Geometrical optical depth profiles of debris disks with a dense parent belt at 30 AU around stars of M_⋆ = 1 M_⊙ (solid lines) and M_⋆ = 2 M_⊙ (dashed lines). The black lines show the end results of the numerical simulations. In green are the maximum τ_geo profiles as given by the analytical model (Eq. (28), with τ_{geo, base}(r) given by Eq. (5), Δr_pile/r_pile = 0.15, r₀ = 30 AU, and β = 0.5).