Rotation, Convection, and Dense-Matter Composition Effects on the Proto-Neutron-Star Collapse Threshold and Black-Hole Formation in Failed Core-Collapse Supernovae

When a massive star's core collapses to a proto-neutron star (PNS) that fails to launch a supernova explosion, the PNS accretes matter until it exceeds the maximum gravitational mass that hot, lepton-rich matter can support and collapses to a black hole (BH). Schneider et al. (2020) established, in spherical symmetry with nucleonic equations of state (EOSs) and an approximate relativistic gravitational potential, that BH formation occurs essentially when the PNS gravitational mass crosses the maximum mass $M_{\mathrm{grav}}^{\max}(\tilde{s})$ of a hot neutron star at constant specific entropy $\tilde{s}$, and that the PNS trajectory in gravitational-mass–entropy space is nearly EOS independent. The open problem, stated by the source and unresolved as of 2026, is to determine how this collapse-threshold picture changes when its simplifications are removed: how stellar rotation, multi-dimensional convection and magnetohydrodynamics, and additional dense-matter degrees of freedom (muons, pions, hyperons, quark phase transitions) alter the PNS mass–entropy trajectory, the maximum supported hot-PNS mass, the BH formation time and initial mass, and the boundary between collapse and explosion — and how accurately approximate effective-gravity treatments track full general relativity through late collapse. The multidimensional and composition-dependent completion of this threshold picture remains open: existing work covers single-progenitor case studies, one-dimensional quark-phase-transition maps, and parameterized long-term mass-ejection models, but no systematic quantification of the rotation/convection/EOS-composition corrections to the collapse threshold or a full-GR calibration of effective gravity in the late-collapse regime beyond the source's own comparison.

Background

A core-collapse supernova (CCSN) begins when the iron core of a massive star exceeds its effective Chandrasekhar mass and collapses, forming a proto-neutron star (PNS) — a hot, lepton-rich — containing an excess of electrons and electron neutrinos trapped during collapse — newborn neutron star with a gravitational mass of roughly 1.2 to over 2 solar masses. The explosion is driven, if at all, by neutrino heating behind the stalled accretion shock. In failed CCSNe the shock is never revived: the PNS accretes the infalling stellar mantle until it becomes gravitationally unstable and collapses to a stellar-mass black hole (BH). The dense-matter equation of state (EOS) — the thermodynamic relation among pressure, energy density, temperature, and composition at supernuclear density — controls how much mass the hot PNS can support. Because the PNS interior evolves through a sequence of quasi-equilibrium states parameterized by its gravitational mass $M_{\mathrm{grav}}^{\mathrm{PNS}}$ and specific entropy per baryon $\tilde{s}$, one can define $M_{\mathrm{grav}}^{\max}(\tilde{s})$, the maximum gravitational mass of a hot neutron star at fixed entropy, and compare it with the PNS trajectory in the $M_{\mathrm{grav}}^{\mathrm{PNS}}$–$\tilde{s}$ plane. Schneider et al. (2020) simulated a large matrix of failing CCSNe in spherical symmetry (over 50 presupernova progenitors and dozens of Skyrme-type and realistic EOSs) and found that BH formation occurs, to within about 1%, once $M_{\mathrm{grav}}^{\mathrm{PNS}}$ exceeds $M_{\mathrm{grav}}^{\max}(\tilde{s})$, while the trajectory itself is well correlated with the progenitor compactness $\xi_{M} = (M/M_{\odot})/(R_{M}/1000\,\mathrm{km})$, where $R_{M}$ is the radius enclosing a baryonic mass $M$, evaluated at $M = 2.5\,M_{\odot}$, and nearly EOS independent. This suggests a route to constrain the temperature dependence of the dense-matter EOS through the neutrino signal of a future failed Galactic supernova, in which an abrupt termination of the neutrino emission marks BH formation. The result rests on four simplifications, each identified by the authors as an open frontier. First, the progenitors were non-rotating and spherically symmetric, whereas real cores rotate and CCSNe are fundamentally multi-dimensional: rotation adds an angular-momentum axis to the mass–entropy diagram (rotating stars can support substantially more mass), and convection and turbulence inside the PNS can remove the negative entropy gradients built by accretion, possibly shifting both the collapse threshold and the $M_{\mathrm{grav}}^{\max}(\tilde{s})$ relations. Second, gravity was modeled with an effective relativistic potential (the 'Case A' prescription of Marek et al. 2006) rather than a dynamical spacetime; Schneider et al. found it underestimates the BH formation time by roughly 4% for the most compact progenitors and up to about 16% for the least compact ones when compared with full general relativity (their GR1D code, a one-dimensional general-relativistic hydrodynamics code), leaving the residual accuracy of approximate gravity in the late collapse regime unsettled. Third, the EOSs studied were essentially nucleonic, whereas at the temperatures and densities inside PNSs additional degrees of freedom are expected: muons (heavy leptons produced by neutrino interactions), pions and hyperons (baryons containing strange quarks, which soften the EOS — the classic 'hyperon puzzle' that hyperons lower the maximum neutron-star mass), and first-order phase transitions to deconfined quark matter. These could lower the maximum hot-PNS mass — Oertel et al. (2016) found the hyperonic LS220+Λ EOS gives $M_{\mathrm{grav}}^{\max}(\tilde{s}=4\,k_{\mathrm{B}}\,\mathrm{baryon}^{-1})$ below the cold maximum, unlike most nucleonic EOSs — and could even produce outcomes absent from the studied EOS sets, such as a strong early neutrino- or magneto-rotational explosion whose remnant PNS still collapses to a BH lighter than the maximum cold neutron-star mass. Fourth, the final BH mass is not fixed at the moment of collapse: the star's outer layers respond over milliseconds to years to the gravitational mass the PNS loses in neutrinos, partially unbinding the envelope, so the ultimate BH mass depends on the long-term accretion and ejecta history, which the source simulations did not follow. Hyperons here mean baryon octet members ($\Lambda$, $\Sigma$, $\Xi$) heavier than nucleons; a first-order phase transition means a discontinuous change in matter properties at a critical density, modeled for instance by hybrid EOSs that switch from hadronic to quark matter. The observational stakes are concrete: the two-solar-mass pulsar mass measurements require any EOS (including its hot sector) to support at least about $2\,M_{\odot}$ cold, and a future Galactic failed supernova would be detected by neutrino and possibly gravitational-wave observatories, whose interpretation depends directly on how BH formation time and PNS evolution depend on rotation, dimensionality, and EOS composition.

