On the Submillimeter Opacity of Protoplanetary Disks

Solid particles with the composition of interstellar dust and power-law size distribution dn/da propto a^{-p} for a < a_max with a_{max} > 3 lambda and 3 < p < 4 will have submm opacity spectral index beta(lambda) = dln(kappa)/dln(nu) approx (p-3) beta_{ism}, where beta_{ism} approx 1.7 is the opacity spectral index of interstellar dust material in the Rayleigh limit. For the power-law index p approx 3.5 that characterizes interstellar dust, and that appears likely for particles growing by agglomeration in protoplanetary disks, grain growth to sizes a > 3 mm will result in beta(1 mm) < ~1. Grain growth can naturally account for beta approx 1 observed for protoplanetary disks, provided that a_{max} > ~ 3 lambda.