The role of secular resonances on trojans of the terrestrial planets

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 ν 1 , ν 12 and ν 5 , and only those with low and high inclinations survived in our simulations. For Venus and Earth trojans, the ν 3 and ν 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 ν 11 resonance with Mercury at high inclinations hence Venus and Earth trojans are stable only at low inclinations. Mars trojans are affected by the ν 3 , ν 4 , ν 13 and ν 14 resonances with the Earth and Mars for low inclinations which increase the eccentricity and suppress oscillations of the inclination. At i =32° , the ν 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.