M_D−Ṁ_* correlation with ϕ = 1, ω = 0.8, and t = 3.5 Myr. The dots are Monte Carlo simulations of 10⁵ proto-planetary disks evolving following Eqs. (4) and (5), where M₀ follows Eq. (9) and t_acc,0 follows a lognormal distribution centered on the natural logarithm of 5 Myr with a spread of 0.13 dex. In addition, M₀ is then generated following a lognormal distribution with the values of t_acc,0 following Eq. (9) with a spread of 0.43 dex. The black line represents Eq. (D.2), the gray line is the best-fitting power law for the population, resulting in ${\dot{M}}_{*}\propto M_D^{0.83}$, while the red line shows Eq. (B.2). For illustrative purposes, we also plot in blue the best-fitting power law for a population for which ${\sigma }_{M_0}=0.26$ dex, which results in ${\dot{M}}_{*}\propto M_D^{0.65}$.