Democratic heliocentric coordinates underestimate the rate of instabilities in long-term integrations of the Solar System

TL;DR

This paper demonstrates that democratic heliocentric coordinates tend to underestimate the rate of instabilities in long-term Solar System simulations, due to artificial numerical effects, and suggests Jacobi coordinates are more reliable at larger timesteps.

Contribution

It reveals the limitations of DHC coordinates in long-term integrations and advocates for using Jacobi coordinates for more accurate planetary stability assessments.

Findings

DHC coordinates suppress Mercury's instabilities at typical timesteps.

Artificial numerical precession depends on eccentricity in DHC splitting.

Jacobi coordinates remain reliable at longer timesteps for eccentric orbits.

Abstract

Wisdom-Holman (WH) integrators are symplectic operator-splitting methods widely used for long-term N-body simulations of planetary systems. Most implementations use either Jacobi coordinates or democratic heliocentric coordinates (DHC) for the Hamiltonian splitting, resulting in slightly different algorithms. In this paper we report results from numerical experiments, which show that integrations of the Solar System using DHC coordinates with typical timesteps of a few days suppress instabilities of the planet Mercury. We further show that this is due to an eccentricity dependent artificial numerical precession introduced by the DHC splitting. While the DHC splitting converges to the correct results at shorter timesteps of ~0.6 days, we argue that Jacobi coordinates remain reliable to significantly longer timesteps when orbits become moderately eccentric, and are thus a better choice when the innermost planet can reach high eccentricities.