Orateur
Description
While two-planet mean-motion resonances have been extensively studied, the dynamics of three-planet resonances are comparatively less understood, especially when tidal dissipation is involved. We present a one-parameter one-degree-of-freedom pendulum-like model of three-planet zeroth-order resonances of the form $pn_1 − (p + q) n_2 + qn_3 = 0$ based on a Hamiltonian expansion carried out to second order in the planetary masses.
Using the new software Aptidal, we produce stability maps of three-planet zeroth-order resonances, where chaos is measured from the diffusion of the fundamental frequencies. The position of the resonance separatrix predicted by our pendulum-like model accurately matches the regions of mild chaos in our stability maps.
In presence of tides, we analytically compute the eigenvalue governing the evolution of the Laplace angle near the 180° equilibrium. We find that its real part is positive regardless of the masses and tidal parameters, demonstrating the instability of three-planet zeroth-order resonances with tides. The inverse of the eigenvalue’s real part gives the instability timescales, that is accurately reproduced by simulations of the complete system with Aptidal.