# Circular Orbit Flip Trajectories Generated by E-Sail
## 3 Numerical Simulations and Discussion
### Case of a Transfer with a Single SWA  
Figure 6 shows the minimum flight time for a range of variation of \(\beta\) compatible with a medium-performance E-sail. As expected, bearing in mind the results obtained in a Solar System escape scenario [42] or in a more challenging interstellar mission [63], the presence of a SWA allows the E-sail to reduce the flight time compared with the DT case discussed in the previous subsection. Moreover, if a single SWA is included in the trajectory design, the same time of flight as in the DT case can be obtained, but with a reduced value of \(\beta\). For example, a comparison between the curves of Figures 3 and 6 indicates that a flight time of about \(8\,T_{0}\) requires a propulsive acceleration \(\beta\simeq 0.21\) for a DT case, while the same flight time is obtained with \(\beta\simeq 0.185\) in a transfer with a single SWA. The presence of a SWA obviously complicates the shape of the transfer trajectory, as confirmed by the schemes in Figure 2 and, more importantly, introduces a perihelion point at a solar distance \(r_{p}\) less than the radius of the circular parking orbit; see Figure 7. Note, however, that Figure 7 does not give the actual value of \(r_{p}\), but it only reports the ratio \(r_{p}/r_{0}\) as a function of the E-sail propulsive characteristics, so that the actual distance of \(r_{p}\) from the Sun depends on the value of \(r_{0}\). This is an important consideration, because in an E-sail mission scenario the value of \(r_{p}\) is inferiorly limited by the maximum operating temperature of the conducting tethers. For example, an E-sail structure made of aluminum tethers must operate at a solar distance greater than 0.5 au, while using copper tethers in the sail design reduces the minimum solar distance to about 0.33 au [42]. In the latter case, if the parking orbit radius is \(r_{0}=r_{\oplus}\), Figure 7 shows that SWA transfers with \(\beta>0.2\) are unfeasible. If, instead, \(r_{0}=0.9\) au, Figure 7 states that \(\beta\) should be less than about 0.18. Figure 6: Case of a transfer with a single SWA: variation of the dimensionless time \(\Delta t/T_{0}\) with \(\beta\). Figure 7: Case of a transfer with a single SWA: dimensionless radial distance \(r_{p}/r_{0}\) of the perihelion point as a function of the propulsive acceleration \(\beta\). An illustrative set of COFTs is summarized in Figure 8 for six different values of the dimensionless propulsive acceleration. In particular, the figure shows how the shape of the COFT changes as the value of \(\beta\) increases, and in this regard, an interesting behavior concerns the position of the aphelion point, marked by a red star. In fact, Figure 8 shows that \(r_{a}\) decreases as \(\beta\) increases, while, in the selected range of \(\beta\), its angular position is essentially in opposition to the starting point. This behavior is confirmed by Figure 9, which shows the coordinates of the aphelion points as a function of \(\beta\). The presence in the optimal COFT of a heliostationary aphelion point with a solar distance on the order of a few multiples of \(r_{0}\), in both a DT- and SWA-based scenario, suggests a potential (and additional) interesting application of this advanced two-dimensional trajectory concept. In fact, as proposed by the authors a few years ago, a propellantless propulsion system can be used to achieve a linear trajectory (i.e., a Keplerian orbit with a null value of the semilatus rectum) that allows the spacecraft to move along a straight, radial path to regions near the Sun to obtain in situ measurements of the space surrounding our star. Building on that mission concept, in a COFT-based scenario, a \"piggy-back\" science probe could be released at the aphelion point with a heliocentric velocity equal to zero. Upon release, the main spacecraft (i.e., the vehicle propelled by the E-sail subsystem) would return to the circular parking orbit by retracing the heliocentric path, while the scientific probe would follow the linear trajectory to final destruction in the Sun's inner regions.