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

TL;DR

The study tests whether the 7/4 mean-motion resonance between Miranda and Ariel could raise Ariel’s eccentricity enough to avoid entrapment in the subsequent 5/3 resonance with Umbriel. Using ~2000 forward N-body simulations with tidal migration consistent with a weak-friction model (e.g., $Q_0=8000$, $k_{2,0}\tau_0=0.064$) and starting near the nominal 7/4 value, the authors find divergent migration precludes resonant capture and only modestly excites Ariel’s eccentricity to about $e_A \approx 3.1\times10^{-4}$ (mean), far below the $\sim 0.01$ target. Miranda shows a larger, often bimodal eccentricity response, but inclinations remain largely unchanged. The results are robust across acceleration factors and configurations including five satellites; thus the 7/4 Miranda–Ariel resonance is not a viable mechanism to boost Ariel’s eccentricity to bypass the 5/3 MMR, pointing to alternative pathways such as specific three-body resonances or resonance locking with Uranus. Data supporting the figures are publicly available on Zenodo.

Abstract

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.

Figures (1)

• Figure 1: Distribution of the mean eccentricities (dark shaded bins) and maximum free eccentricities (light shaded bins) for Miranda (orange) and Ariel (blue), after the passage through the 7/4 MMR. The thin yellow and blue lines mark the mean equilibrium eccentricities values before the resonance passage for Miranda and Ariel, respectively.