## 3 Mission Application and Trajectory Simulations  
The optimal control law of Equation (19) was used to obtain the minimum-time orbit-to-orbit transfer in some typical heliocentric mission scenarios. In all the numerical simulations, the differential equation was integrated using a PECE solver [29] based on the Adams-Bashforth method with a tolerance of \(10^{-12}\), while the associated boundary value problem was solved by a shooting procedure [28] with a tolerance of \(10^{-6}\). The initial guess was obtained by adapting the procedure recently described in Ref. [30].  
The optimization procedure was first validated using, as a comparison, the results reported in Ref. [1] which, however, refer to ephemeris-constrained transfer scenarios. Therefore, it was reasonable to expect that the flight times obtained with the technique proposed in this paper (which refer to a less constrained orbit-to-orbit transfer) would be lower than those reported in Ref. [1]. In the case of an unconstrained sail clock angle, consider an Earth-Venus mission scenario, and assume a characteristic acceleration \(a_{c}=0.175\,\mathrm{mm}/\mathrm{s}^{2}\)[1]. The proposed procedure gave a minimum flight time of about 434.8 days, while Ref. [1] indicated a (constrained) flight time of 456.3 days with a departure on 29 April 2021 and arrival on 27 July 2022. The time variation in the sail clock angle during the flight is shown in Figure 4, while Figure 5 shows the optimal transfer trajectory in a heliocentric-ecliptic Cartesian reference frame [31].  
Figure 4: Time variation in the (unconstrained) sail clock angle \(\delta\) in the minimum-time Earth–Venus mission scenario when \(a_{c}=0.175\,\mathrm{mm}/\mathrm{s}^{2}\). Blue dot \(\rightarrow\) starting point; red square \(\rightarrow\) arrival point.  
The second mission scenario illustrated in Ref. [1] is a heliocentric orbit transfer from Earth to asteroid 433 Eros. In that case, considering again a characteristic acceleration of \(0.175\,\mathrm{mm}/\mathrm{s}^{2}\), [PERSON] et al. [1] indicated an optimal (constrained) flight time of 1207.3 days, departing on 11 November 2024 and arriving on 2 March 2028. The procedure proposed in this paper gave a minimum flight time (in an orbit-to-orbit transfer), of 1125 days, with the optimal time variation in the clock angle shown in Figure 6 and the optimal transfer trajectory shown in Figure 7. Note that the function \(\delta=\delta(t)\) drawn in Figure 6 is only apparently discontinuous, due to the range of variation chosen to describe the sail clock angle.  
Figure 5: Ecliptic projection and isometric view of the rapid transfer trajectory in an Earth–Venus mission scenario, when \(a_{c}=0.175\,\mathrm{mm}/\mathrm{s}^{2}\) and the sail clock angle \(\delta\) is unconstrained. The \(z\)-axis of the isometric view is exaggerated to highlight the three-dimensionality of the transfer trajectory. Black line \(\rightarrow\) spacecraft transfer trajectory; blue line \(\rightarrow\) Earth’s orbit; red line \(\rightarrow\) Venus’s orbit; filled star \(\rightarrow\) perihelion; blue dot \(\rightarrow\) starting point; red square \(\rightarrow\) arrival point; orange dot \(\rightarrow\) the Sun.  
Figure 6: Time variation of the (unconstrained) sail clock angle \(\delta\) in a minimum-time Earth–asteroid 433 Eros mission scenario when \(a_{c}=0.175\,\mathrm{mm}/\mathrm{s}^{2}\). The legend is reported in Figure 4.  
The simulations performed demonstrated that the optimization method and control law described above provided numerical results consistent with the literature and therefore could be used to analyze new mission scenarios, as discussed in the next subsection.