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 interstellar medium (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 time-scales of the clumps and an efficiency parameter of around  ε≈ 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 H₂ fraction on total surface density to argue why neither total surface density nor the H  i surface density is a good local indicator 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.