Dynamical evolution of the Uranian satellite system III. The passage through the7/4 MMR between Miranda and Ariel

The passage through the $5/3$ mean-motion resonance between Ariel and Umbriel, two of Uranus’s largest moons, still raises several open questions. Previous studies suggest that, in order to reproduce the current orbital configuration, Ariel must have had an eccentricity of approximately $\sim 0.01$ before the resonance encounter, which would prevent resonant capture. However, the rapid tidal circularization of Ariel’s orbit implies that some prior mechanism must have excited its eccentricity before the resonance encounter. In this work, we performed a large number of simulations using an N-body integrator to assess whether the earlier $7/4$ mean-motion resonance between Miranda and Ariel could serve as a mechanism to increase Ariel’s eccentricity. Our results show that, due to divergent migration, resonance capture does not occur. As the satellites cross the nominal resonance, Ariel’s eccentricity is only excited to $3.4 \times 10^{-4}$, substantially smaller than the required value. Therefore, the $7/4$ mean-motion resonance is not a viable mechanism for increasing Ariel’s eccentricity.

💡 Research Summary

The paper investigates whether the 7/4 mean‑motion resonance (MMR) between Uranus’s inner moon Miranda and the larger moon Ariel could have provided the eccentricity boost required for Ariel to avoid capture in the later 5/3 Ariel‑Umbriel resonance. Previous work has shown that, to reproduce the present orbital architecture, Ariel must have entered the 5/3 resonance with an eccentricity of order 0.01; otherwise, tidal damping would quickly circularize its orbit and lead to a resonant capture that is inconsistent with observations. Because tidal dissipation in the satellites is efficient, a mechanism that excites Ariel’s eccentricity before the 5/3 encounter is needed.

The authors set up a large suite of numerical experiments using the N‑body integrator TIDYMESS, which includes full gravitational interactions, spin dynamics, tidal deformation, and quadrupole moments. They generated 2 000 simulations of the two‑body system (Miranda and Ariel) with initial eccentricities set to zero, inclinations equal to their current values, and random mean‑longitude‑related angles uniformly sampled. The semi‑major axes were placed just interior to the nominal 7/4 period ratio, using a weak‑friction tidal model with planetary quality factor Q₀ = 8000 and Love number k₂,0 = 0.103. To accelerate the integrations, a time‑scaling factor of 1 000 was applied, and each run was integrated for 20 Myr. Additional tests used scaling factors of 10 and 100, and a subset of simulations included all five major Uranian satellites to check for indirect effects.

The results show that the migration of Miranda and Ariel is divergent, meaning the ratio of their mean motions increases with time. Divergent migration precludes resonant capture, which is confirmed by the absence of any capture events in all simulations. As the system passes through the nominal 7/4 commensurability, only a modest excitation of Ariel’s eccentricity occurs. The final distribution of Ariel’s eccentricity is well described by a Gaussian with a mean of 3.1 × 10⁻⁴ and a standard deviation of 7.0 × 10⁻⁵. The maximum “free” eccentricity reached during the passage is about 8 × 10⁻⁴, still two orders of magnitude below the ∼0.01 required to bypass the 5/3 resonance. Miranda’s eccentricity shows a larger response (mean ≈ 3 × 10⁻³), but this does not translate into a significant effect on Ariel. The eccentricity excitation is far smaller than Ariel’s tidal damping timescale (~260 Myr), so any increase would be quickly erased.

Changing the acceleration factor or adding the outer satellites does not materially alter these statistics, indicating that the conclusion is robust against reasonable variations in the tidal parameters and model complexity.

Consequently, the 7/4 Miranda‑Ariel resonance cannot serve as the pre‑excitation mechanism for Ariel’s eccentricity. The authors suggest alternative dynamical pathways, such as three‑body resonances (e.g., 3n₁ − 8n₂ + 4n₃ involving Miranda, Ariel, and Oberon, or 4n₁ − 10n₂ + 5n₃ involving Miranda, Ariel, and Umbriel) that occur 250–300 Myr before the 5/3 event, or resonance locking of Ariel with Uranus that would lower the planetary quality factor below 1 000, dramatically increasing Ariel’s migration rate. Such scenarios could make other resonances (e.g., a 2:1 Ariel‑Umbriel encounter) dynamically feasible, offering a possible route to the present configuration.

Overall, the study provides a thorough numerical assessment that rules out the 7/4 MMR as a viable source of the required eccentricity, and it points toward more complex multi‑body interactions or altered tidal regimes as promising avenues for future research.