The Bridging Expansion

Two classical series expansions dominate planetary dynamics: Laplace’s, which expands in powers of the semi-major axis ratio, and Legendre’s, which expands in inverse powers of mutual distance. Each converges well in its own regime and fails in the other’s. Laplace handles widely separated orbits; Legendre handles crossing or near-crossing ones. The boundary between these regimes is where most interesting planetary configurations live.

arXiv:2603.16528 constructs a hybrid expansion that retains the structure of Laplace’s approach but keeps exact dependence on both eccentricity and the semi-major axis ratio at each order. The result is a series that converges consistently across the configurations where either classical expansion alone would struggle — bridging the domains of applicability rather than choosing between them.

The construction applies this to the first-order secular Hamiltonian of the planar three-body problem, which governs the long-term orbital evolution of planetary systems. The practical value is a single expansion that does not need to be switched when the dynamical regime changes — eliminating the patchwork of different approximations that planetary dynamicists currently manage.

The structural principle: when two approximation schemes have complementary domains of validity, the right response is not to choose one or to average them, but to find a formulation that keeps what makes each one work while relaxing what makes each one fail. The hybrid keeps Laplace’s organizational structure and Legendre’s tolerance for close encounters. What it discards is each method’s restriction to a subset of parameter space. The bridge is not a compromise — it is a construction that inherits strengths from both parents.