3 Astrophysical Sources of Gravitational Wave Emission

The type of collapsing star and the explosion mechanism both determine what sources of GWs can occur in a stellar collapse. Before we discuss the different collapses individually and the detailed simulations, let’s discuss some of the basics behind stellar collapse and core-collapse supernovae. In this review, the term “stellar collapse” refers to any stellar system that collapses down to a neutron star or black hole. These stellar systems are produced by many different scenarios, but they can be loosely grouped into two categories:

For both classes of stellar collapse, the core is supported by a combination of thermal and degeneracy pressures. When the mass is too great for these pressures to support the star/core, it begins to compress and heat up. The compression leads to electron capture, neutrino emission, and ultimately, dissociation of the elements. Electron capture reduces the support from degeneracy pressure while Urca processes and dissociation of elements remove thermal support. With less support, the core compresses further, accelerating the rate of electron capture and dissociation, ultimately leading to a runaway implosion. This collapse continues until the matter reaches nuclear densities and the formation of a proto neutron star, or, in the case of massive stars above 100M ⊙, a proto black hole [115Jump To The Next Citation Point]4.

In stellar collapse, it is the structure (density, entropy, rotation) of the core that determines whether a black hole or neutron star is formed5 For those stellar collapses that form neutron stars, the structure of the star just beyond the collapsed core is critical in defining the fate of the system. As the collapsed material reaches nuclear densities, nuclear forces and neutron degeneracy pressure halt the collapse and send a bounce shock out through the star. For nearly all systems, this bounce shock stalls. The shock can be revived by neutrinos leaked from the core depositing their energy above the proto neutron star. This is the basis of the neutrino-driven supernova mechanism [56, 19].

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Figure 1: Diagram of the convective engine for supernova explosions.

Over the past 15 years, scientists studying the supernova engine have focused on convection above the proto neutron star as a crucial piece of physics needed to unlock powerful supernova explosions (Figure 1View Image). To drive an explosion, the convective engine must overcome the ram pressure at the top of the convective region caused by the infalling star [36, 140Jump To The Next Citation Point, 97Jump To The Next Citation Point, 156, 211]. The infalling material creates a cap or lid on top of the convective engine that must be blown off to produce an explosion. The strength of this “lid” is a function of the infalling accretion rate. Fryer [97Jump To The Next Citation Point] used this simple picture and the varying accretion rates of different stars (Figure 2View Image) to determine the fate of massive stars. A supernova occurs if the engine can blow off the lid.

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Figure 2: Accretion rates of different mass stars. The more massive stars have higher accretion rates and are, hence, harder to explode. Figure 1 from Fryer [97Jump To The Next Citation Point].

Using this simplified picture, Fryer and collaborators [97Jump To The Next Citation Point, 109Jump To The Next Citation Point, 136Jump To The Next Citation Point] were able to estimate the fate of massive stars. Fryer and Kalogera [109Jump To The Next Citation Point] outlined three fates for the star: neutron-star formation, black-hole formation through fallback after a weak supernova explosion, and prompt (or direct) black-hole formation where no supernova explosion is launched. The “prompt” or “direct” name has led to some confusion. These systems still initially form a proto neutron star. However, they are unable to throw off the infalling lid and thus they collapse down to a black hole without launching a supernova shock. Very massive (above ∼ 300M ⊙) stars might collapse directly to a black hole without the formation of a proto neutron star and without the bounce associated with this formation [115Jump To The Next Citation Point, 217Jump To The Next Citation Point, 218Jump To The Next Citation Point]. But rotation can allow the formation of a proto black hole [115Jump To The Next Citation Point]. Because the binding energy of the star rises so quickly with stellar mass as the explosion energy goes down, the line dividing proto–neutron-star and black-hole formation can be determined fairly accurately (Figure 3View Image). Of course, stellar winds will affect the stellar structure and also this fate (Figure 4View Image).

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Figure 3: Supernova explosion energies (dotted lines) based on simulation (solid circles) and observation (solid dot of SN 1987A) as a function of progenitor mass. The binding energy of the envelop (material beyond the 3M ⊙ core) rises dramatically at 20M ⊙ (solid line). This fast rise with mass and the decline based on explosion simulations of the energies from the convective engine mean that even if the errors in these estimates are large, the likely dividing line between neutron star and black hole formation can be determined fairly accurately. Figure 1 of Fryer & Kalogera [109Jump To The Next Citation Point].
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Figure 4: Fate of massive stars as a function of mass and metallicity. Figure 1 from Heger et al. [136Jump To The Next Citation Point].

One can take this one step further. If the energy of the explosion is stored in the convective region, one can estimate the maximum energy of the explosion based on the infalling accretion rate. The pressure in this convective region is [99Jump To The Next Citation Point]

P (r) = [1 ∕4MNSG (Srad∕S0)−1(1∕r − 1∕rshock) + Ps1h∕4ock]4erg cm −3, (29 )
where MNS is the mass of the proto neutron star, G is the gravitational constant, Srad is the entropy in Boltzmann’s constant per nucleon, S0 = 1.5 × 10 −11 and rshock and Pshock are the radius and pressure of the accretion shock produced where the infalling stellar material hits the convective region. If we set Pshock, the ram pressure of the infalling material, ≡ 1∕2ρv2 ff, where ρ is the density of the infalling stellar material and vff is the free-fall velocity at the shock, we can estimate the maximum energy stored in the convective region (when it exceeds this value, the shock will move outward). This allows us to predict the explosion energy as a function of delay time in the explosion (Figure 5View Image).
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Figure 5: Explosion energy as a function of launch time of the supernova shock for four differently-massed stars assuming the energy is stored in the convective region. The energy is then limited by the ram pressure of the infalling stellar material. Note that it is difficult to make a strong explosion after a long delay. For more details, see [99Jump To The Next Citation Point] and Fryer et al. (in preparation).

