Stellar activity index S​_ph as a function of the Rossby number. Left panel: Rossby number evaluated with the semi-empirical relation by Noyes et al. (1984). Right panel: details are similar to the left panel, but using the Rossby number evaluated with the new convective turnover time from Eq. (9). The thick red lines represent the maximum S​_ph of the 95 percentile of each chunk in Rossby number, evaluated in a range centered around the kink. Sun symbols mark the position of the Sun between the minimum and maximum of activity (Ro_Noyes, ⊙ = 1.989, Ro_{LEGACY, ⊙} = 0.496). The position of the kink is marked by the vertical dashed and dot-dashed lines, for the left and right panel, respectively. The left-pointing arrow shows how the position of the kink shifts when switching from τ_c, N84 to τ_c, LEGACY. The density of stars is indicated by the blue color scale.