Simple Physics and Integrators Accurately Reproduce Mercury Instability Statistics

Abstract

The long-term stability of the solar system is an issue of significant scientific and philosophical interest. The mechanism leading to instability is Mercury’s eccentricity being pumped up so high that Mercury either collides with Venus or is scattered into the Sun. Previously, only three five-billion-year N-body ensembles of the solar system with thousands of simulations have been run to assess long-term stability. We generate two additional ensembles, each with 2750 members, and make them publicly available at https://archive.org/details/@dorianabbot. We find that accurate Mercury instability statistics can be obtained by (1) including only the Sun and the eight planets, (2) using a simple Wisdom–Holman scheme without correctors, (3) using a basic representation of general relativity, and (4) using a time step of 3.16 days. By combining our solar system ensembles with previous ensembles, we form a 9601-member ensemble of ensembles. In this ensemble of ensembles, the logarithm of the frequency of a Mercury instability event increases linearly with time between 1.3 and 5 Gyr, suggesting that a single mechanism is responsible for Mercury instabilities in this time range and that this mechanism becomes more active as time progresses. Our work provides a robust estimate of Mercury instability statistics over the next five billion years, outlines methodologies that may be useful for exoplanet system investigations, and provides two large ensembles of publicly available solar system integrations that can serve as test beds for theoretical ideas as well as training sets for artificial intelligence schemes.

A Secular Solar System Resonance that Disrupts the Dominant Cycle in Earth’s Orbital Eccentricity (g 2 − g 5): Implications for Astrochronology

“…Total energy and angular momentum errors were small throughout the 3.5 Gyr integrations (relative errors: 6 × 10 −10 and 7 × 10 −12 , see Figure 2). Our default numerical timestep (|Δt| = 4 days) is close to the previously estimated value of 3.59 days to sufficiently resolve Mercury's perihelion (Wisdom 2015;Hernandez et al 2022;Abbot et al 2023). In additional eight simulations, we tested |Δt| = 2.15625 days (adequate to e ☿  0.4) and found no difference in terms of σ 12 resonances (see below), which occurred in 3/8 solutions.…”

“…Prediction becomes impossible" (Poincaré 1914). In reference to the solar system, the sensitivity to initial conditions is indeed a key feature of the large-scale dynamical chaos in the system, which has been confirmed numerically (Sussman & Wisdom 1988;Laskar 1989;Ito & Tanikawa 2002;Morbidelli 2002;Varadi et al 2003;Batygin & Laughlin 2008;Zeebe 2015a;Brown & Rein 2020;Abbot et al 2023). Dynamical chaos affects the secular frequencies g i and s i (see Appendix A), where the terms (g 4 −g 3 ) and (s 4 −s 3 ), for instance, show chaotic behavior already on a 50 Myr timescale.…”

“…We performed ensemble integrations of the solar system with a total of N = 64 members. Note that a larger N is unnecessary for the current problem, which samples a common phenomenon (∼40% of solutions), not a rare event, which requires large N (Abbot et al 2023). Different solutions were obtained by offsetting Earth's initial position by a small distance (largest offset Δx 0 ; 1 × 10 −12 au), which is within observational uncertainties (Zeebe 2015a(Zeebe , 2017.…”

OrbitN: A Symplectic Integrator for Planetary Systems Dominated by a Central Mass—Insight into Long-term Solar System Chaos

“…The core of orbitN's integrator scheme is based on a second-order symplectic map, which is described at length elsewhere and is not repeated here (e.g., Wisdom & Holman 1991;Yoshida 1990;Saha & Tremaine 1994;Mikkola 1997;Chambers 1999;Murray & Dermott 1999;Rein & Tamayo 2015). (Note that recent studies indicate that higher-order symplectic schemes are not necessarily advantageous; Hernandez et al 2022;Abbot et al 2023. ) A few features deserve attention here, such as the Hamiltonian splitting and mass factors, which are important, for instance, for the implementation of 1PN corrections.…”

“…The system's motion is a superposition of all eigenmodes, although some modes represent the single dominant term for some (mostly outer) planets. The g 1 − g 5 interaction, "(g 1 − g 5 )" for short, can force Mercury's eccentricity to high values and plays a critical role in the long-term stability of the solar system on a Gyr timescale (e.g., Laskar 1990;Batygin & Laughlin 2008;Lithwick & Wu 2011;Zeebe 2015a;Abbot et al 2023;Brown & Rein 2023). Importantly, (g 1 − g 5 ) is affected by GR, as PN corrections move g 1 up (by ∼0 43 yr −1 at present) and away from g 5 (for illustration, see Figure 4 in Zeebe 2015a), thus reducing the tendency for instability on a Gyr timescale.…”