A quantitative theory linking quasi-integral diffusion to the instability times of the inner Solar System

The orbits of the inner planets (Mercury, Venus, Earth, Mars) are chaotic with a Lyapunov time of only about 5 Myr, yet the inner Solar System is statistically stable over billions of years: the probability that Mercury's eccentricity is driven to encounter-producing values within the Sun's remaining lifetime is of order one percent. Mogavero, Hoang & Laskar (2023) traced this stability to three quasi-integrals of motion — slow, nearly conserved combinations of the secular action variables generated by quasi-symmetries of the leading resonances of the forced secular Hamiltonian — and showed, by comparing the full secular model with a simplified one in which one quasi-integral becomes exactly conserved, that the dynamical half-life is tied to the speed of the quasi-integrals' diffusion. That link, however, is established only by model comparison: no quantitative theory derives the distribution of instability times from the diffusion of the quasi-integrals. The problem posed here is to construct and validate such a theory — anchored in the resonant structure of the secular Hamiltonian — including the determination of the appropriate (possibly nonlinear) slow variables, robust ensemble statistics for the resonance half-widths beyond a single 5-Gyr integration, a bound on the role of the higher-order resonances excluded by the degree-10 truncation, and quantitative validation against 100-Gyr numerical ensembles. Existing adjacent work (instanton computations in reduced stochastic models; phenomenological diffusion and subdiffusion models of the precession frequency g1) treats the escape statistics top-down and leaves the mechanism open.

Background

Whether the Solar System is stable is one of the oldest problems of dynamical astronomy. Numerical integrations show that the inner planetary orbits, while chaotic — with a Lyapunov time of only about 5 Myr, i.e. the characteristic time for nearby trajectories to separate exponentially as measured by the largest Lyapunov exponent of the flow — remain statistically stable over billions of years, with only a small probability of catastrophic events (close encounters, collisions, or ejections of planets) within the Sun's remaining lifetime. Explaining this separation of timescales, roughly three orders of magnitude between the Lyapunov time and the instability time, is the central puzzle.

The standard framework is secular celestial mechanics: after averaging the planetary motion over the fast orbital revolutions (Gauss's averaging of the mutual gravitational interaction), one obtains a Hamiltonian system for the slow deformations of the orbits, their eccentricities and inclinations. In Poincare variables this quasi-integrable dynamics (an integrable Hamiltonian plus a non-integrable perturbation) is written in terms of action-angle variables, where the actions $I=(\mathfrak{X},\Psi)\in\mathbb{R}^{8}$ collect the eccentricity contents $\mathfrak{X}_i$ and inclination contents $\Psi_i$ of the four inner planets and the angles are the longitudes of perihelia and nodes. At leading order, the Laplace–Lagrange secular solution is integrable: each orbit precesses at fixed fundamental frequencies $g_i$ (perihelia) and $s_i$ (nodes) and suitable linear combinations of the actions are conserved. An exact invariant of the full secular problem is the angular momentum deficit (AMD), $C_{\mathrm{ecc}}+C_{\mathrm{inc}}=\sum_i(\mathfrak{X}_i+\Psi_i)$, the difference between the angular momentum the planets would carry on circular coplanar orbits and their actual one. Chaos enters through the nonlinear terms of the averaged Hamiltonian and through the quasi-periodic gravitational forcing exerted by the giant planets, whose secular modes ($g_5$, $g_6$, ..., $s_6$, ...) act as fixed external frequencies in a forced secular model $\mathcal{H}_{2n}$, truncated at degree $2n$ in eccentricities and inclinations. The dominant source of chaos is the secular resonance $g_1-g_5$ — a harmonic of the Hamiltonian whose argument combines the precession frequency $g_1$ of Mercury's perihelion with Jupiter's forced mode $g_5$ so that it becomes resonant along the motion — which can pump Mercury's eccentricity to encounter-producing values (Laskar 2008; Laskar & Gastineau 2009). Ensembles of thousands of solutions over 5–100 Gyr quantify the resulting escape statistics (Laskar & Gastineau 2009; Hoang, Mogavero & Laskar 2022).

