# Optimal Trajectories of Diffractive Sail to Highly Inclined Heliocentric Orbits
## 4 Performance Assessment of Two-Phase Strategy
To further quantify the performance of the two-phase strategy, an optimal transfer was calculated using an indirect method [19] with the intermediate constraint of \(r\geq r_{c}\). The results of the simulations are shown in Figure 9, while the minimum-time trajectory is reported in Figure 10.  
In contrast to the two-phase strategy, in this case the spacecraft moves for a fairly long period, approximately 120 days (i.e., almost 10% of the total transfer time) while maintaining a distance from the Sun close to the minimum admissible value, in order to maximize the boosting effect due to the increased solar radiation pressure. The difference between the two-phase strategy and the optimal transfer is clearly visible in Figure 11.  
The total transfer time in the optimal case is 1294 days, so that the two-phase strategy effectively determines a near-optimal solution, with a difference of 67 days (equivalent to an error of only 4.9%) compared to the optimal solution. Figure 12 shows a simple comparison of the various solutions available in the literature, in terms of flight times, including the simple non-optimal solution proposed by Dub  
Figure 7: Time variation of orbital parameters and control variable in a near-optimal transfer to a circular heliocentric orbit of radius \(r_{f}=0.32\) au and inclination \(i_{f}=60\) deg using the two-phase strategy (\(a_{c}=0.183\) mm/s\({}^{2}\)). Black circle \(\rightarrow\) start; black square \(\rightarrow\) arrival; blue triangle \(\rightarrow\) first phase end; dashed red line \(\rightarrow\) minimum solar distance. Figure 8: Near-optimal trajectory to a circular heliocentric orbit of radius \(r_{f}=0.32\,\mathrm{au}\) and inclination \(i_{f}=60\,\mathrm{deg}\) using the two-phase strategy (\(a_{c}=0.183\,\mathrm{mm}/\mathrm{s}^{2}\)). Black circle \(\rightarrow\) start; black square \(\rightarrow\) arrival; blue triangle \(\rightarrow\) first phase end; blue star \(\rightarrow\) perihelion; orange circle \(\rightarrow\) the Sun; blue line \(\rightarrow\) parking orbit; red line \(\rightarrow\) target orbit; black line \(\rightarrow\) transfer trajectory. Figure 9: Time variation of orbital parameters and control variable in an optimal transfer to a circular heliocentric orbit of radius \(r_{f}=0.32\,\mathrm{au}\) and inclination \(i_{f}=60\,\mathrm{deg}\) (\(a_{c}=0.183\,\mathrm{mm}/\mathrm{s}^{2}\)). Black circle \(\rightarrow\) start; black square \(\rightarrow\) arrival; dashed red line \(\rightarrow\) minimum solar distance. Figure 11: Comparison of the two-phase strategy (blue line) and the optimal control law (black line) with \(a_{c}=0.183\,\mathrm{mm}/\mathrm{s}^{2}\). Dashed red line \(\rightarrow\) minimum solar distance; filled circle \(\rightarrow\) start; filled square \(\rightarrow\) arrival. Figure 6:0 Comparison of flight times obtained with various transfer strategies: [PERSON] et al. [21], [PERSON] et al. [30], two-phase strategy, and optimal result (\(a_{c}=0.183\,\mathrm{mm}/\mathrm{s}^{2}\)). Figure 6:1 Optimal trajectory to a circular heliocentric orbit of radius \(r_{f}=0.32\,\mathrm{au}\) and inclination \(i_{f}=60\,\mathrm{deg}\) (\(a_{c}=0.183\,\mathrm{mm}/\mathrm{s}^{2}\)). Black circle \(\rightarrow\) start; black square \(\rightarrow\) arrival; blue star \(\rightarrow\) perihelion; orange circle \(\rightarrow\) the Sun; blue line \(\rightarrow\) parking orbit; red line \(\rightarrow\) target orbit; black line \(\rightarrow\) transfer trajectory.