Numerical Solutions for the Orbital Motion of the Solar System over the Past 100 Myr: Limits and New Results

I report results from accurate numerical integrations of solar system orbits over the past 100 Myr with the integrator package HNBody. The simulations used different integrator algorithms, step sizes, and initial conditions, and included effects from general relativity, different models of the Moon, the Sun's quadrupole moment, and up to 16 asteroids. I also probed the potential effect of a hypothetical Planet 9, using one set of possible orbital elements. The most expensive integration (Bulirsch–Stoer) required 4 months of wall-clock time with a maximum relative energy error $\lesssim 3\times {10}^{-13}$. The difference in Earth's eccentricity ($\Delta e_E$) was used to track the difference between two solutions, considered to diverge at time τ when max $|\Delta e_E|$ irreversibly crossed ∼10% of mean $e_E$  ($\sim 0.028\times 0.1$). The results indicate that finding a unique orbital solution is limited by initial conditions from current ephemerides and asteroid perturbations to ∼54 Myr. Bizarrely, the 4-month Bulirsch–Stoer integration and a symplectic integration that required only 5 hr of wall-clock time (12-day time step, with the Moon as a simple quadrupole perturbation), agree to ∼63 Myr. Internally, such symplectic integrations are remarkably consistent even for large time steps, suggesting that the relationship between time step and τ is not a robust indicator of the absolute accuracy of symplectic integrations. The effect of a hypothetical Planet 9 on $\Delta e_E$ becomes discernible at ∼65 Myr. Using τ as a criterion, the current state-of-the-art solutions all differ from previously published results beyond ∼50 Myr. I also conducted an eigenmode analysis, which provides some insight into the chaotic nature of the inner solar system. The current study provides new orbital solutions for applications in geological studies.