Equilibrium states of a rigid dumbbell test mass in the Sitnikov circular-restricted (2+2)-body problem

Abstract

The three-body problem is a classical problem in celestial mechanics that has been widely studied even up to this day. These systems are chaotic, and as such, there are no general solutions. As a result, we turn to the restricted three-body problem, wherein the third mass is considered to be an infinitesimal point mass, and hence, does not affect the motion of the two other masses. Today, there are different variations to the problem, and as such, a more general statement, the restricted (n+ν)-body problem is used, where n is the number of primaries, and ν is the number of infinitesimal masses. In this paper, we search for the equilibrium states of a rigid dumbbell in the Sitnikov circular-restricted (2+2)-body problem (Sitnikov CR2+2BP). The dumbbell is composed of two infinitesimal bodies, of equal masses, connected by a massless, rigid rod, interacting gravitationally with 2 massive objects, referred to as the primaries, in the framework of Newtonian gravity. In the Sitnikov CR(n+2)-BP, we found oscillatory and equilibrium states. Moreover, we found that when the rod exceeds a critical length ℓ_critical = 1/√2, bounded states exist such that the center of mass of the dumbbell does not cross the plane of the primaries. Finally, we also obtained the criterion for the initial velocity ζ˙₀ that results in unbounded states.