Mogavero and Laskar (2022) used computer-algebra canonical perturbation theory to retrieve the web of secular resonances underlying the chaotic diffusion of the inner Solar System (ISS) planets, and compared the time statistics of resonance half-widths with the ensemble distribution of the finite-time maximal Lyapunov exponent (FT-MLE). The resulting picture - ISS chaos as a high-dimensional resonance-overlap phenomenon - is statistical and rests on unproven pillars that the authors themselves flagged as heuristic: the Lie-transform normal form is truncated at degree ten without a convergence guarantee; the link to the FT-MLE uses only order-of-magnitude relations (2π FT-MLE∼Δω2\pi\,\mathrm{FT\text{-}MLE}\sim\Delta\omega and pendulum-based half-width scaling beyond their strict domain); the statistics rely on an unproven finite-time ergodicity assumption for a single nominal solution; and validation against direct integrations involved only a handful of solutions. The problem is to convert this statistical resonance-web picture into a quantitative, model-independent dynamical theory of ISS chaos: control the truncation, derive a precise relation between ensemble chaos rates and resonance diagnostics, justify (or delimit) the finite-time ergodicity assumption, and validate the resonance web statistically across large full N-body ensembles and proper-mode choices.

Background

The planets of the inner Solar System (Mercury, Venus, Earth, Mars) move chaotically: infinitesimally close orbits separate exponentially with a Lyapunov time (the inverse of the maximal Lyapunov exponent, the asymptotic rate of exponential divergence of nearby trajectories) of only about 5 million years, as first revealed numerically by Laskar (1989) and Sussman and Wisdom (1992). Because trajectories cannot be followed beyond a few tens of millions of years, the long-term behavior of the ISS is necessarily a statistical question about an ensemble of solutions. Mogavero and Laskar (2021) constructed an efficient forced secular model for the inner planets: after averaging out the fast orbital (Keplerian) motions, the outer planets' regular long-term motion is replaced by a predetermined quasi-periodic forcing, leaving a Hamiltonian system in the slow eccentricity-inclination degrees of freedom of the inner planets. In this framework the state is described through proper modes - complex amplitudes whose phases rotate at the fundamental secular frequencies g1,…,g4g_1,\dots,g_4 (perihelion precession) and s1,…,s4s_1,\dots,s_4 (nodal regression). A secular resonance occurs when a linear integer combination of these frequencies (a resonant harmonic, or resonant angle) librates rather than circulates; each resonance traps trajectories in a band of frequency space whose size is measured by its half-width Δω\Delta\omega. Chirikov's overlap criterion heuristically predicts widespread chaos when neighboring resonances overlap. Mogavero and Laskar (2022) implemented canonical perturbation theory - a Lie transform, i.e. a near-identity change of variables generated by a solving Hamiltonian, implemented in the TRIP computer-algebra system - to eliminate non-resonant harmonics of degree four from polynomially truncated versions H2n\mathbb{H}_{2n} of the secular Hamiltonian, and systematically retrieved the resonances at work along a 5-Gyr nominal solution. Because Mercury can collide with Venus or the Sun, the infinite-time Lyapunov exponent is ill-defined; they instead computed a finite-time MLE (FT-MLE) by Benettin's tangent-vector renormalization algorithm over finite spans, treated as a stochastic variable over an ensemble of solutions. Comparing the time statistics of resonance half-widths and fixed-point frequencies along one nominal solution with the ensemble FT-MLE distribution - using the order-of-magnitude relations Δω∼2ωhyp∥k∥−1\Delta\omega\sim 2\omega_{\mathrm{hyp}}\lVert k\rVert^{-1} (the pendulum/Chirikov half-width, with ωhyp\omega_{\mathrm{hyp}} the hyperbolic fixed-point frequency of the reduced resonant phase space and kk the resonance integer wave vector) and 2π FT-MLE∼Δω2\pi\,\mathrm{FT\text{-}MLE}\sim\Delta\omega - they identified a statistical connection and concluded that the interaction of degree-four harmonics, with resonances such as (g1−g5)−(s1−s2)(g_1-g_5)-(s_1-s_2) among the leading terms, is the primary source of the observed chaos. The paper explicitly relies on approximations whose validity is not established: the Lie transform is truncated at degree ten (its slow apparent convergence suggests higher-degree contributions are non-negligible), the FT-MLE connection is only heuristic (the authors call a precise formula for high-dimensional systems 'perhaps hopeless'), the time statistics along one solution stand in for ensemble statistics via a 'finite-time ergodic assumption', and direct-integration checks used only a few solutions of one N-body model. The unresolved question is whether this statistical picture can be turned into a quantitative, model-independent theory of ISS chaotic diffusion.

Problem Statement

Turn the statistical resonance-web picture of Mogavero & Laskar (2022) into a quantitative, model-independent dynamical theory of chaos in the inner Solar System. Concretely, within the forced secular dynamics of the inner planets (Mogavero & Laskar 2021) and its relation to full N-body models, establish the four linked pillars that the 2022 analysis left heuristic: (1) Truncation control - establish convergence, or a controlled asymptotic behaviour (a stated bound on the truncation error, or an explicit growth rate of the neglected high-degree contributions, as a function of truncation degree at fixed energy), of the Lie-transform normal-form construction as the truncation degree grows beyond ten, and quantify how the retrieved resonance statistics (half-widths, fixed-point frequencies, resonance rankings) change with truncation level, given the authors' observation that slow finite-degree convergence suggests non-negligible high-degree contributions. (2) A precise chaos-resonance relation - derive a theoretically grounded relation, replacing the heuristic 2π FT-MLE∼Δω2\pi\,\mathrm{FT\text{-}MLE}\sim\Delta\omega and the extrapolated pendulum half-width scaling, between the ensemble distribution of the finite-time maximal Lyapunov exponent and the measured resonance diagnostics (half-widths and hyperbolic/elliptic fixed-point frequencies) in the high-dimensional forced secular system. (3) Finite-time ergodicity - justify or delimit the assumption that time statistics of resonance observables along a single nominal multi-Gyr solution represent the large-time ensemble statistics of the chaotic ISS. (4) Model independence - statistically validate the predicted resonance web across large ensembles of full N-body integrations and across different choices of proper-mode definitions, establishing the extent to which the identified web is independent of the dynamical formulation. The long-term diffusion suggested by the 2022 results - mainly perpendicular to the resonance planes - and Mercury's very high eccentricity excursions, in which the (g1−g5)(g_1-g_5) harmonic is known to be involved, motivate this program as descriptive context. These components serve one objective - making the resonance-overlap origin of ISS chaos quantitative and predictive: pillar 1, the truncation-error behaviour just described together with the truncation-level dependence of the resonance statistics; pillar 2, a derived, quantitatively testable relation, with its assumptions and domain of validity stated, that predicts the ensemble FT-MLE distribution from the resonance diagnostics; pillar 3, a proof that the finite-time ergodic assumption holds under explicitly stated conditions, or an explicit delimitation of the regime in which it fails; pillar 4, a confirmation or rejection, at a stated statistical significance, of the resonance web across large full N-body ensembles and across multiple distinct proper-mode definitions.

References

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