Relating grain size distributions in circumstellar discs to the spectral index at millimetre wavelengths

The excess emission seen in spectral energy distributions (SEDs) is commonly used to infer the properties of the emitting circumstellar dust in protoplanetary and debris discs. Most notably, dust size distributions and details of the collision physics are derived from SED slopes at long wavelengths. This paper reviews the approximations that are commonly used and contrasts them with numerical results for the thermal emission. The inferred size distribution indexes p are shown to be greater and more sensitive to the observed sub(mm) spectral indexes, α_mm, than previously considered. This effect results from aspects of the transition from small grains with volumetric absorption to bigger grains that absorb and emit near to their surface, controlled by both the real and the imaginary part of the refractive index. The steeper size distributions indicate stronger size-dependence of material strengths or impact velocities or, otherwise, less efficient transport or erosion processes. Strong uncertainties remain because of insufficient knowledge of the material composition, porosity, and optical properties at long wavelengths.