Mean free path through randomly oriented disc absorbers (${\ell }^{-1}=n_{\mathrm{com}}(1+z{)}^3\pi R_d^2/2$)

For an unclustered population of fully covering circular disc absorbers, the inverse mean free path is proper number density times mean projected area. A comoving number density must first be multiplied by $(1+z{)}^3$. The incidence then constrains the proper radius; covering factor and multiplicity of detectable lines can change the inference.