# Optimal Interplanetary Transfer of Solar Wind Ion Focusing Thruster-Based Spacecraft
## 4 Numerical Simulations and Mission Applications
#### Case Study  
Two specific mission applications are considered to estimate the optimal performance of a SWIFT-based spacecraft, the main characteristics of which are detailed in Table 3. In particular, an orbit raising with \(r_{f}=r_{\mathcal{G}}\triangleq 1.524\)  au, and an orbit lowering with \(r_{f}=r_{\mathcal{G}}\triangleq 0.723\)  au have been studied. Recalling that the radius of the initial circular orbit is \(r_{0}=r_{\oplus}=1\)  au, the case of \(r_{f}=r_{\mathcal{G}}\) (or \(r_{f}=r_{\mathcal{G}}\)) is representative of a simplified, ephemeris-free, Earth-Mars (or Earth-Venus) interplanetary transfer in which the eccentricity and the inclination over the Ecliptic of both planetary orbits are neglected. The results of the numerical optimization for the Earth-Mars transfer give a minimum flight time that is slightly shorter than 8.1 years, with an optimal transfer trajectory that completes five revolutions around the Sun before reaching the target Mars orbit. This aspect is confirmed by Figure 9(a), which shows the optimal transfer trajectory in a polar reference frame. According to Figure 9(b), which shows the optimal time variation of the thrust angle, \(\alpha\) reaches six times the maximum value \(\alpha_{\max}\) with a mean value (over the entire transfer) of about 84  deg.  
The time variations of the spacecraft states \(\{r,\,\theta,\,v_{r},\,v_{\theta}\}\) are reassumed in Figure 11. Note that the radial spacecraft velocity \(v_{r}\) remains small during the whole transfer, so that the optimal transfer trajectory can be approximated with a sort of tight spiral connecting the initial and final (circular) orbits. This feature of the transfer trajectory is the result of a propulsive acceleration that is small, in comparison with the (local) Sun's gravitational acceleration. Figure 11 also shows that the time variation of the Sun-spacecraft distance \(r\) (or the transverse component of the inertial spacecraft velocity \(v_{\theta}\)) may be thought of as a superposition of a secular increase (or decrease) and a short period of oscillation. As a result, the SWIFT system's performance is similar to that obtained using a solar sail [52; 53; 54], an E-sail [25], a Heliogyro [55], or a low-performance electric thruster [56; 57] as a primary propulsion system.  
Figure 10: Optimal transfer trajectory and control angle history in an Earth–Mars transfer. Black circle (or square) refers to the start (or arrival) point.  
Similar considerations can be made for an Earth-Venus transfer (when \(r_{f}=r_{\zeta}\)), which is completed with a flight time of slightly above 3.6 years. In that case, the optimal transfer trajectory and the time variation of the thrust angle are shown in Figure 12, while the spacecraft states are reassumed in Figure 13. Note that \(\alpha<0\) during the entire transfer, that is, the steerable thrust \(T\) has a negative circumferential component. In that case, the value of \(-\alpha_{\max}\) is reached five times, with a mean value of the thrust angle of about \(-80\)  deg.  
Figure 11: Time variation of the spacecraft states in an Earth–Mars transfer. Black circle (or square) refers to the start (or arrival) point.  
Figure 12: Optimal transfer trajectory and control angle history in an Earth–Venus transfer. Black circle (or square) refers to the start (or arrival) point.