Analytical study of the proximity of exoplanetary systems to mean-motion resonances

Aims.In a previous paper (Libert & Henrard 2005, Celest. Mech. & Dyn. Astron., 93, 187) we used a twelfth-order expansion of the perturbative potential in powers of eccentricities to represent the secular effects of two coplanar planets. This expansion was applied successfully to non resonant exoplanetary systems. This study was based on a first order (in the masses of the planets) model and will fail for systems too close to a resonance. In this paper we test the effects of the proximity of a mean-motion resonance on the secular motion of the planets. Methods.We analyse the proximity of several exoplanetary systems to a mean-motion resonance zone by using a first-order (in the mass ratios) Lie algorithm on the perturbative potential expanded to the twelfth order in the eccentricities. This perturbation method evaluates the difference between osculating elements and averaged ones. It permits us to decide whether resonant contributions dominate the terms of this difference or not. Results.This study is applied to several exosystems. We find that HD 168443, HD 38529, HD 74156, HD 217107, and HD 190360 are far away from a mean-motion resonance zone. υ Andromedae and HD 12661 are rather close to the 5/1 resonance, HD 169830 to the 9/1 one. Hence, a secular theory is enough to depict correctly the behaviour of all these systems. On the other hand, HD 108874 and HD 202206 suffer from large perturbations in their motion due to the closeness of the 4/1 and 5/1 resonances, respectively. We also perform a complete investigation of the proximity of the υ Andromedae system to mean-motion resonances, by studying the changes in behaviour due to different values of the outer semi-major axis. The υ Andromedae system begins to be really influenced by the 5/1 resonant terms when the value of the outer semi-major axis decreases from 2.53 to 2.445