## 3 Results of Numerical Simulations and Case Studies
### Earth-Based Scenario  
In this case, the small value of \(e_{0}\) makes the results less sensitive to the value of \(v_{0}\). This aspect is evident in Figure 7.  
Figure 6: Minimum flight time for a phasing ahead in a Mercury-based mission scenario when \(a_{c}=0.1\,\mathrm{mm}/\mathrm{s}^{2}\).  
Figure 7: Minimum flight time as a function of \(\Delta\phi\) and \(v_{0}\) in an Earth-based scenario, when \(a_{c}=0.1\,\mathrm{mm}/\mathrm{s}^{2}\). (**a**) Phasing ahead case; (**b**) phasing behind case.  
In fact, Figure 7 indicates that, for a given value of \(\Delta\phi\), the generic function \(\Delta t=\Delta t(v_{0})\) has a substantially constant value with oscillations of a few days, even when the phasing angle reaches high values. This aspect is confirmed by Figure 8, which shows the curve levels of the function \(\Delta t=\Delta t(v_{0},\Delta\phi)\). In particular, the curve levels in the figure essentially collapse into a single line that indicates the variation of the minimum flight time with the phasing angle.  
For example, using the graph reported in Figure 8, one obtains that a phasing maneuver in which \(\Delta\phi=60\deg\) (or \(\Delta\phi=-60\deg\)) requires a minimum flight time of about \(670\,\mathrm{days}\) (or \(600\,\mathrm{days}\)) when the characteristic acceleration of the Sun-facing diffractive sailcraft is \(a_{c}=0.1\,\mathrm{mm}/s^{2}\). This is an interesting result because such a phasing maneuver models a simplified version of an interplanetary transfer towards the Sun-Earth triangular Lagrange points [36]. In these cases of a transfer towards \(L_{4}\) or \(L_{5}\), the trajectory is reported in Figure 9 when the starting true anomaly is \(v_{0}=90\deg\) and \(a_{c}=0.1\,\mathrm{mm}/s^{2}\), while Figure 10 shows the corresponding optimal control law.  
Figure 8: Minimum flight time for a phasing ahead and behind in an Earth-based mission scenario when \(a_{c}=0.1\,\mathrm{mm}/s^{2}\). (**a**) Phasing ahead case; (**b**) phasing behind case.  
The last part of this subsection analyzes the dependence of the flight time on the value of the characteristic acceleration. The value of \(a_{c}\) has been selected in the interval \([0.06,\,0.12]\,\mathrm{mm}/\mathrm{s}^{2}\). In this case, the initial true anomaly has been set to \(v_{0}=0\,\mathrm{deg}\) (i.e., a starting point at the perihelion has been selected) to reduce the computational effort. Note, however, that the previous analysis has highlighted that the flight time is substantially insensitive to the value of \(v_{0}\). The results of this parametric study are reported in Figure 11, which shows that the dependence of the minimum flight time on the value of the characteristic acceleration becomes more marked as the value of the phasing angle \(\Delta\phi\) increases. For example, in the case of \(\Delta\phi=30\,\mathrm{deg}\), the flight time of a sailcraft with \(a_{c}=0.06\,\mathrm{mm}/\mathrm{s}^{2}\) is about 120 days greater than that obtainable with \(a_{c}=0.12\,\mathrm{mm}/\mathrm{s}^{2}\). On the other hand, if \(\Delta\phi=5\,\mathrm{deg}\), the difference in flight time is about 40 days. The plots in Figure 11 allow the reader to quickly estimate the value of \(\Delta t\) for a low-performance diffractive sailcraft in an Earth-based mission scenario, when the phasing angle range is \(\Delta\phi\in[-30,\,30]\,\mathrm{deg}\). However, in this case, the small value of \(e_{0}\) suggests the use of a simplified mathematical model where the reference orbit is circular. This limiting case is discussed in the next subsection.  
Figure 10: Optimal control law \(\tau=\tau(t)\) in an Earth-based scenario when \(v_{0}=90\,\mathrm{deg}\) and \(a_{c}=