C+/H2 gas in star-forming clouds and galaxies

We present analytic theory for the total column density of singly ionized carbon (C+) in the optically thick photon dominated regions (PDRs) of far-UV irradiated (star-forming) molecular clouds. We derive a simple formula for the C+ column as a function of the cloud (hydrogen) density, the far-UV field intensity, and metallicity, encompassing the wide range of galaxy conditions. When assuming the typical relation between UV and density in the cold neutral medium, the C+ column becomes a function of the metallicity alone. We verify our analysis with detailed numerical PDR models. For optically thick gas, most of the C+ column is mixed with hydrogen that is primarily molecular (H2), and this "C+/H2" gas layer accounts for almost all of the `CO-dark' molecular gas in PDRs. The C+/H2 column density is limited by dust shielding and is inversely proportional to the metallicity down to ~0.1 solar. At lower metallicities, H2 line blocking dominates and the C+/H2 column saturates. Applying our theory to CO surveys in low redshift spirals we estimate the fraction of C+/H2 gas out of the total molecular gas to be typically ~0.4. At redshifts 1disc galaxies the C+/H2 gas represents a very small fraction of the total molecular gas (<0.16). This small fraction at high redshifts is due to the high gas surface densities when compared to local galaxies.