A threshold for sequentially self-propagating star formation

When a stellar wind bubble expands into an homogeneous medium, there are two possible outcomes. This is due to the fact that the self-gravity of the swept-up shell acts in two orthogonal directions: tangentially , to promote fragmentation of the shell, and radially , to decelerate expansion of the shell. The outcome depends on whether self-gravity works faster in the tangential or the radial direction.
If the wind power ℒ_o is large and the effective isothermal sound speed a _o in the swept-up gas is small – approximately         – tangential self-gravity works faster. A thin dense shell is swept up and fragments while it is still expanding supersonically. This is the scenario often invoked to explain sequentially self-propagating star formation.
However, if ℒ_o is small and/or a _o is large, radial self-gravity works faster. Expansion of the bubble stalls before the shell can fragment. The expansion speed ceases to be supersonic, the outer shock dissipates, and the shell is neither thin nor dense. Under this circumstance, the shell is unlikely to fragment and star formation will not propagate sequentially.
These conclusions are probably not altered significantly when the medium into which the wind blows is inhomogeneous, provided that the mean density on opposite sides of the bubble does not differ by many orders of magnitude.