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1. On the divergence of first-order resonance widths at low eccentricities

   https://doi.org/10.1093/mnras/staa1751  

   Malhotra, Renu; Zhang, Nan (August 2020, Monthly Notices of the Royal Astronomical Society)

   null (Ed.)

   ABSTRACT Orbital resonances play an important role in the dynamics of planetary systems. Classical theoretical analyses found in textbooks report that libration widths of first-order mean motion resonances diverge for nearly circular orbits. Here, we examine the nature of this divergence with a non-perturbative analysis of a few first-order resonances interior to a Jupiter-mass planet. We show that a first-order resonance has two branches, the pericentric and the apocentric resonance zone. As the eccentricity approaches zero, the centres of these zones diverge away from the nominal resonance location but their widths shrink. We also report a novel finding of ‘bridges’ between adjacent first-order resonances: at low eccentricities, the apocentric libration zone of a first-order resonance smoothly connects with the pericentric libration zone of the neighbouring first-order resonance. These bridges may facilitate resonant migration across large radial distances in planetary systems, entirely in the low-eccentricity regime. 

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2. The Resonant Remains of Broken Chains from Major and Minor Mergers

   https://doi.org/10.3847/1538-3881/adce0c  

   Li, Rixin; Chiang, Eugene; Choksi, Nick; Dai, Fei (May 2025, The Astronomical Journal)

   Abstract TESS and Kepler have revealed that practically all close-in sub-Neptunes form in mean-motion resonant chains, most of which unravel on timescales of 100 Myr. UsingN-body integrations, we study how planetary collisions from destabilized resonant chains produce the orbital period distribution observed among mature systems, focusing on the resonant fine structures remaining post-instability. In their natal chains, planets near first-order resonances have period ratios just wide of perfect commensurability, driven there by disk migration and eccentricity damping. Sufficiently large resonant libration amplitudes are needed to trigger instability. Ensuing collisions between planets (“major mergers”) erode but do not eliminate resonant pairs; surviving pairs show up as narrow “peaks” just wide of commensurability in the histogram of neighboring-planet period ratios. Merger products exhibit a broad range of period ratios, filling the space between relatively closely separated resonances such as the 5:4, 4:3, and 3:2, but failing to bridge the wider gap between the 3:2 and 2:1—a “trough” thus manifests just short of the 2:1 resonance, as observed. Major mergers generate debris that undergoes “minor mergers” with planets, in many cases further widening resonant pairs. With all this dynamical activity, free eccentricities of resonant pairs, and by extension the phases of their transit timing variations, are readily excited. Nonresonant planets, being merger products, are predicted to have higher masses than resonant planets, as observed. At the same time, a small fraction of mergers produce a high-mass tail in the resonant population, also observed. 

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3. Kepler-discovered Multiple-planet Systems near Period Ratios Suggestive of Mean-motion Resonances Are Young

   https://doi.org/10.3847/1538-3881/ad110e  

   Hamer, Jacob H.; Schlaufman, Kevin C. (January 2024, The Astronomical Journal)

   Abstract Before the launch of the Kepler Space Telescope, models of low-mass planet formation predicted that convergent type I migration would often produce systems of low-mass planets in low-order mean-motion resonances. Instead, Kepler discovered that systems of small planets frequently have period ratios larger than those associated with mean-motion resonances and rarely have period ratios smaller than those associated with mean-motion resonances. Both short-timescale processes related to the formation or early evolution of planetary systems and long-timescale secular processes have been proposed as explanations for these observations. Using a thin disk stellar population’s Galactic velocity dispersion as a relative age proxy, we find that Kepler-discovered multiple-planet systems with at least one planet pair near a period ratio suggestive of a second-order mean-motion resonance have a colder Galactic velocity dispersion and are therefore younger than both single-transiting and multiple-planet systems that lack planet pairs consistent with mean-motion resonances. We argue that a nontidal secular process with a characteristic timescale no less than a few hundred Myr is responsible for moving systems of low-mass planets away from second-order mean-motion resonances. Among systems with at least one planet pair near a period ratio suggestive of a first-order mean-motion resonance, only the population of systems likely affected by tidal dissipation inside their innermost planets has a small Galactic velocity dispersion and is therefore young. We predict that period ratios suggestive of mean-motion resonances are more common in young systems with 10 Myr ≲τ≲ 100 Myr and become less common as planetary systems age. 

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4. Non-perturbative investigation of low-eccentricity exterior mean motion resonances

   https://doi.org/10.1093/mnras/stad483  

   Malhotra, Renu; Chen, Zherui (March 2023, Monthly Notices of the Royal Astronomical Society)

   ABSTRACT Mean motion resonances are important in the analysis and understanding of the dynamics of planetary systems. While perturbative approaches have been dominant in many previous studies, recent non-perturbative approaches have revealed novel properties in the low-eccentricity regime for interior mean motion resonances of Jupiter in the fundamental model of the circular planar restricted three-body model. Here, we extend the non-perturbative investigation to exterior mean motion resonances in the low-eccentricity regime (up to about 0.1) and for perturber mass in the range of ∼5 × 10−5 to 1 × 10−3 (in units of the central mass). Our results demonstrate that first-order exterior resonances have two branches at low eccentricity as well as low-eccentricity bridges connecting neighbouring first-order resonances. With increasing perturber mass, higher order resonances dissolve into chaos, whereas low-order resonances persist with larger widths in their radial extent but smaller azimuthal widths. For low-order resonances, we also detect secondary resonances arising from small-integer commensurabilities between resonant librations and the synodic frequency. These secondary resonances contribute significantly to generating the chaotic sea that typically occurs near mean motion resonances of higher mass perturbers. 

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5. New results on orbital resonances

   https://doi.org/10.48550/arXiv.2111.09289  

   Malhotra, Renu (September 2022, Proceedings of the International Astronomical Union)

   Celletti, Alessandra; Beaugé, Cristian; Galeş, Cătălin; Lemaître, Anne (Ed.)

   Perturbative analyses of planetary resonances commonly predict singularities and/or divergences of resonance widths at very low and very high eccentricities. We have recently re-examined the nature of these divergences using non-perturbative numerical analyses, making use of Poincaré sections but from a different perspective relative to previous implementations of this method. This perspective reveals fine structure of resonances which otherwise remains hidden in conventional approaches, including analytical, semi-analytical and numerical-averaging approaches based on the critical resonant angle. At low eccentricity, first order resonances do not have diverging widths but have two asymmetric branches leading away from the nominal resonance location. A sequence of structures called ``low-eccentricity resonant bridges" connecting neighboring resonances is revealed. At planet-grazing eccentricity, the true resonance width is non-divergent. At higher eccentricities, the new results reveal hitherto unknown resonant structures and show that these parameter regions have a loss of some -- though not necessarily entire -- resonance libration zones to chaos. The chaos at high eccentricities was previously attributed to the overlap of neighboring resonances. The new results reveal the additional role of bifurcations and co-existence of phase-shifted resonance zones at higher eccentricities. By employing a geometric point of view, we relate the high eccentricity phase space structures and their transitions to the shapes of resonant orbits in the rotating frame. We outline some directions for future research to advance understanding of the dynamics of mean motion resonances. 

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