Fast and Furious: Long-term orbit integrations with collocation integrator Lobbie

Abstract

Efficient and accurate numerical integration methods are fundamental to studying long-term solar system dynamics, where numerical errors can accumulate over billions of orbital periods. In this work, we investigate the performance of our modification of Lobbie, a Lobatto collocation integrator for long-term simulations of the Solar System. We compare Lobbie against the widely used WHFast symplectic integrator and the high-order adaptive collocation integrator IAS15 implementations in the Rebound package. The performance of the integrator is evaluated using five benchmark problems: a circular two-body orbit integrated over 100 Myrs, two-body orbits with eccentricities up to 1-10^{-6} to test the adaptive step size control during repeated close encounters, the full eight-planet Solar System integrated over 100 Myrs, the four giant planets integrated over 10 Gyrs, and the eight-planet Solar System together with 50 asteroids on Aten-like orbits integrated over 1Myr to test the integrator's consistency in a chaotic, close-encounter-rich environment. Across all tests, Lobbie demonstrates excellent long-term stability and competitive computational performance. For full Solar System integrations, Lobbie achieves relative energy conservation at the level of 10^{-14} while outperforming IAS15 in computational efficiency and approaching the speed of WHFast at comparable accuracy. In the Solar System plus asteroids test, Lobbie also keeps different realizations of Jupiter's orbit consistent with each other to within about 1.5 km after 1 Myr of integration, two orders of magnitude better than IAS15, being twice faster. Our results demonstrate that Lobbie provides a powerful universal alternative to traditional symplectic mappings or Gauss-Radau collocation methods for long-term dynamical simulations in celestial mechanics.

12 pages, 8 figures, 2 tables