The role of secular resonances on trojans of the terrestrial planets

Abstract

We investigate the role of secular resonances on the motion of terrestrial-planet trojans in tadpole motion for 5 Myr in the inclination range i ≤ 90°. We use a Fourier spectrum of the eccentricity vector, e, and angular momentum vector, L. When i > 40°, the trojans are in a Kozai resonance with Jupiter, which increases the eccentricity to planet-crossing values on a time-scale of 0.1 Myr; hence we found no stable trojan motion when i > 40°. Trojans of Mercury are unstable for a large range in inclination due to the resonances v 1 , v 1 2 and v 5 , and only those with low and high inclinations survived in our simulations. For Venus and Earth trojans, the v 3 and v 4 secular resonances with the Earth and Mars respectively increase the eccentricity to planet-crossing values for moderate inclinations. Venus trojans are also affected by the v 1 1 resonance with Mercury at high inclinations hence Venus and Earth trojans are stable only at low inclinations. Mars trojans are affected by the v 3 , v 4 , v 1 3 and v 1 4 resonances with the Earth and Mars for low inclinations which increase the eccentricity and suppress oscillations of the inclination. At i = 32°, the v 5 resonance destabilizes Mars trojans, although its width is less than a degree in inclination. The typical lifetime of inner-planet trojans trapped in a secular resonance is 1 Myr for Venus, Earth and Mars trojans, while it is 10 5 yr for Mercury trojans. Trojans in secular resonances are chaotic on time-scales of 10 3 -10 4 yr for all inner planets, while outside the resonances this becomes typically 10 6 yr.