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

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1. Prologue

Perhaps no approach in a theory of explosive phenomena has made such a profound impact on the development of analytical methods of investigation as the classical self-similar Sedov solution for a blast wave generated by a point explosion in a uniform static medium (Sedov 1946, 1959; Taylor 1950). Known in hydrodynamics long before 1946, self-similarity most impressively demonstrated, in this instant, a lucky opportunity to obtain an exact analytical solution to a complicated hydrodynamic problem described by the nonlinear partial differential equations. This marvellous ability to obtain exact solutions owes its existence to the fact that self-similarity advantageously employs some symmetrical properties of the physical system, namely its ability for scaling. Ruling out those degrees of freedom which are invariant under scaling transformations allows to diminish the dimensionality of the problem in hand and to study the reduced problem by simpler and more available means.

Self-similarity in the Sedov problem was dictated by necessity because the number of the dimensional governing parameters was four and only four which allowed to obtain solution through the solely possible power-law combination of independent variables and dimensional parameters with the rational power index. Later it was understood that self-similarity may arise also in case when dimensional analysis fails so that the power-law similarity indices may become irrational (Zel'dovich & Raizer 1966). These two cases are distinguished now as similarities of the first and second kind respectively (Barenblatt 1979).

A remarkable success of similarity technique inspired and continues to inspire the followers to search for the solutions for blast waves dynamics in all possible situations in a self-similar form.

There are many situations in which self-similarity arises naturally and is dictated by the physics of the problem, because the number of essential physical factors is exactly the required minimum. There are much more situations in which the number of parameters exceeds the required minimum number and self-similarity is generally broken but can be allowed under some particular conditions imposed on these extra parameters. In the former case the self-similar solutions reflect the generic, universal behaviour of the physical system manifesting itself either as an intermediate (an adiabatic stage of supernova remnant expansion into an interstellar medium (Sedov 1959), an adiabatic shock wave in an exponential atmosphere (Raizer 1964; Grover & Hardy 1966), an adiabatic interaction of supernova blast wave with a circumstellar envelope (Chevalier 1982; Nadyozhin 1985), a blast wave in an inhomogeneous medium with evaporating clouds (Chieze & Lazareff 1981; White & Long 1991)), or an end-state asymptotics (cosmological blast waves (Ikeuchi et al. 1983; Bertschinger 1983; Kovalenko & Sokolov 1993), cosmological detonation waves (Bertschinger 1985; Kazhdan 1986), etc.). In the latter case the self-similar solutions describe just very particular, non-generic situations that have no chance to be realized in nature except perhaps under specially prepared initial, boundary or some other conditions. An enormous amount of works, however, which assume self-similarity a priori artificially adjusting the governing parameters to a required form in order to obtain an analytical or semianalytical solution clearly indicate at once popularity and scantiness of available analytical tools.

Indeed, similarity method remains nowadays perhaps a unique well-known systematic method to search for (and to find) exact unsteady solutions for blast waves. At the same time the blast wave dynamics in actual situations is affected by various factors such as the radiative energy losses behind the shock wave, nonuniformity of the unperturbed medium, non-zero ambient pressure, magnetic fields, and many others, which destroy the self-similarity of the problem. In this case, as far as we are aware, there were no precedents of finding an exact unsteady solution - the usual practice is either to make use of the different analytical approximate methods such as the Kompaneets method (Kompaneets 1960), the sector approximation (Laumbach & Probstein 1969), the thin shell approximation (Chernyi 1957), the shell moment approximation (Ostriker & McKee 1988), composite combinations of these four (Koo & McKee 1990), etc., or to study the problem numerically. Meanwhile there exists a more powerful and general method of analytical study - the symmetry analysis of differential equations which incorporates similarity analysis as one of the particular cases. It shares much the same principal idea of reduction of the system of equations under investigation, but it allows to operate with more various symmetries, not only with the simple scaling. If the similarity analysis partly leans upon the intuitive approach, the symmetry analysis provides direct instructions how to find symmetries and their associated solutions. Successfully used in many fields from the field theory to applied hydrodynamics (Vinogradov 1989; Coggeshall & Axford 1986), it is unfortunately still not widely spread in astrophysical science. At the same time the nonlinear astrophysical problems formulated in terms of partial differential equations offer a fertile field for employment of the symmetry analysis where it performs at its best. We believe that this modern mathematical technique extending analytical tools of analysis and complementing power (though still not outperforming flexibility) of the numerical analysis may soon be in common use not only for pure mathematicians but also for applied researchers especially in connection with the development of software for symbolic computations. Indication of a progress becomes increasingly clear (Ibragimov 1994, 1995). The present paper provides an example of how this formalism allows to obtain exact analytical solutions to the complicated nonlinear problems in astrophysics.

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© European Southern Observatory (ESO) 1998

Online publication: June 12, 1998