due to uncertainties in the actual viscous
damping times and because of other dissipative mechanisms, this
scenario also is consistent with somewhat higher initial spins,
such as the suggested initial spin period of several milliseconds
for the X-ray pulsar in the supernova remnant N157B [224
]. Millisecond pulsars with periods less than a few milliseconds
can then only form after the accretion-induced spin-up of old
pulsars and not in the accretion-induced collapse of a white
dwarf.
The precise limit on the angular velocity of newly-born
neutron stars will depend on several factors, such as the
strength of the bulk viscosity, the cooling process,
superfluidity, the presence of hyperons, and the influence of a
solid crust. In the uniform density approximation, the
r
-mode instability can be studied analytically to
in the angular velocity of the star [182]. A study on the issue of detectability of gravitational waves
from the
r
-mode instability was presented in [238] (see Section
3.5.5), while Andersson, Kokkotas, and Stergioulas [17] and Bildsten [35] proposed that the
r
-mode instability is limiting the spin of millisecond pulsars
spun-up in LMXBs and it could even set the minimum observed spin
period of
(see [12]). This scenario is also compatible with observational data, if
one considers strange stars instead of neutron stars [11
] (see Figure
9).
Since the discovery of the
r
-mode instability, a large number of authors have studied in more
detail the development of the instability and its astrophysical
consequences. Unlike in the case of the
f
-mode instability, many different aspects and interactions have
been considered. This intense focus on the detailed physics has
been very fruitful and we now have a much more complete
understanding of the various physical processes that are
associated with pulsations in rapidly rotating relativistic
stars. The latest understanding of the
r
-mode instability is that it may not be a very promising
gravitational wave source (as originally thought), but the
important astrophysical consequences, such as the limits of the
spin of young and of recycled neutron stars are still considered
plausible. The most crucial factors affecting the instability are
magnetic fields [287,
255,
253,
254], possible hyperon bulk viscosity [166,
207,
140] and nonlinear saturation [294
,
210
,
211,
21
]. The question of the possible existence of a continuous
spectrum has also been discussed by several authors, but the most
recent analysis suggests that higher order rotational effects
still allow for discrete
r
-modes in relativistic stars [332,
258] (see Figure
10).
Magnetic fields can affect the r -mode instability, as the r -mode velocity field creates differential rotation, which is both kinematical and due to gravitational radiation reaction (see Figure 11). Under differential rotation, an initially weak poloidal magnetic field is wound-up, creating a strong toroidal field, which causes the r -mode amplitude to saturate. If neutron stars have hyperons in their cores, the associated bulk viscosity is so strong that it could completely prevent the growth of the r -mode instability. However, hyperons are predicted only by certain equations of state and the relativistic mean field theory is not universally accepted. Thus, our ignorance of the true equation of state still leaves a lot of room for the r -mode instability to be considered viable.
The detection of gravitational waves from
r
-modes depends crucially on the nonlinear saturation amplitude. A
first study by Stergioulas and Font [294] suggests that
r
-modes can exist at large amplitudes of order unity for dozens of
rotational periods in rapidly rotating relativistic stars
(Figure
12). The study used 3D relativistic hydrodynamical evolutions in
the Cowling approximation. This result was confirmed by Newtonian
3D simulations of nonlinear
r
-modes by Lindblom, Tohline, and Vallisneri [207,
210]. Lindblom
et al.
went further, using an accelerated radiation reaction force to
artificially grow the
r
-mode amplitude on a hydrodynamical (instead of the secular)
timescale. At the end of the simulations, the
r
-mode grew so large that large shock waves appeared on the
surface of the star, while the amplitude of the mode subsequently
collapsed. Lindblom
et al.
suggested that shock heating may be the mechanism that saturates
the
r
-modes at a dimensionless amplitude of
.
More recent studies of nonlinear couplings between the r -mode and higher order inertial modes [21] and new 3D nonlinear Newtonian simulations [136] seem to suggest a different picture. The r -mode could be saturated due to mode couplings or due to a hydrodynamical instability at amplitudes much smaller than the amplitude at which shock waves appeared in the simulations by Lindblom et al. Such a low amplitude, on the other hand, modifies the properties of the r -mode instability as a gravitational wave source, but is not necessarily bad news for gravitational wave detection, as a lower spin-down rate also implies a higher event rate for the r -mode instability in LMXBs in our own Galaxy [11, 154]. The 3D simulations need to achieve significantly higher resolutions before definite conclusions can be reached, while the Arras et al. work could be extended to rapidly rotating relativistic stars (in which case the mode frequencies and eigenfunctions could change significantly, compared to the slowly rotating Newtonian case, which could affect the nonlinear coupling coefficients). Spectral methods can be used for achieving high accuracy in mode calculations; first results have been obtained by Villain and Bonazzolla [316] for inertial modes of slowly rotating stars in the relativistic Cowling approximation.
For a more extensive coverage of the numerous articles on the r -mode instability that appeared in recent years, the reader is referred to several excellent recent review articles [14, 116, 201, 180, 7].
