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

Jagadish Singh,Oni Leke

doi:10.1155/2013/898794

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.

Highlights

The classical restricted three-body problem (RTBP) constitutes one of the most important problems in dynamical astronomy
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 ρ1, 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 of density ρ3 moving inside the shell and is subject to the attraction of the second primary and the buoyancy force due to the fluid
We have derived the equations of motion and established the possible equilibrium points of the third body of infinitesimal mass in a setup of Robe’s [1] restricted three-body problem when the three participating bodies all vary their masses arbitrarily at the same rate and the density of the fluid and third body vary as the masses

Summary

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. Singh and Leke [12] investigated the existence and stability of equilibrium points in the Robe’s restricted three-body problem with variable masses. The existence and stability of equilibrium points under the frame of the Robe problem [1], when the participating bodies vary their masses at the same rate, is studied. This study may be useful in the investigations of dynamic the problem of water-planetary system discovered by Kepler spacecraft These “ocean planets” are orbiting the star Kepler-62 and are designated Kepler62e and Kepler-62f. This paper is orginzed as follows: Section 2 contains the equations of motion; the equilibrium points are investigated in Section 3; Section 4 investigates the linear stability of the equilibrium points; Section 5 discusses the obtained results and the conclusions

Equations of Motion

The Equilibrium Points

Stability of Equilibrium Points

Discussion and Conclusion