# Dynamics and Control of Typical Orbits around Saturn
## 2 Sun-Synchronous Orbits and Orbital Maintenance
This control strategy suits the spacecraft whose entire lifetime is not too long. We take an initial inclination bias to make the spacecraft located in a quasi-sun-synchronous orbit. Then the inclination of the quasi-sun-synchronous orbit will oscillate around its normal inclination in its lifetime but still maintain the feature of sun-synchronous orbits. When \(t_{e}=-\frac{\frac{\Delta i(t_{0})}{dt}}{\frac{dt}{dt}}\), the local time drift \(\Delta\hat{T}\) in Equation (11) sees its extremum  
\[\Delta\hat{T}(t_{e})=-\frac{3996}{25\pi}\frac{\partial\hat{\Omega}}{\partial i} \frac{\Delta i(t_{0})^{2}}{\frac{di}{dt}}. \tag{12}\]  
If the end of lifetime of the spacecraft is \(t_{f}\), then the limit of local time drift at descending node is \(|\Delta\hat{T}(tf)|\leq\Delta\hat{T}(t_{e})\). Let \(\Delta\hat{T}(t_{f})=-\Delta\hat{T}(t_{e})\), then we find the initial inclination bias:  
\[\Delta i(t_{0})=(1-\sqrt{2})\frac{di}{dt}t_{f}. \tag{13}\]  
From Figure 3c,d, we learn that the prebiased initial strategy damps the local time drift at the descending node in the designed 5 Earth years' lifetime of spacecraft. The second control strategy is designed for the spacecraft which need to execute a long-time mission. This method needs periodic inclination bias to damp the local time drift caused by solar gravitation. The extremum of local time drift Equation (11) in a period \(t_{c}=-2\frac{\Delta i(t_{0})}{\frac{di}{dt}}\) is equal to a fixed local time drift limitation \(\Delta\hat{T}\)  
\[|\Delta\hat{T}(\frac{t_{c}}{2})|=|\Delta\hat{T}|,\]  
use the calculation equation in  
\[\frac{1.776\times 180}{\pi}[\Delta\Omega(t_{0})-\frac{1}{2}(\frac{\partial\hat{ \Omega}}{\partial a}\frac{da}{dt}/\frac{\partial\hat{\Omega}}{\partial i} \frac{di}{dt}+1)\frac{\partial\hat{\Omega}}{\partial i}\frac{\Delta i(t_{0})^ {2}}{\frac{di}{dt}}]=|\Delta\hat{T}|\]  
when \(\Delta\Omega(t_{0})=0\), then we find  
\[-\frac{3996}{25\pi}(\frac{\partial\hat{\Omega}}{\partial a}\frac{da}{dt}/\frac {\partial\hat{\Omega}}{\partial i}\frac{di}{dt}+1)\frac{\partial\hat{\Omega}} {\partial i}\frac{\Delta i(t_{0})^{2}}{\frac{di}{dt}}=|\Delta\hat{T}|\]  
and the value of periodic inclination bias can be derived naturally:  
\[|\Delta i(t_{c_{0}})|=\sqrt{|\frac{25\pi}{3996}\frac{\frac{di}{dt}\Delta\hat {T}}{\frac{\partial\hat{\Omega}}{\partial a}\frac{da}{dt}}/\frac{\partial\hat {\Omega}}{\partial t}\frac{di}{dt}+1)}| \tag{14}\]  
where the sign of \(\Delta i(t_{c_{0}})\) is opposite to the sign of \(\frac{di}{dt}\). From Figure 3c,d, compared with Figure 4a,b, the periodic inclination biased method has the better effect of local time drift limitation. Meanwhile, comparing Figure 4a with Figure 4b, it is obviously also sensitive to the initial inclination deviation. Therefore, regardless of the methods, it is essential to limit the initial inclination deviation \(\Delta i_{0}\). Figure 3: A comparison between \(\Delta T\) considered atmospheric drag and \(\Delta\bar{T}\) simulating the atmospheric drag in 5 Earth years (**a**) \(\Delta a_{0}=1\) km, \(\Delta i_{0}=1\times 10^{-5}\) deg, \(-(\beta_{s}-\Omega)=-135\) deg; (**b**) \(\Delta a_{0}=1\) km, \(\Delta i_{0}=1\times 10^{-4}\) deg, \(-(\beta_{s}-\Omega)=-135\) deg. A comparison between \(\Delta T\) considered atmospheric drag and \(\Delta\bar{T}\) simulating the atmospheric drag with a prebiased initial inclination in 5 Earth years (**c**) \(\Delta a_{0}=1\) km, \(\Delta i_{0}=1\times 10^{-5}\) deg, \(-(\beta_{s}-\Omega)=-135\) deg; (**d**) \(\Delta a_{0}=1\) km, \(\Delta i_{0}=1\times 10^{-4}\) deg, \(-(\beta_{s}-\Omega)=-135\) deg.