Closing the gap in the solutions of the strong explosion problem: An expansion of the family of second-type self-similar solutions
Abstract
Shock waves driven by the release of energy at the center of a cold ideal gas sphere of initial density rho r-omega approach a self-similar (SLS) behavior, with velocity R Rdelta, as R->∞. 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≤ omega≤ omegag(gamma) ``gap'' (ωg=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 rc(t), such that the energy in the region rc(t)<r<R approaches a constant as R->∞, and an asymptotic solution given by the SLS solution at rc(t)<r<R and deviating from it at r<rc may be constructed. The two requirements coincide for omega>omegag and the latter identifies delta=0 solutions as the asymptotic solutions for 3≤ omega≤ omegag (as suggested by Gruzinov03). In these solutions, rc is a C0 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->∞, 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, dR/d R=delta with slowly varying delta(R), eta d delta/d 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≠0) for 3≤ omega≤ omegag. (abridged)