Robe's restricted problem of 2+2 bodies when the bigger primary is a Roche ellipsoid

In the problem of 2+2 bodies in Robe's setup, one of the primaries of mass m1 is a Roche ellipsoid filled with a homogeneous incompressible fluid of density ρ1. The smaller primary of mass m2 is a point mass outside the ellipsoid. The third and the fourth bodies (of mass m3 and m4, respectively), supposed moving inside the ellipsoid, are small solid spheres of density ρ3 and ρ4, respectively, with the assumption that the mass and the radius of the third and the fourth bodies are infinitesimal. We assume that m2 is describing a circle around m1. The masses m3 and m4 mutually attract each other, do not influence the motions of m1 and m2 but are influenced by them. Robe's restricted three body problem is extended to 2+2 body problem under the assumption that the fluid body assumes the shape of the Roche ellipsoid (Chandrashekhar [2]). We take into consideration all the three components of the pressure field in deriving the expression for the buoyancy force viz (i) due to the own gravitational field of the fluid, (ii) that originating in the attraction of m2, and (iii) that arising from the centrifugal force. In this paper, equilibrium solutions of m3 and m4 and their linear stability are analysed. We have proved that there exist only six equilibrium solutions of the system. In a system where the primaries are considered as earth-moon and m3,m4 as submarines, the equilibrium solutions thus obtained are unstable.