The Restricted Six-Body Problem with Stable Equilibrium Points and a Rhomboidal Configuration

Abstract

We explore the central configuration of the rhomboidal restricted six-body problem in Newtonian gravity, which has four primaries m i (where i = 1 , … 4 ) at the vertices of the rhombus a , 0 , − a , 0 , 0 , b , and 0 , − b , respectively, and a fifth mass m 0 is at the point of intersection of the diagonals of the rhombus, which is placed at the center of the coordinate system (i.e., at the origin 0,0 ). The primaries at the rhombus’s opposite vertices are assumed to be equal, that is, m 1 = m 2 = m and m 3 = m 4 = m ˜ . After writing equations of motion, we express m 0 , m , and m ˜ in terms of mass parameters a and b . Finally, we find the bounds on a and b for positive masses. In the second part of this article, we investigate the motion and different features of a test particle (sixth body m 5 ) with infinitesimal mass that moves under the gravitational effect of the five primaries in the rhomboidal configuration. All four cases have 16, 12, 20, and 12 equilibrium points, with case-I, case-II, and case-III having stable equilibrium points. A significant shift in the position and the number of equilibrium points was found in four cases with the variations of mass parameters a and b . The regions for the possible motion of test particles have been discovered. It has also been observed that as the Jacobian constant C increases, the permissible region of motion expands. We also have numerically verified the linear stability analysis for different cases, which shows the presence of stable equilibrium points.