On the time evolution of the $M_{\mathrm{d}}-M_{\star }$ and $\dot{M}-M_{\star }$ correlations for protoplanetary discs: the viscous timescale increases with stellar mass

Abstract

Large surveys of star-forming regions have unveiled power-law correlations between the stellar mass and the disc parameters, such as the disc mass $M_{\mathrm{d}}\propto {M_{\star }}^{{\lambda }_{\mathrm{m}}}$ and the accretion rate $\dot{M}\propto {M_{\star }}^{{\lambda }_{\mathrm{acc}}}$. 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, ${\lambda }_{\mathrm{m}}$ = ${\lambda }_{\mathrm{acc}}$, and that this limit is uniquely determined by the initial slopes ${\lambda }_{\mathrm{m},0}$ and ${\lambda }_{\mathrm{acc},0}$. We recover the increasing trend claimed from the observations when the difference in the initial values, ${\delta }_0={\lambda }_{\mathrm{m},0}-{\lambda }_{\mathrm{acc},0}$, is larger than $1/2$; moreover, we find that this increasing trend is a consequence of a positive correlation between the viscous timescale 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 ${\lambda }_{\mathrm{m},0}\in [1.2,2.1]$, ${\lambda }_{\mathrm{acc},0}\in [0.7,1.5]$.

Accepted for publication in MNRAS. 14 pages, 8 figures

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