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.
