## 5 Model Validation and Numerical Simulations
The solution of the numerical procedure gives the actual value of the triplet \(\{\tilde{l}_{f},\delta,\lambda_{r}(\tilde{t}_{0})\}\), which was compared with the analytical approximation given by Equations (26), (29), and (33) by introducing the ratio \(\{R_{t},R_{\delta},R_{\lambda}\}\) defined as  
\[R_{t}\triangleq\frac{\tilde{l}_{f}\Big{|}_{\text{an}}}{\tilde{l}_{f}\Big{|}_{ \text{num}}}\quad,\quad R_{\delta}\triangleq\frac{\delta\big{|}_{\text{an}}} {\delta\big{|}_{\text{num}}}\quad,\quad R_{\lambda}\triangleq\frac{\lambda_{ r}(\tilde{t}_{0})\big{|}_{\text{an}}}{\lambda_{r}(\tilde{t}_{0})\big{|}_{\text{num}}} \tag{39}\]  
where \(\square\big{|}_{\text{an}}\) indicates the value of \(\square\) calculated through the analytical approximation, while \(\square\big{|}_{\text{num}}\) is the actual value of \(\square\) (calculated numerically) that solves the TPBVP. The results of the numerical simulations for the four exemplary mission scenarios are summarized in Tables 1-4 in terms of \(\{\tilde{l}_{f},\theta(\tilde{l}_{f}),n,R_{t},R_{\delta},R_{\lambda}\}\) as a function of \(\tilde{a}_{m}\). In particular, as also explicitly indicated in the table captions, the values of \(\{\tilde{l}_{f},\theta(\tilde{l}_{f})\}\) are the results of the numerical solutions of the associated TPBVP, the value of \(n\) is obtained analytically from Equation (38), and the three ratios \(\{R_{t},R_{\delta},R_{\lambda}\}\) are calculated according to Equation (39). The last three columns in Tables 1-4 confirm that the analytical approximations proposed in this work are consistent with the actual (numerical) values of the unknown costates. Furthermore, the expression of \(n\) given by Equation (38) gives a very good approximation of the actual number of completed revolutions around the primary body during the transfer. Finally, the value of \(R_{t}\simeq 1\) in all the transfers analyzed indicates that the expression of the flight time given by Equation (29) is well suited to rapidly calculate the minimum transfer time without solving (numerically) the optimization problem, especially for low propulsive acceleration magnitude. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\tilde{a}_{m}\) & \(\tilde{t}_{f}\) & \(\dfrac{\theta(\tilde{t}_{f})}{2\pi\text{ rad}}\) & \(n\) & \(R_{t}\) & \(R_{\delta}\) & \(R_{\lambda}\) \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 1: Earth–Venus transfer scenario: Simulations results and comparison with the analytical approximations. The values of \(\{\tilde{l}_{f},\theta(\tilde{l}_{f})\}\) are obtained through the numerical solution of the TPBVP, while the values of \(\{n,R_{t},R_{\delta},R_{\lambda}\}\) are calculated using Equations (38) and (39). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\tilde{a}_{m}\) & \(\tilde{f}_{f}\) & \(\theta(\tilde{t}_{f})\) & \(n\) & \(R_{t}\) & \(R_{\delta}\) & \(R_{\lambda}\) \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 2: Earth–Mars transfer scenario: Simulations results and comparison with the analytical approximations. The values of \(\{\tilde{f}_{f},\theta(\tilde{t}_{f})\}\) are obtained through the numerical solution of the TPBVP, while the values of \(\{n,R_{t},R_{\delta},R_{\lambda}\}\) are calculated using Equations (38) and (39). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\tilde{a}_{m}\) & \(\tilde{f}_{f}\) & \(\theta(\tilde{t}_{f})\) & \(n\) & \(R_{t}\) & \(R_{\delta}\) & \(R_{\lambda}\) \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 3: Earth–Jupiter transfer scenario: Simulations results and comparison with the analytical approximations. The values of \(\{\tilde{f}_{f},\theta(\tilde{t}_{f})\}\) are obtained through the numerical solution of the TPBVP, while the values of \(\{n,R_{t},R_{\delta},R_{\lambda}\}\) are calculated using Equations (38) and (39).