FHH [106] (and [108
]) compute an upper limit (via Equation (9
)) to the emitted amplitude from their
dynamically unstable model of
(if coherent emission from a bar located at 10 Mpc persists
for 100 cycles). The corresponding frequency and maximum power are
and
. LIGO-II should be able to detect such a signal (see Figure 23
, where FHH’s upper
limit to
for this dynamical bar mode is identified).
The GW emission from proto neutron stars that are secularly unstable to the bar mode has been
examined by Lai and Shapiro [173, 171]. They predict that such a bar located at 10 Mpc would
emit GWs with a peak characteristic amplitude , if the bar persists for 102 – 104
cycles. The maximum
of the emitted radiation is in the range of 102 – 103 Hz. This type
of signal should be easily detected by LIGO-I (although detection may require a technique
like the fast chirp transform method of Jenet and Prince [157], due to the complicated phase
evolution of the emission). Ou et al. [236
] found that a bar instability was maintained for several
orbits before sheer flows disintegrate the instability, producing GW emissions that would have a
signal-to-noise ratio greater than 8 for LIGO-II out to 32 Mpc. A movie of this simulation is shown in
Figure 24
.
Many stellar models models (e.g., [135, 138]) do not produce stars with sufficiently high spin rates to
produce fast-rotating cores [245, 103
, 113, 325, 231, 69]. Fryer & Heger [103] argued that the
explosion phase could eject a good deal of low angular-momentum material along the poles in
their evolutions. Roughly 1 s after the collapse, the angular momentum in the faster cores will
exceed the secular bar instability limit. It is also possible that standard stellar evolution models
underestimate the rotation rate. Many scenarios for GRBs require much higher rotation rates than
these models predict. This requirement has led to a host of new models predicting much faster
rotation rates: mixing in single stars [338, 340], binary scenarios [114, 240, 104
, 319, 42],
and the collaspe of merging white dwarfs [341, 339, 342, 118, 63, 61]. For example, Fryer &
Heger [104] merged helium stars in an effort to increase this angular momentum (Figure 9
). For
Population III
stars, the situation may be better. Figure 25
shows the
value
and growth time for such a
star at bounce and just prior to the collapse to a black
hole. This bounce is caused by a combination of thermal and rotational support and hence
the bounce is very soft. Fryer et al. [115
] argued that bar modes could well develop in these
systems.
First, scientists have learned that dynamical instabilities can excite modes as well as the
well-studied,
bar modes [46
, 258
, 234
, 235
, 260
]. Second, scientists have discovered that
non-axisymmetric instabilities can occur at much lower values of
when the differential rotation is very
high [278
, 279
, 282, 326
, 257, 260
]. For some models, cores with high differential rotation have exhibited
non-axisymmetric instabilities for values of
. These results may drastically change the importance
of these modes in astrophysical observations.
One way to determine whether a specific star collapses to develop bar modes is through equilibrium models
as initial conditions for hydrodynamical simulations (e.g., [289, 237, 221
, 46
]). Such simulations represent
the approximate evolution of a model beginning at some intermediate phase during collapse or the
evolution of a collapsed remnant. These studies do not typically follow the intricate details of
the collapse itself. Instead, their goals include determining the stability of models against the
development of non-axisymmetric modes and estimation of the characteristics of any resulting GW
emission.
Liu and Lindblom [184, 183
] have applied this equilibrium approach to AIC. Their investigation began
with a study of equilibrium models built to represent neutron stars formed from AIC [184
].
These neutron star models were created via a two-step process, using a Newtonian version of
Hachisu’s self-consistent field method [128]. Hachisu’s method ensures that the forces due to
the centrifugal and gravitational potentials and the pressure are in balance in the equilibrium
configuration.
Liu and Lindblom’s process of building the nascent neutron stars began with the construction of
rapidly-rotating, pre-collapse white dwarf models. Their Models I and II are C-O white dwarfs with central
densities and
, respectively (recall this is the range of densities for which AIC
is likely for C-O white dwarfs). Their Model III is an O-Ne-Mg white dwarf that has
(recall this is the density at which collapse is induced by electron capture). All three models are uniformly
rotating, with the maximum allowed angular velocities. The models’ values of total angular momentum are
roughly 3 – 4 times that of Fryer et al.’s AIC progenitor Model 3 [102]. The realistic equation of state used
to construct the white dwarfs is a Coulomb-corrected, zero temperature, degenerate gas equation of
state [261, 51].
In the second step of their process, Liu and Lindblom [184] built equilibrium models of the
collapsed neutron stars themselves. The mass, total angular momentum, and specific angular
momentum distribution of each neutron-star remnant is identical to that of its white dwarf
progenitor (see Section 3 of [184
] for justification of the specific angular momentum conservation
assumption). These models were built with two different realistic neutron-star equations of
state.
