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5.4 Supernova core at subnuclear density

The outer layers of the supernova core, which after a successful explosion will become the envelope of a proto-neutron star, display a similar range of densities ρ ≲ 1014 g cm −3 and are governed by the same nuclear Hamiltonian as the neutron star crust. This is why we include it in the present review.

The subnuclear density layer of supernova cores shows many similarities with that of neutron star crusts. In both cases, the matter is formed of nuclear clusters embedded in a sea of leptons and hadrons. Nevertheless, the conditions are very different since supernova cores are lepton rich (contain many electrons, positrons as well as trapped neutrinos and antineutrinos) with lepton fraction

n − + n − n + − n YL = -e-----νe----e-----¯νe≲ 0.4, (81 ) nb
and very hot with temperatures typically T ≲ 40 MeV ∕kB while the matter in neutron star crusts is cold, neutrino free, and in β-equilibrium. Neutrinos remain trapped with the supernova core for several seconds, which is their diffusion timescale in dense hot matter. The presence of nuclei in dense hot matter is of utmost importance for the neutrino opacity of the supernova core. Namely, the scattering cross section of neutrinos off nuclei of mass number A is A times larger than the sum of the individual cross sections for A nucleons in a nucleon gas (see, e.g., [373Jump To The Next Citation Point]). As long as neutrinos are trapped, YL = const., and collapse is adiabatic.

Of particular importance for supernova simulations is the adiabatic index defined by

| ∂ logP | γ = -∂ logρ-|| , (82 ) s,Ye
where s is the entropy per nucleon in units of kB and Ye is the electron fraction. For low entropies per nucleon on the order of the Boltzmann constant k B, and for densities ρ ≲ 5 × 1013 g cm −3 the value of γ is mainly determined by the relativistic electrons and is thus close to 4/3 (adiabatic index of an ultra-relativistic Fermi gas); see Figure 35View Image. The adiabatic index jumps to larger values when the nuclear clusters dissolve into a uniform mixture of nucleons and leptons, as discussed in Section 5.1. The adiabatic index is shown in Figure 35View Image for three different EoS that have been used in supernova simulations. The impact of the presence of nuclear pasta phases (Section 3.3) on the adiabatic index is illustrated in Figure 36View Image.

The striking differences between the adiabatic index of supernova matter, γSN (ρ ), and that for cold catalyzed matter in neutron stars, γNS (ρ), deserves additional explanation. In the core collapse, compression of the matter becomes adiabatic as soon as ρ ≳ 1011 g cm −3, so that the entropy per nucleon s = const. Simultaneously, due to neutrino trapping, the electron-lepton fraction is frozen, YL = const. The condition s = const.≈ 1kB blocks evaporation of nucleons from the nuclei; the motion of nucleons have to remain ordered. Therefore, the fraction of free nucleons stays small and they do not contribute significantly to the pressure, which is supplied by the electrons, until the density reaches 1014 g cm–3.

At 14 − 3 ρ ≳ 10 g cm, nuclei coalesce forming uniform nuclear matter. Thus, there are two density regimes for γSN(ρ). For 14 − 3 ρ ≲ 10 g cm, pressure is supplied by the electrons, while nucleons are confined to the nuclei, so that γSN ≃ 4 ∕3 ≈ 1.3. Then, for ρ ≳ 1014 g cm −3 nuclei coalesce into uniform nuclear matter, and the supernova matter stiffens violently, with the adiabatic index jumping by a factor of about two, to γ ≈ 2– 3 SN. This stiffening is actually responsible for the bounce of infalling matter. An additional factor stabilizing nuclei at 14 −3 ρ ≲ 10 g cm in spite of a high 10 T > 10 K, is a large lepton fraction, YL ≈ 0.4, enforcing a relatively large proton fraction, SN Yp ≈ 0.3, to be compared with Y NpS ≈ 0.05 for neutron stars.

Finally, for supernova matter we notice the absence of a neutron-drip softening, so well pronounced in γ NS, Figure 31View Image. This is because neutron gas is present in supernova matter also at ρ < 1011 g cm −3, and the increase of the free neutron fraction at higher density is prevented by strong neutron binding in the nuclei (large SN Yp), and low s ≈ 1kB.

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Figure 35: Variation of the adiabatic index γ of supernova matter with mass density ρ for three different EoSs with trapped neutrinos: Lattimer and Swesty [255Jump To The Next Citation Point] (compressible liquid drop model), Shen [374Jump To The Next Citation Point375Jump To The Next Citation Point] (relativistic mean field theory in the local density approximation) and Wolff [197] (Hartree–Fock with Skyrme nucleon-nucleon interaction). The lepton fraction is Y = 0.4 L and the entropy per nucleon is equal to 1kB.
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Figure 36: Variation of the adiabatic index of supernova matter, γ, with pressure P in the nuclear pasta phases for a fixed electron fraction Y = 0.285 e and the entropy per nucleon 1k B, from the Thomas–Fermi calculations with the Skyrme interaction SkM of Lassaut et al. [252].

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