Anomalous diffusion exponents in Solar System chaotic transport: dynamical origin and genericity

Two independently documented anomalous (non-Fickian) diffusion laws in the quasi-integrable dynamics of the Solar System remain without a dynamical explanation: the secular precession frequency of Mercury, $g_1$, spreads subdiffusively with Hurst exponent alpha ~ 0.26 in billion-year N-body ensembles, and asteroids resonant in the exterior 1:2 mean-motion resonance with Mars show eccentricity dispersion growing as sigma ~ t^b with b ~ 0.31-0.43. For the asteroid case, whether the same diffusive character extends beyond this single resonance is also open. The problem asks for the phase-space mechanism that sets these exponents (including the asymmetric boundary structure of $g_1$) and, for the asteroid case, for a determination, empirical or theoretical, of whether the character of the measured chaotic diffusion is shared by other resonances and locations in the inner main belt.

Background

The orbital motion of the planets is chaotic: although the eight-planet problem is close to integrable (Keplerian two-body motion plus weak mutual perturbations), the Lyapunov time of the inner planets — the timescale on which initially nearby orbits diverge exponentially — is only about 5 million years, so exact orbital prediction is impossible beyond roughly 60 Myr and billion-year (Gyr) evolution must be described statistically. The central statistical objects are the secular frequencies. In secular dynamics one averages over the fast orbital revolutions and follows the slow precession of the orbital ellipses; the perihelion-precession rates of the planets are denoted $g_1, ..., g_8$ (in arcsec/yr), with $g_1$ dominated by Mercury and $g_5$ by Jupiter. Mercury's orbit can destabilize when chaotic variation drives $g_1$ down toward the nearly constant $g_5$ ~ 4.26 arcsec/yr; the resulting $g_1$-$g_5$ secular resonance (a near-commensurability of two secular precession rates) pumps Mercury's eccentricity until it collides with Venus or the Sun.

Because exact prediction fails, $g_1(t)$ has been modeled as a one-dimensional stochastic process: standard (Fickian) Brownian motion with constant diffusivity, a reflecting upper boundary, and an absorbing lower boundary at $g_5$ whose first passage (the first time the process reaches that boundary) signals Mercury instability (Mogavero & Laskar 2022), building on the empirical probability density of the secular frequencies obtained from 120,000 solutions of a high-order secular model (Hoang, Mogavero & Laskar 2021). Brown & Rein (2023) developed a related Fokker-Planck advection-diffusion description — a partial differential equation for the probability density of Mercury's precession frequency — including general-relativistic precession. Such diffusive models reproduce instability statistics only on timescales longer than about 10 Gyr. Abbot et al. (2021, 2023) introduced rare-event sampling (quantile diffusion Monte Carlo, which biases an ensemble of trajectories toward rare outcomes so that instability events are sampled efficiently) and public 2750-member 5 Gyr N-body ensembles to reach shorter times. Abbot, Webber, Hernandez, Hadden & Weare (2024) then showed that the diffusive models underpredict the Mercury instability probability by factors of 3-1000 on the physically relevant timescale of the Sun's remaining main-sequence life (<5 Gyr), because $g_1$ makes larger short-timescale excursions than Brownian motion allows. They replaced the Brownian model with a subdiffusive one based on fractional Brownian motion, a Gaussian process $W(t)$ whose increments satisfy

