A Simple Model for the Relationship Between Star Formation and Surface
  Density
  We investigate the relationship between the star formation rate per unit area
and the surface density of the ISM (the local Kennicutt-Schmitt law) using a
simplified model of the ISM and a simple estimate of the star formation rate
based on the mass of gas in bound clumps, the local dynamical timescales of the
clumps, and an efficiency parameter of around 5 per cent. Despite the
simplicity of the approach, we are able to reproduce the observed linear
relation between star formation rate and surface density of dense (molecular)
gas. We use a simple model for the dependence of H_2 fraction on total surface
density to argue why neither total surface density nor the HI surface density
are good local indicators of star formation rate. We also investigate the
dependence of the star formation rate on the depth of the spiral potential. Our
model indicates that the mean star formation rate does not depend significantly
on the strength of the spiral potential, but that a stronger spiral potential,
for a given mean surface density, does result in more of the star formation
occurring close to the spiral arms. This agrees with the observation that grand
design galaxies do not appear to show a larger degree of star formation
compared to their flocculent counterparts.
