INVESTIGATION OF MOTION IN THE CIRCULAR RESTRICTED PROBLEMS OF THREE-BODIES UNDER PERTURBING FORCES WITH VARIABLE AND CONSTANT MASSES
INVESTIGATION OF MOTION IN THE CIRCULAR RESTRICTED PROBLEMS OF THREE-BODIES UNDER PERTURBING FORCES WITH VARIABLE AND CONSTANT MASSES
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Date
2016-01-19
Authors
ONI, LEKE SAMSON
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Abstract
This thesis investigates the motion of a test body around equilibrium points of the circular restricted problems of three-bodies using the model of the general restricted three-body problem (R3BP) and the Robe’s R3BP, and taking into account various chacterizations which include circumbinary disc, oblateness, triaxiality, radiation, deflections due to the Coriolis and centrifugal forces and to whether the masses are varying with time or constant. In the R3BP with variable masses, we assume that the primaries vary isotropically in accordance with the unified Mestschersky law (UML) and their motion governed by the Gylden-Mestschersky problem (GMP). We transform the non-autonomous equations of motion to the autonomized forms using the Mestschersky transformation (MT), the UML, the particular integral and solutions of the GMP. The equilibrium points (EPs) are examined and it is seen that due to the mass variations coupled with other perturbing forces; five collinear , two triangular and four out-of-plane EPs are found. The linear stability of the EPs is investigated. For the autonomized systems, the collinear and triangular EPs are stable while the out-of-plane EPs are unstable. We examine the periodic orbits around stable triangular EPs of the autonomized systems and found that the orbits can be elliptical, parabolic or hyperbolic in nature due to mass variations. The stability of the EPs of the non-autonomous equations is tested using the Lyapunov’s characteristic number and are seen to be unstable. In the Robe’s R3BP with variable masses, we consider effects of the buoyancy force, the gravitational attraction of the fluid and attraction of the second primary. In the first case, we assumed that the primaries vary in accordance with the UML law and their motion described by the GMP. The transformation from the non-autonomous to the autonomized equations is possible only when the first primary contains no fluid. In this regime, an EP at the center of the first primary and a pair of non-collinear EPs exists. When small change in the
Coriolis and centrifugal forces are introduced, every point on the line joining the primaries is an EP and a pair of non-collinear EP also exists. In the second case, the first primary is a fluid in the shape of a sphere with no shell and the masses of the three bodies vary with time. We use the Gelf’gat transformation to cast the non-autonomous system to the autonomized kind. An EP at the center, another near the center, circular points and a pair of non-collinear EP exists. The collinear EPs are stable while the non-collinear and circular EPs are unstable. The study is applied to the motion of a submarine in the Earth-Moon system. In the study of R3BP with constant masses in the presence of a circumbinary disc under different characterizations, it is seen that two additional collinear EPs are found depending on the mass parameter of the system and the mass of the disc. Our numerical evidence reveals that the existence of the additional EPs depend on the system under study. In the range of linearly stable triangular EPs, we examine the periodic orbits around these points and found that the orbits are ellipses. We test their stability using the Poincare Surfaces of Section (PSS) and found them to be stable. Our problem can be used to study the long-term motion of satellites, submarines and planets in binary systems. These bodies are taken to be test particles moving in the field of circular binary system. A simple question of celestial mechanics is, if the EPs of these bodies near a binary star system, loosing mass isotropically or conserving mass, exist, how long can they keep the bodies from escaping? The determination of ranges of semimajor axes may help to know when the body is likely to escape. The obtained EPs may be used in different problems of stellar dynamics, and also in other astrophysical applications.
Description
A THESIS 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
INVESTIGATION,, MOTION,, CIRCULAR RESTRICTED PROBLEMS,, BODIES,, PERTURBING FORCES,, VARIABLE,, CONSTANT MASSES,