The Scale Dependence of the Molecular Gas Depletion Time in M33

We study the Local Group spiral galaxy M33 to investigate how the observed scaling between the (kpc-averaged) surface density of molecular gas (\Sigma_H2) and recent star formation rate (\Sigma_SFR) relates to individual star-forming regions. To do this, we measure the ratio of CO emission to extinction-corrected Halpha emission in apertures of varying sizes centered both on peaks of CO and Halpha emission. We parameterize this ratio as a molecular gas (H_2) depletion time (\tau_dep). On large (kpc) scales, our results are consistent with a molecular star formation law (Sigma_SFR \sim Sigma_H2^b) with b \sim 1.1 - 1.5 and a median \tau_dep \sim 1 Gyr, with no dependence on type of region targeted. Below these scales, \tau_dep is a strong function of adopted angular scale and the type of region that is targeted. Small (\lesssim 300pc) apertures centered on CO peaks have very long \tau_dep (i.e., high CO-to-Halpha flux ratio) and small apertures targeted toward Halpha peaks have very short \tau_dep. This implies that the star formation law observed on kpc scales breaks down once one reaches aperture sizes of \lesssim 300pc. For our smallest apertures (75pc), the difference in \tau_dep between the two types of regions is more than one order of magnitude. This scale behavior emerges from averaging over star-forming regions with a wide range of CO-to-Halpha ratios with the natural consequence that the breakdown in the star formation law is a function of the surface density of the regions studied. We consider the evolution of individual regions the most likely driver for region-to-region differences in \tau_dep (and thus the CO-to-Halpha ratio).