The collapse of the progenitors of core collapse supernovae has been investigated as a source of gravitational radiation for more than three decades. In an early study published in 1971, Ruffini and Wheeler [203] identified mechanisms related to core collapse that could produce GWs and provided order-of-magnitude estimates of the characteristics of such emission.
Quantitative computations of GW emission during the infall phase of collapse were performed by Thuan and Ostriker [245] and Epstein and Wagoner [66, 65], who simulated the collapse of oblate dust spheroids. Thuan and Ostriker used Newtonian gravity and computed the emitted radiation in the quadrupole approximation. Epstein and Wagoner discovered that post-Newtonian effects prolonged the collapse and thus lowered the GW luminosity. Subsequently, Novikov [188] and Shapiro and Saenz [218, 204] included internal pressure in their collapse simulations and were thus able to examine the GWs emitted as collapsing cores bounced at nuclear densities. The quadrupole GWs from the ringdown of the collapse remnant were initially investigated by the perturbation study of Turner and Wagoner [249] and later by Saenz and Shapiro [205, 206].
Müller [171] calculated the quadrupole GW emission from 2D axisymmetric collapse based on the
Newtonian simulations of Müller and Hillebrandt [174] (these simulations used a realistic
equation of state and included differential rotation). He found that differential rotation enhanced
the efficiency of the GW emission. Update
Stark and Piran [233, 193] were the first to compute the GW emission from fully relativistic
collapse simulations, using the ground-breaking formalism of Bardeen and Piran [11]. They
followed the (pressure-cut induced) collapse of rotating polytropes in 2D. Their work focused
in part on the conditions for black hole formation and the nature of the resulting ringdown
waveform, which they found could be described by the quasi-normal modes of a rotating black
hole. In each of their simulations, less than 1% of the gravitational mass was converted to GW
energy.
Seidel and collaborators also studied the effects of general relativity on the GW emission during collapse and bounce [215, 216]. They employed a perturbative approach, valid only in the slowly rotating regime.
The gravitational radiation from non-axisymmetric collapse was investigated by Detweiler and Lindblom, who used a sequence of non-axisymmetric ellipsoids to represent the collapse evolution [57]. They found that the radiation from their analysis of non-axisymmetric collapse was emitted over a more narrow range of frequency than in previous studies of axisymmetric collapse.
For further discussion of the first two decades of study of the GW emission from stellar collapse see [72]. In the remainder of Section 3.4, more recent investigations will be discussed.
The core collapse simulations of Mönchmeyer et al. began with better iron core models and a more
realistic microphysical treatment (including a realistic equation of state, electron capture, and a simple
neutrino transport scheme) than any previous study of GW emission from axisymmetric stellar core
collapse [170]. The shortcomings of their investigation included initial models that were not in rotational
equilibrium, an equation of state that was somewhat stiff in the subnuclear regime, and the use of
Newtonian gravity. Each of their four models had a different initial angular momentum profile. The
rotational energies of the models ranged from 0.1 - 0.45 of the maximum possible rotational
energy.
The collapses of three of the four models of Mönchmeyer et al. were halted by centrifugal
forces at subnuclear densities. This type of low bounce had been predicted by Shapiro and
Lightman [219] and Tohline [246] (in the context of the “fizzler” scenario for failed supernovae;
see also [105, 106, 124]), and had been observed in earlier collapse simulations [175, 238].
Mönchmeyer and collaborators found that a bounce caused by centrifugal forces would last
for several
, whereas a bounce at nuclear densities would occur in
. They also
determined that a subnuclear bounce produced larger amplitude oscillations in density and radius,
with larger oscillation periods, than a bounce initiated by nuclear forces alone. They pointed
out that these differences in timescale and oscillatory behavior should affect the GW signal.
Therefore, the GW emission could indicate whether the bounce was a result of centrifugal or nuclear
forces.
