4.1 Bounce

Most of the original GW studies of core-collapse supernovae focused on the bounce phase. Gravitational radiation will be emitted during the collapse/explosion of a core-collapse SN due to the star’s changing quadrupole moment. A rough description of the possible evolution of the quadrupole moment is given in the remainder of this paragraph. During the first 100 – 250 ms of the collapse, as the core contracts and flattens, the magnitude of the quadrupole moment ℐ jk will increase. The contraction speeds up over the next 20 ms and the density distribution becomes more centrally condensed  [201Jump To The Next Citation Point]. In this phase the core’s shrinking size dominates its increasing deformation and the magnitude of ℐjk decreases. As the core bounces, ℐjk changes rapidly due to deceleration and rebound. If the bounce occurs because of nuclear pressure, its timescale will be < 1 ms. If centrifugal forces play a role in halting the collapse, the bounce can last up to several ms [201Jump To The Next Citation Point]. The magnitude of ℐjk will increase due to the core’s expansion after bounce. As the resulting shock moves outward, the unshocked portion of the core will undergo oscillations, causing ℐjk to oscillate as well. The shape of the core, the depth of the bounce, the bounce timescale, and the rotational energy of the core all strongly affect the GW emission. For further details see [86, 201Jump To The Next Citation Point, 167, 228Jump To The Next Citation Point].

The core collapse simulations of Mönchmeyer et al. [201Jump To The Next Citation Point] 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 ρc 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. [201Jump To The Next Citation Point] 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 AE2 20 for a Type I waveform is shown in Figure 10View Image (see [306Jump To The Next Citation Point, 350Jump To The Next Citation Point] for expressions relating E2 A 20 to h). The waveforms identified as Type II exhibit several maxima, which result from multiple bounces (see Figure 11View Image for an example of a Type II waveform). Note that the waveforms displayed in Figures 10View Image and 11View Image are from the study of Zwerger and Müller [350Jump To The Next Citation Point], discussed below.

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Figure 10: Type I waveform (quadrupole amplitude AE2 20 as a function of time) from one of Zwerger and Müller’s [350Jump To The Next Citation Point] simulations of a collapsing polytrope. The vertical dotted line marks the time at which the first bounce occurred. (Figure 5d of [350Jump To The Next Citation Point]; used with permission.)
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Figure 11: Type II waveform (quadrupole amplitude AE2 20 as a function of time) from one of Zwerger and Müller’s [350Jump To The Next Citation Point] simulations of a collapsing polytrope. The vertical dotted line marks the time at which bounce occurred. (Figure 5a of [350Jump To The Next Citation Point]; used with permission.)

The model of Mönchmeyer et al. [201Jump To The Next Citation Point] that bounced due to nuclear forces had the highest GW amplitude of all of their models, hpk ∼ 10− 23 for a source distance d = 10 Mpc, and the largest emitted energy EGW ∼ 1047 erg. 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 [350Jump To The Next Citation Point]. They simulated the collapse of 78 initial models with varying amounts of rotational kinetic energy (reflected in the initial value of the stability parameter βi), 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 Γ i = 4∕3. Collapse was induced by reducing the adiabatic index to a value Γ r 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 [168Jump To The Next Citation Point]).

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Figure 12: Type III waveform (quadrupole amplitude E2 A 20 as a function of time) from one of Zwerger and Müller’s [350Jump To The Next Citation Point] simulations of a collapsing polytrope. The vertical dotted line marks the time at which bounce occurred. (Figure 5e of [350Jump To The Next Citation Point]; used with permission).

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 Γ r). In their simulations, Type I signals (as labelled by Mönchmeyer et al. [201Jump To The Next Citation Point]) were produced by models with relatively soft equations of state, Γ r ≲ 1.31. Type II signals were produced by the models with stiffer equations of state, Γ r ≳ 1.32. They found a smooth transition between these signal types if Γ r was increased while all other parameters were held fixed. They also observed another class of signal, Type III, for their models with the lowest Γ r (= 1.28). 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 12View Image). Type III signals were not observed in the evolution of strongly differentially-rotating Γ r = 1.28 models and were also not seen in subsequent investigations [168Jump To The Next Citation Point, 230Jump To The Next Citation Point]. Their waveforms were computed with the same technique as used in [201Jump To The Next Citation Point].

