STABILITY ABOUT LIBRATION POINTS FOR RESTRICTED FOUR-BODY PROBLEM

Ismail, M. N.; Ibrahim, A. H.; Zaghrout, A. S.; Younis, S. H.; Elmalky, F. S.; El-Masry, L. E.

doi:10.21608/absb.2019.86756

STABILITY ABOUT LIBRATION POINTS FOR RESTRICTED FOUR-BODY PROBLEM

Document Type : Original Article

Authors

¹ astronomy, faculty of science,al-azhar university

² Astronomy and Meteorology Department, Faculty of Science, Al-Azhar University, Cairo, Egypt.

³ Mathematics Department, Faculty of Science (Girls), Al-Azhar University, Cairo, Egypt

⁴ Mathematics Department, Faculty of Science (Girls), Al-Azhar University, Cairo, Egypt.

⁵ Ph.D student in Math. Dep., Faculty of Science (Girls), Al-Azhar University, Cairo.

10.21608/absb.2019.86756

Abstract

In this work, the Restricted Four-Body Problem is formulated in Hamiltonian form. The canonical form for the system is obtained which represents the equations of motion. The collinear libration points are obtained, we have five collinear libration points. The non-collinear libration points are found which are three non collinear libration points, they are obtained for different angles between the sight of Sun and the plane of Earth-Moon. The periodic orbits around each of these libration points are studied using two methods. The first method depends on the reduction of order of differential equations and the second method depends on the Eigen values of the characteristic equation. Two codes of MATHEMATICA are constructed to apply these two methods on the Sun- Earth-Moon-Spacecraft. The Poincare sections are obtained using the first method, these sections are used to illustrate the intersect points of the trajectories with the plane perpendicular to the plane of motion about each of the collinear libration points. Mirror symmetry is explored about each of these points. The Lyapunov orbits, and the Lissajous orbits about each of the collinear libration points are the results obtained by the second method. The eccentricities and the periods of each orbit are obtained. This study illustrates that the motion about the libration point L2 is more stable than the motion about any other collinear libration points.

Keywords