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

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.

Simple Physics and Integrators Accurately Reproduce Mercury Instability Statistics

“…We perform all simulations using the REBOUND N-body code's (Rein & Liu 2012) WHFAST integration scheme (Rein & Tamayo 2015), which is a Wisdom-Holman scheme (WH; Wisdom & Holman 1991). We do not use REBOUND's WHCKL scheme, as Brown & Rein (2020, Abbot et al (2021) did, based on results from Hernandez et al (2020Hernandez et al ( , 2022, suggesting there is no clear advantage to a higher-order scheme for this problem. We emphasize that we have not used symplectic correctors (Wisdom et al 1996).…”

“…We emphasize that we have not used symplectic correctors (Wisdom et al 1996). An integration improved by correctors indicates numerical instability is not present (Wisdom 2015;Hernandez et al 2022), but correctors do not appreciably affect the existence of numerical instability. As in Abbot et al (2021), we define a Mercury instability event as occurring when Mercury passes within 0.01 au of Venus, and we stop the simulations at that point.…”

“…We produce two ensembles of simulations (Var dt and Fix dt) that both contain 2750 members. In Var dt, we use the heartbeat functionality in REBOUND to halve the time step if the Wisdom (2015) and Hernandez et al (2022) criterion (Equation ( 1)) is about to be violated. Specifically, Mercury's heliocentric distance, r M , is monitored, and the time step, dt, is halved whenever r M < a M (1 − e crit (dt)) where a M ≈ 0.387 au is Mercury's semimajor axis, which we approximate as fixed, and e crit (dt) is the critical eccentricity satisfying Equation (1) for the current time step.…”

“…We will show in Section 3 that the Mercury instability statistics of the Laskar & Gastineau (2009) and Zeebe (2015) ensembles do not differ statistically, and that the Mercury instability statistic of the Abbot et al (2021) ensemble is significantly larger in the range of 4-4.75 billion years. Hernandez et al (2022) showed that all existing ensembles are subject to numerical chaos due to time stepping, and this artificial instability can be mitigated by using smaller time steps. They postulated that the resulting artificial diffusion could affect phase-space statistics such as the Mercury instability probability.…”

Switching integrators reversibly in the astrophysical $N$-body problem

“…Finally, we consider a system in which one of the planets has an eccentric orbit. We use a Wisdom-Holman method (Wisdom & Holman 1991) with step sizes suggested by Wisdom (2015); Hernandez et al (2022). We obtain excellent error behavior even while adapting step sizes.…”