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. 1. Introduction The classical restricted three-body problem (RTBP) constitutes one of the most important problems in dynamical astronomy. The study of this problem is of great theoretical, practical, historical, and educational relevance. The investigation of this problem in its several versions has been the focus of continuous and intense research activity for more than two hundred years. The study of this problem in its many variants has had important implications in several scientific fields including, among others, celestial mechanics, galactic dynamics, chaos theory, and molecular physics. The RTBP is still a stimulating and active research field that has been receiving considerable attention of scientists and astronomers because of its applications in dynamics of the solar and stellar systems, lunar theory, and artificial satellites. A different kind of restricted three-body problem was formulated by Robe [1], a set up in which the first primary is a rigid spherical shell filled with homogenous, incompressible fluid of density , and the second primary is a mass point outside the shell and moving around the first primary in a Keplerian orbit, while the infinitesimal mass is a small sphere
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