The time evolution of dusty protoplanetary disc radii: observed and physical radii differ

Abstract: Proto-planetary disc surveys conducted with ALMA are measuring disc radii in multiple star forming regions. The disc radius is a fundamental quantity to diagnose whether discs undergo viscous spreading, discriminating between viscosity or angular momentum removal by winds as drivers of disc evolution. Observationally, however, the sub-mm continuum emission is dominated by the dust, which also drifts inwards, complicating the picture. In this paper we investigate, using theoretical models of dust grain growth and radial drift, how the radii of dusty viscous proto-planetary discs evolve with time. Despite the existence of a sharp outer edge in the dust distribution, we find that the radius enclosing most of the dust $\textit{mass}$ increases with time, closely following the evolution of the gas radius. This behaviour arises because, although dust initially grows and drifts rapidly onto the star, the residual dust retained on Myr timescales is relatively well coupled to the gas. Observing the expansion of the dust disc requires using definitions based on high fractions of the disc $\textit{flux}$ (e.g. 95 per cent) and very long integrations with ALMA, because the dust grains in the outer part of the disc are small and have a low sub-mm opacity. We show that existing surveys lack the sensitivity to detect viscous spreading. The disc radii they measure do not trace the mass radius or the sharp outer edge in the dust distribution, but the outer limit of where the grains have significant sub-mm opacity. We predict that these observed radii should shrink with time.