Mathematical Theory of One-dimensional Isothermal Blast Waves in a Magnetic Field

Abstract

An investigation is made of the self-similar flow behind a one-dimensional blast wave from a planar explosion (situated on z = 0) in a medium whose density and magnetic field vary with distance as Z-W ahead of the blast front, with the assumption that the flow is isothermal. It is found that; if OJ < 0 a continuous, single-valued solution does not exist; if OJ = 0 the solution is singular and piecewise continuous with an inner region where no fluid flow occurs and an outer region where the fluid flow gradually increases; if t > OJ > 0 the governing equation possesses a set of movable critical points. For a weak, but nonzero, magnetic field it is shown that the value of the smallest critical point does not lie in the physical domain z > O. The post-shock fluid flow then cannot intersect the critical point, and is smoothly continuous. It is shown that to be physically acceptable, the fluid flow speed must pass through the origin. It is also shown that OJ must be less than t for the magnetic energy swept up by the blast wave to remain finite. The overall conclusion from the investigation is that the behaviour of isothermal blast waves in the presence of an ambient magnetic field differs substantially from the behaviour calculated for no magnetic field. These results point to the inadequacy of previous attempts to apply the theory of self-similar flows to evolving supernova remnants without making any allowance for the dynamical influence of magnetic field pressure.