# Basins of Convergence in a Multi-Perturbed CR3 BP +
## 5 Conclusions  
In this paper, different modifications to the classical CR3 BP were made in order to increase the precision of the model. These modifications, which were added in a cumulative way, included the following: the effects of radiation forces, the effects of the oblateness of the primary masses, relativistic effects, and the contribution of satellites to the effective potential. The iterative, numerical NR method was implemented to solve the equations of the modified complex problem. The algorithm designed allows us to calculate the basins of convergence and the Lagrange points in a considerably versatile way. It can be configured according to the problem conditions and the desired accuracy: the analyzed domain, the precision of the calculations, the primary bodies considered, etc. It has been seen that when reducing the number of effects included in our code, we obtained results similar to the results obtained by other authors. Once the feasibility of the algorithm was proven, it was applied to the Sun-Mars system. In this case, it has been observed that the convergence basins and the libration points are mainly affected by the effects of radiation. Oblateness and the presence of natural satellites barely affect the results since the mass of the Sun is much bigger than the mass of Mars, so the terms of the effective potential associated with the star are more important than those for Mars. Moreover, this application shows that the number of iterations required to  
Figure 19: Iterations needed until convergence for the Sun–Mars system when triaxiality effects are included [zooming in on Mars]. Figure 18: Iterations needed until convergence for the Sun–Mars system when triaxiality effects are included [reduced domain]. achieve convergence towards collinear points is significantly lower than that required for convergence towards triangular ones. All of the authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript. This research was partially funded by the European Union Next Generation EU/PRTR, Programa de Planes Complementarios I+D+I, ref.ASFAE/2022/014, and by Generalitat Valenciana, Spain, grant CIAICO/2022/252. Not applicable. Data Availability Statement: Data is contained within the article. The authors declare no conflicts of interest. The following abbreviations are used in this manuscript:  
C.R.B.C. Note that inverse transformation provides the positions of the primary bodies in the \(OXY\) system as follows:  
\[X_{1}=-A_{1}\quad X_{2}=A_{2} \tag{10}\]  
\[Y_{1}=0\qquad Y_{2}=0\]  
Substituting expressions (11) into Equations (10) and (12), the equations for the motion of the infinitesimal particle expressed in the rotating reference frame are the following:  
\[\frac{d^{2}X}{dt^{2}}-2n_{D}\frac{dY}{dt}-n_{D}^{2}X=-Gm_{1}\frac{X-X_{1}}{d_{1 }^{3}}-Gm_{2}\frac{X-X_{2}}{d_{2}^{3}} \tag{11}\]  
\[\frac{d^{2}Y}{dt^{2}}+2n_{D}\frac{dX}{dt}-n_{D}^{2}Y=-Gm_{1}\frac{Y-Y_{1}}{d_{1 }^{3}}-Gm_{2}\frac{Y-Y_{2}}{d_{2}^{3}} \tag{12}\]  
where \(d_{i}\) is calculated as follows:  
\[d_{i}=\sqrt{(X-X_{i})^{2}+(Y-Y_{i})^{2}}=\sqrt{(\xi-\xi_{i})^{2}+(\eta-\eta_{i })^{2}}. \tag{13}\]  
Then, defining the potential function in this reference frame as  
\[\hat{\Omega}(X,Y)=\frac{n_{D}^{2}}{2}(X^{2}+Y^{2})+G\frac{m_{1}}{d_{1}}+G\frac {m_{2}}{d_{2}}, \tag{14}\]  
Equations (11) and (12) can be written as  
\[ \begin{array}{l}\frac{d^{2}X}{dt^{2}}-2n_{D}\frac{dY}{dt}=\frac{\hat{\partial }\hat{\Omega}}{\partial X}\\ \frac{d^{2}Y}{dt^{2}}+2n_{D}\frac{dX}{dt}=\frac{\hat{\partial}\hat{\Omega}}{ \partial Y}.\end{array}  \tag{15}\]  
In order to write the dimensionless versions of these equations, the distance \(R\) between the two primary masses is taken as the reference length, the sum \(m_{1}+m_{2}\) as the reference mass, and the inverse of the mean movement \(t_{0}=1/\Omega\) as the time unit.