On the time evolution of the Md−M⋆ and Ṁ–M⋆ correlations for protoplanetary discs: the viscous time-scale increases with stellar mass

Large surveys of star-forming regions have unveiled power-law correlations between the stellar mass and the disc parameters, such as the disc mass M-d -M-* and the accretion rate. (M)over dot-M-*. The observed slopes appear to be increasing with time, but the reason behind the establishment of these correlations and their subsequent evolution is still uncertain. We conduct a theoretical analysis of the impact of viscous evolution on power-law initial conditions for a population of protoplanetary discs. We find that, for evolved populations, viscous evolution enforces the two correlations to have the same slope, lambda(m) =lambda(acc,) and that this limit is uniquely determined by the initial slopes. lambda(m,0) and lambda(acc,0). We recover the increasing trend claimed from the observations when the difference in the initial values, delta(0) =lambda(m,0)-lambda(acc,0), is larger than 1/2; moreover, we find that this increasing trend is a consequence of a positive correlation between the viscous time-scale and the stellar mass. We also present the results of disc population synthesis numerical simulations, that allow us to introduce a spread and analyse the effect of sampling, which show a good agreement with our analytical predictions. Finally, we perform a preliminary comparison of our numerical results with observational data, which allows us to constrain the parameter space of the initial conditions to lambda(m,0 )is an element of [1.2, 2.1],lambda(acc,0) is an element of [0.7, 1.5].