3.1 Core-collapse supernovae

Stars more massive than ∼ 10 – 12M ⊙ and less massive than ∼ 20M ⊙ are the primary candidates for what is considered the standard formation scenario behind core-collapse supernovae. For these stars, the current belief is that the collapse proceeds through bounce and proto neutron star formation. Convection, or perhaps an alternate mechanism like magnetic fields, then deposits enough energy above the proto neutron star to drive an explosion. Scientists presently think that this process is responsible for most type Ib/c and type II supernovae7. This is the best studied, and probably best-understood, core-collapse scenario in astronomy.

3.1.1 Core-collapse supernovae rate

The rate of core-collapse supernovae can be determined by simply multiplying the star-formation rate times the fraction of stars in the ∼ 10 –12 and ∼ 20M ⊙ mass range. The uncertainties in such a calculation lie in determining the limits on either side of this range (determined by theory), the power in the initial mass function and the star formation rate (both beyond theory at this point). The latter two quantities, both set by observations, dominate the errors in this rate estimate.

Indeed, especially at high redshift, the supernova rate is better constrained by observations than the rate of star formation and it is often used to determine the star-formation history. The local rate of supernovae as a function of galaxy type has been studied [43, 193Jump To The Next Citation Point]. The rate of supernovae in a Milky-Way–sized galaxy is roughly 1 per 100 y for type II and type Ib/c supernovae. It is believed that the rate has been decreasing since a redshift of roughly 1 – 1.5 (Figure 6View Image).

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Figure 6: Left-hand panel: Evolution of the core-collapse supernova rate with redshift. The dashed line shows the intrinsic total supernova rate derived from the star formation density. The supernova rate that can be recovered by optical searches is shown in light gray, and is compared with the data points from Cappellaro et al. [44] (squares) and Dahlen et al. [59] (circles). Right-hand panel: Fraction of CC SNe that are not present in the optical and near-IR searches as a function of redshift. (Figure 2 of Mannucci et al. [193]; used with permission.)

3.1.2 Core-collapse evolution

The evolution of a core-collapse supernova passes through a number of phases: collapse and bounce, post-bounce and convection and neutron-star cooling. The runaway collapse of the core continues until the core reaches nuclear densities. At this point, nuclear forces and neutron degeneracy pressure sharply increase the core pressure, causing the core to bounce and sending a shock wave throughout the star. The nature of the bounce depends upon the equation of state: a stiffer equation of state causes the star to bounce more quickly but also more weakly than a softer equation of state. Within the current uncertainties, the bounce still occurs at an enclosed mass (mass zone measured from the center of the star) of 0.55 ± 0.2M ⊙. By the time the bounce shock stalls, the mass of the core is closer to 0.9 ± 0.2M ⊙. Including the effects of general relativity acts to effectively soften the equation of state. Rotation can cause the bounce to occur at lower densities (as centrifugal support contributes to the bounce), but for the current fastest-rotating cores produced in stellar models, this effect is less than 20% [113Jump To The Next Citation Point, 69Jump To The Next Citation Point].

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Figure 7: Six snapshots in time of the convection in the collapse of a 23M ⊙ star. The plot displays slices of the data in the x-z plane. The vectors denote direction and magnitude of the particle motion. The colors denote entropy. The convection is far from symmetric, but we do not get the single-sided downflows seen in many recent 2-dimensional calculations. Figure 4 of [116Jump To The Next Citation Point].

As the bounce shock moves outward, it becomes optically thin to neutrinos and the shock loses its internal energy. For these “standard” core-collapse explosions, the shock then does not have enough energy to throw off the infalling star and the shock stalls. But it leaves behind an entropy profile that is convectively unstable. Furthermore, heating from the hot core also drives convection8. If the convective region can develop enough energy to blow off the infalling star, a successful supernova explosion (Figure 7View Image) results. Especially for explosions with long delays (greater than a few hundred ms), the SASI may play a strong role in the convection (see the review at the beginning of this section).

