## 5 Model Validation and Numerical Simulations
_Case Study: Transfer towards Comet 29P/Schwassmann-Wachmann 1_  
The proposed method is now used to rapidly solve the TPBVP associated with the (heliocentric) minimum-time transfer towards comet 29P/Schwassmann-Wachmann 1 [24; 25; 26; 27]. The latter was very recently considered by the author [28] as a potential target for a solar-sail-based (rendezvous) interplanetary mission. The interested reader can refer to the Introduction section of reference [28] to obtain a snapshot of the recent research concerning comet 29P/Schwassmann-Wachmann 1 (indicated as \"comet SW1\" in the rest of this paper). The current, heliocentric orbit of comet SW1 is nearly circular (the actual eccentricity is \(4.4526\times 10^{-2}\)), with an inclination of about 9 deg. The comet orbit is between Jupiter and Saturn [29]. The numerical results discussed in reference [28] indicate that a simplified circle-to-circle, two-dimensional transfer scenario can be used to approximate the real, three-dimensional Earth-comet SW1 trajectory in a classical interplanetary rendezvous mission. In this context, the comet orbit around the Sun is considered circular, with radius \(r_{\rm SW1}=6.0499\) AU, and coplanar to the heliocentric (assumed circular with radius \(r_{0}=1\) AU) orbit of Earth. Therefore, in this heliocentric scenario (in which \(\mu=\mu_{\odot}\)), the dimensionless target radius is \(\bar{r}_{f}=r_{\rm SW1}/r_{0}=\bar{r}_{\rm SW1}=6.0499\), and Equation (38) gives the function \(n=n(\bar{a}_{m})\), sketched in Figure 4. According to Figure 4, the condition \(n\geq 2\) is approximately satisfied when \(\bar{a}_{m}\leq 0.02\), so the optimal Earth-comet SW1 transfer is studied parametrically considering (again) a dimensionless maximum propulsive acceleration in the range \(\bar{a}_{m}\in[0.001,0.02]\). Note that, in this case, one has \(\mu_{\odot}/r_{0}^{2}\simeq 5.93\) mm/s\({}^{2}\) so that a value \(\bar{a}_{m}=0.02\) (or \(\bar{a}_{m}=0.001\)) corresponds to a dimensional propulsive acceleration of \(0.1186\) mm/s\({}^{2}\) (or \(0.00593\) mm/s\({}^{2}\)). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\bar{a}_{m}\) & \(\bar{t}_{f}\) & \(\frac{\theta(\bar{t}_{f})}{2\pi\) rad} & \(n\) & \(R_{t}\) & \(R_{g}\) & \(R_{\lambda}\) \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 4: LEO-GEO transfer scenario: Simulations results and comparison with the analytical approximations. The values of \(\{\bar{t}_{f},\theta(\bar{t}_{f})\}\) are obtained through the numerical solution of the TPBVP, while the values of \(\{n,R_{t},R_{g},R_{\lambda}\}\) are calculated using Equations (38) and (39). The TPBVP associated with the optimization problem was solved using the procedure previously described, and the results of the numerical simulations are summarized in Table 5, which can be considered as an extension (to this scenario) of the results of Tables 1-4. As expected, the values in Table 5 confirm the accuracy of the proposed analytical model based on Equations (26), (29), and (33). The time variation in the thrust angle and the corresponding optimal transfer trajectory are two outputs of the optimization process. Assuming, for example, \(\tilde{a}_{m}=\{0.001,0.005,0.01,0.02\}\), the polar form of the optimal transfer trajectory is sketched in Figure 5, while the (dimensionless) time variation in the thrust angle \(\alpha\) is shown in Figure 6. Note that, according to Figure 6, when the magnitude of the propulsive acceleration is sufficiently low (as in the case of \(\tilde{a}_{m}=0.001\)), the value of the thrust angle during the entire transfer is close to \(\pi/2\) rad, as modeled by  
Figure 4: Earth–comet SW1 scenario: number \(n\) of completed revolutions around the Sun as a function of \(\tilde{a}_{m}\); see also Equation (38).