Celmech II: A universal integrable model for mean motion resonances in closely packed systems

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.

A complication in the general case of two massive planets is that there are always multiple strong cosine terms near a given MMR involving different combinations of the two orbits’ pericenters. An important, long-standing result is that for first-order (j:j-1) MMRs, it is possible to combine these cosine terms into one through a rotation of variables, and recover the simple integrable models mentioned above. To my knowledge this is a coincidence, in the sense that it is not rigorously extendable to higher order MMRs. Additionally, different j:j-1 MMRs have different Fourier amplitudes with no obvious relationship to one another.

In the limit where the two planetary orbits are very close to one another (Hill’s approximation), there is an additional conserved quantity in the problem, which can be thought of as a center-of-mass eccentricity vector. Hadden (2019) realized that this implies that in the coplanar problem, all MMR cosine terms should be collapsible into a single term. He additionally showed that while this is not rigorously true at wider separations, it is true to excellent approximation for MMRs closer than the 2:1.

We will explore the valuable physical intuition this simplifying approximation provides, and how that can guide our naming and thinking about the various dynamical variables and conserved quantities in the general MMR problem. We also show how thinking about the problem in the tightly spaced limit motivates a natural, universal definition of variables in which all MMRs of a given order share the same Fourier amplitude to lowest order in the eccentricities.

We are implementing these transformations in the open-source celmech package to easily go back and forth with orbital elements / N-body integrations.