Stepsize errors in the $N$-body problem: discerning Mercury's true possible long-term orbits

TL;DR

This paper demonstrates that numerical chaos significantly impacts long-term orbital predictions in the Solar System, especially Mercury's orbit, and proposes methods to eliminate these errors for more accurate simulations.

Contribution

It introduces a technique to remove numerical chaos effects by resolving Mercury's pericentre passage, challenging the reliability of higher order symplectic maps for long-term orbit calculations.

Findings

Numerical chaos affects Mercury's orbit predictions.

Resolving Mercury's pericentre passage reduces numerical errors.

Higher order symplectic maps may not offer advantages in long-term orbit resolution.

Abstract

Numerical integrations of the Solar System have been carried out for decades. Their results have been used, for example, to determine whether the Solar System is chaotic, whether Mercury's orbit is stable, or to help discern Earth's climate history. We argue that all of the past studies we consider in this work are affected by numerical chaos to different degrees, affecting the possible orbits and instability probability of Mercury, sometimes significantly. We show how to eliminate the effects of numerical chaos by resolving Mercury's pericentre passage. We also show that several higher order symplectic maps do not exhibit significant differences in resolving pericentre passage of Mercury (at fixed time step), making their advantages suspect for calculating long-term orbits. Resolving pericentre passage affects a wide array of orbital numerical studies, like exoplanet studies, studies of the galactic centre, and other $N$-body problems.