Go to previous page Go up Go to next page

3.4 Numerical predictions of GW emission

3.4.1 Historical investigations

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 [6665], 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 [218204] 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 [205206].

Müller [171Jump To The Next Citation Point] 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. UpdateJump To The Next Update Information

Stark and Piran [233Jump To The Next Citation Point193] 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 [215216]. 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.

3.4.2 Axisymmetric simulations

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 [170Jump To The Next Citation Point]. 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 rc bounce had been predicted by Shapiro and Lightman [219] and Tohline [246] (in the context of the “fizzler” scenario for failed supernovae; see also [105106124]), and had been observed in earlier collapse simulations [175238]. 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. 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 [17174]) 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 E2 A 20 for a Type I waveform is shown in Figure 3View Image (see [241Jump To The Next Citation Point271Jump 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 4View Image for an example of a Type II waveform). Note that the waveforms displayed in Figures 3View Image and 4View Image are from the study of Zwerger and Müller [271Jump To The Next Citation Point], discussed below.

View Image

Figure 3: Type I waveform (quadrupole amplitude AE220 as a function of time) from one of Zwerger and Müller’s [271Jump 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 [271Jump To The Next Citation Point]; used with permission.)
View Image

Figure 4: Type II waveform (quadrupole amplitude AE220 as a function of time) from one of Zwerger and Müller’s [271Jump To The Next Citation Point] simulations of a collapsing polytrope. The vertical dotted line marks the time at which bounce occurred. (Figure 5a of [271Jump To The Next Citation Point]; used with permission.)
The model of Mönchmeyer et al. that bounced due to nuclear forces had the highest GW amplitude of all of their models, -23 hpk ~ 10 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 [271Jump 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 bi), 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 Gi = 4/3. Collapse was induced by reducing the adiabatic index to a value Gr 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 [144Jump To The Next Citation Point]).UpdateJump To The Next Update Information

View Image

Figure 5: Type III waveform (quadrupole amplitude AE220 as a function of time) from one of Zwerger and Müller’s [271Jump To The Next Citation Point] simulations of a collapsing polytrope. The vertical dotted line marks the time at which bounce occurred. (Figure 5e of [271Jump 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 Gr). In their simulations, Type I signals (as labelled by Mönchmeyer et al. [170Jump To The Next Citation Point]) were produced by models with relatively soft equations of state, Gr <~ 1.31. Type II signals were produced by the models with stiffer equations of state, Gr >~~ 1.32. They found a smooth transition between these signal types if Gr was increased while all other parameters were held fixed. They also observed another class of signal, Type III, for their models with the lowest Gr (= 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 5View Image). Type III signals were not observed in the evolution of strongly differentially rotating Gr = 1.28 models and were also not seen in subsequent investigations [144190Jump To The Next Citation Point].UpdateJump To The Next Update Information Their waveforms were computed with the same technique used in [170Jump To The Next Citation Point].

In contrast to the results of Mönchmeyer et al. [170], in Zwerger and Müller’s investigation the value of rc at bounce did not determine the signal type. Instead, the only effect on the waveform due to rc was a decrease in hpk in models that bounced at subnuclear densities. The effect of the initial value of bi on hpk was non-monotonic. For models with bi <~ 0.1, hpk increased with increasing bi. This is because the deformation of the core is larger for faster rotators. However, for models with larger bi, hpk decreases as bi 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 hpk for a given sequence was reached when rc at bounce was just less than rnuc. 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 bi < 0.01; and those with soft equations of state, bi > 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 model with the highest h is identified in Figure 2View Image). The corresponding energies ranged from 1044 <~ EGW <~ 1047 erg. 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. Magnetic fields lower these amplitudes by ~ 10% [145], but realistic stellar profiles can lower the amplitudes by a factor of >~~ 4- 10 [190177Jump To The Next Citation Point], restricting the detectability of supernovae to within our Galaxy (<~ 10 kpc). UpdateJump To The Next Update Information

Yamada and Sato [264] used techniques very similar to those of Zwerger and Müller [271Jump 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 rnuc. The characteristics of the GW emission from their models were similar to those of Zwerger and Müller.

