The path to instability in compact multi-planetary systems

Abstract

The dynamical stability of tightly packed exoplanetary systems remains poorly understood. While a sharp stability boundary exists for a two-planet system, numerical simulations of three-planet systems and higher show that they can experience instability on timescales up to billions of years. Moreover, an exponential trend between the planet orbital separation measured in units of Hill radii and the survival time has been reported. While these findings have been observed in numerous numerical simulations, little is known of the actual mechanism leading to instability. Contrary to a constant diffusion process, planetary systems seem to remain dynamically quiescent for most of their lifetime before a very short unstable phase. In this work, we show how the slow chaotic diffusion due to the overlap of three-body resonances dominates the timescale leading to the instability for initially coplanar and circular orbits. While the last instability phase is related to scattering due to two-planet mean motion resonances (MMRs), for circular orbits the two-planets MMRs are too far separated to destabilise systems initially away from them. The studied mechanism reproduces the qualitative behaviour found in numerical simulations very well. We develop an analytical model to generalise the empirical trend obtained for equal-mass and equally spaced planets to general systems on initially circular orbits. We obtain an analytical estimate of the survival time consistent with numerical simulations over four orders of magnitude for the planet-to-star-mass ratio ε, and 6 to 8 orders of magnitude for the instability time. We also confirm that measuring the orbital spacing in terms of Hill radii is not adapted and that the right spacing unit scales as ε1∕4. We predict that beyond a certain spacing, the three-planet resonances are not overlapped, which results in an increase of the survival time. We confirm these findings with the aid of numerical simulations of three-planet systems with different masses. We finally discuss the extension of our result to more general systems, containing more planets on initially non-circular orbits.

An integrable model for first-order three-planet mean motion resonances

“…which is the equation of the locus of the unperturbed resonance defined by k 1 and k 3 . One can remark that except for −k −1 3 on the right-hand side, it is similar to the zeroth-order resonance relationship used in Petit et al (2020). However, unlike the zeroth-order case, there are three different families of resonances depending on the signs of k 1 and k 3 .…”

“…An important difference with the case of zeroth-order resonances treated in Petit et al (2020) is that the resonance locus equation depends specifically on the two integers k 1 and k 3 rather than on their ratio. This property forbids to create a continuous coordinate constant on the resonance loci.…”

“…Such averaging is possible if the system is situated 'far enough' from any two-planet MMR using the classical approach from perturbation theory, the Lie series method (Deprit 1969). In order to perform the averaging and compute the resonant coefficients, I follow the same method and notations as in Petit et al (2020) and I refer the reader to this work for a more extensive description. Nevertheless, it is convenient to recall the notations used.…”

“…The question has gained a renewed interest thanks to the discovery of close-in exoplanet systems trapped into zeroth-order resonances such as Trappist-1 (Gillon et al 2017;Agol et al 2021) or more recently TOI-178 (Leleu et al 2021). Three-planet resonances are also thought to be the main driver of the early instability of tightly packed systems (Quillen 2011;Petit et al 2020).…”

“…A general integrable model for zeroth-order three-planet MMRs is directly obtained from a second-order averaging since no transformation of the coordinates is needed besides the introduction of the resonant angle (see Petit et al 2020). However, this result is not immediately generalized to higher-order MMRs.…”