A new method for obtaining the star formation law in galaxies
  We present a new observational method to evaluate the star formation law as
formulated by Schmidt: the power-law expression assumed to relate the rate of
star formation in a volume of space to the local total gas volume density.
Volume densities in the clouds surrounding an OB association are determined
with a simple model which considers atomic hydrogen as a photodissociation
product on cloud surfaces. The photodissociating flux incident on the cloud is
computed from the far-UV luminosity of the OB association and the geometry. We
have applied this "PDR Method" to a sample of star-forming regions in M33 using
VLA 21-cm data for the HI and GALEX imagery in the far-UV. It provides an
estimate of the total volume density of hydrogen (atomic + molecular) in the
gas clouds surrounding the young star cluster. A logarithmic graph of the
cluster UV luminosity versus the surrounding gas density is a direct measure of
the star formation law. However, this plot is severely affected by
observational selection, rendering large areas of the diagram inaccessible to
the data. An ordinary least-squares regression fit therefore gives a strongly
biased result. Its slope primarily reflects the boundary defined when the 21-cm
line becomes optically thick, no longer reliably measuring the HI column
density. We use a maximum-likelihood statistical approach which can deal with
truncated and skewed data, taking into account the large uncertainties in the
derived total gas densities. The exponent we obtain for the Schmidt law in M33
is 1.4 \pm 0.2.
