Relation between the power-law indices of the grain size distributions and the total opacities between λ₁ = 1 mm and λ₂ = 7 mm for four materials at r = 100 au around a star with L = L_⊙. Solid coloured lines show numerical results. Dotted lines of the same colours show linear fits for 3.2 ≤ p ≤ 3.8, with fit coefficients indicated in the figure keys. Dashed lines show the analytic approximation according to Eq. (30). The solid grey lines trace the approximate Eq. (31).