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

Jagadish Singh,Abubakar Umar Sandah

doi:10.1155/2012/679063

Jagadish Singh, Abubakar Umar Sandah

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https://doi.org/10.1155/2012/679063

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Abstract

This paper investigates the positions and linear stability of an infinitesimal body around the equilibrium points in the framework of the Robe’s circular restricted three-body problem, with assumptions that the hydrostatic equilibrium figure of the first primary is an oblate spheroid and the second primary is an oblate body as well. It is found that equilibrium point exists near the centre of the first primary. Further, there can be one more equilibrium point on the line joining the centers of both primaries. Points on the circle within the first primary are also equilibrium points under certain conditions and the existence of two out-of-plane points is also observed. The linear stability of this configuration is examined and it is found that points near the center of the first primary are conditionally stable, while the circular and out of plane equilibrium points are unstable.

Highlights

Robe 1 considered a new kind of restricted three-body problem in which, one of the primaries of mass m1 is a rigid spherical shell, filled with homogenous, incompressible fluid of density ρ1; the second one is a point mass m2 located outside the shell and moving around the mass m1 in a Keplerian orbit; the infinitesimal mass m3 is a small sphere of density ρ3, moving inside the shell and is subject to the attraction of m2 and the buoyancy force due to the fluid of the first primary
Robe 1 assumed that the pressure field of the fluid ρ1 has a spherical symmetry around the center of the shell and he took into account only one out of the three components of the pressure field which is due to the own gravitational field of the fluid ρ1
When 2πρ1A1 n2 1 − μ, points on the circle 1 − x1 2 x22 r2, x3 0 lying within the first primary are equilibrium points

Summary

Introduction

Robe 1 considered a new kind of restricted three-body problem in which, one of the primaries of mass m1 is a rigid spherical shell, filled with homogenous, incompressible fluid of density ρ1; the second one is a point mass m2 located outside the shell and moving around the mass m1 in a Keplerian orbit; the infinitesimal mass m3 is a small sphere of density ρ3, moving inside the shell and is subject to the attraction of m2 and the buoyancy force due to the fluid of the first primary He discussed the linear stability of an equilibrium point obtained in two cases. They found that when the density parameter D is taken as zero, every point inside the fluid is an equilibrium point; otherwise the center of the ellipsoid is the only equilibrium point and it is linearly stable

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