## 3. Self-similar solutionUsing 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 where is the average molecular weight and
is the number density of particles in the
ambient medium. The expansion velocity of the shell Inserting eqs. (10) and (11) in eq. (3) we get the instability parameter : The temporal evolution of defines an 'instability time' , i.e. the onset of gravitational instability, which can be evaluated by the condition , or Eq. (12) shows that the system is always gravitationally stable prior to and becomes unstable at later times. The radius of the shell at is 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. , and , according to formulae (13), (14) and (15) are shown together with the results of 3D computer simulations (see below) in Fig. 1.
In order to calculate the 'fragmentation time' , i.e. the time when the fragmentation becomes strongly nonlinear, we can rewrite the growth rate (1) as and apply the relation together with the Sedov solution (10). From the condition , we get with . It should be noted that the ratio of the fragmentation time to the instability time 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 . 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 ,
the fragmentation time is of the order of (5-6)
. The corresponding radii of the shells are
shown in Fig. 2: for we obtain radii of
600 pc to 1.1 kpc for sound speeds within the shell
Since gravitational instability is a condition for the onset of star formation, radii from Fig. 2 and from Fig. 3 give a lower limit for distances between the subsequent star-forming regions and a lower limit for the required in an OB association to propagate star formation at a given density of the ambient medium . 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 , radii at the fragmentation time shrink to 10-20 pc which are smaller than typical tidal radii for a - stellar system in a 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). © European Southern Observatory (ESO) 1997 Online publication: March 24, 1998 |