Why do stars form in clusters? An analytic model for stellar correlation functions

Recently, we have shown that if the interstellar medium is governed by supersonic turbulent flows, the excursion-set formalism can be used to calculate the statistics of self-gravitating objects over a wide range of scales. On the largest self-gravitating scales ('first crossing'), these correspond to giant molecular clouds (GMCs), and on the smallest non-fragmenting self-gravitating scales ('last crossing'), to protostellar cores. Here, we extend this formalism to rigorously calculate the autocorrelation and cross-correlation functions of cores (and by extension, young stars) as a function of spatial separation and mass, in analogy to the cosmological calculation of halo clustering. We show that this generically predicts that star formation is very strongly clustered on small scales: stars form in clustered regions, themselves inside GMCs. Outside the binary-star regime, the projected correlation function declines as a weak power law, until a characteristic scale which corresponds to the characteristic mass scale of GMCs. On much larger scales the clustering declines such that star formation is not strongly biased on galactic scales, relative to the actual dense gas distribution. The precise correlation function shape depends on properties of the turbulent spectrum, but its qualitative behaviour is quite general. The predictions agree well with observations of young star and core autocorrelation functions over similar to 4 dex in radius. Clustered star formation is a generic consequence of supersonic turbulence if most of the power in the velocity field, hence the contribution to density fluctuations, comes from large scales similar to h. The distribution of self-gravitating masses near the sonic length is then imprinted by fluctuations on larger scales. We similarly show that the fraction of stars formed in 'isolated' modes should be small, less than or similar to 10 per cent.