The path to instability in compact multi-planetary systems

TL;DR

This paper investigates the mechanisms behind the long-term instability of tightly packed multi-planet systems, emphasizing the role of three-body resonances and developing an analytical model to predict survival times.

Contribution

It introduces an analytical model for planetary system stability that accounts for chaotic diffusion due to three-body resonances, extending empirical trends to general systems.

Findings

Chaotic diffusion from three-body resonance overlap dominates instability timescales.

Analytical estimates of survival time match simulations over broad parameter ranges.

Proper orbital spacing scales as the fourth root of the planet-star mass ratio.

Abstract

The dynamical stability of tightly packed exoplanetary systems remains poorly understood. While for a two-planet system a sharp stability boundary exists, numerical simulations of three and more planet systems 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 (MMR), for circular orbits the two-planets MMR are too far separated to destabilize systems initially away from them. We develop an analytical model to generalize the empirical trend obtained for equal mass and equally-spaced planets to general systems. We obtain an analytical estimate of the survival time consistent with simulations over four orders of magnitude for the planet to star mass ratio $\epsilon$, 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 $\epsilon^{1/4}$. We predict that beyond a certain spacing, the three-planet resonances are not overlapped, which results in an increase of the survival time. We finally discuss the extension of our result to more general systems, containing more planets on initially non circular orbits.