# Dynamics and Control of Typical Orbits around Saturn
## 3 Repeating Ground Track Orbits and Orbital Maintenance
To estimate the maximum of repetition parameter, we use the rotational period of Saturn \(T_{s}\) and the period of spacecraft in a low orbit \(T_{l}\). Then the upper bound of \(Q\) can be presented as:  
\[Q\leq\frac{T_{s}}{T_{l}}.\]  
Using the data from the planet model of Saturn, we find the meaningful range of repetition parameter for engineering application: \(2.4701\leq Q\leq 2.5191\). For different chosen values of repetition parameter \(Q\), we see a function \(f(\sin i)\) that describes the relation between the inclination \(i\) and the semimajor axis \(a\) of repeating ground track orbits. The relation between \(i\) and \(a\) are shown in Figure 5 for three given meaningful values of \(Q\). Using Equation (21), when initial semimajor axis and eccentricity are given, the corresponding inclination is the root of \(f(\sin i)\). In Figure 6, we present the repeating ground track orbit around Saturn when initial orbital elements and repetition parameter \(Q\) are given. Figure 5: The relation between \(a\) and \(i\) of repeating ground track orbits for repetition parameter \(Q=2.48,2.49\), and \(2.50\). An inevitable perturbation which can cause the semimajor decay in repeating ground track orbits is atmospheric drag. The semimajor axis decay due to atmospheric drag leads to a shorter period and damage of the recursive feature. The difference between actual angular velocity and nominal angular velocity can be presented as [29]:  
\[\Delta n=-\frac{3\bar{n}}{2\bar{n}}(a_{0}+\frac{da}{dt}t-\bar{a}),\]  
where \(a_{0}\) is the initial semimajor axis, \(\bar{a}\) is the nominal semimajor axis, and \(\bar{n}\) is nominal angular speed. The increase in angular speed results in the longitude drift of ground track. Obviously, we can learn from Equation (7) that \(\dot{a}<0\). Therefore, if \(a_{0}\leq\bar{a}\), the longitude will drift eastward monotonously. To maintain the feature of repeating ground track orbits, we should take a maneuver to make initial semimajor axis \(a_{0}>\bar{a}\). Then, in the first coming time interval \(t_{s}=-\frac{\Delta t}{\bar{a}}\), the longitude will drift westward. When \(\frac{d\Delta\lambda}{dt}=-\frac{3\bar{n}}{\bar{a}}(\Delta a+at)=0\), the longitude of the ground track would reach the western boundary. In the second time interval \(t_{s}\), the longitude will drift eastward and finally return back to the initial longitude. Then we can find the longitude drift compared to the nominal repeating ground track orbits:  
\[\Delta\lambda =-\frac{\omega_{s}}{\bar{n}}\int_{0}^{t}\frac{3\bar{n}}{2\bar{a} }(a_{0}+\frac{da}{dt}t-\bar{a})dt \tag{22}\] \[+\frac{9 nJ_{2}^{2}R_{e}^{4}}{4p^{4}}\sqrt{1-e^{2}}(-\frac{3}{2} \sqrt{1-e^{2}}-\frac{19}{3}-\frac{26}{3}e^{2}-\frac{35}{4}\frac{e^{4}}{1-e^{2}} +\frac{25J_{4}}{4J_{2}^{2}}e^{2}),\]0  
where \(\omega_{s}\) is the angular rotation speed of Saturnian. In the whole control period, when we fix the limitation of longitude drift \(\Delta\lambda_{L}\), then we can find the control period \(\Delta T_{\text{C}}\) and compensation \(\Delta a\) of semimajor axis [29]:  
\[+\frac{9 nJ_{2}^{2}R_{e}^{4}}{4p^{4}}\sqrt{1-e^{2}}(-\frac{3}{2} \sqrt{1-e^{2}}-\frac{19}{3}-\frac{26}{3}e^{2}-\frac{35}{4}\frac{e^{4}}{1-e^{2}} +\frac{25J_{4}}{4J_{2}^{2}}e^{2}),\]1 \[+\frac{9 nJ_{2}^{2}R_{e}^{4}}{4p^{4}}\sqrt{1-e^{2}}(-\frac{3}{2} \sqrt{1-e^{2}}-\frac{19}{3}-\frac{26}{3}e^{2}-\frac{35}{4}\frac{e^{4}}{1-e^{2}} +\frac{25J_{4}}{4J_{2}^{2}}e^{2}),\]2 \[+\frac{9 nJ_{2}^{2}R_{e}^{4}}{4p^{4}}\sqrt{1-e^{2}}(-\frac{3}{2} \sqrt{1-e^{2}}-\frac{19}{3}-\frac{26}{3}e^{2}-\frac{35}{4}\frac{e^{4}}{1-e^{2}} +\frac{25J_{4}}{4J_{2}^{2}}e^{2}),\]3  
Thus, in each period, when the longitude of ground track drift to the eastern boundary, we have to take a maneuver \(\Delta a\) to compensate the decay of semimajor axis. Here we assumed projected area of spacecraft \(S=20\) m\({}^{2}\), \(C_{d}=2.1\), \(\Delta\lambda_{L}=10\) km, and mass \(m=3000\) kg.