Systematic survey of the dynamics of Uranus Trojans

Abstract

Context. The discovered Uranus Trojan (UT) 2011 QF99 and several candidate UTs have been reported to be in unstable orbits. This implies that the stability region around the triangular Lagrange points L4 and L5 of Uranus should be very limited. Aims. In this paper, we aim to locate the stability region for UTs and find out the dynamical mechanisms responsible for the structures in the phase space. The null detection of primordial UTs also needs to be explained. Methods. Using the spectral number as the stability indicator, we constructed the dynamical maps on the (a0, i0) plane. The proper frequencies of UTs were determined precisely with a frequency analysis method that allows us to depict the resonance web via a semi-analytical method. We simulated radial migration by introducing an artificial force acting on planets to mimic the capture of UTs. Results. We find two main stability regions: a low-inclination (0° −14°) and a high-inclination regime (32° −59°). There is also an instability strip in each of these regions at 9° and 51°, respectively. These strips are supposed to be related with g − 2g5 + g7 = 0 and ν8 secular resonances. All stability regions are in the tadpole regime and no stable horseshoe orbits exist for UTs. The lack of moderate-inclined UTs is caused by the ν5 and ν7 secular resonances, which could excite the eccentricity of orbits. The fine structures in the dynamical maps are shaped by high-degree secular resonances and secondary resonances. Surprisingly, the libration centre of UTs changes with the initial inclination, and we prove it is related to the quasi 1:2 mean motion resonance (MMR) between Uranus and Neptune. However, this quasi-resonance has an ignorable influence on the long-term stability of UTs in the current planetary configuration. About 36.3% and 0.4% of the pre-formed orbits survive fast and slow migrations with migrating timescales of 1 and 10 Myr, respectively, most of which are in high inclination. Since low-inclined UTs are more likely to survive the age of the solar system, they make up 77% of all such long-life orbits by the end of the migration, making a total fraction up to 4.06 × 10−3 and 9.07 × 10−5 of the original population for fast and slow migrations, respectively. The chaotic capture, just like depletion, results from secondary resonances when Uranus and Neptune cross their mutual MMRs. However, the captured orbits are too hot to survive until today. Conclusions. About 3.81% UTs are able to survive the age of the solar system, among which 95.5% are on low-inclined orbits with i0 < 7.5°. However, the depletion of planetary migration seems to prevent a large fraction of such orbits, especially for the slow migration model. Based on the widely adopted migration models, a swarm of UTs at the beginning of the smooth outward migration is expected and a fast migration is favoured if any primordial UTs are detected.