Boundary between chaotic and regular motion in N-planet systems: full phase-space structure and the transition from local chaos to orbital instability

For two planets orbiting a star, the Chirikov-Wisdom resonance-overlap criterion gives a sharp boundary between chaotic and regular motion at a relative separation of order the planet-to-star mass ratio to the power 2/7. Zhou and Sun (2007) showed that in 'EMS' systems (equal planet masses on initially circular, coplanar orbits spaced in mutual Hill radii) with three or more planets no such transition region exists in initial-separation space: the velocity dispersion diffuses roughly as the square root of time, the Lyapunov time is empirically about 0.01 times the orbit-crossing time, and apparently regular orbits gradually turn chaotic, while leaving open whether a structured boundary exists in the full phase space of general N-planet systems and when weak local chaos becomes macroscopic instability. Subsequent work established chaos-onset criteria, instability mechanisms, and probabilistic lifetime predictions for compact Kepler-like architectures, but a characterization of the chaotic/regular boundary for general masses, eccentricities, and inclinations, together with a general theory of the chaos-to-instability conversion, remains open.

Background

A planetary system is a Newtonian N-body problem: one star and $n$ planets interacting gravitationally. Poincaré proved that such systems are generically non-integrable (no closed-form global solution exists), and Arnold diffusion implies that in near-integrable systems with more than two degrees of freedom the conserved quantities of the approximating system can drift over long times; Laskar's (1989) numerical experiment demonstrated that even the Solar System is chaotic. The standard local diagnostic of chaos is the Lyapunov exponent: one integrates the variational equations alongside the orbits to obtain $\chi(t) = t^{-1}\log(|\xi(t)|/|\xi(0)|)$ for the separation growth of an infinitesimally displaced neighboring trajectory; regular (quasi-periodic) orbits have $\chi \to 0$, chaotic orbits have $\chi > 0$, and the inverse Lyapunov time $\tau_L = 1/\chi$ measures how quickly nearby trajectories diverge.

A 'boundary between chaotic and regular motion' is a surface in the system's phase space (the space of all positions and velocities) separating these two behaviors. For two planets, Wisdom (1980) showed, via the Chirikov resonance-overlap criterion — the principle that motion becomes chaotic where the libration zones of adjacent mean-motion resonances (commensurabilities of orbital periods, MMRs) overlap — that the chaotic zone around a perturbing planet has a width scaling as $\mu^{2/7}$ in orbital separation, where $\mu$ is the planet-to-star mass ratio; Gladman (1993) confirmed a sharp borderline for two massive planets. Measured in mutual Hill radii, $R_H \approx ((m_1+m_2)/(3M_\star))^{1/3}(a_1+a_2)/2$ (the distance within which the planets' mutual attraction dominates over the star's differential pull), this boundary sits at $k_c \approx 2.3\,\mu^{-1/21}$.

Zhou and Sun (2007) asked whether analogous boundaries exist for $n \ge 3$ planets. They studied EMS systems — equal planet masses, initially circular and coplanar orbits, equal spacing $k_0$ in mutual Hill radii — an idealization of the roughly Moon-to-Mars-mass planetary embryos left after the oligarchic (few-large-bodies) stage of planet formation. Integrating the variational equations up to $10^7$ years, they found: for $n=2$ the Wisdom–Gladman borderline is reproduced ($k_c \approx 5.2$ for $\mu = 10^{-8}$); for $n \ge 3$ there is no transition region in initial-separation space — $\chi(t)$ plateaus at intermediate values for a long time before growing, the planets' velocity dispersion diffuses as $\sim t^{1/2}$, and the empirical relation $\log \chi \approx 2 - \log T_c$ holds, i.e. the Lyapunov time is roughly two orders of magnitude shorter than the orbit-crossing time $T_c$ (the time to the first close encounter or orbit crossing). Their interpretation was that slow diffusion of eccentricity gradually converts regular orbits into chaotic ones, so no sharp boundary appears in the separation projection, although boundaries may still exist in the full phase space. The study was restricted to equal masses and initially circular, coplanar orbits, and left open both the structure of the chaotic/regular boundary in the full phase space of general N-planet systems and the classic prediction problem of when weak local chaos becomes significant instability — how long a given system can survive.

Since 2007 this problem has become central to exoplanet science, because the Kepler mission revealed a large population of compact multiplanet systems (several sub-Neptune planets on tightly packed orbits) whose stability boundaries govern both their long-term fate and their use as probes of planet formation. Subsequent work introduced tools the 2007 paper lacked: three-body resonances (commensurabilities involving three orbital periods, 3BRs) as the dominant chaos driver in multiplanet systems, the angular momentum deficit (AMD, a measure of the orbital eccentricity content available to fuel chaotic diffusion), and machine-learning stability classifiers.

Problem Statement

For a planetary system of one star and $n \ge 3$ planets with arbitrary masses, eccentricities, and inclinations, characterize the structure of the boundary between chaotic and regular motion in the full phase space of positions and velocities — determine whether a well-defined transition boundary exists, what its geometry is (including its projections onto separations in mutual Hill radii, eccentricity/AMD, and inclination), and how it depends on the number of planets and the mass distribution — and, as the linked second facet of the same transition, establish the general relation between the local Lyapunov time of initially weakly chaotic or regular-looking motion and the macroscopic instability time (time to orbit crossing, close encounter, or ejection): the conditions under which slow chaotic diffusion converts quasi-regular motion into significant instability, and the resulting predictive law (deterministic or explicitly probabilistic) for how long a given system can remain stable. The question generalizes Zhou & Sun (2007), whose conclusions were restricted to equal-mass, initially circular, coplanar (EMS) systems with spacings measured in mutual Hill radii and integration times up to $10^7$ years; an acceptable answer must cover general initial conditions and mass distributions, and must jointly address the boundary structure and the chaos-to-instability conversion, since these are two views of the same transition.

References

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