2020ApJ...892..136S

It is known that the millimeter dust thermal emission of protoplanetary disks is affected by scattering, such that for optically thick disks, the emission decreases with respect to the pure absorption case and the spectral indices can reach values below 2. The latter can also be obtained with temperature gradients. Using simple analytical models of radiative transfer in thin slabs, we quantify the effect of scattering, vertical temperature gradients, and dust settling on the emission and spectral indices of geometrically thin face-on accretion disks around young stars. We find that, in vertically isothermal disks with large albedo (ω_ν≳0.6), the emergent intensity can increase at optical depths between 10^–2 and 10^–1. We show that dust settling has important effects on the spectral indices in the optically thick regime, since the disk emission mainly traces small dust grains in the upper layers of the disk. The λ = 870 µm emission of these small grains can hide large grains at the disk midplane when the dust surface density is larger than ∼3.21 g cm^–2. Finally, because of the change of the shape of the spectral energy distribution, optically thick disks at 1.3 mm and grains with sizes between 300µm< a_{max}< 1 mm have a 7 mm flux ∼60% higher than the extrapolation from higher millimeter frequencies, assumed when scattering is neglected. This effect could provide an explanation for the excess emission at λ=7mm reported in several disks.