3.4.3 Non-axisymmetric simulations

The GW emission from non-axisymmetric hydrodynamics simulations of stellar collapse was first studied by Bonazzola and Marck [16328]. 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 hpk 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 [198Jump To The Next Citation Point]. These authors started their Newtonian simulations with the only model (A4B5G5) of Zwerger and Müller [271Jump To The Next Citation Point] that had a post-bounce value for the stability parameter b = 0.35 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 EOS (G = 1.28 r), highest b = 0.04 i, 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 b 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 m = 4. The growth of this particular mode was instigated by the cubical nature of the computational grid. When an m = 3 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 m = 2 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. [49Jump To The Next Citation Point] indicate it is possible that some of the post-bounce configurations of Zwerger and Müller, which have lower values of b than the model studied by Rampp, Müller, and Ruffert [198Jump To The Next Citation Point], may also be susceptible to non-axisymmetric instabilities. Centrella et al. have performed 3D hydrodynamics simulations of G = 1.3 polytropes to test the stability of configurations with off-center density maxima (as are present in many of the models of Zwerger and Müller [271Jump To The Next Citation Point]). 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 b >~~ 0.14. The observed instability had a dominant m = 1 mode. Centrella et al. estimate that if a stellar core of mass M ~ 1.4 Mo . and radius R ~ 200 km encountered this instability, the values of hpk from their models would be -24 -23 ~ 2× 10 - 2× 10, for d = 10 Mpc. The frequency at which hpk 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 [35Jump To The Next Citation Point] 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 [198Jump To The Next Citation Point]. He performed 3D hydrodynamical simulations of the post-bounce configurations resulting from 2D simulations of core collapse. His pre-collapse initial models are G = 4/3 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 [109Jump To The Next Citation Point]. 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 [198Jump To The Next Citation Point] had much stronger differential rotation than any of Brown’s models. To induce collapse, Brown reduced the adiabatic index of his models to G = 1.28, the same value used by [198Jump To The Next Citation Point].

Brown found that b 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 [198Jump To The Next Citation Point]. 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 _O_24 and _O_20, both of which had b > 0.27 after bounce) and of the model of Rampp et al. (which, although it starts out with b = 0.35, has a sustained b < 0.2). 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 m = 1 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 [198Jump To The Next Citation Point]).

Brown’s simulations determined that both his most rapidly rotating model _O_24 (with post-bounce b > 0.35) and model RMR are unstable to growth of the m = 2 bar-mode. However, his model _O_20 (with post-bounce b > 0.3) was stable. Brown observed no dominant m = 3 or m = 4 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 m = 2 mode begins to grow in model RMR at about the same time as Rampp et al. stopped their evolutions. No substantial m = 1 growth was observed.

The results of Brown’s study indicate that the overall b of the post-bounce core may not be a good diagnostic for the onset of instability. He found, as did Rampp, Müller, and Ruffert [198Jump To The Next Citation Point], that only the innermost portion of the core (with 10 -3 r > 10 g cm) is susceptible to the bar-mode. This is evident in the stability of his model _O_20. This model had an overall b > 0.3, but an inner core with bic = 0.15. Brown also observed that the b of the inner core does not have to exceed 0.27 for the model to encounter the bar-mode. Models _O_24 and RMR had bic ~~ 0.19. 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 bd < 0.27 for such configurations.

Fryer and Warren [91Jump 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 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 h of this emission, computed in the quadrupole approximation, was ~ 3 × 10-26, for d = 10 Mpc [88Jump To The Next Citation Point]. In later work, Fryer & Warren [92Jump To The Next Citation Point] included full Newtonian gravity through a tree algorithm and studied the rotating progenitors from Fryer & Heger [83Jump 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 [83Jump 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 [87Jump To The Next Citation Point]. The fastest rotating models achieved a signal of - 24 h ~ 2 × 10 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. UpdateJump To The Next Update Information

Fryer and collaborators have also modeled asymmetric collapse and asymmetric explosion calculations in 3 dimensions [80Jump To The Next Citation Point90]. 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 fGW 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, E GW must be ~ 10 -3- 10-2M c2 o.. It is unlikely that this much energy is stored in these modes (the collapse itself may only emit -7 2 ~ 10 Mo. c in gravitational waves [60Jump To The Next Citation Point]).

