Letter to the Editor

An analytic formalism to describe the N_eff(H)−n_H relationship in molecular clouds

Faculty of Physics, University of Duisburg-Essen, Lotharstraße 1, 47057 Duisburg, Germany

^⋆ Corresponding author: brandt.gaches@uni-due.de

Received: 13 June 2025
Accepted: 22 July 2025

Abstract

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.