The core collapse simulations of Mönchmeyer et al. [201] 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. 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 [274] and Tohline [312] (in the context of the “fizzler” scenario for failed supernovae;
see also [133, 134, 153]), and had been observed in earlier collapse simulations [210, 299].
Mönchmeyer and collaborators found that a bounce caused by centrifugal forces would last
for several ms, whereas a bounce at nuclear densities would occur in
1 ms. 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. [201] identified two different types of waveforms in their models (computed using
the numerical quadrupole approximation discussed in Section 2.1). The waveforms they categorized as
Type I (similar to those observed in previous collapse simulations [204, 89]) 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 GW amplitude
for a Type I waveform is shown in Figure 10
(see [306
, 350
] for expressions relating
to
). The waveforms identified as Type II exhibit several maxima, which result from
multiple bounces (see Figure 11
for an example of a Type II waveform). Note that the waveforms
displayed in Figures 10
and 11
are from the study of Zwerger and Müller [350
], discussed
below.
The model of Mönchmeyer et al. [201] that bounced due to nuclear forces had the highest GW
amplitude of all of their models,
for a source distance
, and the largest emitted
energy
. The accompanying power spectrum peaked in the frequency range
5 × 102 – 103 Hz.
The most extensive Newtonian survey of the parameter space of axisymmetric, rotational core collapse is
that of Zwerger and Müller [350]. 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 [79]. 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 [168
]).
The major result of Zwerger and Müller’s investigation was that the signal type of the
emitted gravitational waveform in their runs was determined by the stiffness of the equation of
state of the collapsing core (i.e., the value of ). In their simulations, Type I signals (as
labelled by Mönchmeyer et al. [201
]) were produced by models with relatively soft equations of
state,
. Type II signals were produced by the models with stiffer equations of state,
. They found a smooth transition between these signal types if
was increased while all
other parameters were held fixed. They also observed another class of signal, Type III, for their
models with the lowest
(
). Type III waveforms have a large positive peak just
prior to bounce, a smaller negative peak just after bounce, and smaller subsequent oscillations
with very short periods (see Figure 12
). Type III signals were not observed in the evolution
of strongly differentially-rotating
models and were also not seen in subsequent
investigations [168
, 230
]. Their waveforms were computed with the same technique as used
in [201
].
In contrast to the results of Mönchmeyer et al. [201], 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 corresponding energies ranged from
. The peaks of their power
spectra were between 500 Hz and 1 kHz. Such signals would fall just outside of the range of
LIGO-II.
Yamada and Sato [337] used techniques very similar to those of Zwerger and Müller [350] 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.
A series of new results (e.g., [168, 169
, 232
, 233
, 68
, 69
]) have essentially confirmed these
results.
Kotake et al. [169] found that magnetic fields can lower these amplitudes (they lower the rotation rate)
by 10%, but realistic stellar profiles can lower the amplitudes by a factor of
4 – 10 [230, 209
],
restricting the detectability of supernovae to within our galaxy (
10 kpc). Obviously, this effect
depends on the strength of the magnetic fields and Obergaulinger et al. [227
, 226
] studied this effect by
using a variety of magnetic-field strengths. General-relativistic magneto-hydrodynamic calculations of
the bounce phase have confirmed these results [48, 47]. Magnetohydrodynamic calculations
are becoming more common, gradually increasing our intuition about this important piece of
physics [267, 266, 300].
Fryer and Warren [112] 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 gray diffusion approximation for neutrino transport, and Newtonian spherical gravity.
The gray approximation is a limiting assumption and a multi-group calculation can produce
different results [34, 325
]. Their initial model was non-rotating. 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
[108
]. In later work,
Fryer & Warren [113
] included full Newtonian gravity through a tree algorithm and studied the
rotating progenitors from Fryer & Heger [103
]. 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 [103
], the explosion occured quickly for their models (
100 ms) 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 [107
]. 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. Dimmelmeier et al. [68
, 69
] and Ott et al. [232
, 233
] have studied these rotating
progenitors, producing amplitudes on par with these results, but providing much more detailed
information, including spectra. These models included full general relativity and are discussed in
Section 4.1.1.
