Open issues of Bethe's analytic supernova-shock theory: the breakout energy, its progenitor dependence, and the crossover between visible and failed supernovae

Core-collapse supernovae are thought to explode by the delayed neutrino-heating mechanism: neutrinos radiated by the newborn proto-neutron star reheat the matter behind the stalled bounce shock. In the sixth paper of his analytic series, Bethe (The Supernova Shock. VI., ApJ 473, 343, 1996) estimated that the shock breaks out at about 300 km, powered by the recombination of free nucleons into alpha particles, carrying roughly $10^{51}$ erg accumulated in the convective gain region, and conjectured that this energy depends only weakly on progenitor mass - implying a crossover progenitor mass $M_1$ beyond which the net energy is negative, the star gives no visible supernova, and a black hole forms. He left five linked issues open: the neutrino luminosity and heating history; the two-dimensional fate of infalling matter at the stable proto-neutron-star atmosphere; the robustness of the convective-region mass to the strong-shock and 1 MeV assumptions; the envelope-binding and fallback corrections that determine the final energy, remnant mass, and $M_1$; and a survey of one-dimensional collapse histories to test the weak progenitor dependence. The neutrino calculation was absorbed into standard radiation-hydrodynamics simulations; the multidimensional and mass-estimate questions were superseded by two- and three-dimensional simulations that compute the gain region directly; the progenitor survey was carried out and refuted weak dependence in its simple form, replacing it with a structure-dependent landscape of explodability. That landscape, however, comes from calibrated one-dimensional and parameterized population studies (O'Connor & Ott 2011; Pejcha & Thompson 2015; Ertl et al. 2016; Sukhbold et al. 2016): they fix which progenitors explode and estimate explosion energies and remnant masses, but they are not validated against multidimensional simulations or failed-supernova observations, and so do not by themselves deliver the validated mapping of final kinetic energy, remnant masses, and the neutron-star/black-hole boundary that the surviving objective requires. What remains open is the source's central objective: a validated, quantitative mapping from progenitor structure to explosion outcome - final kinetic energy, remnant masses, and the neutron-star/black-hole crossover - including the mantle-binding and fallback corrections, a mapping that the field's 2021 review still flags as the unsolved detail of the explosion mechanism.

Background

A massive star ends its life when its iron core, no longer supported by electron degeneracy pressure, exceeds the Chandrasekhar mass (about 1.45 solar masses, the stability limit of a degenerate electron-supported core) and collapses gravitationally. The inner part of the core forms a proto-neutron star (PNS), a hot, compact object tens of kilometers in radius; the stiffening of nuclear matter at supranuclear density drives a bounce shock into the still-infalling outer core. This prompt shock stalls within a few tens of milliseconds because it loses energy to the photodisintegration of infalling nuclei and to neutrino emission. The favored revival route is delayed neutrino heating (Bethe & Wilson 1985): the PNS radiates its gravitational binding energy as a seconds-long burst of neutrinos (the neutrino sphere is the surface of last scattering from which they escape), and a small fraction of that flux is reabsorbed by free nucleons behind the stalled shock. The gain radius marks where the net neutrino energy exchange changes sign; between the gain radius and the shock, matter gains energy, and the resulting entropy gradient (entropy decreasing outward) makes this gain layer convectively unstable. Two-dimensional hydrodynamic simulations (Herant et al. 1994; Burrows, Hayes & Fryxell 1995; Janka & Müller 1995) showed that this overturn strengthens neutrino energy deposition and can restart the shock. Bethe cast the mechanism into a series of analytic papers, The Supernova Shock (Papers I-VIII, 1979-1997), alongside his review of supernova mechanisms (Bethe 1990).

The source, The Supernova Shock. VI. (Bethe 1996), analyzes the moment the shock breaks out of the stalled phase. Bethe divides the star just before breakout into four regions: N, the PNS inside the neutrino sphere; A, a stably stratified atmosphere above the PNS in which entropy increases outward and which he argues acts as a barrier to accretion; C, the convective, neutrino-heated gain region between A and the shock; and F, the freely infalling matter outside the shock. His companion paper (Bethe 1996, ApJ 469, 737, Paper V) argued that breakout occurs at a shock radius of about 300 km, triggered when the postshock temperature falls to about 1 MeV and free nucleons recombine into alpha particles, releasing roughly 8 MeV per nucleon of nuclear binding; this recombination energy is the dominant reservoir of the hot bubble. In the source, the specific internal energy of region-C matter at breakout is $e_t = D + e_{\rm th} - V_g \approx 3.8\times10^{18}$ erg per gram, where $D \approx 8.2\times10^{18}$ erg per gram is the recombination energy per gram (measured relative to a standard oxygen-16 nucleus), $e_{\rm th} \approx 2\times10^{18}$ erg per gram is the thermal energy of matter and radiation, and $V_g \approx 6.4\times10^{18}$ erg per gram is the gravitational binding at 300 km around a PNS of 1.5 solar masses (baryonic, i.e. counting nucleons; the gravitationally measured mass is smaller by the binding energy). The mass accumulated in region C, $M_C \approx 1.8\times10^{32}$ g (about 0.09 solar masses), follows from the Rankine-Hugoniot jump conditions (the conservation relations for mass, momentum, and energy across a shock discontinuity) in the strong-shock limit, written in the simplified Hugoniot form $P_2 \approx \rho_1 v_1^2$ (postshock pressure set by the infall density $\rho_1$ and velocity $v_1$), evaluated on a one-dimensional infall profile quoted from contemporary collapse calculations and closed with the assumptions $T_{\rm shock} \approx 1$ MeV and a temperature span $T_{\max}/T_{\min} \approx 2$ across region C. The product gives a shock energy of about 0.7 foe (one foe, now often called one Bethe, is $10^{51}$ erg, the standard supernova unit), roughly half the observed kinetic energy $1.4\pm0.4$ foe of SN 1987A (Bethe & Pizzochero 1990).

