Closing the gap in the solutions of the strong explosion problem: An expansion of the family of second-type self-similar solutions
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 behavior, with velocity \dot{R}\propto R^delta, as R\rightarrow\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<=omega<=\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)omega_g, and the latter identifies delta=0 solutions as
the asymptotic solutions for 3<=omega<=omega_{g} (as suggested by Gruzinov 2003). 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 self-similar behavior as R\rightarrow\infty, due to the slow convergence to self-similarity for omega~3. We show that in this case the flow may be described by a modified self-similar 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 self-similar solution corresponding to the instantaneous value of
delta and a self-similar correction linear in eta. The modified self-similar solutions provide an excellent approximation to numerical solutions obtained for omega~3 at large R, with delta\rightarrow0 (and eta
eq0) for 3<=omega<=omega_{g}.