# Dynamics and Control of Typical Orbits around Saturn
## 5 Stationary Orbits and Orbital Maintenance
\begin{cases}\frac{\Delta i_{x}}{\Delta t}=&\frac{1}{2\pi}\int_{0}^{2\pi}\frac{di_{x }}{dt}d\beta_{s}\\ &=&\frac{3n_{s}^{2}}{n}[(-\frac{1}{16}-\frac{3}{16}\cos^{2}i_{s}+ \frac{1}{4}\sin^{2}i_{s})i_{y}+\frac{1}{4}\sin i_{s}\cos i_{s}],\\ \frac{\Delta i_{y}}{\Delta t}=&\frac{1}{2\pi}\int_{0}^{2\pi}\frac{di_{y}}{dt}d \beta_{s}\\ &=&\frac{3n_{s}^{2}}{n}(\frac{1}{16}+\frac{3}{16}\cos^{2}i_{s}- \frac{1}{4}\sin^{2}i_{s})i_{x}.\end{cases}  \tag{32}\]  
After solving this first order linear ordinary differential equation system Equation (32), we find the \(i_{x}(t),i_{y}(t)\) for any time \(t\) with initial condition \(i_{x}(t_{0}),i_{y}(t_{0})\). \[ \begin{split} i_{x}(t)=& i_{x}(t_{0})\cos[\frac{3n_{ s}^{2}}{32n}(7\cos(2i_{s})+1)t]\\ &-(i_{y}(t_{0})-\frac{4\sin(2i_{s})}{7\cos(2i_{s})+1})\sin[\frac{3 n_{s}^{2}}{32n}(7\cos(2i_{s})+1)t]\\ =&(0.621896-i_{y}(t_{0}))\sin(1.35686\times 10^{-13}t)+i_{x}(t_{ 0})\cos(1.35686\times 10^{-13}t),\\ i_{y}(t)=& i_{x}(t_{0})\sin[\frac{3n_{s}^{2}}{32n}(7\cos(2i_{s})+1)t] \\ &+(i_{y}(t_{0})-\frac{4\sin(2i_{s})}{7\cos(2i_{s})+1})\cos[\frac{3n_{s}^{2 }}{32n}(7\cos(2i_{s})+1)t]+\frac{4\sin(2i_{s})}{7\cos(2i_{s})+1}\\ =& 0.621896+i_{x}(t_{0})\sin(1.35686\times 10^{-13}t)\\ &+(i_{y}(t_{0})-0.621896)\cos(1.35686\times 10^{-13}t).\end{split}  \tag{33}\]  
Before we analyze the perturbation of inclination vector, we firstly set the maximum of inclination perturbation \(|\Delta i_{max}|=0.2865^{\circ}\). Then we could draw the Figure 9(a),b of \(i_{x},i_{y}\) separately for each initial condition value \(i_{x_{0}}=i_{x}(t_{0}),i_{y_{0}}=i_{y}(t_{0})\) in the limited circle. Once we see Figure 9(a),b, what we know immediately is that the inclination perturbation is so small that the magnitude is smaller than \(1\times 10^{-3}\) in 5 Earth years. Furthermore, comparing Figure 9(a) with Figure 9(b), we learn evidently that the magnitude of \(\Delta i_{y}\) is two order magnitude smaller than \(\Delta i_{x}\) in a given time of 5 Earth years. This means when we plot the perturbation of inclination vector of stationary orbit around Saturn over a long time, we could hardly see a sight difference in \(i_{y}\). Though the inclination perturbation caused by solar gravity could hardly influence the spacecraft in stationary orbit around Saturn, we still analyze the control strategy of \(i_{x},i_{y}\) in a preliminary way. When we know how the solar gravity perturbation acts, we could calculate \(\Delta i\) for any initial \((i_{x_{0}},i_{y_{0}})\) to stay in the limited circle. The area of the cool  
Figure 10: (a) \(\Delta i_{x}\) (b) \(\Delta i_{y}\) after 5 Earth years with initial inclination vector point \((i_{x_{0}},i_{y_{0}})\) in the inclination limited circle whose radius \(r=0.2865\) deg perturbed by solar gravity. tone circle overlap limited circle in Figure 11 indicates the initial point \((i_{x_{0}},i_{y_{0}})\) in this area are still remained in the limited circle after 100 Earth days. By observing the perturbation feature of stationary orbit around Saturn, we know that minimum maneuver \(\overrightarrow{\Delta i}\) to keep \(i_{x},i_{y}\) staying in the limited circle is from the boundary of limited circle pointing towards the center of the cool tone circle. The value of \(\overrightarrow{\Delta i}\) of \((i_{x_{0}},i_{y_{0}})\) in warm tone area in Figure 11 indicates the minimum inclination maneuver of \((i_{x_{0}},i_{y_{0}})\) to keep \((i_{x}(t),i_{y}(t))\) remained in the limited circle after 100 Earth days. After analyzing the effect caused by solar gravity perturbation, we now turn to the solar radiation pressure which may lead to eccentricity drift. First, the mean solar irradiance around Saturn is \(P=15.04\) W/m\({}^{2}\) while the distance from the Sun to Saturn is \(1.426\times 10^{12}\) m [38]. Then we find the intensity of solar radiation pressure is \(p=\frac{p}{c}=5.0168\times 10^{-8}\) N/m\({}^{2}\), where \(c=299\),792,458 m/s is the speed of light.