# Basins of Convergence in a Multi-Perturbed CR3 BP +
## 4 Application to the Sun-Mars System: Adding the Effects of Phobos and Deimos
### Triaxiality Effects  
Triaxiality effects are now introduced. It is assumed that the semi-axes, contained in the \(\{x,y\}\) plane, are equal, while the \(z\) semi-axis means that the primary masses are not perfectly spherical. The semi-axes of Mars were extracted from NASA's data sheets [32]. However, this information is not available for the Sun, so an estimation of the corresponding semi-axes was made using its mean volumetric radius and its ellipticity [31]. This information is reflected in Table 10.  
Using these data, it is possible to calculate the oblateness parameters, which modify the equations according to the model shown in Section 2.2. Note that the terms of the potential associated with the Sun hardly change when introducing the effects of triaxiality due to its very low ellipticity. However, the oblateness of Mars is significantly greater than that of the Sun, which implies a substantial variation in the terms associated with Mars.  
In Figures 11-13, the BCs of the Sun-Mars problem are shown, taking into account their oblateness. Table 11 shows the coordinates of the libration points in this case.  
As can be seen in Table 11, the location of the Lagrange points has not changed due to the inclusion of the effects of oblateness. Nor have the BCs in the environment around Mars (Figures 12 and 13). This can be explained because the terms associated with the Sun have greater weight than those associated with this planet, and the star is practically spherical. However, an appreciable change in the shape of the BCs can be observed in far-field solutions (see Figure 11) since the effects of the oblateness of Mars begin to become more noticeable.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Sun [31]** & **Mars [32]** \\ \hline Ellipticity [-] & 0.00005 & 0.00589 \\ Average volumetric radius [km] & \(6.95700\times 10^{5}\) & \(3.38950\times 10^{3}\) \\ Equatorial semi-axis [km] & \(6.95712\times 10^{5}\) & \(3.39620\times 10^{3}\) \\ Polar semi-axis [km] & \(6.95677\times 10^{5}\) & \(3.37620\times 10^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 10: Data on the primary masses used for the generation of the BCs. They allow us to calculate the triaxiality parameters using Equation (10).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Libration Points** &
\begin{tabular}{c} **Obtained Coordinates** \\ \(\left(\mathbf{x_{L},y_{L}}\right)\left[-\right]\) \\ \end{tabular} \\ \hline \(L_{1}\) & \((0.99525,\,0)\) \\ \(L_{2}\) & \((1.00476,\,0)\) \\ \(L_{3}\) & \((-1.00000,\,0)\) \\ \(L_{4}\) & \((0.50000,\,0.86603)\) \\ \(L_{5}\) & \((0.50000,\,-0.86603)\) \\ \hline \hline \end{tabular}
\end{table}
Table 9: Coordinates of the Sun–Mars libration points in the classical formulation.  
Figure 11: Basins of convergence for the Sun–Mars system considering triaxiality effects and a broad-size domain.  
Figure 12: Basins of convergence for the Sun–Mars system considering triaxiality effects and a reduced size domain.  
Figure 13: Basins of convergence for the Sun–Mars system considering triaxiality effects, zooming in on Mars.