Liu and Lindblom’s cold neutron-star remnants had values of the stability parameter ranging from
0.23 – 0.26. It is interesting to compare these results with those of Villain et al. [323
] or of Zwerger and
Müller [350
]. Villain et al. [323] found maximum
values of 0.2 for differentially-rotating models and
0.11 for rigidly-rotating models. Zwerger and Müller performed axisymmetric hydrodynamics
simulations of stars with polytropic equations of state (
). Their initial models were
polytropes, representative of massive white dwarfs. All of their models started with
. Their model that was closest to being in uniform rotation (A1B3) had 22% less total
angular momentum than Liu and Lindblom’s Model I. The collapse simulations of Zwerger
and Müller that started with model A1B3 all resulted in remnants with values of
.
Comparison of the results of these two studies could indicate that the equation of state may play a
significant role in determining the structure of collapsed remnants. Or it could suggest that
the assumptions employed in the simplified investigation of Liu and Lindblom are not fully
appropriate.
In a continuation of the work of Liu and Lindblom, Liu [183] used linearized hydrodynamics to perform
a stability analysis of the cold neutron-star AIC remnants of Liu and Lindblom [184]. He found that only
the remnant of the O-Ne-Mg white dwarf (Liu and Lindblom’s Model III) developed the dynamical bar
mode (
) instability. This model had an initial
. Note that the
mode, observed
by others to be the dominant mode in unstable models with values of
much lower than
0.27 [314, 332, 237, 46
], did not grow in this simulation. Because Liu and Lindblom’s Models I and II had
lower values of
, Liu identified the onset of instability for neutron stars formed via AIC as
.
Liu estimated the peak amplitude of the GWs emitted by the Model III remnant to be
and the LIGO-II signal-to-noise ratio (for a persistent signal like that seen in the work
of [221] and [30]) to be
(for
). These values are for a source located at
100 Mpc. He also predicted that the timescale for gravitational radiation to carry away enough angular
momentum to eliminate the bar mode is
(
cycles). Thus,
. (Note
that this value for
is merely an upper limit as it assumes that the amplitude and frequency of the GWs
do not change over the 7 s during which they are emitted. Of course, they will change as angular
momentum is carried away from the object via GW emission.) Such a signal may be marginally detectable
with LIGO-II (see Figure 23
). Details of the approximations on which these estimates are based can be
found in [183].
Liu cautions that his results hold if the magnetic field of the proto neutron star is . If the
magnetic field is larger, then it may have time to suppress some of the neutron star’s differential rotation
before it cools. This would make bar formation less likely. Such a large field could only result if the white
dwarf progenitor’s
field was
. Observation-based estimates suggest that about 25% of white
dwarfs in interacting close binaries (cataclysmic variables) are magnetic and that the field strengths for
these stars are
[330]. Strong toroidal magnetic fields of
will also suppress
the bar instabilities [119].
The GW emission from non-axisymmetric hydrodynamics simulations of stellar collapse was first studied by
Bonazzola and Marck [194, 23]. They used a Newtonian, pseudo-spectral hydrodynamics code to follow the
collapse of polytropic models. Their simulations covered only the pre-bounce phase of the collapse. They
found that the magnitudes of in their 3D simulations were within a factor of two of those from
equivalent 2D simulations and that the gravitational radiation efficiency did not depend on the equation of
state.
The first use of 3D hydrodynamics collapse simulations to study the GW emission well beyond the core
bounce phase was performed by Rampp, Müller, and Ruffert [245]. These authors started their Newtonian
simulations with the only model (A4B5G5) of Zwerger and Müller [350
] that had a post-bounce value for
the stability parameter
that significantly exceeded 0.27 (recall this is the value at which the
dynamical bar instability sets in for MacLaurin spheroid-like models). This model had the softest equation
of state (
), highest
, and largest degree of differential rotation of all of Zwerger and
Müller’s models. The model’s initial density distribution had an off-center density maximum (and therefore
a torus-like structure). Rampp, Müller, and Ruffert evolved this model with a 2D hydrodynamics
code until its
reached
0.1. At that point, 2.5 ms prior to bounce, the configuration
was mapped onto a 3D nested cubical grid structure and evolved with a 3D hydrodynamics
code.
Before the 3D simulations started, non-axisymmetric density perturbations were imposed to seed the
growth of any non-axisymmetric modes to which the configuration was unstable. When the imposed
perturbation was random (5% in magnitude), the dominant mode that arose was . The growth of
this particular mode was instigated by the cubical nature of the computational grid. When an
perturbation was imposed (10% in magnitude), three clumps developed during the post-bounce
evolution and produced three spiral arms. These arms carried mass and angular momentum away
from the center of the core. The arms eventually merged into a bar-like structure (evidence of
the presence of the
mode). Significant non-axisymmetric structure was visible only
within the inner 40 km of the core. Their simulations were carried out to
14 ms after
bounce.
The amplitudes of the emitted gravitational radiation (computed in the quadrupole approximation) were
only 2% different from those observed in the 2D simulation of Zwerger and Müller. Because of low
angular resolution in the 3D runs, the energy emitted was only 65% of that emitted in the corresponding 2D
simulation.
