Perturbed Robe’s restricted problem of 2+2 bodies when the primaries form a Roche ellipsoid-triaxial system

Abstract

Abstract The aim of this paper is to study the effect of perturbations in the Coriolis and centrifugal forces on the location and stability of the equilibrium solutions in the Robe’s restricted problem of 2+2 bodies under the assumption that the hydrostatic equilibrium figure of the first primary is a Roche ellipsoid and the shape of the second primary is triaxial. The third and the fourth bodies (of mass m3 and m4 respectively) are small solid spheres of density ρ3 and ρ4 respectively inside the ellipsoid, 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 m1. The masses m3 and m4 mutually attract each other, do not influence the motion of m1 and m2 but are influenced by them. We have taken 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 (iii) that arising from the centrifugal force. The linear stability of this configuration is examined. It is observed that there exist only six equilibrium solutions of the system, provided they lie within the Roche ellipsoid. The equilibrium solutions of m3 and m4 lying on x1-axis are unstable for ε > 0, ε′ > 0 and ε < 0, ε′ > 0 and stable for ε > 0, ε′ < 0 and ε < 0, ε′ < 0 ,using the data of submarines in the Earth -Moon system. The equilibrium solutions of m3 and m4 respectively when the displacement is given in the direction of x2 or x3− axis are conditionally stable.We observe that the conditions of stability are influenced by the small perturbations in the Coriolis and centrifugal forces.