Mönchmeyer et al. identified two different types of waveforms in their models (computed using the
numerical quadrupole approximation discussed in Section 2.4). The waveforms they categorized as Type I
(similar to those observed in previous collapse simulations [171, 74]) are distinguished by a large amplitude
peak at bounce and subsequent damped ringdown oscillations. They noted that Type I signals were
produced by cores that bounced at nuclear densities (or bounced at subnuclear densities if the cores had
small ratios of radial kinetic to rotational kinetic energies). The quadrupole gravitational wave
amplitude for a Type I waveform is shown in Figure 3
(see [241
, 271
] for expressions
relating
to
). The waveforms identified as Type II exhibit several maxima, which result
from multiple bounces (see Figure 4
for an example of a Type II waveform). Note that the
waveforms displayed in Figures 3
and 4
are from the study of Zwerger and Müller [271
], discussed
below.
The most extensive Newtonian survey of the parameter space of axisymmetric, rotational core collapse is
that of Zwerger and Müller [271]. They simulated the collapse of 78 initial models with varying amounts of
rotational kinetic energy (reflected in the initial value of the stability parameter
), differential rotation,
and equation of state stiffness. In order to make this large survey tractable, they used a simplified equation
of state and did not explicitly account for electron capture or neutrino transport. Their initial models were
constructed in rotational equilibrium via the method of Eriguchi and Müller [67]. The models had a
polytropic equation of state, with initial adiabatic index
. Collapse was induced by reducing the
adiabatic index to a value
in the range 1.28 - 1.325. The equation of state used during
the collapse evolution had both polytropic and thermal contributions (note that simulations
using more sophisticated equations of state get similar results [144
]).Update
In contrast to the results of Mönchmeyer et al. [170], in Zwerger and Müller’s investigation the value
of at bounce did not determine the signal type. Instead, the only effect on the waveform due to
was a decrease in
in models that bounced at subnuclear densities. The effect of the initial value of
on
was non-monotonic. For models with
,
increased with increasing
. This
is because the deformation of the core is larger for faster rotators. However, for models with
larger
,
decreases as
increases. These models bounce at subnuclear densities.
Thus, the resulting acceleration at bounce and the GW amplitude are smaller. Zwerger and
Müller found that the maximum value of
for a given sequence was reached when
at
bounce was just less than
. The degree of differential rotation did not have a large effect
on the emitted waveforms computed by Zwerger and Müller. However, they did find that
models with soft equations of state emitted stronger signals as the degree of differential rotation
increased.
The models of Zwerger and Müller that produced the largest GW signals fell into two categories: those
with stiff equations of state and ; and those with soft equations of state,
, and large
degrees of differential rotation. The GW amplitudes emitted during their simulations were in
the range
, for
(the model with the highest
is
identified in Figure 2
). The corresponding energies ranged from
. The
peaks of their power spectra were between
and
. Such signals would fall just
outside of the range of LIGO-II. Magnetic fields lower these amplitudes by
10% [145], but
realistic stellar profiles can lower the amplitudes by a factor of
[190, 177
], restricting
the detectability of supernovae to within our Galaxy (
). Update
Yamada and Sato [264] used techniques very similar to those of Zwerger and Müller [271] in their core
collapse study. Their investigation revealed that the
for Type I signals became saturated when the
dimensionless angular momentum of the collapsing core,
, reached
. They also
found that
was sensitive to the stiffness of the equation of state for densities just below
. The
characteristics of the GW emission from their models were similar to those of Zwerger and
Müller.
The GW emission from non-axisymmetric hydrodynamics simulations of stellar collapse was first studied by
Bonazzola and Marck [163, 28]. 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 3D hydrodynamics collapse simulations to study the GW emission well beyond the core bounce
phase were performed by Rampp, Müller, and Ruffert [198]. These authors started their Newtonian
simulations with the only model (A4B5G5) of Zwerger and Müller [271
] that had a post-bounce value for
the stability parameter
that significantly exceeded
(recall this is the value at which the
dynamical bar instability sets in for MacLaurin spheroid-like models). This model had the softest EOS
(
), 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
. At that point,
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
of the core. Their simulations were carried out to
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. [49] 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 [198
], 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 [271
]). 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
. This instability would have to persist for at least
cycles to be detected with
LIGO-II.
