# Effect of the Planetesimal Belt on the Dynamics of the Restricted Problem of 2 + 2 Bodies
## 1 Introduction  
In the field of celestial mechanics, the circular restricted three-body problem (CRTBP), where an infinitesimal body moves under the influence of two primaries without affecting the motion of the primaries, has been extensive used. For details and investigations, see [1, 2, 3, 4]. The dynamics of the combined system of two primaries and two minor bodies, called the restricted 2 + 2 body problem (R2+2 BP), was analyzed in [5, 6]. In this model, two equations of the motion of the CRTBP are coupled with each other by mutual gravitational interactions. This model, or the dual-satellite-like model, helps with studying the dynamic behavior of binary asteroids in the presence of two primaries. [PERSON] studied R2+2 BP and found 14 pairs of equilibrium points [5]. The study of R2+2 BP with different disturbing forces produced more precise and accurate data about the system's dynamic behavior. Many researchers incorporated different effects of R2+2 BP to analyze the perturbed dynamics of the system. For example, [PERSON] et al. studied the generalized R2+2 BP by considering the second primary as a straight segment, and showed that length parameter has a subsequent effect on the location of all equilibrium points [7]. [PERSON] and [PERSON] [8] found R2+2 BP dynamics in the presence of the photo gravitational effect, and [PERSON] [9] studied the impact of oblate in R2+2 BP. The families of a periodic orbit in R2+2 BP were discussed by [PERSON] [10], who found that the orbits are stable. The stability of R2+2 BP was evaluated by [PERSON] and [PERSON] [11], who showed that the integral of the system, which is similar to RTBP, does not result in hill stability. The restricted 2 + 2 problem with a homogeneous axis-symmetric ellipsoid was described by [PERSON] [12], who also found 16 solutions in the neighborhood of triangular equilibrium points. A ring-shaped disk-like region formed from dust, comets, asteroids, etc., in the space having considerable mass exerts some gravitational force and affects the dynamic behavior of infinitesimal bodies. In our solar system, the asteroid belt and Kuiper belt are dust-belt-like structures. Dust-belt-like structures are also present in the Proxima Centauri system. Many researchers have studied the effect of the asteroid belt in CRTBP [13; 14; 15; 16], and found that these perturbations exhibit significant changes in the equilibrium position. In this paper, we investigate some new aspects of R2+2 BP, along with the disk-like belt effect on the potential function; as such, we found the change in the equilibrium positions. Moreover, we analyze the variation in the distance of the equilibrium location of a minor body \(P_{1}\) and \(P_{2}\) for \(\mu_{1}=\mu_{2}\) from its neighboring equilibrium point \(L_{i},i=1,2,3,4,5\)) versus mass ratio \(\mu\). The effects of perturbed equilibrium points are compared with those of the unperturbed 2 + 2 body problem. We discuss the stability of the perturbed system with the help of eigenvalues for the particle \(P_{1}\). In general, the restricted 2 + 2 bodies problem is formulated as a credible model to show the capture of small bodies by a planet. In particular, two primaries are considered to revolve in a circular mutual orbit and two infinitesimal bodies, where neither of them affects the primaries' motion. If the small bodies are temporarily captured in the Hill sphere of the smaller primary, they may become close enough to each other to exchange energy so that one of them becomes regularly and permanently captured. The aforementioned descriptions are considered the major applications of the restricted 2 + 2 bodies problem, which motivated us to study a more generalized model for this problem. This paper is organized as follows: In Section 2, we formulate the restricted 2 + 2 body problem in the presence of the planetesimal belt effect. In Section 3, the variation in equilibrium points against mass ratio \(\mu\) is described and we compare the perturbed equilibrium points with unperturbed equilibrium points.