Core Bounce and Shock Formation

Core bounce converts the subsonic inner core’s infall into an outward pressure wave when the nuclear equation of state stiffens; that wave steepens into a shock near the sonic boundary. The launch is violent but not normally an immediate explosion. Dissociation of heavy nuclei, neutrino losses, continued accretion, and the gravitational binding of overlying matter drain the shock until it stalls. Establishing this sequence quantitatively is a coupled hydrodynamics, gravity, equation-of-state, and weak-interaction benchmark.

Helpful background. Stress–Energy and Conservation supplies flux balances; the Einstein equations supply relativistic gravity; Relativistic Matter and Compact Objects supplies the finite-temperature equation of state; Black-Hole Dynamics supplies generic collapse geometry; Computational Relativity supplies conservative shock evolution; and Black-Hole Perturbation Theory supplies stability diagnostics.

Continuum problem at bounce

A minimal spherical calculation evolves baryon number, stress–energy, and electron fraction,

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

where $\rho=m_b n_b$ is rest-mass density, $T^{\mu\nu}$ is the material stress tensor, and ${\cal R}_{Y_e}=u^\mu\nabla_\mu Y_e$ is a rate per fluid proper time. The equations use a finite-temperature composition-dependent closure $P=P(\rho,T,Y_e)$ and $\epsilon=\epsilon(\rho,T,Y_e)$. The neutrino–matter exchange four-vector $Q^\nu_{\rm weak}$ and lepton rate ${\cal R}_{Y_e}$ must be derived from a declared weak-interaction and transport approximation. The neutrino stress tensor has the opposite source, $\nabla_\mu T_{\rm rad}^{\mu\nu}=-Q^\nu_{\rm weak}$, so the combined system conserves stress–energy. Inputs also include the precollapse progenitor profile, nuclear equation-of-state table and version, gravity model, outer accretion boundary, atmosphere treatment, and definitions of bounce and shock position.

Hydrodynamic conserved variables and fluxes must be tied to a specified Eulerian frame. A dimension check must restore one power of mass per baryon in specific energies and one power of inverse length in rates. Across a geometrically thin shock whose internal sources are negligible, the covariant jump conditions are

$$\left[\rho u^\mu s_\mu\right]=0,\qquad \left[T^{\mu\nu}s_\mu\right]=0,$$

where $s_\mu$ points from upstream to downstream and $[A]=A_{\rm down}-A_{\rm up}$. These relations independently test the numerical compression front’s mass, momentum, and energy fluxes.

Nuclear stiffening and shock launch

During infall, subnuclear matter is soft enough for the homologous inner core to accelerate inward. As the center approaches nuclear density, short-range nuclear repulsion and the symmetry-energy response stiffen the pressure. The inner core decelerates and rebounds. The outer core is still supersonically infalling, so the pressure wave steepens near the sonic point into a shock. The shock initially encloses approximately the homologous-core mass, not the entire iron core.

The exact peak density and rebound strength depend on the finite-temperature equation of state, trapped lepton fraction, general-relativistic gravity, rotation, and progenitor structure. A true bounce diagnostic therefore combines a central-density maximum, a velocity reversal in the inner core, and formation of a discontinuity satisfying the jump conditions. A local pressure spike alone can be an interpolation or reconstruction artifact. Janka describes the inner-core rebound and shock formation sequence in Janka 2012, §IV.B and Fig. 4.

The Newtonian weak-field limit provides a useful analytic check: the momentum equation must reduce to Euler flow with gravitational acceleration $-Gm(r)/r^2$, and the shock must propagate outward in radius while upstream matter continues inward. A homologous test should recover $v\propto r$ before bounce and a regular center.

Why the prompt shock stalls

As the shock enters iron-rich matter, it heats and dissociates nuclei. The binding energy per baryon is of order several MeV, so processing even a modest mass consumes energy on the scale of a supernova explosion. Electron captures on newly liberated protons produce a prompt electron-neutrino burst once the shock crosses matter from which neutrinos can escape. Continued accretion adds ram pressure and continually presents fresh bound material. Burrows compares these sinks and the resulting stalled-shock configuration in Burrows 2013, §§II.C–II.E, Fig. 3.

An energy budget should distinguish kinetic, internal, gravitational, nuclear binding, and neutrino terms using one zero of energy. Mixing equation-of-state tables or reaction networks with different nuclear mass conventions can create a spurious gain or loss. The shock’s positive kinetic and thermal energy after launch is not the asymptotic explosion energy: it must still pay dissociation and the binding energy of overlying layers. Consequently, “shock formed”, “shock reached a large radius”, and “unbound explosion” are separate diagnostics.

Benchmark and convergence tests

A computational result should begin from a published spherical progenitor and pin the code commit, compiler and dependency versions, floating-point precision, gravity formulation, equation-of-state table checksum, weak-rate set, hydrodynamic reconstruction and Riemann solver, radial grid, time-step control, and atmosphere. A suitable reference benchmark reports bounce time, central density, homologous-core mass, shock-formation radius and enclosed mass, and baryon, energy, and lepton-number residuals. Mezzacappa and collaborators describe the coupled physical and numerical problem in Mezzacappa et al. 2020, §§5.3, 6.1.3, and 6.5.4.

Convergence requires more than a finer radial grid. Refine space and time, reconstruction order, gravity tolerance, and equation-of-state interpolation; compare shock position at matched physical times; and state error norms and acceptance tolerances before inspecting the answer. Key outputs should approach stable values, while conservation residuals decrease at the expected order away from the discontinuity. Cross-check with an independently implemented homologous-collapse or spherical benchmark.

Each failure mode must be diagnosed explicitly. Important examples include artificial-atmosphere heating, primitive-variable recovery outside the equation-of-state domain, inconsistent binding-energy offsets, entropy or lepton drift, a numerically broadened unresolved sonic point, and mistaking a compression wave for the physical shock. A moving shock also reduces formal convergence order, so trajectory uncertainty should be quoted rather than concealed.

No simulation was executed for this page, and the page does not provide a software environment or claim to reproduce a published run. A rerun-capable companion artifact would need one documented command that regenerates machine-readable benchmark and refinement tables from the pinned environment and exits unsuccessfully when conservation, admissibility, benchmark, or convergence thresholds are violated.

Check the dissociation scale

Estimate the energy needed to dissociate $0.10\,M_\odot$ of iron-group matter if the mean nuclear binding cost is $8.8\,{\rm MeV}$ per baryon. Use $M_\odot=1.99\times10^{33}\,{\rm g}$, $m_b=1.66\times10^{-24}\,{\rm g}$, and $1\,{\rm MeV}=1.602\times10^{-6}\,{\rm erg}$.

Solution

The number of baryons is

$$N_b=\frac{0.10(1.99\times10^{33})}{1.66\times10^{-24}} \simeq1.20\times10^{56}.$$

Therefore

$$E_{\rm diss}=N_b(8.8)(1.602\times10^{-6}) \simeq1.69\times10^{51}\ {\rm erg}.$$

This one sink is already of order $10^{51}\,{\rm erg}$. The estimate explains why a newly launched shock can stall, but it is not a complete shock budget: the processed composition, recombination, neutrino exchange, accretion, and gravitational binding must also be included consistently.

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.