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Advances in Astronomy
Volume 2016, Article ID 9897681, 11 pages
http://dx.doi.org/10.1155/2016/9897681
Research Article

Planar Central Configurations of Symmetric Five-Body Problems with Two Pairs of Equal Masses

1Department of Mathematics, University of Ha’il, P.O. Box 2440, Ha’il 81451, Saudi Arabia
2Abu Dhabi Men’s College, Higher Colleges of Technology, P.O. Box 25035, Abu Dhabi, UAE
3Applied Mathematics and Sciences, Khalifa University of Science Technology and Research, P.O. Box 127788, Abu Dhabi, UAE

Received 2 November 2015; Revised 11 January 2016; Accepted 18 January 2016

Academic Editor: Elmetwally Elabbasy

Copyright © 2016 Muhammad Shoaib et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study central configuration of a set of symmetric planar five-body problems where the five masses are arranged in such a way that , , and are collinear and , , and are collinear; the two sets of collinear masses form a triangle with at the intersection of the two sets of collinear masses; four of the bodies are on the vertices of an isosceles trapezoid and the fifth body can take various positions on the axis of symmetry both outside and inside the trapezoid. We form expressions for mass ratios and identify regions in the phase space where it is possible to choose positive masses which will make the configuration central. We also show that the triangular configuration is not possible.