# Effect of the Planetesimal Belt on the Dynamics of the Restricted Problem of 2 + 2 Bodies
## 3 Equilibrium Points
### Noncollinear Equilibrium Points
The positions of the noncollinear equilibrium points were calculated numerically and are shown in Tables 7-10 for \(\mu_{1}=10^{-10}\), \(\mu_{2}=10^{-12}\), \(M_{b}=3.7\times 10^{-7}\), and \(T=0.11\). Table 7 shows the perpendicular equilibrium points, where the coordinates were taken as \((PxL_{4+}^{P_{i}},PyL_{4-}^{P_{i}})\) and \((PxL_{4+}^{P_{i}},\,PyL_{4+}^{P_{i}})\), \(i=1,2\). We observed that the coordinates \(PxL_{4+}^{P_{i}}\), \(PyL_{4+}^{P_{i}}\), and \(PxL_{4-}^{P_{i}}\) increased, and \(PyL_{4-}^{P_{i}}\) decreased as we increased the value of \(\mu\). The inline equilibrium points \((lxL_{4-}^{P_{i}},lyL_{4-}^{P_{i}})\) and \((lxL_{4+}^{P_{i}},lyL_{4+}^{P_{i}})\) are shown in Table 8.  
Furthermore, we observed that \(lxL_{4-}^{P_{i}},\,lyL_{4-}^{P_{i}}\), and \(lxL_{4+}^{P_{i}}\) increased as \(\mu\) increased, whereas \(lyL_{4+}^{P_{i}}\) decreased as \(\mu\) increased. Table 9 represents the perpendicular equilibrium points near \(L_{5}\), and \((PxL_{5+}^{P_{i}},PyL_{5+}^{P_{i}})\) and \((PxL_{5-}^{P_{i}},\,PyL_{5-}^{P_{i}})\) are coordinates in which \(PxL_{5-}^{P_{i}}\), \(PyL_{5-}^{P_{i}}\), and \(PxL_{5+}^{P_{i}}\) increased and \(PyL_{5+}^{P_{i}}\) decreased as \(\mu\) increased. Finally, the coordinates of the inline equilibrium points are shown in Table 10. \((lxL_{5+}^{P_{i}},\,lyL_{5+}^{P_{i}})\) and \((lxL_{5-}^{P_{i}},\,lyL_{5-}^{P_{i}})\)  
Figure 5: Distance versus \(\mu\) in the presence of the planetesimal belt effect for noncollinear equilibrium points. (**a**) Perpendicular equilibrium points. (**b**) Inline equilibrium points. Figure 4:0 Comparison of distance versus \(\mu\) between perturbed (red line) and unperturbed (green line) noncollinear equilibrium points in R2+2 BP. (**a**) Perpendicular equilibrium points. (**b**) Inline equilibrium points. are the coordinates, and we observed that \(IxL_{5+}^{P_{1}}\), \(IyL_{5+}^{P_{1}}\)) and \((IxL_{5-}^{P_{1}}\)) increased and \(IyL_{5-}^{P_{1}}\) decreased as the value of \(\mu\) increased. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{\(\mu\)} & \(IxL_{4-}^{P_{1}}\) & \(IyL_{4-}^{P_{1}}\) & \(IxL_{4+}^{P_{1}}\) & \(IyL_{4+}^{P_{1}}\) \\  & \(IxL_{4-}^{P_{1}}\) & \(IyL_{4-}^{P_{1}}\) & \(IxL_{4+}^{P_{1}}\) & \(IyL_{4+}^{P_{1}}\) \\ \hline
0.0001 & \(-\)0.49978629067686 & 0.86595962036838 & \(-\)0.50001370932314 & 0.86609090745337 \\  & \(-\)0.48852906768613 & 0.85946090966168 & \(-\)0.51127093231387 & 0.87258961816007 \\
0.0010 & \(-\)0.49894714879015 & 0.86599478074186 & \(-\)0.49905285120985 & 0.86605574695864 \\  & \(-\)0.49371487901505 & 0.86297695301132 & \(-\)0.50428512098495 & 0.86907357468918 \\
0.0100 & \(-\)0.48997516675820 & 0.86601107095682 & \(-\)0.49002483324180 & 0.86603945554077 \\  & \(-\)0.48751667582044 & 0.86460603405128 & \(-\)0.49248332417956 & 0.86744449244630 \\
0.1000 & \(-\)0.39998667432169 & 0.86601844124247 & \(-\)0.400013325678307 & 0.86603207428765 \\  & \(-\)0.39866743216923 & 0.86534360550600 & \(-\)0.40133256783077 & 0.86670691002412 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: In-ine equilibrium solution of the restricted 2 + 2 body problem near \(L_{4}\).