An analytic formalism to describe the <i>N</i>_<i>eff</i>(H)−n_H relationship in molecular clouds

Context. Astrochemical modeling requires, as input, the effective column density of gas (or extinction) that attenuates an external, isotropic, far-ultraviolet radiation field. In three-dimensional simulations, this can be calculated through ray-tracing schemes, while in zero-dimensional chemical models it is often treated as a free parameter. Aims. We aim to produce an analytic, physically motivated formalism to predict the average relationship between the effective hydrogen-nucleus column density, N eff (H), and the local hydrogen-nucleus number density, n H . Methods. We constructed an analytic model utilizing characteristic length scales that connects the turbulence-dominated regime and the gravitational-dominated regime at high densities. Results. The model reproduces a previous analytic fit to simulation results well and is consistent with the high-density power-law indices, for example N eff ( H )∝ n γ , where γ ≈ 0.4 − 0.5, found in previous numerical simulations utilizing ray-tracing. Conclusions. We present an analytic model that relates the average effective column density, N eff , to the local number density, n H , and reproduces the behaviors found in three-dimensional simulations. The analytic model can be utilized as a sub-grid prescription for shielded molecular gas or in astrochemical models for a physically motivated estimation of the attenuating column density.