Computational Studies in the Few-Body Problem : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Applied Mathematics at Massey University, Albany, New Zealand
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2019
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Massey University
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Abstract
In this research we investigate the gravitational four-body problem which describes the motion in the system of four stars moving under their mutual gravitational forces. It involves studies of the dynamics of few-body systems and finding periodic orbits, and subsequently an analysis of their stability. One of the challenges of the problem is the necessity of using regularisation algorithms in order to avoid singularities when there is a possibility of collision or close encounter between stars.
One of the featured solutions of the problem is the collinear Schubart orbit discovered for systems of three [59] and four [70] bodies. This orbit has been shown to form families of collinear and planar orbits for three bodies by Henon [32]. Sweatman discovered Schubart orbits in the collinear symmetric four-body problem [70, 71].
In this work we generate the families of Schubart orbits starting from the planar orbit obtained by Sweatman [73]. Utilising the symmetries of the four-body Schubart orbits, we solve the Caledonian symmetric four-body problem (CS4BP). Initially we consider the case of equal masses. This is later extended to the case of pairwise symmetric masses.
The family is parametrised by two parameters: the mass of the outer bodies and the distance between the two closest non-symmetric bodies. The collinear orbits are collisional, but there are no collisions when the orbits evolve to planar motion.
The planar family of pairwise symmetric masses is bounded by the line of the collinear Schubart orbits. Within its boundaries, there are four regions separated by two special types of orbits present in the family: the equal-mass orbits and the double choreography orbits. Two of the regions are symmetrical to the other two.
We perform a linear stability analysis of the discovered solutions both in and out of the plane. We also distinguish the influence of symmetrical and non-symmetrical perturbations on the orbits. An algorithm for the orbit search and orbital stability analysis is presented.