A Unified, Physical Framework for Mean Motion Resonances

Abstract

The traditional approach to analyzing mean motion resonances is through canonical perturbation theory. While this is a powerful method, its generality leads to complicated combinations of variables that are challenging to interpret and require looking up numerical coefficients particular to every different resonance. In this paper we develop simpler scaling relations in the limit where orbits are closely spaced (period ratios $\lesssim 2$) and interplanetary interactions can be approximated by only considering the close-approaches each time the inner planet overtakes the outer at conjunction. We develop geometric arguments for several powerful results: (i) that $p$:$p-q$ MMRs of the same order $q$ are all rescaled versions of one another (ii) that the general case of two massive planets on closely spaced, eccentric, co-planar orbits can be approximately mapped onto the much simpler case of an eccentric test particle perturbed by a massive planet on a co-planar circular orbit and (iii) that while the effects of consecutive conjunctions add up coherently for first-order ($p$:$p-1$) MMRs, they partially cancel for $p$:$p-q$ MMRs with order $q>1$, providing a physical explanation for why these higher order MMRs are weaker and can often be ignored. Finally, we provide simple expressions for the widths of MMRs and their associated oscillation frequencies that are universal to all closely spaced MMRs of a given order $q$, in the pendulum approximation.