# Optimal Trajectories of Diffractive Sail to Highly Inclined Heliocentric Orbits
## 3 Transfer Strategy  
As mentioned in the previous section, it is not possible to determine a priori whether or not the minimum-time trajectory requires a close pass to the Sun in order to reduce the time necessary to increase the orbital inclination and reach the plane of the target orbit. To better analyze this problem, unconstrained optimal transfers to the target orbit were preliminarily calculated, without taking into account the minimum distance from the Sun reached during the transfer. These transfers were obtained by imposing to reach in the minimum time a circular orbit of a radius equal to that of the target orbit (i.e., \(r=r_{f}=0.32\,\text{au}\)), but varying the final orbit inclination in the interval \(i_{f}\in[0,60]\) deg with a step of \(5\,\text{deg}\). Simulations were carried out using a characteristic acceleration \(a_{c}=0.26\,\text{mm}/\text{s}^{2}\), which is compatible with the performance of the reflective sail used for the Solar Polar Imager mission study [29]. The results can be seen in Figure 2, where the optimal transfer times \(\Delta t\), the total number of revolutions around the Sun \(n\), and the minimum distance from the Sun reached during the transfer are illustrated. The figure shows that up to an inclination of about 20 deg, the trajectory tends to the target orbit without the minimum distance dropping below \(r_{f}\), while for larger inclinations, the minimum-time trajectory would tend to approach the Sun up to distances less than \(r_{f}\) and then rise to the final desired value. Assuming that the minimum distance from the Sun compatible with the maximum thermal load supportable by the sail is \(r_{c}=0.3\,\mathrm{au}\), which corresponds to the perihelion distance of Mercury from the Sun, it is clear that unconstrained transfers with inclinations greater than about 20 deg are not feasible. Since the computation of minimum-time trajectories with a constraint on the minimum distance from the Sun usually presents considerable numerical convergence difficulties [27], it was decided to study near-optimal trajectories, using a two-step strategy. In more detail, having fixed the value of \(r_{c}\), with the help of the results in Figure 2 it is possible to determine the maximum inclination, denoted by \(i^{*}\), achievable with an unconstrained optimal transfer such that the satellite is taken to the final distance \(r_{f}\) without violating the constraint on \(r_{c}\). Clearly, it follows from the definition that \(i^{*}\leq i_{f}\). The near-optimal trajectory consists of a first phase (referred to as approach phase) in which the spacecraft reaches a circular orbit with radius \(r_{f}\) and inclination \(i^{*}\) using a minimum-time trajectory. The second phase consists of an optimal cranking trajectory, i.e., a minimum-time trajectory in which the initial and final distances coincide and the orbital plane is rotated until the final orbital inclination is obtained. At the end of the second phase, the spacecraft has thus reached the circular orbit of radius \(r_{f}\) and inclination \(i_{f}\). The strategy adopted is qualitatively illustrated in Figure 3 in which the blue triangle represents the instant at which the first phase ends and the second begins. Figure 2: Simulations of optimal trajectories with \(a_{c}=0.26\,\mathrm{mm/s^{2}}\) to a circular heliocentric orbit of radius \(r=0.32\,\mathrm{au}\) for different values of final orbital inclination. Using this strategy in the present case, the first phase consists of reaching a circular orbit of radius \(r_{f}=0.32\,\mathrm{au}\) and inclination \(i^{*}=20\,\mathrm{deg}\), while the cranking phase consists of the minimum-time trajectory that transfers the spacecraft to the orbit with inclination \(i_{f}=60\,\mathrm{deg}\) and the radius still equal to \(0.32\,\mathrm{au}\). The results of the simulations are summarized in Figure 4, which shows the time variation of the semimajor axis, eccentricity, inclination, distance from the Sun, and sail clock angle.