From Coasting to Energy-conserving: New Self-similar Solutions to the Interaction Phase of Strong Explosions

Abstract

Astrophysical explosions that contain dense and ram-pressure-dominated ejecta evolve through an interaction phase, during which a forward shock (FS), contact discontinuity (CD), and reverse shock (RS) form and expand with time. We describe new self-similar solutions that apply to this phase and are most accurate in the limit that the ejecta density is large compared to the ambient density. These solutions predict that the FS, CD, and RS expand at different rates in time and not as single temporal power laws, are valid for explosions driven by steady winds and homologously expanding ejecta, and exist when the ambient density profile is a power law with a power-law index shallower than ∼3 (specifically when the FS does not accelerate). We find excellent agreement between the predictions of these solutions and hydrodynamical simulations, both for the temporal behavior of the discontinuities and for the variation of the fluid quantities. The self-similar solutions are applicable to a wide range of astrophysical phenomena and—although the details are described in future work—can be generalized to incorporate relativistic speeds with arbitrary Lorentz factors. We suggest that these solutions accurately interpolate between the initial “coasting” phase of the explosion and the later, energy-conserving phase (or, if the ejecta is homologous and the density profile is sufficiently steep, the self-similar phase described in R. A. Chevalier).

Keywords

• Analytical mathematics
• Core-collapse supernovae
• Transient sources
• Hydrodynamics
• Shocks