Long-term stability of horseshoe orbits

Unlike Trojans, horseshoe co-orbitals are not generally considered to be long-term stable. As the lifetime of Earth’s and Venus’s horseshoe co-orbitals is expected to be about a Gyr, we investigated the possible contribution of late-escaping inner planet co-orbitals to the lunar Late Heavy Bombardment. Contrary to analytical estimates, we do not find many horseshoe objects escaping after the first 100 Myr. In order to understand this behaviour, we ran a second set of simulations featuring idealized planets on circular orbits with a range of masses. We find that horseshoe co-orbitals are generally long lived (and potentially stable) for systems with primary-to-secondary mass ratios larger than about 1200. This is consistent with the results of Laughlin & Chambers for equal-mass pairs of co-orbital planets and the instability of Jupiter’s horseshoe companions. Horseshoe orbits at smaller mass ratios are unstable because they must approach within 5 Hill radii of the secondary. In contrast, tadpole orbits are more robust and can remain stable even when approaching within 4 Hill radii of the secondary.