A New Method for Obtaining the Star Formation Law in Galaxies

We present a new observational method to evaluate the exponent of the star formation law as initially formulated by Schmidt, i.e., the power-law expression assumed to relate the rate of star formation in a volume of space to the local total gas volume density present there. Total volume densities in the gas clouds surrounding an OB association are determined with a simple model which considers the atomic hydrogen as a photodissociation product on the cloud surfaces. The photodissociating photon flux incident on the cloud is computed from the far-UV luminosity of the OB association and the geometry. As an example, we have applied this "PDR Method" to a sample of star-forming regions in M33 using Very Large Array (VLA) 21 cm data for the Hi and Galaxy Evolution Explorer (GALEX) imagery in the far-UV. With these two observables, our approach provides an estimate of the total volume density of hydrogen (atomic + molecular) in the gas clouds surrounding the young star cluster. A graph in logarithmic coordinates of the cluster UV luminosity versus the total density in the surrounding gas provides a direct measure of the exponent of the star formation law. However, we show that this plot is severely affected by observational selection, which renders large areas of the diagram inaccessible to the data. An ordinary least-squares regression fit to a straight line, therefore, gives a strongly biased result. In the present case, the slope of such a fit primarily reflects the boundary defined when the 21 cm line becomes optically thick and is no longer a reliable measure of the H I column density. We use a maximum likelihood statistical approach which can deal with truncated and skewed data, and also takes account of the large uncertainties in the total gas densities which we derive. The exponent we obtain for the Schmidt law in M33 is 1.4 +/- 0.2.