Laplace-like resonances with tidal effects

The paper establishes via a resonant Hamiltonian model plus extensive numerics that: (i) with a fast-rotating central body, first-order Laplace-like chains keep the Laplace angle in libration while mixed-/second-order chains circulate, (ii) with a slowly rotating central body, all resonant angles circulate, (iii) in first-order chains all three inner semimajor axes increase, whereas in mixed (only the inner two) and second-order (only the innermost) they do not, and (iv) adding a fourth body yields capture only for first-order chains; moreover the conclusions are robust under scaling the tidal strength by α up to 10^5. These findings are stated and/or illustrated throughout Sections 3–6, including the tide models (Eqs. 4 and 6), the survival/circulation of the three-body angle, the migration patterns, the α-robustness of stability indices, and the S4-capture experiments (e.g., 3.2.1, 3.2.2, 3.3, 4.2, 5, 6) . The candidate solution reaches the same conclusions but via a structural lemma about the order at which the three-body angle enters an O(e^2,s^2) truncation (explaining why L is dynamically active at first order only) and a resonant-invariants argument for semimajor-axis balances. Aside from a minor slip in the explicit three-body angle for the 2:1&3:1 case and a notational mismatch for S_B, the model’s reasoning is consistent with the paper’s framework and outcomes.