Core Collapse and Supernovae

Core collapse is a sequence of coupled decisions rather than a single catastrophe. Electron capture and nuclear composition destabilize an iron core; nuclear stiffening launches a shock; dissociation and neutrinos stall it; heating and multidimensional flow may revive it; and the accreting proto-neutron star either survives or crosses an evolving stability limit. This chapter follows that material history while keeping generic horizon geometry and detector-likelihood machinery in their dedicated treatments.

Helpful background. Stress–Energy and Conservation supplies energy and lepton budgets; the Einstein equations supply relativistic gravity; Relativistic Matter and Compact Objects supplies equations of state and transport; Black-Hole Dynamics develops generic collapse and horizon geometry; Computational Relativity supplies conservative evolution; and Black-Hole Perturbation Theory supplies stability diagnostics.

Route from iron core to remnant

The six pages should be read in order because the output of each becomes the initial data or boundary condition of the next.

1. Onset of Iron-Core Collapse explains why electron capture, evolving composition, and nuclear dissociation reduce the effective pressure response of a nearly marginal degenerate core.
2. Core Bounce and Shock Formation follows nuclear stiffening, inner-core rebound, shock launch, and the energy sinks that produce stagnation.
3. Neutrino Heating and Transport develops the gain radius, charged-current deposition, transport closures, and the advection-to-heating timescale diagnostic.
4. Explosion Mechanisms and Multidimensional Instabilities compares convection, SASI, turbulence, rotation, magnetic fields, and competing criteria for a converged explosion.
5. Proto-Neutron-Star Evolution and Black-Hole Formation tracks accretion, cooling, deleptonization, fallback, the evolving supported mass, and collapse diagnostics.
6. Supernova Multimessenger Signals connects neutrinos, gravitational waves, photons, and ejecta to calibrated, selection-aware inference.

The ordering prevents two common shortcuts: a formed shock is not yet an explosion, and a cold maximum mass does not by itself decide the fate of a hot, lepton-rich, accreting remnant.

One coupled conservation problem

The continuum description combines baryon number, stress–energy, electron lepton number, an equation of state, and neutrino transport,

$$\nabla_\mu(\rho u^\mu)=0,\qquad \nabla_\mu T^{\mu\nu}=Q^\nu_{\rm weak},\qquad \nabla_\mu(\rho Y_e u^\mu)=\rho{\cal R}_{Y_e}.$$

Here $Q^\nu_{\rm weak}$ is the neutrino–matter exchange four-vector, and ${\cal R}_{Y_e}$ is the weak-interaction source. A calculation must specify the progenitor, finite-temperature EOS, weak rates and opacity set, neutrino species and energy treatment, gravity, boundary conditions, and spatial dimension. Mezzacappa and collaborators describe why the physical, numerical, and computational parts cannot be validated independently in Mezzacappa et al. 2020, §§3–5 and 6.5.

The conventions are $(-,+,+,+)$, $u^\mu u_\mu=-1$, explicit $G$, and $c=\hbar=k_B=1$. Rates have dimension $L^{-1}$ and $GM/R$ is a dimensionless compactness. Restoring units is an important check when neutrino energy per time, specific heating, and gravitational binding are combined. Nuclear binding must use the same energy zero in the equation of state and reaction network.

Three dynamical thresholds

The first threshold is collapse. A relativistic degenerate electron gas has a pressure response close to $\Gamma=4/3$ and an ideal supported-mass scaling $M_{\rm Ch}\propto Y_e^2$. Electron capture lowers $Y_e$ and therefore support. Once the collapse is dynamical, the inner core remains approximately homologous until nuclear stiffening produces bounce. Janka reviews the onset, bounce, gain region, and multidimensional mechanism in Janka 2012, §§II.B, IV.B, and IV.C, Eqs. (1)–(11).

The second threshold is shock revival. A useful local indicator is

$$\frac{\tau_{\rm adv}}{\tau_{\rm heat}} \sim \frac{M_g/\dot M}{|E_{\rm bind,g}|/\dot Q_g}.$$

A value near or above unity favors net unbinding, but it is definition- and model-dependent. Convection and SASI can increase residence time and nonradial support. A credible explosion verdict additionally requires sustained shock expansion and an energy trend that accounts for recombination and overlying binding. Burrows places these criteria in the broader mechanism comparison in Burrows 2013, §§II.E, III, and IV.A.

The third threshold is remnant stability. The actual baryon mass rises through accretion or fallback while the supported hot mass changes with entropy, lepton profile, rotation, and cooling. A turning point is a warning; a resolved dynamical evolution with conservation and an apparent-horizon diagnostic establishes black-hole formation. In the limit of vanishing accretion and long cooling, the model should approach an appropriate cold stellar sequence.

How to compare calculations

Different explosion outcomes are interpretable only after matching progenitor, EOS and table version, opacity set, gravity, transport and closure, dimensionality, perturbations, resolution, and runtime. Two-dimensional axisymmetry and three-dimensional turbulence have different cascades, so a finite-resolution ordering is not a theorem. A code comparison should expose these formulation axes before attributing a difference to one mechanism.

Computational pages in this chapter state continuum inputs, a spherical or cell benchmark, refinement variables, output tolerances, and characteristic failure modes. No page claims to have executed a simulation. Reproducible work would pin code and data versions and regenerate machine-readable conservation and convergence tables. Observational conclusions add instrument calibration, selection, likelihoods, priors, systematics, and a fixed evidence date.

Across the sequence, independent checks include the $Y_e^2$ supported-mass scaling, regular homologous flow, covariant shock jumps, detailed balance in an equilibrium transport cell, energy and lepton closure, matched-resolution shock trajectories, and synthetic-signal recovery. Passing one check does not validate the others.

Review the chapter

A stalled-shock model has $M_g=0.015\,M_\odot$, $\dot M=0.10\,M_\odot\,{\rm s}^{-1}$, $|E_{\rm bind,g}|=3.0\times10^{50}\,{\rm erg}$, and $\dot Q_g=1.5\times10^{51}\,{\rm erg\,s}^{-1}$. Compute the advection and heating times and state whether the diagnostic favors revival.

Solution

The estimates are

$$\tau_{\rm adv}=\frac{0.015}{0.10}\ {\rm s}=0.15\ {\rm s}, \qquad \tau_{\rm heat}=\frac{3.0\times10^{50}}{1.5\times10^{51}}\ {\rm s} =0.20\ {\rm s}.$$

Hence $\tau_{\rm adv}/\tau_{\rm heat}=0.75<1$. In this averaged snapshot, matter is advected through the gain layer before absorbing its full binding scale, so revival is not favored. This is not a proof of failure: turbulence changes residence-time distributions, progenitor interfaces change accretion, and both times evolve. A verdict requires sustained shock and total-energy diagnostics at controlled resolution.

References

• Burrows, Adam. “Colloquium: Perspectives on Core-Collapse Supernova Theory.” Reviews of Modern Physics 85 (2013): 245–261. DOI.
• Janka, Hans-Thomas. “Explosion Mechanisms of Core-Collapse Supernovae.” Annual Review of Nuclear and Particle Science 62 (2012): 407–451. arXiv.
• Mezzacappa, Anthony, Eirik Endeve, O. E. Bronson Messer, and Stephen W. Bruenn. “Physical, Numerical, and Computational Challenges of Modeling Neutrino Transport in Core-Collapse Supernovae.” Living Reviews in Computational Astrophysics 6 (2020): 4. DOI.