Canceling Effects of Conjunctions Render Higher Order Mean Motion Resonances Weak

TL;DR

This work addresses why higher-order mean motion resonances (MMRs) are weak by developing a physically motivated, conjunction-based picture in the Hill limit. By mapping the problem to the circular restricted three-body problem and treating the dynamics as a sequence of impulsive kicks at conjunctions, the authors derive a Fourier expansion for the mean-motion changes and show that multiple, equally spaced conjunctions per resonant cycle cause cancellations that suppress higher-order terms. The key result is that only Fourier modes with $m$ a multiple of the resonance order $q$ survive, leading to a leading term that scales as $\tilde{e}^q$ and thus reproducing the traditional $e^q$ scaling of MMR strength; the approach also yields a pendulum-like reduced model for both first- and higher-order resonances. The findings provide a clear, geometric intuition for resonance structure in closely spaced planetary systems and offer a framework applicable to other close-encounter dynamical problems.

Abstract

Mean motion resonances (MMRs) are a key phenomenon in orbital dynamics. The traditional disturbing function expansion in celestial mechanics shows that, at low eccentricities, $p$:$p-q$ MMRs exhibit a clear hierarchy of strengths, scaling as $e^q$, where $q$ is the order of the resonance. This explains why first-order MMRs (e.g., 3:2 and 4:3) are important, while the infinite number of higher order integer ratios are not. However, this relationship derived from a technical perturbation series expansion provides little physical intuition. In this paper, we provide a simple physical explanation of this result for closely spaced orbits. In this limit, interplanetary interactions are negligible except during close encounters at conjunction, where the planets impart a gravitational "kick" to each other's mean motion. We show that while first-order MMRs involve a single conjunction before the configuration repeats, higher order MMRs involve multiple conjunctions per cycle, whose effects cancel out more precisely the higher the order of the resonance. Starting from the effects of a single conjunction, we provide an alternate, physically motivated derivation of MMRs' $e^q$ strength scaling.

Figures (5)

• Figure 1: Illustration of the coplanar circular restricted three body problem, including the variables we use. Panel (a): the inner planet is on a circular orbit of radius $a_p$ at an angle $\lambda_p$. The outer test particle is on an elliptical orbit offset $ae$ from the center of the circle, where $e$ is the eccentricity of the ellipse. The orbit’s pericenter is located at $\varpi$ and the test particle is at the angular position $\lambda$. Panel (b): When the planets are at conjunction, they are both at the same longitude $\lambda_{\mathrm{conj}}$. The resonant angle is given by $\theta = \lambda_{\mathrm{conj}} - \varpi$.
• Figure 2: Fourier decomposition of the fractional change in mean motion $\delta n / n$ as a function of conjunction angle $\theta$, compared to results from direct $N$-body integrations. The N-body line (solid blue) simulates a suite of individual conjunctions at various angles $\theta$, and plots the resulting change to the mean motion. Systems parameters are given in the main text. The first four individual Fourier terms are plotted as dotted lines in orange, green, red, and purple, respectively; the dashed brown line shows their sum. Top panel: For small eccentricity ($\tilde{e} = 0.1$), the first-order term (orange) is a good approximation for the numerical result. Bottom panel: At higher eccentricity ($\tilde{e} = 0.4$), higher-order terms are comparable in magnitude to the first term and contribute significantly to the numerical result.
• Figure 3: Angular locations of the conjunction angles multiplied by Fourier mode number $m$, shown for the 27:22 mean motion resonance. In all panels, the five original conjunction angles $\theta_k$ (for $k = 0$ to 4) are plotted around the orbit, with the ordering of each $m \theta_k$ indicated by color and label. Panel (a): For $m = 1$, the configuration of angles is equally spaced around the orbit, with a separation of $\Delta \theta = \frac{2 \pi p}{q}$ between subsequent conjunctions. Panels (b–d): For $m = 2, 3, 4$, the angular locations $m \theta_k$ remain evenly spaced around the orbit, but their order is permuted within the cycle, as shown by the changing sequence of colors. Panel (e): For $m = q = 5$, all $m \theta_k$ coincide at the angle $\theta$.
• Figure 4: Geometric interpretation of the Fourier sum over kicks in a mean motion resonance cycle, shown for the case where the Fourier mode number $m$ is not divisible by $q$ (specifically, the 27:22 MMR with $q=5$). Panel (a): Each arrow represents a complex vector with angle $m\theta_k$, corresponding to one term in the sum of Eq. \ref{['e:kick_for_mode']}. The five vectors are spaced uniformly around the unit circle by $2\pi/q$ and point in directions determined by the angles $m \theta_k$. Panel (b): Adding the vectors head-to-tail yields a closed regular polygon. As a result, their vertical components cancel exactly and the total contribution to the mean motion kick from this mode vanishes. This configuration corresponds to a sum over the $q$th roots of unity, which always vanishes when $m$ is not a multiple of $q$rootsofunity.
• Figure 5: Comparison between the Fourier coefficients of the effective potential computed via the Celmech function $\mathrm{s}_m(\tilde{e})$ (solid blue lines) and the truncated eccentricity expansions of the corresponding $W_m$ terms from Namouni's formalism Namouni96. Left: The coefficient of $\cos(\theta)$ (i.e., $n=1$) is plotted as a function of normalized eccentricity $\tilde{e}$. The leading-order term $W_1^{1,0}\tilde{e}$ dominates, with higher-order terms $W_1^{3,0}\tilde{e}^3$, $W_1^{5,0}\tilde{e}^5$, and $W_1^{7,0}\tilde{e}^7$ contributing minimally. Their partial sum (dashed orange) closely tracks $-2\pi \cdot \mathrm{s}_1(\tilde{e})$. Right: Same comparison for $n=2$, corresponding to $\cos(2\theta)$. Again, the leading term $W_2^{2,0}\tilde{e}^2$ dominates, with higher-order corrections providing only small deviations. Even at $\tilde{e} = 0.7$, the leading-order approximations remain accurate to within a few percent, justifying their use throughout this work.