3.4.4 General relativistic simulations

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 [239271Jump To The Next Citation Point198Jump To The Next Citation Point37].

The full collapse simulations of Fryer and Heger [83Jump To The Next Citation Point] are the most sophisticated axisymmetric simulations from which the resultant GW emission has been studied [86Jump To The Next Citation Point88Jump To The Next Citation Point]. 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 [83Jump To The Next Citation Point] 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) [107Jump 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 (49 2 - 1 0.95- 1.9× 10 g cm s) 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 [3232110].UpdateJump To The Next Update Information

FHH’s [86Jump To The Next Citation Point] numerical quadrupole estimate of the GWs from polar oscillations in the collapse simulations of Fryer and Heger [83Jump To The Next Citation Point] predicts a peak dimensionless amplitude hpk = 4.1 × 10-23 (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.

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 1 s after the collapse, b becomes high enough in their models to exceed the secular bar instability limit. The b 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 [88Jump To The Next Citation Point]) compute an upper limit (via Equation (2View Equation)) to the emitted amplitude from their dynamically unstable model of h ~ 3 × 10-22 (if coherent emission from a bar located at 10 Mpc persists for 100 cycles). The corresponding frequency and maximum power are 3 fGW ~~ 10 Hz and 53 -1 PGW = 10 erg s. LIGO-II should be able to detect such a signal (see Figure 2View Image, where FHH’s upper limit to h 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 [148146Jump To The Next Citation Point]. 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 [146Jump To The Next Citation Point] 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 10 Mpc would emit GWs with a peak characteristic amplitude h ~ 10 -21, if the bar persists for 102- 104 cycles. The maximum fGW of the emitted radiation is in the range 2 3 10 - 10 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 [134] due to the complicated phase evolution of the emission).

Alternatively, Ou et al. [191Jump To The Next Citation Point] 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 32 Mpc. A movie of this simulation is shown in Figure 6Watch/download Movie. UpdateJump To The Next Update Information

Watch/download Movie

Figure 6: Movie showing the evolution of a secular bar instability, see Ou et al. [191] for details.
FHH predict that a fragmentation instability is unlikely to develop during core collapse SNe because the cores have central density maxima (see also [88Jump To The Next Citation Point]). However, they do give estimates [calculated via Equations (4View Equation, 5View Equation)] for the amplitude, power, and frequency of the emission from such an instability: hpk ~ 2 × 10 -22, PGW = 1054 erg s-1, fGW ~~ 2 × 103 Hz. Again, this signal would fall just beyond the upper limit of LIGO-II’s frequency range. UpdateJump To The Next Update Information

The GW emission from r-mode unstable neutron star remnants of core collapse SNe would be easily detectable if a ~ 1 max (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 r-mode growth (note that a single r-mode episode can have multiple amplitude peaks [153]). FHH calculate that the characteristic amplitude of the GW emission from this r-mode evolution tracks from 6- 1 × 10-22, over a frequency range of 103- 102Hz (see Section 2.4 for details). They estimate the emitted energy to exceed 1052 erg.

General relativity has been more fully accounted for in the core collapse studies of Dimmelmeier, Font, and Müller [585960Jump To The Next Citation Point] and Shibata and collaborators [230],UpdateJump To The Next Update Information which build on the Newtonian, axisymmetric collapse simulations of Zwerger and Müller [271Jump 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 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 [271Jump 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). 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. [60Jump To The Next Citation Point] are shown in Figures 7Watch/download Movie, 8Watch/download Movie, 9Watch/download Movie, and 10Watch/download Movie. The four evolutions shown include a regular collapse (Movie 7Watch/download Movie), a rapid collapse (Movie 8Watch/download Movie), a multiple bounce collapse (Movie 9Watch/download Movie), and a very rapidly and differentially rotating collapse (Movie 10Watch/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 gravitational wave amplitude and the maximum density, respectively. These movies can also be viewed at [165].

