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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 integratororbitN(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 massM₀. Among other features,orbitN-1.0includesM₀’s quadrupole moment, a lunar contribution, and post-Newtonian corrections (1PN) due toM₀(fast symplectic implementation). To reduce numerical round-off errors, Kahan compensated summation was implemented. I useorbitNto 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,orbitNis about as fast as or faster (factor 1.15–2.6) than comparable integrators, depending on hardware.¹¹The orbitN source code (C) is available athttp://github.com/rezeebe/orbitN. 

Abstract The dynamical evolution of the solar system is chaotic with a Lyapunov time of only ∼5 Myr for the inner planets. Due to the chaos it is fundamentally impossible to accurately predict the solar system’s orbital evolution beyond ∼50 Myr based on present astronomical observations. We have recently developed a method to overcome the problem by using the geologic record to constrain astronomical solutions in the past. Our resulting optimal astronomical solution (called ZB18a) shows exceptional agreement with the geologic record to ∼58 Ma (Myr ago) and a characteristic resonance transition around 50 Ma. Here we show that ZB18a and integration of Earth’s and Mars’ spin vector based on ZB18a yield reduced variations in Earth’s and Mars’ orbital inclination and Earth’s obliquity (axial tilt) from ∼58 to ∼48 Ma—the latter being consistent with paleoclimate records. The changes in the obliquities have important implications for the climate histories of Earth and Mars. We provide a detailed analysis of solar system frequencies (gandsmodes) and show that the shifts in the variation in Earth’s and Mars’ orbital inclination and obliquity around 48 Ma are associated with the resonance transition and caused by changes in the contributions to the superposition ofsmodes, plusg–smode interactions in the inner solar system. Theg–smode interactions and the resonance transition (consistent with geologic data) are unequivocal manifestations of chaos. Dynamical chaos in the solar system hence not only affects its orbital properties but also the long-term evolution of planetary climate through eccentricity and the link between inclination and axial tilt.