The K+09 derived their column density calculations from the analytic model from krumholz08 of formation and photodissociation of a spherical cloud bathed in a uniform ISRF.\@ See krumholz09 and krumholz08 for details in the derivation, and a summary of the results in lee12. Here we quickly summarize the important assumptions and results.

\noindent where is the dust optical depth of the cloud if and dust is well-mixed, is the dust-to-molecule absorption ratio, and is the ratio of number densities between the cloud molecular component and CNM component. They modeled using empircal results from PDR models as a function of the ratio of the rate at which Lyman-Werner photons are absorbed by dust grains to the rate at which LW photons are absorbed by , . K+09 relates to the CNM and WNM properties. They define

\noindent where is the CNM number density, and is the minimum CNM number density required for the CNM to remain in pressure equilibrium with the WNM.\@ Written in terms of

We can calculate the CNM temperature, from EQ (19) in K+09. See this module for the solution of given a value. We can plot as a function of galactic longitude:

where we can see that Perseus and Taurus have much lower predicted values than California. This is expected given the low HI surface density thresholds seen in Perseus and Taurus. The locations of cores in galactic coordinates is shown here (taken from Lombardi et al. (2007))