Dynamical origin of Mercury's subdiffusive secular evolution: why the secular eigenfrequency $g_1$ subdiffuses, why its boundaries are asymmetric, and what produces intermittency

Direct N-body simulations show that Mercury's orbit can destabilize within the Sun's remaining main-sequence lifetime, and the accepted trigger is the slow chaotic wandering of the secular eigenfrequency $g_1$ (the precession frequency of Mercury's perihelion) down to Jupiter's nearly constant frequency $g_5$. Abbot, Webber, Hernandez, Hadden and Weare (2024) showed that a stochastic model in which $g_1$ undergoes subdiffusion — fractional Brownian motion with Hurst index $\alpha\approx0.26$, a reflecting upper boundary, and an absorbing boundary at $g_5$ — reproduces Mercury instability statistics from N-body ensembles over 1–40 Gyr, whereas the earlier diffusive model underpredicts the instability probability by a factor of 3–10,000 below 5 Gyr. The parameters of this successful model are tuned, not derived, and the authors pose as open challenges why $g_1$ subdiffuses rather than diffuses and what mechanism produces the restoring upper boundary while no analogous lower boundary prevents $g_1$ from reaching $g_5$; in the same Discussion they identify as further limitations how the intermittency of N-body $g_1$ trajectories coexists with a stationary fractional-Brownian phenomenology, and how the model can be validated at timescales below 2 Gyr where N-body instability events are too rare. No published work derives the subdiffusion exponent or the boundary asymmetry from the Solar System's dynamics, constructs a mechanistic intermittency model, or tests the subdiffusive model at short times. The questions remain open.

Background

The long-term orbital evolution of the inner planets is chaotic, so their future can only be described statistically. In secular dynamics — the description obtained by averaging planetary motion over the fast orbital revolutions, so that planets are tracked through the slow precession of their orbital ellipses — the eccentricity-type variables of the eight planets oscillate, in the linear Laplace–Lagrange approximation, at fixed perihelion-precession frequencies $g_1,\dots,g_8$ and nodal-regression frequencies $s_1,\dots,s_8$. In the real Solar System these secular frequencies are not constant: the fundamental frequencies wander chaotically. The frequency $g_1\approx5.59$ arcsec/yr is associated mainly with the precession of Mercury's perihelion, while $g_5\approx4.257$ arcsec/yr, associated with Jupiter, is effectively constant.

Mercury is the fragile part of the system. Direct numerical integrations show that if chaotic evolution drives $g_1$ down to $g_5$, the resulting secular resonance (a resonant term in the averaged Hamiltonian) pumps Mercury's eccentricity until its orbit crosses that of Venus, generally ending in a collision with Venus or the Sun. Laskar and Gastineau (2009) integrated an ensemble of 2501 N-body solutions and found a probability of about 1% that Mercury's orbit destabilizes within 5 Gyr, roughly the Sun's remaining main-sequence lifetime. Because such ensembles are expensive, reduced stochastic descriptions were developed in which $g_1$ is modeled as a random process between an upper reflecting boundary (the process is turned back when it reaches it) and a lower absorbing boundary at $g_5$ (reaching it ends the process and marks a Mercury instability event): Woillez and Bouchet (2020) built such a diffusive description for a simplified secular Hamiltonian with Mercury treated as massless and the other planets quasiperiodic (Batygin, Morbidelli and Holman 2015), and Mogavero and Laskar (2021) applied a diffusive model of $g_1$ itself, tuned to reproduce secular-simulation instability statistics beyond 10 Gyr. In a diffusive model the mean-square displacement of $g_1$ grows linearly with time. Hoang, Mogavero and Laskar (2021), measuring the chaotic diffusion of all fundamental frequencies in large ensembles of high-order secular simulations, found instead that the standard deviation of $g_1$ grows as a power law of time with exponent about 0.26 — subdiffusion.

