On the dynamics of the exoplanetary system Kepler-413

This study is concerned with the stability of motion of the circumbinary exoplanet Kepler-413b. The analysis is performed within the framework of a flat restricted three-body problem. The stability diagram is plotted in the plane of initial conditions “pericentric distance—eccentricity” using mass calculations of Lyapunov exponents. According to the diagram, the Kepler-413b planet is located in a stable resonance cell, confined by the mean-motion resonances 6: 1 and 7: 1 with a central binary star, which agrees with the conclusions of Kostov et al. (2014) based on calculations of the MEGNO parameter. It is shown that the value of the critical semimajor axis acquired from the empirical formula of Holman and Wiegert (1999) almost coincides with the value obtained directly from the stability diagram; at low and moderate eccentricities of the planetary orbit, the position of the calculated boundary of the chaos zone is in close agreement with the boundary predicted by Shevchenko’s theory (2015). If the planet were in the instability zone, its characteristic Lyapunov time would be only ~1 year. In accordance with the conclusions of Kostov et al. (2014), it has been shown that the planet Kepler-413b is outside the habitability zone of the system.