# Effect of the Planetesimal Belt on the Dynamics of the Restricted Problem of 2 + 2 Bodies
## 3 Equilibrium Points
### Collinear Equilibrium Points  
The collinear equilibrium points appear on the \(X\)-axis. These points can be calculated using Equations (3) and (6) with conditions \(y_{1}=0\) and \(y_{2}=0\). Then, we have,  
\[ \begin{array}{c}\mu_{1}\Big{(}nx_{1}-\frac{(1-\mu)(x_{1}-\mu)}{|x_{1}-\mu|^{3 }}-\frac{\mu(x_{1}-\mu+1)}{|x_{1}-\mu+1|^{3}}-\frac{\mu_{2}(x_{1}-x_{2})}{|x_{ 1}-x_{2}|^{3}}\Big{)}\\ \qquad\qquad\qquad-\frac{M_{b}}{x_{1}^{3}}\bigg{(}1-\frac{3}{2}\frac{T^{2}}{x_ {1}^{2}}+\frac{15}{8}\frac{T^{4}}{x_{1}^{4}}\bigg{)}=0,\end{array}  \tag{10}\]  
\[ \begin{array}{c}\mu_{1}\Big{(}nx_{1}-\frac{(1-\mu)(x_{1}-\mu)}{|x_{1}-\mu|^{ 3}}-\frac{\mu(x_{1}-\mu+1)}{|x_{1}-\mu+1|^{3}}-\frac{\mu_{2}(x_{1}-x_{2})}{|x_ {1}-x_{2}|^{3}}\Big{)}\\ \qquad\qquad\qquad\qquad-\frac{M_{b}}{x_{1}^{3}}\bigg{(}1-\frac{3}{2}\frac{T^{2 }}{x_{1}^{2}}+\frac{15}{8}\frac{T^{4}}{x_{1}^{4}}\bigg{)}=0.\end{array}  \tag{11}\]  
The solution of the present model obtained by the perturbation method was proposed by [5]. The solutions of Equations (10) and (11) are \(x_{1}\) and \(x_{2}\), expressed in a power series with small parameters [5; 8]. \[x_{1} = L_{j}+a_{11}\varepsilon_{2}+a_{12}\varepsilon_{2}^{2}+\ldots,\] \[x_{2} = L_{j}+a_{21}\varepsilon_{1}+a_{22}\varepsilon_{1}^{2}+\ldots,\]where \(\epsilon_{i}=\dfrac{\mu_{i}}{\left(\mu_{1}+\mu_{2}\right)^{\frac{2}{3}}}\) and \(L_{j};j=1,2,3\) are the collinear Lagrangian points of CRTBP. Hence, Equations (10) and (11) can be written as:  
\[a_{11}W_{xx}^{0}\epsilon_{2}-\dfrac{\mu_{2}(x_{1}-x_{2})}{|x_{1} -x_{2}|^{3}}=0, \tag{12}\] \[a_{21}W_{xx}^{0}\epsilon_{1}-\dfrac{\mu_{1}(x_{2}-x_{1})}{|x_{1} -x_{2}|^{3}}=0, \tag{13}\]  
where  
\[W=\dfrac{1}{2}n^{2}\left(x^{2}+y^{2}\right)+\dfrac{1-\mu}{r_{1}}+\dfrac{\mu}{ r_{2}}+\dfrac{M_{b}}{\left(R^{2}+T^{2}\right)^{\frac{1}{2}}}. \tag{14}\]  
Using Equations (12) and (13), we have \(a_{11}=-a_{21}\) and \(a_{11}=\pm\dfrac{1}{\left(W_{xx}\right)^{\frac{1}{3}}}\). Thus, the equilibrium points of R2+2 BP are:  
\[L_{j\pm}^{P_{1}}=L_{j}\pm\dfrac{\mu_{2}}{\left[(\mu_{1}+\mu_{2}) ^{2}W_{xx}\right]^{\frac{1}{3}}}, \tag{15}\] \[L_{j\pm}^{P_{2}}=L_{j}\pm\dfrac{-\mu_{1}}{\left[(\mu_{1}+\mu_{2} )^{2}W_{xx}\right]^{\frac{1}{3}}}. \tag{16}\]  
Equations (15) and (16) yield paired equilibrium positions near the collinear equilibrium points \(L_{j};j=1,2,3\) of R2+2 BP. We denote the collinear equilibrium points as \(L_{j\pm}^{P_{1}}\), \((i=1,2,j=1,2,3)\). Superscript \(P_{i}\) denotes the equilibrium points for an infinitesimal body; \(P_{i}\), \(i=1,2\) denotes the equilibrium points near \(L_{1},L_{2}\), and \(L_{3}\). Subscript \(j\pm\) denotes the relative position of equilibrium points, where \(+\) indicates the right and \(-\) indicates the left position with respect to \(L_{j}\); \(j=1,2,3\). Figure 2 shows the positions of the equilibrium points when \(\mu=0.1\), \(\mu_{1}=0.01\), and \(\mu_{2}=0.001\) with the planetesimal belt effect \(M_{b}=3\times 10^{-7}\) and parameter \(T=0.11\). The positions of the equilibrium points are shown as \(P_{1}\) (green) and \(P_{2}\) (red). In the presence of the planetesimal belt effect, we can observe that the first equilibrium position of \(P_{1}\) \(\left(L_{1-}^{P_{1}}\right)\) is to the left of \(L_{1}\). The equilibrium position of \(P_{2}\) \(\left(L_{1+}^{P_{2}}\right)\) is to the right of \(L_{1}\). Similarly, the second equilibrium position of \(P_{1}\), i.e., \(\left(L_{1+}^{P_{1}}\right)\), is to the right of \(L_{1}\). The equilibrium position of \(P_{2}\), i.e., \(\left(L_{1-}^{P_{2}}\right)\), is to the left of \(L_{1}\). As such, we also found four collinear equilibrium points near \(L_{2}\) and \(L_{3}\). The positions of the collinear equilibrium points were also calculated numerically, as shown in Tables 1-3 for \(\mu_{1}=10^{-10}\), \(\mu_{2}=10^{-12}\), \(M_{b}=3.7\times 10^{-7}\), and \(T=0.11\) with the variation in \(\mu\). Table 1 shows that when \(\mu\) increases \(L_{1\pm}^{P_{1}}\), \(L_{1\pm}^{P_{2}}\) decreases.