## 2 Thrust Vector Description and Mathematical Model
### Analytical Thrust Vector Model of Sun-Facing Diffractive Sailcraft  
Consider a two-dimensional heliocentric mission scenario in which the Sun-facing diffractive sailcraft moves in the plane of the reference elliptical orbit. The vehicle orbits in the interplanetary space under the effect of the gravitational attraction of the Sun and the thrust provided by the propellantless propulsion system. The latter has an array of electro-optically controlled panels, formed from a grid of binary metamaterial. In this case, paralleling the approach proposed by [PERSON] [2] and using the analytical results obtained in [32], the propulsive acceleration vector \(\mathbf{a}\) can be expressed as a function of the solar distance \(r\), which is appropriately dimensionless by means of the reference distance \(r_{\oplus}\triangleq 1\,\mathrm{AU}\); the Sun-sailcraft radial unit vector \(\mathbf{\hat{r}}\); the transverse unit vector \(\mathbf{\hat{t}}\) in the direction of the sailcraft's heliocentric velocity vector \(\mathbf{v}\), i.e., \(\mathbf{v}\cdot\mathbf{\hat{t}}>0\) during all the interplanetary flight; the dimensionless binary control term \(\tau\in\{-1,\,1\}\), which allows the verse of the transverse component of the thrust to be changed; and the performance parameter \(a_{c}\) commonly indicating as \"characteristic acceleration\", in analogy with the case of an RSS-based spacecraft. In particular, \(a_{c}\) is the maximum value of \(\|\mathbf{a}\|\) when the sail solar distance is equal to \(r_{\oplus}\). Therefore, according to [32], the compact form of the propulsive acceleration vector \(\mathbf{a}\) of a Sun-facing diffractive sailcraft is \[\mathbf{a}=\frac{a_{c}}{\sqrt{2}}\left(\frac{r_{\oplus}}{r}\right)^{2}\left(\mathbf{\hat{r}} -\tau\mathbf{\hat{f}}\right) \tag{1}\]  
The latter equation is valid when the grating momentum unit vector, which is fixed in a body reference frame, is aligned with the direction of \(\mathbf{\hat{t}}\). In this case, during the interplanetary flight, the direction of the vector \(\mathbf{a}\) remains in the plane of the reference (parking) elliptical orbit, and the motion of the Sun-facing diffractive sailcraft is two-dimensional. Note that \(\tau^{2}\equiv 1\) so that when \(r=r_{\oplus}\), one has \(\|\mathbf{a}\|=a_{c}\), as expected. In particular, \(\tau=1\) indicates that the transverse component of the thrust is opposite to the direction of the vehicle's heliocentric velocity vector \(\mathbf{v}\), while the case of \(\tau=-1\) is consistent with an increasing in the sailcraft's kinetic energy. However, as highlighted in [32], the choice of the sign of the binary control variable \(\tau\) is rather arbitrary, since it depends on the direction of the grating momentum unit vector whose position in the membrane plane is fixed. The important aspect that emerges from Equation (1) is that the binary variable \(\tau\) is actually able to vary the verse of the transverse component of the thrust vector, as sketched in Figure 1.  
Furthermore, Equation (1) indicates that the magnitude of \(\mathbf{a}\) depends linearly on the characteristic acceleration and varies inversely quadratically with the solar distance. In other words, once the value of \(a_{c}\) and \(r\) have been assigned, the magnitude of the propulsive acceleration vector \([\|\mathbf{a}\|=a_{c}\left(r_{\oplus}/r\right)^{2}]\) is fixed as are its radial \([\mathbf{a}\cdot\mathbf{\hat{r}}=\left(a_{c}/\sqrt{2}\right)\left(r_{\oplus}/r\right) ^{2}]\) and transverse \([\mathbf{a}\cdot\mathbf{\hat{t}}=-\tau(a_{c}/\sqrt{2})\left(r_{\oplus}/r\right)^{2}]\) components. In particular, the verse of the transverse component of \(\mathbf{a}\) depends on the value of \(\tau\). This aspect clearly indicates a reduced maneuverability of the Sun-facing diffractive sailcraft ultimately linked to the choice of having a fixed orientation during flight and, therefore, to exploit the possibility of passively maintaining this specific attitude.