Out-of-plane dynamics: a study within the circular restricted eight-body framework

This manuscript thoroughly explores the dynamics of a test particle around out-of-plane equilibrium points within the circular restricted eight-body problem. This particular scenario features a central primary emitting radiation, and it is a specific case derived from Kalvouridis and Hadjifotinou's analysis of Maxwell's ring problem in 2011. Our investigation uncovers two symmetrical out-of-plane equilibrium points denoted as E_1,2(0, 0, z₀), where z₀ is determined by the equation z₀ = ±a tanυ; υ = arcsin[(‒q/6)^1/3], with q falling within the range (‒6, 0). Here, a denotes the radius of the circular orbit of peripheral primaries around the radiating central primary, and q signifies the radiation factor due to the central primary. Significantly, for a critical radiation factor value, q_c = ‒3/√2, the equilibrium points E_1,2 precisely align along the z-axis on the sphere of radius a and centered at the central primary. Within the intervals of ‒6 < q < q_c and q_c < q < 0, equilibrium points E₁ and E₂ are situated outside and inside the mentioned sphere on the z-axis, respectively. Specifically, for q ≤ q_c, | z₀ | ≤ a, while for q > q_c, | z₀ | > a. The study further explores the linear stability of E_1,2. By analyzing characteristic curves derived from the variational equations of motion for infinitesimal mass around these equilibrium points, particularly for q values of ‒3/4, ‒3/√2, and ‒9√3/4, we observe that these out-of-plane equilibria, E_1,2, demonstrate linear instability. This insight provides a comprehensive understanding of the intricate dynamics in this specific multi-body problem. Finally, the research illustrates periodic orbits surrounding the out-of-plane equilibrium point for specific values of q.