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This paper examines the motion of a dust grain around a triaxial primary and an oblate companion orbiting each other in elliptic orbits about their common barycenter in the neighborhood of collinear libration points. The positions and stability of these points are found to be affected by the triaxiality and oblateness of the primaries, and by the semi-major axis and eccentricity of their orbits. The stability behavior of the collinear points however remains unchanged; they are unstable in the Lyapunov sense.

The famous restricted three-body problem (R3PB) has been receiving considerable attention of scientists and astronomers because of its application in the dynamics of the solar and stellar systems, lunar theory and planetary sciences [

The ER3BP generalizes the original CR3BP, and improves its applicability, while some outstanding and useful properties of the circular model still hold true or can be adapted to the elliptic case. In particular, possible po- sitions of equilibrium occur when the three bodies form equilateral triangles. An application of this model can be seen in the motion of the Trojan asteroids around the triangular point

The bodies in the R3BP are strictly spherical in shape, but in nature, celestial bodies are not perfect spheres. They are either oblate or triaxial. The Earth, Jupiter, Saturn, Regulus, Neutron stars and black dwarfs are oblate spheroids [

Our aim is to study the effect of the triaxiality of the bigger primary and oblateness of the smaller one on the positions and stability of the collinear libration points. This system can be applied to the Earth-Moon system and to double pulsars. This paper is organized as follows: in Section 2, the governing equations of motion are presented; Section 3 describes the positions of the collinear points, while their linear stability is analyzed in Section 4; finally Section 5 concludes the paper.

The equations of motion of a dust grain particle in the ER3BP with a triaxial primary and an oblate secondary in dimensionless-pulsating coordinate system (ξ, η, ζ) are given by

with the force function

and

and the mean motion

where the prime represents differentiation w.r.t. the eccentric anomaly e and r_{i} (i = 1, 2) are the distances between the third body and the primaries; n, a, e, A and

The collinear points are the solutions of equations Ω_{ξ} = Ωη = Ω_{ζ} = 0 and η = ζ = 0; i.e.

From Equation (3), with η = ζ = 0 and the first of Equation (5), we have

To locate the collinear points on the ξ-axis, we divide the orbital plane into three parts; ξ < ξ_{1}, ξ_{1} < ξ < ξ_{2} and ξ_{2} < ξ with respect to the primaries.

Let

Since the distance between the primaries is unity, i.e.

Now, substituting Equation (7) in Equation (6) and clearing the fractions, we obtain

Solving, we get

Now,

Equation (10) in (6) yields,

Solving, we get

Since

Substituting Equation (13) in (6) and solving, we obtain

Equations (9), (12) and (14) are seventh degree equations and according to Descartes’ rule of sign, there exist only one real root each corresponding to the three collinear points_{2} for constant oblateness A = 0.001; a = 1.2, e = 0.25, μ = 0.35.

It is seen from _{2} as triaxiality decreases is a shift away from the origin (λ_{3}). This is shown in _{1}, σ_{2}, λ_{1}, λ_{2}, λ_{3} and σ_{1}, σ_{2}, λ_{3} respectively. An interesting result of the numerical computation is the existence of three real roots when σ_{1} = σ_{2} = 0.04 (i.e. oblateness of the primary) and σ_{1} = 0.04; σ_{2} = 0.03.

In order to study the stability of the collinear points, we consider the characteristic equation of the system [

The second partial derivatives are:

Considering the last interval, ξ_{2} < ξ.

The collinear points exist on the ξ-axis, i.e. η = ζ = 0, which means

Equation (17) in the first of (16) gives

. Effect of triaxiality on the position of the inner collinear point L_{2}

Triaxiality Factors | Characteristics Roots | |||
---|---|---|---|---|

σ_{1} | σ_{2} | λ_{1} | λ_{2} | λ_{3} |

0.04 | 0.040 | 0.489138 | 0.590313 | 1.55752 |

0.04 | 0.030 | 0.472706 | 0.617405 | 1.54967 |

0.03 | 0.025 | - | - | 1.56224 |

0.02 | 0.015 | - | - | 1.57147 |

0.01 | 0.075 | - | - | 1.58266 |

Effect of triaxiality on the position of the inner colli- near point L_{2}

Effect of triaxiality on the position of the inner colli- near point L_{2}

From the second and third of Equation (16) with η = 0, we get

and

Using Equation (18),

but,

Neglecting higher order terms in e^{2}, a, A and σ_{i} (i = 1, 2), we get

Since _{1} > 1, r_{2} < 1,

Also, the last of Equation (16) by virtue of η = 0 gives

Similarly, for the collinear points lying in (ξ_{1} < ξ < ξ_{2}) and (ξ < ξ_{1}),

Since

The positions of the collinear equilibrium points when the primary is a triaxial rigid body and the secondary is an oblate spheroid have been obtained (Equations (9), (12), (14) and