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

Abstract

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 M 0. Among other features, orbitN-1.0 includes M 0’s quadrupole moment, a lunar contribution, and post-Newtonian corrections (1PN) due to M 0 (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 1 The orbitN source code (C) is available at http://github.com/rezeebe/orbitN.

A Secular Solar System Resonance that Disrupts the Dominant Cycle in Earth’s Orbital Eccentricity (g 2 − g 5): Implications for Astrochronology

“…Solar system integrations were performed following our earlier work (Zeebe 2017;Zeebe & Lourens 2019;Zeebe 2022) with our integrator package orbitN (v1.0) (Zeebe 2023), using a symplectic integrator and Jacobi coordinates (Wisdom & Holman 1991;Zeebe 2015b). 3 The methods used here and our integrator package have been extensively tested and compared against other studies (Zeebe 2017;Zeebe & Lourens 2019;Zeebe 2022Zeebe , 2023. For the present study, we also included simulations with an independent integrator package (HNBody; Rauch & Hamilton 2002) and found the same dynamical behavior.…”

“…All simulations include contributions from general relativity, available in orbitN as post-Newtonian effects due to the dominant mass. The Earth-Moon system was modeled as a gravitational quadrupole (Quinn et al 1991;Varadi et al 2003;Zeebe 2017Zeebe , 2023. Initial conditions for the positions and velocities of the planets and Pluto were generated from the latest JPL ephemeris DE441 (Park et al 2021) using the SPICE toolkit for Matlab.…”

“…Our integrations included a lunar contribution, i.e., a gravitational quadrupole model of the Earth-Moon system (Quinn et al 1991;Varadi et al 2003;Zeebe 2017Zeebe , 2023. In the present context, the lunar contribution has a relatively small effect on the overall dynamics, yet the integration requires the Earth-Moon distance (R) as parameter at a given time in the past.…”

“…For the present study, we performed state-of-the-art solar system integrations, including the eight planets and Pluto, a lunar contribution, general relativity, the solar quadrupole moment, and solar mass loss (see Section 2; Zeebe 2017; Zeebe & Lourens 2019;Zeebe 2022Zeebe , 2023. Initial conditions at time t 0 were taken from the latest JPL ephemeris DE441 (Park et al 2021) and the equations of motion were numerically integrated to t = −3.5 Gyr (beyond −3.5 Gyr parameters such as the lunar distance have large uncertainties, see Section 2).…”

“…integrator package have been extensively tested and compared against other studies(Zeebe 2017;Zeebe & Lourens 2019;Zeebe 2022Zeebe , 2023. (ii) The σ 12 resonance was recognized previously, although to our knowledge only by two studies(Lithwick & Wu 2011;Mogavero & Laskar 2022) and not its effect on g 25 /LEC (see below).…”