## 3 Sun-Synchronous Orbits  
The oblate nature of the primary body can lead to a secular variation of the ascending node of the orbit. However, we can use the orbit perturbations to keep the orientation of the Sun line direction fixed with respect to the orbital plane (for example, perpendicular to it) during one revolution of Jupiter around the Sun. Based on the mean element theory, the mean nodal precession rate coming from the secular perturbations of the first and second order [36; 37] can be described as  
\[ \begin{array}{l}\hat{\Omega}=-\frac{3 ht_{2}R^{2}}{2a^{2}(1-e^{2})}\cos i- \frac{9 nt_{2}^{2}R^{4}}{4 pt^{2}}\cos i\left\{\frac{3}{2}+\frac{e^{2}}{6}+ \sqrt{1-e^{2}}-\sin^{2}i\left(\frac{5}{3}-\frac{5e^{2}}{2M}+\frac{3}{2}\sqrt{ 1-e^{2}}\right)\right.\\ \left.-\frac{35f_{4}}{18f_{2}^{2}}\left[\frac{6}{7}+\frac{9e^{2}}{7}-\sin^{2 }i\left(\frac{3}{2}+\frac{9e^{2}}{4}\right)\right]\right\}.\end{array}  \tag{15}\]  
For sun-synchronous orbits, the mean nodal precession rate is equal to the mean motion of Jupiter orbiting around the Sun (\(n_{s}\)). Thus, we get the following equation:  
\[g(\cos i)=\Gamma_{1}\cos^{3}i+\Gamma_{2}\cos i+\Gamma_{3}=0, \tag{16}\]  
where  
\[\Gamma_{1}=\frac{9 nt_{2}^{2}R^{4}}{4p^{4}}\left[\frac{5}{3}-\frac{5}{24}e^{2} +\frac{3}{2}\sqrt{1-e^{2}}-\frac{35f_{4}}{6f_{2}^{2}}\left(\frac{1}{2}+\frac {3}{4}e^{2}\right)\right], \tag{17}\]  
\[\Gamma_{2}=\frac{3 nt_{2}R^{2}}{2p^{2}}\Bigg{\{}1+\frac{3f_{2}R^{2}}{2p^{2}} \Bigg{[}-\frac{1}{6}+\frac{3e^{2}}{8}-\frac{1}{2}\sqrt{1-e^{2}}+\frac{5f_{4}}{ 2f_{2}^{2}}\left(\frac{1}{2}+\frac{3e^{2}}{4}\right)\Bigg{]}\Bigg{\}}, \tag{18}\]  
\[\Gamma_{3}=n_{s}. \tag{19}\]  
Here, we remark that in Equations (17) and (18), \(p\) is equal to \(a(1-e^{2})\). First, setting \(x=\cos i\), we note that the equation \(g(x)=0\) has three real roots if the discriminant satisfies the following inequality:  
\[\Delta=-\left(\frac{\Gamma_{3}^{2}}{4\Gamma_{1}^{2}}+\frac{\Gamma_{2}^{3}}{2 7\Gamma_{1}^{3}}\right)\geq 0. \tag{20}\]Then, it is also necessary to avoid impact with Jupiter. However, the surface of Jupiter cannot be unambiguously defined since it is a gas giant. The critical perijovian distance, \(d_{c}\), is usually much larger than the radius of Jupiter, \(R\), due to radiation safety issues. However, let us take for convenience \(d_{c}=R\), so that the semi-major axis and eccentricity have to satisfy  
\[d_{c}=R<a(1-e). \tag{21}\]  
The variation of the discriminant with respect to \(a\) and \(e\) is shown in Figure 4a. We see that the discriminants are always negative for \(a\in[R,2R]\) and \(e\in[0,1-\frac{R}{a}]\), therefore, it can be concluded that there usually exists one meaningful inclination for sun-synchronous orbits when the semi-major axis and eccentricity are given. The inclinations for different semi-major axis and eccentricities are shown in Figure 4b. It was shown that the inclination monotonically increases with respect to the semi-major axis. For the same semi-major axis, the inclination decreases as the eccentricity increases. Moreover, the inclinations with the same altitude ratio, \(a/R\), are relatively smaller than those of near-Earth sun-synchronous orbits. For satellites on sun-synchronous orbits with medium altitude, perturbations from the gravity of the third body are usually smaller compared to satellites on stationary orbits. However, the risk coming from the Jupiter rings and the magnetic field may increase greatly. For sun-synchronous orbits, when \(a=1.4373R=102,755.451\) km and \(e=0.4\), we have \(i=90.183\) deg; when \(a=1.5308R=109,439.953\) km and \(e=0.1\), the corresponding inclination is 90.321 deg. The evolution of \(\Omega\) and of the difference between \(\Delta\Omega\) and \(n_{s}t\) over 25\(T\) for these two cases are presented in Figure 5. In the first case, this difference is about 0.018 deg; and in the second case, it is about 0.008 deg.