Fryer and collaborators have also modeled asymmetric collapse and asymmetric explosion
calculations in three dimensions [98, 110]. These calculations will be discussed in Sections 4.2 and
4.4.
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 [302, 350
, 245
, 33].
The full collapse simulations of Fryer and Heger [103] followed the axisymmetric evolution of the
collapse, bounce and explosion including equation of state and neutrino transport from which the resultant
GW emission has been studied [106
, 108
]. Fryer and Heger only include the effects of general relativity in
the monopole approximation. This ignores many feedback mechanisms. The GW emission from these
simulations was evaluated with either the quadrupole approximation or simpler estimates (see
below).
The work of Fryer and Heger [103] was 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) [135
], 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 [2, 290, 138
].
FHH’s [106] numerical quadrupole estimate of the GWs from oscillations observed in the
collapse simulations of Fryer and Heger [103
] predicts an upper limit of the peak dimensionless
amplitude
(for
), emitted at
. The radiated
energy
. This signal would be just out of the detectability range of the
LIGO-II detector. However, most simulations predict lower amplitudes peaking at different
frequencies.
Dimmelmeier and collaborators [68, 69] have now completed a very extensive study of axisymmetric
collapse including modern equations of state and a recipe for the evolution of the electron fraction. The
range of their results was consistent with the upper limits placed by FHH, with their strongest signal have a
peak dimensionless amplitude
(for
).
General relativity has been more fully accounted for in the core collapse studies of Dimmelmeier,
Font, and Müller [65, 66, 67] and Shibata and collaborators [281, 283], which build on the
Newtonian, axisymmetric collapse simulations of Zwerger and Müller [350
]. 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 spacetime geometry [57] 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 [350
] 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). Obergaulinger et al. [227, 226] identified a fourth
type caused by the effects of magnetic fields. 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)11.
They found that multiple bounce collapse is much rarer in general-relativistic simulations (occurring in only
two of their models). When multiple bounces do 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. [67] are
shown in Figures 13
, 14
, 15
, and 16
. The four evolutions shown include a regular collapse
(Movie 13
), a rapid collapse (Movie 14
), a multiple bounce collapse (Movie 15
), and a very rapidly-
and differentially-rotating collapse (Movie 16
). 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 GW amplitude and the maximum density, respectively. These movies can also be viewed
at [124].
Get Flash to see this player.
Get Flash to see this player.
Get Flash to see this player.
Get Flash to see this player.
Dimmelmeier et al. found that models for which the collapse type was the same in both Newtonian and
relativistic simulations had lower GW amplitudes in the relativistic case. This is because the
Newtonian models were less compact at bounce and thus had material with higher densities and velocities
at larger radii. Both higher and lower values of
were observed in models for which the collapse type
changed. Overall, the range of
(4 × 10–24 – 3 × 10–23, for a source located at 10 Mpc) seen in the
relativistic simulations was quite close to the corresponding Newtonian range. The average
was
somewhat higher in the relativistic case (1.5 × 1047 erg compared to the Newtonian value of
6.4 × 1046 erg). The overall range of GW frequencies observed in their relativistic simulations
(60 – 1000 Hz) was close to the Newtonian range. They did note that relativistic effects always caused the
characteristic frequency of emission,
, to increase (up to five-fold). Studies of non-linear pulsations in
neutron stars expect high frequencies between 1.8 – 3.6 kHz [293]. For most of their models, this increase in
was not accompanied by an increase in
. This means that relativistic effects could decrease the
detectability of GW signals from some core collapses. However, the GW emission from the models of
Dimmelmeier et al. could be detected by the first generation of ground-based interferometric
detectors if the sources were fortuitously located in the Local Group of galaxies. A catalog
containing the signals and spectra of the GW emission from all of their models can be found
at [292].
Ott et al. [232, 233
] argue that the GW signal from the collapse, bounce, and early postbounce phases
of the core collapse evolution is much more generic than many of these past results show, arguing that
variations in the rotation (within the limits studied in their calculations) do not alter the signal
significantly. If these results are confirmed, firm estimates can be provided to the GW detector
community.
http://www.livingreviews.org/lrr-2011-1 |
Living Rev. Relativity 14, (2011), 1
![]() This work is licensed under a Creative Commons License. E-mail us: |