Orbital evolution of two fictitious Trojans over some period at the end of the integration (except the top right plot). The orbits in the left and right panels are initialised with (a₀, i₀) = (19.225 AU, 56.5° ) and (19.317 AU, 5° ),  respectively. We illustrate the temporal evolution of the eccentricity e, inclination i, and resonant angle of the 1:1 MMR with Uranus σ for both orbits. The semi-major axis ratio of the UT and Uranus a/a_U, resonant angle of the 1:2 MMR with Neptune σ_1 : 2 and apsidal difference with Neptune Δϖ₈ are also shown in the left panel, while the apsidal differences with Jupiter Δϖ₅ and Uranus Δϖ₇ over the last few million years (bottom right) and the whole integration time (top right) are shown in the right panel, respectively. The vertical dashed lines in the left panel indicate the time 3.2716 × 10⁷ yr, 3.2762 × 10⁷ yr and 3.2922 × 10⁷ yr, when the UT enters and leaves the quasi-satellite orbit, and is captured by the Kozai mechanism.