# E-Sail Three-Dimensional Interplanetary Transfer with Fixed Pitch Angle
## 4 Numerical Simulations and Results
An interesting aspect that emerges from the figure is that the sail pitch angle moves, during the interplanetary flight, around the average value of \(45\,\mathrm{deg}\) that maximizes the transversal component of the propulsive acceleration \(a_{\mathrm{TN}}\) (see also Figure 5) both in the case of an Earth-Mars transfer, i.e., in the case of an orbit raising, and in the case of an Earth-Venus transfer, which can be considered a sort of heliocentric orbit lowering. In these two mission scenarios, the difference is made by the value of the clock angle \(\delta\), which oscillates around \(180\,\mathrm{deg}\) for the Venus-based transfer, or zero for the transfer to Mars. As expected, according to Figures 10 and 11, the cases of a medium or high performance E-sail, i.e., the cases where \(a_{c}=0.5\,\mathrm{mm}/\mathrm{s}^{2}\) or \(a_{c}=1\,\mathrm{mm}/\mathrm{s}^{2}\), show a reduced number of revolutions around the Sun and a simpler optimal control law of \(\delta\) and \(\alpha_{n}\) when compared to the case of \(a_{c}=0.1\,\mathrm{mm}/\mathrm{s}^{2}\). In any case, the mean value of the sail pitch angle remains close to \(45\,\mathrm{deg}\), i.e., the value that maximizes \(a_{\mathrm{TN}}\) even in an interplanetary transfer with a medium-high performance E-sail. Figure 8: Minimum flight time \(\Delta t_{u}\) in an unconstrained (from the point of view of the sail pitch angle \(\alpha_{n}\)) orbit-to-orbit interplanetary transfer as a function of the spacecraft characteristic acceleration \(a_{c}\). (a) Case of an Earth–Venus transfer; (b) Case of an Earth–Mars transfer. Figure 9: Optimal spacecraft trajectory and time variation of the two control angles in an unconstrained interplanetary transfer when \(a_{c}=0.1\,\mathrm{mm}/\mathrm{s}^{2}\). Black line \(\rightarrow\) spacecraft trajectory; blue line \(\rightarrow\) Earth orbit; red line \(\rightarrow\) target planet orbit; blue star \(\rightarrow\) Earth’s perihelion point; red star \(\rightarrow\) target planet’s perihelion point; orange dot \(\rightarrow\) the Sun; blue dot \(\rightarrow\) start; red square \(\rightarrow\) arrival. **(a)** Case of an Earth–Venus transfer; **(b)** Case of an Earth–Mars transfer. Figure 10: Case of an unconstrained interplanetary transfer when \(a_{c}=0.5\,\mathrm{mm}/s^{2}\). The legend is reported in Figure 9. (**a**) Case of an Earth–Venus transfer; (**b**) Case of an Earth–Mars transfer. Figure 11: Case of an unconstrained interplanetary transfer when the E-sail has a canonical value of the characteristic acceleration, i.e., \(a_{c}=$1\,\mathrm{mm}\mathrm{/}\mathrm{s}^{2}$\). The legend is reported in Figure 9. **(a)** Case of an Earth–Venus transfer; **(b)** Case of an Earth–Mars transfer. At this point, we consider an interplanetary transfer with a fixed value of the sail pitch angle, which is selected within the interval \(\alpha_{n}\in[35,\,55]\,\mathrm{deg}\). Note that this interval substantially corresponds to taking a range of \(\pm 10\,\mathrm{deg}\) with respect to the value (precisely \(45\,\mathrm{deg}\)) that maximizes the transversal component of propulsive acceleration; see Figure 5. The results of the optimization of the spacecraft transfer trajectory, in terms of variation of the (dimensionless) minimum flight time \(\Delta t\) as a function of \(\alpha_{n}\in[35,\,55]\,\mathrm{deg}\) and \(a_{c}\in\{0.1,\,0.5,\,1\}\,\mathrm{mm}/\mathrm{s}^{2}\), are summarized in Figure 12.  
The curves in Figure 12 clearly indicate that, in the considered range of the fixed-sail-pitch angle, the minimum flight time \(\Delta t\) remains close to the value \(\Delta t_{u}\) obtained in the unconstrained case. In fact, both in the Earth-Venus and Earth-Mars mission scenarios, the ratio \(\Delta t/\Delta t_{u}\) remains well below \(1.1\) and, in the specific case of the Earth-Venus transfer with \(a_{c}=0.1\,\mathrm{mm}/\mathrm{s}^{2}\), the difference between \(\Delta t\) and \(\Delta t_{u}\) is only a few days.