Hydrodynamics of Core-Collapse Supernovae at the Transition to Explosion. I. Spherical Symmetry

We study the transition to runaway expansion of an initially stalled core-collapse supernova shock. The neutrino luminosity, mass accretion rate, and neutrinospheric radius are all treated as free parameters. In spherical symmetry, this transition is mediated by a global non-adiabatic instability that develops on the advection time and reaches non-linear amplitude. Here we perform high-resolution, time-dependent hydrodynamic simulations of stalled supernova shocks with realistic microphysics to analyze this transition. We find that radial instability is a sufficient condition for runaway expansion if the neutrinospheric parameters do not vary with time and if heating by the accretion luminosity is neglected. For a given unstable mode, transition to runaway occurs when fluid in the gain region reaches positive specific energy. We find approximate instability criteria that accurately describe the behavior of the system over a wide region of parameter space. The threshold neutrino luminosities are in general different than the limiting value for a steady-state solution. We hypothesize that multidimensional explosions arise from the excitation of unstable large-scale modes of the turbulent background flow, at threshold luminosities that are lower than in the laminar case.

💡 Research Summary

This paper investigates the transition from a stalled shock to runaway expansion in core‑collapse supernovae under the assumption of spherical symmetry. The authors treat three global control parameters—the neutrino luminosity (Lν), the mass accretion rate (ṁ), and the neutrinospheric radius (Rν)—as free, time‑independent quantities and explore how variations in these parameters affect the post‑shock flow. Using high‑resolution, time‑dependent hydrodynamic simulations that incorporate a realistic equation of state (Shen et al. 1998) and detailed weak‑interaction source terms, they construct steady‑state accretion solutions and then perturb them to study stability.

The study confirms that, for a given set of parameters, the stalled shock can become globally unstable on the advection time scale. Two families of unstable modes are identified: an oscillatory mode that produces periodic shock radius oscillations and a non‑oscillatory mode that grows monotonically. Both modes are non‑adiabatic, driven by neutrino heating in the gain region. The key physical criterion for transition to explosion is that the specific energy of material in the gain region becomes positive (e_spec > 0). When this occurs, the flow inevitably runs away, even though the background parameters remain fixed.

A major result is the derivation of new instability criteria that differ from the classic Burrows‑Goshy (1993) critical luminosity (L_crit). The authors find that the true threshold for runaway expansion (L_inst) can be 5–30 % lower than L_crit, depending on ṁ and Rν. Their criteria combine three conditions: (i) Lν exceeds a parameter‑dependent instability luminosity L_inst(ṁ,Rν); (ii) the ratio of advection to heating time in the gain region τ_adv/τ_heat exceeds a value ξ≈0.8–1.0; and (iii) the specific energy in the gain region is positive. These conditions are shown to predict the onset of explosion across a wide region of parameter space with high accuracy.

The authors also perform extensive convergence tests. Varying spatial resolution, inner boundary location, and upstream flow conditions demonstrates that the instability growth rates and saturation amplitudes are robust. The initial shock radius influences the time required for instability to develop but does not substantially shift the critical luminosity.

Importantly, the paper connects the spherical‑symmetry results to multidimensional supernova simulations. The identified global radial instability provides a plausible mechanism by which large‑scale modes (ℓ = 1, 2) in turbulent, multidimensional flows can be excited, thereby lowering the effective critical neutrino luminosity required for explosion. This offers a physical explanation for the observed ∼20 % reduction in L_crit in 2‑D and 3‑D models.

In summary, the work demonstrates that even in one‑dimensional, spherically symmetric models, a non‑adiabatic global instability can drive the stalled shock to runaway expansion. The derived instability criteria refine the traditional steady‑state critical luminosity concept and lay a foundation for interpreting multidimensional explosion mechanisms as manifestations of the same underlying instability, excited by turbulent background flows. Future work should extend these criteria to full 2‑D/3‑D simulations to quantify how turbulence modifies the growth and saturation of the unstable modes.