Orbital inclinations of Laplace state P₁ and its twin P₋₁ (colorbar, given by Eqs. (18), (20)) as functions of a_m/r_M and the planet’s obliquity θ. In both panels, the location of a_m = r_M is denoted with a vertical black dashed line, and the E₁ region (where P₁ and P₋₁ are unstable) is shown by a white hatched zone. Across the two panels, a representative evolutionary history is shown via the three cyan points and arrows (S1, S2, S3). While the inclinations of P₁ and P₋₁ only depend on θ and a_m/r_M, we also illustrate the evolution using the following concrete parameters for interpretability: a_p = 0.4 au, e_p = 0, R_p = 2 R_⊕, and m_p = 10 M_⊕. We then denote the corresponding values of the planet’s spin rate along the top-left axis (relevant to planetary tidal evolution) and the lunar semi-major axis along the top-right axis (relevant to lunar migration). In the left panel, the planet is initially rapidly rotating with a period of 7.5 hours, and the moon is located at a_m = 5R_p (S1). Then, as the planet despins and the obliquity damps, r_M decreases while the moon is still located at a_m = 5R_p, and the system reaches S2 (we have neglected lunar tidal migration for illustrative purposes). Going from S1 to S2, the inclination of the lunar orbit smoothly flips from prograde to retrograde (assuming that the moon occupies P₋₁). In the right panel, as the moon inspirals, starting from S2, a_m decreases until the moon is disrupted by the planet when a_m ≃ R_p ≳ r_M (S3).