Symmetric constellations of satellites moving around a central body of large mass

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.