Astrophysical blastwaves

Abstract

The authors present a general discussion of spherical, nonrelativistic blastwaves in an astrophysical context. A variety of effects has been included: expansion of the ambient medium, gravitation, and an embedded fluid of clouds capable of exchanging mass, energy, or momentum with the medium. The authors also consider cases of energy injection due either to a central source or to detonations. Cosmological solutions are extensively treated. Most attention is devoted to problems in which it is permissible to assume self-similarity, as in the prototype Sedov-Taylor blastwave. A general virial theorem for blastwaves is derived. For self-similar blastwaves, the radius varies as a power of the time, ${R}_{s}\ensuremath{\propto}{t}^{\ensuremath{\eta}}$. The integral properties of the solution are completely specified by two dimensionless numbers measuring the relative importance of thermal and kinetic energy. The authors find certain exact kinematical relations and a variety of analytic approximations to determine these numbers with varying degrees of accuracy. The approximations may be based on assumptions about the internal density distributions (e.g., shell-like), pressure distribution, or velocity distribution. In many cases exact conditions from, for example, boundary conditions or other constraints may be used to determine unspecified parameters. One new set of exact integral constraints has been derived. The various approximation schemes are tested with known solutions. The authors find that for blastwaves in which the flow extends to the origin, the assumption that the internal velocity is linear with radius is reasonably accurate. For blastwaves in which an interior vacuum develops, the equally simple approximation of constant interior velocity is accurate. These lowest-order approximations are shown to give numerical coefficients in the relation $R=\mathrm{const}\ifmmode\times\else\texttimes\fi{}{t}^{\ensuremath{\eta}}$ which are accurate to about 1-2%. The higher-order approximations show an accuracy that in some cases equals that obtained, to date, by direct numerical integration. In addition to the new methods presented, the authors have obtained new results for evaporative blastwaves, impeded blastwaves, blastwaves with cloud crushing, bubbles, cosmological blastwaves (self-similar and non-self-similar, radiative and nonradiative), blastwaves in a wind, and detonations. Some of the new results found are exact. Included are the radiative, cosmological self-similar solution, appropriate to the universe ($z>10$) when inverse Compton cooling is efficient [$\mathrm{ln}R=\mathrm{const}+\frac{(\mathrm{ln}t)(15+\sqrt{17})}{24}$], and certain properties of the solutions mentioned above. In a series of appendixes several related issues are treated: energy conservation for multicomponent fluid in an expanding universe; central and edge derivatives of physical quantities in self-similar adiabatic blastwaves; shock jump conditions including energy input (detonations), and a variety of other matters.