r_cross = a_p(1 − e_cross,p) vs. e_cross obtained by the N-body simulations. e_cross and e_cross,p are calculated by taking the average of eccentricities of planetesimals whose orbits are just before crossing to the planet’s orbit. The dashed lines show e_cross obtained from Eqs. (40) and (41). The thick dashed lines mean Eq. (44) is achieved. When Eq. (44) is achieved, e_cross becomes smaller than the analytical models predict. As indicated in the legend, the different colors correspond to the different eccentricity excitation timescales τ_ecc.