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12.1 Supernovae and the physics of hot dense inhomogeneous matter

The stellar evolution of massive stars with a mass M ∼ 10– 20M ⊙ ends with the catastrophic gravitational collapse of the degenerate iron core (for a recent review, see, for instance, [219Jump To The Next Citation Point] and references therein). Photodissociation of iron nuclei and electron captures lead to the neutronization of matter. As a result, the internal pressure resisting the gravitational pull drops, thus accelerating the collapse, which proceeds on a time scale of ∼ 0.1 s. When the matter density inside the core reaches ∼ 1012 g cm–3, neutrinos become temporarily trapped, thus hindering electron captures and providing additional pressure to resist gravity. However, this is not sufficient to halt the collapse and the core contraction proceeds until the central density reaches about twice the saturation density ρ ≃ 2.8 × 1014 g cm −3 inside atomic nuclei. After that, due to the stiffness (incompressibility) of nuclear matter, the collapse halts and the core bounces, generating a shock wave. The shock wave propagates outwards against the infalling material and eventually ejects the outer layers of the star, thus spreading heavy elements into the interstellar medium. A huge amount of energy, ∼ 1053 erg, is released, almost entirely (99%) in the form of neutrinos and antineutrinos of all flavors. The remaining energy is lost into electromagnetic and gravitational radiation. This scenario of core-collapse supernova explosion proved to be consistent with dense matter theory and various observations of the supernova 1987A in the Large Magellanic Cloud (discovered on February 23, 1987). In particular, the observation of the neutrino outburst provided the first direct estimate of the binding energy of the newly-born neutron star. With the considerable improvement of neutrino detectors and the development of gravitational wave interferometers, future observations of galactic supernova explosions would bring much more restrictive constraints onto theoretical models of dense matter. Supernova observations would indirectly improve our knowledge of neutron star crusts despite very different conditions, since in collapsing stellar cores and neutron star crusts the constituents are the same and are therefore described by the same microscopic Hamiltonian.

In spite of intense theoretical efforts, numerical simulations of supernovae still fail to reproduce the stellar explosion, which probably means that some physics is missing and more realistic physics input is required [280]. One of the basic ingredients required by supernova simulations is the equation of state of hot dense matter for both the inhomogeneous and homogeneous phases, up to a few times nuclear saturation densities (Section 5.4). The equation of states (EoS) plays an important role in core collapse, the formation of the shock and its propagation [402Jump To The Next Citation Point219]. The key parameter for the stability of the star is the adiabatic index defined by Equation (82View Equation). The stellar core becomes unstable to collapse when the pressure-averaged value of the adiabatic index inside the core falls below some critical threshold γ c. A stability analysis in Newtonian gravitation shows that γc = 4∕3. The effects of general relativity increase the critical value above 4/3. The precise value of the adiabatic index in the collapsing core depends on the structure and composition of the hot dense matter and, in particular, on the presence of nuclear pastas, as can be seen in Figure 36View Image. The composition of the collapsing core and its evolution into a proto-neutron star depend significantly on the EoS. The mass fractions of the various components present inside the stellar core during the collapse are shown in Figure 65View Image for two different EoS, the standard Lattimer & Swesty [255Jump To The Next Citation Point] EoS (L&S) based on a compressible liquid drop model and the recent relativistic mean field EoS of Shen et al. [374Jump To The Next Citation Point375Jump To The Next Citation Point] (note however that the treatment of the inhomogeneous phases is not quantal but is based on the semi-classical Thomas–Fermi approximation, discussed in Section 3.2.2). As seen in Figure 65View Image, the L&S EoS predicts a larger abundance of free protons than the Shen EoS. As a consequence, the L&S EoS enhances electron captures compared to the Shen EoS and leads to a stronger deleptonization of the core, thus affecting the formation of a shock wave. The effects of the EoS are more visible during the late period of the propagation of the shock wave as shown in Figure 66View Image. The L&S EoS leads to a more compact proto-neutron star, which is therefore hotter and has higher neutrino luminosity, as can be seen in Figure 67View Image.

