Insights from the Hill Problem for Understanding Mean Motion Resonances

Our analytical understanding of the dynamics between a co-planar pair of planets in mean motion resonances (MMRs) typically relies on models involving a single cosine term in the classical disturbing function expansion, e.g., the pendulum model and the 'second fundamental model for resonance'). This falls out naturally at leading order in the eccentricities in the circular restricted three-body problem, i. e., when one body is a test particle. In the general case of two massive planets on eccentric orbits, the extra degrees of freedom and additional resonant terms that need to be considered are typically tackled in a Hamiltonian formalism through a long chain of canonical transformations. An important, long-standing result is that for first-order (j:j-1) MMRs, to leading order in the eccentricities it is possible to combine the separate resonant terms into a single one, and recover a one degree-of-freedom Hamiltonian identical to that in the much simpler circular restricted problem. Hadden (2019) realized that this must also be approximately true for higher order MMRs, as long as the orbits have period ratios ≲ 2, in order to satisfy the well-known properties of the solution in the Hill limit where the orbits are very close to one another. In this pedagogically aimed talk, we will explore this deep connection to elucidate why the properties of the Hill problem imply a powerful approximate mapping from the general two-body resonance problem to the much simpler circular restricted case. We also show how thinking about the problem in the tightly spaced limit motivates a natural, universal definition of variables in which all j:j-k MMRs of a given order share the same Fourier amplitude to lowest order in the eccentricities (independent of j). We are implementing these transformations in the open-source celmech package to easily go back and forth with orbital elements / N-body integrations.