Analytic H I-to-H2 Photodissociation Transition Profiles

We present a simple analytic procedure for generating atomic (H I) to molecular ({{{H}}}₂) 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/{{{H}}}₂ 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}{-}{{{H}}}₂ 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 {{{H}}}₂ density profiles depend primarily on the Sternberg “α G 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}{-}{{{H}}}₂ transition points as a function of just α G. Our formula will be useful for interpreting emission-line observations of H I/{{{H}}}₂ interfaces, for estimating star formation thresholds, and for sub-grid components in hydrodynamics simulations.