## 4 Simplified Form of the Control Law  
The problem of studying the heliocentric phasing maneuver, in the case where the reference orbit is circular and the spacecraft is equipped with a low-performance RSS, was studied in 2003 by [PERSON] [42] using an elegant mathematical model that describes the linearized dynamics of the sailcraft in the case where its solar distance remains, during the flight, close to the radius of the circular reference orbit. This linearized model, which is essentially a sort of extension of the classical Clohessy-Wiltshire model, approximates well the actual heliocentric dynamics of the sailcraft even when its phasing angle at the virtual point is large. Therefore, for this reason, it is well suited to quickly study the phasing performance in a heliocentric circular case. In this context, assume that \(e_{0}=0\), \(p_{0}=r_{\oplus}=1\,\mathrm{AU}\), and observe that the angle \(\phi\) between the Sun-sailcraft line and the Sun-virtual point line is given by  
\[\phi=\theta-\sqrt{\frac{\mu_{\odot}}{r_{\oplus}^{3}}}\,t\equiv\theta-\widetilde {t} \tag{17}\]  
with \(\theta(\widetilde{t}_{0})=0\,\mathrm{deg}\), and introduce a dimensionless auxiliary variable \(\rho\) defined as  
\[\rho\triangleq\frac{r}{p_{0}}-1\equiv\widetilde{r}-1 \tag{18}\]Keeping in mind that the dimensionless form of the propulsive acceleration vector is given by Equation (5), and in parallel with the procedure described in [42], it is possible to obtain the following set of linear differential equations describing the (linearized) dynamics of the Sun-facing diffractive sailcraft in the vicinity of the reference circular orbit, as follows:  
\[\rho^{\prime\prime}=2\,\phi^{\prime}+3\,\rho+\frac{\widetilde{a}_{c}}{\sqrt{2}},\quad\phi^{\prime\prime}=-2\,\rho^{\prime}-\tau\,\frac{\widetilde{a}_{c}}{ \sqrt{2}} \tag{19}\]  
with zero initial conditions, viz. \[\rho(\widetilde{t}_{0})=0,\quad\rho^{\prime}(\widetilde{t}_{0})=0,\quad\phi (\widetilde{t}_{0})=0,\quad\phi^{\prime}(\widetilde{t}_{0})=0 \tag{20}\]  
For a given temporal variation of the binary control parameter \(\tau\), the linear differential system (19) with the initial conditions (20) can be solved using a standard method to obtain the function \(\phi=\phi(\widetilde{t})\), which gives the temporal variation of the angle between the position vectors of the sailcraft and the virtual point. However, the final constraints of the phasing maneuver require that the diffractive sail return to the reference circular orbit at the end of the transfer, i.e., when \(\widetilde{t}=\widetilde{t}_{f}\). This corresponds to obtaining, at \(\widetilde{t}=\widetilde{t}_{f}\), the following conditions:  
\[\rho(\widetilde{t}_{f})=0,\quad\rho^{\prime}(\widetilde{t}_{f})=0,\quad\phi^ {\prime}(\widetilde{t}_{f})=0 \tag{21}\]  
while the value \(\phi(\widetilde{t}_{f})\) gives the actual phasing angle obtained through the maneuver, that is, \(\phi(\widetilde{t}_{f})\equiv\Delta\phi\). In his original work [42] on an RSS-based scenario with a piecewise-constant attitude, [PERSON] suggests using a very simple and effective control law that allows for naturally satisfying the final constraints given by Equation (21). In particular, [PERSON] indicates that a control law in which the pitch angle of the RSS is kept constant for the first half of the flight (i.e., for \(\widetilde{t}\in[\widetilde{t}_{0},\,\widetilde{t}_{f}/2]\)) and then reversed for the remaining second half (i.e., for \(\widetilde{t}\in[\widetilde{t}_{f}/2,\,\widetilde{t}_{f}]\)) allows the sailcraft to meet the final constraints of Equation (21) if \(\widetilde{t}_{f}=2m\pi\), where \(m\) is a positive integer.