A STUDY ON EQUILIBRIUM POINTS AND STABILITY OF THE ROBES R3BP WITH DENSITY VARIATION

Abstract

The paper investigates equilibrium points and stability of an infinitesimal mass under the Robe restricted three-body problem with density variation. The equations of motion have been stated and the equilibrium points investigated. It is seen that axial equilibrium point at the centre and axial equilibrium points near the centre of the first primary were found. The axial points near the centre were further explored and it seen that five types of axial equilibrium points namely; minimal axial points, primary axial points of first kind, interior points, primary axial points of second kind and the terminal point, exists. These points are either located inside the first primary or on the points at which the circle bounding the first primary is coincident with the axis. Further a pair of non-collinear equilibrium points and circular equilibrium points which are points on the circle bounding the first primary, exists. The linear stability of these equilibrium points are investigated and it is seen that the axial points at the centre and those near the centre are conditionally stable analytically but our numerical explorations of the stability reveals that these points can be unstable and stable, a condition which depends on the mass parameter and the density variation of the fluid. The circular points are unstable due to double roots of the characteristic equation while the non-collinear equilibrium points are also unstable due to a positive root which induces instability at the non-collinear equilibrium points. For our numerical explorations of the equilibrium points and the stability outcomes, we in particular use the Submarine-Earth and Moon system and the general case for all mass ratios. The Robe’s problem has been useful in discussing the long time stability of the Earth’s core and for describing the motion of underwater vehicles under the Earth’s and Moon’s attraction.