A Optical Depth in a Porous Medium

where τb = κρH is the optical depth of individual clumps or blobs, withH = l/f the “porosity length” for clumps of scale l and volume filling factor f . Note that optically thin blobs (τb ≪ 1) recover the non-porous limit, κeff → κ, whereas the limit of optically thick blobs (τb ≫ 1) has an effective opacity given by the ratio of their area l to mass mb(= l ρ/f = lρH), i.e. κeff → 1/ρH. The effective opacity (2) thus retains the proper asymptotic limits, with however a somewhat simpler general form than posited by Owocki et al. (2004) (cf. their eqn. 35). Indeed a key advantage of this simpler form is that it can be integrated analytically to obtain the effective optical depth