  We study the dynamics of an elastic body whose shape and position evolve due
to the gravitational forces exerted by a pointlike planet. The main result is
that, if all the deformations of the satellite dissipate some energy, then
under a suitable nondegeneracy condition there are only three possible outcomes
for the dynamics: (i) the orbit of the satellite is unbounded, (ii) the
satellite falls on the planet, (iii) the satellite is captured in synchronous
resonance i.e. its orbit is asymptotic to a motion in which the barycenter
moves on a circular orbit, and the satellite moves rigidly, always showing the
same face to the planet. The result is obtained by making use of LaSalle's
invariance principle and by a careful kinematic analysis showing that energy
stops dissipating only on synchronous orbits. We also use in quite an extensive
way the fact that conservative elastodynamics is a Hamiltonian system invariant
under the action of the rotation group.
