Multi-wavelength continuum sizes of protoplanetary discs: scaling relations and implications for grain growth and radial drift

We analyse spatially resolved ALMA observations at 0.9, 1.3, and 3.1 mm for the 26 brightest protoplanetary discs in the Lupus star-forming region. We characterise the discs multi-wavelength brightness profiles by fitting the interferometric visibilities in a homogeneous way, obtaining effective disc sizes at the three wavelengths, spectral index profiles and optical depth estimates. We report three fundamental discoveries: first, the millimeter continuum size - luminosity relation already observed at 0.9 mm is also present at 1.3 mm with an identical slope, and at 3.1 mm with a steeper slope, confirming that emission at longer wavelengths becomes increasingly optically thin. Second, when observed at 3.1 mm the discs appear to be only 9% smaller than when observed at 0.9 mm, in tension with models of dust evolution which predict a starker difference. Third, by forward modelling the sample of measurements with a simple parametric disc model, we find that the presence of large grains ($a_\mathrm{max}>1 $mm) throughout the discs is the most favoured explanation for all discs as it reproduces simultaneously their spectral indices, optical depth, luminosity, and radial extent in the 0.9-1.3 mm wavelength range. We also find that the observations can be alternatively interpreted with the discs being dominated by optically thick, unresolved, substructures made of mm-sized grains with a high scattering albedo.