Evolution of Neptune with the presence of a planetesimal ring. The horizontal and vertical axes represent time and a_N, respectively. Lines of different colors correspond to different planetesimal ring masses. The horizontal gray line represents the initial center of the planetesimal ring (34.5 au). The hollow triangles, circles, and squares mark the positions where Neptune crosses R_3N:2U, R_2N:1U and R_5N:2U, respectively.