Resonant Hamiltonian Dynamics in the CR3BP: Bistability and Stochastic Resonance in Binary Planetary Systems

Context: The Circular Restricted Three-Body Problem provides a fundamental framework for understanding resonant dynamics in binary star systems. Aims: We develop a unified Hamiltonian formulation for mean-motion resonances that encompasses both circumstellar and circumbinary planetary orbits within the CR3BP. Unlike the Solar System case where the perturbing body is a planet of negligible mass, here the perturber (a stellar companion) has a non-negligible, finite mass, a crucial difference that we fully incorporate. Methods: Starting from the full Hamiltonian in each configuration, we perform canonical transformations to resonant action angle variables and derive reduced one-degree-of-freedom Hamiltonians through systematic averaging over the fast orbital motion. Leading-order scaling laws for the Fourier coefficients of the resonant perturbation are obtained, revealing their dependence on the binary mass ratio and the planet's orbital distance. Results: The resulting effective potential is shown to exhibit bistability under the well-defined condition epsilon2/epsilon1 > 1/4, where epsilon1 and epsilon2 are the amplitudes of the first two resonant harmonics. This bistability creates the essential dynamical setting for stochastic resonance. Scaling laws for the Fourier coefficients are derived for both S-type and P-type configurations. Estimates for known binary-planet systems show that while currently observed systems lie below the bistability threshold, the theory predicts that extreme configurations (a/a b < 1.5 for P-type, almost equal mass binary) could host bistable resonances accessible to future observations. Conclusions: This work provides a natural Hamiltonian framework for studying stochastic resonance in binary planetary systems, bridging analytical celestial mechanics and the nonlinear dynamics of exoplanetary systems subject to realistic perturbations.