$$⟨|W(t)-W(s)|^2⟩ = |t-s|^{2\alpha},$$

where alpha in (0,1) is the Hurst exponent: alpha = 1/2 gives standard diffusion, while alpha < 1/2 gives subdiffusion, with larger short-timescale and smaller long-timescale dispersion. Their tuned model (alpha ~ 0.26, an effective reflecting upper boundary near 5.85 arcsec/yr, an absorbing lower boundary) reproduces Mercury instability statistics from 1 to 40 Gyr. The same paper poses, but cannot answer, the dynamical questions: why subdiffusion rather than standard diffusion; what physical structure acts as the restoring upper boundary (which blocks the $g_1$-$g_2$ resonance with $g_2$ ~ 7.45 arcsec/yr); and why there is no analogous restoring lower boundary preventing $g_1$ from reaching $g_5$. It further notes that N-body $g_1$ trajectories show intermittency — sustained quiescent and active periods — that its subdiffusive model does not reproduce, speculating that quiescent periods could be associated with proximity to islands of regularity. Candidate mechanisms (stickiness near remnant KAM tori, partial barriers, fractal-like transport) are unverified: KAM tori are the invariant quasi-periodic manifolds that, in Hamiltonian systems with two degrees of freedom, partition phase space into regular islands and a chaotic sea, and their boundaries transiently trap chaotic orbits ('stickiness'); in the Solar System's high-dimensional secular phase space, such tori no longer partition phase space, and no analogue of the two-dimensional stickiness theory has been established.

Independently, anomalous power-law diffusion has been measured in asteroid dynamics. The inner main belt (semimajor axes 2.1-2.5 au) is bounded by the ν6 secular resonance (in which an asteroid's perihelion-precession rate equals Saturn's, $g_6$) and the 1:3 mean-motion resonance with Jupiter. A mean-motion resonance (MMR) is a commensurability of orbital periods; the exterior 1:2 resonance with Mars (near 2.42 au, inside the inner main belt) is the configuration in which the asteroid completes one orbit for every two Mars orbits. For asteroids larger than about 30 km the Yarkovsky effect — a size-dependent radiation-reaction drift in semimajor axis — is negligible, so orbital mobility is purely gravitational. Christou, Dermott & Li (2022) integrated controlled ensembles of test particles near this resonance and measured the growth of the dispersion of proper elements (the averaged, nearly constant orbital elements that label quasi-periodic orbits): sigma(t) = C·t^b with b = 0.306-0.434 for eccentricity and 0.350-0.518 for inclination, i.e., slower than the random-walk value b = 1/2. They also found that at high eccentricity resonance trapping protects asteroids from close encounters with Mars, lengthening their residence in the belt, and they explicitly left open whether the character of chaotic diffusion measured in the vicinity of this resonance is also shared by other resonances and locations in the inner main belt.

Problem Statement

The problem is to explain and generalize the anomalous (non-Fickian) power-law diffusion exponents measured in the chaotic transport of the quasi-integrable Solar System, as documented in two anchor studies. It has two components, each corresponding to an open question posed by one of the two anchor studies.

(1) Dynamical origin. Identify the phase-space mechanism or mechanisms in the high-dimensional Hamiltonian dynamics of the Solar System that produce: (a) the subdiffusive scaling of Mercury's secular precession frequency $g_1$ measured in Gyr-scale N-body ensembles (Hurst exponent alpha ~ 0.26), together with the observed asymmetric boundary structure — an effective restoring upper boundary near 5.85 arcsec/yr that blocks the $g_1$-$g_2$ resonance, and the absence of any restoring lower boundary preventing $g_1$ from descending to $g_5$ ~ 4.26 arcsec/yr; and (b) the sub-Fickian eccentricity-dispersion exponent b ~ 0.31-0.43 (inclination 0.35-0.52) of asteroids resonant in the exterior 1:2 mean-motion resonance with Mars. The target is a dynamical explanation of these measured behaviors, rather than a further stochastic phenomenology fitted to them.

(2) Genericity. Determine whether the character of chaotic diffusion measured in the vicinity of the exterior 1:2 mean-motion resonance with Mars is also shared by other resonances and locations in the inner main belt: measure and/or predict from theory the dispersion exponents for other mean-motion-resonance contexts of the inner main belt (for example, other martian or jovian resonances), establishing whether values such as b ~ 0.4 are generic properties of chaotic diffusion there or peculiar to the documented resonance.

References

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