The Dynamics of Neptune Trojan: I. the Inclined Orbits

The stability of Trojan type orbits around Neptune is studied. As the first part of our investigation, we present in this paper a global view of the stability of Trojans on inclined orbits. Using the frequency analysis method based on the FFT technique, we construct high resolution dynamical maps on the plane of initial semimajor axis a0 versus inclination i0. These maps show three most stable regions, with i0 in the range of (0◦, 12◦), (22◦, 36◦) and (51◦, 59◦) respectively, where the Trojans are most probably expected to be found. The similarity between the maps for the leading and trailing triangular Lagrange points L4 and L5 confirms the dynamical symmetry between these two points. By computing the power spectrum and the proper frequencies of the Trojan motion, we figure out the mechanisms that trigger chaos in the motion. The Kozai resonance found at high inclination varies the eccentricity and inclination of orbits, while the ν8 secular resonance around i0 ∼ 44 ◦ pumps up the eccentricity. Both mechanisms lead to eccentric orbits and encounters with Uranus that introduce strong perturbation and drive the objects away from the Trojan like orbits. This explains the clearance of Trojan at high inclination (> 60◦) and an unstable gap around 44◦ on the dynamical map. An empirical theory is derived from the numerical results, with which the main secular resonances are located on the initial plane of (a0, i0). The fine structures in the dynamical maps can be explained by these secular resonances.