Lissajous Orbits at the Collinear Libration Points in the Restricted Three-Body Problem with Oblateness

In the present work, the collinear equilibrium points of the restricted three-body problem are studied under the effect of oblateness of the bigger primary using an analytical and numerical approach. The periodic orbits around these points are investigated for the Earth-Moon system. The Lissajous orbits and the phase spaces are obtained under the effect of oblateness.


Introduction
One of the most important object in astrodynamics is the restricted three-body problem, which has many applications in space missions.It deals with the motion of an infinitesimal mass under the effect of gravitational attraction of two bodies, called primaries, which move in a Keplerian orbit around their common center of mass.This problem has five libration points: three of them are called the collinear points 1 2 , L L and 3 L , which lie on the line joining the two primaries; the other two are called the triangular points 4 L and 5 L [1] [2].Several studies have been carried out on the collinear libration points by considering the oblateness of one or two primaries for the circular restricted three-body problem.Sharma and Subbarao [3] investigated the the locations of the five libration points under the effect of oblateness of the more massive primary for some systems of celestial A. H. Ibrahim et al.
bodies.Ibrahim A. [4] investigated the libration points for the Sun-Earth-Moon System.Ismail M. [5] studied the effect of solar radiation pressure on the libration points of the restricted four-body problem.
The existence of the periodic orbits near the collinear libration points was treated by many authors.Grgory A. [6] determined a class of Eight Lissajous orbits near collinear libration points by using Lindstedt Poincares technique.
Celletti A. [7] analyzed the Lissajous and halo orbits near the collinear libration points by using the classical perturbation theory.In this paper, the restricted three-body problem is studied by considering the more massive primary as an oblate spheroid.The Lissajous orbits in this case are presented, and also the phase spaces are obtained.

Equations of Motion
Using a barycentric-synodic coordinate system ( ) ξ η ζ and dimensionless variables, the equations of motion of a test particle in the circular restricted three-body problem under the effects of oblateness of the bigger primary can be expressed as 2n ) ( ) ( ) ( ) where The mean motion n of the primaires is given by ( ) , where is the oblateness coefficient of 1 m having the equatorial and polar radii as e r and p r , respectively.

Location of the Libration Points
The collinear equilibrium points can be obtained by solving Equations ( 4), ( 5) and ( 6) when 0 ) ( ) The coordinate of the collinear Points 1 2 , L L and 3 L are where 1 2 , x x and 3 x satisfy seventh degree polynomials: ) Each of Equations ( 8), ( 9) and (10) has three complex pairs roots, whose equal imaginary parts in magnitude and only one real number represents the position of corresponding collinear point.The intersections of curves for Figure 1 with horizontal axis represent the three collinear libration points under the effect of oblateness.

The Motion around Collinear Libration Points
To study the motion of an infinitesimal neighborhood around the libration points the variationally variables ( ) , , where, , , .
Then characteristic equation corresponding to Equations ( 14) and ( 15) is [8] ( ) ( ) where i A and i β are constant coefficients.and ( ) To get the values of constants i A and i β .Let 0 0 0 , , ξ η ξ  and 0 η be the initial coordinates and components of velocity then Equations ( 17) and (18) give at t = 0 ( ) Then Equations ( 17) and ( 18 this system using Implist Runge Kutta method.Figure 5 displays the phase space for the motion near L1, also Figure 6 and Figure 7 display the phase space near L2 and L3 for the Earth-Moon-spacecraft system.Table 1 displays the collinear libration points and the eigen values at each point when the coefficient of oblateness A = 0, and A = 0.001.

Conclusion
Periodic orbits around collinear points in the restricted three-body problem have been studied under the effect of oblateness due to the bigger primary which enables the uses of these effects in the space missions.The obtained Lissijous orbit is one aim of the maneuvers through the path of any space craft.

Figure 1 .
Figure 1.The positions of collinear libration points.
.Ibrahim et al.Because of all the libration points are in-plane, the partial derivatives containing z-components are vanished.Therefore, Equations (11) through (13) roots of the characteristic Equation (16) are found to be , λ and s are real.Let the variational of elements depend on time given by