# E-Sail Three-Dimensional Interplanetary Transfer with Fixed Pitch Angle
## 4 Numerical Simulations and Results  
The first step to determine the impact on the flight time of a constrained control law that provides for the maintenance of a constant value of the sail pitch angle during the interplanetary transfer is to calculate the mission time in the classic case in which \(\alpha_{n}\) can be freely varied during the flight. This specific (reference) situation is indicated in the rest of the paper, with the subscript \(u\). In other words, the first step is to solve an unconstrained (with regard to the sail pitch angle) minimum-time optimization problem, in order to obtain a set of reference values for the flight time \(\Delta t=\Delta t_{u}\) to be used as a term of comparison for the results that will then be obtained in the next step, i.e., the one in which the value of \(\alpha_{n}\) will be fixed during the transfer. To this end, the procedure described in ref. [45] is used, leaving the sail pitch angle free to vary during the flight according to the corresponding optimal guidance law, in order to minimize the orbit-to-orbit transfer time in a given mission scenario (in this case, an Earth-Venus or an Earth-Mars) and for a given value of the spacecraft characteristic acceleration \(a_{c}\). More precisely, in this kind of unconstrained sail pitch angle-case, the control law described by Equation (30) of [45] is used for \(\alpha_{n}\) and, considering \(a_{c}\in[0.1,\,1]\) mm/s\({}^{2}\), the curves drawn in Figure 8 are obtained. In particular, the graphs in that figure describe the usual shape of the function \(\Delta t_{u}=\Delta t_{u}(a_{c})\) which has a vertical asymptote when the characteristic acceleration is reduced to very small values. On the other hand, the variation (i.e., the reduction) in the optimal flight time is less marked when the propulsive performance of the E-sail increases, and the value of \(a_{c}\) reaches (or exceeds) the canonical one of 1 mm/s\({}^{2}\). Using the curves in Figure 8, it is possible to extrapolate the reference values of the minimum flight time \(\Delta t_{u}\), in the two interplanetary mission scenarios, when \(a_{c}\in\{0.1,\,0.5,\,1\}\) mm/s\({}^{2}\). The values of such flight times, which will be used to compare the results obtained in the sail pitch angle-constrained case, are summarized in Table 1.  
The optimal spacecraft transfer trajectory and the optimal time variation of the two control angles \(\delta\) and \(\alpha_{n}\) in the two mission scenarios, when \(a_{c}=0.1\) mm/s\({}^{2}\) and the sail pitch angle is unconstrained, is reported in Figure 9. Moreover, Figures 10 and 11 show the unconstrained results when \(a_{c}=0.5\) mm/s\({}^{2}\) and \(a_{c}=1\) mm/s\({}^{2}\), respectively. \begin{table}
\begin{tabular}{c c c c} \hline
**Case** & \(\mathbf{a_{c}=0.1\,\text{mm/s}^{2}}\) & \(\mathbf{a_{c}=0.5\,\text{mm/s}^{2}}\) & \(\mathbf{a_{c}=1\,\text{mm/s}^{2}}\) \\ \hline Earth–Venus & 2332 days & 532 days & 296 days \\ Earth–Mars & 3584 days & 657 days & 461 days \\ \hline \end{tabular}
\end{table} 
Table 1: Numerical value of the minimum flight time \(\Delta t_{u}\) in an unconstrained (from the sail pitch angle point of view) interplanetary orbit-to-orbit transfer when \(a_{c}\in\{0.1,\,0.5,\,1\}\) mm/s\({}^{2}\). In particular, Figure 9 clearly indicates that, in the unconstrained case of a low-performance E-sail (i.e., for \(a_{c}=0.1\,\mathrm{mm}/\mathrm{s}^{2}\) and \(\alpha_{n}\in[0,\,90]\) deg), the three-dimensional interplanetary transfer is performed by completing a significative number of revolutions around the Sun. This involves a continuous variation of both the two control angles \(\delta\) and \(\alpha_{n}\), which perform a sort of oscillatory motion around their average values.