Symmetric Constellations of Satellites Moving Around a Central Body of Large Mass

• Published in: Journal of dynamics and differential equations 2023-06, Vol.35 (2), p.1511-1559
• Main Authors: Fenucci, Marco, Gronchi, Giovanni Federico
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Asymptotic methods; Asymptotic properties; Convergence; Mathematics; Mathematics and Statistics; Orbits; Ordinary Differential Equations; Partial Differential Equations; Satellite constellations; Symmetry; Variational methods
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-021-10083-5

We consider a ( 1 + N ) -body problem in which one particle has mass m 0 ≫ 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 ∼ 1 r α , where α ∈ [ 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 Γ -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 Γ -converges to the action functional of a Kepler problem, defined on a suitable set of loops. In some cases, minimizers of the Γ -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 × Z 2 and the rotation groups of Platonic polyhedra on the set of loops.