Dynamical constraints on the three-planet system Kepler-51

While many multi-planetary systems possess planets in (or near) mean-motion resonances (MMRs) or resonant chains and although a number of efficient fitting methods has been developed, the optimum deduction of the orbital elements can be a complex process. Oftentimes, the observational data are provided with very large uncertainties, which locate the exoplanets in chaotic domains and as a result, hinder their stable evolution for long-time spans. Here, we suggest that dynamical analyses guided by the periodic orbits be performed in parallel to data fitting, in order to address the observational limitations. We present novel results in the General 4-Body Problem and the 1:2:3 resonant symmetric periodic orbits. Stable periodic orbits were only found in the low eccentricity regime. We demonstrate our method by applying it to the three-planet system Kepler-51. Guided by the stable orbits, we either validate or put possible constraints on the observational eccentricities, mean anomalies and apsidal differences by unravelling the stable regions, i.e. possible boundaries of the orbital elements, in phase space. Three main dynamical mechanisms could secure the long-term stability of Kepler-51bcd, namely i) the 2/1 and 3/2 two-body MMRs, ii) the 1:2:3 three-body Laplace resonance and iii) the combination of the 1/1 secondary resonance for the inner pair and the apsidal difference oscillation of the outer pair of planets.