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

Abstract

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.

Nekhoroshev Estimates for the Survival Time of Tightly Packed Planetary Systems

“…However, we note that some numerical experiments agree well with the ∆a a M m 1/4 scaling (Chambers 2001;Faber and Quillen 2007). Similarly, this scaling is born out from theoretical works estimating the regions for the onset of chaos either from three-body resonance overlap (Quillen 2011) or two-body resonance overlap for eccentric planets (Hadden and Lithwick 2018). A more exhaustive study on the mass scaling of instability times will shed light on this issues.…”

“…We shall start by assuming that the evolution of a single planetary orbit is mainly determined by its two closest neighbors, so the relevant perturbative Hamiltonian involves the gravitational interactions of only three planets. This choice is justified by the fact that the strength of interactions, which is proportional to the Laplace coefficients and decays exponentially with interplanetary separation (Quillen 2011). Furthermore, we focus on a possible minimal perturbative Hamiltonian that may be responsible for destroying the KAM tori.…”

Detectable close-in planets around white dwarfs through late unpacking

“…No analytical criterion exists like for the two-planet Hill stability case. Consequently, several authors (Chambers et al 1996;Faber & Quillen 2007;Zhou et al 2007;Chatterjee et al 2008;Smith & Lissauer 2009;Quillen 2011) have attempted to fit empirical instability timescales to systems of three or more planets from numerical simulations. Because these estimates are primarily based on simulations with equal-mass planets, here we also adopt equal-mass planets in order to utilize these estimates.…”

Traditional formation scenarios fail to explain 4:3 mean motion resonances

“…Nesvorný & Morbidelli (, ) do so, but treat one of the bodies as small. Quillen () consider three equal mass bodies, but on near‐circular orbits with semi‐major axes in a geometrical ratio. As mentioned above, other studies consider the possible formation of resonant chains due to the presence of a disc.…”