Discrete Self-Similarity in Type-II Strong Explosions

Abstract

We present new solutions to the strong explosion problem in a non-power law density profile. The unperturbed self-similar solutions discovered by Waxman & Shvarts describe strong Newtonian shocks propagating into a cold gas with a density profile falling off as $r^{-\omega }$, where $\omega >3$ (Type-II solutions). The perturbations we consider are spherically symmetric and log-periodic with respect to the radius. While the unperturbed solutions are continuously self-similar, the log-periodicity of the density perturbations leads to a discrete self-similarity of the perturbations, i.e. the solution repeats itself up to a scaling at discrete time intervals. We discuss these solutions and verify them against numerical integrations of the time dependent hydrodynamic equations. Finally we show that this method can be generalized to treat any small, spherically symmetric density perturbation by employing Fourier decomposition.

References in corpus (4)

• First and second-type self-similar solutions of implosions and explosions containing ultra-relativistic shocks
• Self-similar solutions for relativistic shocks emerging from stars with polytropic envelopes
• The stability of decelerating shocks revisited
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Cited by in corpus (6)

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• Closing the gap in the solutions of the strong explosion problem: An expansion of the family of second-type self-similar solutions
• Slightly Two or Three Dimensional Self-Similar Solutions
• Discrete Self-Similarity in Ultra-Relativistic Type-II Strong Explosions
• Discrete Self Similarity in Filled Type I Strong Explosions
• Angular Density Perturbations to Filled Type I Strong Explosions