Observed sizes of planet-forming disks trace viscous spreading

Context. The evolution of protoplanetary disks is dominated by the conservation of angular momentum, where the accretion of material onto the central star is fed by the viscous expansion of the outer disk or by disk winds extracting angular momentum without changing the disk size. Studying the time evolution of disk sizes therefore allows us to distinguish between viscous stresses or disk winds as the main mechanism of disk evolution. Observationally, estimates of the size of the gaseous disk are based on the extent of CO submillimeter rotational emission, which is also affected by the changing physical and chemical conditions in the disk during the evolution.Aims. We study how the gas outer radius measured from the extent of the CO emission changes with time in a viscously expanding disk. We also investigate to what degree this observable gas outer radius is a suitable tracer of viscous spreading and whether current observations are consistent with viscous evolution.Methods. For a set of observationally informed initial conditions we calculated the viscously evolved density structure at several disk ages and used the thermochemical codeDALIto compute synthetic emission maps, from which we measured gas outer radii in a similar fashion as observations.Results. The gas outer radii (R-CO,R- 90%) measured from our models match the expectations of a viscously spreading disk: R-CO,R- 90% increases with time and, for a given time, R-CO,R- 90% is larger for a disk with a higher viscosity alpha(visc). However, in the extreme case in which the disk mass is low (M-disk <= 10(-4) M-circle dot) and alpha(visc) is high (>= 10(-2)), R-CO,R- 90% instead decreases with time as a result of CO photodissociation in the outer disk. For most disk ages, R-CO,R- 90% is up to similar to 12x larger than the characteristic size R-c of the disk, and R-CO,R- 90%/R-c is largest for the most massive disk. As a result of this difference, a simple conversion of R-CO,R- 90% to alpha(visc) overestimates the true alpha(visc) of the disk by up to an order of magnitude. Based on our models, we find that most observed gas outer radii in Lupus can be explained using viscously evolving disks that start out small (R-c(t = 0) similar or equal to 10 AU) and have a low viscosity (alpha(visc) = 10(-4)-10(-3)).Conclusions. Current observations are consistent with viscous evolution, but expanding the sample of observed gas disk sizes to star-forming regions, both younger and older, would better constrain the importance of viscous spreading during disk evolution.