Abbot, Webber, Hernandez, Hadden and Weare (2024) showed that the diffusive model of $g_1$ underpredicts the Mercury instability probability by a factor of 3–10,000 on timescales shorter than 5 Gyr — the physically relevant regime — because it produces variations of $g_1$ that are too small on timescales below about 0.3 Gyr. They built a five-parameter subdiffusive model: $g_1$ follows fractional Brownian motion, a Gaussian process with stationary, correlated increments and self-similarity (Hurst) index $\alpha$; ordinary diffusion corresponds to $\alpha=1/2$, and $\alpha<1/2$ produces larger short-time excursions and smaller long-time dispersion, that is, subdiffusion with mean-square displacement growing as $t^{2\alpha}$. With $\alpha\approx0.26$, a reflecting upper boundary, and an absorbing boundary at $g_5$, tuned to the mean-square displacement, probability density, and instability fraction of a 2750-member N-body ensemble (Abbot et al. 2023), the model reproduces Mercury instability statistics over 1–40 Gyr. Its parameters, however, are fitted rather than derived.

The paper's Discussion then poses the questions this problem records. The authors' own hypothesis sketch comes from conservative Hamiltonian dynamics: in two-degree-of-freedom systems the phase space is a mixture of chaotic trajectories and regular regions bounded by KAM tori (invariant tori that survive perturbation of an integrable system, by the Kolmogorov–Arnold–Moser theorem), and chaotic orbits can be "sticky" near the borders of regular islands — a classic source of anomalous (non-Brownian) transport; in higher dimensions surviving KAM tori no longer partition phase space but might act as a soft, spring-like boundary. The authors explicitly call this sketch highly speculative. They also note that N-body $g_1$ trajectories show intermittency — sustained quiescent periods alternating with sustained active periods — which the stationary fractional-Brownian model does not reproduce, and that too few N-body Mercury instability events exist below 2 Gyr to test the model at the shortest times; they suggest ensembles of order $10^6$ high-order secular simulations, or rare-event sampling techniques such as Diffusion Monte Carlo (here a rare-event trajectory-cloning and sampling scheme, not the quantum-chemistry method) and action-minimization path sampling, possibly with machine-learned importance samplers.

Problem Statement

Explain, from the Hamiltonian dynamics of the Solar System, the subdiffusive phenomenology of the secular precession frequency $g_1$ that the successful stochastic model of Mercury's instability currently encodes as tuned parameters, and validate the resulting description. Concretely:

(1) Subdiffusion mechanism. Identify the dynamical structures and transport mechanisms in the Solar System's high-dimensional chaotic phase space that make the evolution of $g_1$ subdiffusive — with an effective Hurst index $\alpha\approx0.26$ rather than the $\alpha=1/2$ of standard diffusion — so that the scaling of the mean-square displacement (equivalently, the power-law growth of the standard deviation) of $g_1$ over the simulated time range is predicted rather than fitted. Candidate pictures, such as stickiness near remnant KAM structures or the resonance web of the secular Hamiltonian, have been suggested but not demonstrated for this system.

(2) Asymmetric boundaries. Determine the physical origin of the effective restoring upper boundary on $g_1$ inferred from the probability-density fits — which keeps $g_1$ from resonating with $g_2\approx7.45$ arcsec/yr — and explain why there is no analogous restoring lower boundary that would prevent $g_1$ from descending to $g_5$ and thereby suppress Mercury instability events.

(3) Intermittency. Establish to what extent the intermittency of N-body $g_1$ trajectories — alternation of sustained quiescent and active periods — invalidates a stationary fractional-Brownian phenomenology, and either derive a mechanistic model that reproduces the intermittency and the non-Gaussian jump behavior near resonances, or show that the stationary subdiffusive model remains adequate for instability statistics despite them.

(4) Short-time validation. Test the subdiffusive model on timescales below about 2 Gyr, where existing N-body ensembles contain too few Mercury instability events, with integrator artifacts controlled.

Items (1)–(3) constitute the explanatory core: they ask for the dynamical causes of the empirical regularities that the phenomenological model represents by tuned parameters ($\alpha$, boundary locations, diffusivity). Item (4) is the empirical validation of the same model at short times. The overarching objective is a single one: to replace the tuned stochastic description of Mercury's chaotic secular evolution with one derived from, or quantitatively constrained by, the actual dynamics of the Solar System, which items (1)–(4) jointly address.