This picture does not change even if the standing accretion shock instability contributes to the convective instabilities. Many of the recent simulations have focused on the SASI and it is worth reviewing how it effects the basic convective picture of supernovae. This instability was originally discussed in the context of white-dwarf and neutron-star accretion in scenarios under the assumption that the accretion envelope was stable to Rayleigh–Taylor instabilities [144Jump To The Next Citation Point]. Asymmetries and strong Rayleigh–Taylor convection in actual models [101Jump To The Next Citation Point] coupled with the long predicted growth times for the SASI (∼ 3 s [144Jump To The Next Citation Point]) in neutron star accretion, led the accretion community to limit such instabilities to late times. Blondin et al. [21Jump To The Next Citation Point] introduced this instability into the core-collapse supernovae by setting up conditions that were stable to Rayleigh–Taylor instabilities to study the SASI. In this case, SASI dominates and, as Blondin et al. [21Jump To The Next Citation Point] found, can produce low-mode convection.

There are two dominant questions currently under discussion with regard to this instability. First, how dominant is SASI when entropy gradients do exist? Recall, the analytic estimates of the SASI were originally derived under the assumption that the envelope is Rayleigh–Taylor stable [144] and in neutron-star fallback; simulations argue that Rayleigh–Taylor instabilities dominate6. Low-mode convection is not a sure indicator of the SASI, as Rayleigh–Taylor instabilities also predict a growth toward low modes [139Jump To The Next Citation Point]). Second, is the SASI driven by an advective-acoustic instability or is it simply an acoustic instability? Blondin and collaborators, who first mentioned the possibility of advected vortices in the supernova context, later argued for an acoustic scenario [20]. Answering these question has spawned a great deal of both simulation and analytic theory work [38, 39Jump To The Next Citation Point, 82, 81, 94, 93, 92, 116Jump To The Next Citation Point, 195Jump To The Next Citation Point, 196Jump To The Next Citation Point, 229Jump To The Next Citation Point, 264, 263, 164, 336].

Our analysis of explosion energy holds regardless of the dominant instability or the cause of that instability if the assumption that the energy is stored in the convective region remains true. In such a case, it is difficult to construct a strong explosion from a mechanism that has a long delay. But other sources of energy exist. The neutron star itself can store energy in oscillations to add to the explosion energy [40Jump To The Next Citation Point, 39]. Fallback may also drive additional explosions [101, 100Jump To The Next Citation Point]. We will study both of these in more detail below. This has repercussions on the explosion engine and the resultant GW mechanisms.

Alternative mechanisms for the core-collapse supernova mechanism exist, most notably the magnetic-field mechanism [175Jump To The Next Citation Point, 299Jump To The Next Citation Point, 2Jump To The Next Citation Point, 3Jump To The Next Citation Point, 6Jump To The Next Citation Point, 35Jump To The Next Citation Point, 61Jump To The Next Citation Point, 62, 265]. The idea here is that magnetic fields strengthened in the collapse and subsequent convection (especially if the star is rotating rapidly) will drive an explosion. How the magnetic fields affect the GW signal will depend on how and when the magnetic field develops. For most magnetic-field generation schemes, the field grows after the bounce of the core, and signal from the collapse/bounce phase (and indeed some of the convective phase) from supernovae will be the same whether the mechanism is this alternate magnetic-field driven mechanism or the standard convective engine. For some magnetar models, the magnetic field develops after the launch of a weak, convectively-driven explosion [304]. In such a case, the GW signal will be identical to the convective-driven mechanism.

Finally, if the core collapses to a black hole and the infalling material has enough angular momentum to prevent its infall directly into the event horizon, explosions might be produced. An accretion disk forms and, either through the wind-up of magnetic fields or neutrino annihilation above the disk, an explosion may result. This type of explosion has been posited as the engine behind GRBs [219Jump To The Next Citation Point, 333Jump To The Next Citation Point]. The association of GRBs with supernova-like outbursts has argued strongly for a massive-star origin for at least long-duration GRBs [335, 117].

In this section, we review the various progenitors of core collapse, describing their physical properties, its occurrence rate, evolution, and likely GW emission mechanisms. Rather than follow the order of progenitor mass, we first discuss the fate of stars in the 12 –20M ⊙ range, the likely progenitors for core-collapse supernovae. These are the most-studied gravitational-collapse systems. We then discuss the AIC of white dwarfs and the very similar collapse of stars in the ∼ 9– 12M ⊙ range. We then move upwards in mass studying massive, very massive (20 –500M ⊙) and supermassive (5 > 10 M ⊙) stars.

 3.1 Core-collapse supernovae
  3.1.1 Core-collapse supernovae rate
  3.1.2 Core-collapse evolution
  3.1.3 Core collapse: Sources of gravitational wave emission
 3.2 Accretion-induced collapse
  3.2.1 AIC rates
  3.2.2 Evolution of AIC
  3.2.3 AIC: Sources of gravitational wave emission
 3.3 Electron-capture supernovae and core-collapse supernovae below 10 – 12 solar masses: Low mass stars
  3.3.1 Rate of supernovae from low-mass stars
  3.3.2 Low-mass star collapse: Evolution
  3.3.3 Low-mass star collapse: Gravitational wave emission mechanisms
 3.4 Core-collapse supernovae from stars above 20 solar masses: Massive star collapse
  3.4.1 Massive star collapse: Evolution
  3.4.2 Rates of massive star collapse
  3.4.3 Gravitational waves from massive stars
 3.5 Supermassive stars
  3.5.1 Supermassive stars: Evolution
  3.5.2 Rates of supermassive stars
  3.5.3 Gravitational-wave emission mechanisms of supermassive stars

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