The column density distribution in turbulent interstellar media

Our ability to correct the observational photometry of galaxies depends on our knowledge of the attenuation of light produced by the dust contained in the interstellar medium. The magnitude and wavelength dependence of the attenuation produced by a turbulent dust screen depends primarily on the distribution of the column density, which is itself determined by the distribution of the local density. Here we provide quantitative estimates of the dependence of the variance σ_N of the column densities N on the structure of the local density ρ and show how this variance depends on the thickness Δ of a turbulent, dusty screen. We provide an analytical approximation for the variance in the limit of a thick slice in which the thickness Δ is larger than the maximum scale L_max of the turbulent medium. Provided that the turbulence is Kolmogorov and the minimum scale is much smaller than L _max, the variance in the limit of a thick slice is given by σ_N/〈N〉 = σ _ρ/〈ρ〉[L_max/(8Δ)]^1/2, where 〈N〉 and 〈ρ〉 are the means of the column density and the local density, respectively. We show that the density distribution can be well approximated by a lognormal function, provided that the variance σρ/_〈ρ〉 of the lognormal density distribution of the local density with the mean density 〈ρ〉 is not much larger than ∼2.5.