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 $M_{\mathrm{d}} \!-\! {M_{\star }}$ and the accretion rate $\dot{M} \!-\! {M_{\star }}$. 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, λm = λacc, and that this limit is uniquely determined by the initial slopes λm, 0 and λacc, 0. We recover the increasing trend claimed from the observations when the difference in the initial values, δ0 = λm, 0−λacc, 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 λm, 0 ∈ [1.2, 2.1], λacc, 0 ∈ [0.7, 1.5].