# Venus Magnetotail Long-Term Sensing Using Solar Sails
## 2 Mission Description and Simplified Mathematical Model
### Solar Sail Thrust Components with Sun-Facing Orientation
Note that the Sun belongs to the apse line of the science orbit at a distance approximately equal to \(r_{SV}\), which is considered a constant of motion. the apse line should be rotated about \(1.6\deg/\mathrm{day}\) to maintain the apocytherion inside the planet's induced magnetotail (recall that the orbital period of Venus is roughly 225 days, so \(360/225=1.6\deg/\mathrm{day}\)), while in a geocentric case, the required rotation rate is slightly less than \(1\deg/\mathrm{day}\). The effects on the spacecraft's planetocentric dynamics of the simplified control law given by Equations (14) and (15), which are shown in Figure 6, have been accurately studied in Ref. [26]. In particular, [PERSON] et al. [26] indicated that the simple control law (14) and (15) allowed the authors to obtain an averaged value equal to zero regarding both the semimajor axis and the eccentricity over a complete revolution of the spacecraft around the planet. This result can be easily obtained by substituting Equations (8), (14), and (15) into Equations (9) and (10) and observing that \(\mathrm{d}\bar{a}/\mathrm{d}\
u\) and \(\mathrm{d}e/\mathrm{d}\
u\) are odd functions of the true anomaly. On the other hand, that proposed control law gives a non-zero net variation (over a single revolution around the planet) considering the argument of pericytherion \(\omega\). Therefore, this specific characteristic of the simple control law given by Equations (14) and (15) can be used to obtain a (target) value of \(\omega\), which allows the science orbit apse line to be aligned again with the Venus-spacecraft direction at the end of the generic revolution around the planet. Bearing in mind that the initial value of \(\omega\) is zero and observing that the azimuthal angle \(\Delta\theta\) traveled by Venus's center of mass along its heliocentric (circular) orbit in a time interval \(\Delta t\) is simply \(\Delta\theta=\Delta t\,\sqrt{\mu_{S}/\tau_{SV}^{3}}\), the following final condition is derived at the true anomaly \(\
u=v_{f}\triangleq 2\pi\,\mathrm{rad}\):  
\[\omega(v_{f})=\Delta\omega\triangleq t_{f}\sqrt{\frac{\mu_{S}}{\tau_{SV}^{3}} }\equiv\bar{t}_{f}\,\sqrt{\left(\frac{\mu_{S}}{\mu_{V}}\right)\left(\frac{R_ {V}}{\tau_{SV}}\right)^{3}} \tag{16}\]  
where \(t_{f}=t(v_{f})\) is the time instant at the end of the spacecraft revolution around the planet, and Equation (7) is used to express the dimensionless version of the flight time. In fact, recall that the final values of both \(\bar{a}\) and \(e\), i.e., the values at the end of the revolution around the planet, correspond to their initial value [26]. In other terms, for a given value of the dimensionless pair \(\{\bar{r}_{p},\,\bar{r}_{a}\}\), which defines the characteristics of the planetocentric science orbit, the scalar constraint given by Equation (16) can be reached by selecting a suitable value of the (dimensionless) sail characteristic acceleration \(\bar{a}_{c}\). In this respect, the required value of \(\bar{a}_{c}\) can be obtained through a simple numerical procedure described in Section 3, where the Cauchy problem is transformed to a sort of root-finding problem, which is then solved using a standard numerical procedure. Figure 6: Variation of the dimensionless components of the solar sail-induced propulsive acceleration vector with the spacecraft true anomaly \(\
u\), according to Equations (14) and (15). However, the specific characteristics of the spacecraft's planetocentric dynamics and the simple form of the proposed control law given by Equations (14) and (15), allow the designer to obtain a rapid estimate of the required (dimensionless) sail characteristic acceleration \(\bar{a}_{c}\) by using an approximate, elegant, analytical approach. This simplified mathematical model is illustrated in the next subsection.