[Paper Review] Special mean motion resonance pairs: Mimas-Tethys and Titan-Hyperion

Jing Luan|arXiv (Cornell University)|Oct 9, 2014

Astro and Planetary Science1 references3 citations

TL;DR

This paper explains why the Mimas-Tethys and Titan-Hyperion mean motion resonances are uniquely deep and exhibit large libration amplitudes: negligible long-term damping of Mimas’s inclination and Hyperion’s eccentricity allows sustained resonance capture. Using Hamiltonian dynamics and capture probability models, it shows that Titan-Hyperion’s resonance was almost certain to form via convergent migration, while Mimas-Tethys had only ~6% capture probability, with initial conditions inferred from current libration amplitudes.

ABSTRACT

Five pairs of large solar system satellites occupy first order mean-motion resonances (MMRs). Among these, the pairs of Mimas-Tethys and Titan-Hyperion are special. They are located much deeper in resonance than the others and their critical arguments librate with much greater amplitudes. These characteristics are traced to the insignificant damping, over $\Gyr$ timescales, of Mimas's orbital inclination and Hyperion's orbital eccentricity. Absent that, these resonances would not survive. Instead their librations would be overstable and escape from resonance would occur on the relevant damping time. Unlike the aforementioned MMRs, those involving Enceladus-Dione, Io-Europa, and Europa-Ganymede are limited by eccentricity damping. They must either remain at the shallow depths they currently occupy, or, if they venture deeper, retreat after a limited time. The latter seems almost certain for Enceladus-Dione and quite likely for the others, We examine the MMRs involving Mimas-Tethys and Titan-Hyperion under the assumption that they formed as a result of convergent migration. Capture probabilities are $\sim 6\%$ for the former and $100\%$ for the latter. The possibility of collisional excitation of their large librations is investigated but largely discounted.

Motivation & Objective

• To explain why the Mimas-Tethys and Titan-Hyperion mean motion resonances are uniquely deep and exhibit large libration amplitudes compared to other satellite pairs.
• To investigate the role of dissipative processes—specifically inclination damping for Mimas and eccentricity damping for Hyperion—in enabling long-term resonance survival.
• To assess the likelihood of resonant capture via convergent migration, using improved capture probability models.
• To evaluate whether collisional excitation could have produced the observed large libration amplitudes, and rule it out as the primary mechanism.
• To infer initial orbital conditions (inclination and eccentricity) prior to resonance capture using current libration amplitudes and phase-space dynamics.

Proposed method

• Modeling the resonant dynamics using Hamiltonian mechanics with the resonant argument φ and conserved quantities k (leading-order mass term) and H (Hamiltonian), which define phase-space topology.
• Applying the capture probability formalism of Borderies & Goldreich (1984), derived from Henrard (1982) and Yoder (1979), to estimate capture likelihoods under convergent migration.
• Using the phase-space area A_f of the resonant island to infer capture probability: A_f < A_c implies 100% capture for Titan-Hyperion, while A_f ≈ 0.06A_c implies ~6% for Mimas-Tethys.
• Estimating damping timescales for Mimas’s inclination and Hyperion’s eccentricity using tidal dissipation models (τ ∝ a^{13/2}/m_s), showing they are >10^7 Gyr, thus negligible.
• Inferring initial free eccentricity e′_i ≈ 0.019 and initial free inclination I₀ ≈ 0.36° from phase-space area and current libration amplitudes.
• Assessing alternative mechanisms such as collisional excitation by comparing expected excitation levels with observed libration amplitudes, concluding it is unlikely to be the cause.

Experimental results

Research questions

• RQ1 Why are the Mimas-Tethys and Titan-Hyperion mean motion resonances significantly deeper than other first-order resonances in the solar system?
• RQ2 What role does the negligible long-term damping of Mimas’s orbital inclination and Hyperion’s eccentricity play in enabling sustained resonance capture?
• RQ3 What is the probability of resonant capture for Mimas-Tethys and Titan-Hyperion under convergent migration, and how do these compare to other resonant pairs?
• RQ4 Could collisional excitation of orbital elements have produced the observed large libration amplitudes in these resonances?
• RQ5 What were the initial orbital conditions (inclination and eccentricity) of Mimas and Hyperion prior to resonance capture, based on current dynamical state?

Key findings

• The Mimas-Tethys resonance is located at k ≈ 5.5k_c, indicating a deep resonance, with critical argument librating around 0° with amplitude ~36°.
• The Titan-Hyperion resonance is also deep (k ≈ 5.5k_c), with libration amplitude ~36° around 180°, and has a phase-space area A_f ≈ 0.092A_c, implying 100% capture probability.
• Mimas-Tethys has a capture probability of ~6% due to its smaller resonant island area, consistent with earlier estimates (~4%) by Sinclair (1972).
• Hyperion’s eccentricity damping timescale is estimated at τ_e′ ≈ 6×10^7 Gyr, making damping negligible over Gyr timescales, which explains its long-term resonance stability.
• The initial free inclination of Mimas is inferred to be I₀ ≈ 0.36°, and the initial free eccentricity of Hyperion is e′_i ≈ 0.019, based on phase-space area and current libration amplitudes.
• Collisional excitation is ruled out as the primary cause of large libration amplitudes, as it would require implausibly high-velocity impacts to produce such effects without disrupting the system.