Three Body Resonance Overlap in Closely Spaced Multiple Planet Systems

TL;DR

This paper analyzes three-body resonances in closely spaced multi-planet systems, deriving an overlap criterion and demonstrating their role in system instability and orbital wandering.

Contribution

It introduces a resonance overlap criterion for three-body resonances in low eccentricity, coplanar multi-planet systems and links resonance density to system instability.

Findings

Three-body resonances overlap when interplanetary separation is less than a factor times planet mass to the quarter power.

Resonance overlap accounts for semi-major axis wander and system instability.

Stability timescales increase outside the resonance overlap region.

Abstract

We compute the strengths of zero-th order (in eccentricity) three-body resonances for a co-planar and low eccentricity multiple planet system. In a numerical integration we illustrate that slowly moving Laplace angles are matched by variations in semi-major axes among three bodies with the outer two bodies moving in the same direction and the inner one moving in the opposite direction, as would be expected from the two quantities that are conserved in the three-body resonance. A resonance overlap criterion is derived for the closely and equally spaced, equal mass system with three-body resonances overlapping when interplanetary separation is less than an order unity factor times the planet mass to the one quarter power. We find that three-body resonances are sufficiently dense to account for wander in semi-major axis seen in numerical integrations of closely spaced systems and they are likely the cause of instability of these systems. For interplanetary separations outside the overlap region, stability timescales significantly increase. Crudely estimated diffusion coefficients in eccentricity and semi-major axis depend on a high power of planet mass and interplanetary spacing. An exponential dependence previously fit to stability or crossing timescales is likely due to the limited range of parameters and times possible in integration and the strong power law dependence of the diffusion rates on these quantities.