The concentric Sitnikov problem: Circular case

A new configuration in the sense of concentric Sitnikov problem is introduced in this paper. This configuration consists of four primaries Pi(i=1,2,3,4) of masses mi, respectively. Here, all the primaries are placed on a straight line and moving in concentric circular orbits around their common center of mass under the restrictions of masses as m1=m2=m, m3=m4=ḿ and m>ḿ. The objective of the present manuscript is to study the existence of equilibrium points and their linear stability, first return map, periodic orbits, and N-R BoA in the proposed model. Under the restriction of mass parameter 0<μ́<1/4, it is found that there exist three equilibrium points and all the equilibrium points are linearly unstable for all values of mass parameter 0<μ́<1/4. Graphically, the nature of orbits have been examined with the help of first return map. The families of periodic orbits of infinitesimal mass m5 around the primaries and equilibrium points have been obtained for different values of mass parameter μ́. Finally, we explore the Newton–Raphson Basins of Attraction (N-R BoA), related to the equilibrium points in the concentric problem of Sitnikov.