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Figure 65: Mass fractions of different particles in a supernova core as a function of baryon mass coordinate at the time when the central density reaches 1011 g cm–3. Solid, dashed, dotted, and dot-dashed lines show mass fractions of protons, neutrons, nuclei, and alpha particles, respectively. The results are given for two equations of state: the compressible liquid drop model of Lattimer & Swesty [255Jump To The Next Citation Point] (thin lines) and the relativistic mean field theory in the local density approximation of Shen et al. [374Jump To The Next Citation Point375Jump To The Next Citation Point] (thick lines). See [402Jump To The Next Citation Point] for details.
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Figure 66: Radial positions of shock waves as a function of time after bounce (the moment of greatest compression of the central core corresponding to a maximum central density) for two different equations of states: the compressible liquid drop model of Lattimer & Swesty [255Jump To The Next Citation Point] (thin line) and the relativistic mean field theory in the local density approximation of Shen et al. [374Jump To The Next Citation Point375Jump To The Next Citation Point] (thick line). See [402Jump To The Next Citation Point] for details. Notice that these particular models failed to produce a supernova explosion.

The collapse of the supernova core and the formation of the proto-neutron star are governed by weak interaction processes and neutrino transport [250]. Numerical simulations generally show that as the shock wave propagates outwards, it loses energy due to the dissociation of heavy elements and due to the pressure of the infalling material so that it finally stalls around ∼ 102 km, as can be seen, for instance, in Figure 66View Image. According to the delayed neutrino-heating mechanism, it is believed that the stalled shock is revived after ∼ 100 ms by neutrinos, which deposit energy in the layers behind the shock front. The interaction of neutrinos with matter is therefore crucial for modeling supernova explosions. The microscopic structure of the supernova core has a strong influence on the neutrino opacity and, therefore, on the neutrino diffusion timescale. In the relevant core layers, neutrinos form a nondegenerate gas, with a de Broglie wavelength λν = 2πℏc ∕Eν, where E ν ∼ 3kBT ∼ 5 –10 MeV. If λ ν > 2RA, where RA is the radius of a spherical cluster, then thermal neutrinos “do not see” the individual nucleons inside the cluster and scatter coherently on the A nucleons. Putting it differently, a neutrino couples to a single weak current of the cluster of A nucleons. If the neutrino scattering amplitude on a single nucleon is f, then the scattering amplitude on a cluster is Af, and the scattering cross section is coh 2 2 σA = A |f| ([150], for a review, see [373Jump To The Next Citation Point]). Consider now the opposite case of λν ≪ 2RA. Neutrinos scatter on every nucleon inside the cluster. As a result, the scattering amplitudes add incoherently, and the neutrino-nucleus scattering cross section σinAcoh = A |f |2, similar to that for a gas of A nucleons. In this way, σcoh∕σincoh ≈ A ∼ 100 A A. One therefore concludes, that the presence of clusters in hot matter can dramatically increase the neutrino opacity. The neutrino transport in supernova cores depends not only on the characteristic size of the clusters, but also on their geometrical shape and topology. In particular, the presence of an heterogeneous plasma (due to thermal statistical distribution of A and Z) in the supernova core [65] or the existence of nuclear pastas instead of spherical clusters [201384] have a sizeable effect on the neutrino propagation. The outcome is that the neutrino opacity of inhomogeneous matter is considerably increased compared to that of uniform matter.

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Figure 67: Luminosities of ν e, ¯ν e, and ν μ∕τ as a function of time after bounce (the moment of greatest compression of the central core corresponding to a maximum central density) for two different equations of state: the compressible liquid drop model of Lattimer & Swesty [255] (thin lines) and the relativistic mean field theory in the local density approximation of Shen et al. [374375] (thick lines). See [402] for details.

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