When the stalled bounce shock of a collapsing massive star is revived—hundreds of milliseconds or more after bounce—by neutrino heating aided by the standing accretion shock instability (SASI) and convection, the explosion energy is not fixed at the moment of runaway: accretion downflows and neutrino-heated outflows coexist for many hundreds of milliseconds to seconds while the blast builds. Marek and Janka's two-dimensional simulations demonstrated such delayed explosions for 11.2 M⊙11.2\,M_\odot and 15 M⊙15\,M_\odot progenitors but had to be stopped within ~700 ms of bounce, leaving the final explosion energies undetermined and the mechanism's robustness to dimensionality, resolution, microphysics, and competing channels untested. That determination comprises the final explosion energies of delayed neutrino-driven supernovae, their dependence on progenitor structure and uncertain input physics, and the conditions under which the mechanism succeeds or fails. Later work resolved major parts of this problem—three-dimensional first-principles explosions now exist, multi-second models converge to canonical energies of order 105110^{51} erg with quantified energy budgets, and the alternative acoustic mechanism based on proto-neutron-star g-modes is damped below energetic relevance—but the first-principles prediction of which progenitors explode and with what final energy across the stellar mass range, including the still-unsettled role of neutrino flavor conversion, remains open.

Background

A core-collapse supernova ends the life of a massive star (roughly 8 M⊙8\,M_\odot and above on the main sequence). Its core—iron for most of this mass range, but oxygen–neon–magnesium for stars near the ∼8\sim8–10 M⊙10\,M_\odot lower end—collapses to nuclear density, where the nuclear equation of state (EoS, the thermodynamic relation of pressure to density, temperature, and composition for dense matter) stiffens the material and launches a 'bounce' shock; this shock stalls within milliseconds as it loses energy to neutrino emission and nuclear dissociation. The newly formed proto-neutron star radiates several 105310^{53} erg of gravitational binding energy in neutrinos. In the prevailing delayed neutrino-heating mechanism, a small fraction of these neutrinos is reabsorbed in the gain region—the zone between the proto-neutron star and the stalled accretion shock where net heating exceeds cooling—until buoyant runaway revives the shock. Because spherical-symmetric heating is too weak for most progenitors, multi-dimensional fluid instabilities are decisive: neutrino-driven convection, and the standing accretion shock instability (SASI), a global sloshing or spiral oscillation of the shock and postshock flow fed by a feedback cycle in which perturbations advected from the shock to the neutron star couple to sound waves that travel back out to the shock.

Marek and Janka (2009) presented two-dimensional simulations with energy-dependent neutrino transport in the 'ray-by-ray-plus' approximation, in which the transport is solved along radial rays and lateral (non-radial) neutrino fluxes are neglected, closing the moment equations with variable Eddington factors. They found SASI-aided, neutrino-driven explosions beginning about 200 ms after bounce for an 11.2 M⊙11.2\,M_\odot progenitor and about 600 ms after bounce for a 15 M⊙15\,M_\odot progenitor (model s15s7b2), with a softer high-density EoS favoring and rotation disfavoring the explosion. Crucially, their runs ended at most ~700 ms after bounce. The explosion energy is not set at shock revival: accretion downflows continue to feed the gain layer with fresh gas while neutrino-heated, buoyant gas expands outward, and this simultaneous accretion–outflow phase lasts many hundreds of milliseconds to seconds. The final energies—their analytic estimates suggested several 105010^{50} erg for the 11.2 M⊙11.2\,M_\odot star and up to the canonical value of 105110^{51} erg (1 bethe) for the 15 M⊙15\,M_\odot star—could therefore only be determined by much longer computations. They explicitly left open: the sensitivity of the outcome to the high-density nuclear EoS, neutrino opacities and inelastic processes, relativistic gravity (they used an effective relativistic potential rather than full general relativity), rotation, magnetic fields, and neutrino flavor oscillations; the robustness to grid resolution, stochastic SASI fluctuations, and the two-dimensional approximation itself, since three dimensions admit non-axisymmetric SASI spiral modes absent in 2D; and a discrepancy with contemporaneous simulations by Burrows et al. (2006), in which the shock was revived only after more than a second by acoustic power from large-amplitude gravity-mode (g-mode) oscillations of the proto-neutron star—oscillations whose restoring force is buoyancy—rather than by neutrino heating. Whether such g-modes could reach dynamically relevant amplitudes at post-bounce times beyond about a second, under any physical or progenitor conditions, was unresolved; Weinberg and Quataert (2008) had argued that nonlinear coupling of the primary g-mode to higher-order modes damps it far below explosion requirements, but the damping estimate and its applicability were not settled by observation or converged simulation.

Problem Statement

Determine the final explosion energies of delayed, SASI- and convection-aided neutrino-driven supernova explosions—the genuinely asymptotic energies reached after the entire simultaneous, multi-second accretion–outflow phase, once accretion has ceased—and establish the robustness and domain of validity of this mechanism. The open problem is the completion of this determination for iron-core progenitors (roughly 1010–25 M⊙25\,M_\odot on the main sequence; the ∼8\sim8–10 M⊙10\,M_\odot oxygen–neon–magnesium stars, which may explode via electron capture rather than by iron-core collapse, form a distinct channel outside this scope), on the finite model set comprising Marek and Janka's 11.2 M⊙11.2\,M_\odot (s11.2) and 15 M⊙15\,M_\odot (s15s7b2) progenitors together with any one published, solar-metallicity, non-rotating iron-core progenitor of about 25 M⊙25\,M_\odot. For each model of that set, the first-principles determination comprises: (1) whether the star explodes or collapses to a black hole and, if it explodes, its asymptotic final explosion energy and its energy budget (neutrino heating, nucleon recombination, accretion power, overburden and fallback); and (2) the sensitivity of the onset time and of the final energy to the high-density nuclear equation of state, taken as any two standard finite-temperature equations of state of differing high-density stiffness, and to the dimensionality (two versus three dimensions, including non-axisymmetric SASI spiral modes). Together, (1) and (2) determine whether the delayed neutrino-driven mechanism produces the observed explosion energies of roughly 105010^{50} to 2×10512\times10^{51} erg within this model set and where it fails.

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