# Solar Sail-Based Mars-Synchronous Displaced Orbits for Remote Sensing Applications
## 4 Preliminary Study of the Heliocentric Transfer Trajectory
Note that, in this case, the spacecraft lightness number is obtained from Equation (9). The trajectory optimization problem consists of finding \(\alpha=\alpha(t)\) and \(\delta=\delta(t)\) that maximize the performance index  
\[\dot{r} =v_{r} \tag{23}\]1  
while driving the spacecraft states to their target values at the (unknown) time \(t_{f}\), viz. \[\dot{r} =v_{r} \tag{23}\]2  
Note that the final value of \(\theta\) is left free during the optimization process. In particular, the optimal (i.e., minimum-time) trajectory was obtained with an indirect approach by enforcing the necessary conditions for optimality according to [PERSON]'s maximum principle [49]. The mathematical model used for the optimization procedure, that is, the expression of the Hamiltonian function, the expression of the transversality condition, and the Euler-Lagrange equations, is detailed in Ref. [35]. The differential equations governing the two-point boundary-value problem associated with the optimization problem were integrated in double precision using a variable order Adams-Bashforth-Moulton solver scheme [50] with absolute and relative errors equal to \(10^{-12}\). In particular, the two-point boundary-value problem associated with the indirect-approach-based optimization procedure was solved using a shooting method [51] with a tolerance of \(10^{-8}\). In this context, the first guess of the initial unknown costates was obtained by adapting the procedure  
\[\dot{r} =v_{r} \tag{23}\]3 Solar sail thrust vector control angles \(\{\alpha,\,\delta\}\). Consistent with an insertion in a circular DNKO, the velocity components \(\{v_{r},\,v_{\gamma}\}\) are zero at the end of both transfers, while the final values of \(v_{\theta}\) are both equal to \(\rho\,\omega\), that is, \(v_{\theta}\simeq 24.101\,\mathrm{km/s}\) for scenario \(A\) and \(v_{\theta}\simeq 24.115\,\mathrm{km/s}\) for scenario \(B\). As far as the time variation of the control angle \(\alpha\) is concerned, note that its value at the end of the transfer does not coincide (in general) with the value necessary to maintain the desired DNKO. Therefore, at the end of the transfer, it will be necessary to implement a new control law for maintaining the DNKO, i.e., the one provided by Equation (8) with unit vector \(\hat{n}\) belonging to the \((\hat{\mathbf{p}}_{r},\hat{\mathbf{p}}_{\gamma})\) plane. Finally, note the trend of the clock angle \(\delta\) over time. In both the analyzed scenarios, the \(\delta\) value is close to \(0\,\mathrm{deg}\) or \(360\,\mathrm{deg}\), which means that the \((\hat{\mathbf{p}}_{r},\hat{\mathbf{n}})\) plane is almost parallel to the \((\hat{\mathbf{p}}_{r},\hat{\mathbf{p}}_{\theta})\) plane for the entire duration of the orbit-to-orbit transfer. This interesting result derives from the fact of having considered target  
\[\dot{r} =v_{r} \tag{23}\]4 Optimal control and resulting trajectory for scenario \(B\), where \(\beta\simeq 0.05\). (**a**) Variation of \(\{\alpha,\,\delta\}\) with \(t\); (**b**) minimum-time optimal transfer trajectory. \[\dot{r} =v_{r} \tag{23}\]5 Time variation of spacecraft states for scenario \(B\), where \(\beta\simeq 0.05\). (**a**) Variation of \(\{r,\,\rho,\,H\}\) with \(t\); (**b**) variation of \(\{v_{r},\,v_{\gamma},\,v_{\theta}\}\) with \(t\). DNKOs very close to the orbital plane of the parking orbit and, consequently, from having quasi-two-dimensional transfers between coaxial circular orbits of assigned radii.