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Astron. Astrophys. 328, 121-129 (1997)

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3. Self-similar solution

Using a dimensional analysis, Sedov (1959) derived a solution for supersonic, spherical expansion of a shell into the homogeneous medium in case of a single instantaneous energy input and vanishing external pressure. For the case of interstellar wind bubbles Castor et al. (1975) and Weaver et al. (1977) derived the solution which describes the structure and evolution of the shocked stellar wind region (their zone b). This case is characterized by steady energy and mass input rates. Since this should be more appropriate for the temporal energy injection by type II supernovae events in an OB association, we applied the same method here. The solution gives the relation between the radius of the shell R, the supernova rate [FORMULA], the mass density of the ambient medium [FORMULA], and the expansion time t as follows ([FORMULA] is the energy release per supernova):

[EQUATION]

where [FORMULA] is the average molecular weight and [FORMULA] is the number density of particles in the ambient medium. The expansion velocity of the shell v is given by the time derivative of Eq. (10). Assuming that the total mass [FORMULA] is swept up in the shell, the surface density is given by

[EQUATION]

Inserting eqs. (10) and (11) in eq. (3) we get the instability parameter [FORMULA]:

[EQUATION]

The temporal evolution of [FORMULA] defines an 'instability time' [FORMULA], i.e. the onset of gravitational instability, which can be evaluated by the condition [FORMULA], or

[EQUATION]

Eq. (12) shows that the system is always gravitationally stable prior to [FORMULA] and becomes unstable at later times. The radius of the shell at [FORMULA] is

[EQUATION]

and the expansion velocity

[EQUATION]

The expansion velocity at the fragmentation time is independent of the density of the ambient medium. This may be one of the reasons why the random motion in the interstellar medium is almost constant over a broad range of densities. [FORMULA], [FORMULA] and [FORMULA], according to formulae (13), (14) and (15) are shown together with the results of 3D computer simulations (see below) in Fig. 1.

[FIGURE] Fig. 1. The instability time [FORMULA] (a), the shell radius [FORMULA] (b) and the expansion velocity [FORMULA] (c) as a function of [FORMULA] for different [FORMULA]. Continuous lines are the solution as given by formulae (13), (14) and (15), symbols show results of computer simulations. [FORMULA] corresponds to [FORMULA], [FORMULA] to [FORMULA], [FORMULA] to [FORMULA], and [FORMULA] to [FORMULA].

In order to calculate the 'fragmentation time' [FORMULA], i.e. the time when the fragmentation becomes strongly nonlinear, we can rewrite the growth rate (1) as

[EQUATION]

and apply the relation [FORMULA] together with the Sedov solution (10). From the condition [FORMULA], we get

[EQUATION]

with [FORMULA]. It should be noted that the ratio of the fragmentation time [FORMULA] to the instability time [FORMULA] is independent of the density of the ambient medium and of the energy injection rate and the sound speed in the shell. For a homogeneous medium a numerical solution of Eq. (17) gives [FORMULA]. This value will be compared with the results of 3D computer simulations below.

The timescales themselves are also almost independent on the energy injection rate, but depend on the density of the ambient medium. For a typical density in the solar neighbourhood of [FORMULA], the fragmentation time is of the order of (5-6) [FORMULA]. The corresponding radii of the shells are shown in Fig. 2: for [FORMULA] we obtain radii of 600 pc to 1.1 kpc for sound speeds within the shell c between [FORMULA] and [FORMULA], respectively. In Fig. 3 we give the [FORMULA] as a function of the ambient medium density [FORMULA], which is necessary to reach the critical value of column density [FORMULA] (9) at times [FORMULA] and [FORMULA] required for molecularization. This diagram gives the lower limit for [FORMULA] at given [FORMULA] for the formation of molecular clouds from fragmenting shells.

[FIGURE] Fig. 2. The contour lines show the radius of the shell (in pc) at the fragmentation time [FORMULA] as a function of the supernova rate [FORMULA] and the number density n in a homogeneous ambient ISM. The sound speed inside the shell was assumed to be [FORMULA] km/s and the molecular weight was set to [FORMULA].
[FIGURE] Fig. 3. Solid lines show the value of [FORMULA] required by condition (9) at [FORMULA] and at [FORMULA] as a function of the density [FORMULA] of the ambient medium. Dashed lines indicate constraints on the [FORMULA] implied by the non-zero sound speed in the ambient medium (see Sect. 6).

Since gravitational instability is a condition for the onset of star formation, radii from Fig. 2 and [FORMULA] from Fig. 3 give a lower limit for distances between the subsequent star-forming regions and a lower limit for the required [FORMULA] in an OB association to propagate star formation at a given density of the ambient medium [FORMULA]. If the propagating star formation is induced by instabilities in the shells, we may also estimate the speed of propagation.

If the OB stars are exploding inside a molecular cloud of [FORMULA], radii at the fragmentation time shrink to 10-20 pc which are smaller than typical tidal radii for a [FORMULA] - [FORMULA] stellar system in a [FORMULA] galaxy. Therefore, at high densities in the ambient medium the system of fragments can form quickly enough in order to prevent a tidal disruption of the shell. Finally, these clumps might recollapse and form a gravitationally bound system as suggested by Brown et al. (1991).

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998

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