# Venus Magnetotail Long-Term Sensing Using Solar Sails
## 2 Mission Description and Simplified Mathematical Model  
In this section, the dynamics of the solar sail-based spacecraft in its motion around Venus are described using the simplified approach proposed by [PERSON] et al. [26]. More precisely, assuming a two-dimensional scenario in which the spacecraft and Venus move around the Sun along the same plane \(\mathcal{P}\) introduced in the previous section, the solar sail dynamics along the planetocentric, elliptic, science orbit is described by the following four simplified Lagrange planetary equations (in Gaussian form) [44]:  
\[\frac{\mathrm{d}a}{\mathrm{d}\
u}=\frac{2\,p\,r^{2}}{\mu_{V}\,(1-e^{2})^{2}}\, \left[a_{p_{r}},\,e\,\sin\
u+\frac{a_{p_{t}}\,p}{r}\right] \tag{1}\]  
\[\frac{\mathrm{d}e}{\mathrm{d}\
u}=\frac{r^{2}}{\mu_{V}}\left[a_{p_{r}},\,\sin \
u+a_{p_{t}}\left(\cos\
u+\frac{r\,\cos\
u+e\,r}{p}\right)\right] \tag{2}\]  
\[\frac{\mathrm{d}\omega}{\mathrm{d}\
u}=\frac{r^{2}}{\mu_{V}\,e}\left[-a_{p_{r} }\,\cos\
u+a_{p_{t}},\,\sin\
u\bigg{(}1+\frac{r}{p}\bigg{)}\right] \tag{3}\]  
\[\frac{\mathrm{d}t}{\mathrm{d}\
u}=\frac{r^{2}}{\sqrt{\mu_{V}\,p}}\left\{1+ \frac{r^{2}}{\mu_{V}\,e}\bigg{[}a_{p_{r}},\,\cos\
u-a_{p_{t}}\,\sin\
u\left(1 +\frac{r}{p}\right)\bigg{]}\right\} \tag{4}\]  
which are consistent with the set of equations used in Ref. [31], where \(\mu_{V}\) is Venus's gravitational parameter, while  
\[p=a\Big{(}1-e^{2}\Big{)}\quad,\quad r=\frac{p}{1+e\,\cos\
u} \tag{5}\]  
In particular, Equations (1)-(4) give the variation with the true anomaly \(\
u\in[0,\,2\pi]\) rad of the spacecraft's osculating orbit semimajor axis \(a>0\), eccentricity \(e\in(0,\,1)\), time \(t\geq 0\) of passage from the pericytherion, and argument of pericytherion \(\omega\in[0,\,2\pi]\) rad. The latter is measured counterclockwise from a fixed direction, which coincides with the Venus-Sun line at the initial time \(t=0\). In Equations (1)-(4), the term \(r\) is the Venus-spacecraft distance, while \(a_{p_{t}}\) (or \(a_{p_{t}}\)) is the radial (or transverse) component of the solar sail-induced propulsive acceleration vector \(\mathbf{a}_{p}\). In particular, the radial component of vector \(\mathbf{a}_{p}\) is directed along the Venus-spacecraft line, while the transverse component \(a_{p_{t}}\) is positive when \(\mathbf{a}_{p}\cdot\mathbf{v}>0\), where \(\mathbf{v}\) is the spacecraft's velocity vector in its motion around Venus. The four first-order differential Equations (1)-(4) are completed by four initial conditions, that is, four scalar conditions at the initial true anomaly \(\
u=v_{0}\triangleq 0\). In this context, without losing generality, we assume that, initially, the science orbit apse line is aligned with the Venus-Sun direction, so that the initial value of \(\omega\) is zero; this situation is illustrated in Figure 4, where the pericytherion radius \(r_{p}\), the apocytherion radius \(r_{a}\), and the mean radius of Venus \(R_{V}\) are indicated. Accordingly, the four initial conditions are as follows:  
\[a(v_{0})=\frac{r_{a}+r_{p}}{2}\,\ e(v_{0})=\frac{r_{a}-r_{p}}{r_{a}+r_{p}}\,\ \omega(v_{0})=0\,\ t(v_{0})=0 \tag{6}\]  
The spacecraft equations of motion and the initial conditions given by the previous equation are more conveniently rewritten in a dimensionless form by introducing the following (dimensionless) terms:  
\[[\bar{r}_{p},\bar{r}_{a},\bar{a},\bar{p},\bar{r}]=\frac{[r_{p},r_{a},a,p,r]}{R _{V}}\,\ \bar{t}=t\sqrt{\frac{\mu_{V}}{R_{V}^{3}}}\,\ [\bar{a}_{p_{r}},\bar{a}_{p_{t}}]= \frac{[a_{p_{r}},a_{p_{t}}]}{\mu_{V}/r_{V}^{2}} \tag{7}\]  
with  
\[\bar{r}=\frac{\bar{a}\big{(}1-e^{2}\big{)}}{1+e\,\cos\
u} \tag{8}\]In the remainder of the paper, the dimensionless version of a generic variable will be indicated with the tilde superscript.