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

Hanno Rein, Kavi Dey, Daniel Tamayo

TL;DR

This work shows that the choice of Hamiltonian-splitting coordinates in Wisdom-Holman integrators strongly affects long-term Solar System stability results. Through 5 Gyr simulations with Jacobi and democratic heliocentric coordinates (DHC), the authors demonstrate that DHC induces an eccentricity-dependent artificial precession that suppresses Mercury-instability outcomes at typical timesteps, while Jacobi remains accurate at much larger timesteps. They establish that DHC only converges to the correct instability rate when the timestep is as small as about $0.6$ days, whereas Jacobi maintains correct rates across a wide range of timesteps. The findings reconcile conflicting instability rates reported in the literature and provide practical guidance: prefer Jacobi coordinates for high-accuracy, long-term integrations, and switch to hybrid or non-symplectic methods when close encounters are possible, ensuring the timestep respects the pericenter-crossing timescale $T_f$ (e.g., $dt \lesssim T_f/4$, ideally $T_f/17$).

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.

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

TL;DR

This work shows that the choice of Hamiltonian-splitting coordinates in Wisdom-Holman integrators strongly affects long-term Solar System stability results. Through 5 Gyr simulations with Jacobi and democratic heliocentric coordinates (DHC), the authors demonstrate that DHC induces an eccentricity-dependent artificial precession that suppresses Mercury-instability outcomes at typical timesteps, while Jacobi remains accurate at much larger timesteps. They establish that DHC only converges to the correct instability rate when the timestep is as small as about days, whereas Jacobi maintains correct rates across a wide range of timesteps. The findings reconcile conflicting instability rates reported in the literature and provide practical guidance: prefer Jacobi coordinates for high-accuracy, long-term integrations, and switch to hybrid or non-symplectic methods when close encounters are possible, ensuring the timestep respects the pericenter-crossing timescale (e.g., , ideally ).

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.
Paper Structure (7 sections, 1 equation, 2 figures, 2 tables)

This paper contains 7 sections, 1 equation, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Tests
  3. Instability rate in 5 Gyr integrations
  4. Artificial numerical precession
  5. A physical cause for the different instability rates
  6. Discussion
  7. Conclusions

Figures (2)

  • Figure 1: Error in Mercury's precession rate as a function of timestep for different eccentricities of Mercury. The blue curve shows the error in simulations using Jacobi coordinates. The red curve shows the error in simulations using democratic heliocentric coordinates. Also shown are the pericenter timescale of Mercury $T_f$ as a vertical line and the precession rate of Mercury due to general relativity.
  • Figure 2: Left panel shows Mercury's eccentricity as a function of time in 640 integrations using Jacobi coordinates, exhibiting near vertical take-offs when the frequency dominantly associated with Mercury's eccentricity precession $g_1$ becomes resonant with that of Jupiter $g_5$. Right panel shows the same for 640 integrations using democratic heliocentric coordinates, where the vertical takeoffs are suppressed by artificial precession (see text). Both sets of integrations use a 6 day timestep.