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Figure 8: Images of the gas entropy (red is higher than the equilibrium value, blue is lower) illustrate the instability of a spherical standing accretion shock. This model has γ = 4∕3 and is perturbed by placing overdense rings into the infalling preshock gas. The shock is kept stalled by using a cooling function. Note that with the scaling for a realistic supernova model, the last image on the right corresponds to 300 ms. These simulations are axisymmetric, forcing a reflection symmetry about the vertical axis. (Figure 6 of [21]; used with permission.)

In any event, both instabilities, if driven to late times, will naturally produce explosions with asymmetries that may explain the observational evidence for asphericities in the core-collapse explosion: pulsar velocities, mixing, polarization (Figure 8View Image). It may also produce stronger gravitational wave signals than the higher-mode neutrino-driven convection. Fryer [99] has argued that the long delays required for SASI to dominate leads to low-energy explosions that do not eject much 56Ni and produce dim supernovae (Figure 5View Image). If true, any extremely delayed (more than 400 ms) explosion mechanism does not match supernova observations and cannot be the dominant fate of stellar collapse (although given the rough observed limits on the supernova rate and the difficulty in observing dim supernovae, we cannot constrain the importance of this mechanism with any certainty).

As the proto neutron star cools, it may also develop strong convection and this could be another source of GWs [160Jump To The Next Citation Point, 34Jump To The Next Citation Point]. But the extent of this convection ranges from growing throughout the entire proto neutron star [160Jump To The Next Citation Point], to select regions in the star [60], to small regions in the star [140, 112Jump To The Next Citation Point, 113Jump To The Next Citation Point, 116Jump To The Next Citation Point]. Only when the convection is in the entire proto neutron star [160] is it considered to play a large role in the supernova explosion. Although a small contribution to the GW signal, this convection may not be negligible in calculating the full GW signal from core collapse [228Jump To The Next Citation Point].

Another possible source of GWs is the oscillations in the proto neutron star [40Jump To The Next Citation Point]. But, like the strong proto–neutron-star convection, there are arguments against such strong oscillations [116Jump To The Next Citation Point, 85Jump To The Next Citation Point, 343Jump To The Next Citation Point, 195Jump To The Next Citation Point, 327Jump To The Next Citation Point].

In strongly magnetized neutron stars, when neutrino oscillations into sterile neutrinos in the core occur, we also expect asymmetries in the neutrino emission to lead to asymmetries in the neutrinos escaping the core, producing GWs [170Jump To The Next Citation Point, 120Jump To The Next Citation Point].

But, by far, the most studied source of GWs from proto neutron stars are the rotationally-induced bar-mode instabilities. All of these can contribute to the outflows from the cooling neutron star. If magnetic fields develop or there is a high mass-infall rate [305], these outflows can add significantly to the explosion energy and at least one supernovae seems to have experienced such a secondary explosion [191].

3.1.3 Core collapse: Sources of gravitational wave emission

In Section 2, we reviewed the wide variety of mechanisms that lead to GW emission, i.e., time varying quadrupole moments in the matter or the radiation. The complex dynamics and physics within the evolution of a standard core-collapse supernova allow for a variety of scenarios for GW emission at different phases in the collapse.

Bounce: At core bounce, when the infalling material reaches nuclear densities and the collapse halts, the matter reaches its peak acceleration. If the collapse phase is asymmetric, either by asymmetries in the stellar structure or through rotation, this phase can lead to the strongest GW emission. The primary source of this emission is the rapidly changing quadrupole moment in the matter as the asymmetries evolve (Section 4.1). But the asymmetries in the bounce also produce the initial asymmetry in the neutrino emission and this asymmetry may lead to a strong GW signal as well (Section 4.4).

Post-bounce and Convection: The convection above the proto neutron star can also develop strong asymmetries as the convective cells merge into low-mode convection. This can lead to rapidly varying quadrupole moments in both the matter (Section 4.2) and the core (Section 4.4).

In the Neutron Star: Convection in the cooling neutron star could produce strong GW emission. So can asymmetric neutrino emission produced by neutrino oscillations to stellar neutrinos in the core9. Pulsations in the newly formed proto neutron star may also produce a strong GW signal [85Jump To The Next Citation Point, 229Jump To The Next Citation Point]. But the most-studied GW source arises from bar-mode instabilities, in part because, if they develop, they may produce a GW signal that rivals both the bounce and convective GW signals.

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