# Optimal Trajectories of Diffractive Sail to Highly Inclined Heliocentric Orbits
## 4 Performance Assessment of Two-Phase Strategy  
The previous section described the two-phase strategy for generating near-optimal trajectories that combine the peculiarities of locally optimal methodologies (in terms of simplicity and effectiveness) with those of globally optimal approaches (ability to satisfy boundary conditions and obtain accurate solutions). Such a two-phase strategy, consisting of an approach phase and a cranking phase, is simple to implement numerically and is thus able to provide an approximate solution that is particularly useful in the initial phase of a mission analysis, where many feasible trajectories can be generated at a low computational cost. It is interesting to verify whether the solutions obtained with the above strategy are indeed near-optimal, that is, whether they deviate by a few percentage points from an optimal solution. To answer this question and quantify this difference, we will compare the results that can be obtained in a reference case similar to the one described in the previous section, but using different data in terms of characteristic acceleration and constraint on the minimum distance from the Sun. More precisely, the reference case is taken from the mission scenario discussed in Ref. [21] and recently analyzed by [PERSON] et al. [30] through the use of a local optimization approach. The mission objective is again to place the satellite in a target circular orbit around the Sun with a radius \(r_{f}=0.32\,\mathrm{au}\) and an inclination of \(i_{f}=60\,\mathrm{deg}\) with respect to the Ecliptic, but in this case the spacecraft is propelled by an ideal diffractive sail with a characteristic acceleration \(a_{c}=0.183\,\mathrm{mm}/\mathrm{s}^{2}\). For the sake of comparison, the distance of the minimum approach from the Sun was set as equal to \(r_{c}=0.22\,\mathrm{au}\), the same as that used in Ref. [30], although this value is probably too small to be used in a real mission [27]. We begin this study by calculating minimum-time trajectories to a circular orbit of radius \(r_{f}=0.32\,\mathrm{deg}\) and different values of final inclination, increasing the latter by \(5\,\mathrm{deg}\) with each new simulation. The results obtained are summarized in Figure 6.  
Figure 6: Simulations of optimal trajectories with \(a_{c}=0.183\,\mathrm{mm}/\mathrm{s}^{2}\) to a circular heliocentric orbit of radius \(r=0.32\,\mathrm{au}\) for different values of final orbital inclination. Dashed red line \(\rightarrow\) minimum solar distance. As in the case discussed in the previous section, the minimum distance from the Sun tends to decrease for final inclinations greater than about 20 deg, and the minimum distance equal to \(r_{c}=0.22\) au is reached for final inclinations of about 35 deg. Therefore, the two-phase strategy in this case consists of a first approaching phase that ends on an osculating orbit with \(r_{f}=0.32\) au and inclination \(i^{*}=35\) deg, and is connected with the second cranking phase in which the final distance still remains equal to 0.32 au, but the inclination rises to \(i_{f}=60\) deg. The results are summarized in Figure 7.  
During the first phase, the distance from the Sun decreases until it reaches the minimum distance constraint after about 910 days, and then rises again to the distance of 0.32 au. During the second cranking phase, the spacecraft tends to slightly decrease its distance from the Sun, without approaching the minimum value, while the orbital inclination increases monotonically with small oscillations. The transfer trajectory found is shown in Figure 8. The duration of the first phase is 1070 days, and that of the second phase is 291 days for a total of 1361 days. These results are close to those reported by [PERSON] et al. [30], who found a total time of 1324 days for the same transfer, albeit using a more complex locally optimal control law than the one proposed in this paper.