Spatial statistics in star forming regions: is star formation driven by column density alone?

Star formation is known to occur more readily where more raw materials are available. This is often expressed by a 'Kennicutt-Schmidt' relation where the surface density of Young Stellar Objects (YSOs) is proportional to column density to some power, $\mu$. The aim of this work was to determine if column density alone is sufficient to explain the locations of Class 0/I YSOs within Serpens South, Serpens Core, Ophiuchus, NGC1333 and IC348, or if there is clumping or avoidance that would point to additional influences on the star formation. Using the O-ring test as a summary statistic, 95 per cent confidence envelopes were produced for different values of $\mu$ from probability models made using the Herschel column density maps. The YSOs were tested against four distribution models: the best-estimate of $\mu$ for the region, $\mu=0$ above a minimum column density threshold and zero probability elsewhere, $\mu=1$, and the power-law that best represents the five regions as a collective, $\mu=2.05 \pm 0.20$. Results showed that $\mu=2.05$ model was consistent with the majority of regions and, for those regions, the spatial distribution of YSOs at a given column density is consistent with being random. Serpens South and NGC1333 rejected the $\mu = 2.05$ model on small scales of $\sim 0.15 \mathrm{pc}$ which implies that small-scale interactions may be necessary to improve the model. On scales above 0.15 pc, the positions of YSOs in all five regions can be well described using column density alone.