The Relationship Between Gas Content and Star Formation in Molecule-Rich Spiral Galaxies

We investigate the relationship between HI, H_2, and the star formation rate (SFR) using azimuthally averaged data for seven CO-bright spiral galaxies. Contrary to some earlier studies based on global fluxes, we find that the SFR surface density exhibits a much stronger correlation with the H_2 than with the HI, as the HI surface density saturates at a value of $\sim$10 Msol pc^{-2} or even declines at large SFR surface densities. Hence the good correlation between the SFR surface density and the total (HI+H_2) gas surface density is driven by the molecular component in these galaxies. We find no clear evidence for a link between the gravitational instability parameter for the gas disk (Q_g) and the SFR, and suggest that Q_g be considered a measure of the gas fraction. This implies that for a state of marginal gravitational stability to exist in galaxies with low gas fractions, it must be due to instability of the stellar disk. In regions where we have both HI and CO measurements, the ratio of HI to H_2 surface density scales with radius R as roughly R^{1.5}, and we suggest that the balance between HI and H_2 is determined primarily by the midplane interstellar pressure. These results favor a "law" of star formation in quiescent disks in which the ambient pressure and metallicity control the formation of molecular clouds from HI, with star formation then occurring at a roughly constant rate per unit H_2 mass. (abstract abridged)