Bethe's Bakerian Lecture (1993) reduced the delayed neutrino mechanism of core-collapse supernovae to a quantitative analytic model resting on three self-acknowledged conjectures: that convection equalizes the entropy of the neutrino-heated region; that about 0.04 M⊙0.04\,M_\odot falls back onto the newborn neutron star under assumed uniform density and radius-proportional velocity; and that kaon condensation limits the neutron-star maximum mass to about 1.5 M⊙1.5\,M_\odot, so that 1818-30 M⊙30\,M_\odot progenitors leave black holes. Later literature has refuted the kaon conjecture — two-solar-mass pulsars and GW170817 exclude such a soft high-density equation of state, and no dominant low-mass black-hole population is observed — and has superseded the fallback and convection estimates with systematic one- and three-dimensional simulations that now robustly explode many progenitors. The surviving open core is the fully predictive, first-principles mapping from progenitor structure to explosion energy, fallback mass, and compact-remnant mass and fate, including a closure theory for neutrino-driven turbulent convection; recent work on explosion criteria and remnant-mass distributions explicitly continues this problem, which remains open.

Background

Massive stars (roughly ≥10 M⊙\geq 10\,M_\odot at birth, where M⊙M_\odot denotes the mass of the Sun) end their lives as core-collapse supernovae: the iron core grown at the center exceeds the Chandrasekhar mass (the maximum mass an electron-degenerate configuration can support, about 1.4 M⊙1.4\,M_\odot), loses pressure support through electron capture and photodisintegration of iron, and collapses. Collapse halts when the inner core reaches nuclear density, launching a shock that stalls almost immediately because dissociating iron into free nucleons drains its energy. Nearly all of the released gravitational energy (about 3×10533\times10^{53} erg) leaves the newly formed proto-neutron star as neutrinos. In the delayed neutrino mechanism proposed by Bethe and Wilson (1985), a small fraction (of order 1%) of this neutrino energy is reabsorbed behind the stalled shock — mainly by charged-current absorption of electron neutrinos and antineutrinos on free nucleons in the gain region, the layer where neutrino heating exceeds cooling — eventually reviving the shock and ejecting the stellar mantle with the canonical 105110^{51} erg (1 'foe') of kinetic energy. The neutrino detections from SN 1987A, the supernova that exploded in the Large Magellanic Cloud in February 1987, at the Kamiokande-II and Irvine-Michigan-Brookhaven (IMB) water-Cherenkov neutrino detectors, remain the only direct neutrino observations of a supernova (Hirata et al. 1987; Bionta et al. 1987).

Bethe's Bakerian Lecture (delivered 1993, published 1994) compressed this mechanism into a transparent analytic model: from the SN 1987A neutrino events it inferred a νe+νˉe\nu_e+\bar\nu_e luminosity of L=3×1052L=3\times10^{52} erg s−1^{-1} averaged over the first 2 s; assuming hydrostatic equilibrium and convection-driven entropy equalization in the nucleon region it derived the temperature profile T=2.89/r7−0.39T=2.89/r_7-0.39 MeV (r7r_7 = radius in units of 100 km); a gain radius Rg=136R_g=136 km; a total neutrino-absorption efficiency of 2.14% (about 1.4% effectively transferred to the shock, with convection credited for part of the transfer); and a total explosion energy of 1.2×10511.2\times10^{51} erg, consistent with the observed 1.4±0.41.4\pm0.4 foe of SN 1987A. The lecture then followed the compact-remnant consequences: after accretion onto the proto-neutron star ceases, the neutrino luminosity drops and the inner part of the shocked matter, losing pressure support, falls back; assuming uniform density in the shocked region and outward velocity proportional to radius, the point where kinetic plus gravitational energy vanishes lies at r=0.4Rsr=0.4R_s (with RsR_s the shock radius) and about 0.04 M⊙0.04\,M_\odot falls back onto the neutron star. Invoking the scenario of Brown and Bethe (1994), in which kaon condensation — the formation of a Bose condensate of negatively charged K−K^- mesons in dense nuclear matter that replaces electrons as carriers of negative charge, softens the equation of state (the pressure-density relation) above nuclear density, and lowers the maximum stable neutron-star mass to about 1.5 M⊙1.5\,M_\odot — the lecture conjectured that compact cores of stars with initial main-sequence masses 1818-30 M⊙30\,M_\odot exceed this limit after the explosion and collapse to black holes, while lighter progenitors leave neutron stars.

Each load-bearing step was flagged by the lecture itself as conjecture: the fallback estimate rests on assumed density and velocity profiles; the black-hole channel rests on the then-unconfirmed kaon-condensation hypothesis for the dense-matter equation of state; and the convection treatment rests on the assumption that turbulent mixing equalizes the entropy of the neutrino-heated region, with the actual efficiency of turbulent mixing and its effect on energy transport not quantified from first principles. These three conjectures are not independent questions: the convective transport efficiency fixes the explosion energy, which together with the progenitor's binding-energy profile fixes the fallback, which together with the equation-of-state-limited maximum neutron-star mass fixes the final remnant mass and whether it is a neutron star or a black hole.

Problem Statement

For the delayed neutrino mechanism of core-collapse supernovae as formulated in Bethe's Bakerian Lecture (1993), determine — from first principles and/or observation — the physics that the 1994 analytic model had to assume, with the common objective of predicting the compact-remnant outcome of a massive star. Specifically: (i) the efficiency of convective/turbulent mixing and energy transport in the neutrino-heated region, replacing the isentropic (entropy-equalizing) assumption — how much of the deposited neutrino energy is transported to the shock, and how does this depend on the neutrino luminosity and the progenitor structure? (ii) The amount and late-time dynamics of the mass that falls back onto the proto-neutron star after shock revival, replacing the uniform-density, velocity-proportional estimate of 0.04 M⊙0.04\,M_\odot, as a function of progenitor structure and explosion energy. (iii) The equation-of-state limit on the maximum stable neutron-star mass and the resulting boundary between neutron-star and black-hole formation across the progenitor mass range, given the established refutation of the kaon-condensation conjecture that predicted black-hole remnants for 1818-30 M⊙30\,M_\odot progenitors; reproducing that refutation alone does not satisfy this sub-question. These three questions together are the progenitor-to-remnant mapping that the lecture's program left open and that must be tested against observation: the convective transport efficiency fixes the explosion energy, which together with the progenitor binding-energy profile fixes the fallback, which together with the maximum neutron-star mass fixes the final remnant mass and its identity as a neutron star or black hole. The resulting progenitor-to-remnant mapping is to be confronted with the observed neutron-star and black-hole mass distributions and with individual remnants — including the compact object in SN 1987A, for which JWST spectroscopy has provided evidence (an unresolved, blueshifted source ionizing the innermost ejecta) without yet settling whether it is a pulsar with a wind nebula or a cooling neutron star (Larsson et al. 2025).

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