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A&A 506, 1215-1228 (2009)
DOI: 10.1051/0004-6361/200912602
The propagation of the shock wave from a strong explosion in a plane-parallel stratified medium: the Kompaneets approximation
C. A. OlanoUniversidad Nacional de La Plata, Facultad de Ciencias Astronómicas y Geofísicas, Paseo del Bosque, 1900 La Plata, Argentina
e-mail: colano@fcaglp.fcaglp.unlp.edu.ar
Received 29 May 2009 / Accepted 30 July 2009
Abstract
Context. Using certain simplifications, Kompaneets derived a
partial differential
equation that states the local geometrical and kinematical conditions that each surface element of a
shock wave, created by a point blast in a stratified gaseous medium, must
satisfy. Kompaneets could solve his equation analytically for the case of
a wave propagating in an exponentially stratified medium, obtaining
the form of the shock front at progressive evolutionary stages. Complete analytical solutions of
the Kompaneets equation for shock wave motion in further plane-parallel stratified media were not
found, except for radially stratified media.
Aims. We aim to analytically solve the Kompaneets equation for the motion of a shock wave in different plane-parallel
stratified media that can reflect a wide variety of
astrophysical contexts. We were
particularly interested in solving the Kompaneets equation for a strong explosion in the
interstellar medium of the Galactic disk, in which, due to intense winds and
explosions of stars, gigantic gaseous structures known as superbubbles and
supershells are formed.
Methods. Using the Kompaneets approximation, we derived a pair of equations that we call
adapted Kompaneets equations, that
govern the propagation of a shock wave in a stratified medium and that
permit us to obtain solutions in parametric form. The solutions provided by
the system of adapted
Kompaneets equations are equivalent to those of the Kompaneets equation. We
solved the adapted Kompaneets equations for shock wave propagation in a generic stratified medium by
means of a power-series method.
Results. Using the series solution
for a shock wave in a generic medium, we obtained the series solutions for four specific
media whose respective density distributions in the direction perpendicular to the
stratification plane are of an exponential, power-law
type (one with exponent k=-1 and the other with k =-2) and a quadratic hyperbolic-secant. From these series solutions, we deduced exact solutions for the four
media in terms of elemental functions. The exact solution for shock wave propagation in a medium of quadratic hyperbolic-secant density distribution is very appropriate to describe the growth of
superbubbles in the Galactic disk.
Key words: hydrodynamics -- shock waves -- Galaxy: disk -- ISM: bubbles -- ISM: supernova remnants
© ESO 2009
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