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

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6. Exhaustion of the energy supplies and the sound speed in the ambient medium

The energy supply from SN into the bubble is limited by the finite number of massive stars in an OB association. In our Galaxy this number has been estimated by Tenorio-Tagle & Bodenheimer (1988) to 40 - 100. When the energy input stops, the supersonic expansion speed decelerates more rapidly than according to the self-similar solution (10), reaching the value of the sound speed in the ambient medium earlier at lower shell surface densities. This implies that the fragmentation conditions (3) and (7) may not be fulfilled. This is particularly important in the low density interstellar medium, where the deceleration of expanding shells proceeds slowly and the value of [FORMULA] is large.

Another constraint is given by the value of the sound speed in the ambient medium [FORMULA]. The accumulation of mass into the shell stops when v decelerates to [FORMULA], which happens at time [FORMULA] (for the uninterrupted energy input given by the following formula):


The shell motion continues as a sound wave. Such subsonic expansion dilutes the shell decreasing its surface density [FORMULA] and increasing the value of the instability parameter [FORMULA], which may imply that the fragmentation conditions (3) and (7) are again not fulfilled.

We distinguish three cases:

  • for [FORMULA] the shell never fragments;
  • for [FORMULA] the fragments form if at [FORMULA] the value of the fragmentation integral [FORMULA] is close to 1. If [FORMULA] is close to 0 the fragments reexpand again;
  • for [FORMULA] the fragments detach from the shell moving along individual ballistic orbits in galaxy.

It has been shown in the previous section that for spherical shells with continuous energy supply [FORMULA] is lower than 10 km/s. When the energy supply stops before [FORMULA] the velocity [FORMULA] is even lower.

As it is shown in Fig. 3 for [FORMULA] the fragments form at [FORMULA] or [FORMULA] when the values of [FORMULA] and [FORMULA] are to the right of the solid lines. They are molecular if [FORMULA] and [FORMULA] are above the dashed lines. For a homogeneous, non-stratified distribution of the ambient medium, and with OB associations composed of 40 - 100 massive stars, we can separate: [FORMULA], which gives [FORMULA] and the shell fragments; [FORMULA], which implies [FORMULA], and the shell will fragment only in some cases; [FORMULA], which yields [FORMULA], and the fragments will never form.

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© European Southern Observatory (ESO) 1997

Online publication: March 24, 1998