Self-similarity of the third type in ultra relativistic blastwaves
Abstract: A new type of self-similarity is found in the problem of a plane-parallel, ultra-relativistic blast wave, propagating in a powerlaw density profile of the form $\rho \propto z{-k}$. Self-similar solutions of the first kind can be found for $k<7/4$ using dimensional considerations. For steeper density gradients with $k>2$, second type solutions are obtained by eliminating a singularity from the equations. However, for intermediate powerlaw indices $7/4<k<2$ the flow does not obey any of the known types of self-similarity. Instead, the solutions belong to a new class in which the self-similar dynamics are dictated by the non self-similar part of the flow. We obtain an exact solution to the ultra-relativistic fluid equations and find that the non self-similar flow is described by relativistic expansion into vacuum, composed of (1) an accelerating piston that contains most of the energy and (2) a leading edge of fast material that coincides with the interiors of the blastwave and terminates at the shock. The dynamics of the piston itself are self-similar and universal, and do not depend on the external medium. The exact solution of the non self-similar flow is used to solve for the shock in the new class of solutions.
- Y. B. Zel’dovich and Y. P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena (1967).
- G. I. Barenblatt, “Self-similarity - Similarity and intermediate asymptotics,” Radiofizika 19, 902–931 (1976).
- G. Taylor, “The Formation of a Blast Wave by a Very Intense Explosion. II. The Atomic Explosion of 1945,” Proceedings of the Royal Society of London Series A 201, 175–186 (1950).
- J. von Neumann, “Blast Waves,” Los Alamos Science Laboratory Technical Series, Los Alamos 7 (1947).
- L. Sedov, “Similarity and Dimensional Methods in Mechanics,” Academic, New York (1969).
- E. Waxman and D. Shvarts, “Second-type self-similar solutions to the strong explosion problem,” Physics of Fluids A 5, 1035–1046 (1993).
- A. Gruzinov, “Self-similarity of the Third Type in the Strong Explosion Problem,” arXiv e-prints , astro-ph/0303242 (2003), arXiv:astro-ph/0303242 [astro-ph] .
- D. Kushnir and E. Waxman, “Closing the Gap in the Solutions of the Strong Explosion Problem: An Expansion of the Family of Second-type Self-similar Solutions,” Astrophys. J. 723, 10–19 (2010), arXiv:1002.3869 [astro-ph.HE] .
- R. D. Blandford and C. F. McKee, “Fluid dynamics of relativistic blast waves,” Physics of Fluids 19, 1130–1138 (1976).
- P. Best and R. Sari, “Second-type self-similar solutions to the ultrarelativistic strong explosion problem,” Physics of Fluids 12, 3029–3035 (2000).
- R. Perna and M. Vietri, ‘‘A Self-similar Solution for the Propagation of a Relativistic Shock in an Exponential Atmosphere,” The Astrophysical Journal Letters 569, L47–L50 (2002), arXiv:astro-ph/0203208 [astro-ph] .
- J. C. Hidalgo and S. Mendoza, “Self-similar imploding relativistic shock waves,” Physics of Fluids 17, 096101–096101 (2005), arXiv:astro-ph/0406603 [astro-ph] .
- R. Sari, “First and second type self-similar solutions of implosions and explosions containing ultrarelativistic shocks,” Physics of Fluids 18, 027106–027106 (2006), arXiv:astro-ph/0505174 [astro-ph] .
- T. Faran and R. Sari, “The non-relativistic interiors of ultra-relativistic explosions: Extension to the Blandford-McKee solutions,” Physics of Fluids 33, 026105 (2021), arXiv:2011.05990 [astro-ph.HE] .
- M. H. Johnson and C. F. McKee, “Relativistic Hydrodynamics in One Dimension,” Phys. Rev. D 3, 858–863 (1971).
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