  We consider a $(1+N)$-body problem in which one particle has mass $m_0 \gg 1$
and the remaining $N$ have unitary mass. We can assume that the body with
larger mass (central body) is at rest at the origin, coinciding with the center
of mass of the $N$ bodies with smaller masses (satellites). The interaction
force between two particles is defined through a potential of the form $U \sim
\frac{1}{r^\alpha},$ where $\alpha \in [1,2)$ and $r$ is the distance between
the particles. Imposing symmetry and topological constraints, we search for
periodic orbits of this system by variational methods. Moreover, we use
$\Gamma$-convergence theory to study the asymptotic behaviour of these orbits,
as the mass of the central body increases. It turns out that the Lagrangian
action functional $\Gamma$-converges to the action functional of a Kepler
problem, defined on a suitable set of loops. In some cases, minimizers of the
$\Gamma$-limit problem can be easily found, and they are useful to understand
the motion of the satellites for large values of $m_0$. We discuss some
examples, where the symmetry is defined by an action of the groups $Z_4$ , $Z_2
\times Z_2$ and the rotation groups of Platonic polyhedra on the set of loops.
