# E-Sail Three-Dimensional Interplanetary Transfer with Fixed Pitch Angle
## 5 Conclusions  
This paper has studied the performance of a deep space vehicle equipped with an Electric Solar Wind Sail (E-sail) in the case where the sail attitude is partially constrained during the orbital transfer between Earth and Venus or Mars. In particular, the study proposed in this paper considers a constant value of the sail pitch angle, allowing a \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & \(a_{c}=0.1\) mm/s\({}^{2}\) & \(a_{c}=0.5\) mm/s\({}^{2}\) & \(a_{c}=1\) mm/s\({}^{2}\) \\ \hline Earth–Venus & 2334 days & 538 days & 301 days \\ Earth–Mars & 3628 days & 664 days & 469 days \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Minimum flight time \(\Delta t\) in a constrained orbit-to-orbit interplanetary transfer with \(a_{n}=45\deg\). Figure 14: Constrained Earth–Venus interplanetary transfer with \(a_{n}=45\deg\) and \(a_{c}=0.5\) mm/s\({}^{2}\). The legend is reported in Figure 9.  
reduced thrust vectoring of the propulsion system thanks to the variation of the clock angle. Furthermore, the presence of possible coasting arcs in the interplanetary trajectory of the spacecraft is ensured by the presence of an appropriate switching function that models the possible shutdown of the propulsion system. The presence of the constraint on the sail pitch angle value does not substantially influence the transfer performance, if the flight time is considered as a performance index and an appropriate pitch angle value is used. In fact, the numerical simulations carried out considering three different performance levels of the E-sail, defined by three different values of the characteristic acceleration, have clearly indicated that the increase in flight time by moving from an unconstrained reference model to one in which the sail pitch angle is fixed during the flight, is rather limited and results in a few percentage points if a pitch angle of forty-five degrees is selected. This specific value of the sail pitch angle is, in fact, the one that maximizes the transversal component of the E-sail-induced propulsive acceleration vector. This aspect confirms the semi-analytical results of the recent literature which, however, were obtained considering a two-dimensional (and purely circular) model of the interplanetary transfer. In this work instead, the interplanetary orbital transfer was studied in its actual three-dimensionality, with the only simplifying hypothesis regarding the use of planetary ephemerides, in order to obtain globally optimal results from the point of view of the time of flight. The obtained results indicate that a simplified guidance law is indeed possible for a spacecraft equipped with an E-sail, provided that the value of the sail pitch angle is appropriately selected in the assigned mission scenario. In this sense, the procedure and the model discussed in this paper can be considered an intermediate way between the classical unconstrained case in which the sail nominal plane is freely steerable and the much more stringent one, in which the inertial attitude of the E-sail propulsion system is kept fixed during the heliocentric flight. This research received no external funding. Not applicable. The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. The author declares that he has not used any kind of generative artificial intelligence in the preparation of this manuscript, nor in the creation of images, graphs, tables, or related captions. The author declares no conflicts of interest.