Extrasolar planetary systems commonly exhibit planets on eccentric orbits, with many systems located near or within mean-motion resonances, showcasing a wide diversity of orbital architectures. Such complex systems challenge traditional secular theories, which are limited to first-order approximations in planetary masses or rely on expansions in orbital elements—eccentricities, inclinations, and semi-major axis ratios—that are subject to convergence issues, especially in highly eccentric, inclined, or tightly-packed systems. In this talk, we present a numerical approach to second-order perturbation theory, developed using the Lie series formalism, to address these limitations. We first outline the Hamiltonian framework for the three-body planetary problem and apply a canonical transformation to eliminate fast angle dependencies, deriving the secular Hamiltonian up to second order in the mass ratio. We then use the fast Fourier transform algorithm to numerically simulate, with high precision, the long-term evolution of planetary systems near or away from mean-motion resonances. Finally, we validate our methods against well-known planetary configurations, such as the Sun–Jupiter–Saturn system, as well as exoplanetary systems like WASP-148, TIC 279401253, and GJ 876, demonstrating the applicability of our model across a wide range of orbital architectures.
A. Albouy, A. Chenciner, J. Laskar
