Since convection was suggested as a key ingredient in the explosion, it has been postulated that asymmetries in the convection can produce the large proper motions observed in the pulsar population . 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  proposed that asymmetries in the collapse could produce 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 [13, 172]. These asymmetries will then drive asymmetries in the convection and ultimately, the supernova explosion. Burrows and Hayes  found that not only could they produce strong motions in the nascent neutron star, but detectable GW signals. The peak amplitude calculated was , for a source located at 10 Mpc.
Fryer  was unable to produce the large neutron star velocities seen by Burrows and Hayes  even after significantly increasing the level of asymmetry in the initial star in excess of 25%. More recent results by Burrows’ group  suggest that the Burrows and Hayes results were not quantitatively correct. However, the GW signal produced by the more recent and the original simulations is comparable. Figure 17 shows the gravitational waveform from the Burrows & Hayes simulation (including separate matter and neutrino contributions). Note that the neutrinos dominate the amplitude of the GW signal. As we shall see in Section 4.4, the neutrino contribution may dominate the GW signal from many asymmetric collapse simulations. Figure 18 shows the matter contribution to the gravitational waveforms for the Fryer results.
The study by Nazin and Postnov  predicts a lower limit for 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 (, where and are the mass and velocity of the neutron star) can be set as a lower limit for (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 present in the collapse and the corresponding characteristic GW amplitude (). This amplitude is 3 × 10 –25 for a source located at 10 Mpc and emitting at .
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. 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 ).
For typical iron core masses, the convectively unstable region in the proto neutron star extends over the inner 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 . 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  for justification); however, a sophisticated equation of state was utilized. Figure 19 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  and 3 – 5 × 10–26 in 3D . The emitted energy was 9.8 × 1044 erg in 2D and 1.3 × 1042 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 also have played a role. The quadrupole GW amplitude from the 2D simulation is shown in the upper left panel of Figure 20 (see [350, 306] for expressions relating to ).
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 21 shows a movie of the development of this entropy-driven convection. This unstable region extends over the inner mass range (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).
The peak GW amplitude resulting from these 2D simulations of convective mass motions in the hot bubble region was , for . The emitted energy was . 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 . 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  for details). They found that the amplitude of the GWs emitted can be a factor of 5 – 10 times higher than the GW amplitudes resulting from convective mass motion. More recent simulations by Müller et al.  argue that the GWs produced by asymmetric neutrino emission is weaker than that of the convective motions. But Marek et al.  have argued strongly that calculating the GW signal from asymmetries in neutrinos is extremely difficult and detailed neutrino transport is required to determine the GW signal from neutrinos.
Our understanding of the convective engine is deepening with time. The latest focus of attention has been the SASI instability, which produces, at late times, extremely low mode convection. This topic is currently a matter of heated debate. Whether this instability dominates at late times, or whether the late-time, low-mode convection is simply the merger of convective cells  is, in the opinion of these authors, yet to be conclusively determined. Convection is very difficult to simulate and has been studied for many decades on a variety of applications from the combustion engine to astrophysics, with no accepted resolution. But many groups are now finding low-mode convection above the proto neutron star (which can heighten the GW emission) and all agree that this occurs at late times (more than a few hundred milliseconds after bounce). As we have shown in Figure 5, such late explosions will be weak and, if the assumptions of that analysis are correct, these late explosion mechanisms cannot explain the observed core-collapse supernovae. Marek & Janka  believe otherwise. Certainly, if material can continue to accrete onto the neutron star after the launch of the explosion, which it does in some of the recent results of the Mezzacappa team (in preparation), stronger explosions may be produced.
In addition, Burrows et al.  found that the downstreams in the SASI can drive oscillations in the neutron star, which may also be a source for GWs (see Figure 22). Many groups have shown that the convection does not, and can not in semi-analytic studies, excite the strong oscillations observed [116, 343, 195], but see Weinberg & Quataert . When low-mode convection does occur as in the SASI, a GW signal is produced and has been modeled by several groups [85, 229, 228, 196, 165]. Most find that the actual matter signal is quite weak: , for , but the SASI can drive oscillations in the neutron star that produce a potentially important neutrino signal.
Living Rev. Relativity 14, (2011), 1
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