Protoplanetary discs are the site of star and planet formation, and their evolution and consequent dispersal deeply affect the formation of planetary systems. In the standard scenario they evolve on timescales ∼ Myr due to the viscous transport of angular momentum. The analytical self-similar solution for their evolution predicts also specific disc isochrones in the accretion rate -disc mass plane. However, photoevaporation by radiation emitted by the central star is likely to dominate the gas disc dispersal of the innermost region, introducing another (shorter) timescale for this process. In this paper, we include the effect of internal (X and EUV) photoevaporation on the disc evolution, finding numerical solutions for a population of protoplanetary discs. Our models naturally reproduce the expected quick dispersal of the inner region of discs when their accretion rates match the rate of photoevaporative mass loss, in line with previous studies. We find that photoevaporation preferentially removes the lightest discs in the sample. The net result is that, counter-intuitively, photoevaporation increases the average disc mass in the sample, by dispersing the lightest discs. At the same time, photoevaporation also reduces the mass accretion rate by cutting the supply of material from the outer to the inner disc. In a purely viscous framework, this would be interpreted as the result of a longer viscous evolution, leading to an overestimate of the disc age. Our results thus show that photoevaporation is a necessary ingredient to include when interpreting observations of large disc samples with measured mass accretion rates and disc masses. Photoevaporation leaves a characteristic imprint on the shape of the isochrone. Accurate data in the accretion rate -disc mass plane in the low disc mass region therefore give clues on the typical photoevaporation rate.

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, 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 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 λm, 0 ∈ [1.2, 2.1], λacc, 0 ∈ [0.7, 1.5].