Problem Statement

Determine, quantitatively and against the one-dimensional nucleonic baseline of Schneider et al. (2020), how the removal of that baseline's four simplifications changes the collapse threshold of failing core-collapse supernovae. Specifically: (1) how stellar rotation and multi-dimensional convection, turbulence, and magnetohydrodynamics modify the proto-neutron-star trajectory in gravitational-mass–specific-entropy space, the maximum gravitational mass $M_{\mathrm{grav}}^{\max}(\tilde{s})$ supported by hot dense matter, the black-hole formation time and initial black-hole mass, and whether the criterion that collapse follows once $M_{\mathrm{grav}}^{\mathrm{PNS}}$ exceeds $M_{\mathrm{grav}}^{\max}(\tilde{s})$ survives in multiple dimensions — including whether these effects can shift individual progenitors across the boundary between black-hole formation and a (neutrino-driven or magneto-rotational) explosion; (2) how additional dense-matter degrees of freedom — muons, pions and hyperons, and first-order transitions to quark matter — modify the temperature and lepton-fraction dependence of $M_{\mathrm{grav}}^{\max}(\tilde{s})$ and the collapse-versus-explosion outcome when included self-consistently in proto-neutron-star evolution, in particular whether for any realistic EOS and progenitor the maximum hot proto-neutron-star mass falls enough to produce outcomes absent from the nucleonic picture (such as an exploded star whose remnant still collapses to a black hole lighter than the maximum cold neutron-star mass); (3) how accurately effective-potential (approximate general-relativistic) gravity treatments reproduce full general relativity through the late collapse and black-hole-formation regime beyond the single-EOS, one-dimensional comparison of Schneider et al. (2020), across the progenitor compactness range, EOS variants, and multidimensional or rotating conditions; and (4) how the final black-hole mass is set by the interplay of the early collapse trajectory, neutrino-driven gravitational-mass loss and envelope unbinding, and fallback accretion over the milliseconds-to-years aftermath, beyond the one-dimensional parameterized treatment already available. Answers should identify which effects are degenerate with, and which are separable from, progenitor compactness, so that a future Galactic failed-supernova neutrino and gravitational-wave detection can still be inverted for the equation of state. Each of the four parts (1)–(4) is an independent sub-question of this single objective. Where a part lists several effects (as in parts (1) and (2)), the list delimits the class of admissible effects.

References

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