Periodic and Secular Solutions in the Restricted Three–Body Problem under the Effect of Zonal Harmonic Parameters

Authors

Elbaz I. Abouelmagd, Mathematics Department, Faculty of Science and Arts , University of Jeddah, Saudi Arabia.\\ National Research Institute of Astronomy and Geophysics, Cairo, Egypt.\\ Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.Follow
M. S. Alhothuali, Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.Follow
Juan L. G. Guirao, Departamento de Matem´atica Aplicada y Estad´ıstica. Universidad Polit´ecnica de Cartagena, Hospital de Marina, 30203–Cartagena, Regi´on de Murcia, SpainFollow
H. M. Malaikah, Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.Follow

Author Country (or Countries)

Saudi Arabia.

Abstract

The aim of the present work is to study the periodic structure of the restricted three–body problem considering the effect of the zonal harmonics J2 and J4 for the more massive body. We show that the triangular points in the restricted three–body problem have long or short periodic orbits in the range 0 ≤ µ < µc. We also present a graphical analysis for the variations of the angular frequencies for the long and short periodic orbits computing explicitly the expressions of the lengths of the semi–major and semi–minor axes and determining the orientations of the principal axes for the ellipses that represent periodic orbits. Moreover, the secular solution when µ = µc is stated and it is proved that the triangular points have periodic orbits in this case too. This model has special significance in space missions either to place telescopes or for dispatching satellites and exploring vehicles.

Recommended Citation

I. Abouelmagd, Elbaz; S. Alhothuali, M.; L. G. Guirao, Juan; and M. Malaikah, H. (2015) "Periodic and Secular Solutions in the Restricted Three–Body Problem under the Effect of Zonal Harmonic Parameters," Applied Mathematics & Information Sciences: Vol. 09: Iss. 4, Article 1.
Available at: https://digitalcommons.aaru.edu.jo/amis/vol09/iss4/1