# Effect of the Planetesimal Belt on the Dynamics of the Restricted Problem of 2 + 2 Bodies
## 3 Equilibrium Points
### Noncollinear Equilibrium Points  
The noncollinear equilibrium points of R2+2 BP can be found by solving Equations (3), (4), (6) and (7) with \(y_{1}\
eq 0\) and \(y_{2}\
eq 0\). The solution can be obtained by the power series perturbation method. Let \(x_{i}\) and \(y_{i}\), \(i=1,2\) be the solutions; then,  
\[x_{i} =xL_{j}+a_{i1}\varepsilon_{3-i}+a_{i2}\varepsilon_{3-i}^{2}+\ldots,\] \[y_{i} =yL_{j}+a_{i1}\varepsilon_{3-1}+a_{i2}\varepsilon_{3-i}^{2}+\ldots.\] \begin{table}
\begin{tabular}{c c c c c} \hline \(\mu\) & \(L_{3+}^{\mathbf{P_{i}}}\) & \(L_{3-}^{\mathbf{P_{i}}}\) & \(L_{3+}^{\mathbf{P_{i}}}\) & \(L_{3-}^{\mathbf{P_{i}}}\) \\ \hline
0.0002860 & 1.00011913765467 & 1.00004978229365 & 1.00011899908252 & 1.00018835444353 \\
0.0000310 & 1.00001290410545 & 0.99994354530304 & 1.00001276552642 & 1.00008212432883 \\
0.0000003 & 1.0000019376614 & 0.99993083453848 & 1.00000005518627 & 1.00006941441393 \\ \hline \end{tabular}
\end{table} 
Table 6: Equilibrium solution near \(L_{3}\) of different planetary systems. Figure 4: Comparison of distance versus \(\mu\) between perturbed (red line) and unperturbed (green line) collinear equilibrium points in R2+2 BP. **(a)** For \(L_{1}\) equilibrium point. **(b)** For \(L_{2}\) equilibrium point. **(c)** For \(L_{2}\) equilibrium point. **(d)** For \(L_{3}\) equilibrium point. Solving the above, we obtain  
\[x_{i}^{P}=\mu-\frac{1}{2}\pm\frac{\alpha_{j}(-1)^{i}\mu_{3-i}}{[(\mu_{1}+\mu_{2}) ^{2}(W_{xy}\alpha_{j}+W_{yy})]^{\frac{1}{3}}[1+\alpha_{j}^{2}]^{\frac{1}{2}}}, \tag{17}\]  
\[y_{i}^{p}=\frac{(-1)^{k+1}\sqrt{3}}{2}\pm\frac{\mu_{3-i}}{((\mu_{1}+\mu_{2})^{2 }(W_{xy}^{0}\alpha_{j}+W_{yy}^{0})^{\frac{1}{3}}(1+\alpha_{j}^{2})^{\frac{1}{2} }}, \tag{18}\]  
\[x_{i}^{I}=\mu-\frac{1}{2}\pm\frac{\alpha_{j}\mu_{3-i}}{((\mu_{1}+\mu_{2})^{2}( W_{xx}^{0}+\frac{W_{yy}^{0}}{\alpha_{j}}))^{\frac{1}{3}}(1+\frac{1}{\alpha_{j} ^{2}})^{\frac{1}{2}}}, \tag{19}\]  
\[y_{i}^{I}=\frac{(-1)^{k+1}\sqrt{3}}{2}\pm\frac{(-1)^{i+1}\mu_{3-i}}{[(\mu_{1}+ \mu_{2})^{2}(W_{xx}+\frac{W_{xy}}{\alpha_{j}})]^{\frac{1}{3}}[1+\frac{1}{ \alpha_{j}^{2}}]^{\frac{1}{2}}}, \tag{20}\]  
where, \(j=1,k=1\) at \(L_{4}\) and \(j=2,k=2\) at \(L_{5}\)  
\[\alpha_{1,2}=\frac{(-1)^{k+1}\pm(-1)^{k}\sqrt{1-12(\mu-0.5)}}{2\sqrt{3(\mu-0. 5)}}.\]  
Equations (17)-(20) represent the noncollinear equilibrium points in R2+2 BP. In this section, we use different notations to represent equilibrium points near \(L_{4}\) and \(L_{5}\). The prefixes \(x\) and \(y\) are used in \(L_{i\pm}^{P_{i}}\) to denote the \(X\) and \(Y\) coordinates of equilibrium points, respectively. The noncollinear equilibrium points can be distinguished as the perpendicular and inline equilibrium solutions in [9]. Again, the prefixes \(I\) and \(P\) are used for inline and perpendicular equilibrium points, i.e., \(IX_{4-}^{P_{i}}\) is the \(X\)-coordinate of the equilibrium point of the first infinitesimal body toward the origin. The distance versus \(\mu\) of the perpendicular and inline equilibrium positions to \(L_{4,5}\) with the planetesimal belt effect \(M_{b}=3.7\times 10^{-7}\), \(T=0.11\) and \(\mu_{1}=\mu_{2}=10^{-2}\) are shown in Figure 5a,b. In the case of the perpendicular equilibrium position \(d(L_{4,5},PL_{4,5\pm}^{P_{i}})\) increases as \(\mu\) increases, as shown in Figure 5a. In Figure 5b, in the case of the inline equilibrium position, the distance \(d(L_{4,5},IL_{4,5\pm}^{P_{i}})\) increases rapidly when \(\mu\) approaches \(\frac{1}{2}\). Figure 6a shows that \(d_{1}(L_{4,5},PL_{4,5\pm}^{P_{i}})>d_{2}(L_{4,5},PL_{4,5\pm}^{P_{i}})\), i.e., due to perturbation, the distance increases with respect to the unperturbed distance in the case of perpendicular noncollinear equilibrium points. Here, we use \(d_{1}\) and \(d_{2}\) for the perturbed and unperturbed distances, respectively. Furthermore, when the noncollinear equilibrium points are in the perpendicular position, \(d_{1}(L_{4,5},IL_{4,5\pm}^{P_{i}})>d_{2}(L_{4,5},IL_{4,5\pm}^{P_{i}})\), as shown in Figure 6b. Hence, we concluded that the distance increases due to the perturbation.