Star Formation Laws: the Effects of Gas Cloud Sampling
  Recent observational results indicate that the functional shape of the
spatially-resolved star formation-molecular gas density relation depends on the
spatial scale considered. These results may indicate a fundamental role of
sampling effects on scales that are typically only a few times larger than
those of the largest molecular clouds. To investigate the impact of this
effect, we construct simple models for the distribution of molecular clouds in
a typical star-forming spiral galaxy, and, assuming a power-law relation
between SFR and cloud mass, explore a range of input parameters. We confirm
that the slope and the scatter of the simulated SFR-molecular gas surface
density relation depend on the size of the sub-galactic region considered, due
to stochastic sampling of the molecular cloud mass function, and the effect is
larger for steeper relations between SFR and molecular gas. There is a general
trend for all slope values to tend to ~unity for region sizes larger than 1-2
kpc, irrespective of the input SFR-cloud relation. The region size of 1-2 kpc
corresponds to the area where the cloud mass function becomes fully sampled. We
quantify the effects of selection biases in data tracing the SFR, either as
thresholds (i.e., clouds smaller than a given mass value do not form stars) or
backgrounds (e.g., diffuse emission unrelated to current star formation is
counted towards the SFR). Apparently discordant observational results are
brought into agreement via this simple model, and the comparison of our
simulations with data for a few galaxies supports a steep (>1) power law index
between SFR and molecular gas.
