CLOSING THE GAP IN THE SOLUTIONS OF THE STRONG EXPLOSION PROBLEM: AN EXPANSION OF THE FAMILY OF SECOND-TYPE SELF-SIMILAR SOLUTIONS
- Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 76100 (Israel)
Shock waves driven by the release of energy at the center of a cold ideal gas sphere of initial density {rho} {proportional_to} r {sup -}{omega} approach a self-similar behavior, with velocity R-dot {proportional_to}R{sup {delta}}, as R {yields} {infinity}. For {omega}>3 the solutions are second-type solutions, 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}{sub g}({gamma}) 'gap' ({omega}{sub g} = 3.26 for adiabatic index {gamma} = 5/3). We argue that in general second-type solutions should not be required to include a sonic point. Rather, it is sufficient to require the existence of a characteristic line r{sub c} (t), such that the energy in the region r{sub c} (t) < r < R approaches a constant as R {yields} {infinity}, and an asymptotic solution given by the self-similar solution at r{sub c} (t) < r < R and deviating from it at r < r{sub c} may be constructed. The two requirements coincide for {omega}>{omega}{sub g} and the latter identifies {delta} = 0 solutions as the asymptotic solutions for 3 {<=} {omega} {<=} {omega}{sub g} (as suggested by Gruzinov). In these solutions, r{sub c} is a C{sub 0} characteristic. Using numerical solutions of the hydrodynamic equations, it is difficult to check whether the flow indeed approaches a {delta} = 0 self-similar behavior as R {yields} {infinity}, due to the slow convergence to self-similarity for {omega} {approx} 3. We show that in this case the flow may be described by a modified self-similar solution, d ln R-dot/d ln R = {delta} with slowly varying {delta}(R), {eta} {identical_to} d{delta}/dln 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} {approx} 3 at large R, with {delta} {yields} 0 (and {eta} {ne} 0) for 3 {<=} {omega} {<=} {omega}{sub g}.
- OSTI ID:
- 21467218
- Journal Information:
- Astrophysical Journal, Vol. 723, Issue 1; Other Information: DOI: 10.1088/0004-637X/723/1/10; ISSN 0004-637X
- Country of Publication:
- United States
- Language:
- English
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