Existence and Stability of Equilibrium Points in the Robe’s Restricted Three-Body Problem with Variable Masses

Jagadish Singh, Oni Leke

Department of Mathematics, Faculty of Science, Ahmadu Bello University, Zaria, Nigeria.
Department of Mathematics, Statistics and Computer Science, College of Science, University of Agriculture, Makurdi, Nigeria.

DOI: 10.4236/ijaa.2013.32013                

Abstract

The positions and linear stability of the equilibrium points of the Robe’s circular restricted three-body problem, are generalized to include the effect of mass variations of the primaries in accordance with the unified Meshcherskii law, when the motion of the primaries is determined by the Gylden-Meshcherskii problem. The autonomized dynamical system with constant coefficients here is possible, only when the shell is empty or when the densities of the medium and the infinitesimal body are equal. We found that the center of the shell is an equilibrium point. Further, when k﹥1; k being the constant of a particular integral of the Gylden-Meshcherskii problem; a pair of equilibrium point, lying in the -plane with each forming triangles with the center of the shell and the second primary exist. Several of the points exist depending on k; hence every point inside the shell is an equilibrium point. The linear stability of the equilibrium points is examined and it is seen that the point at the center of the shell of the autonomized system is conditionally stable; while that of the non-autonomized system is unstable. The triangular equilibrium points on the -plane of both systems are unstable.

Keywords

Robe’s Problem; Meshcherskii Law; GMP; Equilibruim Points

Conflicts of Interest

The authors declare no conflicts of interest.

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