Watch/download Movie

Figure 7: Movie showing the evolution of the regular collapse model A3B2G4 of Dimmelmeier et al. [60Jump 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.
Watch/download Movie

Figure 8: Movie showing the same as Movie 7Watch/download Movie, but for rapid collapse model A3B2G5 of Dimmelmeier et al. [60Jump To The Next Citation Point].
Watch/download Movie

Figure 9: Movie showing the same as Movie 7Watch/download Movie, but for multiple collapse model A2B4G1 of Dimmelmeier et al. [60Jump To The Next Citation Point].
Watch/download Movie

Figure 10: Movie showing the same as Movie 7Watch/download Movie, but for rapid, differentially rotating collapse model A4B5G5 of Dimmelmeier et al. [60].
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 h pk (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 (47 1.5 × 10 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 [235]. 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 [164].
UpdateJump To The Next Update Information

3.4.5 Simulations of convective instabilities

UpdateJump To The Next Update Information

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 [41Jump To The Next Citation Point173Jump To The Next Citation Point]. 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 [111Jump To The Next Citation Point]. 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 [41Jump To The Next Citation Point] 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 [20147]. These asymmetries will then drive asymmetries in the convection and ultimately, the supernova explosion. Burrows and Hayes [41Jump To The Next Citation Point] found that not only could they produce strong motions in the nascent neutron star, but detectable gravitational wave signals. The peak amplitude calculated was -24 hpk ~ 3× 10, for a source located at 10 Mpc.

Fryer [80] was unable to produce the large neutron star velocities seen by Burrows and Hayes [41Jump To The Next Citation Point] 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 11View Image shows the gravitational waveform from the Burrows & Hayes simulation (including separate matter and neutrino contributions). Figures 12View Image13View Image 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 hpk ~ 6 × 10- 24, for a source located at 10 Mpc in Fryer’s most extreme example.

View Image

Figure 11: The gravitational waveform (including separate matter and neutrino contributions) from the collapse simulations of Burrows and Hayes [41Jump To The Next Citation Point]. The curves plot the gravitational wave amplitude of the source as a function of time. (Figure 3 of [41]; used with permission.)
View Image

Figure 12: The gravitational waveform for matter contributions from the asymmetric collapse simulations of Fryer et al. [87Jump To The Next Citation Point]. The curves plot the the gravitational wave amplitude of the source as a function of time. (Figure 3 of [87Jump To The Next Citation Point]; used with permission.)
View Image

Figure 13: The gravitational waveform for neutrino contributions from the asymmetric collapse simulations of Fryer et al. [87Jump To The Next Citation Point]. The curves plot the product of the gravitational wave amplitude to the source as a function of time. (Figure 8 of [87Jump To The Next Citation Point]; used with permission.)
The study by Nazin and Postnov [183] predicts a lower limit for E GW emitted during an asymmetric core collapse SN (where such asymmetries could be induced by both aspherical mass motion and neutrino emission). They assume that observed pulsar kicks are solely due to asymmetric collapse. They suggest that the energy associated with the kick (M v2/2, where M and v are the mass and velocity of the neutron star) can be set as a lower limit for EGW (which can be computed without having to know the mechanism behind the asymmetric collapse). From observed pulsar proper motions, they estimate the degree of asymmetry e present in the collapse and the corresponding characteristic GW amplitude (V~ - h oc e). This amplitude is 3× 10-25 for a source located at 10 Mpc and emitting at fGW = 1 kHz.

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 [176Jump To The Next Citation Point]. 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).

