STABILITY ABOUT LIBRATION POINTS FOR RESTRICTED FOUR-BODY PROBLEM
Ibrahim, A., Ismail, M., Ibrahim, A., Zaghrout, A., Younis, S., Elmalky, F., El-Masry, L. (2019). STABILITY ABOUT LIBRATION POINTS FOR RESTRICTED FOUR-BODY PROBLEM. EKB Journal Management System, 2019(12), 1-11. doi: 10.21608/absb.2019.86756
A. Ibrahim; M. N. Ismail; A. H. Ibrahim; A. S. Zaghrout; S. H. Younis; F. S. Elmalky; L. E. El-Masry. "STABILITY ABOUT LIBRATION POINTS FOR RESTRICTED FOUR-BODY PROBLEM". EKB Journal Management System, 2019, 12, 2019, 1-11. doi: 10.21608/absb.2019.86756
Ibrahim, A., Ismail, M., Ibrahim, A., Zaghrout, A., Younis, S., Elmalky, F., El-Masry, L. (2019). 'STABILITY ABOUT LIBRATION POINTS FOR RESTRICTED FOUR-BODY PROBLEM', EKB Journal Management System, 2019(12), pp. 1-11. doi: 10.21608/absb.2019.86756
Ibrahim, A., Ismail, M., Ibrahim, A., Zaghrout, A., Younis, S., Elmalky, F., El-Masry, L. STABILITY ABOUT LIBRATION POINTS FOR RESTRICTED FOUR-BODY PROBLEM. EKB Journal Management System, 2019; 2019(12): 1-11. doi: 10.21608/absb.2019.86756

STABILITY ABOUT LIBRATION POINTS FOR RESTRICTED FOUR-BODY PROBLEM

Al-Azhar Bulletin of Science
Article 7, Volume 2019, Issue 12, December 2019, Page 1-11  XML PDF (606.33 K)
Document Type: Original Article
DOI: 10.21608/absb.2019.86756
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Authors
A. Ibrahim1; M. N. Ismail email 2; A. H. Ibrahim1; A. S. Zaghrout3; S. H. Younis4; F. S. Elmalky4; L. E. El-Masry5
1Astronomy and Meteorology Department, Faculty of Science, Al-Azhar University, Cairo, Egypt.
2astronomy, faculty of science,al-azhar university
3Mathematics Department, Faculty of Science (Girls), Al-Azhar University, Cairo, Egypt
4Mathematics Department, Faculty of Science (Girls), Al-Azhar University, Cairo, Egypt.
5Ph.D student in Math. Dep., Faculty of Science (Girls), Al-Azhar University, Cairo.
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
Lissajous orbits; Poincare surface sections; Libration points; Stability of libration points; restricted four-body; motion near libration points
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