# Venus Magnetotail Long-Term Sensing Using Solar Sails
## 4 Conclusions  
The literature results indicate that a low-performance photonic solar sail is able to rotate the apse line of a geocentric elliptic orbit to obtain the continuous monitoring of the Earth's magnetotail. This study extends that interesting result to a Venus-based mission scenario by considering a simple guidance law in which the sail nominal plane is perpendicular to the direction of the incoming sun rays. The mathematical model proposed in this work allows the required solar sail performance to be evaluated with a simple numerical procedure, while an analytical (approximate) equation was obtained for a quick estimation of the sail's characteristic acceleration as a function of the elements of the desired planetocentric (science) orbit. Numerical simulations indicate that the presence of an eclipse period slightly increases the value of the required characteristic acceleration compared to the ideal case of full illumination during flight. In this regard, an interesting extension of this work could be to consider an optimal control law (which can be even combined with the employment of a diffractive or a refractive membrane film) that considers variations in the sail attitude during flight, in order to reduce the performance degradation due to the presence of the period of eclipse.  
Figure 12: The required solar sail’s characteristic acceleration \(a_{c}\) as a function of the pericytherion \(\tilde{r}_{p}\) and apocytherion \(\tilde{r}_{a}\) dimensionless radii in the presence of a period of eclipse: (**a**) contour plot of \(a_{c}=a_{c}(\tilde{r}_{p},\tilde{r}_{a})\); (**b**) case of \(a_{c}\in\{0.1,0.15,0.2\}\) mm/s\({}^{2}\).  
Figure 13: Function \(\omega=\omega(v)\), in the presence of a period of eclipse, when \(r_{p}=1.3\,R_{V}\), \(r_{a}=6\,R_{V}\), and \(a_{c}\simeq 0.42\) mm/s\({}^{2}\). The black line indicates the output of an orbit simulator, while the red dashed line indicates the rotation of the Venus–Sun line during the spacecraft revolution around the planet.  
Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding author. Conflicts of Interest: The author declares no conflicts of interest. Notation  
\begin{tabular}{p{142.3 pt} p{142.3 pt}}
**Notation** \\ \(a\) & osculating orbit semimajor axis [km] \\ \(a_{c}\) & characteristic acceleration [mm/s\({}^{2}\)] \\ \(a_{p_{r}}\) & radial component of the propulsive acceleration vector [mm/s\({}^{2}\)] \\ \(a_{p_{t}}\) & transverse component of the propulsive acceleration vector [mm/s\({}^{2}\)] \\ \(e\) & osculating orbit eccentricity \\ \(p\) & semilatus rectum [km] \\ \(R_{V}\) & Venus's mean radius [km] \\ \(r\) & Venus-spacecraft distance [km] \\ \(r_{a}\) & apocytherion radius [km] \\ \(r_{p}\) & pericytherion radius [km] \\ \(r_{SE}\) & reference distance [1 au] \\ \(r_{SV}\) & Sun-Venus distance [au] \\ \(t\) & time [hours] \\ \(\mu_{V}\) & Venus's gravitational parameter [km\({}^{3}\)/s\({}^{2}\)] \\ \(\mu_{S}\) & Sun's gravitational parameter [km\({}^{3}\)/s\({}^{2}\)] \\ \(\
u\) & true anomaly [rad] \\ \(\omega\) & osculating orbit apse line rotation angle [rad] \\ Subscripts \\ \(0\) & initial, parking orbit \\ \(f\) & final \\ Superscripts & \\ \(\sim\) & dimensionless version \\ \end{tabular}