The findings of Centrella et al. [46] indicate it is possible that some of the post-bounce configurations
of Zwerger and Müller, which have lower values of
than the model studied by Rampp,
Müller, and Ruffert [245
], may also be susceptible to non-axisymmetric instabilities. Centrella et
al. have performed 3D hydrodynamics simulations of
polytropes to test the stability of
configurations with off-center density maxima (as are present in many of the models of Zwerger and
Müller [350]). The simulations carried out by Centrella and collaborators were not full collapse
simulations, but rather began with differentially-rotating equilibrium models. These simulations
tracked the growth of any unstable non-axisymmetric modes that arose from the initial 1%
random density perturbations that were imposed. Their results indicate that such models can
become dynamically unstable at values of
. The observed instability had a dominant
mode. Centrella et al. estimate that if a stellar core of mass
and radius
encountered this instability, the values of
from their models would be
, for
. The frequency at which
occurred in their simulations
was
200 Hz. This instability would have to persist for at least
15 cycles to be detected with
LIGO-II.
Brown [31] carried out an investigation of the growth of non-axisymmetric modes in post-bounce cores
that was similar in many respects to that of Rampp, Müller, and Ruffert [245]. He performed 3D
hydrodynamical simulations of the post-bounce configurations resulting from 2D simulations of core
collapse. His pre-collapse initial models are
polytropes in rotational equilibrium. The
differential rotation laws used to construct Brown’s initial models were motivated by the stellar
evolution study of Heger, Langer, and Woosley [137
]. The angular velocity profiles of their
pre-collapse progenitors were broad and Gaussian-like. Brown’s initial models had peak angular
velocities ranging from 0.8 – 2.4 times those of [137]. The model evolved by Rampp, Müller, and
Ruffert [245
] had much stronger differential rotation than any of Brown’s models. To induce
collapse, Brown reduced the adiabatic index of his models to
, the same value used
by [245
].
Brown found that increased by a factor
2 during his 2D collapse simulations. This is much less
than the factor of
9 observed in the model studied by Rampp, Müller, and Ruffert [245
].
This is likely a result of the larger degree of differential rotation in the model of Rampp et
al.
Brown performed 3D simulations of the two most rapidly-rotating of his post-bounce models (models
and
, both of which had
after bounce) and of the model of Rampp et
al. (which, although it starts out with
, has a sustained
). Brown refers to the
Rampp et al. model as model RMR. Because Brown’s models do not have off-center density
maxima, they are not expected to be unstable to the
mode observed by Centrella et
al. [46
]. He imposed random 1% density perturbations at the start of all three of these 3D
simulations (note that this perturbation was of a much smaller amplitude than those imposed
by [245
]).
Brown’s simulations determined that both his most rapidly-rotating model (with post-bounce
) and model RMR are unstable to growth of the
bar mode. However, his
model
(with post-bounce
) was stable. Brown observed no dominant
or
modes growing in model RMR at the times at which they were seen in the simulations
of Rampp et al. This suggests that the mode growth in their simulations was a result of the
large perturbations they imposed. The
mode begins to grow in model RMR at about
the same time as Rampp et al. stopped their evolutions. No substantial
growth was
observed.
The results of Brown’s study indicate that the overall of the post-bounce core may not be a good
diagnostic for the onset of instability. He found, as did Rampp, Müller, and Ruffert [245], that only
the innermost portion of the core (with
) is susceptible to the bar mode.
This is evident in the stability of his model
. This model had an overall
, but
an inner core with
. Brown also observed that the
of the inner core does not
have to exceed 0.27 for the model to encounter the bar mode. Models
and RMR had
. He speculates that the inner cores of these later two models may be bar-unstable because
interaction with their outer envelopes feeds the instability or because
for such
configurations.
The GW emission from non-radial quasinormal mode oscillations in proto neutron stars has been
examined by Ferrari, Miniutti, and Pons [85]. They found that the frequencies of emission during
the first second after formation (600 – 1100 Hz for the first fundamental and gravity modes) are
significantly lower than the corresponding frequencies for cold neutron stars and thus reside
in the bandwidths of terrestrial interferometers. However, for first generation interferometers
to detect the GW emission from an oscillating proto neutron star located at 10 Mpc, with a
signal-to-noise ratio of 5,
must be
. It is unlikely that this much energy
is stored in these modes (the collapse itself may only emit
in gravitational
waves [67]).
Shibata et al. [278, 279] found that, with extremely differentially-rotating cores, a bar mode instability
can occur at values of 0.01. They found that such an instability was weakly dependent on the
polytropic index describing the equation of state and on the velocity profile (as long as the
differential rotation is high). They predict an effective amplitude of roughly 10–22 at a distance of
100 Mpc.
Studies of systems with extreme differential rotation have also discovered the development of
one-arm () instabilities. The work of Centrella et al. [46
] has been followed by a large set
of results, varying the density and angular velocity profiles [46, 258, 234, 235
, 260]. Ou &
Tohline [235] argued that these instabilities are akin to the Rossby wave instabilities studied in
black-hole accretion disks [176]. These one-armed spirals could also produce a considerable GW
signal.
Watts et al. [326] argue that these low- instabilities can be explained if the corotating f-mode
develops a dynamical shear instability. This occurs when the corotating f-mode enters the corotation band
and when the degree of differential rotation exceeds a threshold value (that is within those produced in
some collapse progenitors). These new instabilities are drastically changing our view of non-axisymmetric
modes.
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