Journal of Astrophysics

Volume 2013 (2013), Article ID 898794, 11 pages

http://dx.doi.org/10.1155/2013/898794

## On Robe's Circular Restricted Problem of Three Variable Mass Bodies

^{1}Department of Mathematics, Faculty of Science, Ahmadu Bello University Zaria, PMB 2222, Samaru-Zaria, Kaduna, Nigeria^{2}Department of Mathematics, College of Science, University of Agriculture, PMB 2373, North-Bank, Makurdi, Nigeria

Received 25 March 2013; Accepted 25 August 2013

Academic Editors: M. Biesiada, K. Bolejko, M. S. Dimitrijevic, M. Jamil, and E. Saridakis

Copyright © 2013 Jagadish Singh and Oni Leke. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper investigates the motion of a test particle around the equilibrium points under the setup of the Robe’s circular restricted three-body problem in which the masses of the three bodies vary arbitrarily with time at the same rate. The first primary is assumed to be a fluid in the shape of a sphere whose density also varies with time. The nonautonomous equations are derived and transformed to the autonomized form. Two collinear equilibrium points exist, with one positioned at the center of the fluid while the other exists for the mass ratio and density parameter provided the density parameter assumes value greater than one. Further, circular equilibrium points exist and pairs of out-of-plane equilibrium points forming triangles with the centers of the primaries are found. The out-of-plane points depend on the arbitrary constant , of the motion of the primaries, density ratio, and mass parameter. The linear stability of the equilibrium points is studied and it is seen that the circular and out-of-plane equilibrium points are unstable while the collinear equilibrium points are stable under some conditions. A numerical example regarding out-of-plane points is given in the case of the Earth, Moon, and submarine system. This study may be useful in the investigations of dynamic problem of the “ocean planets” Kepler-62e and Kepler-62f orbiting the star Kepler-62.