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Astron. Astrophys. 335, 370-378 (1998)

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3. Model

We base our consideration on the model of ideal inviscid fluid with a constant specific heats ratio [FORMULA] which is described by the hydrodynamic equations

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] are the density, the velocity and the pressure of the fluid respectively. The unperturbed medium is assumed to be freely expanding according to the law

[EQUATION]

The moment t corresponds to the beginning of expansion. Suppose at the moment t a point explosion occurs in the origin of coordinates with an instantaneous release of energy E. This results in formation of the spherical expanding blast wave. The goal of our paper is to find the dynamics and structure of the resulting flow.

Our consideration focuses on the case of the power-law medium. We also suggest that the mass ejected by a source of explosion is neglible compared to the swept-up mass.

In the motionless cold unperturbed medium with the power-law radial density distribution

[EQUATION]

and negligible pressure

[EQUATION]

the strong blast wave dynamics is described by the self-similar Sedov solution (Sedov 1959). The similarity arises naturally because the dynamics in this case is unambigiously determined by four dimensional parameters: t and E. Particularly, the shock front position R is related with the other three variables through the power-law combination:

[EQUATION]

In case of the freely expanding medium the power-law density distribution evolves in time as

[EQUATION]

where A is a constant, and similarity dies away because an additional essential dimensional parameter t appears. Now R and t cannot be unambigiously related from the dimensional arguments. Nevertheless, an exact solution can be obtained by using the symmetry analysis.

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998

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