INVESTIGATION OF THE STABILITY OF EQUILIBRIUM POINTS IN THE RELATIVISTIC RESTRICTED THREE-BODY PROBLEM WITH PERTURBATIONS

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Date
2016-09
Authors
NAKONE, BELLO
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Abstract
In the work by Bhatnagar and Hallan (1998), linear stability of the relativistic triangular 4 5 L and L points was studied and it was shown that these points were unstable for the whole range 1 0 2    despite the well-known fact that the non-relativistic 4 L and 5 L are stable for 0    , where 0  is the Routh critical mass ratio. The same problem was later investigated by Douskos and Perdios (2002) and Ahmed et al. (2006) and they obtained two different ranges of mass ratios in which the relativistic triangular points are linearly stable in contradiction with the result of Bhatnagar and Hallan (1998). In this thesis we reconsider and generalize the problem investigated by these authors in that perturbations in the Coriolis and centrifugal forces, radiation pressure, oblateness and triaxiality factors of the primaries have been considered in our investigation. The locations of equilibrium points are obtained and their stability are analyzed by using variational method and Lyapunov‟s criteria. The triangular points of the relativistic three-body problem (R3BP) are studied from various aspects of perturbations such as oblateness, triaxiality and radiation pressure of the primaries as well as the small perturbations in the centrifugal and Coriolis forces. It is found that the locations of the triangular points are affected by the asphericity of the primaries, the relativistic terms and a small change in the centrifugal force. It is also found that the triangular points are stable for c 0    and unstable for 2 1     c , where c  is the critical mass parameter depending on the perturbation parameters and relativistic terms. It is further found that the Coriolis force has stabilizing tendency, while the centrifugal force, radiation pressure forces, oblateness, triaxiality of the primaries and relativistic terms have destabilizing effects. The motion of an infinitesimal mass near the collinear equilibrium points when the smaller primary is a triaxial body is also studied. It is observed that the positions of the collinear points are affected by the relativistic and triaxiality factors. The collinear points are found to remain unstable. Numerical studies in this connection with the Sun-Earth, Sun-Pluto and Earth-Moon systems have been carried out to show the relativistic and triaxiality effects. The motion of an infinitesimal mass near the collinear equilibrium points when the smaller primary is oblate is also investigated. The collinear points are found to be unstable. A numerical exploration in this connection, with some members of our solar system reveals that the locations of the collinear points 1 2 L , L are affected prominently by the relativistic factor in the absence of oblateness and they are also affected significantly by the oblateness factor in the absence of relativistic terms. It is also found that in most of the cases, the position of 3 L is negligibly affected by the relativistic and oblateness factors. More specifically, all parameters involved have no effect on the position of 3 L of the Sun-Mars system. The results show that the oblateness and relativistic factors have the same but separate effect on the position of 1 L of the Sun-Uranus system and have also the same effect on the position of 1 L of the Sun-Neptune system. It is also found that in the presence of relativistic terms, the effect of oblateness on the Sun-Planet pairs does not show physically. Also, the frequencies of the long and short orbit of the periodic motion, eccentricities, axes and the orientation of the orbits around the stable triangular points when the bigger primary is triaxial are determined and found to be affected by the triaxiality and relativistic effects. The results of this study generalise the classical relativistic restricted three-body problem (R3BP) and the results of Douskos and Perdios (2002) can be deduced from this study while the present results differ with the results of Bhatnagar and Hallan (1998) and differ also with the results of Ahmed et al. (2006).
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A THESIS SUBMITTED TO THE SCHOOL OF POSTGRADUATE STUDIES, AHMADU BELLO UNIVERSITY ZARIA, NIGERIA IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF DOCTOR OF PHILOSOPHY IN MATHEMATICS
Keywords
INVESTIGATION,, STABILITY,, EQUILIBRIUM POINTS,, RELATIVISTIC RESTRICTED THREE-BODY,, PERTURBATIONS,
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