In contrast to the results of Mönchmeyer et al. [201], in Zwerger and Müller’s investigation the value of ρc at bounce did not determine the signal type. Instead, the only effect on the waveform due to ρc was a decrease in h pk in models that bounced at subnuclear densities. The effect of the initial value of βi on hpk was non-monotonic. For models with βi ≲ 0.1, hpk increased with increasing βi. This is because the deformation of the core is larger for faster rotators. However, for models with larger βi, hpk decreases as βi 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 h pk for a given sequence was reached when ρ c at bounce was just less than ρnuc. 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 βi < 0.01; and those with soft equations of state, βi ≥ 0.018, and large degrees of differential rotation. The GW amplitudes emitted during their simulations were in the range 4 × 10−25 ≲ h ≲ 4 × 10− 23, for d = 10 Mpc. The corresponding energies ranged from 1044 ≲ E ≲ 1047 erg GW. 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 [350Jump To The Next Citation Point] in their core collapse study. Their investigation revealed that the hpk for Type I signals became saturated when the dimensionless angular momentum of the collapsing core, q = J∕ (2GM ∕c), reached ∼ 0.5. They also found that hpk was sensitive to the stiffness of the equation of state for densities just below ρnuc. 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, 169Jump To The Next Citation Point, 232Jump To The Next Citation Point, 233Jump To The Next Citation Point, 68Jump To The Next Citation Point, 69Jump To The Next Citation Point]) 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, 209Jump To The Next Citation Point], 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. [227Jump To The Next Citation Point, 226Jump To The Next Citation Point] 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 [112Jump To The Next Citation Point] 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, 325Jump To The Next Citation Point]. 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 h of this emission, computed in the quadrupole approximation, was ∼ 3 × 10−26, for d = 10 Mpc [108Jump To The Next Citation Point]. In later work, Fryer & Warren [113Jump To The Next Citation Point] included full Newtonian gravity through a tree algorithm and studied the rotating progenitors from Fryer & Heger [103Jump To The Next Citation Point]. 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 [103Jump To The Next Citation Point], 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 [107Jump To The Next Citation Point]. The fastest-rotating models achieved a signal of h ∼ 2 × 10−24 for d = 10 Mpc and characteristic frequencies of fGW ∼ 1000 Hz. For supernovae occuring within the galaxy, such a signal is detectable by LIGO-II. Dimmelmeier et al. [68Jump To The Next Citation Point, 69Jump To The Next Citation Point] and Ott et al. [232Jump To The Next Citation Point, 233Jump To The Next Citation Point] 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 [98Jump To The Next Citation Point, 110]. These calculations will be discussed in Sections 4.2 and 4.4.

4.1.1 General relativistic calculations

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 [302Jump To The Next Citation Point, 350Jump To The Next Citation Point, 245Jump To The Next Citation Point, 33].

The full collapse simulations of Fryer and Heger [103Jump To The Next Citation Point] 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 [106Jump To The Next Citation Point, 108Jump To The Next Citation Point]. 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 [103Jump To The Next Citation Point] 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) [135Jump To The Next Citation Point], 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 (0.95 – 1.9 × 1049 g cm2 s−1) 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, 138Jump To The Next Citation Point].

FHH’s [106Jump To The Next Citation Point] numerical quadrupole estimate of the GWs from oscillations observed in the collapse simulations of Fryer and Heger [103Jump To The Next Citation Point] predicts an upper limit of the peak dimensionless amplitude −23 hpk = 4.1 × 10 (for d = 10 Mpc), emitted at fGW ≈ 20 Hz. The radiated energy EGW ∼ 2 × 1044 erg. 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, 69Jump To The Next Citation Point] 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 h = 1.0 × 10−23 pk (for d = 10 Mpc).

General relativity has been more fully accounted for in the core collapse studies of Dimmelmeier, Font, and Müller [65, 66, 67Jump To The Next Citation Point] and Shibata and collaborators [281, 283], which build on the Newtonian, axisymmetric collapse simulations of Zwerger and Müller [350Jump To The Next Citation Point]. 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 [350Jump To The Next Citation Point] 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. [67Jump To The Next Citation Point] are shown in Figures 13Watch/download Movie, 14Watch/download Movie, 15Watch/download Movie, and 16Watch/download Movie. The four evolutions shown include a regular collapse (Movie 13Watch/download Movie), a rapid collapse (Movie 14Watch/download Movie), a multiple bounce collapse (Movie 15Watch/download Movie), and a very rapidly- and differentially-rotating collapse (Movie 16Watch/download Movie). 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].

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Figure 13: mpeg-Movie (11557 KB) The evolution of the regular collapse model A3B2G4 of Dimmelmeier et al. [67Jump To The Next Citation Point]. The left frame contains 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.

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Figure 14: mpeg-Movie (10308 KB) Same as Movie 13Watch/download Movie, but for rapid collapse model A3B2G5 of Dimmelmeier et al. [67Jump To The Next Citation Point].

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Figure 15: mpeg-Movie (12174 KB) Same as Movie 13Watch/download Movie, but for multiple collapse model A2B4G1 of Dimmelmeier et al. [67Jump To The Next Citation Point].

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Figure 16: mpeg-Movie (8722 KB) Same as Movie 13Watch/download Movie, but for rapid, differentially-rotating collapse model A4B5G5 of Dimmelmeier et al. [67Jump To The Next Citation Point].

Dimmelmeier et al. found that models for which the collapse type was the same in both Newtonian and relativistic simulations had lower GW amplitudes hpk 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 hpk were observed in models for which the collapse type changed. Overall, the range of hpk (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 EGW 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, fGW, 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 fGW was not accompanied by an increase in hpk. 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. [232Jump To The Next Citation Point, 233Jump To The Next Citation Point] 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.


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