Brown [35] 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 [198
]. 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 [109
]. 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 [109]. The model evolved by Rampp, Müller, and
Ruffert [198
] 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 [198
].
Brown found that increased by a factor
during his 2D collapse simulations. This is much less
than the factor of
observed in the model studied by Rampp, Müller, and Ruffert [198
].
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. [49]. 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 [198
]).
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 [198
], 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
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.
Fryer and Warren [91] performed the first 3D collapse simulations to follow the entire collapse through
explosion. They used a smoothed particle hydrodynamics code, a realistic equation of state, the
flux-limited diffusion approximation for neutrino transport, and Newtonian spherical gravity.
Their initial model was nonrotating. Thus, no bar-mode instabilities could develop during their
simulations. The only GW emitting mechanism present in their models was convection in the core.
The maximum amplitude
of this emission, computed in the quadrupole approximation,
was
, for
[88
]. In later work, Fryer & Warren [92
] included full
Newtonian gravity through a tree algorithm and studied the rotating progenitors from Fryer &
Heger [83
]. By the launch of the explosion, no bar instabilities had developed. This was because of
several effects: they used slowly rotating, but presumably realistic, progenitors [83
], the explosion
occured quickly for their models (
) and, finally, because much of the high angular
momentum material did not make it into the inner core. These models have been further studied
for the GW signals [87
]. The fastest rotating models achieved a signal of
for
and characteristic frequencies of
. For supernovae occuring within
the Galaxy, such a signal is detectable by LIGO-II. Update
Fryer and collaborators have also modeled asymmetric collapse and asymmetric explosion calculations in
3 dimensions [80, 90]. These calculations will be discussed in Section 3.4.5.
The GW emission from nonradial quasinormal mode oscillations in proto-neutron stars has been
examined by Ferrari, Miniutti, and Pons [70]. They found that the frequencies of emission during
the first second after formation (
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
, 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 [60
]).
General relativistic effects oppose the stabilizing influence of rotation in pre-collapse cores. Thus, stars
that might be prevented from collapsing due to rotational support in the Newtonian limit may
collapse when general relativistic effects are considered. Furthermore, general relativity will
cause rotating stars undergoing collapse to bounce at higher densities than in the Newtonian
case [239, 271, 198
, 37].
The full collapse simulations of Fryer and Heger [83] are the most sophisticated axisymmetric
simulations from which the resultant GW emission has been studied [86
, 88
]. Fryer and Heger include the
effects of general relativity, but assume (for the purposes of their gravity treatment only) that the mass
distribution is spherical. The GW emission from these simulations was evaluated with either the quadrupole
approximation or simpler estimates (see below).
The work of Fryer and Heger [83] is an improvement over past collapse investigations because it starts
with rotating progenitors evolved to collapse with a stellar evolution code (which incorporates angular
momentum transport via an approximate diffusion scheme) [107
], incorporates realistic equations of state
and neutrino transport, and follows the collapse to late times. The values of total angular momentum of
the inner cores of Fryer and Heger (
) are lower than has often
been assumed in studies of the GW emission from core collapse. Note that the total specific
angular momentum of these core models may be lower by about a factor of 10 if magnetic fields
were included in the evolution of the progenitors [3, 232, 110].Update
FHH’s [86] numerical quadrupole estimate of the GWs from polar oscillations in the collapse simulations
of Fryer and Heger [83
] predicts a peak dimensionless amplitude
(for
),
emitted at
. The radiated energy
. This signal would be just out of the
detectability range of the LIGO-II detector.
The cores in the simulations of Fryer and Heger [83] are not compact enough (or rotating rapidly
enough) to develop bar instabilities during the collapse and initial bounce phases. However, the explosion
phase ejects a good deal of low angular momentum material along the poles in their evolutions. Therefore,
about after the collapse,
becomes high enough in their models to exceed the secular bar
instability limit. The
of their model with the least angular momentum actually exceeds the dynamical
bar instability limit as well (it contracts to a smaller radius and thus has a higher spin rate than the model
with higher angular momentum). FHH (and [88
]) compute an upper limit (via Equation (2
)) to
the emitted amplitude from their dynamically unstable model of
(if coherent
emission from a bar located at
persists for 100 cycles). The corresponding frequency and
maximum power are
and
. LIGO-II should be able to detect
such a signal (see Figure 2
, where FHH’s upper limit to
for this dynamical bar-mode is
identified).
