A Multi-Scale Study of Star Formation in Messier 33

For the Local Group Scd galaxy M 33 this paper presents a multi-scale study of the relationship between the monochromatic star formation rate (SFR) estimator based on 12 $\mu$m emission and the total SFR estimator based on a combination of far-ultraviolet and 24 $\mu$m emission. We show the 12 $\mu$m emission to be a linear estimator of total SFR on spatial scales from 782 pc down to 49 pc, over almost four magnitudes in SFR. These results therefore extend to sub-kpc length scales the analogous results from other studies for global length scales. We use high-resolution HI and $^{12}\mathrm{CO}(J=2-1)$ image sets from the literature to compare the star formation to the neutral gas. For the full range of length scales we find well-defined power-law relationships between 12 $\mu$m-derived SFR surface densities and neutral gas surface densities. For the H$_\mathrm{2}$ gas component almost all correlations are consistent with being linear. No evidence is found for a breakdown in the star formation law at small length scales in M 33 reported by other authors. We show that the average star formation efficiency in M 33 is roughly $10^{-9}$ yr$^{-1}$ and that it remains constant down to giant molecular cloud length scales. Toomre and shear-based models of the star formation threshold are shown to inaccurately account for the star formation activity in the inner disc of M 33. Finally, we clearly show that the HI saturation limit of $\approx 9$ M$_{\odot}$ pc$^{-2}$ reported in the literature for other galaxies is not an intrinsic property of M 33 - it is systematically introduced as an artefact of spatially smoothing the data.