On the submillimeter opacity of protoplanetary disks

Solid particles with the composition of interstellar dust and power-law size distribution dn/da ∝ a^-p (for a ≤ a_max with a_max ≳ 3λ and 3 < p < 4) will have submillimeter opacity spectral index β(λ) = d ln κ/d ln ν ≈ (p - 3)β_ISM, where β_ISM ≈ 1.7 is the opacity spectral index of interstellar dust material in the Rayleigh limit. For the power-law index p ≈ 3.5, which characterizes interstellar dust and may apply for particles growing by agglomeration in protoplanetary disks, grain growth to sizes a ≳ 3 mm will result in β(1 mm) ≳ 1. Grain growth can naturally account for β ≈ 1 observed for protoplanetary disks, provided that a_max ≳ 3λ.