Critical Condition of Core-Collapse Supernovae I: One Dimensional Models

When the core of a massive star collapses, neutrino heating can energize the stalled accretion shock, leading to a successful supernova. The critical condition that characterizes the transition from accretion to explosion is a central topic of study and is often characterized by a critical proto-neutron star (PNS) neutrino luminosity $L_\nu^{\rm crit}$, which depends on the post-collapse mass accretion rate $\dot{M}$ from the progenitor. We examine the critical condition by solving the spherically symmetric time-dependent Euler equations with a general equation of state and realistic microphysics for a range of $\dot{M}$, average neutrino energy $\left< \epsilon_{\nu}\right>$, luminosity $L_{\nu}$, PNS radius $R_{\star}$, mass $M_{\star}$, and pre-shock Mach number $\mathcal{M}$ for a fixed neutrino optical depth from the PNS surface of $2/3$. We derive $L_{\nu}^{\mathrm{crit}}$ as a function of the input parameters. We show that pressurized pre-shock flow, as parameterized by low $\mathcal{M}$, changes the normalization of the critical condition because accretion of higher entropy shells at later times after collapse leads to lower $L_{\nu}^{\mathrm{crit}}$. We connect this finding to the onset of explosion due to compositional interface accretion. Across our parameter space, we test critical conditions that have been proposed in the literature, including the ``antesonic" condition, the ``force explosion condition," and the heuristic heating-advection timescale condition. We discuss how shock oscillations impact these critical conditions. Compared to other explosion conditions, we find that the antesonic ratio shows the least variation across the model space we explore. This work is preparatory for similar experiments in 2D axisymmetry and 3D.