A Unified, Physical Framework for Mean Motion Resonances

TL;DR

This paper introduces a simplified physical framework for analyzing mean motion resonances in planetary systems, focusing on closely spaced orbits and providing universal scaling relations and insights into resonance strengths.

Contribution

It develops geometric and scaling relations that simplify the understanding of mean motion resonances, especially for closely spaced planetary orbits, reducing reliance on complex perturbation theory.

Findings

Resonances of the same order are rescaled versions of each other.

Two massive planets can be approximated by a test particle model.

Higher order resonances are weaker due to partial cancellation of conjunction effects.

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.