Numerical Solutions for the orbital motion of the Solar System over the Past 100 Myr: Limits and new results

Abstract

I report results from accurate numerical integrations of Solar System orbits over the past 100Myr with the integrator package HNBody. The simulations used different integrator algorithms, step sizes, initial conditions, and included effects from general relativity, different models of the Moon, the Sun's quadrupole moment, and up to sixteen 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 wall-clock time with a maximum relative energy error <~3e{-13}. The difference in Earth's eccentricity (DeE) was used to track the difference between two solutions, considered to diverge at time tau when max|DeE| irreversibly crossed ~10\% of mean eE (~0.028x0.1). The results indicate that finding a unique orbital solution is limited by initial conditions from current ephemerides and asteroid perturbations to ~54Myr. Bizarrely, the 4-month Bulirsch-Stoer integration and a symplectic integration that required only 5~hours wall-clock time (12-day time step, Moon as a simple quadrupole perturbation), agree to ~63Myr. Internally, such symplectic integrations are remarkably consistent even for large time steps, suggesting that the relationship between time step and tau is not a robust indicator for the absolute accuracy of symplectic integrations. The effect of a hypothetical Planet~9 on DeE becomes discernible at ~65Myr. Using tau as a criterion, the current state-of-the-art solutions all differ from previously published results beyond ~50Myr. 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.