In the final paper of his 'Supernova Shock' series, Hans Bethe (1997) distilled the analytic theory of the revived, neutrino-driven supernova shock and listed the questions it could not settle: the early-time postshock density compression (ρ2/ρ1\rho_2/\rho_1 of order 8) and its dependence on the rate of energy supply; whether the difference between the electron-neutrino and electron-antineutrino fluxes can be small enough to yield the electron fraction YeY_e near 0.495 in the convective zone — above the infalling value of about 0.488 — avoiding overproduction of 58Ni relative to the observed 56Ni; the fallback of heavy elements that blocks a determination of the neutron-star mass in SN 1987A and of the maximum neutron-star mass; and whether convection can be captured without one-dimensional mixing-length prescriptions or the artificial angular-momentum conservation of two-dimensional calculations. The method questions are now addressed by three-dimensional neutrino-hydrodynamics with self-consistently transported YeY_e, the standard treatment, while three questions remain open: the Fe-group yields and YeY_e history of SN 1987A, whose nucleosynthesis favors rapid explosions (energy-growth timescales of at most about 250 ms) that current ab-initio neutrino-driven models, still under-energetic at simulation end, do not reach; the fallback mass and gravitational mass of the compact remnant remain undetermined despite JWST/ALMA evidence for ionizing radiation from a neutron star kicked at about 510 km/s; and the maximum neutron-star mass sits between conflicting observational bounds (GW170817-era upper limits near 2.17–2.4 solar masses versus the PSR J0952-0607 lower limit above 2.27 solar masses). The problem, in its surviving form, is to make the neutrino-driven account of SN 1987A quantitatively self-consistent — jointly predicting explosion energy, Fe-group yields, fallback, and remnant mass — and thereby to fix the maximum neutron-star mass.

Background

When a massive star's iron core collapses to a proto-neutron star, a rebounding shock forms, stalls within about 100 ms as it loses energy to nuclear dissociation and neutrino emission, and — in the delayed neutrino-heating mechanism proposed by Bethe & Wilson (1985) — is revived by partial reabsorption of the neutrinos diffusing out of the newly formed neutron star. The 'gain radius' is the radius inside the shocked region beyond which net neutrino heating exceeds cooling; the matter between it and the shock, the 'gain region', controls shock revival. Bethe (1997), the last paper of his eight-part series 'Supernova Shock', gave an analytic theory of the shock's propagation after revival, calibrated mainly by numerical results of J. Wilson (as characterized by Janka 2001), and confronted it with Supernova 1987A — the explosion of the blue supergiant Sanduleak -69 202 in the Large Magellanic Cloud, whose roughly two dozen neutrinos detected by Kamiokande-II, IMB and Baksan established the core-collapse picture, whose optical light curve was powered by about 0.07 solar masses of radioactive 56Ni synthesized in the explosion, and whose total kinetic energy is estimated at 1.4±0.4×10511.4 \pm 0.4 \times 10^{51} erg (one 'foe' = 105110^{51} erg, the standard supernova energy unit).

Bethe's analysis concluded that the shock breaks out of the stellar core at a radius of roughly 300 km because the infall velocity of the material it must sweep up decreases outward; that about 1.15 foe is deposited by neutrino heating during the first 0.05-0.3 s, consistent with SN 1987A's observed energy; but that the explosion synthesizes about 0.225 solar masses of iron-group nuclei, roughly three times the observed 56Ni mass, implying that the excess must fall back — 'fallback' denotes shocked material that fails to reach escape velocity and re-accretes onto the neutron star. Turbulent convection below the shock was argued to cease once the shock radius exceeds about 1000 km, leaving the electron fraction YeY_e — the number of protons (electrons) per baryon, which in nuclear statistical equilibrium (the high-temperature regime in which nuclear reactions are so fast that isotope abundances settle to chemical equilibrium) decides which iron-group nuclei form: Ye=0.5Y_e = 0.5 gives 56Ni, while slightly lower YeY_e gives more neutron-rich isotopes such as 58Ni — close to the value of the infalling gas: about 0.50 in the laminar zone (essentially pure 56Ni) and about 0.488 in the convective zone. On this basis Bethe listed the questions his theory could not answer, which form the present problem, restated in the problem statement.

Two of his concerns were about method. 'Mixing-length theory' is the classical one-dimensional parametrization of convection as heat-carrying parcels of a characteristic mixing length; Bethe doubted it captures the real three-dimensional flow, while the two-dimensional calculations of his era suffered from artificial conservation of angular momentum (a numerical artifact of axisymmetry that spuriously transports angular momentum and distorts the flow). The 'maximum neutron-star mass' is the largest gravitational mass a non-rotating neutron star can support, set by the dense-matter equation of state (EOS); Bethe noted that uncertain fallback prevents a precise neutron-star mass for SN 1987A and hence a tight statement of this maximum. These method questions are now addressed by three-dimensional neutrino-hydrodynamics with self-consistently transported YeY_e, and the modern descendant of Bethe's 'rate of energy supply' remark is the explosion-energy growth-timescale problem.

Problem Statement

The problem is the list of unresolved questions Bethe (1997, 'Supernova Shock. VIII.') attached to his analytic account of the neutrino-driven explosion, all serving one objective: making the predicted final shock energy, iron-group nucleosynthesis yields, and remnant mass of SN 1987A reliable within the neutrino-driven mechanism. As stated in the source, the questions are: (1) Can the early-time postshock compression (the ratio ρ2/ρ1\rho_2/\rho_1 of the postshock to the preshock density, about 8 at early times), which Bethe could not derive from first principles, be obtained from a self-consistent treatment of the shock and gain region, together with its dependence on the rate at which neutrino energy is supplied to the shocked gas? (2) Can the difference between the electron-neutrino and electron-antineutrino fluxes be small enough that the electron fraction in the convective zone is YeY_e of about 0.495 — above the infalling value of about 0.488 — so that nucleosynthesis does not overproduce 58Ni relative to the observed 56Ni (a calculation attributed by Bethe to R. D. Hoffman)? (3) How much fallback of heavier elements (Si, S, and iron-group material) occurs onto the neutron star, and what are the resulting gravitational mass of the neutron star in SN 1987A and the maximum neutron-star mass? (4) Can convection be treated without one-dimensional mixing-length prescriptions or the artificial angular-momentum conservation of two-dimensional calculations, so that the predicted nucleosynthesis yields and final shock energy are trustworthy? The objective is a quantitatively self-consistent neutrino-driven account of SN 1987A whose explosion energy, Fe-group yields (including the 58Ni/56Ni ratio and the YeY_e history that sets it), fallback mass, and compact-remnant mass are simultaneously confronted with the observational constraints, together with a determination of the maximum neutron-star mass. The Fe-group yields and YeY_e history of (2), the explosion-energy growth timescale of (1), and the fallback and compact-remnant masses of (3) are physically coupled.

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