# Optimal Trajectories of Diffractive Sail to Highly Inclined Heliocentric Orbits
## 2 Mission Description and Models Used  
The mission scenario considered is the same as that proposed by [PERSON] and [PERSON] [21]. It is a mission using a 6U CubeSat [25] with a mass of 14 kg, similar to that used by the NASA Near-Earth Asteroid Scout [26], but in our study the CubeSat is propelled by a Sun-facing diffractive (solar) sail. The mission objective is to place the satellite in a target circular orbit around the Sun with a radius \(r_{f}=0.32\,\mathrm{au}\) and an inclination \(i_{f}=60\,\mathrm{deg}\) with respect to the Ecliptic. Thrust control is achieved by rotating the sail plane by an angle \(\delta\) around the radial direction. This rotation angle, called the clock angle, is the only control variable of the trajectory and can be generated by means of a suitable reaction wheel, as discussed in Ref. [21]. As the clock angle varies, the thrust vector describes a cone with a half angle of \(45\,\mathrm{deg}\) with respect to the Sun-spacecraft direction. Figure 1 shows the diffractive sail in its motion along the osculating orbit and highlights the radial-tangential-normal reference frame with unit vectors \(\hat{\mathbf{i}}_{R}\), \(\hat{\mathbf{i}}_{T}\), and \(\hat{\mathbf{i}}_{N}\) used to write the components of the propulsive acceleration vector. Figure 1: Definition of clock angle \(\delta\), radial–tangential–normal reference frame, and direction of propulsive acceleration \(a_{\mathrm{P}}\). Figure 1 also shows that \(\delta\), measured in the nominal plane of the sail, is the angle between \(\hat{\mathbf{t}}_{T}\) and \(\hat{\mathbf{t}}_{x}\), where the latter is in the opposite direction of the (diffractive film) grating momentum vector. The diffractive sail thrust model used in the simulations is that described in Ref. [19] in which the propulsive acceleration \(\mathbf{a}_{\text{p}}\) is a function of the characteristic acceleration \(a_{c}\) (defined as the maximum value of \(\|\mathbf{a}_{\text{p}}\|\) at the reference distance \(r=r_{\oplus}\equiv 1\,\text{au}\)), the distance \(r\), and the clock angle \(\delta\) as  
\[\mathbf{a}_{\text{p}}=\frac{a_{c}}{\sqrt{2}}\left(\frac{r_{\oplus}}{r}\right)^{2} \left(\hat{\mathbf{t}}_{R}+\hat{\mathbf{t}}_{T}\,\cos\delta+\hat{\mathbf{t}}_{N}\,\sin \delta\right) \tag{1}\]  
Note that, unlike a classical reflective sail, the thrust generated by a diffractive sail cannot be zeroed out, and thus the spacecraft trajectory cannot contain any coasting arc. The mission studied in the simulations consists of a heliocentric transfer trajectory between the orbit of the Earth and the circular orbit of arrival of a fixed radius and inclination. The spacecraft is subject only to the thrust generated by the diffractive sail and the Sun's gravitational pull, while all other sources of perturbation are neglected. The objective of this work is to determine the minimum-time trajectory that allows the spacecraft to be transferred to the target orbit starting from the Earth with zero hyperbolic excess velocity relative to the planet. In this study, the heliocentric orbit of the Earth is considered in its actual shape, that is, an ellipse with semimajor axis \(a=1\,\text{au}\) and eccentricity \(e=0.0167\). Previous studies concerning minimum-time trajectories to achieve a highly inclined orbit around the Sun with reflective sails [11; 27] have shown that depending on the value of the radius and inclination of the final orbit, as well as the value of the characteristic acceleration of the solar sail spacecraft, the optimal trajectory may steadily decrease the distance from the Sun until \(r_{f}\) is reached or, more frequently, may initially decrease the distance to values of \(r<r_{f}\), so as to take advantage of the increased solar radiation pressure to facilitate the change in orbital inclination, and then increase the distance from the Sun again until the target orbit is reached. In the latter case, the distance from the Sun must be constrained not to fall below a minimum value, which typically depends on the maximum tolerable temperature of the sail film [28].