The long-term orbital evolution of the solar system is chaotic, yet Mercury's instability statistics in ensembles of N-body simulations with, without, and with time-varying general relativistic (GR) precession are reproduced by a one-dimensional Fokker–Planck advection–diffusion model for the secular precession frequency g1g_1 with no free parameters (Brown & Rein 2023). That work explicitly poses, as future work, whether this one-dimensional stochastic reduction is representative of other secularly evolving planetary systems or special to the solar system; and its central finding — that removing GR precession raises the instability rate mainly by increasing the rate of chaotic diffusion of g1g_1, not merely by moving g1g_1 closer to the g1g_1–g5g_5 secular resonance — is stated without a demonstrated dynamical mechanism. The open problem is to give the stochastic description a dynamical foundation: identify the secular resonances and interaction terms that set the diffusion coefficient and its dependence on relativistic precession in the solar system, and determine for which planetary architectures long-term secular chaos admits a one-dimensional advection–diffusion description and when higher-dimensional or anomalous (e.g., subdiffusive) stochastic models are required.

Background

Secular dynamics describes planetary motion after averaging over the orbital periods: each planet's eccentricity and perihelion direction decompose into eigenmodes whose perihelia precess at characteristic rates, the secular frequencies gig_i (conventionally quoted in arcseconds per year for the solar system). A linear secular resonance occurs when two of these precession rates approach each other; for the inner solar system the dominant one is between Mercury's principal frequency g1g_1 (with g1,planets≃5.323′′ yr−1g_{1,\mathrm{planets}}\simeq5.323''\,\mathrm{yr}^{-1} set by Newtonian planetary interactions) and Jupiter's nearly constant mode g5g_5. General relativity (GR) contributes a post-Newtonian perihelion precession (the leading first-order relativistic correction to Newtonian motion) of 0.4298′′ yr−10.4298''\,\mathrm{yr}^{-1} for Mercury — more than three orders of magnitude larger than all other effects commonly omitted (solar oblateness, moons, asteroids, tides, together \sim2.8\times10^{-4}''\,\mathrm{yr}^{-1}). The planetary orbits are chaotic, with a Lyapunov time (the timescale on which nearby trajectories diverge) of only about 5 Myr, yet Mercury's expected dynamical lifetime is tens of Gyr: chaotic diffusion of g1g_1 toward the g1g_1–g5g_5 resonance can drive Mercury's eccentricity up until close encounters destabilize the inner planets.

Laskar & Gastineau (2009) showed with ensemble integrations that omitting GR precession raises the fraction of solar-system replicas that destabilize within 5 Gyr by roughly a factor of 60 (from about 1% to about 60%), establishing that relativistic precession is essential when modeling the solar system's future. Mogavero & Laskar (2021) subsequently showed that Mercury's long-term evolution is well captured by diffusion among coupled secular resonances, and Hoang, Mogavero & Laskar (2021) analyzed the chaotic diffusion of the solar system's fundamental frequencies. Brown & Rein (2023) unified these results into a simple one-dimensional Fokker–Planck advection–diffusion model, in which the probability density of g1g_1 evolves as

∂ρ∂t=−μ ∂ρ∂g1+D ∂2ρ∂g12\frac{\partial\rho}{\partial t}=-\mu\,\frac{\partial\rho}{\partial g_1}+D\,\frac{\partial^2\rho}{\partial g_1^2}

with advection rate μ\mu and diffusion coefficient DD, between an upper reflecting boundary near g1g_1's initial value and an absorbing boundary at the g1g_1–g5g_5 resonance where instability ensues; the model reproduces the instability statistics of N-body ensembles with constant, absent, and smoothly time-varying GR precession using parameters fixed entirely by earlier work, with no free parameters, and was validated against 1280 new 12.5 Gyr integrations. Two features of their results frame the open problem. First, moving g1g_1 closer to the g1g_1–g5g_5 resonance explains only part of the increased instability rate without GR; a significant increase in the diffusion rate is also necessary, and the dynamical origin of that increase is not demonstrated. Second, the response of the system to the GR precession strength is smooth, with no critical value below which the system becomes markedly more unstable. Whether such a one-dimensional reduction is generic — Hussain & Tamayo (2020) characterized instability-time distributions of compact exoplanet systems through ensemble statistics, but not via a secular-frequency diffusion model — or whether the solar system is special in this respect is posed by the source itself as an open question. Adjacent work sharpens the issue: Abbot et al. (2024) found that plain diffusion underpredicts Mercury's instability probability on timescales below 5 Gyr and that a subdiffusive model — one in which the spread of g1g_1 grows more slowly with time than normal diffusion predicts — fits better, indicating that even the solar-system case may require an anomalous-diffusion refinement.

Problem Statement

Give the advection–diffusion description of planetary secular chaos a dynamical foundation, in two linked parts that jointly answer when and why the long-term chaotic evolution of a planetary system reduces to one-dimensional stochastic evolution of a single secular frequency.

(1) Mechanism in the solar system. Identify the specific secular resonances and interaction terms — at the relevant order of the secular Hamiltonian — by which the presence of general relativistic perihelion precession reduces the chaotic diffusion rate of Mercury's secular frequency g1g_1, and from them derive quantitatively how the diffusion coefficient (equivalently the diffusion timescale) depends on the GR precession strength. The identification should explain why removing GR increases the stochastic forcing of g1g_1 beyond what the shift of g1g_1 toward the g1g_1–g5g_5 resonance alone predicts, and should be consistent with the finding that the instability response to GR strength is smooth, with no critical precession rate.

(2) Generality across systems. Determine whether the one-dimensional advection–diffusion reduction — instability statistics reproduced by drift plus diffusion of a single secular frequency between its upper bound and a resonant absorbing boundary, without free parameters — is representative of other secularly evolving planetary systems or special to the solar system: for which planetary architectures (planet numbers, masses, spacings, eccentricities, separations of precession frequencies) the reduction holds, and when higher-dimensional or anomalous stochastic models — e.g., subdiffusive ones, in which the diffusing frequency spreads more slowly than normal diffusion predicts — are required instead.

The mechanism's predicted GR-dependence of the diffusion rate must be quantitatively testable against existing and new ensemble simulations, and the class of systems for which the reduction holds or fails must be delineated.

References

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