Is one-dimensional advection-diffusion in a secular frequency representative of planetary systems beyond the Solar System, and what are their expected instability times?

The long-term chaotic evolution of the Solar System is remarkably well captured by a simple one-dimensional stochastic model: the perihelion-precession secular frequency of Mercury, $g_1$ , undergoes advection and diffusion until it reaches the $g_1-g_5$ secular resonance with Jupiter, driving Mercury's eccentricity to instability. Brown & Rein (2023) showed this Fokker-Planck description reproduces ensemble N-body instability statistics with and without general relativistic precession, with no new free parameters. They explicitly left open whether such a one-dimensional advection-diffusion model is representative of other secularly evolving planetary systems, or whether the Solar System is special and other systems require higher-dimensional stochastic descriptions, and correspondingly what the expected instability times of other secularly evolving systems are.

Background

When the fast orbital phases of the planets are averaged out, their eccentricities and inclinations evolve under 'secular' (slow) interactions, described at leading order by the Laplace-Lagrange linear theory. This theory assigns each planet a set of normal modes whose frequencies, conventionally $g_1,\dots,g_9$ for perihelion precession, are constant. In the real Solar System these frequencies wander chaotically because the secular Hamiltonian is non-integrable at higher order, and if two frequencies drift into a secular resonance, angular-momentum exchange can pump eccentricities until planetary orbits cross. The innermost planet, Mercury, is the vulnerable one: its precession mode $g_1$ can diffuse downward toward Jupiter's nearly constant frequency $g_5$ , and entry into the $g_1-g_5$ secular resonance excites Mercury's eccentricity until it encounters Venus or the Sun. General relativity (GR) advances Mercury's perihelion, shifting $g_1$ away from $g_5$ and, as the landmark ensemble study of Laskar & Gastineau (2009) showed, reducing the fraction of Solar-System realizations that destabilize within 5 Gyr from about 60% without GR to about 1% with it. Mogavero & Laskar (2021) showed that the observed statistics of Mercury's destabilization can be reproduced by modeling $g_1$ as a stochastic process (a Wiener process, the idealized continuous random walk, with drift), i.e. a Fokker-Planck advection-diffusion equation with an absorbing boundary at resonance. Brown & Rein (2023) extended this phenomenology with 1280 direct N-body integrations over 12.5 Gyr in which the GR precession rate was smoothly varied in time; a one-dimensional advection-diffusion model in $g_1$ , calibrated only on the two limiting cases (GR fully on or off, with a no-GR diffusion timescale of 3.22 Gyr versus roughly 30 Gyr with GR), reproduced the survival fractions across all intermediate regimes without new free parameters, and showed there is no critical GR precession rate at which instability abruptly rises. An analogous phenomenon, 'secular chaos' driven by diffusing secular variables, is thought to operate in exoplanetary systems (Wu & Lithwick 2011; O'Connor et al. 2022), and ensemble studies of compact multi-planet systems characterize their instability-time distributions (Hussain & Tamayo 2020). What has not been done is to test whether the Solar System's simple one-dimensional secular-frequency transport model — one diffusing coordinate, one absorbing resonance — describes any other planetary system, or is an accident of the Solar System's architecture.

Problem Statement

Determine whether the one-dimensional advection-diffusion (Fokker-Planck) description of secular-frequency evolution that reproduces the Solar System's instability statistics — a single diffusing coordinate, Mercury's precession frequency $g_1$ , with drift and an absorbing boundary at the $g_1-g_5$ secular resonance — is representative of other secularly evolving planetary systems, or whether the Solar System is special and other systems require higher-dimensional stochastic models; and correspondingly determine the expected time to instability for other secularly evolving planetary systems. Concretely, the task is resolved by establishing, for at least one class of non-Solar-System architectures in which instability is driven by secular (not mean-motion, i.e. orbital-period) chaos, whether the one-dimensional advection-diffusion description suffices or higher-dimensional stochastic models are necessary, together with the expected time to instability for that class. This requires identifying the slowly diffusing secular variable(s) and resonance condition analogous to $g_1$ and $g_1-g_5$ , constructing and calibrating (or deriving from first principles) the advection and diffusion coefficients of the stochastic transport model, and validating the model's predicted instability-time statistics against ensembles of direct N-body integrations. The problem spans secularly evolving planetary systems; compact multi-planet systems, eccentric giant-planet systems, and hierarchical systems driven by secular chaos are all admissible instances.

References

Brown, G. & Rein, H. 2023, 'General relativistic precession and the long-term stability of the Solar system', Monthly Notices of the Royal Astronomical Society 521(3):4349-4355, DOI 10.1093/mnras/stad719, https://doi.org/10.1093/mnras/stad719 (preprint arXiv:2303.05567, https://arxiv.org/abs/2303.05567)
Laskar, J. & Gastineau, M. 2009, 'Existence of collisional trajectories of Mercury, Mars and Venus with the Earth', Nature 459:817-819, DOI 10.1038/nature08096, https://doi.org/10.1038/nature08096
Mogavero, F. & Laskar, J. 2021, 'Long-term dynamics of the inner planets in the Solar System', Astronomy & Astrophysics 655:A1, DOI 10.1051/0004-6361/202141007, https://doi.org/10.1051/0004-6361/202141007
Hussain, N. & Tamayo, D. 2020, 'Fundamental limits from chaos on instability time predictions in compact planetary systems', Monthly Notices of the Royal Astronomical Society 491:5258-5267, DOI 10.1093/mnras/stz3402, https://doi.org/10.1093/mnras/stz3402
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Wu, Y. & Lithwick, Y. 2011, 'Secular Chaos and the Production of Hot Jupiters', The Astrophysical Journal 735(2):109, DOI 10.1088/0004-637X/735/2/109, https://doi.org/10.1088/0004-637X/735/2/109 (preprint arXiv:1012.3475, https://arxiv.org/abs/1012.3475)