Semi-analytical model for planetary resonances: application to planets around single and binary stars

In spite of planetary resonances being a common dynamical mechanism acting on planetary systems, no general model exists for describing their properties, particularly for commensurabilities of any order and arbitrary values of the eccentricities and inclinations. The present work presents a semi-analytical model that describes the resonance strength, width, location and stability of fixed points, as well as periods of small-amplitude librations. The model is valid to any two gravitationally interacting massive bodies, and thus applicable to planets around single or binary stars. Using a theoretical framework in Poincar\'e and Jacobi reference system we develop a semi-analytical method that employs a numerical evaluation of the averaged resonant disturbing function. Validations of the model are presented comparing its predictions with dynamical maps for real and fictitious systems. The model is shown to describe very well many dynamical features of planetary resonances. Notwithstanding the good agreements found in all cases, a small deviation is noted in the location of the resonance centers for circumbinary systems. As a consequence of its application to the HD31527 system we have found that the updated best-fit solution leads to a high-eccentricity stable libration between the middle and outer planets inside the 16/3 mean-motion resonance. This is the first planetary system whose long-term dynamics appears dominated by such a high-order commensurability. In the case of circumbinary planets, the overlap of N/1 mean-motion resonances coincides very well with the size of the global chaotic region close to the binary, as well as its dependence with the mutual inclination.