The onset of instability in resonant chains

Abstract

There is evidence that most chains of mean motion resonances of type $k$:$k-1$ among exoplanets become unstable once the dissipative action from the gas is removed from the system, particularly for large $N$ (the number of planets) and $k$ (indicating how compact the chain is). We present a novel dynamical mechanism that can explain the origin of these instabilities and thus the dearth of resonant systems in the exoplanet sample. It relies on the emergence of secondary resonances between a fraction of the synodic frequency $2\pi (1/P_1-1/P_2)$ and the libration frequencies in the mean motion resonance. These secondary resonances excite the amplitudes of libration of the mean motion resonances thus leading to an instability. We detail the emergence of these secondary resonances by carrying out an explicit perturbative scheme to second order in the planetary masses and isolating the harmonic terms that are associated with them. Focusing on the case of three planets in the 3:2 -- 3:2 mean motion resonance as an example, a simple but general analytical model of one of these resonances is obtained which describes the initial phase of the activation of one such secondary resonance. The dynamics of the excited system is also briefly described. This scheme shows how one can obtain analytical insight into the emergence of these resonances, and into the dynamics that they trigger. Finally, a generalisation of this dynamical mechanism is obtained for arbitrary $N$ and $k$. This leads to an explanation of previous numerical experiments on the stability of resonant chains, showing why the critical planetary mass allowed for stability decreases with increasing $N$ and $k$.

Submitted to MNRAS

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• TOI-1670 b and c: An Inner Sub-Neptune with an Outer Warm Jupiter Unlikely to have Originated from High-Eccentricity Migration
• The origins of nearly coplanar, non-resonant systems of close-in super-Earths
• An integrable model for first-order three-planet mean motion resonances
• Dissipative Capture of Planets Into First-Order Mean-Motion Resonances
• Forming the Trappist-1 system in two steps during the recession of the disc inner edge
• The distribution of mutual inclinations arising from the stellar quadrupole moment
• Periodic orbits in the 1:2:3 resonant chain and their impact on the orbital dynamics of the Kepler-51 planetary system
• Close-in compact super-Earth systems emerging from resonant chains: slow destabilization by unseen remnants of formation
• The Dynamical History of the Kepler-221 Planet System
• Higher-Order Mean-Motion Resonances Can Form in Type-I Disk Migration
• Orbital stability of compact three-planet systems III. The role of three-body resonances