  We provide sufficient conditions for the occurrence of time-periodic Hopf
bifurcation for the coupled system constituted by a rigid sphere, $\mathscr S$,
freely moving under gravity in a Navier-Stokes liquid. Since the region of flow
is unbounded (namely, the whole space outside $\mathscr S$), the main
difficulty consists in finding the appropriate functional setting where general
theory may apply. In this regard, we are able to show that the problem can be
formulated as a suitable system of coupled operator equations in Banach spaces,
where the relevant operators are Fredholm of index 0. In such a way, we can use
the theory recently introduced by the author, and give sufficient conditions
for time-periodic bifurcation to take place.
