Interpreting the mean surface density of companions in star-forming regions

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We study the interpretation of the mean surface density of stellar companions as a function of separation (or, equivalently, 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 dependences of the separation at which a transition from the binary to the large-scale clustering regime occurs. Larson 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 the parameters of the binaries. Furthermore, the transition separation evolves with time. We also note that in young star-forming regions, binaries with separations greater than the transition separation may exist, while in older unbound clusters that have expanded significantly the transition contains a record of the stellar density when the stars formed. 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 caution against overinterpreting the mean surface density of stellar companions; while Larson showed that Taurus–Auriga is consistent with the stars having a fractal large-scale distribution, we show that Taurus–Auriga is also consistent with stars being grouped in non-hierarchical clusters. We also argue that to make a meaningful study of the stellar distribution in a star-forming region requires a relatively complete stellar survey over a large area. Such a survey does not currently exist for Ophiuchus. Finally, we show that there is no evidence for subclustering 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.