Elliptic concentric restricted five-body problem: application to binary star systems with circumbinary planets

This study explores the elliptic restricted five-body problem involving two pairs of primary bodies. The first pair, identical in shape and size, has a greater mass than the second pair, which is also identical in shape and size. These four primary bodies move along elliptical orbits around their common center of mass. The system features seven equilibrium points within the orbital plane of the primaries: two symmetric pairs of collinear points, a symmetric pair of non-collinear points, and one located at the center of mass. The influence of the mass parameter and eccentricity parameter on the positions of these equilibrium points is investigated through numerical and graphical analyses. Stability analysis reveals that only the non-collinear equilibrium points are stable. Notably, for a specific value of the mass parameter, there exists a range of the eccentricity parameter within which the non-collinear equilibrium points exhibit linear stability. Time series and phase-space analyses confirm the nature of particle motion near both stable and unstable equilibrium configurations. Additionally, effective potential contours are used to construct orbital safety zones, transitional boundaries, and hazard regions. The theoretical results are applied to circumbinary systems such as Kepler-34 and Kepler-35, providing insight into possible equilibrium configurations and dynamically safe regions for test particle motion. This study offers a generalized framework for modeling multi-body gravitational systems and contributes to the understanding of orbital dynamics in astrophysical contexts.