Interpreting the Mean Surface Density of Companions in Star-Forming   Regions
  We study the interpretation of the mean surface density of stellar companions as a function of separation (or the two point correlation function of stars) in star-forming regions. First, we consider the form of the functions for various simple stellar distributions (binaries, global density profiles, clusters, and fractals) and the effects of survey boundaries.   Following this, we study the dependencies of the separation at which a transition from the binary to the large-scale clustering regime occurs. Larson (1995) found that the mean surface density of companions follows different power-law functions of separation in the two regimes. He identified the transition separation with the typical Jeans length in the molecular cloud. However, we show that this is valid only for special cases. In general, the transition separation depends on the volume density of stars, the depth of the star-forming region, the volume-filling nature of the stellar distribution, and on the parameters of the binaries. Furthermore, the transition separation evolves with time.   We then apply these results to the Taurus-Auriga, Ophiuchus, and Orion Trapezium star-forming regions. We find that while the transition separation in the Taurus-Auriga star-forming region may indicate a typical Jeans length, this is not true of the Orion Trapezium Cluster. We also show that there is no evidence for sub-clustering or fractal structure in the stars of the Orion Trapezium Cluster. This is consistent with the fact that, if such structure were present when the stars formed, it would have been erased by the current age of the cluster due to the stellar velocity dispersion. [Abstract abridged] 
Matthew R. Bate, Cathie J. Clarke, and Mark J. McCaughrean