Critical Condition of Core-collapse Supernovae. I. One-dimensional Models

Abstract 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 protoneutron star (PNS) neutrino luminosity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ν</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">crit</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> , which depends on the post-collapse mass-accretion rate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mover accent="true"> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>̇</mml:mo> </mml:mrow> </mml:mover> </mml:math> , average neutrino energy 〈 ϵ ν 〉, luminosity L ν , PNS radius R ⋆ , mass M ⋆ , and pre-shock Mach number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> </mml:math> for a fixed neutrino optical depth from the PNS surface of 2/3. We derive <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ν</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">crit</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> as a function of the input parameters. We show that increasing the pressure of the pre-shock flow, as parameterized by lowering <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">M</mml:mi> </mml:math> , lowers the normalization of the critical condition. We connect this finding with realistic massive progenitors and show that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>ν</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">crit</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> can decrease by ∼5%–8% due to thermal changes during accretion of compositional interfaces onto the stalled shock. 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.