In this paper we study the existence and linear stability of the equilibrium points in the Robe’s restricted three-body problem when one of the primaries is a finite straight segment. We observe that the center of the first primary is always an equilibrium point for all values of the density parameter k , mass parameter μ and length parameter l . There is an additional equilibrium point lying on the line joining the center of the first primary and the second primary, provided k > 1 + l 2 . When k + μ = 1 + l 2 ( 1 − μ ) , there are infinite number of equilibrium points lying on a circle with radius 1 − 2 3 l 2 and center as the second primary, provided they lie within the spherical shell. When k 0 and k + μ + 2 μ l 2 > 0 , there are two more equilibrium points lying in the x z -plane forming triangles with the center of the shell and the second primary. We analyze the linear stability of the equilibrium points.

Robe's restricted three-body problem when one of the primaries is a finite straight segment

Effect of oblateness and viscous force in the Robe’s circular restricted three-body problem

This paper investigates the motion of a test particle (third body) in the frame work of Robe’s circular restricted three-body problem (RCR3BP) when it also experiences the viscous force of the fluid and small perturbations of Coriolis and centrifugal forces. The existence of equilibrium points and their stability are analysed under certain conditions. A pair of axial and an infinite number of circular equilibrium points are observed to exist in the orbital plane of motion, while two equilibrium points, making triangles with the line joining the centre of the spherical shell and the second primary exist in the ‘off-plane’ of motion and thereby are known as ‘out-of-plane’ equilibrium points. The positions of all these equilibrium points are affected by the Viscous force and a small perturbation of the Coriolis force. As regards their stability, both axial points are seen to be asymptotically stable, while the circular and the ‘out-of-plane’ points are unstable. It is noticed that the viscous force improves the results of stability.

Perturbed Robe’s CR3BP with viscous force

Effect of perturbations in the Coriolis and centrifugal forces in the Robe‐finite straight segment model with arbitrary density parameter

Kind of Robe’s restricted problem with heterogeneous irregular primary of N-layers when outer most layer has viscous fluid

Effects of perturbations in Coriolis and centrifugal forces on the locations and stability of libration points in Robe’s circular restricted three-body problem under oblate-triaxial primaries

This paper studies the motion of an infinitesimal mass in the framework of Robe’s circular restricted three-body problem in two cases; the first case is when the hydrostatic equilibrium figure of the first primary is an oblate spheroid, the shape of the second primary is considered as an oblate spheroid with oblateness coefficients up to the second zonal harmonic, while the first primary is a Roche ellipsoid in the second case and the full buoyancy of the fluid is taken into account. In case one; it is observed that there are two axial libration points on the line joining the centres of the primaries, points on the circle within the first primary are also libration points under certain conditions. It is further found that the first axial point is stable, while the second one is conditionally stable, and the circular points are unstable. It is found in case two that there is exist only one libration point (0,0,0) this point is stable.

Robe’s circular restricted three-body problem with zonal harmonics

Effects of zonal harmonics on the out-of-plane equilibrium points in the generalized Robe's circular restricted three-body problem

This paper investigates Robe’s circular restricted three-body problem for two cases: with a Roche ellipsoid-triaxial system and with a Roche ellipsoid-oblate system. Without ignoring any component in both problems, a full treatment is given of the buoyancy force. The relevant equations of motion are established, and the special case where the density of the fluid and that of the infinitesimal mass are equal (D=0) is discussed. The location of the libration point and its stability when the infinitesimal mass is denser than the medium (D>0) are studied and it is found that the point (0,0,0) is the only libration point and this point is stable.

Achonu Joseph Omale

Papers

Effects of perturbations in Coriolis and centrifugal forces on the locations and stability of libration points in Robe’s circular restricted three-body problem under oblate-triaxial primaries

Robe’s circular restricted three-body problem with zonal harmonics

Effects of zonal harmonics on the out-of-plane equilibrium points in the generalized Robe's circular restricted three-body problem

Robe’s circular restricted three-body problem with a Roche ellipsoid-triaxial versus oblate system