Constraints on star formation from the close packing of protostars in clusters

The millimetre-wave continuum sources (MCS) in Ophiuchus have mutual collision rates less than their collapse rates by a factor of 10 to 100, suggesting that most will form stars without further interactions. However, the ratio of these rates would have exceeded unity in the past if they were only 2.5 times larger than they are now. Such a high previous ratio suggests three possible scenarios: (1) the MCS contracted from lower densities and acquired their present masses through collisional agglomeration, (2) they contracted independently from lower densities elsewhere and moved to the cluster core recently, or (3) they grew from smaller sizes at a constant high density. The third of these is most likely, implying that the MCS formed in the shocked regions of a supersonically turbulent fluid. The first scenario gives the wrong mass function and the second does not give the observed hierarchical clustering. The ratio of rates also exceeds unity today if the MCS have envelopes with smooth profiles that end in pressure balance with the ambient cloud cores; this suggests again that turbulent flows define their outer layers. Proximity constraints like this are even more important in massive clusters, including globular clusters, in which 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 else significant coalescence will occur in the protostellar phase, possibly forming massive black holes. 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 (IMF) are also discussed. Warm cloud cores can produce a top-heavy IMF because of a simultaneous increase in the thermal Jeans mass and the collisional destruction rate of low-mass MCS.