View Image

Figure 14: Convective instabilities inside the proto-neutron star in the 2D simulation of Müller and Janka [176Jump To The Next Citation Point]. The evolutions of the temperature (left panels) and logarithmic density (right panels) distributions are shown for the radial region 15 - 95 km. The upper and lower panels correspond to times 12 and 21 ms, respectively, after the start of the simulation. The temperature values range from 10 2.5 × 10 to 11 1.8× 10 K. The values of the logarithm of the density range from 10.5 to -3 13.3 g cm. The temperature and density both increase as the colors change from blue to green, yellow, and red. (Figure 7 of [176Jump To The Next Citation Point]; used with permission.)
For typical iron core masses, the convectively unstable region in the proto-neutron star extends over the inner 0.7- 1.20 Mo . of the core mass (this corresponds to a radial range of ~ 10- 50 km). The convection in this region, which begins approximately 10 - 20 ms after the shock forms and may last for ~~ 20 ms -1 s, is caused by unstable gradients in entropy and/or lepton number resulting from the stalling of the prompt shock and deleptonization outside the neutrino sphere. Müller and Janka’s simulations of convection in this region began with the 1D, non-rotating, 12 ms post-bounce model of Hillebrandt [113]. This model included general relativistic corrections that had to be relaxed away prior to the start of the Newtonian simulations. Neutrino transport was neglected in these runs (see Section 2.1 of [176Jump To The Next Citation Point] for justification); however, a sophisticated equation of state was utilized. Figure 14View Image shows the evolution of the temperature and density distributions in the 2D simulation of Müller and Janka.

The peak GW amplitude resulting from convective mass motions in these simulations of the proto-neutron star was ~~ 3× 10-24 in 2D and ~~ 2 × 10-25 in 3D, for d = 10 Mpc. More recent calculations get amplitudes of ~~ 10 -26 in 2D [177Jump To The Next Citation Point] and ~~ 3- 5 × 10 -26 in 3D [87].UpdateJump To The Next Update Information The emitted energy was 9.8 × 1044 erg in 2D and 42 1.3 × 10 erg in 3D. The power spectrum peaked at frequencies of 200 -600 Hz in 2D and 100 -200 Hz 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 AE2 20 from the 2D simulation is shown in the upper left panel of Figure 15View Image (see [271241] for expressions relating E2 A 20 to h).

View Image

Figure 15: Quadrupole amplitudes E2 A20 [cm] from convective instabilities in various models of [176Jump To The Next Citation Point]. The upper left panel is the amplitude from a 2D simulation of proto-neutron star convection. The other three panels are amplitudes from 2D simulations of hot bubble convection. The imposed neutrino flux in the hot bubble simulations increases from the top right model through the bottom right model. (Figure 18 of [176Jump To The Next Citation Point]; used with permission.)
Convection in the hot bubble region between the shock and neutrino sphere arises because of an unstable entropy gradient resulting from the outward moving shock and subsequent neutrino heating. Figure 16Watch/download Movie shows a movie of the development of this entropy-driven convection. This unstable region extends over the inner mass range 1.25 -1.40Mo. (corresponding to a radial range of ~~ 100 -1000 km). Convection in the hot bubble begins ~~ 50- 80 ms after shock formation and lasts for ~~ 100 - 500 ms. Only 2D simulations were performed in this case. These runs started with a 25 ms post-bounce model provided by Bruenn. A simple neutrino transport scheme was used in the runs and an imposed neutrino flux was located inside the neutrino sphere. Due to computational constraints, the computational domain did not include the entire convectively unstable region inside the proto-neutron star (thus this set of simulations only accurately models the convection in the hot bubble region, not in the proto-neutron star). UpdateJump To The Next Update Information
Watch/download Movie

Figure 16: Movie showing the isosurface of material with radial velocities of 1000 km s-1 for 3 different simulation resolutions. The isosurface outlines the outward moving convective bubbles. The open spaces mark the downflows. Note that the upwelling bubbles are large and have very similar size scales to the two-dimensional simulations.
The peak GW amplitude resulting from these 2D simulations of convective mass motions in the hot bubble region was -25 hpk ~~ 5× 10, for d = 10 Mpc. The emitted energy was 42 <~ 2× 10 erg. The energy spectrum peaked at frequencies of 50 - 200 Hz. As the explosion energy was increased (by increasing the imposed neutrino flux), the violent convective motions turn into simple rapid expansion. The resultant frequencies drop to fGW ~ 10 Hz. The amplitude of such a signal would be too low to be detectable with LIGO-II.

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. 17Watch/download Movie). UpdateJump To The Next Update Information

Watch/download Movie

Figure 17: Movie showing the oscillation of the proto-neutron star caused by acoustic instabilities in the convective region above the shock.

  Go to previous page Go up Go to next page