# Venus Magnetotail Long-Term Sensing Using Solar Sails
## 3 Numerical Simulations and Results  
The mathematical model described in the previous section is employed to determine the required solar sail propulsive performance in a reference mission scenario. In this respect, the characteristics of the science orbit described in Ref. [48] are used as a potential case study. In particular, [PERSON] et al. [48] considered an elliptic (science) orbit around Venus with a pericytherion radius \(r_{p}=1.3R_{V}\) and an apocytherion radius \(r_{a}=6R_{V}\), so the dimensionless (reference) initial values of the semimajor axis \(\tilde{a}_{0}\) and the eccentricity \(e_{0}\) are as follows:  
\[\tilde{a}_{0}=\frac{1.3+6}{2}=3.65\quad,\quad e_{0}=\frac{6-1.3}{6+1.3}\simeq 0.644 \tag{22}\]  
According to Equation (21), the approximate value of the solar sail dimensionless characteristic acceleration required to obtain the desired rotation of the apse line is \(\tilde{a}_{c}\simeq 4.106\times 10^{-5}\), which corresponds to a dimensional value \(a_{c}\simeq 0.363\,\mathrm{mm/s^{2}}\). For example, NASA's proposed Solar Cruiser mission [49] was planned to employ a solar sail-propelled spacecraft with a characteristic acceleration of about \(0.12\,\mathrm{mm/s^{2}}\). Such a value of \(\tilde{a}_{c}\) is used to initialize a numerical procedure through which the actual value of the characteristic acceleration is determined using a routine based on the classical shooting method [50]. In particular, the procedure involves the numerical integration of Equations (9)-(12) with the initial conditions (13) in the range \(\
u\in[0,\,2\pi]\,\mathrm{rad}\), by using a PECE solver with an absolute and relative tolerance of \(10^{-12}\). The output of the numerical procedure gives the actual value of the dimensionless characteristic acceleration \(\tilde{a}_{c}\simeq 4.09\times 10^{-5}\), which corresponds to \(a_{c}\simeq 0.3628\,\mathrm{mm/s^{2}}\). The numerical results indicate that the analytical approximation given by Equation (20) allows the actual value of \(\tilde{a}_{c}\) to be accurately estimated. The numerical solution (i.e., the numerical integration) of the Cauchy problem with the right value of \(\tilde{a}_{c}\) also gives the \(\
u\)-variation in the spacecraft's osculating orbit characteristics. In this context, Figure 7 shows the variation in \(\
u\) considering the argument of pericytherion \(\omega\) during the spacecraft's revolution around Venus (the black line). The same graph also indicates, as shown with the red dashed line, the angle of rotation of the Sun-Venus line given by the product \(t\sqrt{\mu_{S}/r_{SV}^{3}}\). Note how the two lines coincide both at the beginning (by construction) and at the end (as required) of the spacecraft's revolution around the planet. Finally, the \(\
u\)-variation considering the dimensionless semimajor axis and the eccentricity is reported in Figure 8, where one can observe that both \(\tilde{a}\) and \(e\) (1) remain close to the reference value \(\{\tilde{a}_{0},e_{0}\}\) during the entire flight and (2) return to their initial values  
Figure 7: Variation of \(\omega\) with \(\
u\) obtained by the numerical integration of the Lagrange planetary equations (black line), when \(r_{p}=1.3\,R_{V}\), \(r_{a}=6\,R_{V}\), and \(a_{c}\simeq 0.363\,\mathrm{mm/s^{2}}\). The red dashed line indicates the rotation of the Venus–Sun line during the spacecraft’s revolution around the planet. Note how the black and red dashed lines overlap when \(\
u=2\pi\,\mathrm{rad}\). at the end of the revolution around Venus. This aspect confirms the effectiveness of the simple (Sun-facing) solar sail control law given by Equations (14) and (15) and the validity of the hypotheses underlying the approximate expression of \(\tilde{a}_{c}\) given by Equation (20). Finally, the numerical procedure was used to perform a parametric study of the variation in the required (dimensional) characteristic acceleration \(a_{c}\) with the value of the pericibretion and apocytherion radii.