# Dynamics and Control of Typical Orbits around Saturn
## 5 Stationary Orbits and Orbital Maintenance
Using other supposed parameters: the reflection parameter \(K=1\), area of spacecraft vertical to the Sun \(A=20\) m\({}^{2}\), the mass of spacecraft \(m=3000\) kg, then the solar radiation pressure of spacecraft on stationary orbits around Saturn is [34]:  
\[\mathbf{F_{s}}=-\,Kp(\frac{A}{m})\mathbf{S}=3.3445\times 10^{-10}\mathbf{S} \tag{34}\]  
where \(\mathbf{S}\) is the unit vector from the centroid of Saturn pointing at the Sun. Using the Lagrange perturbation equations, we could find the average time derivatives of the eccentricity vector \(e_{x}=e\cos(\omega+\Omega),e_{y}=e\sin(\omega+\Omega)\) in Equation (35). \[ \begin{split}\frac{de_{x}}{dt}&=\frac{1}{2\pi}\int _{0}^{2\pi}\frac{1}{na}(F_{r}\sin l+2F_{t}\cos l)dl\\ &=-\frac{3F_{s}}{2 na}\sin l_{s}\cos i_{s},\\ \frac{de_{y}}{dt}&=\frac{1}{2\pi}\int_{0}^{2\pi} \frac{1}{na}(-F_{r}\cos l+2F_{t}\sin l)dl\\ &=\frac{3F_{s}}{2 na}\cos l_{s},\end{split}  \tag{35}\]  
Figure 11: \(\Delta i\) for control period \(\Delta T_{\rm C}=100\) Earth days with initial inclination vector point \((i_{x_{0}},i_{y_{0}})\) in the inclination limited circle whose radius \(r=4.58\times 10^{-5}\) deg perturbed by solar gravity. \[F_{r}= -\frac{F_{s}}{2}[(1-\cos{i_{s}})\cos(\Omega+\omega+f+l_{s})+(1+\cos{i_ {s}})\cos(\Omega+\omega+f-l_{s})], \tag{36}\] \[F_{t}= \frac{F_{s}}{2}[(1-\cos{i_{s}})\sin(\Omega+\omega+f+l_{s})+(1+\cos{ i_{s}})\sin(\Omega+\omega+f-l_{s})].\]  
where \(F_{r}\) and \(F_{t}\) are the radial component and tangential component of solar radiation pressure \(\mathbf{F_{s}}\). When we analyze the eccentricity perturbation of spacecraft in stationary orbits over a long time period, we learn that the motion of the Sun in a time period could be described as \(\Delta l_{s}=l_{s}(t)-l_{s}(t_{0})=n_{s}(t-t_{0})\). Then we find the time integral of the eccentricity vector for any given \(t\). \[e_{x}(t) =e_{x}(t_{0})+\frac{3F_{s}}{2 na}\frac{1}{n_{s}}[\cos{l_{s}(t)}- \cos{l_{s}(t_{0})}]\cos{i_{s}}, \tag{37}\] \[e_{y}(t) =e_{y}(t_{0})+\frac{3F_{s}}{2 na}\frac{1}{n_{s}}[\sin{l_{s}(t)}- \sin{l_{s}(t_{0})}].\]  
We learn from Equation (37) that the curve of the eccentricity vector is an ellipse due to the existence of \(\cos{i_{s}}\), and initial condition \((e_{x_{0}},e_{y_{0}})\) and \(l_{s_{0}}\) influence the position of this eccentricity ellipse. From Figure 12, we learn that \(l_{s_{0}}\), which means the angular between initial solar radiation direction and \(x\) axis, can influence the perturbing direction. The centre of eccentricity limitation circle is \((e_{x_{0}},e_{y_{0}})\). Here, we fix the \((e_{x_{0}},e_{y_{0}})=(0,0)\). When \((e_{x_{0}},e_{y_{0}})\
eq(0,0)\), we can first take a maneuver to make the point \((e_{x_{0}},e_{y_{0}})=(0,0)\) in the curve of eccentricity perturbation. Then, we can design a maintenance strategy. When taking an eccentricity control, we used to set \(\Delta V_{t}>0\), then we could suppose \(B=-\frac{\Delta V_{t}}{T}\), and find the maneuver function [39]:  
\[\frac{de_{x}}{dt} =-\frac{2}{na}B\cos(l_{s}-\frac{\pi}{2}),\] \[\frac{de_{y}}{dt} =-\frac{2}{na}B\sin(l_{s}-\frac{\pi}{2}).\]  
Combined with solar radiation pressure perturbation Equation (35), the rate of eccentricity vector \((e_{x},e_{y})\) after a maneuver could be rewritten as  
Figure 12: A total of 72 curves of \(f(e_{x},e_{y})\) perturbed by solar radiation pressure for \(l_{0}=5n\) deg, the integral \(n\in[0,72]\), in a Saturnian year with initial eccentricity vector \((e_{x_{0}},e_{y_{0}})=(0,0)\) in the eccentricity perturbation limited circle whose radius \(r=3\times 10^{-7}\).