Mogavero & Laskar (2021, 2022) developed a computer-algebra pipeline (the TRIP system) that expands the secular Hamiltonian and enumerates its Fourier harmonics, ranking those that become resonant along numerical integrations. Mogavero, Hoang & Laskar (2023), the source of this problem, then computed the entire Lyapunov spectrum of the forced secular model: its eight positive finite-time Lyapunov exponents span two orders of magnitude, revealing a slow–fast structure with characteristic times from about 5 Myr to several hundred Myr. A systematic analysis of the resonant harmonics along a 5-Gyr integration disclosed three quasi-symmetries — patterns in the wave vectors of the strongest resonances that leave particular combinations of the actions invariant. These generate three quasi-integrals of motion (QIs), quantities that are not exactly conserved but diffuse orders of magnitude more slowly than the individual actions: $C_{\mathrm{inc}}=\Psi_1+\Psi_2+\Psi_3+\Psi_4$ (the inclination AMD, interpretable as a remnant of the conservation of the total secular AMD), $C_2=-\mathfrak{X}_3-\mathfrak{X}_4+\Psi_1+\Psi_2+2\Psi_3+2\Psi_4$ (discovered algebraically from the wave vectors of the leading resonances; the source offers a partial interpretation in terms of a decoupling between the Earth–Mars and Mercury–Venus subsystems), and $\mathcal{E}_{2n}$, a nonlinear quantity associated with invariance under the shift that removes the forcing frequency $g_5$ (its linear approximation is $E_2=\gamma_3\cdot I$ for an explicit vector $\gamma_3$). The quasi-symmetries are broken only by weak resonances, whose half-widths (the size of the resonant regions, computable from the harmonic amplitudes) are small. A principal component analysis (PCA), a linear dimensionality-reduction technique that finds the directions of smallest statistical variance in data, independently confirms that the QIs are among the slowest variables of the dynamics, since the last principal components of long integrations align with them. Crucially, in a simplified Hamiltonian $\mathbb{H}^{\bullet}_{2n}$ that retains only the forcing through Jupiter's $g_5$ mode (dropping the forcing mainly exerted by Saturn, Uranus, and Neptune), $\mathcal{E}^{\bullet}_{2n}$ becomes an exact integral of motion, the diffusion of the QIs is strongly slowed, and the 1% instability time grows from roughly 15 Gyr ($\mathcal{H}_4$) to beyond 100 Gyr ($\mathbb{H}^{\bullet}_4$). The paper concludes that the long dynamical half-life of the inner Solar System is linked, in a critical way, to the speed of QI diffusion — but this link is established only by comparing the full and simplified models, not by any quantitative theory that derives the escape-time distribution from the diffusion of the QIs.

Two adjacent strands make the gap concrete. Woillez & Bouchet (2020) computed the instanton — in large-deviation theory, the most probable fluctuation path leading to a rare event — for Mercury's destabilization within a reduced stochastic model of the $g_1$ dynamics. Abbot and collaborators (2021, 2024) fitted phenomenological random-walk models of the frequency $g_1$ to N-body escape statistics, finding that ordinary diffusion underpredicts the instability probability at times below 5 Gyr by factors of 3–10,000, while a subdiffusive model (with larger short-time and smaller long-time displacements than diffusion) reproduces the statistics over 1–40 Gyr. These are top-down stochastic descriptions; no existing work derives the escape-time distribution from the quasi-integral structure of the secular Hamiltonian.

Problem Statement

Construct and validate a quantitative theory that derives the statistical distribution of dynamical instability times of the inner Solar System — concretely, the distribution of the first times at which Mercury's eccentricity reaches encounter-producing values (about 0.7) in the forced secular dynamics — from the chaotic diffusion of the quasi-integrals of motion identified by Mogavero, Hoang & Laskar (2023). The objective comprises four components:

1. The correct slow variables: determine whether the linear quasi-integrals ($C_{\mathrm{inc}}$, $C_2$, $E_2$) suffice or whether genuinely nonlinear slow quantities are required, and clarify the origin of the three quasi-symmetries — conservation-law remnants or accidental features of the resonance network.

2. The effective slow dynamics: derive the drift and diffusion of these variables, or the appropriate large-deviation rate functional, from the resonant structure of the secular Hamiltonian, with resonance identification and half-width statistics that are robust beyond the single 5-Gyr integration used in the source (ensemble estimates over independent solutions), and with a quantitative bound on the contribution of resonances of order 12 and higher that the degree-10 truncation of the harmonic tables excludes.

3. The passage to escape statistics: connect the slow-variable dynamics to the instability times, predicting the escape-time distribution and explaining quantitatively the difference between the full model $\mathcal{H}_{2n}$ and the simplified model $\mathbb{H}^{\bullet}_{2n}$ in which one quasi-integral is exactly conserved and instability is pushed beyond 100 Gyr.

4. Validation: quantitative comparison of the predictions against direct numerical ensembles, reproducing the measured escape-time distribution and the power-law growth of the quasi-integrals' dispersion.

These four components jointly constitute the overarching objective — a quantitative, predictive account of the long-term instability of the inner Solar System in terms of quasi-integral diffusion. Extensions of the framework to other planetary systems or quasi-integrable Hamiltonians, and the replacement of principal component analysis by nonlinear dimensionality-reduction methods, are future directions acknowledged in the source but lie outside the scope of this problem.

References

1. Federico Mogavero, Nam H. Hoang, and Jacques Laskar (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://doi.org/10.1103/PhysRevX.13.021018 (preprint arXiv:2305.01683, https://arxiv.org/abs/2305.01683)
2. Federico Mogavero and Jacques Laskar (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 (preprint arXiv:2205.03298, https://arxiv.org/abs/2205.03298)
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