4.2 Convection

As we discussed at the beginning of Section 3, it is becoming increasingly accepted that convection above the proto neutron star plays a major role in the supernova explosion mechanism (e.g., Figure 1View Image). 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 [37Jump To The Next Citation Point, 206Jump To The Next Citation Point]. Inside the proto neutron star, 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. But the SASI and low-mode convection do produce large-scale accelerated mass motions [165Jump To The Next Citation Point, 213]. Note that Rayleigh–Taylor instabilities in the exploding star 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 (with low velocities) [206].

4.2.1 Asymmetric collapse

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 [139Jump 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 [37Jump To The Next Citation Point] 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 [37Jump To The Next Citation Point] found that not only could they produce strong motions in the nascent neutron star, but detectable GW signals. The peak amplitude calculated was hpk ∼ 3 × 10−24, for a source located at 10 Mpc.

Fryer [98] was unable to produce the large neutron star velocities seen by Burrows and Hayes [37Jump To The Next Citation Point] even after significantly increasing the level of asymmetry in the initial star in excess of 25%. More recent results by Burrows’ group [212] 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 17View Image 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 18View Image shows the matter contribution to the gravitational waveforms for the Fryer results.

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Figure 17: The gravitational waveform (including separate matter and neutrino contributions) from the collapse simulations of Burrows and Hayes [37Jump To The Next Citation Point]. The curves plot the GW amplitude of the source as a function of time. (Figure 3 of [37]; used with permission.)
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Figure 18: The gravitational waveform for matter contributions from the asymmetric collapse simulations of Fryer et al. [107Jump To The Next Citation Point]. The curves plot the GW amplitude of the source as a function of time. (Figure 3 of [107Jump To The Next Citation Point]; used with permission.)

The study by Nazin and Postnov [220] predicts a lower limit for EGW 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 𝜖 present in the collapse and the corresponding characteristic GW amplitude (√ - h ∝ 𝜖). This amplitude is 3 × 10 –25 for a source located at 10 Mpc and emitting at f = 1 kHz GW.

4.2.2 Proto neutron star convection

Müller and Janka [208Jump To The Next Citation Point] 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 [208Jump To The Next Citation Point]).

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Figure 19: Convective instabilities inside the proto neutron star in the 2D simulation of Müller and Janka [208Jump 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 2.5 × 1010 to 1.8 × 1011 K. The values of the logarithm of the density range from 10.5 to 13.3 g cm–3. The temperature and density both increase as the colors change from blue to green, yellow, and red. (Figure 7 of [208Jump 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– 1.20M ⊙ 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 [142]. 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 [208Jump To The Next Citation Point] for justification); however, a sophisticated equation of state was utilized. Figure 19View 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 [209Jump To The Next Citation Point] and ∼ 3 – 5 × 10–26 in 3D [107Jump To The Next Citation Point]. 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 AE2 20 from the 2D simulation is shown in the upper left panel of Figure 20View Image (see [350Jump To The Next Citation Point, 306] for expressions relating E2 A 20 to h).

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Figure 20: Quadrupole amplitudes AE220 [cm] from convective instabilities in various models of [208Jump 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 [208Jump To The Next Citation Point]; used with permission.)

4.2.3 Convection above the proto neutron star

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 21Watch/download Movie shows a movie of the development of this entropy-driven convection. This unstable region extends over the inner mass range 1.25 –1.40M ⊙ (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 [32]. 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).

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Figure 21: gif-Movie (15979 KB) Isosurface of material with radial velocities of 1000 km s–1 for three 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. From Fryer & Warren [112].

The peak GW amplitude resulting from these 2D simulations of convective mass motions in the hot bubble region was hpk ≈ 5 × 10−25, for d = 10 Mpc. The emitted energy was ≲ 2 × 1042 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 [208] 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. [209Jump To The Next Citation Point] argue that the GWs produced by asymmetric neutrino emission is weaker than that of the convective motions. But Marek et al. [196Jump To The Next Citation Point] 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.

4.2.4 Low-mode convection and the standing accretion-shock instability

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 [139] 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 5View Image, 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 [195Jump To The Next Citation Point] 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. [40Jump To The Next Citation Point] found that the downstreams in the SASI can drive oscillations in the neutron star, which may also be a source for GWs (see Figure 22Watch/download Movie). Many groups have shown that the convection does not, and can not in semi-analytic studies, excite the strong oscillations observed [116, 343, 195Jump To The Next Citation Point], but see Weinberg & Quataert [327]. When low-mode convection does occur as in the SASI, a GW signal is produced and has been modeled by several groups [85Jump To The Next Citation Point, 229, 228Jump To The Next Citation Point, 196, 165Jump To The Next Citation Point]. Most find that the actual matter signal is quite weak: −25 hpk ≈ 5 × 10, for d = 10 Mpc, but the SASI can drive oscillations in the neutron star that produce a potentially important neutrino signal.

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Figure 22: avi-Movie (4081 KB) The oscillation of the proto neutron star caused by downstreams in the SASI-induced convective region above the proto neutron star. From Burrows et al. [40].


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