Stability Analysis of the Rhomboidal Restricted Six-Body Problem

We discuss the restricted rhomboidal six-body problem (RR6BP), which has four positive masses at the vertices of the rhombus, and the fifth mass is at the intersection of the two diagonals. These masses always move in rhomboidal CC with diagonals $2a$ and $2b$ . The sixth body, having a very small mass, does not influence the motion of the five masses, also called primaries. The masses of the primaries are $m_1=m_2=m_0=m$ and $m_3=m_4=\tilde{m}$ . The masses $m$ and $\tilde{m}$ are written as functions of parameters $a$ and $b$ such that they always form a rhomboidal central configuration. The evolution of zero velocity curves is discussed for fixed values of positive masses. Using the first integral of motion, we derive the region of possible motion of test particle $m_5$ and identify the value of Jacobian constant $C$ for different energy intervals at which these regions become disconnected. Using semianalytical techniques, we show the existence and uniqueness of equilibrium solutions on the axes and off the axes. We show that, for $b\in \left(1/\sqrt{3},1.1394282249562009\right)$ , there always exist 12 equilibrium points. We also show that all 12 equilibrium points are unstable.