# Dynamics and Control of Typical Orbits around Saturn
## 5 Stationary Orbits and Orbital Maintenance  
Stationary orbits are more complex than those we have considered before. Therefore, we investigate stationary orbits in a special condition. When we investigate the stationary orbits around terrestrial planets, we must take the \(J_{22}\) term into our gravitational field model and analyze the longitude drift caused by aspheric perturbation. Here we only analyze aspheric perturbation caused by the \(J_{2}\), \(J_{4}\) terms, as Saturn is exactly a gaseous planet which means the \(J_{22}\) term could be neglected. The spherical coordinates \(O-r,\lambda,\varphi\), where \(O\) is the centroid of Saturn, \(r\) is the distant from the spacecraft instant position to \(O\), \(\lambda\) and \(\varphi\) represent the longitude and latitude of the spacecraft separately. The equations of motion of stationary orbits in the spherical coordinates can be written as [28]  
Figure 9: The eccentricity \(e\) for different value of semi major axis \(a\) and inclination \(i\) when \(\omega=270\) deg. \[\left\{ \begin{aligned} \ddot{r}-r\cos^{2}\varphi\dot{\lambda}^{2}-r \dot{\varphi}^{2}=&-\frac{\mu}{r^{2}}-\frac{3\mu J_{2}R_{c}^{2}}{2r ^{4}}(3\sin^{2}\varphi-1)\\ &+\frac{5\mu J_{4}R_{c}^{4}}{8r^{6}}(35\sin^{4}\varphi-30\sin^{2} \varphi+3),\\ \frac{d}{dt}(r^{2}\lambda\cos^{2}\varphi)=& 0,\\ r\ddot{\varphi}+2\dot{r}\dot{\varphi}+\frac{1}{2}r^{2}\sin(2 \varphi)\dot{\lambda}^{2}=&-[\frac{3\mu J_{2}R^{2}}{2r^{3}}+ \frac{\mu J_{4}R^{4}}{8r^{5}}(70\sin^{2}\varphi-30)]\sin 2\varphi.\end{aligned} \right. \tag{27}\]  
Spacecraft staying in stationary orbits around Saturn must satisfy these conditions [28]:  
\[\left\{ \begin{aligned} \dot{r}&=\ddot{r}=0,\\ \dot{\lambda}&=n,\ddot{\lambda}=0,\\ \dot{\varphi}&=\ddot{\varphi}=0.\end{aligned} \right.\]  
Then we gain the equation of stationary orbits at \(\varphi=0\) as  
\[\frac{\mu}{r^{3}}+\frac{3\mu J_{2}R^{2}}{2r^{5}}-\frac{15\mu J_{4}R^{4}}{8r^{ 7}}=n^{2}. \tag{28}\]  
Solving Equation (28), we find the radius of stationary orbits around Saturn is \(r_{s}=112\),\(506.0294\) km. After learning the structure of Saturn's rings [12], we know that spacecraft in the stationary orbit around Saturn may have a collision with the B-ring. Using nonsingular elements \(\{a,e_{x}=e\cos(\Omega+\omega),e_{y}=e\sin(\Omega+\omega),i_{x}=\sin i\sin\Omega,i_{y}=\sin i\cos\Omega,\,\lambda=\Omega+\omega+M\}\), we first calculate the secular solar gravitation perturbation of \(i_{x},i_{y}\) and \(e_{x},e_{y}\). When \(\varphi=0\), the normal perturbation force caused by the solar gravitation can be written as [33]:  
\[F_{n}=3 rn_{s}^{2}(\sin\alpha\sin\beta_{s}\cos i_{s}+\cos\alpha\cos\beta_{s}) \sin\beta_{s}\sin i_{s}, \tag{29}\]  
where \(\alpha\) is the longitude of spacecraft in the stationary orbit, \(n_{s}\) is the angular speed of Saturn around the Sun, \(\beta_{s}\) is the ecliptic longitude of the Sun, \(i_{s}\) is the obliquity of the ecliptic of Saturn. The derivative of \(i_{x},i_{y}\) perturbed by normal solar gravitation in a Saturnian rotation period are [33]  
\[ \begin{aligned} \frac{di_{x}}{dt}=&\frac{3n_{s}^{2}}{n}[ \frac{1}{4}\sin^{2}\beta_{s}\sin 2i_{s}+(\frac{1}{8}\sin 2\beta_{s}\cos i_{s})i_{x},\\ &+(-\frac{1}{8}\cos^{2}\beta_{s}-\frac{3}{8}\sin^{2}\beta_{s} \cos i_{s}+\frac{1}{2}\sin\beta_{s}\sin i_{s})i_{y}],\\ \frac{di_{y}}{dt}=&\frac{3n_{s}^{2}}{n}[\frac{1}{4 }\sin 2\beta_{s}\sin i_{s}-(\frac{1}{8}\sin 2\beta_{s}\cos i_{s})i_{x} \\ &+(\frac{1}{8}\cos^{2}\beta_{s}+\frac{3}{8}\sin^{2}\beta_{s}\cos i _{s}-\frac{1}{2}\sin\beta_{s}\sin i_{s})i_{y}].\end{aligned}  \tag{30}\]  
Here, we apply the Equations (30) and (31) to analyze the perturbation of the stationary orbit caused by solar gravitation, then we find the average rate of inclination vector perturbed by solar gravity in a sidereal orbit period:\[