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

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5. Discussion

It is convenient to interpret the solution derived above in terms of an effective power-law index


From Eq. (30) we see that [FORMULA] grows monotonically from 2 at the initial moment and asymptotically tends to 1 at late times. Thus the invariant solution presents a continuous joining of two self-similar asymptotics - a really rare event in mathematical physics (Barenblatt 1979). We recall that the solution is true only for [FORMULA] but this case is the most interesting one in astrophysical applications. Another merit of the solution is its possible reversibility in time. If we replace t by t and v by - everywhere, the formulas will describe explosion and the subsequent blast wave evolution on a contracting background. Obviously, the moment t will then correspond to the instant of collapse.

The solution found is fundamentally based on two assumptions of adiabaticity and infinite intensity of the blast wave. Let us consider the conditions at which our assumptions become invalid.

First discuss the range of validity of a strong blast approximation. It is clear that as the area occupied by a blast wave expands the strength of the blast should fall off so that our initial assumption may fail with time. The blast wave is considered as strong when the ratio of the postshock p and preshock p pressures is large, that is,


The ambient pressure drops adiabatically. In case of the uniform medium it has the form


Then the intensity of the blast wave according to Eqs. (30), (31), (33), (40), (42), (60), (66) is determined by the formula


This ratio is a monotonically decreasing function of time asymptotically converging to a non-zero constant at infinity. If


the blast wave will remain strong the whole time of expansion and our neglect of the ambient pressure will be consistent. Let us make estimates for a supernova exploded inside a galactic fountain. The typical magnitudes of density and sound speed in the rarefied hot gas of the fountain close to the plane of the galactic disk are [FORMULA] g[FORMULA]cm-3, c km[FORMULA]s. The fountain begins to form when several supernovae burst within a short period of time a short distance from each other. For the frequency of supernova explosions it is usually adopted [FORMULA] yr-1 (MacLow & McCray 1988; Tenorio-Tagle et al. 1990) therefore for the well-formed fountain we have t yr. For the sake of clarity let us assume k which corresponds to the Mach number Ma. Then from inequality (68) we find E erg. Thus, a shock produced by a single Type II supernova with the mean energy release E erg strongly decays before it reaches the edge of the cavity. This estimate is, however, too uncertain because of the sensitivity of inequality (68) to c variation.

Another constraint imposed on our solution arises from the radiative energy losses. The influence of radiative cooling can be crudely estimated depending on whether the cooling time t exceeds the characteristic time of expansion t or not. If the cooling time is much shorter than t, the blast wave enters the cooling phase well before the overall expansion of a background will be dynamically important. In this case our adiabatic solution extended for large times is no longer valid. We do not cite here the estimation of the cooling time in a static power-law medium referring an interested reader to the recent comprehensive discussion of this question in Franco et al. (1994). If t exceeds t, the radiative cooling ceases to be a significant factor for dynamics. Indeed, at the moment t, the thermal energy is lowered by a factor 4 (see Eq. (58)). Its fraction in the total energy budget decreases much more since the kinetic energy increases [FORMULA] times (Eq. (57)). So by the time t the blast wave, being initially for the most part pressure-driven wave, becomes a momentum-driven one. Taking away some of the thermal energy by emerging radiation under these circumstances cannot anywhere affect motion of the blast wave.

Cosmology is a more promising field as far as application of the found solution goes. Cosmological blast waves from a point explosion were studied in connection with the model of the explosive galaxy formation (Ostriker & Cowie 1981; Ikeuchi 1981). Ikeuchi et al. (1983) and Bertschinger (1983) developed a self-similar adiabatic ([FORMULA]) solution (hereafter ITO-B) for a blast wave in an unperturbed flat universe in Friedmann cosmology for which the density parameter [FORMULA]. The most distinctive feature of this solution is that the postshock material is totally confined to a very thin dense shell with a thickness [FORMULA] for [FORMULA]. Kovalenko and Sokolov (1993) showed that this solution describes in fact a final stage of any spherically symmetric positive energy perturbation in the flat universe. In the flat universe the influence of gravity is never small; the background therefore expands decelerating and our present invariant solution cannot be applied to this case. However it can be applied to a model of an open universe with [FORMULA] for which expansion becomes inertial at late times when [FORMULA]. Ikeuchi et al. (1983) and Ostriker and McKee (1988) considered the final stage of a blast wave against a freely expanding Friedmann background and properly predicted its asymptotical self-similar law [FORMULA] and asymptotical comoving with the ambient fluid. At the same time they were mistaken inferring that the postshock material eventually concentrates to even denser shell, in the limit, an infinitely thin shell just behind a shock front. This prediction contradicts the numerical calculations (Hausman et al. 1983; Hoffman et al. 1983). Bertschinger (1983) was more accurate stating that the postshock density distribution may be arbitrarily dependent on the history of a blast wave evolution at the intermediate non-self-similar stages. Based themselves upon the fact that [FORMULA] all these authors fell victims to a common mistake coming to the conclusion that the shock discontinuity asymptotically degenerates into a contact discontinuity. The true solution, as we can see, shows that in none of the moments discontinuity ceases to be a shock and thus the postshock gas has no tendency to pile up into an infinitely thin shell behind a shock front.

All troubles with the asymptotic self-similar solution arose from the stringent assumption of the exact self-similarity with [FORMULA] which eventually led to the condition [FORMULA]. To correctly resolve the internal structure of the blast wave one has to allow for a small deviation from the exact self-similar law. Luckily, with our new method we are now in a position to follow the asymptotical approach to the final self-similar state and moreover we are able to trace the blast wave evolution from start to finish at least in the limit of negligible gravity. To gain a better understanding of how a blast evolves from the moment of explosion in the universe let us briefly review the general case.

In the [FORMULA] universe the blast wave evolution goes on in the following sequence. Just after an explosion the Sedov profile is established. After several Hubble times the matter gradually escapes the central parts and accumulates to a thin shell behind a shock front; the postshock density distribution eventually acquires a well-pronounced shell-like profile prescribed by a self-similar adiabatic ITO-B cosmological blast wave solution.

In case of the [FORMULA] universe the blast wave behaves similarly but stops the postshock density enhancement earlier either delaying the shell formation or not achieving the final ITO-B state with a thin shell at all. One can discern two limiting cases dependent on the relation between the moment of explosion t measured from the big bang and the Hubble time (we now return to the ususal time scale where the beginning of the background expansion, the big bang, is associated with the moment t). Let us define the Hubble time t as the time at which [FORMULA] becomes noticeably small, say [FORMULA]. In case of explosion in an early epoch, t, the blast wave in an open universe experiences three consecutive self-similar episodes of its evolution:

  • Sedov stage ([FORMULA], [FORMULA]) [FORMULA] quasi-ITO-B cosmological blast wave stage ([FORMULA], [FORMULA]) [FORMULA] "freezing-out" ([FORMULA], [FORMULA]).

The profile of the blast wave gradually rearranges itself according to the above mentioned scheme.

If explosion occurs at late times t when the gravity forces have already fallen off, an intermediate ITO-B stage is lacking. In this case the shell does not form and the blast wave evolution can be described with great accuracy by the present invariant solution.

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Online publication: June 12, 1998