A STUDY ON PERIODIC ORBITS IN THE GENERALIZED RESTRICTED THREE-BODY PROBLEM
A STUDY ON PERIODIC ORBITS IN THE GENERALIZED RESTRICTED THREE-BODY PROBLEM
No Thumbnail Available
Date
2017-10
Authors
GYEGWE, JESSICA MRUMUN
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
In the first problem, a modification of the restricted three-body problem has been considered where the primary (more massive body) is a triaxial rigid body and the secondary (less massive body) is an oblate spheroid and study periodic motions around the collinear equilibrium points. The locations of these points are firstly determined for ten combinations of the parameters of the problem. In all ten cases, the collinear equilibrium points are found to be unstable, as in the classical problem, and the Lyapunov periodic orbits around them have been computed accurately by applying known corrector-predictor algorithms. An extensive study on the families of three-dimensional periodic orbits emanating from these points has also been done. To find suitable starting points, for all the computed families, semi-analytical solutions have been obtained, for both two- and three-dimensional cases, around the collinear equilibrium points using the Lindstedt-Poincaré method. Finally, the stability of all computed periodic orbits has been studied. The second problem presents a third order analytic approximation solution of Lyapunov orbits around the collinear equilibrium points in the planar restricted three-body problem by utilizing the Lindstedt Poincaré method. The primaries are oblate bodies and sources of radiation pressure. The theory has been applied to the binary -Centuari system in six cases. Also, the positions of the collinear equilibrium points have been determined numerically and the effects of the parameters concerned with these equilibrium points shown graphically.
In the third problem, an investigation of three-dimensional periodic orbits and their stability emanating from the collinear equilibrium points of the restricted three-body problem with oblate and radiating primaries is presented. A numerical simulation is done by using five binary systems: Sirius, Procyon, Luhman 16, -Centuari and Luyten 726-8. Firstly, based on the topological degree theory, the total number of the collinear equilibrium points for the five binary systems were obtained and then, their positions were determined numerically. The linear stability of these equilibrium points was also examined and found to be unstable in the Lyapunov sense. An analytical approximation of three-dimensional periodic solutions around them was established via the Lindstedt-Poincaré local analysis. Finally, using the analytical solution to obtain starting orbits, the families of three-dimensional periodic orbits emanating from these equilibria have been continued numerically. Additionally, the collinear equilibrium points and periodic motion around them are studied in the framework of the restricted three-body problem where the two primaries are triaxial rigid bodies which emit radiation in the fourth problem. Firstly, the positions and stability of the collinear equilibria are studied for HD 191408, Kruger 60 and HD 155876 binary systems. Then, the planar and three-dimensional periodic motion about these points is considered. The study includes both semi-analytical and numerical determination of these motions. It is found that all families of planar periodic orbits emanating from these points terminate with asymptotic periodic orbits at the triangular equilibrium points while the corresponding families of three-dimensional periodic orbits terminate with planar periodic orbits.
Description
A DISSERTATION SUBMITTED TO THE SCHOOL OF POSTGRADUATE STUDIES, AHMADU BELLO UNIVERSITY, ZARIA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE AWARD OF DOCTOR OF PHILOSOPHY IN MATHEMATICS
Keywords
STUDY,, PERIODIC,, ORBITS,, GENERALIZED,, RESTRICTED,, THREE-BODY,, PROBLEM