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 $$\begin{aligned} U \sim \frac{1}{r^\alpha }, \end{aligned}$$ 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 $${\mathbb {Z}}_4$$ , $${\mathbb {Z}}_2 \times {\mathbb {Z}}_2$$ and the rotation groups of Platonic polyhedra on the set of loops.