Robe’s restricted problem of 2+2 bodies with one of the primaries an oblate body

Abstract

In this problem, one of the primaries of mass $m^{*}_{1}$ is a rigid spherical shell filled with a homogeneous incompressible fluid of density ρ₁. The smaller primary of mass m₂ is an oblate body outside the shell. The third and the fourth bodies (of mass m₃ and m₄ respectively) are small solid spheres of density ρ₃ and ρ₄ respectively inside the shell, with the assumption that the mass and the radius of the third and the fourth body are infinitesimal. We assume that m₂ is describing a circle around $m^{*}_{1}$. The masses m₃ and m₄ mutually attract each other, do not influence the motions of $m^{*}_{1}$ and m₂ but are influenced by them. We also assume that masses m₃ and m₄ are moving in the plane of motion of mass m₂. In the paper, equilibrium solutions of m₃ and m₄ and their linear stability are analyzed. There are two collinear equilibrium solutions for the system. The non collinear equilibrium solutions exist only when ρ₃=ρ₄. There exist an infinite number of non collinear equilibrium solutions of the system, provided they lie inside the spherical shell. In a system where the primaries are considered as earth-moon and m₃,m₄ as submarines, the collinear equilibrium solutions thus obtained are unstable for the mass parameters μ,μ₃,μ₄ and oblateness factor A. In this particular case there are no non-collinear equilibrium solutions of the system.