2009ApJ...694..978H

A statistical equilibrium treatment of a simplified three-level hydrogen molecule has been used by several authors to predict the volume densities of hydrogen nuclei produced by photodissociation in cold low-density clouds. This procedure does not require knowledge of the excitation conditions in the molecular gas, but the derived densities depend linearly on an integral function,, which approaches a constant value as the molecular hydrogen column (N₂) becomes infinite. Previous studies presumed this function could be replaced by that constant without determining the function's actual value in locations of interest. Numerical integrations of have been performed to assess the impact of that assumption using four different published forms of the H₂self-shielding function embedded in its integrand. In each case, these calculations showed that N₂ must exceed ~6x10²⁰/cm2 for to be constant when solar neighborhood values are assumed for the dust-to-gas mass ratio (δ₀), but that constant's actual value,, depends on the function chosen to describe self-shielding. When the latter is represented by N₂^–1/2 as often assumed in the literature for an isolated H₂ absorption line, scales as ()₀(δ/δ₀)^1/2. However, use of a more exact self-shielding function derived previously for an ensemble of overlapping lines shows this quantity actually scales as (δ/δ₀)^0.7. The volume density of photodissociated hydrogen nuclei found in this case, n_E, is related to the value computed by assuming self-shielding arises solely from a single line, n_S, by n_E= n_S(δ/δ₀)^0.2.