Analytical approach to the secular behaviour of exoplanetary systems

We analyse the secular interactions of two coplanar planets which are not in mean motion resonances. The analysis is based on a high order (order 12) expansion of the perturbative potential in powers of the eccentricities. The model depends on only two parameters (the ratio of semi-major axis and the mass ratio of the planets) and can be reduced to a one degree of freedom system, allowing for an exhaustive parametric analysis. Following Pauwels [Pauwels T.: 1983, Celet. Mech. & Dyn. Astro. 30, 229-247] we map the phase space on a sphere, avoiding in this way the artificial singularities introduced by other mappings. We show that the 12 order expansion is able to describe correctly most of the exosolar planetary systems discovered so far, even if the eccentricities of these planets are considerably larger than the eccentricities of our own solar system. The expansion is even able to reproduce, at moderate eccentricities, the secular resonances discovered numerically by Michtchenko and Malhotra [Michtchenko, T. A. and Malhotra, R.: 2004, Icarus 168, 237-248] at moderate to large eccentricities.