# Dynamics and Control of Typical Orbits around Saturn
## 3 Repeating Ground Track Orbits and Orbital Maintenance  
The repeating ground track is significant for remote sensing satellites. The condition for spacecraft achieving repeating ground track is \(R\Delta\lambda=2\pi N\), where \(R\) and \(N\) are relatively positive prime numbers. The interval of the adjacent ground track on the equator is [28; 29]  
\[\Delta\lambda=T_{N}(\omega_{S}-\hat{\Omega}), \tag{15}\]  
which \(\omega_{S}\) is the angular rotational velocity of Saturn. The nodal period of the motion of spacecraft is  
\[T_{N}=\frac{2\pi}{\dot{M}+\dot{\omega}}. \tag{16}\]  
According to the mean element theory [16], the average perturbing rates of \(M\) and \(\omega\) is [37]  
\[ \begin{split}\dot{M}&=n+\dot{M}_{1}+\dot{M}_{2},\\ \dot{\omega}&=\dot{\omega}_{1}+\dot{\omega}_{2},\end{split}  \tag{17}\]  
where  
\[ \begin{split}\dot{M}_{1}&=\frac{3 nI_{2}R_{c}^{2}}{2p^ {2}}(1-\frac{3}{2}\sin^{2}i)\sqrt{1-e^{2}},\\ \dot{\omega}_{1}&=-\frac{3 nI_{2}R_{c}^{2}}{2p^{2}} (\frac{5}{2}\sin^{2}i-2),\end{split}  \tag{18}\]  
\[ \begin{split}\dot{M}_{2}&=\frac{9 nI_{2}^{2}R_{c}^{4 }}{8p^{4}}(1-e^{2})(1-\frac{3}{2}\sin^{2}i)^{2}\\ &+\frac{9 nI_{2}^{2}R_{c}^{4}}{4p^{4}}\{\sqrt{1-e^{2}}(\frac{5}{2 }-\frac{19}{3}\sin^{2}i+\frac{233}{48}\sin^{4}i)\\ &+\frac{e^{4}}{\sqrt{1-e^{2}}}(\frac{35}{12}-\frac{35}{4}\sin^{2 }i+\frac{315}{32}\sin^{4}i)\\ &+\sqrt{1-e^{2}}e^{2}[(\frac{10}{3}-\frac{5I_{4}}{4I_{2}^{2}})+ (-\frac{26}{3}+\frac{25I_{4}}{4I_{2}^{2}})\sin^{2}i+(\frac{103}{12}-\frac{175 I_{4}}{32I_{2}^{2}})\sin^{4}i]\},\end{split}  \tag{19}\]  
\[ \begin{split}\dot{\omega}_{2}&=\frac{9 nI_{2}^{2}R_{ c}^{4}}{4p^{4}}\{(4-\frac{10I_{4}}{3I_{2}^{2}})+(-\frac{103}{12}+\frac{155I_{4}}{1 2I_{2}^{2}})\sin^{2}i+(\frac{215}{48}-\frac{245I_{4}}{24I_{2}^{2}})\sin^{4}i \\ &+\sqrt{1-e^{2}}(2-\frac{11}{2}\sin^{2}i+\frac{15}{4}\sin^{4}i)+ e^{2}[(\frac{7}{12}-\frac{15I_{4}}{4I_{2}^{2}})+(-\frac{3}{8}+\frac{105I_{4}}{8 \int_{2}^{2}})\sin^{2}i\\ &+(-\frac{15}{32}-\frac{315I_{4}}{32I_{2}^{2}})\sin^{4}i]\}.\end{split}  \tag{20}\]  
In this paper, we use the ground track repetition parameter \(Q=R/N\) to describe different repeating ground track orbits. Here, we should pay attention to the meaningful range of repetition parameter \(Q\)[29]. To find a suitable repetition parameter, we should firstly find the lower bound of \(Q\) by using the discriminant of \(f(\sin i)\):  
\[f(\sin i)=A_{1}\sin^{4}i+B_{1}\sin^{2}i+C_{1}=0, \tag{21}\]  
where  
\[ \begin{split} A_{1}=&\frac{9 nI_{2}^{2}R_{c}^{4}}{4 p^{4}}([\frac{215}{48}-\frac{15}{32}e^{2}+\frac{15}{4}\sqrt{1-e^{2}})-\frac{35I_{4}}{ 18I_{2}^{2}}(\frac{21}{4}+\frac{81}{16}e^{2})]+\frac{9 nI_{2}^{2}R_{c}^{4}}{4p^ {4}}\sqrt{1-e^{2}}(\frac{9}{8}\sqrt{1-e^{2}}\\ &+\frac{233}{48}+\frac{103}{12}e^{2}+\frac{315}{32}\frac{e^{4}}{1 -e^{2}}-\frac{105I_{4}}{32I_{2}^{2}}e^{2}),\end{split}  \tag{22}\]\[B_{1} =\frac{3 nJ_{2}R_{e}^{2}}{4p^{2}}\{-5+\frac{3J_{2}R_{e}^{2}}{p^{2}}[- \frac{103}{12}-\frac{3}{8}e^{2}-\frac{11}{2}\sqrt{1-e^{2}}+\frac{35J_{4}}{6J_{2} ^{2}}(\frac{31}{14}+\frac{9}{4}e^{2})]\}-\frac{9 nJ_{2}R_{e}^{2}}{4p^{2}}\sqrt{1- e^{2}}\] \[+\frac{9 nJ_{2}^{2}R_{e}^{4}}{4p^{4}}\sqrt{1-e^{2}}(-\frac{3}{2} \sqrt{1-e^{2}}-\frac{19}{3}-\frac{26}{3}e^{2}-\frac{35}{4}\frac{e^{4}}{1-e^{2}} +\frac{25J_{4}}{4J_{2}^{2}}e^{2}),\] \[C_{1} =\frac{3 nJ_{2}R_{e}^{2}}{4p^{2}}\{4+\frac{3J_{2}R_{e}^{2}}{p^{2}}[ (4+\frac{7}{12}e^{2}+2\sqrt{1-e^{2}})-\frac{5J_{4}}{6J_{2}^{2}}(4+\frac{9}{2}e ^{2})]\}+\frac{3 nJ_{2}R_{e}^{2}}{2p^{2}}\sqrt{1-e^{2}}\] \[+\frac{9 nJ_{2}R_{e}^{4}}{4p^{4}}\sqrt{1-e^{2}}(\frac{1}{2}\sqrt{ 1-e^{2}}+\frac{5}{2}+\frac{10}{3}e^{2}+\frac{35}{12}\frac{e^{4}}{1-e^{2}}- \frac{5J_{4}}{4J_{2}^{2}}e^{2})+n+Q(n_{s}-\omega_{s}).\]  
To promise \(f(\sin i)\) has four real roots, the discriminant should be non-negative, i.e.,  
\[B_{1}^{2}-4A_{1}[\bar{C}_{1}-Q(\omega_{s}-n_{s})]\geq 0.\]  
Solving the inequality, we find the lower bound of \(Q\):  
\[Q\geq\frac{\bar{C}_{1}-\frac{B_{1}^{2}}{4A_{1}}}{\omega_{s}-n_{s}},\]  
where \(\bar{C}_{1}=C_{1}-Q(n_{s}-\omega_{s})\).