Effects of Viscosity and Oblateness on the Perturbed Robe’s Problem with Non-Spherical Primaries

We here analyzed the effects of viscosity, oblateness of the primary m₁, length parameter l, and perturbations in the Coriolis and centrifugal forces on the stability of the equilibrium points of the Robe’s problem. In the setting, it is assumed that the two primaries m₁, an oblate spheroid of incompressible homogeneous viscous fluid of density ρ₁ and m₂, a finite straight segment of length 2l revolve around their common center of mass in circular orbits while third body m₃ (a small solid sphere of density ρ₃) moves inside m₁. Two collinear {L₁, L₂} and infinite non-collinear equilibrium points are evaluated and found that the location of equilibrium points remain unaffected by viscosity. However, the effects of oblateness and perturbation in the centrifugal force are quite noticeable from the expressions of the equilibrium points. The stability criterion for L₁ and L₂ are stated whereas the non-collinear equilibrium points are found to be unstable. It is observed that the viscosity has a substantial effect on the stability as it changes the nature of stability from marginal stability to asymptotic stability. The perturbations do not affect the stability of L₁ but affect the stability of L₂. Moreover, the effect of oblateness on the stability of the equilibrium points is quite evident. A very important observation of the study is that the oblateness parameter A neutralizes the effects of the length parameter l and perturbation ε₂, on the stability of equilibrium point L₁. The results obtained are applied on Earth-Moon, Jupiler-Amalthea, Jupiler-Ganymede systems (astrophysical problems) to predict the stability of L₁.