# Dynamics and Control of Typical Orbits around Saturn
## 2 Sun-Synchronous Orbits and Orbital Maintenance
\[\frac{di}{dt} = \frac{r\cos(\omega+f)}{na^{2}\sqrt{1-e^{2}}}F_{n}\] \[\approx \frac{3n_{s}^{2}\cos(\omega+f)\cos\xi}{n}(\cos\beta_{s}\sin i\sin \Omega-\sin\beta_{s}\cos i_{s}\sin i\cos\Omega+sin\beta_{s}\sin i_{s}\cos i),\] \[\cos\xi = \cos\beta_{s}[\cos(\omega+f)\cos\Omega-\cos i\sin(\omega+f)\sin \Omega]+\sin\beta_{s}\cos i_{s}[\cos(\omega+f)\sin\Omega\] \[+\cos i\sin(\omega+f)\cos\Omega]+\sin\beta_{s}\sin i_{s}\sin i\sin( \omega+f),\]  
where \(F_{n}\) is the normal perturbation force caused by the solar gravitation, \(n_{s}\) 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, and \(\xi\) is the angle between the unit vector from center of mass of Saturn pointing the spacecraft and the unit vector from center of mass of Saturn pointing at the Sun. Then we average the secular derivative of inclination in a period of circular sun-synchronous orbits [35]. We could see the secular inclination perturbation as:  
\[\frac{\overline{di}}{dt}=-\frac{3n_{s}^{2}}{16n}\sin i(1+\cos i_{s})^{2}\sin( 2\beta_{s}-2\Omega) \tag{6}\]  
It is important to emphasize that we only consider the secular effect of perturbations. We can substitute \(\frac{di}{dt}\) for \(\frac{\overline{di}}{dt}\) for convenience. \((2\beta_{s}-2\Omega)\) in Equation (6) is associated with the local time drift at the descending node. The semimajor axis decay due to atmospheric drag leads to a shorter period and damage of the recursive feature. Here we simply assume the atmosphere of Saturn is stationary. According to the preliminary study of Saturn atmosphere model [8], we could estimate the acceleration of semi-major axis caused by atmospheric drag. We describe the variation of semimajor axis due to atmospheric drag of circular orbits by [29; 36]  
\[\frac{da}{dt}=-\frac{SC_{d}\rho na^{2}}{m}, \tag{7}\]  
where \(C_{d}\) is the drag coefficient, \(S\) is the projected area, \(\rho\) is the neutral atmosphere density, and \(m\) is the mass of spacecraft. When we analyze the inclination perturbation, we can calculate the local time drift at the descending node instead. Meanwhile, it is a fact that the local time drift at the descending node is also the local time drift at ascending node. Thus, we could continue to analyze the evolution of the ascending node. According to Equation (3), the first-order approximation of \(\hat{\Omega}\) by using first-order Taylor expansion is  
\[\Delta\hat{\Omega}=\frac{\partial\Delta\hat{\Omega}}{\partial a}\Delta a+ \frac{\partial\Delta\hat{\Omega}}{\partial i}\Delta i \tag{8}\]  
where  
\[\left\{\begin{aligned} \Delta a&=\Delta a_{0}+ \frac{da}{dt}(t-t_{0  
For Saturn, because the rotation period is 10.656 h, 1 deg drift of the ascending node results in 1.776 min drift of spacecraft. Then according to Equation (8), the local time drift of spacecraft at ascending node is:  
\[ \begin{split}\Delta T=&\frac{1.776\times 180}{\pi} \Delta\Omega(t)\\ =&\frac{1.776\times 180}{\pi}[\Delta\Omega(t_{0})+ \frac{\partial\hat{\Omega}}{\partial a}\Delta a(t_{0})(t-t_{0})+\frac{1}{2} \frac{\partial\hat{\Omega}}{\partial a}\frac{da}{dt}(t-t_{0})^{2}\\ &+\frac{\partial\hat{\Omega}}{\partial i}\Delta i(t_{0})(t-t_{0} )+\frac{1}{2}\frac{\partial\hat{\Omega}}{\partial i}\frac{di}{dt}(t-t_{0})^{2 }].\end{split}  \tag{9}\]  
The unit of spacecraft running time is second. The local time drift is counted in minutes. When we only investigate the local time drift caused by solar gravitation, Equation (9) could be rewritten as  
\[ \begin{split}\Delta T=&\frac{1.776\times 180}{\pi} [\Delta\Omega(t_{0})+\frac{\partial\hat{\Omega}}{\partial a}\Delta a(t_{0})(t- t_{0})\\ &+\frac{\partial\hat{\Omega}}{\partial i}\Delta i(t_{0})(t-t_{0} )+\frac{1}{2}\frac{\partial\hat{\Omega}}{\partial i}\frac{di}{dt}(t-t_{0})^{2 }].\end{split}  \tag{10}\]  
There are four initial variables, \(\Delta\Omega(t_{0})\), \(\Delta a(t_{0})\), \(\Delta i(t_{0})\), and \(-(\beta_{s}-\Omega)\) in Equation (10).