# Venus Magnetotail Long-Term Sensing Using Solar Sails
## 2 Mission Description and Simplified Mathematical Model
### Performance Approximation through a Simple Analytical Approach  
This section illustrates an analytical procedure to obtain an accurate approximation of the value of the dimensionless characteristic acceleration required to reach the desired rotation \(\Delta\omega\) of the science orbit apse line; see the scalar constraint described by Equation (16).  
The analytical procedure is based on the assumption that, during the generic revolution of the spacecraft around Venus, the values of \(a\) and \(e\) remain sufficiently close to the assigned \(a_{0}\) and \(e_{0}\), respectively. In fact, the specific form of the control law given by Equations (14) and (15) allows the semimajor axis and the eccentricity of the osculating orbit to return to their initial values at the end of the spacecraft's revolution around the planet. Bearing in mind Equations (8), (14) and (15), when the conditions \(\bar{a}\simeq\bar{a}_{0}\) and \(e\simeq e_{0}\) are enforced in Equation (11), the result is as follows:  
\[\frac{\mathrm{d}\omega}{\mathrm{d}\
u}\simeq\bar{a}_{c}\left(\frac{r_{SE}}{r_ {SV}}\right)^{2}\frac{\bar{a}_{0}^{2}\left(1-e_{0}^{2}\right)^{2}}{e_{0}}\, \frac{2+e_{0}\,\cos\
u-\cos^{2}\
u}{\left(1+e_{0}\,\cos\
u\right)^{3}} \tag{17}\]  
The previous equation can be easily integrated into the interval \(\
u\in[0,\,2\pi]\,\mathrm{rad}\) to obtain an analytical approximation of \(\Delta\omega\), that is, the variation in the argument of per-cytherion during a single spacecraft's revolution around Venus. In this case, one has the following:  
\[\Delta\omega\simeq 3\,\pi\,\bar{a}_{c}\left(\frac{r_{SE}}{r_{SV}}\right)^{2} \frac{\bar{a}_{0}^{2}\sqrt{1-e_{0}^{2}}}{e_{0}} \tag{18}\]  
which is, indeed, a linear function of the sail performance parameter \(\bar{a}_{c}\). Observing that \(\bar{t}_{f}\) is approximated by the (dimensionless) orbital period of the science orbit, viz.,  
\[\bar{t}_{f}\simeq 2\,\pi\,\sqrt{\bar{a}_{0}^{3}} \tag{19}\]  
from Equations (16), (18), and (19), one easily obtains the analytical approximation of the required (dimensionless) characteristic acceleration as follows:  
\[\bar{a}_{c}\simeq\frac{2\,e_{0}\,\sqrt{\left(\frac{H_{S}}{\mu_{V}}\right) \left(\frac{R_{V}}{r_{SV}}\right)^{3}}}{3\left(\frac{r_{SE}}{r_{SV}}\right)^{ 2}\sqrt{\bar{a}_{0}\,(1-e_{0}^{2})}}\simeq(9.324\times 10^{-5})\,\frac{e_{0} }{\sqrt{\bar{a}_{0}\,(1-e_{0}^{2})}} \tag{20}\]  
The dimensional value of the required characteristic acceleration \(a_{c}\), in \(\,\mathrm{mm}/s^{2}\), is then obtained from the previous equation as follows:  
\[a_{c}\simeq 0.827\frac{e_{0}}{\sqrt{\bar{a}_{0}\,(1-e_{0}^{2})}}\qquad(\mathrm{ in}\,\,\mathrm{mm}/s^{2}) \tag{21}\]  
Note that the denominator of Equations (20) and (21) coincides with the square root of the dimensionless semilatus rectum of the (elliptic) science orbit. The expression of \(\bar{a}_{c}\) (or equivalently that of \(a_{c}\)) given by Equation (20) can be used to quickly estimate the required sail performance as a function of the characteristics of the spacecraft's science orbit around Venus. The accuracy of the proposed approximate model is investigated in the next section.