Laplace–Legendre expansion of the planar planetary Hamiltonian

We explore a hybrid expansion of the disturbing function in planetary dynamics that combines elements of the classical Laplace and Legendre developments. This formulation retains the structure of the Laplace expansion, but expresses the inverse of the mutual distance as a series whose terms keep an exact dependence on both the eccentricity and the semi-major axis ratio. We use it to construct the first-order secular Hamiltonian of the planar 3-body problem, relevant for modeling the long-term evolution of planetary systems. We assess the convergence of the new expansion numerically and compare it with that of the Laplace and Legendre series across a range of orbital configurations. The results show that the new expansion provides consistent performance across diverse dynamical regimes, bridging the domains of applicability of the two classical approaches.