# Solar Sail-Based Mars-Synchronous Displaced Orbits for Remote Sensing Applications
## 3 Numerical Results
### Evaluation of the Mars-Spacecraft Distance  
The effectiveness of a scientific mission to monitor the high-latitude regions of Mars depends on the distance \(d\) of the spacecraft from the planet; see the scheme in Figure 1. Note that the monitoring of Mars polar region can be also obtained with a Keplerian orbit with a suitable inclination. However, in that case, the observation of the high-latitude regions of the planet is carried out only in a part of the spacecraft (planetocentric) orbit, while a DNKO ensures continuous monitoring. By replicating the procedure described in Ref. [27], in this section we quantify the Mars-spacecraft distance, showing that it depends on four parameters, namely, the DNKO radius \(\rho\), the DNKO displacement \(H\), the true anomaly \(\
u\) of Mars, and the point in the planetary orbit (described by the angle \(\overline{\
u}\)) at which the spacecraft and Mars have the same angular position in their (always parallel) planes of motion. For this purpose, consider the components of the Sun-spacecraft position vector \(\mathbf{r}\) in the spherical reference frame \(\mathcal{T}_{O}\), that is  
\[\left[\mathbf{r}\right]_{\mathcal{T}_{O}}= \begin{bmatrix}\rho\,\cos\theta\\ \rho\,\sin\theta\\ H\end{bmatrix}  \tag{12}\]  
Figure 5: Required values of \(\{\alpha,\,\beta\}\) as a function of \(\{\rho/a,\,H/a\}\) for scenarios \(A\) and \(B\), which are consistent with a high-performance and a medium-performance ideal solar sail, respectively. where \(\theta\) is the angle between the projection of \(\mathbf{r}\) into the orbital plane of Mars and axis \(p\), as it is sketched in Figures 1 and 2. The components of the Sun-Mars position vector in \(\mathcal{T}_{O}\), instead, are given by  
\[\left[\mathbf{r}_{\mathcal{G}^{\prime}}\right]_{\mathcal{T}_{O}}= \begin{bmatrix} \rho_{\mathcal{G}^{\prime}}\,\cos\
u\\ \rho_{\mathcal{G}^{\prime}}\,\sin\
u\\ 0\end{bmatrix}  \tag{13}\]  
where \(\rho_{\mathcal{G}^{\prime}}\) is given by Equation (11). In particular, the value of \(\theta\) may also be written as  
\[\theta=\omega\left(t-t_{0}\right)+\theta_{0} \tag{14}\]  
where \(t\) is the time and \(\theta_{0}\) is the angular offset between the spacecraft and axis \(p\) at the initial time \(t_{0}\triangleq 0\). The latter is here defined, without loss of generality, as the time instant at which Mars passes the perihelion (i.e., \(\
u(t_{0})\triangleq 0\)), from which it follows that \(\theta_{0}\) also describes the angular offset between the spacecraft and Mars at the reference time \(t_{0}\). Therefore, for net of multiples of one period of revolution of Mars (corresponding to about 687 days), the time \(t\) can be written as a function of \(\
u\) as  
\[t=\sqrt{\frac{a^{3}}{\mu_{\odot}}}\left(E-e\,\sin E\right) \tag{15}\]  
where \(E\in[0,\,2\pi)\) rad is the eccentric anomaly of Mars, given by the well-known formula  
\[E\triangleq 2\,\arctan\left(\sqrt{\frac{1-e}{1+e}}\,\tan\frac{\
u}{2}\right) \tag{16}\]  
The value of \(\theta_{0}\) may be determined backward from the value of \(\
u\) (namely, \(\overline{\
u}\)) such that \(\theta=\overline{\
u}\), that is, the value of \(\
u\) (assumed to exist) at which the solar sail and Mars have the same angular position in their planes of motion around the Sun. Accordingly, the value of \(E\) when \(\
u=\overline{\
u}\) is given by  
\[\overline{E}=2\,\arctan\left(\sqrt{\frac{1-e}{1+e}}\,\tan\frac{\overline{\
u }}{2}\right) \tag{17}\]  
the corresponding time instant is  
\[\overline{t}=\sqrt{\frac{a^{3}}{\mu_{\odot}}}\left(\overline{E}-e\,\sin \overline{E}\right) \tag{18}\]  
while \(\theta_{0}\) is given by  
\[\theta_{0}=\overline{\
u}-\omega\,\overline{t} \tag{19}\]  
Bearing in mind that \(\overline{t}=\overline{t}(\overline{\
u})\) (see Equations (17) and (18)), then Equation (19) gives \(\theta_{0}=\theta_{0}(\overline{\
u})\), which is plotted in Figure 6. Note that \(|\theta_{0}|\) is maximum when \(\overline{\
u}\simeq\{94.02,\,265.98\}\) deg and its value is about 10.71 deg.