Exponential growth of round-trip error and instability of orbits of small bodies in the outer solar system

Abstract

Exponential growth behavior of the round-trip orbital error in numerical integrations of the orbits for periodic comets and comet-like bodies was examined in relation to the time scales of their orbits becoming chaotic and unstable. For the majority of the orbits of 159 short-period (SP) comets that we integrated for 4400 yr, the round trip error for each of the Keplerian elements increased approximately exponentially from a very small value (say, 10 −29) to the order of unity within 1000–2000 yr; this period of time was found to be well correlated with the time scale of substantial orbital changes leading to capture/ejection for the SP comets. Similarly as a result of 0.2–0.4 Myr integration of the orbits for giant comet-like objects such as Chiron and Pholus, we found that their time scale of the round-trip orbital error growth is consistent with the time scale for the divergence of the ensemble of orbits with slightly different orbital elements while the range of orbital error growth is less than the order of 1 AU. Assuming that this approximate parallelism between the time scale of the round-trip error growth and that for the orbital divergence is valid for the objects with much slower orbital evolution, we here estimated from the round-trip error curve the time scale for the orbit of 1992 QB1, a candidate of the Kuiper belt members, to become unstable by the order of 1 AU. The time scale is found to be 24–46 Myr. Some resonant cases are discussed in which the round-trip error growth behavior may not be a measure of the dynamical instability.