Title:Hierarchical Star-Formation in M33: Fundamental properties of the star-forming regions

Abstract: Star-formation within galaxies appears on multiple scales, from spiral structure, to OB associations, to individual star clusters, and often sub-structure within these clusters. This multitude of scales calls for objective methods to find and classify star-forming regions, regardless of spatial size. To this end, we present an analysis of star-forming groups in the local group spiral galaxy M33, based on a new implementation of the Minimum Spanning Tree (MST) method. Unlike previous studies which limited themselves to a single spatial scale, we study star-forming structures from the effective resolution limit (~20pc) to kpc scales. We find evidence for a continuum of star-forming group sizes, from pc to kpc scales. We do not find a characteristic scale for OB associations, unlike that found in previous studies, and we suggest that the appearance of such a scale was caused by spatial resolution and selection effects. The luminosity function of the groups is found to be well represented by a power-law with an index, -2, similar to that found for clusters and GMCs. Additionally, the groups follow a similar mass-radius relation as GMCs. The size distribution of the groups is best described by a log-normal distribution and we show that within a hierarchical distribution, if a scale is selected to find structure, the resulting size distribution will have a log-normal distribution. We find an abrupt drop of the number of groups outside a galactic radius of ~4kpc, suggesting a change in the structure of the star-forming ISM, possibly reflected in the lack of GMCs beyond this radius. (abridged)