Closing the gap in the solutions of the strong explosion problem: An expansion of the family of second-type self-similar solutions

Details Closing the gap in the solutions of the strong explosion problem: An expansion of the family of second-type self-similar solutions Closing the gap in the solutions of the strong explosion problem: An expansion of the family of second-type self-similar solutions

2010-02-20

arxiv.org/abs/1002.3869v2

Astronomy and Astrophysics

Astrophysics

High Energy Astrophysical Phenomena

10 pages, 11 figures, somewhat revised, version accepted to ApJ

Scientific paper

Shock waves driven by the release of energy at the center of a cold ideal gas sphere of initial density rho\propto r^{-omega} approach a self-similar (SLS) behavior, with velocity \dot{R}\propto R^delta, as R->\infty. For omega>3 the solutions are of the second-type, i.e., delta is determined by the requirement that the flow should include a sonic point. No solution satisfying this requirement exists, however, in the 3\leq omega\leq omega_{g}(gamma) ``gap'' (\omega_{g}=3.26 for adiabatic index gamma=5/3). We argue that second-type solutions should not be required in general to include a sonic point. Rather, it is sufficient to require the existence of a characteristic line r_c(t), such that the energy in the region r_c(t)\infty, and an asymptotic solution given by the SLS solution at r_c(t)omega_g and the latter identifies delta=0 solutions as the asymptotic solutions for 3\leq omega\leq omega_{g} (as suggested by Gruzinov03). In these solutions, r_c is a C_0 characteristic. It is difficult to check, using numerical solutions of the hydrodynamic equations, whether the flow indeed approaches a delta=0 SLS behavior as R->\infty, due to the slow convergence to SLS for omega~3. We show that in this case the flow may be described by a modified SLS solution, d\ln\dot{R}/d\ln R=delta with slowly varying delta(R), eta\equiv d delta/d\ln R<<1, and spatial profiles given by a sum of the SLS solution corresponding to the instantaneous value of delta and a SLS correction linear in eta. The modified SLS solutions provide an excellent approximation to numerical solutions obtained for omega~3 at large R, with delta->0 (and eta\neq0) for 3\leq omega\leq omega_{g}. (abridged)

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