## 2 Trajectory Design and Optimization
### Thrust Vector Model of the Gradient-Index Sail with a Sun-Facing Attitude  
Consider an SFGIS and neglect the billowing of the sail membrane due to the interaction of the sail exposed surface with the solar radiation pressure [13; 14; 15]. In this specific configuration, we define the sail nominal plane \(\mathcal{P}\) as the plane that (ideally) contains the sail membrane and observe that the \(\mathcal{P}\) plane is perpendicular to the Sun-spacecraft line, because the sail attitude is Sun-facing; see the conceptual sketch in Figure 1.  
Keeping in mind the scheme of Figure 1, we introduce the unit vector \(\hat{\mathbf{n}}\) normal to the plane \(\mathcal{P}\) in the direction opposite to the Sun, and the unit vector \(\hat{\mathbf{m}}\) indicating a fixed  
Figure 1: Sketch of the SFGIS with the normal unit vector \(\hat{\mathbf{n}}\) and the reference unit vector \(\hat{\mathbf{m}}\), whose directions are fixed in a (spacecraft) body reference frame. Note that the sail nominal plane \(\mathcal{P}\) is perpendicular to the Sun–spacecraft line, and the unit vector \(\hat{\mathbf{m}}\) belongs to the \(\mathcal{P}\) plane. The conceptual scheme of the waveguide array was adapted from Ref. [1], courtesy of Dr. [PERSON]. direction along the sail nominal plane. In particular, the unit vector \(\hat{n}\) belongs to the Sun-spacecraft line, while the direction of \(\hat{m}\) (which is fixed in \(\mathcal{P}\) by assumption) depends on the arrangement of the waveguide arrays covering the membrane of the solar sail [1]. According to [PERSON] et al. [1], the SFGIS-induced propulsive acceleration vector \(a_{p}\) can be written as  
\[a_{p}=a_{c}\left(\frac{r_{\oplus}}{r}\right)^{2}\left[\eta_{n}\,\hat{n}+\eta_{m }\,\hat{m}\right] \tag{1}\]  
where \(r\) is the spacecraft's distance from the Sun, \(r_{\oplus}=1\,\mathrm{AU}\) is a reference distance that coincides with the average Earth-Sun distance, and \(a_{c}\) is the characteristic acceleration (which is the typical performance parameter in solar sail trajectory design) defined as the maximum value of \(\|a_{p}\|\) when the spacecraft's distance from the Sun is equal to \(r_{\oplus}\). In Equation (1), the two terms \(\{\eta_{n},\,\eta_{m}\}\) are dimensionless coefficients that depend on the specific design of the waveguide array covering the sail membrane [1]. According to Ref. [1], the value of the pair \(\{\eta_{n},\,\eta_{m}\}\) can be obtained by using a semi-analytical procedure employing the ray tracing technique. The result is [1]  
\[\eta_{n}=0.6299\quad,\quad\eta_{m}=0.7767 \tag{2}\]  
with \(\sqrt{\eta_{n}^{2}+\eta_{m}^{2}}=1\), according to the definition of the characteristic acceleration \(a_{c}\). The expression of the propulsive acceleration vector in Equation (1) allows us to make some important considerations. First, note that the thrust vector is fixed in a body (spacecraft) reference frame. Consequently, it is not possible to obtain the condition \(\|a_{p}\|=0\) during the flight. In other terms, the heliocentric trajectory of a spacecraft propelled by an SFGIS contains no coasting arcs. This interesting aspect is similar to the one found in the mathematical model describing the thrust vector of a diffractive sail [4; 5; 16; 17] with a Sun-facing attitude, as discussed in Ref. [18]. More precisely, the main difference between the thrust model described by Equation (1) and the one proposed by [PERSON] and [PERSON] [18] is the value of the so-called \"cone angle\" \(\alpha\), i.e., the angle between the direction of the propulsive acceleration vector and the Sun-spacecraft line. In fact, in a Sun-facing diffractive sail, the value of the cone angle is \(45^{\circ}\), while for the case of an SFGIS, according to Equations (1) and (2), we have  
\[\alpha=\arctan\left(\frac{\eta_{m}}{\eta_{n}}\right)\simeq 51^{\circ} \tag{3}\]  
which is therefore a constant of motion.