Abstract
We present a simple analytic procedure for generating atomic-to-molecular
(HI-to-H$_2$) density profiles for optically thick clouds illuminated by
far-ultraviolet radiation. Our procedure is based on the analytic theory for
the structure of 1D HI/H$_2$ photon-dominated regions, presented by Sternberg
et al. (2014). Depth-dependent HI and H$_2$ density fractions may be computed
for arbitrary gas density, far-ultraviolet field intensity, and the metallicity
dependent H$_2$ formation rate coefficient, and dust absorption cross section.
We use our procedure to generate a set of HI-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 HI and H$_2$ density profiles depend primarily on the
Sternberg "$G$ parameter" (dimensionless) that determines the dust
optical depth associated with the total photodissociated HI column. We derive a
universal analytic formula for the HI-to-H$_2$ transition points as a function
of just $G$. Our formula will be useful for interpreting emission-line
observations of HI/H$_2$ interfaces, for estimating star-formation thresholds,
and for sub-grid components in hydrodynamics simulations.
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