Two conjectures make the estimate consequential. First, because the infall conditions at 300 km are set mainly by the Chandrasekhar-mass iron core rather than by the extended envelope, Bethe conjectured that the original shock energy depends only weakly on the progenitor star. Second, since the binding energy of the progenitor envelope outside about 2 solar masses grows steadily with progenitor mass (Woosley & Weaver 1995), the net final energy should become negative beyond a crossover mass $M_1$: such stars would give no visible supernova, would still emit a comparable neutrino signal, and would collapse to black holes. The neutron-star mass should be close to the enclosed mass at the entropy maximum (the mass coordinate where the post-bounce entropy profile peaks; about 1.5 solar masses baryonic), with late fallback of infalling matter onto the remnant (estimated at about 0.15 solar masses by Chevalier 1993) able to push marginal events over the neutron-star maximum mass. Modern work has replaced the analytic framework with radiation-hydrodynamics simulations using energy-dependent, so-called multi-group neutrino transport in two and three dimensions, and with population studies of explodability across progenitor libraries; the crossover question survives as the problem of mapping progenitor structure onto explosion outcome.

Problem Statement

The problem, as posed by the source, is the validation and quantitative completion of Bethe's analytic theory of the energy delivered to the revived supernova shock. Its five components are linked by this single objective - a quantitative account of what sets the explosion energy of a core-collapse supernova and which progenitors explode rather than collapse to black holes:

(1) Neutrino input. Compute the neutrino luminosity and heating history from the proto-neutron star that supplies the energy later released by nucleon recombination in the convective region - a calculation the source deliberately defers as a separate problem, leaving the thermodynamic evolution of the convective region hostage to assumed neutrino fluxes and spectra.

(2) Multidimensional accretion. Determine, by two-dimensional calculation carried to a shock radius of about 300 km, whether infalling blobs arrive at the surface of the stably stratified atmosphere (region A) cold and penetrate through it to the neutron star, or are neutrino-heated in the convective region and assembled on top of region A - the answer controls the mass and entropy budget of the convective region and the accretion luminosity, and the source notes it may be sensitive to details not yet known.

(3) Robustness of the convective-region mass. Assess quantitatively how the accumulated mass $M_C$ (about $1.8\times10^{32}$ g) changes when the simplifying assumptions are relaxed: the strong-shock Hugoniot form $P_2 \approx \rho_1 v_1^2$, the fixed postshock temperature of about 1 MeV, and the uncertain temperature span $T_{\max}/T_{\min}$ across the convective region.

(4) Energy corrections and the crossover. Evaluate the corrections to the breakout energy - chiefly the binding energy of the progenitor envelope exterior to the mass coordinate of the ejecta and the late-time fallback onto the compact remnant, both dependent on progenitor structure - in order to determine the final observable kinetic energy, the remnant mass, and the crossover progenitor mass $M_1$ above which the net energy is negative, the supernova fails, and the star collapses to a black hole (with a neutrino signal still expected).

(5) Progenitor survey. Test the claimed weak dependence of the original shock energy on progenitor mass against a set of one-dimensional collapse histories for a range of progenitor models, up to the time the shock reaches about 300 km.

External work has already discharged part of this load: the neutrino-luminosity calculation has been absorbed into standard radiation-hydrodynamics methodology (component 1); the multidimensional fate of infalling matter and the gain-region mass are now computed directly by multi-dimensional simulations, superseding the analytic estimates of components 2 and 3; and the progenitor survey of component 5 has been carried out, with the weak-dependence conjecture refuted in its simple form. What survives as the live target is the central objective as instantiated by component 4: the validated, quantitative mapping from progenitor structure to explosion outcome - final kinetic energy, remnant masses, and the neutron-star/black-hole crossover that generalizes $M_1$ - including the envelope-binding and fallback corrections.

References

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