Comparing the $M_{gas}-N_{yso}$ Relation inside a Giant Molecular Cloud

In this paper we present a simple analysis around scaling relations derived from the Schmidt conjecture for star-forming molecular clouds, at the intra-cloud scale. Using a hierarchical tree (dendrograms) above a constant threshold ($A_V$ = 7 mag), we separate individual gas structures in a column density map of the nearby Giant Molecular Cloud Orion A, constructed from Herschel far-infrared maps. These structures define regions of dense molecular gas that can actively form stars. We also estimate their current embedded population using a list of known young stars. From the combined analysis of the column density map and the young star catalog, we construct a series of plots that show the intra-cloud level behavior of three well-known scaling relations: $N_{yso}$ vs. $M_{gas}$, $\Sigma_{SFR}$ vs. $\Sigma_{gas}$ and $R_{eq}$ vs. $M_{gas}$. Our dataset, along with other sets from literature, show the validity of a linear relation for $N_{yso}$ vs. $M_{gas}$, from intra-cloud to inter-cloud scales, over three orders of magnitude. We also especulate on the possibility that the relation could be valid over an even larger scale range. Additionally, our data are consistent with the $R_{eq}$ vs. $M_{gas}$ discussed in previous studies. However, our data is not quite in agreement with previously proposed fits for the $\Sigma_{SFR}$ vs. $\Sigma_{gas}$ relation, and we discuss the implications of using the free-fall timescale as the main parameter defining the star-forming efficiency in dense gas regions.