OrbitN: A Symplectic Integrator for Planetary Systems Dominated by a Central Mass—Insight into Long-term Solar System Chaos

Reliable studies of the long-term dynamics of planetary systems require numerical integrators that are accurate and fast. The challenge is often formidable because the chaotic nature of many systems requires relative numerical error bounds at or close to machine precision (∼10−16, double-precision arithmetic); otherwise, numerical chaos may dominate over physical chaos. Currently, the speed/accuracy demands are usually only met by symplectic integrators. For example, the most up-to-date long-term astronomical solutions for the solar system in the past (widely used in, e.g., astrochronology and high-precision geological dating) have been obtained using symplectic integrators. However, the source codes of these integrators are unavailable. Here I present the symplectic integrator orbitN (lean version 1.0) with the primary goal of generating accurate and reproducible long-term orbital solutions for near-Keplerian planetary systems (here the solar system) with a dominant mass M0. Among other features, orbitN-1.0 includes M0's quadrupole moment, a lunar contribution, and post-Newtonian corrections (1PN) due to M0 (fast symplectic implementation). To reduce numerical round-off errors, Kahan compensated summation was implemented. I use orbitN to provide insight into the effect of various processes on the long-term chaos in the solar system. Notably, 1PN corrections have the opposite effect on chaoticity/stability on a 100 Myr versus Gyr timescale. For the current application, orbitN is about as fast as or faster (factor 1.15–2.6) than comparable integrators, depending on hardware.1