Three‐body resonance overlap in closely spaced multiple‐planet systems

We compute the strengths of zeroth 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 semimajor 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 uniformly 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 semimajor 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 time-scales significantly increase. Crudely estimated diffusion coefficients in eccentricity and semimajor axis depend on a high power of planet mass and interplanetary spacing. An exponential dependence previously fit to stability or crossing time-scales 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.