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At present, the main satellites of Uranus are not involved in any low order mean motion resonance (MMR). However, owing to tides raised in the planet, Ariel and Umbriel most likely crossed the 5/3 MMR in the past. Previous studies on this resonance passage relied on limited time-consuming $N-$body simulations or simplified models focusing solely on the effects of the eccentricity or the inclination. In this paper, we aim to provide a more comprehensive view on how the system evaded capture in the 5/3 MMR. For that purpose, we developed a secular resonant two-satellite model with low eccentricities and low inclinations, including tides using the weak friction model. By performing a large number of numerical simulations, we show that capture in the 5/3 MMR is certain if the initial eccentricities of Ariel, $e_1$, and Umbriel, $e_2$, are related through $(e_1^2 + e_2^2)^{1/2} < 0.007$. Moreover, we observe that the eccentricity of Ariel is the key variable to evade the 5/3 MMR with a high probability. We determine that for $e_1 > 0.015$ and $e_2 < 0.01$, the system avoids capture in at least $60\%$ of the cases. We also show that, to replicate the currently observed system, the initial inclinations of Ariel and Umbriel must lay within $I_1 \le 0.05^{\circ}$ and $0.06^{\circ} \le I_2 \le 0.11^{\circ}$, respectively. We checked these results using a complete $N-$body model with the five main satellites and did not observe any significant differences., Comment: Published in Icarus. arXiv admin note: text overlap with arXiv:2305.08794

Mutual gravitational interactions between the five major Uranian satellites raise small quasi-periodic fluctuations on their orbital elements. At the same time, tidal interactions between the satellites and the planet induce a slow outward drift of the orbits, while damping the eccentricities and the inclinations. In this paper, we revisit the current and near past evolution of this system using a $N-$body integrator, including spin evolution and tidal dissipation with the weak friction model. We update the secular eigenmodes of the system and show that it is unlikely that any of the main satellites were recently captured into a high obliquity Cassini state. We rather expect that the Uranian satellites are in a low obliquity Cassini state and compute their values. We also estimate the current variations in the eccentricities and inclinations, and show that they are not fully damped. We constrain the modified quality factor of Uranus to be $Q_U' = ( 1.2 \pm 0.4 ) \times 10^5$, and that of Ariel to be $ Q_A' = (7 \pm 3) \times 10^4 $. We find that the system most likely encountered the 5/3 mean motion resonance between Ariel and Umbriel in the past, at about $(0.7 \pm 0.2)$ Gyr ago. We additionally determine the eccentricities and inclinations of all satellites just after the resonance passage that comply with the current system. We finally show that, from the crossing of the 5/3 MMR to the present, the evolution of the system is mostly peaceful and dominated by tides raised on Uranus by the satellites., Comment: Published at Icarus

The evolution of the five largest satellites of Uranus during the crossing of the 5/3 mean motion resonance between Ariel and Umbriel is strongly affected by chaotic motion. Studies with numerical integrations of the equations of motion and analysis of Poincar\'e surface sections provided helpful insights to the role of chaos on the system. However, they lack of a quantification of this chaos in the phase-space. Here, we construct stability maps using the frequency analysis method. We determine that for low energies (small eccentricity and/or inclinations), the phase-space is mainly stable. As the energy increases, the chaotic regions replace the stable motion, until only small, localized libration regions remain stable., Comment: arXiv admin note: text overlap with arXiv:2305.08794

The orbits of the main satellites of Uranus are expected to slowly drift away owing to tides raised in the planet. As a result, the 5/3 mean motion resonance between Ariel and Umbriel was likely encountered in the past. Previous studies have shown that, in order to prevent entrapment in this resonance, the eccentricities of the satellites must be larger than $\sim 0.01$ at the epoch, which is hard to explain. On the other hand, if the satellites experience some temporary capture and then escape, the inclinations rise to high values that are not observed today. We have revisited this problem both analytically and numerically focussing on the inclination, using a secular two-satellite model with circular orbits. We show that if the inclination of Umbriel was around $0.15^{\circ}$ at the time of the 5/3 resonance encounter, capture can be avoided in about $60\%$ of the cases. Moreover, after the resonance crossing, the inclination of Umbriel drops to a mean value around $0.08^{\circ}$, which is close to the presently observed one. The final inclination of Ariel is distributed between $0.01^{\circ}$ and $0.25^{\circ}$ with a nearly equal probability, which includes the present mean value of $0.02^{\circ}$., Comment: 19 pages, 10 figures