## 2 Mathematical Preliminaries
### Equations of Motion  
The propulsive acceleration vector model of Equation (1) can be used to describe the two-dimensional motion of the diffractive sail in a heliocentric scenario, in which the spacecraft initially (i.e., at time \(t_{0}=0\)) traces a circular parking orbit of assigned radius \(r_{0}\). To that end, we introduce a heliocentric polar reference frame \(\mathcal{T}(O;r,\theta)\), whose origin coincides with the Sun's center of mass \(O\), where \(\theta\) is the polar angle measured counterclockwise from the Sun-spacecraft line at time \(t=t_{0}\) (see the scheme of Figure 7), \(\hat{\boldsymbol{i}}_{r}\) is the radial unit vector, and \(\hat{\boldsymbol{i}}_{\theta}\) is the transverse unit vector.  
Assume that at the initial time, the (body-fixed) unit vector \(\hat{\boldsymbol{p}}\) coincides with \(\hat{\boldsymbol{i}}_{\theta}\), and recall that in a Sun-facing configuration, the direction of the normal unit vector \(\hat{\boldsymbol{n}}\) is always aligned with the Sun-spacecraft line, that is, \(\hat{\boldsymbol{n}}\equiv\hat{\boldsymbol{i}}_{r}\) at any time instant. In this case, \(\hat{\boldsymbol{p}}\equiv\hat{\boldsymbol{i}}_{\theta}\) for \(t\geq t_{0}\), and according to Equation (2), the propulsive acceleration vector \(\boldsymbol{a}\) belongs to the plane \((\hat{\boldsymbol{i}}_{r},\hat{\boldsymbol{i}}_{\theta})\) for \(t\geq t_{0}\). Note that the latter plane coincides with the parking orbit plane. Equation (2) can be therefore rewritten as  
\[\boldsymbol{a}=\frac{a_{c}}{\sqrt{2}}\left(\frac{r_{\oplus}}{r}\right)^{2}\left( \hat{\boldsymbol{i}}_{r}-\tau\,\hat{\boldsymbol{i}}_{\theta}\right) \tag{4}\]  
which shows that the spacecraft heliocentric motion is two-dimensional, and the vehicle's dynamics can be described by means of a set of polar equations. More precisely, using Equation (4) to write the components of \(\boldsymbol{a}\) in \(\mathcal{T}\), the spacecraft equations of motion become  
\[\dot{r} =u \tag{5}\] \[\dot{\theta} =\frac{v}{r}\] (6) \[\dot{u} =-\frac{\mu_{\odot}}{r^{2}}+\frac{v^{2}}{r}+\frac{a_{c}}{\sqrt{2 }}\left(\frac{r_{\oplus}}{r}\right)^{2}\] (7) \[\dot{v} =-\frac{u\,v}{r}-\frac{a_{c}\,\tau}{\sqrt{2}}\left(\frac{r_{ \oplus}}{r}\right)^{2} \tag{8}\]  
Figure 7: Polar reference frame, parking orbit, and spacecraft state variables \(r\) and \(\theta\).  
where the dot symbol represents a derivative taken with respect to the time \(t\), \(\mu_{\odot}\) is the Sun's gravitational parameter, and \(u\) (or \(v\)) is the radial (or transverse) component of the spacecraft velocity vector. The first-order differential Equations (5)-(8) are completed by four initial conditions that model the spacecraft motion along the circular parking orbit, viz.  
\[r(t_{0})=r_{0}\quad,\quad\theta(t_{0})=0\quad,\quad u(t_{0})=0\quad,\quad v(t_{ 0})=\sqrt{\frac{\mu_{\odot}}{r_{0}}} \tag{9}\]  
where the initial value of the polar angle \(\theta\) has been set equal to zero without loss of generality because of the polar symmetry of the problem.  
The value of the binary-switching parameter \(\tau\), which appears in the last term of Equation (8), is obtained by solving an optimization problem in which the spacecraft trajectory is calculated in such a way as to minimize the time necessary for the spacecraft to reach a circular target orbit of assigned radius \(r_{f}\), starting from a circular (coplanar) parking orbit of radius \(r_{0}\). This mission scenario, which models a typical circle-to-circle orbit raising (or orbit lowering) when \(r_{f}>r_{0}\) (or \(r_{f}<r_{0}\)), is a typical application of a continuous-thrust propulsion system and is analyzed in the next section.