Dynamical structures associated with high-order and secondary resonances in the spin-orbit problem

In our Solar system, spin-orbit resonances are common under Sun--planet, planet--satellite and binary asteroid configurations. In this work, high-order and secondary spin-orbit resonances are investigated by taking numerical and analytical approaches. Poincar\'e sections as well as two types of dynamical maps are produced, showing that there are complicated structures in the phase space. To understand numerical structures, we adopt the theory of perturbative treatments to formulate resonant Hamiltonian for describing spin-orbit resonances. Results show that there is an excellent agreement between analytical and numerical structures. It is concluded that the main V-shape structure arising in the parameter space $(\dot\theta,\alpha)$ is sculpted by the synchronous primary resonance, those minute structures inside the V-shape region are dominated by secondary resonances and those structures outside the V-shape region are governed by high-order resonances. At last, the analytical approach is applied to binary asteroid systems (65803) Didymos and (4383) Suruga to reveal their phase-space structures.