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

Abstract

Recent works on three-planet mean motion resonances (MMRs) have highlighted their importance for understanding the details of the dynamics of planet formation and evolution. While the dynamics of two-planet MMRs are well understood and approximately described by a one-degree-of-freedom Hamiltonian, little is known of the exact dynamics of three-body resonances besides the cases of zeroth-order MMRs or when one of the bodies is a test particle. In this work, I propose the first general integrable model for first-order three-planet mean motion resonances. I show that one can generalize the strategy proposed in the two-planet case to obtain a one-degree-of-freedom Hamiltonian. The dynamics of these resonances are governed by the second fundamental model of resonance. The model is valid for any mass ratio between the planets and for every first-order resonance. I show the agreement of the analytical model with numerical simulations. As examples of application, I show how this model could improve our understanding of the capture into MMRs as well as their role in the stability of planetary systems.

Three-Body Resonances in the Saturnian System

“…While we were able to automate the search for the locations of first and second-order three-body resonances among the mid-sized moons of Saturn, we were not able to predict the relative strength of these resonances with any confidence without running direct numerical integrations. In other contexts three-body resonances have been studied analytically with good success (Quillen 2011;Charalambous et al 2018;Petit et al 2020;Petit 2021), but more work is needed to determine if a practicable model of TBRs among Saturnian moons can be constructed.…”

Is the orbital distribution of multiplanet systems influenced by pure three-planet resonances?

“…Both Charalambous et al (2018) and Petit (2021) have shown that resonance capture in zero-order and first-order 3P-MMRs far from double resonances is possible as long as the orbital decay timescale is sufficiently long. It is thus possible that the observed correlation could have occurred during the last stages of the primordial disc when its surface density was very low and orbital excursions very limited.…”

“…They found that for most planet-triplets, migration would halt once the chain was reached. Furthermore, Petit (2021) showed that a system can even be trapped in a first-order resonance, although this was only achieved for controlled conditions and robustness is uncertain.…”

“…As of today these multi-resonant systems appear as resonance chains whose building blocks are either two-planet resonances (2P-MMR) or zero-order three-planet resonances (3P-MMR). However, N-body simulations by Charalambous et al (2018) and Petit (2021) have shown that capture in pure 3P-MMR configurations are also possible provided the migration rate is sufficiently low. We denote pure 3P-MMRs as those where no adjacent pair of planets lie in 2-planet commensurabilities.…”

New results on orbital resonances

“…8. Several recent analyses of planetary mean motion resonances in the context of exo-planetary systems have been based on the conventional critical resonant angle (e.g., Deck et al, 2013;Ramos et al, 2015;Hadden & Lithwick, 2018;Hadden, 2019;Petit, 2021). These could potentially also benefit from validation with the lens of the sub-multiple of the critical resonant angle used in our work, and/or employing physically-motivated stroboscopic records to generate pseudo-Poincaré sections for comparison with the semi-analytic approaches.…”