Why Do Stars Form In Clusters? An Analytic Model for Stellar Correlation
  Functions
  Recently, we have shown that if the ISM is governed by super-sonic 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 GMCs, and on
the smallest non-fragmenting self-gravitating scales ('last crossing'), to
protostellar cores. Here, we extend this formalism to rigorously calculate the
auto 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
clusters, 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 behavior is quite general. The predictions agree well with
observations of young star and core autocorrelation functions over ~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. 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 (\lesssim10%).