References

1. Abbot, D. S., Webber, R. J., Hernandez, D. M., Hadden, S., & Weare, J. (2024). Mercury's Chaotic Secular Evolution as a Subdiffusive Process. The Astrophysical Journal, 967, 121. DOI: 10.3847/1538-4357/ad3e5f. https://arxiv.org/abs/2306.11870
2. Hoang, N. H., Mogavero, F., & Laskar, J. (2021). Chaotic diffusion of the fundamental frequencies in the Solar System. Astronomy & Astrophysics, 654, A156; corrigendum: Astronomy & Astrophysics, 671, C9 (2023). DOI: 10.1051/0004-6361/202140989. https://arxiv.org/abs/2106.00584
3. 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
4. Hoang, N. H., Mogavero, F., & Laskar, J. (2022). Long-term instability of the inner Solar System: numerical experiments. Monthly Notices of the Royal Astronomical Society, 514, 1342. DOI: 10.1093/mnras/stac1299. https://arxiv.org/abs/2205.04170
5. Mogavero, F., & Laskar, J. (2022). The origin of chaos in the Solar System through computer algebra. Astronomy & Astrophysics, 662, L3. DOI: 10.1051/0004-6361/202243327. https://doi.org/10.1051/0004-6361/202243327
6. Mogavero, F., Hoang, N. H., & Laskar, J. (2023). Timescales of Chaos in the Inner Solar System: Lyapunov Spectrum and Quasi-integrals of Motion. Physical Review X, 13, 021018. DOI: 10.1103/PhysRevX.13.021018. https://arxiv.org/abs/2305.01683
7. Laskar, J. (2008). Chaotic diffusion in the Solar System. Icarus, 196, 1. DOI: 10.1016/j.icarus.2008.02.017. https://doi.org/10.1016/j.icarus.2008.02.017
8. Laskar, J., & Gastineau, M. (2009). Existence of collisional trajectories of Mercury, Mars and Venus with the Earth. Nature, 459, 817. DOI: 10.1038/nature08096. https://doi.org/10.1038/nature08096
9. Woillez, E., & Bouchet, F. (2020). Instantons for the Destabilization of the Inner Solar System. Physical Review Letters, 125, 021101. DOI: 10.1103/PhysRevLett.125.021101. https://doi.org/10.1103/PhysRevLett.125.021101
10. Batygin, K., Morbidelli, A., & Holman, M. J. (2015). Chaotic Disintegration of the Inner Solar System. The Astrophysical Journal, 799, 120. DOI: 10.1088/0004-637X/799/2/120. https://doi.org/10.1088/0004-637X/799/2/120
11. Abbot, D. S., Hernandez, D. M., Hadden, S., Webber, R. J., Afentakis, G. P., & Weare, J. (2023). Simple Physics and Integrators Accurately Reproduce Mercury Instability Statistics. The Astrophysical Journal, 944, 190. DOI: 10.3847/1538-4357/acb6ff. https://arxiv.org/abs/2212.14844
12. Kaib, N. A., & Raymond, S. N. (2025). The Influence of Passing Field Stars on the Solar System's Dynamical Future. Icarus, 439, 116632. DOI: 10.1016/j.icarus.2025.116632. https://doi.org/10.1016/j.icarus.2025.116632
13. Hernandez, D. M. (2026). Artificial Precession and Instability in Solar System and Planetary Simulations: Analytic and Numerical Results. The Astrophysical Journal, 1005(1), 25. DOI: 10.3847/1538-4357/ae6dad. https://arxiv.org/abs/2603.24456
14. Rein, H., Dey, K., & Tamayo, D. (2026). Democratic Heliocentric Coordinates Underestimate the Rate of Instabilities in Long-term Integrations of the Solar System. The Astrophysical Journal, 1004(2), 197. DOI: 10.3847/1538-4357/ae7363. https://arxiv.org/abs/2601.08019