Occurrence and Stability of Apsidal Resonance in Multiple Planetary Systems

With the help of the Laplace-Lagrange solution of the secular perturbation theory in a double-planet system, we study the occurrence and the stability of apsidal secular resonance between the two planets. The explicit criteria to predict whether two planets are in apsidal resonance is derived. The pobability of two planets falling in apsidal resonance is given in the initial element space. We verify the criteria with numerical integrations for the HD12661 system, and find they give good predictions except at the boundary of the criteria or when the planet eccentricities are too large. The nonlinear stability of the two planets in HD12661 system are studied by calculating the Lyapunov exponents of their orbits in a general three-body model. The applications of the criteria to the other multiple planetary systems are discussed.