## 3 Simulations
### Simulation Settings and Results
For this algorithm, the following settings were considered for the iterations of the solver: a step tolerance of \(T_{\rm s}=10^{-15}\), an optimality tolerance of \(T_{\rm o}=10^{-12}\), a constraint tolerance of \(T_{\rm c}=10^{-12}\), an interior point algorithm, a scheme of central finite differences, and a maximum number of iterations equal to 2400. The value of the maximum number of iterations was chosen in order to never be reached. The choice of the settings was made observing the precision of the solution. In particular, the selected case for these tests was the one with the emitter position in \(\Lambda=30^{\circ}\), \(\lambda=0^{\circ}\), and \(h=0\) m. The altitudes of the satellites were \(h_{\rm S}=400\) km and the angular position of the first satellite was \(\Omega_{1}=\delta\big{(}h_{\rm S}\big{)}\Big{/}\Big{(}1+\sqrt{2}\Big{)}\approx 8.198361918^{\circ}\), \(v_{1}=\Lambda-\delta\big{(}h_{\rm S}\big{)}\Big{/}\Big{(}1+\sqrt{2}\Big{)} \approx 21.80163808^{\circ}\). The choice of the first satellite at the border of the constraints was made to stress the solver not only on the step tolerance \(T_{\rm s}\) and the optimality tolerance \(T_{\rm o}\) but also on the tolerance related to the constraints \(T_{\rm c}\). The selected cases of the tests were on step tolerance \(T_{\rm s}\), \(T_{\rm o}\), and \(T_{\rm c}\) from \(10^{-3}\) to \(10^{-12}\). The results of the tests were for the precision of the numerical figures of the solution and are reported in Table 1. For simulations, the step tolerance was further decreased to \(T_{\rm s}=10^{-15}\) because it was observed that this was the main reason that the convergence of the solver stopped. Even if the numerical convergence analysis was conducted for up to 15 figures and beyond, the solutions needed to be considered physically valid for up to 7 figures because the limitation was on the input data: in particular, the mean Earth radius \(r_{\oplus}\) considered in this study was precise for up to 7 figures. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Settings (\(T_{\rm s},T_{\rm o},T_{\rm c}\))** & **Solution [*]** & **Precision** \\ \hline \(\big{(}10^{-3},10^{-3},10^{-3}\big{)}\) & \(\Delta\Omega_{1}=7.79988394977692678367021\) & \(10^{-5}\) \\  & \(\Delta v_{1}=-22.2001160502229595294921\) & \(10^{-6}\) \\  & \(\gamma_{1,2}=46.956746998389434112318\) & \(10^{-5}\) \\ \hline \(\big{(}10^{-6},10^{-6},10^{-6}\big{)}\) & \(\Delta\Omega_{1}=7.79989639975224235968199\) & \(10^{-7}\) \\  & \(\Delta v_{1}=-22.2001036002477576403180\) & \(10^{-8}\) \\  & \(\gamma_{1,2}=46.9567650909294656003112\) & \(10^{-8}\) \\ \hline \(\big{(}10^{-9},10^{-9},10^{-9}\big{)}\) & \(\Delta\Omega_{1}=7.79989630705961189960362\) & \(10^{-10}\) \\  & \(\Delta v_{1}=-22.2001036929402744135587\) & \(10^{-11}\) \\  & \(\gamma_{1,2}=46.9567649562266637985886\) & \(10^{-10}\) \\ \hline \(\big{(}10^{-12},10^{-12},10^{-12}\big{)}\) & \(\Delta\Omega_{1}=7.79989630740783468354493\) & \(<\)\(10^{-40}\) \\  & \(\Delta v_{1}=-22.2001036925921653164551\) & \(<\)\(10^{-40}\) \\  & \(\gamma_{1,2}=46.9567649567326341752960\) & \(<\)\(10^{-40}\) \\ \hline \(\big{(}10^{-15},10^{-12},10^{-12}\big{)}\) & \(\Delta\Omega_{1}=7.79989630740783468354493\) & Reference \\  & \(\Delta v_{1}=-22.2001036925921653164551\) & Reference \\  & \(\gamma_{1,2}=46.9567649567326341752960\) & Reference \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Analysis of the precision of the results with different settings of the solver. In particular, 17 evenly spaced values of the latitude of the emitter \(\Lambda\) from \(-80^{\circ}\) to \(80^{\circ}\) and 9 evenly spaced values of the orbit radius \(r\) with \(h\) from 400 km to 2000 km were considered; 14 evenly spaced values were considered for both \(\Omega_{1}\) and \(\
u_{1}\). Since, as observed in a previous work [26], in this kind of problem, multiple values of local minimum occur, a pre-evaluation of the objective function with a fine grid of \(0.01^{\circ}\times 0.01^{\circ}\) was considered for the selection of the initial value of the solver.