Weak Shock Propagation with Accretion. II. Stability of Self-similar Solutions to Radial Perturbations

Abstract

Coughlin et al. (2018b) (Paper I) derived and analyzed a new regime of self-similarity that describes weak shocks (Mach number of order unity) in the gravitational field of a point mass. These solutions are relevant to low energy explosions, including failed supernovae. In this paper, we develop a formalism for analyzing the stability of shocks to radial perturbations, and we demonstrate that the self-similar solutions of Paper I are extremely weakly unstable to such radial perturbations. Specifically, we show that perturbations to the shock velocity and post-shock fluid quantities (the velocity, density, and pressure) grow with time as t α , where α ≤ 0.12, implying that the ten-folding timescale of such perturbations is roughly ten orders of magnitude in time. We confirm these predictions by performing high-resolution, time-dependent numerical simulations. Using the same formalism, we also show that the Sedov-Taylor blastwave is trivially stable to radial perturbations provided that the self-similar, Sedov-Taylor solutions extend to the origin, and we derive simple expressions for the perturbations to the post-shock velocity, density, and pressure. Finally, we show that there is a third, self-similar solution (in addition to the the solutions in Paper I and the Sedov-Taylor solution) to the fluid equations that describes a rarefaction wave, i.e., an outward-propagating sound wave of infinitesimal amplitude. We interpret the stability of shock propagation in light of these three distinct self-similar solutions.

Numerical Simulations of the Random Angular Momentum in Convection: Implications for Supergiant Collapse to Form Black Holes

“…Panel (a) shows that ṁ is predicted extremely well and serves as independent confirmation of the Coughlin et al (2019) rarefaction wave solutions. While there are some small differences between the two curves in panels (b)-(d), the magnitude and direction angles of j acc are well-predicted using the mean profiles and the assumption of ballistic infall from rest.…”

“…There is, in fact, a small pressure gradient behind the rarefaction wave (Coughlin et al 2019), so the time for the shell to fall back after the arrival of the rarefaction wave is actually 1.14 t ff (r) for our simulated values of b and γ (E. R. Coughlin, private communication). The total time required for the shell at r to reach the sink is thus tacc(r) = 1.14 t ff (r) + twave(r).…”

“…(A7). The factor of 1.14 (supplied by Eric R. Coughlin and based on the self-similar rarefaction wave solutions of Coughlin et al 2019), comes from the fact that infall begins from a profile in hydrostatic equilibrium. There is thus a pressure gradient behind the rarefaction wave, so the infall is not true zero-pressure 'free-fall' as assumed by t ff (r).…”

Partial stellar explosions – ejected mass and minimal energy

“…Our models assumed a point explosion at the centre of the star, but different energy injection mechanisms (such as the outward expansion during a failed SN, Coughlin et al 2018b, or spatially extended wave heating in a stellar envelope) can allow for the self-similar solution to arise. As discussed in Coughlin et al (2019) and Ro et al (2019), despite being weakly unstable, their self-similar solution is likely to prevail in the envelopes of supergiants where density scales as ρ ∝ r −n over several orders of magnitude. As the weak shock approaches the stellar surface, it eventually transitions to a strong shock and starts accelerating, as dictated by the GFKS solution.…”

Partial Stellar Explosions -- Ejected Mass and Minimal Energy

“…The propagation of weak shocks in stellar envelopes has been studied analytically by Coughlin et al (2018), who found a selfsimilar solution describing the propagation of a spherical weak shock wave in a hydrostatic medium with a point mass gravitational field, and later analyzed the solution's stability in Coughlin et al (2019) and Ro et al (2019). Their solution exists when the density scales as 𝜌 ∝ 𝑟 −𝑛 , where 2 < 𝑛 < 3.5, with the shock propagating with a constant and order-unity Mach number, 𝑣 sh = 𝑉 √︁ 𝐺 𝑀 0 /𝑟, where 𝑉 (𝑛) 1 is a constant and 𝑀 0 is the point mass dominating the gravitational field.…”