# Dynamics and Control of Typical Orbits around Saturn
## 2 Sun-Synchronous Orbits and Orbital Maintenance
In this paper, we assume \(\Delta\Omega(t_{0})=0\) and only consider the effect of other three variables of a given sun-synchronous orbit around Saturn. The initial orbital elements of the circular sun-synchronous orbits given in this paper are \(a=62,\)\(268\) km, \(e=0.01,i=90.0483\) deg, \(\Omega=-45\) deg, \(\omega=30\) deg, and \(M=0\) deg. The ecliptic of longitude of the Sun is fixed at \(\beta_{s}=90\) deg. When we choose small initial orbital deviation \(\Delta a_{0}=1\) km, \(\Delta i_{0}=0.001\) deg, and arbitrary \(\Omega\in[0,360]\) deg, we could know the local time drift at any descending node after 10 Earth years in Figure 1(a). In Figure 1(a), in the range of \(-(\beta_{s}-\Omega)\in[0,90]\cup[180,270]\) deg, the local time drift \(\Delta T\) is positive, which means the local time at descending node has been delayed. In the range \(-(\beta_{s}-\Omega)\in[90,180]\cup[-90,0]\) deg, the local time drift \(\Delta T\) is negative, which means the local time at the descending node has been brought forward. When the initial inclinaiton deviation is slightly greater, the local time in a larger range of \(-(\beta_{s}-\Omega)\) in Figure 1(b) is delayed. On the contrary, when we choose a negative initial inclination deviation, the absolute value is equal to the initial inclination deviation of Figure 1(b), the local time of more area of Figure 1(c) is brought forward. Furthermore, we could still observe from Figure 1(a)-c that the distribution of the extremum of local time drift is fixed, regardless of whether and how the value and sign of \(-(\beta_{s}-\Omega)\) change. To reduce the influence of different choices of initial condition, we choose the value of \(-(\beta_{s}-\Omega)\) where the local time drift will reach the extremum after 10 Earth years. Therefore, in this paper, we fix the value \(-(\beta_{s}-\Omega)=-135\) deg, which means the local time at the descending node is 15:00. When we fix the initial inclination deviation and choose different initial semimajor axis deviation, comparing Figure 1(b) with Figure 1(d), the extremum of local time drift of these two figures almost have the same distribution. This means the first-order approximation of local time drift is sensitive to the initial inclination deviation and insensitive to the initial semimajor axis deviation. Thus, we fix the semimajor axis deviation in this paper \(\Delta a_{0}=1\) km. When we consider the solar gravitation perturbation and the atmospheric drag at the same time, we should use Equation (9) to calculate the local time drift at the descending node. The fact we should notice is that the changing trend of \(\frac{da}{dt}\) and \(\frac{di}{dt}\) is quadratic function of time \(t\)[29]. Therefore, we try to simulate the effect of atmospheric drag by using the inclinaiton drift caused by solar gravity. In this way, we only need to choose inclination prebiased methods to offset the local time drift caused by two different kinds of force. The explicit calculation of local time drift:  
\[\Delta\hat{T}= \frac{1.776\times 180}{\pi}\Delta\Omega(t)\] \[= \frac{1.776\times 180}{\pi}\{\Delta\Omega(t_{0})+(\frac{\partial \hat{\Omega}}{\partial a}\frac{da}{dt}/\frac{\partial\hat{\Omega}}{\partial i} \frac{di}{dt}+1) \tag{11}\] \[[\frac{\partial\hat{\Omega}}{\partial i}\Delta i(t_{0})(t-t_{0}) +\frac{1}{2}\frac{\partial\hat{\Omega}}{\partial i}\frac{di}{dt}(t-t_{0})^{2}]\}.\]  
We find two figures to test the effect of simulating method used in Equation (11). In each figure, there are two curves representing the local time drift in 5 years in different calculation methods. The curves which have considered atmospheric drag use Equation (9). The curves simulating the atmospheric drag use Equation (11). Comparing Figure 3a with Figure 3b, we learn that the circular sun-synchronous orbits with the smaller initial inclination deviation has the better simulating effect of atmospheric drag. Then we use two methods in this paper to offset the local time drift at the descending node. The first control strategy is using initial inclination biased method.