7:2 Mean-Motion Resonance (MMR) Dynamics

• 7:2 MMR is defined by a 7:2 orbital ratio, with its resonant angle librating around a fixed center and identifiable by stripe patterns from geometric analysis.
• The FAIR method confirms the resonance by detecting nearly-vertical and horizontal stripes and ensuring persistent libration with narrow amplitude.
• Hamiltonian approaches yield scaling laws for resonance strength and width, highlighting the high-order sensitivity to eccentricity and dynamical perturbations.

A 7:2 mean-motion resonance (MMR) describes the commensurability between the orbital periods of two bodies, most commonly a small object (e.g., asteroid, TNO) and a massive planet, such that the ratio of their orbital periods approaches 7:2—i.e., the small body completes 7 revolutions for every 2 of the planet. High-order MMRs such as 7:2 play a critical role in sculpting orbital distributions throughout planetary systems and provide stringent tests of resonance theory due to their inherently weak nonlinear coupling and sharply-defined structure. The identification, dynamical properties, scaling laws, and physical origin of the 7:2 MMR have been elucidated through both geometrical and Hamiltonian approaches (Forgács-Dajka et al., 2018, Tamayo et al., 29 Oct 2024).

1. Formal Definition and Critical Angle Structure

A $(p+q):p = 7:2$ mean-motion resonance has $p=2$, $q=5$. Its critical (resonant) angle for inner-type resonances (when the test particle is interior to the perturber) is constructed as

$$\sigma_{7:2} = 7\lambda' - 2\lambda - 5\varpi \quad (\mathrm{or}\; 7\lambda' - 2\lambda - 5\varpi')$$

where $\lambda$ and $\lambda'$ are the mean longitudes of the test particle and planet, and $\varpi$, $\varpi'$ their longitudes of perihelion (Forgács-Dajka et al., 2018). Resonance occurs when $\sigma_{7:2}(t)$ librates—oscillates around a center $\bar{\sigma}$, typically near $0^\circ$ or $180^\circ$, with amplitude $\Delta\sigma < 180^\circ$.

2. Identification and Geometric Verification via FAIR Method

The FAST Identification of Resonances (FAIR) method utilizes the geometric signature of high-order commensurabilities to efficiently detect resonance without a priori specification (Forgács-Dajka et al., 2018). For a 7:2 MMR:

• Integrate the orbits and record the instantaneous values of $\lambda(t)$, $M(t)$, $\varpi(t)$, $\lambda'(t)$.
• Construct plots of $M$ (mean anomaly of the test particle) vs.\ $\Delta\lambda = \lambda' - \lambda$.
• A body in exact $(p+q):p = 7:2$ resonance shows 5 (=$q$) nearly-vertical stripes and 7 (=$p+q$) nearly-horizontal stripes in the plot.
• Count stripes and verify that the libration of $\sigma_{7:2}(t)$ persists for at least $\sim 100$ outer-body periods with amplitude $<180^\circ$ and fixed center.

Numerical confirmation requires:

1. The mean-motion ratio $|n/n' - 7/2| < \delta_n$ with $\delta_n \sim 10^{-3} - 10^{-4}$;
2. Stripe counts as above;
3. Persistent libration of $\sigma_{7:2}(t)$;
4. No secular drift into circulation under small changes in initial conditions (Forgács-Dajka et al., 2018).

3. Hamiltonian Dynamics and Resonance Strength Scaling

In the vicinity of a general $p:(p-q)$ MMR in the Hill (closely-spaced) limit, the motion is governed by a pendulum-like Hamiltonian (Tamayo et al., 29 Oct 2024): $$H(\phi, J) = \frac{1}{2} J^2 + \epsilon_q \cos{\phi}$$ where $\phi = q\theta$ is the resonant phase and $J$ is proportional to the deviation of mean motion from resonance. The strength of the $q$th-order resonance is encapsulated in the coefficient $\epsilon_q$, which for the 7:2 resonance is (Tamayo et al., 29 Oct 2024): $$\epsilon_5 \simeq 25 A_5^2 \mu \left( \frac{e}{e_c} \right)^5 \frac{n^2}{e_c^2}$$ with $A_5 \simeq 0.83$ (order-unity coefficient), $\mu = (m_p + m_\mathrm{test})/M_\star$, $e$ the eccentricity, $e_c = 2q/(3p)$ the crossing eccentricity, and $n$ the mean motion.

4. Resonance Width and Libration Frequency

The half-width in semimajor axis ($\Delta a$) and the small-amplitude libration frequency ($\omega_\mathrm{lib}$) for the 7:2 MMR follow precise scaling relations (Tamayo et al., 29 Oct 2024): $$\Delta a_\mathrm{max} \simeq 2 A_5 a \sqrt{ \mu \left( \frac{e}{e_c} \right)^5 }$$

$$\omega_\mathrm{lib} = 5 A_5 n \sqrt{ \frac{\mu e^5}{e_c^7} }$$

where $a$ is semimajor axis. For typical small-body eccentricities $e \ll e_c$, the width is extremely narrow. For comparison, the width of 7:2 is smaller by a factor $\simeq (A_5/A_1) (e/e_c)^2$ than a first-order resonance at the same location, e.g., for $e \sim 0.1$, $e_c \simeq 0.48$, $e/e_c \sim 0.2$, leading to suppression by at least a factor $\sim 0.04$ and thus $\gtrsim 25\times$ narrower (Tamayo et al., 29 Oct 2024).

5. Physical Origin of Weakness in High-Order (e.g., 7:2) Resonances

The intrinsic weakness of $q>1$ MMRs such as 7:2 arises from the cancellation of effects at successive conjunctions. In a $q$th-order resonance there are $q$ distinct conjunctions per cycle, each producing an impulsive change; the net result after summing over all $q$ encounters in one period yields a scaling $\propto e^q$, as the leading-order (linear in $e$) terms cancel by symmetry (Tamayo et al., 29 Oct 2024). Therefore, for $q=5$ the residual is $\propto e^5$, producing extremely narrow and dynamically subtle resonance zones.

6. Applications in Dynamical Surveys and Resonant Object Identification

Although explicit examples of the 7:2 MMR are not provided in the application section of the original FAIR method paper, the identification steps extend without modification to the 7:2 case. The method has been used in large-scale surveys to systematically catalog objects in high-order MMRs within the asteroid belt and trans-Neptunian region, emphasizing the necessity of robust geometric and dynamical confirmation—especially given the subtle dynamical imprint and strong chaos boundaries of the 7:2 commensurability (Forgács-Dajka et al., 2018).

Researchers employ the above criteria to classify TNOs and asteroids as locked or temporarily captured within the 7:2 MMR, informing population studies and dynamical mapping of resonance structures.

7. Comparative Perspective and Theoretical Significance

The 7:2 MMR exemplifies the generic structure of high-order commensurabilities: extreme sensitivity to eccentricity, sharply reduced widths, and complex phase-space topologies characterized by narrow resonance islands embedded in seas of chaotic and regular motions. The recent development of physically unified scaling laws provides a transparent framework for comparing resonance strength and dynamics across order–$q$ families, mapping them onto rescaled versions of the test-particle/planet paradigm (Tamayo et al., 29 Oct 2024). A plausible implication is that resonance capture and retention at 7:2 are rare without considerable excitation in eccentricity and/or inclination, and thus the resonance serves as a sensitive probe of dynamical histories in planetary and minor-body systems.