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 m 1 , 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 m 1 , an oblate spheroid of incompressible homogeneous viscous fluid of density ρ 1 and m 2 , a finite straight segment of length 2 l revolve around their common center of mass in circular orbits while third body m 3 (a small solid sphere of density ρ 3 ) moves inside m 1 . Two collinear { L 1 , L 2 } 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 L 1 and L 2 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 L 1 but affect the stability of L 2 . 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 ε 2 , on the stability of equilibrium point L 1 . The results obtained are applied on Earth-Moon, Jupiler-Amalthea, Jupiler-Ganymede systems (astrophysical problems) to predict the stability of L 1 .