As mentioned above, the proto-neutron stars of Fryer and Heger are likely to be unstable
to the development of secular bar instabilities. The GW emission from proto-neutron stars
that are secularly unstable to the bar-mode has been examined by Lai and Shapiro [148, 146].
Because the timescale for secular evolution is so long, 3D hydrodynamics simulations of the
nonlinear development of a secular bar can be impractical. To bypass this difficulty, Lai [146
]
considers only incompressible fluids, for which there are exact solutions for (Dedekind and
Jacobi-like) bar development. He predicts that such a bar located at
would emit GWs
with a peak characteristic amplitude
, if the bar persists for
cycles. The
maximum
of the emitted radiation is in the range
. 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 [134] due to the complicated phase evolution of the
emission).
Alternatively, Ou et al. [191] bypassed the long secular timescale by increasing the driving force of the
instability. They found that a bar instability was maintained for several orbits before sheer flows, producing
GW emission that would have a signal-to-noise ratio greater than 8 for LIGO-II out to
. A
movie of this simulation is shown in Figure 6
. Update
|
The GW emission from -mode unstable neutron star remnants of core collapse SNe would be easily
detectable if
(which is likely not physical; see Section 2.3). Multiple GW bursts will occur as
material falls back onto the neutron star and results in repeat episodes of
-mode growth (note that a
single
-mode episode can have multiple amplitude peaks [153]). FHH calculate that the characteristic
amplitude of the GW emission from this
-mode evolution tracks from
, over a frequency
range of
(see Section 2.4 for details). They estimate the emitted energy to exceed
.
General relativity has been more fully accounted for in the core collapse studies of Dimmelmeier, Font,
and Müller [58, 59, 60] and Shibata and collaborators [230],Update
which
build on the Newtonian, axisymmetric collapse simulations of Zwerger and Müller [271
]. In all, they have
followed the collapse evolution of 26 different models, with both Newtonian and general relativistic
simulations. As in the work of Zwerger and Müller, the different models are characterized by
varying degrees of differential rotation, initial rotation rates, and adiabatic indices. They use
the conformally flat metric to approximate the space time geometry [56] in their relativistic
hydrodynamics simulations. This approximation gives the exact solution to Einstein’s equations in the
case of spherical symmetry. Thus, as long as the collapse is not significantly aspherical, the
approximation is relatively accurate. However, the conformally flat condition does eliminate
GW emission from the spacetime. Because of this, Dimmelmeier, Font, and Müller used the
quadrupole approximation to compute the characteristics of the emitted GW signal (see [271
] for
details).
The general relativistic simulations of Dimmelmeier et al. showed the three different types of collapse
evolution (and corresponding gravitational radiation signal) seen in the Newtonian simulations of Zwerger
and Müller (regular collapse - Type I signal; multiple bounce collapse - Type II signal; and rapid collapse
- Type III signal). However, relativistic effects sometimes led to a different collapse type than in the
Newtonian case. This is because general relativity did indeed counteract the stabilizing effects of rotation
and led to much higher bounce densities (up to 700% higher). They found that multiple bounce collapse is
much rarer in general relativistic simulations (occurring in only two of their models). When
multiple bounce does occur, relativistic effects shorten the time interval between bounces by
up to a factor of four. Movies of the simulations of four models from Dimmelmeier et al. [60]
are shown in Figures 7
, 8
, 9
, and 10
. The four evolutions shown include a regular collapse
(Movie 7
), a rapid collapse (Movie 8
), a multiple bounce collapse (Movie 9
), and a very rapidly
and differentially rotating collapse (Movie 10
). The left frames of each movie contain the 2D
evolution of the logarithmic density. The upper and lower right frames display the evolutions of the
gravitational wave amplitude and the maximum density, respectively. These movies can also be viewed
at [165].
