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 1 < z < 3 in massive disc 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.