Constraints on Star Formation from the Close Packing of Protostars in   Clusters
  The mm-wave continuum sources (MCS) in Ophiuchus have mutual collision rates less than their collapse rates by a factor of 10 to 100, suggesting most will form stars without further interactions. However, this ratio of rates would have exceeded unity in the past if the MCS were 2.5 times larger than they are now. Such a high previous ratio was not likely, suggesting that the MCS grew from smaller sizes at a constant high density, possible behind turbulent shocks. Proximity constraints are even more important in massive clusters, where massive stars with the same or greater space density are more strongly interacting than the Ophiuchus MCS. As a result, the density contrast for MCS must be larger in massive clusters than it is in Ophiuchus, or significant coalescence will occur in the protostellar phase, possible forming a massive black hole. A proportionality to the second power of the Mach number allows the MCS cores to collapse independently. These results suggest that stars in dense clusters generally form on a dynamical time by the continuous collection and rapid collapse of turbulence-shocked gas. Implications of proximity constraints on the initial stellar mass function are also discussed. 
Bruce G. Elmegreen (1), Mohsen Shadmehri (2) ((1) IBM Watson Research   Center; (2) Ferdowsi University, Mashhad, Iran)