# Effect of the Planetesimal Belt on the Dynamics of the Restricted Problem of 2 + 2 Bodies
## 3 Equilibrium Points
### Collinear Equilibrium Points
In Tables 2 and 3, \(L_{2\pm}^{P_{1}}\), \(L_{2\pm}^{P_{2}}\), \(L_{3\pm}^{P_{1}}\), and \(L_{3\pm}^{P_{2}}\) increase with increasing \(\mu\). Moreover, we analyzed the effect of the planetesimal belt on R2+2 BP. The equilibrium positions of two minor bodies are the same if the masses are equal, i.e., \(\mu_{1}=\mu_{2}\). Consequently, the distance from \(L_{j}\) to \(L_{j\pm}^{P_{1}}\) for \(i=1,2\) is equal, i.e., if \(d(x,y)\) is the distance between points \(x\) and \(y\), then \(d(L_{j},L_{j\pm}^{P_{1}})=d(L_{j},L_{j\pm}^{P_{2}})\). Therefore, we plot the distance versus \(\mu\) in Figure 2 by considering \(M_{b}=3.7\times 10^{-7}\), \(T=0.11\) and \(\mu_{1}=\mu_{2}=10^{-2}\); in Figure 3, \(d(L_{1},L_{1\pm}^{P_{1}})\) increases as \(\mu\) increases in \(0<\mu<0.5\). Concurrently, \(d(L_{2},L_{2\pm}^{P_{1}})\), and \(d(L_{3},L_{3\pm}^{P_{1}})\) decrease as \(\mu\) increases from \(0\) to \(0.5\). Figure 3: Distance versus mass ratio \(\mu\) plot in the presence of the planetesimal belt effect for collinear equilibrium points. Figure 2: The position of equilibrium points when \(\mu=0.1\), \(\mu_{1}=0.01\), \(\mu_{2}=0.001\), \(M_{a}=3\times 10^{-7}\), and parameter \(T=0.11\). The positions of the primaries are \(M_{1}\) and \(M_{2}\), represented by an asterisk, and blue dot, respectively. \(L_{1}\), \(L_{2}\), \(L_{3}\), \(L_{4}\), and \(L_{5}\) with black dots are the Lagrangian points of CRTBP. The equilibrium points of R2+2 BP with the planetesimal belt effect are shown in green and red dots, for \(P_{1}\) and \(P_{2}\), respectively. In Figure 4, we compare the effect of planetesimal belt perturbation on the collinear equilibrium points with unperturbed collinear equilibrium points in consideration of a distance function. In Figure 4a-d, the red line shows the distance \(d_{1}(L_{j},L_{j\pm}^{P_{1}})\), \((j=1,2,3)\) and \((i=1,2)\) with the planetesimal belt effect, and the green line shows the distance \(d_{2}(L_{j},L_{j\pm}^{P_{1}})\) without the planetesimal belt effect. Here, \(d_{1}\) is used to show the effect on distance with planetesimal belt perturbation, whereas \(d_{2}\) is used for unperturbed collinear points. Figure 4a,d shows that \(d_{1}(L_{1},L_{1\pm}^{P_{1}})>d_{2}(L_{1},L_{1\pm}^{P_{1}})\) and \(d_{1}(L_{3},L_{3\pm}^{P_{1}})>d_{2}(L_{3},L_{3\pm}^{P_{1}})\) when \(0<\mu<0.5\). We used a step length \(h=0.001\) for the variation in \(\mu\). We found that \(d_{1}(L_{2},L_{2\pm}^{P_{1}})<d_{2}(L_{2},L_{2\pm}^{P_{1}})\) for \(\mu<0.153\) and \(d_{1}(L_{2},L_{2\pm}^{P_{1}})>d_{2}(L_{2},L_{2\pm}^{P_{1}})\) for \(\mu>0.153\), as shown in Figure 4a-c. Tables 4-6 show the collinear equilibrium points of the system: Sun-Saturn with the Kuiper belt (\(\mu=0.000286\), \(M_{b}=3.00\times 10^{-7}\), \(T=0.11\)), Sun-Mars with an asteroid belt (\(\mu=0.000003\), \(M_{b}=1.6\times 10^{-9}\), \(T=0.11\)), and the Proxima Centauri system with a dust disc (\(\mu=0.000031\), \(M_{b}=2.50\times 10^{-7}\), \(T=0.11\)), as described by the authors in [16; 23; 24] for the restricted 2 + 2 body problem having \(\mu_{1}=10^{-10}\) and \(\mu_{2}=10^{-12}\). Table 4 shows the equilibrium points near \(L_{1}\), Table 5 represents the equilibrium points near \(L_{2}\), and the equilibrium points near \(L_{3}\) are depicted in Table 6. \begin{table}
\begin{tabular}{c c c c c} \hline \(\mu\) & \(L_{1+}^{P_{1}}\) & \(L_{1-}^{P_{1}}\) & \(L_{1+}^{P_{1}}\) & \(L_{1-}^{P_{1}}\) \\ \hline
0.0001 & \(-1.03220039257577\) & \(-1.03264998998316\) & \(-1.03242743926651\) & \(-1.03242294329243\) \\
0.0010 & \(-1.0696877632157\) & \(-1.07014443189694\) & \(-1.06991838089976\) & \(-1.06991381421295\) \\
0.0100 & \(-1.14652917405563\) & \(-1.14700090922027\) & \(-1.14676740031377\) & \(-1.14676268296213\) \\
0.1000 & \(-1.25944627441467\) & \(-1.25995339005807\) & \(-1.25970236781459\) & \(-1.25969729665815\) \\ \hline \end{tabular}
\end{table} 
Table 1: Equilibrium solution of R2+2 BP near \(L_{1}\).