ANALYTIC H I-to-H₂ PHOTODISSOCIATION TRANSITION PROFILES

We present a simple analytic procedure for generating atomic (H I) to molecular (H₂) density profiles for optically thick hydrogen gas clouds illuminated by far-ultraviolet radiation fields. Our procedure is based on the analytic theory for the structure of one-dimensional H I/H₂ photon-dominated regions, presented by Sternberg et al. Depth-dependent atomic and molecular density fractions may be computed for arbitrary gas density, far-ultraviolet field intensity, and the metallicity-dependent H₂ formation rate coefficient, and dust absorption cross section in the Lyman-Werner photodissociation band. We use our procedure to generate a set of H I-to-H₂ transition profiles for a wide range of conditions, from the weak- to strong-field limits, and from super-solar down to low metallicities. We show that if presented as functions of dust optical depth, the H I and H₂ density profiles depend primarily on the Sternberg " parameter" (dimensionless) that determines the dust optical depth associated with the total photodissociated H I column. We derive a universal analytic formula for the H I-to-H₂ transition points as a function of just αG. Our formula will be useful for interpreting emission-line observations of H I/H₂ interfaces, for estimating star formation thresholds, and for sub-grid components in hydrodynamics simulations.