Three families of 5-body central configurations in the plane

Abstract

We study the planar 5-body central configurations with different masses in the plane, in the limit cases when one of the masses is very large or very small with respect to the other four different positive masses. The case of a very small mass reduces to find the planar central configuration relative position of a satellite with respect to a computed 4-body planar central configuration. We will numerically find several examples of nine different positions of the satellite allowed by each example of the 4-body central configuration. The case of a large mass leads us to the co-orbital configuration with the very large mass at the center and the other four different positive masses on a circle. This case leads for a computed configuration to a two-dimensional plane sector in the positive mass space compatible with each geometrical configuration. The third studied case is a kite configuration with three different positive masses on the axis of symmetry and two equal masses located symmetrically with respect to that axis on a perpendicular line; we give new cases where a two-dimensional plane sector in the positive mass space is compatible with the geometric configuration. Algorithms for computation are rather explicit.