  The time-dependent restricted $(n+1)$-body problem concerns the study of a
massless body (satellite) under the influence of the gravitational field
generated by $n$ primary bodies following a periodic solution of the $n$-body
problem. We prove that the satellite has periodic solutions close to the
large-amplitude circular orbits of the Kepler problem (comet solutions), and in
the case that the primaries are in a relative equilibrium, close to
small-amplitude circular orbits near a primary body (moon solutions). The comet
and moon solutions are constructed with the application of a Lyapunov-Schmidt
reduction to the action functional. In addition, using reversibility technics,
we compute numerically the comet and moon solutions for the case of four
primaries following the super-eight choreography.
