## 2 Thrust Vector Description and Mathematical Model
### Heliocentric Dynamics of a Sun-Facing Diffractive Sailcraft  
The heliocentric two-dimensional motion of the Sun-facing diffractive sailcraft can be conveniently studied by introducing a typical polar reference frame \(\mathcal{T}\) of unit vectors \(\left\{\boldsymbol{\hat{r}},\boldsymbol{\hat{t}}\right\}\), in which the polar angle \(\theta\) is measured from the direction of the eccentricity vector of the reference elliptical orbit, that is, from the line connecting the Sun and the perihelion point of that orbit. The latter is a Keplerian (closed) trajectory with a given value of both the semimajor axis \(a_{0}\) and the eccentricity \(e_{0}\) so that the semilatus rectum is \(p_{0}=a_{0}\left(1-e_{0}^{2}\right)\). In the special case of circular reference orbit (i.e., when \(e_{0}=0\) and \(p_{0}\equiv a_{0}\)), the polar angle  
Figure 2: Variation of the propulsive acceleration magnitude of a Sun-facing diffractive sailcraft with \(r\) and \(a_{c}\). The red (or green) area refers to the range of solar distance of the orbit of Mercury (or the Earth). is measured from the line connecting the Sun and the sailcraft at the beginning of the flight, i.e., at the initial time \(t=t_{0}\triangleq 0\). At that time, the state of the space vehicle is given by the following four scalar terms:  
\[r(t_{0})=\frac{p_{0}}{1+e_{0}\,\cos\upsilon_{0}},\quad\theta(t_{0})= \upsilon_{0},\quad\upsilon_{r}(t_{0})=\sqrt{\frac{\mu_{\odot}}{p_{0}}}\,e_{0} \,\sin\upsilon_{0},\\ v_{t}(t_{0})=\sqrt{\frac{\mu_{\odot}}{p_{0}}}(1+\,e_{0}\,\cos \upsilon_{0}) \tag{2}\]  
where \(\upsilon_{0}\) is the true anomaly of the sailcraft along the reference orbit at time \(t_{0}\), \(\upsilon_{r}\) (or \(v_{t}\)) is the radial (or transverse) component of the heliocentric velocity vector \(\mathbf{v}\), and \(\mu_{\odot}\) is the gravitational parameter of the primary body (i.e., the Sun). The initial conditions reported in the previous equation and the spacecraft dynamics can be more conveniently expressed in a dimensionless form by introducing the following terms:  
\[\widetilde{r}\triangleq\frac{r}{p_{0}},\quad\widetilde{\upsilon}_{r}\triangleq \frac{\upsilon_{r}}{\sqrt{\mu_{\odot}/p_{0}}},\quad\widetilde{\upsilon}_{t} \triangleq\frac{\upsilon_{t}}{\sqrt{\mu_{\odot}/p_{0}}},\quad\widetilde{t} \triangleq\frac{t}{\sqrt{p_{0}^{3}/\mu_{\odot}}},\quad\widetilde{a}_{c} \triangleq\frac{a_{c}}{\mu_{\odot}/p_{0}^{2}} \tag{3}\]  
so that Equation (2) becomes  
\[\widetilde{r}(\widetilde{t}_{0})=\frac{1}{1+e_{0}\,\cos\upsilon_{0}}\,\, \theta(\widetilde{t}_{0})=\upsilon_{0}\,\ \widetilde{\upsilon}_{r}(\widetilde{t}_{0})=e_{0}\,\sin \upsilon_{0}\,\ \widetilde{\upsilon}_{t}(\widetilde{t}_{0})=1+\,e_{0}\,\cos \upsilon_{0} \tag{4}\]  
Using the terms in Equation (3) and taking Equation (1) into account, the dimensionless form of the propulsive acceleration vector \(\widetilde{\mathbf{a}}\) is simply  
\[\widetilde{\mathbf{a}}=\frac{\widetilde{a}_{c}}{\sqrt{2}}\left(\frac{\widetilde{ \tau}_{\oplus}}{\widetilde{\tau}}\right)^{2}(\mathbf{\hat{r}}-\tau\,\mathbf{\hat{f}}) \tag{5}\]  
where \(\widetilde{\tau}_{\oplus}\triangleq r_{\odot}/p_{0}\). The expression of \(\widetilde{\mathbf{a}}\) given by the previous equation is used to write the equations of motion of the Sun-facing diffractive sailcraft in \(\mathcal{T}\), and the result is  
\[\widetilde{\tau}^{\prime}=\widetilde{\upsilon}_{r},\quad\theta^{\prime}=\frac {\widetilde{\upsilon}_{t}}{\widetilde{\tau}},\quad\widetilde{\upsilon}^{ \prime}_{r}=\frac{\widetilde{\upsilon}_{t}^{2}}{\widetilde{\tau}}+\frac{ \widetilde{a}_{c}\widetilde{\tau}_{\oplus}^{2}-\sqrt{2}}{\sqrt{2}\,\widetilde {\tau}^{2}},\quad\widetilde{\upsilon}^{\prime}_{t}=-\frac{\widetilde{\upsilon} _{r}\,\widetilde{\upsilon}_{t}}{\widetilde{\tau}}-\frac{\tau\,\widetilde{a}_{c }\,\widetilde{\tau}_{\oplus}^{2}}{\sqrt{2}\,\widetilde{\tau}^{2}} \tag{6}\]  
where the apex indicates the derivative with respect to the dimensionless time \(\widetilde{t}\) defined in Equation (3).