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Effects of scattering, temperature gradients, and settling on the derived dust properties of observed protoplanetary disks

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 ($ω_ν \gtrsim 0.6$), 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 $λ= 870 \ μ$m emission of these small grains can hide large grains at the disk mid plane when the dust surface density is larger than $\sim$ 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 300 $μ$m $< a_{\rm max} <$ 1 mm have a 7 mm flux $\sim 60$\% higher than the extrapolation from higher millimeter frequencies, assumed when scattering is neglected. This effect could provide an explanation to the excess emission at $λ= 7$ mm reported in several disks.