# Venus Magnetotail Long-Term Sensing Using Solar Sails
## 2 Mission Description and Simplified Mathematical Model
### Solar Sail Thrust Components with Sun-Facing Orientation  
To complete the Cauchy problem given by Equations (9)-(13), it is necessary to assign the control law that provides the variation with the true anomaly \(\
u\) of the two components \(\{\tilde{a}_{p_{t}},\tilde{a}_{p_{t}}\}\) of the dimensionless propulsive acceleration vector \(\mathbf{a}_{p}\) given by the solar sail. In this respect, taking into account the numerical results of Ref. [26], we consider a fixed orientation of the solar sail nominal plane (i.e., the plane of the thin reflective membrane when sail billowing is neglected) with respect to the Sun-spacecraft line during the entire  
Figure 4: Scheme of the solar sail-based spacecraft’s science orbit around Venus, in which the initial direction of the planetocentric orbit apse line coincides with the Venus–Sun line. In particular, initially, the Sun belongs to the positive direction of the first axis of a classical perifocal reference frame. The Sun–Venus distance \(r_{SV}\) is a constant of motion, and its value is high enough to assume that the Sun rays arrive parallel to the solar sail-based spacecraft. flight along the science orbit. In this context, paralleling the procedure described in Ref. [31], we assume that the solar sail's orientation is Sun-facing, that is, the sail nominal plane is perpendicular to the Sun-spacecraft line at any time instant [26]. This situation is schematized in Figure 5, which also shows the two dimensionless components (i.e., the radial and transverse components) of the propulsive acceleration vector \(\mathbf{a}_{p}\) in a generic point of the spacecraft's science orbit. In this case, bearing in mind that during the flight, the Sun-spacecraft distance is substantially constant and equal to \(r_{SV}\), neglecting the eclipse period experienced during the flight, and observing that the direction of the sun rays is parallel to the science orbit apse line (recall that the Sun belongs to that line; see the scheme in Figure 4), one has the following expressions of the two components \(\{\tilde{a}_{p},\tilde{a}_{p_{t}}\}\):  
\[\tilde{a}_{p_{t}} =-\tilde{a}_{c}\left(\frac{r_{SE}}{r_{SV}}\right)^{2}\cos\
u \tag{14}\] \[\tilde{a}_{p_{t}} =\tilde{a}_{c}\left(\frac{r_{SE}}{r_{SV}}\right)^{2}\sin\
u \tag{15}\]  
where \(r_{SE}\triangleq 1\,\mathrm{AU}\) is a reference distance that coincides with the Sun-Earth mean distance, and \(\tilde{a}_{c}\triangleq a_{c}/(\mu_{V}/R_{V}^{2})\) is the dimensionless version of the sail characteristic acceleration \(a_{c}\), which is the typical performance parameter in the trajectory design of a solar sail-based space vehicle when the optical degradation of the reflective film is neglected [45]. Note that Equations (14) and (15) indicate the effect of the solar radiation pressure (i.e., the thrust vector induced by the solar sail) on the motion of the spacecraft around the planet, according to the typical results in the literature. In particular, recall that \(a_{c}\) is defined [46; 47] as the maximum value of the magnitude of the solar sail-induced propulsive acceleration vector when the distance from the Sun is equal to \(r_{SE}\). Note that, given the same sail design characteristics (as, for example, the sail area, the mass breakdown, and the reflective film performance), the use of a solar sail propulsion system in a Venus-centered mission scenario allows for an increase of \((r_{SE}/r_{SV})^{2}=1/0.7233^{2}\simeq 1.91\) the maximum magnitude of the propulsive acceleration vector compared to the case of an Earth-centered application. However, in a Venus-centered scenario, the solar sail must be able to rotate the science orbit apse line faster than in a classical Earth-centered scenario. In fact, in a Venus case,  
Figure 5: Sketch of a Sun-facing solar sail in a Venus-centered mission scenario in which the direction of the incoming Sun rays is coincident with the elliptic orbit apse line.