  The aim of this article is to propose a model, that is a planar version of
the Full Two-Body Problem, and discuss the existence and stability of a
relevant periodic solution. Consider two homogeneous ellipsoids orbiting around
each other in fixed coplanar Keplerian orbits. Moreover, their respective spin
axes are assumed to be perpendicular to the orbital plane, that is also a
common equatorial plane. The spin-spin model deals with the coupled rotational
dynamics of both ellipsoids. For a non-zero orbital eccentricity, it has the
structure of a non-autonomous system of coupled pendula. This model is a
natural extension of the classical spin-orbit problem for two extended bodies.
In addition, we consider dissipative tidal torques, that can trigger the
capture of the system into spin-orbit and spin-spin resonances. In this paper
we give some theoretical results for both the conservative model and the
dissipative one. The conservative model has a Hamiltonian structure. We use
properties of Hamiltonian systems to give some sufficient conditions in the
space of parameters of the model, that guarantee existence, uniqueness and
linear stability of an odd periodic solution. This solution represents a double
synchronous resonance in the conservative regime. Such solution can be
continued to the dissipative regime, where it becomes asymptotically stable. We
see asymptotic stability as a dynamical mechanism for the capture into the
double synchronous resonance. Finally we apply our results to several cases
including the Pluto-Charon binary system and the Trojan binary asteroid 617
Patroclus, target of the LUCY mission.
