The Stellar Initial Mass Function from Random Sampling in Hierarchical Clouds. II. Statistical Fluctuations and a Mass Dependence for Starbirth Positions and Times

Observed variations in the slope of the stellar initial mass function (IMF) are shown to be consistent with a previously introduced model in which the protostellar gas is randomly sampled from clouds with a self-similar hierarchical structure. Root mean square variations in the IMF slope around the Salpeter value are ±0.4 when only 100 stars are observed, and ±0.1 when 1000 stars are observed. Similar variations should be present in other stochastic models as well. The hierarchical sampling model reproduces the tendency for massive stars to form closer to the center of a cloud at a time somewhat later than the formation time of the lower mass stars. The systematic variation in birth position results from the tendency for the trunk and larger branches of the hierarchical tree of cloud structure to lie closer to the cloud center, while the variations in birth order result from the relative infrequency of stars with larger masses. The hierarchical cloud sampling model has now reproduced most of the reliably observed features of the cluster IMF. The power-law part of the IMF comes from cloud hierarchical structure that is sampled during various star formation processes with a relative rate proportional to the square root of the local density. These processes include turbulence compression, magnetic diffusion, gravitational collapse, and clump or wavepacket coalescence, all of which have about this rate dependence. The low-mass flattening comes from the inability of gas to form stars below the thermal Jeans mass at typical temperatures and pressures. The thermal Jeans mass is the only relevant scale in the problem. Considerations of heating and cooling processes indicate why the thermal Jeans mass should be nearly constant in normal environments and why this mass might increase in starburst regions. In particular, the relative abundance of high-mass stars should increase where the average density of the interstellar medium is very large; accompanying this increase should be an increase in the average total efficiency of star formation. Alternative models in which the rate of star formation is independent of density and the local efficiency decreases systematically with increasing stellar mass can also reproduce the IMF, but this is an adjustable result and not a fundamental property of hierarchical cloud structure, as is the preferred model. The steep IMF in the extreme field is not explained by the model, but other origins are suggested, including one in which massive stars in low-pressure environments halt star formation in their clouds. In this case, the slope of the extreme field IMF is independent of the slope of each component cluster IMF and is given by (γ-1)/α for a cloud mass function slope, -γ~-2, and a power-law relation, ML∝Mαc, between the largest star in a low-pressure cloud, ML, and the cloud mass, Mc. A value of α~1/4 is required to explain the extreme field IMF as a superposition of individual cluster IMFs; cloud destruction by ionizing has this property. We note that the similarity between cluster IMFs and the average IMF from global studies of galaxies implies that most stars form in clusters and that massive stars do not generally halt star formation in the same cloud.