|
Convectively driven inhomogeneities in the density distribution of the outer regions of the nascent
neutron star and anisotropic neutrino emission are other sources of GW emission during the
collapse/explosion [41, 173
]. GW emission from these processes results from small-scale asphericities, unlike
the large-scale motions responsible for GW emission from aspherical collapse and non-axisymmetric global
instabilities. Note that Rayleigh-Taylor instabilities also induce time-dependent quadrupole moments at
composition interfaces in the stellar envelope. However, the resultant GW emission is too weak to be
detected because the Rayleigh-Taylor instabilities occur at very large radii [173].
Since convection was suggested as a key ingredient into the explosion, it has been postulated that
asymmetries in the convection can produce the large proper motions observed in the pulsar
population [111]. Convection asymmetries can either be produced by asymmetries in the progenitor star
that grow during collapse or by instabilities in the convection itself. Burrows and Hayes [41
] proposed that
asymmetries in the collapse could produce the pulsar velocities. The idea behind this work was that
asymmetries present in the star prior to collapse (in part due to convection during silicon and oxygen
burning) will be amplified during the collapse [20, 147]. These asymmetries will then drive asymmetries in
the convection and ultimately, the supernova explosion. Burrows and Hayes [41
] found that not only
could they produce strong motions in the nascent neutron star, but detectable gravitational
wave signals. The peak amplitude calculated was
, for a source located at
.
Fryer [80] was unable to produce the large neutron star velocities seen by Burrows and
Hayes [41] even after significantly increasing the level of asymmetry in the initial star in excess of
25%. This discrepancy is now known to be due to the crude 2-dimensional model and gravity
scheme used by Burrows and Hayes [40]. However, the gravitational wave signal produced by
both simulations is comparable. Figure 11
shows the gravitational waveform from the Burrows
& Hayes simulation (including separate matter and neutrino contributions). Figures 12
, 13
show the matter and neutrino contributions respectively to the gravitational wave forms for
the Fryer results. The gravitational wave amplitude is dominated by the neutrino component
and can exceed
, for a source located at
in Fryer’s most extreme
example.
Müller and Janka performed both 2D and 3D simulations of convective instabilities in the
proto-neutron star and hot bubble regions during the first second of the explosion phase of a Type II
SN [176]. They numerically computed the GW emission from the convection-induced aspherical mass
motion and neutrino emission in the quadrupole approximation (for details, see Section 3 of their
paper).
|
The peak GW amplitude resulting from convective mass motions in these simulations of
the proto-neutron star was in 2D and
in 3D, for
.
More recent calculations get amplitudes of
in 2D [177
] and
in
3D [87].Update
The emitted energy was
in 2D and
in 3D. The power spectrum peaked at frequencies of
in 2D and
in 3D. Such signals would not be detectable with LIGO-II. The reasons for the
differences between the 2D and 3D results include smaller convective elements and less under- and
overshooting in 3D. The relatively low angular resolution of the 3D simulations may have also
played a role. The quadrupole gravitational wave amplitude
from the 2D simulation is
shown in the upper left panel of Figure 15
(see [271, 241] for expressions relating
to
).
|
The case for GWs from convection induced asymmetric neutrino emission has also varied with time. Müller and Janka estimated the GW emission from the convection induced anisotropic neutrino radiation in their simulations (see [176] for details). They found that the amplitude of the GWs emitted can be a factor of 5 - 10 higher than the GW amplitudes resulting from convective mass motion. Müller et al. (2004) [177] argue now that the GWs produced by asymmetric neutrino emission is less than that of the convective motions.
Our understanding of the convective engine is evolving with time. Scheck et al. [212] found that the
convective cells could merge with time, producing a single lobe convective instability, fulfilling the
prediction by Herant [111]. Blondin et al. [26] argue that a standing shock instability could develop to
drive low-mode convection and Burrows et al. [43] argue that it is the shocks produced in this convection
that truly drives the supernova explosion. This convection can drive oscillations in the neutron star which
may also be a source for GWs (see Fig. 17). Update
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