## 4 Discussion and Conclusions  
A diffractive sail can be considered as a sort of evolution of the classical concept of a reflecting solar sail, which received much attention in the last decade thanks to various space missions successfully flown. A solar sail with a diffractive film has interesting features; for example, it simplifies the spacecraft guidance by maintaining a Sun-facing configuration, and it guarantees good transfer performance. This paper discusses a simplified thrust model for a flat diffractive sail equipped with an electro-optically controlled panel that allows the direction of the grating momentum unit vector to be changed during the flight. Using such a feature, the in-plane component of the diffractive sail thrust vector can be easily flipped without changing the sail attitude with respect to a classical orbital reference frame. In this sense, the proposed thrust model can be thought of as a simplified version of the model for interplanetary trajectory analysis recently discussed by the authors [53]. In particular, the mathematical model proposed in this paper can be used to perform a preliminary design of an optimal interplanetary trajectory in a two-dimensional mission case, whose main limitation is related to the eccentricity of the initial and target orbits, which are both set to zero. The interesting aspect that emerges from the numerical results is that the performance of a Sun-facing diffractive sail with an active (binary) electro-controlled film is comparable or, in some cases, even better than those obtained with a more conventional \begin{table}
\begin{tabular}{c c c c} \hline
**Target Planet** & **Reflective Sail** & **Diffractive Sail** & **Variation** \\ \hline Venus (\(\lx@sectionsign\)) & 205 days & 189 days & \(-\)8\% \\ Mars (\(\lx@sectionsign\)) & 408 days & 365 days & \(-\)10\% \\ Jupiter (\(\lx@sectionsign\)) & 3777 days & 2420 days & \(-\)36\% \\ \hline \end{tabular}
\end{table} 
Table 1: Minimum flight time in a classical Earth–target planet, circle-to-circle, coplanar transfer for a diffractive or a reflective sail with \(a_{c}=1\,\mathrm{mm}/\,\mathrm{s}^{2}\). Figure 13: Time variation of the control variable \(\tau\) for the three mission cases using a Sun-facing diffractive sail. Black circle \(\rightarrow\) start, black square \(\rightarrow\) arrival. reflective solar sail. More precisely, the numerical simulations have shown that the flight time saving obtainable with a diffractive sail, when compared with a reflective sail of equal (canonical) characteristic acceleration, is in the order of 10% for a transfer toward Mars or Venus, and it increases up to 36% in an Earth-Jupiter scenario. However, a thorough comparison between a reflective and a diffractive sail-based spacecraft cannot be confined to a simple evaluation of the optimal flight time. Indeed, although the two sails share the same concept of transforming the solar radiation pressure in propulsive acceleration without the use of propellant, the specific technology at the base of a diffractive sail concept (with electro-optically controlled panels) is different from the more conventional metalized film installed on a reflective sail surface. In this sense, a more accurate comparison between a diffractive (with active panels) and reflective sail should include a detailed mass breakdown model of the two propulsion systems. Finally, the numerical results indicate that using an optimal control law, that is, the optimal time variation of the switching parameter, a circle-to-circle orbit transfer requires a very simple guidance scheme that potentially permits finding an analytical solution to the trajectory design. The latter point is beyond the scope of this paper and is left to future research. Conceptualization, A.A.Q. ; methodology, A.A.Q. ; software, A.A.Q. ; writing--original draft preparation, A.A.Q. ; writing--review and editing, G.M. All authors have read and agreed to the published version of the manuscript. This work is partly supported by the University of Pisa, Progetti di Ricerca di Ateneo (Grant no.