Titan's Tidal Migration and Saturnian Resonance Cascades: Numerical Tests of Applicability Conditions

Hyperion, the only confirmed chaotic satellite of the Solar System, occupies a 4:3 near-resonance with Titan. We present a numerical study of the conditions governing resonant capture, instability, and escape in the Saturn–Titan–Hyperion–Dione system, based on >100 high-precision integrations with a numba-accelerated Wisdom–Holman integrator and IAS15 cross-checks. Our principal result is a segmented escape-time scaling law: the median escape time of Hyperion obeys three distinct power-law regimes as the Titan mass multiplier m is varied — t_esc ∝ m^{≈−1.2} at low m (weakly constrained), a precise plateau t_esc ∝ m^{−10/3} for 1 ≲ m ≲ 3.2 (matching the standard first-order resonance-width scaling Δa_res ∝ m^{2/3} combined with t_esc ∝ Δa_res^{−5}), and a steep high-m cutoff. We further show that (i) the Hyperion 4:3 resonance is so narrow (Δa_res ≈ 2.6×10^{−8} AU) that capture fails at any migration rate ≳ 1.6×10^{−14} AU yr^{−1}, including the real Titan rate (ε = 0.021); and (ii) the three-body stabilization of resonance chains is not supported: adding Dione destabilizes the Titan–Hyperion resonance, with physical escape of Hyperion in 3/3 IAS15 trials versus 0/3 in the two-body case. These results place concrete boundary conditions on the resonance-cascade scenario for Saturn's rings (Čuk et al. 2016)