Resonant Chains of Exoplanets: Libration Centers for Three-body Angles

• Published in: The Astronomical journal Vol. 161; no. 6; pp. 290 - 298
• Main Authors: Siegel, Jared C., Fabrycky, Daniel
• Format: Journal Article
• Language: English
• Published: Madison The American Astronomical Society 01.06.2021; IOP Publishing
• Subjects: Astronomy; Diagnostic systems; Dynamic models; Exoplanet dynamics; Exoplanet evolution; Exoplanet tides; Exoplanets; Extrasolar planets; Libration; Orbital resonances (celestial mechanics); Orbits; Planet formation; Planetary systems
• ISSN: 0004-6256; 1538-3881
• DOI: 10.3847/1538-3881/abf8a6

Abstract Resonant planetary systems contain at least one planet pair with orbital periods librating at a near-integer ratio (2/1, 3/2, 4/3, etc.) and are a natural outcome of standard planetary formation theories. Systems with multiple adjacent resonant pairs are known as resonant chains and can exhibit three-body resonances—characterized by a critical three-body angle. Here we study three-body angles as a diagnostic of resonant chains through tidally damped N -body integrations. For each combination of the 2:1, 3:2, 4:3, and 5:4 mean motion resonances (the most common resonances in the known resonant chains), we characterize the three-body angle equilibria for several mass schemes, migration timescales, and initial separations. We find that under our formulation of the three-body angle, which does not reduce coefficients, 180° is the preferred libration center, and libration centers shifted away from 180° are associated with nonadjacent resonances. We then relate these angles to observables, by applying our general results to two transiting systems: Kepler-60 and Kepler-223. For these systems, we compare N -body models of the three-body angle to the zeroth order in e approximation accessible via transit phases, used in previous publications. In both cases, we find the three-body angle during the Kepler observing window is not necessarily indicative of the long-term oscillations and stress the role of dynamical models in investigating three-body angles. We anticipate our results will provide a useful diagnostic in the analysis of resonant chains.