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 for predicting whether two planets are in apsidal resonance is derived, which shows that the occurrence of the apsidal resonance depends only on the mass ratio (m1/m2), semimajor axis ratio (a1/a2), initial eccentricity ratio (e10/e20), and the initial relative apsidal longitude (ϖ20 - ϖ10) between the two planets. The probability of two planets falling in apsidal resonance is given in the initial element space. We verify the criteria with numerical integrations for the HD 12661 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 HD 12661 system are studied by calculating the Lyapunov exponents of their orbits in a general three-body model. We find that two planets in large-eccentricity orbits could be stable only when they are in aligned apsidal resonance. When the planets are migrated under the planet-disk interactions, for more than half of the studied cases, the configurations of the apsidal resonances are preserved. We find the two planets of the HD 12661 system could be in aligned resonance and thus more stable, provided they have Ω2 - Ω1 ≈ 180°. The applications of the criteria to the other multiple planetary systems are discussed.