Going further If rotating stars with very high compactness exist, then w -modes can also become unstable, as was recently found by Kokkotas, Ruoff, and Andersson [183]. The possible astrophysical implications are still under investigation.
Since
and
,
, a mode will grow only if
is shorter than the viscous timescales, so that
.
In normal neutron star matter, shear viscosity is dominated by
neutron-neutron scattering with a temperature dependence of
[101], and computations in the Newtonian limit and post-Newtonian
approximation show that the CFS instability is suppressed for
-
[160
,
159
,
326
,
199]. If neutrons become a superfluid below a transition temperature
, then mutual friction, which is caused by the scattering of
electrons off the cores of neutron vortices could significantly
suppress the
f
-mode instability for
[204], but the
r
-mode instability remains unaffected [205]. The superfluid transition temperature depends on the
theoretical model for superfluidity and lies in the range
-
[240].
In a pulsating fluid that undergoes compression and expansion,
the weak interaction requires a relatively long time to
re-establish equilibrium. This creates a phase lag between
density and pressure perturbations, which results in a large bulk
viscosity [263]. The bulk viscosity due to this effect can suppress the CFS
instability only for temperatures for which matter has become
transparent to neutrinos [191,
41
]. It has been proposed that for
, matter will be opaque to neutrinos and the neutrino phase space
could be blocked ([191
]; see also [41
]). In this case, bulk viscosity will be too weak to suppress the
instability, but a more detailed study is needed.
In the neutrino transparent regime, the effect of bulk
viscosity on the instability depends crucially on the proton
fraction
. If
is lower than a critical value (
), only modified URCA processes are allowed. In this case bulk
viscosity limits, but does not completely suppress, the
instability [160,
159,
326]. For most modern EOSs, however, the proton fraction is larger
than
at sufficiently high densities [194], allowing direct URCA processes to take place. In this case,
depending on the EOS and the central density of the star, the
bulk viscosity could almost completely suppress the CFS
instability in the neutrino transparent regime [337]. At high temperatures,
, even if the star is opaque to neutrinos, the direct URCA
cooling timescale to
could be shorter than the growth timescale of the CFS
instability.
Lai and Shapiro [191] have studied the development of the
f
-mode instability using Newtonian ellipsoidal models [189,
190]. They consider the case when a rapidly rotating neutron star is
created in a core collapse. After a brief dynamical phase, the
proto-neutron star becomes secularly unstable. The instability
deforms the star into a nonaxisymmetric configuration via the
l
=2 bar mode. Since the star loses angular momentum via the
emission of gravitational waves, it spins down until it becomes
secularly stable. The frequency of the waves sweeps downward from
a few hundred Hz to zero, passing through LIGO's ideal
sensitivity band. A rough estimate of the wave amplitude shows
that, at
, the gravitational waves from the CFS instability could be
detected out to the distance of 140 Mpc by the advanced LIGO
detector. This result is very promising, especially since for
relativistic stars the instability will be stronger than the
Newtonian estimate [296]. Whether
r
-modes should also be considered a promising gravitational wave
source depends crucially on their nonlinear saturation amplitude
(see Section
3.5.3).
Going further The possible ways for neutron stars to emit gravitational waves and their detectability are reviewed in [44, 45, 121, 100, 307, 266, 80].
In Newtonian polytropes, the instability occurs only for stiff
polytropes of index
N
<0.808 [164,
283]. For relativistic models, the situation for the instability
becomes worse, since relativistic effects tend to suppress the
viscosity-driven instability (while the CFS instability becomes
stronger). According to recent results by Bonazzola
et al.
[39], for the most relativistic stars, the viscosity-driven bar mode
can become unstable only if
N
<0.55. For
stars, the instability is present for
N
<0.67.
These results are based on an approximate computation of the instability in which one perturbs an axisymmetric and stationary configuration, and studies its evolution by constructing a series of triaxial quasi-equilibrium configurations. During the evolution only the dominant nonaxisymmetric terms are taken into account. The method presented in [39] is an improvement (taking into account nonaxisymmetric terms of higher order) of an earlier method by the same authors [41]. Although the method is approximate, its results indicate that the viscosity-driven instability is likely to be absent in most relativistic stars, unless the EOS turns out to be unexpectedly stiff.
An investigation by Shapiro and Zane [269] of the viscosity-driven bar mode instability, using
incompressible, uniformly rotating triaxial ellipsoids in the
post-Newtonian approximation, finds that the relativistic effects
increase the critical
T
/
W
ratio for the onset of the instability significantly. More
recently, new post-Newtonian [88] and fully relativistic calculations for uniform density
stars [129] show that the viscosity-driven instability is not as strongly
suppressed by relativistic effects as suggested in [269]. The most promising case for the onset of the viscosity-driven
instability (in terms of the critical rotation rate) would be
rapidly rotating strange stars [130], but the instability can only appear if its growth rate is
larger than the damping rate due to the emission of gravitational
radiation - a corresponding detailed comparison is still
missing.
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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 |