# Circular Orbit Flip Trajectories Generated by E-Sail
## 3 Numerical Simulations and Discussion
### Case of DT Scenario  
For a DT case, Figure 3 shows the ratio of the minimum flight time \(\Delta t\) to the period of the circular parking orbit \(T_{0}=2\,\pi\sqrt{r_{0}^{3}/\mu_{\odot}}\) as a function of \(\beta\). Note that  
\[\frac{\Delta t}{T_{0}}=\frac{\Delta t}{2\,\pi\sqrt{r_{0}^{3}/\mu_{\odot}}} \equiv\frac{\Delta\widetilde{t}}{2\,\pi} \tag{28}\]  
so that \(\Delta t/T_{0}\) is independent of the radius \(r_{0}\). Additionally, note that  
\[\frac{\Delta t}{T_{0}}=-\frac{\widetilde{J}}{2\,\pi} \tag{29}\]  
where \(\widetilde{J}\) is the performance index defined in Equation (19).  
The curve plotted in Figure 3 can be used to rapidly estimate the transfer performance in an orbit flip mission for an E-sail of given characteristics, starting from a circular orbit of assigned radius. For example, Figure 3 indicates that an E-sail with a propulsive acceleration of \(\beta=0.25\) travels through the optimal COFT in a time interval of about \(6\,T_{0}\). In that case, assuming \(r_{0}=r_{\oplus}=1\,\mathrm{au}\) (a scenario compatible with the deployment of the sail just outside the Earth's sphere of influence when the escape trajectory is parabolic and the eccentricity of the Earth's orbit is neglected), the period \(T_{0}\) is equal to \(1\) year, while \(\mu_{\odot}/r_{0}^{2}\simeq 5.93\,\mathrm{mm}/\mathrm{s}^{2}\). The orbital flip is completed in about \(6\) years using an E-sail with a characteristic acceleration of about \(0.25\times 5.93\,\mathrm{mm}/\mathrm{s}^{2}\simeq 1.5\,\mathrm{mm}/\mathrm{s}^{2}\), corresponding to a medium-high performance propulsion system [62]. As a second example, if the mission is to be completed in less than \(4\) years, Figure 3 states that \(\beta\geq 0.35\), that is, it is necessary an E-sail with \(a_{c}\geq 2.1\,\mathrm{mm}/\mathrm{s}^{2}\) when \(r_{0}=r_{\oplus}\).  
Since in a DT case the perihelion coincides with the starting point, the position of the COFT aphelion is an output of the optimization process. The simulation results are summarized in Figure 4. The upper part of Figure 4 shows the aphelion distance \(r_{a}\), while the bottom part shows the angular position (i.e., the polar angle \(\theta_{a}\)) of the aphelion point.  
Figure 3: Case of a DT: ratio of the minimum flight time \(\Delta t\) to the orbital period of the parking orbit \(T_{0}\) as a function of the dimensionless propulsive acceleration \(\beta\).  
For example, assuming again \(r_{0}=r_{\oplus}\), an optimal COFT obtained with \(\beta=0.25\) has an aphelion point at a distance of about 4 au from the Sun, and the E-sail sweeps a polar angle of about 170 deg to move from the starting to the aphelion point. This same result is clearly visible in the polar COFT depicted in Figure 5, which also shows the optimal transfer trajectories for different values of \(\beta\).  
Figure 4: Case of a DT: radial (\(r_{a}\)) and angular (\(\theta_{a}\)) position on the aphelion point as a function of \(\beta\).  
Figure 5: Polar form of the COFT (black line) for a set of values of the propulsive acceleration \(\beta\) in a DT scenario. Black circle \(\rightarrow\) starting point; red star \(\rightarrow\) aphelion; orange circle \(\rightarrow\) Sun; blue line \(\rightarrow\) parking orbit. The radial distance is normalized with the parking orbit radius \(r_{0}\).