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The aim of this article is to propose a model, that is a planar version ofthe Full Two-Body Problem, and discuss the existence and stability of arelevant periodic solution. Consider two homogeneous ellipsoids orbiting aroundeach other in fixed coplanar Keplerian orbits. Moreover, their respective spinaxes are assumed to be perpendicular to the orbital plane, that is also acommon equatorial plane. The spin-spin model deals with the coupled rotationaldynamics of both ellipsoids. For a non-zero orbital eccentricity, it has thestructure of a non-autonomous system of coupled pendula. This model is anatural extension of the classical spin-orbit problem for two extended bodies.In addition, we consider dissipative tidal torques, that can trigger thecapture of the system into spin-orbit and spin-spin resonances. In this paperwe give some theoretical results for both the conservative model and thedissipative one. The conservative model has a Hamiltonian structure. We useproperties of Hamiltonian systems to give some sufficient conditions in thespace of parameters of the model, that guarantee existence, uniqueness andlinear stability of an odd periodic solution. This solution represents a doublesynchronous resonance in the conservative regime. Such solution can becontinued to the dissipative regime, where it becomes asymptotically stable. Wesee asymptotic stability as a dynamical mechanism for the capture into thedouble synchronous resonance. Finally we apply our results to several casesincluding the Pluto-Charon binary system and the Trojan binary asteroid 617Patroclus, target of the LUCY mission.

The spin-spin model and the capture into the double synchronous resonance