# Dynamics and Control of Typical Orbits around Saturn
## 2 Sun-Synchronous Orbits and Orbital Maintenance  
The first order secular term for the precession rate of ascending node is [15; 16; 32]:  
\[\dot{\Omega}_{1}=-\frac{3 nI_{2}R_{e}^{2}}{2p^{2}}\cos i. \tag{1}\]  
Here, the second order secular perturbation considered \(J_{4}\) for the precession rate is [16]  
\[ \begin{split}\dot{\Omega}_{2}=&-\frac{9 nI_{2}^{2}R_ {e}^{4}}{4p^{4}}\{[\frac{3}{2}-\frac{5}{3}\sin^{2}i-\frac{35J_{4}}{18I_{2}^{2} }(\frac{6}{7}-\frac{3}{2}\sin^{2}i)]\\ &+e^{2}[\frac{1}{6}+\frac{5}{24}\sin^{2}i-\frac{35J_{4}}{18I_{2}^ {2}}(\frac{9}{7}-\frac{9}{4}\sin^{2}i)]+\sqrt{1-e^{2}}(1-\frac{3}{2}\sin^{2}i )\}\cos i.\end{split}  \tag{2}\]  
Sun-synchronous orbits require the precession rate of ascending node to be equal to the Saturn rotational angular speed around the Sun. Hence, the average nodal precession rate is \begin{table}
\begin{tabular}{l l} \hline \(\mu\) (km\({}^{3}\)s\({}^{-2}\)) & 37,931,207.7 \(\pm\) 1.1 \\ Equatorial radius \(R_{e}\) (km) & 60,268 \\ \(J_{2}\) (\(\times 10^{8}\)) & 1,629,057.33 \(\pm\) 2.8 \\ \(J_{3}\) (\(\times 10^{8}\)) & 5.89 \(\pm\) 2.3 \\ \(J_{4}\) (\(\times 10^{8}\)) & \(-93,531.36\pm 3.7\) \\ Obliquity to orbit \(i_{s}\) & 26.73 deg \\ Sidereal orbit period (days) & 10,759.22 \\ Sidereal rotation period (hours) & 10.656 \\ \hline \end{tabular}
\end{table} 
Table 1: Some parameters for basic model of Saturn. \[\hat{\Omega}=\hat{\Omega}_{1}+\hat{\Omega}_{2}=n_{\rm s}. \tag{3}\]  
Here \(n_{\rm s}\) is the mean angular velocity around the Sun. Thus, we gained the equation from Equation (3):  
\[f(\cos i)=A\cos^{3}i+B\cos i+C=0, \tag{4}\]  
where  
\[A= \frac{9 nJ_{2}^{2}R_{\varepsilon}^{4}}{4p^{4}}[e^{2}(-\frac{5}{24} -\frac{35J_{4}}{8J_{2}^{2}})+\frac{3}{2}\sqrt{1-e^{2}}+(\frac{5}{3}-\frac{35J_ {4}}{12J_{2}^{2}})],\] \[B= \frac{3 nJ_{2}R_{\varepsilon}^{2}}{2p^{2}}\{1+\frac{3J_{2}R_{ \varepsilon}^{2}}{2p^{2}}[e^{2}(\frac{3}{8}+\frac{15J_{4}}{8J_{2}^{2}})-\frac {1}{2}\sqrt{1-e^{2}}-(\frac{1}{6}-\frac{5J_{4}}{4J_{2}^{2}})]\},\] \[C= n_{\rm s}.\]  
For sun-synchronous orbits, we can derive the inequality systems (5) to calculate the inclination from theoretical analysis. The following inequalities are the requirements to find the roots of \(\hat{\Omega}=n_{\rm s}\). \[\left\{ \begin{aligned} 1-R_{\varepsilon}/a&>e\\ \frac{C^{2}}{4A^{2}}+\frac{B^{3}}{27A^{3}}&\leq 0 \end{aligned} \right. \tag{5}\]  
The first inequality in (5) prevents the spacecraft from crashing into the outer atmosphere of Saturn. The second inequality is the discriminant of Equation (4). Then we obtain the plot of the discriminant, Figure 1.  
We learn from Figure 1 that the discriminant is almost always positive for any chosen \(a\) and \(e\). It means Equation (4) only has one real root, and there is only one sun-synchronous orbit for any chosen semimajor axis and eccentricity. Here, we choose the orbital elements \(a=62\),\(268\) km and \(e=0.01\), substitute them into Equation (4), and solve the equation. Then the inclination \(i=90.0483\) deg can be obtained naturally. Here we continue to analyze the perturbations of spacecraft in circular sun-synchronous orbits. The spacecraft located in circular sun-synchronous orbits around Saturn would be perturbed by the Sun. The derivative of inclination is presented by using the Lagrange equations as [33; 34]:  
Figure 1: The discriminant of sun-synchronous orbits for different values of semimajor axis and eccentricity.