On the submillimeter opacity of protoplanetary disks

Solid particles with the composition of interstellar dust and power-law size distribution dn/da alpha a(-p) ( for a <= a(max) with a(max) greater than or similar to 3 lambda and 3 < p < 4) will have submillimeter opacity spectral index beta(lambda) equivalent to d 1n kappa/d 1n approximate to (p - 3) beta(ISM), where beta(ISM) approximate to 1: 7 is the opacity spectral index of interstellar dust material in the Rayleigh limit. For the power-law index p approximate to 3: 5, which characterizes interstellar dust and may apply for particles growing by agglomeration in protoplanetary disks, grain growth to sizes a greater than or similar to 3 mm will result in beta(1 mm) less than or similar to 1. Grain growth can naturally account for beta approximate to 1 observed for protoplanetary disks, provided that a(max) greater than or similar to 3 lambda.