12.1 Supernovae and the physics of hot dense inhomogeneous matter
The stellar evolution of massive stars with a mass ends with the catastrophic
gravitational collapse of the degenerate iron core (for a recent review, see, for instance, [219] 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 10^{12} 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 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, 10^{53} 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 [402, 219]. The key parameter for the
stability of the star is the adiabatic index defined by Equation (82). The stellar core becomes
unstable to collapse when the pressure-averaged value of the adiabatic index inside the core falls
below some critical threshold . A stability analysis in Newtonian gravitation shows that
. 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 36. 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 65 for two different EoS, the standard
Lattimer & Swesty [255] EoS (L&S) based on a compressible liquid drop model and the recent
relativistic mean field EoS of Shen et al. [374, 375] (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 65, 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 66. 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 67.
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 10^{2} km, as can be seen, for instance, in
Figure 66. 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 , where . If
, where is the radius of a spherical cluster, then thermal neutrinos “do not
see” the individual nucleons inside the cluster and scatter coherently on the nucleons.
Putting it differently, a neutrino couples to a single weak current of the cluster of nucleons. If
the neutrino scattering amplitude on a single nucleon is , then the scattering amplitude
on a cluster is , and the scattering cross section is ([150], for a review,
see [373]). Consider now the opposite case of . Neutrinos scatter on every nucleon inside
the cluster. As a result, the scattering amplitudes add incoherently, and the neutrino-nucleus
scattering cross section , similar to that for a gas of nucleons. In this way,
. 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
and ) in the supernova core [65] or the existence of nuclear pastas instead of spherical
clusters [201, 384] 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.