The revival of the stalled bounce shock by neutrino heating, aided by neutrino-driven convection and the standing accretion shock instability (SASI), is the standard model for ordinary core-collapse supernova explosions, and two-dimensional parametric simulations show that explosion requires the formation and long-enough survival of at least one large-scale, high-entropy bubble in the gain region (Fernández, Müller, Foglizzo & Janka 2014). That work left four interrelated questions open: the quantitative multi-dimensional explosion threshold (what fraction of the gain-region volume or mass must exceed given entropy or specific-energy values, and how those thresholds depend on heating rate, accretion rate, nuclear dissociation, and progenitor structure); the detailed lifecycle of the large bubbles (seeding, growth, survivability, and turbulent shredding); the generalization from two to three dimensions, where the inverse turbulent cascade of 2D is absent; and a robust predictive explosion criterion combining this phenomenology with the mean-flow convection parameter. Later literature resolved part of the dimensionality question within the same idealized setup (Fernández 2015: SASI-dominated models explode with up to about 20% lower heating rate in 3D; convection-dominated models differ by less than 8%), and produced turbulence-modified critical conditions and calibrated 1D turbulence-aided surrogates. However, the volume/mass-fraction thresholds have never been quantified as functions of the system parameters, no quantitative theory of the bubble lifecycle exists, and the renewed first-principles critical-condition program remains one-dimensional and explicitly preparatory for multi-dimensional experiments (Pochik & Thompson 2025). The problem — a quantitative, validated characterization of the multi-dimensional explosion threshold and a predictive criterion valid across SASI- and convection-dominated regimes and transferable to full physics — remains open.

Background

A core-collapse supernova begins when the iron core of a massive star collapses to a proto-neutron star. The stiffening of the nuclear equation of state launches a shock at bounce, but the shock stalls within tens of milliseconds because energy is lost to neutrino emission and to nuclear dissociation of infalling material. In the prevailing neutrino mechanism, a small fraction of the neutrinos streaming from the hot proto-neutron star is reabsorbed behind the shock: in the gain region — the shell between the gain radius, where net neutrino heating begins, and the stalled shock — matter is heated, and if heating is strong enough the shock runs away and explodes the star. In spherical symmetry this transition has a classical characterization: above a critical neutrino luminosity no stationary accretion solution exists (Burrows & Goshy 1993), equivalently expressed as the ratio of the advection time of matter through the gain region to the time needed to unbind the gain region by heating.

In multiple spatial dimensions two hydrodynamic instabilities operate in the gain region. Neutrino-driven convection is buoyant overturn fed by the negative entropy gradient that heating establishes. The standing accretion shock instability (SASI) is a global oscillatory instability of the stalled shock, fed by the advective-acoustic cycle: sound waves generated by advected post-shock perturbations reflect and amplify at the shock and at the proto-neutron-star surface. A useful regime discriminator is the convection parameter χ\chi, the ratio of the advection time through the gain region to the buoyancy growth time: χ>3\chi>3 favors convection, χ<3\chi<3 leaves the flow SASI-dominated. Fernández, Müller, Foglizzo & Janka (2014) studied the approach to explosion with two-dimensional, time-dependent parametric simulations of the stalled shock — a light-bulb-like setup with a simple equation of state including nuclear dissociation at a rate set by a parameter ε\varepsilon, constant volumetric heating at a rate BB, and steady accretion, for which the linear stability properties are known — complemented by selected general-relativistic radiation-hydrodynamic 2D models. They found that in both regimes explosion requires the formation of at least one large-scale, high-entropy bubble that survives long enough to expand against the ram pressure of the infalling material; they tracked bubbles with the diagnostic fV(s>s0)f_V(s>s_0), the fraction of the post-shock volume with entropy above a threshold s0s_0, and showed that sudden decreases of fVf_V (bubble destruction) precede large-amplitude shock sloshings. They also found that the initial value of χ\chi predicts which instability will dominate, and introduced χˉ\bar{\chi}, the same parameter computed from time-averaged flow quantities, which behaves differently from a naive average because χ\chi is a nonlinear function.

Two features of the two-dimensional setting shape the open questions. First, 2D turbulence possesses an inverse cascade: kinetic energy flows toward the largest angular scales, which helps convective bubbles merge into the large structures that the explosions need; in 3D the cascade runs forward to small scales, so bubble merging and survival must be re-examined. Second, the parametric setup omits physics known to matter — the negative feedback on the heating rate from the drop in accretion luminosity when the shock expands, time-dependent accretion, and a realistic equation of state with nuclear recombination — so thresholds measured in it must be treated with caution before becoming an explosion criterion. The source paper closed by posing the four interrelated open questions that define this problem: quantify the multi-dimensional threshold (the volume or mass fraction of the gain region that must reach given entropy or positive specific energy, and the parameter dependence of those thresholds); characterize the bubble lifecycle quantitatively (seeding by shock displacements, survivability against lateral SASI flows and disruption by low-entropy plumes, neutrino-heating and buoyancy feedback, turbulent shredding, and the bubbles' feedback on SASI modes); determine how the absence of the 2D inverse cascade modifies the picture in 3D; and combine this phenomenology into a robust, predictive explosion criterion valid for both SASI- and convection-dominated systems and transferable from parametric to full-physics simulations.

Problem Statement

Determine, quantitatively, how multi-dimensional hydrodynamics set the threshold for shock revival in neutrino-driven core-collapse supernovae, and construct from that understanding a robust predictive explosion criterion. With the gain region defined as the shell of net neutrino heating behind the stalled shock, the problem comprises three linked parts:

  1. Threshold quantification. Measure what fraction of the gain-region volume or mass must reach a given specific entropy or a positive specific total energy for runaway shock expansion to ensue, and determine how these fractions and the required entropy/energy thresholds depend on the dominant system parameters — the neutrino heating rate, the mass accretion rate, the nuclear dissociation at the shock, and the proto-neutron-star and progenitor structure — in both the SASI-dominated and the convection-dominated regime.

  2. Bubble lifecycle. Develop a quantitative account of the large-scale high-entropy bubbles whose survival the two-dimensional simulations identified as a necessary condition for explosion: how bubbles are seeded by shock displacements (SASI sloshing or spiral modes, or convective mergers); how neutrino heating, buoyancy, and the turbulent cascade govern their growth; what determines their survivability against lateral SASI flows and disruption by low-entropy plumes; and what sets their breakup rates — including how all of these change in three dimensions, where the inverse turbulent cascade that merges bubbles in 2D is absent.

  3. Predictive criterion. Combine the threshold and bubble phenomenology, together with time-averaged flow quantities such as the mean-flow convection parameter χˉ\bar{\chi}, into an explosion criterion that predicts explosion versus failure across both instability regimes and transfers from parametric simulations to full-physics simulations — accounting at least for the accretion-luminosity feedback on the heating rate, time-dependent accretion, and a realistic equation of state with nuclear recombination.

The three parts are linked as stated by the source: the criterion (part 3) is the integrating product of the threshold measurement (part 1) and the bubble-lifecycle theory (part 2), and the 2D-versus-3D threshold comparison already performed in the idealized setup leaves the problem open.

References

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