Proto-Neutron-Star Evolution and Black-Hole Formation

A proto-neutron star is hot, lepton rich, and often still accreting. Its supported mass is therefore not the cold, beta-equilibrated maximum quoted for an old neutron star. Survival is decided by the intersection of two evolving quantities: the star’s baryon-mass and angular-momentum history, and the finite-temperature, composition-dependent stability limit. Continued accretion, deleptonization, cooling, or fallback can move that intersection toward dynamical collapse and black-hole formation.

Helpful background. Stress–Energy and Conservation supplies mass and energy budgets; the Einstein equations supply relativistic gravity; Relativistic Matter and Compact Objects supplies hot stellar sequences; Black-Hole Dynamics develops horizon formation and generic collapse geometry; Computational Relativity supplies evolution and horizon-finding methods; and Black-Hole Perturbation Theory supplies stability diagnostics.

An evolving hot stellar sequence

The continuum state requires a finite-temperature equation of state

$$P=P(n_b,s,Y_e),\qquad \epsilon=\epsilon(n_b,s,Y_e),$$

for the material component, coupled to general-relativistic hydrodynamics and neutrino transport. Trapped neutrinos carry additional energy, pressure, and lepton number: either their distribution is evolved separately, or an equilibrium closure must supply its neutrino chemical potentials and total lepton fraction. They must not be counted in both the material EOS and a separate neutrino stress tensor. Initial data include the post-bounce density, entropy and composition profiles; accretion history; rotation and magnetic assumptions; and outer boundary.

For a slowly evolving spherical model, one can compare snapshots with hot TOV sequences at fixed entropy and lepton profiles. That comparison is diagnostic, not an exact equilibrium reduction: accretion, shocks, neutrino flux, and composition gradients violate strict barotropy. The relevant supported mass is the maximum for the instantaneous thermal, compositional, rotational, and accretion state. Thermal pressure can increase support in some layers, while trapped neutrinos and composition change both pressure and the equilibrium sequence. Cooling does not imply that the maximum mass moves monotonically in every equation-of-state model.

Burrows describes the proto-neutron-star and failed-supernova evolution in Burrows 2013, §IV.B. A dimension check distinguishes physical mass $M$ from the length $GM$; the compactness is $GM/R$. Any comparison of mass thresholds must also state whether it uses baryonic mass or gravitational mass.

Accretion, cooling, and deleptonization

Baryon mass and total energy answer different bookkeeping questions. A schematic balance is

$$\frac{dM_b}{dt}=\dot M_{\rm acc}-\dot M_{\rm ej},$$

whereas the gravitational mass changes through accreted energy, neutrino luminosity, electromagnetic or gravitational radiation, and ejecta:

$$\frac{dM_g}{dt} =\dot E_{\rm acc}-L_\nu-L_\gamma-L_{\rm GW}-\dot E_{\rm ej}.$$

These are control-volume balances: use the same boundary and time coordinate for every term, and include gravitational redshift when comparing a local luminosity with an energy flux at infinity. In restored units the right-hand side of the second equation is divided by $c^2$. Neutrino diffusion removes energy and lepton number, the star contracts, and its spectra and luminosities evolve. Accretion can simultaneously increase $M_b$ and deposit entropy. Janka reviews the connection between stalled-shock accretion, proto-neutron-star contraction, and possible collapse in Janka 2012, §§V.A and VI.C.

Fallback is the later return of material that was initially moving outward but remained bound or was decelerated by the envelope. It can push a remnant across its stability limit after an explosion. A calculation must therefore extend long enough, or couple consistently to an exterior model, before declaring permanent survival.

Crossing the stability threshold

At each time define a model-dependent supported baryon mass

$$M_{b,\max}=M_{b,\max}[s(m_b),Y_l(m_b),J,B,\mathrm{EOS}],$$

where the bracket emphasizes dependence on profiles rather than only central temperature. Collapse becomes possible when the actual $M_b$ exceeds this limit and no nearby stable configuration with the conserved quantities exists. A turning point along a deliberately constructed equilibrium sequence is a useful warning; dynamical radial-mode analysis or time evolution is the stronger test.

The collapse diagnostic should combine invariant or geometrically controlled signals. Rapid central contraction and lapse collapse are coordinate-sensitive alone. A resolved apparent horizon or trapped surface, together with constraint and conservation monitoring, supplies a physical black-hole diagnostic. Detailed horizon geometry and the subsequent spacetime evolution are treated in Black-Hole Dynamics; this page concerns how microphysics and accretion bring the material remnant to that transition.

Different equations of state can change the hot maximum, contraction history, and threshold time. Rotation adds centrifugal support but accretion and magnetic or hydrodynamic torques change angular momentum. The correct statement is conditional: a named progenitor with a named EOS, transport model, gravity formulation, and resolution crossed a specified diagnostic—not that one zero-temperature maximum mass universally predicts the outcome.

Computational validation

A reference calculation should pin the progenitor and post-bounce mapping, EOS table version and checksum, opacity and weak-rate set, neutrino species and energy grid, gravity formulation, hydrodynamic discretization, atmosphere, outer accretion boundary, grid, time-step and nonlinear tolerances, and the definitions of radius, mass, collapse time, and horizon. The software environment must record code commit, compiler or interpreter, dependency lock, floating-point precision, hardware-relevant solver settings, and deterministic seeds.

A suitable spherical benchmark closes baryon and total-energy budgets while tracking radius, luminosities, mean energies, $M_b$, $M_g$, and contraction. For an accreting case it should recover a published collapse time and enclosed mass within declared uncertainty and verify that the evolving supported-mass threshold is crossed. Mezzacappa and collaborators detail the transport and coupling difficulties that enter such a benchmark in Mezzacappa et al. 2020, §§4.5, 5.3, and 6.5.4.

Convergence studies refine hydrodynamic grid, neutrino energy groups, time step, EOS interpolation, atmosphere floor, gravity solve, and apparent-horizon resolution. Report error norms and tolerances for luminosity, radius, baryon mass, collapse time, enclosed mass, constraints, and horizon location. An independent implementation or matched spherical reference is a valuable cross-check.

Each failure mode must be separated: EOS-table-edge extrapolation, accretion-boundary dependence, missing weak reactions, transport-closure bias, atmosphere mass loading, nonconservative remapping, and a gauge-only collapse indicator can all move the verdict. Unresolved horizon formation is not cured by quoting more digits in collapse time.

No evolution was run for this page. A reproducible companion would expose one documented rerun command that regenerates machine-readable benchmark and refinement tables from a pinned environment and fails when conservation, EOS-admissibility, reference, constraint, horizon, or convergence thresholds are missed.

Check a toy threshold crossing

A model has $M_b(0)=2.20\,M_\odot$ and $M_{b,\max}(0)=2.45\,M_\odot$. Assume constant net accretion $dM_b/dt=0.25\,M_\odot\,{\rm s}^{-1}$ and, in this toy thermal sequence, $dM_{b,\max}/dt=-0.05\,M_\odot\,{\rm s}^{-1}$. When do the two curves meet?

Solution

The initial gap is $0.25\,M_\odot$. It closes at the relative rate

$$\frac{d}{dt}(M_b-M_{b,\max}) =0.25-(-0.05) =0.30\,M_\odot\,{\rm s}^{-1}.$$

Thus

$$t_{\rm cross}=\frac{0.25}{0.30}\ {\rm s}\simeq0.83\ {\rm s}.$$

This is only a bookkeeping forecast. The actual instability time requires profile-dependent equilibria or radial modes, changing accretion, transport, rotation, and a resolved dynamical-collapse diagnostic.

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.