TNO reservoirs at 4.5 Gyr for different Tisserand parameters, T. Hot TNOs are shown in red and cold TNOs in blue. Unstable TNOs are shown in black at the last sampling epoch before instability. The first and third columns display the orbital distributions in the (a, I) and (a, e) planes. The second and fourth columns show I, e, and a at minimum semimajor axis, where a vertical black line denotes Neptune’s semimajor axis. The solid orange and blue curves in the inclination panels are, respectively, the Tisserand inclination pathway I_Tiss(a, T, a_N) (2) and the circular orbits’ initial inclination, I_circ(a, T, a_N) (5), with respect to Neptune. The solid green curves for retrograde and prograde inclinations respectively are the Tisserand inclinations pathways with respect to Saturn I_Tiss(a, −2, a_S) (135^° inclination) and I_Tiss(a, 2, a_S) (45^° inclination). In the eccentricity panels, the curves indicate perihelia that match the four giant planets’ semimajor axes. The Tisserand inclination pathway with respect to Neptune is not shown for T = 2.8 because expression (2) is not valid for T > 2.7 (see Paper III for details).