# Self-Organizing Control of Mega Constellations for Continuous Earth Observation
## 2 Materials and Methods
### Continuous Coverage Constraint of Mega Constellation
#### 2.1.3 Intersatellite Motion Constraints  
Due to perturbations and initial orbit errors, the coplanar satellites will drift from each other, and thus the RAAN differences between coplanar satellites will get larger over time without consideration of constellation control. The drift range of the RAAN difference \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Constellation** & **Altitude** & **Inclination** & **Planes** & **Satellites per Plane** &
\begin{tabular}{c} **Whether Continuous Coverage** \\ **Constraint Is Satisfied** \\ \end{tabular} \\ \hline \multirow{4}{*}{Starlink Genaration 1} & 550 km & 53\({}^{\circ}\) & 72 & 22 & Yes \\  & 540 km & 53.2\({}^{\circ}\) & 72 & 22 & Yes \\  & 570 km & 70\({}^{\circ}\) & 36 & 20 & Yes \\  & 560 km & 97.6\({}^{\circ}\) & 6 & 58 & No \\ \hline \multirow{4}{*}{OneWeb} & 1120 km & 87.9\({}^{\circ}\) & 18 & 40 & Yes \\  & 1150 km & 53\({}^{\circ}\) & 32 & 50 & Yes \\  & 1110 km & 53.8\({}^{\circ}\) & 32 & 50 & Yes \\ \hline \multirow{4}{*}{GW-A59} & 590 km & 85\({}^{\circ}\) & 16 & 30 & No \\  & 600 km & 50\({}^{\circ}\) & 40 & 50 & Yes \\  & 508 km & 55\({}^{\circ}\) & 60 & 60 & Yes \\ \hline \multirow{4}{*}{GW-2} & 1145 km & 30\({}^{\circ}\) & 48 & 36 & Yes \\  & 1145 km & 40\({}^{\circ}\) & 48 & 36 & Yes \\ \cline{1-1}  & 1145 km & 50\({}^{\circ}\) & 48 & 36 & Yes \\ \cline{1-1}  & 1145 km & 60\({}^{\circ}\) & 48 & 36 & Yes \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of several mega constellation missions. between adjacent co-rotating orbital planes \(\delta\Omega\), and the drift range of the geocentric angle between adjacent coplanar satellites \(\theta\) under the continuous coverage constraint, are worth investigating. The SOC geometry of the perturbed coplanar satellites is shown in Figure 4, where  
\[\angle\mathrm{AC}=\arccos\biggl{(}\frac{\cos 2\alpha_{\mathrm{max}}}{\cos\theta} \biggr{)} \tag{7}\]  
\[\angle\mathrm{DE}=\theta \tag{8}\]  
\[\angle\mathrm{EF}=\delta\Omega \tag{9}\]  
It can be calculated that the coverage range of two adjacent coplanar satellites along the track direction is  
\[\angle\mathrm{DF}=\arccos\biggl{(}\frac{\cos\theta}{\cos\delta\Omega}\biggr{)} \tag{10}\]  
The coverage range along the normal direction of the orbital plane is  
\[\angle\mathrm{BC}=\frac{\sin\theta\cos 2\alpha_{\mathrm{max}}}{\sqrt{\sin^{2} \!\theta\!\cos^{2}\!\theta-(\cos^{2}\!\theta-\cos^{2}\!2\alpha_{\mathrm{max}}) \!\sin^{2}\!\delta\Omega}} \tag{11}\]  
To ensure the SOC of every two adjacent co-rotating orbital planes overlap with each other, there exists  
\[\angle\mathrm{BC}\geq\frac{2\pi}{P} \tag{12}\]  
To ensure the coverage range of every two adjacent coplanar satellites along the track direction overlap with each other, there exists  
\[\angle\mathrm{DF}_{\mathrm{max}}\geq\frac{2\pi}{S} \tag{13}\]  
According to Equations (11) and (12), the drift range of \(\theta\) can be obtained, which is related to the drift range of \(\delta\Omega\). \[\theta_{\mathrm{min}}\leq\theta\leq\theta_{\mathrm{max}} \tag{14}\]  
where  
\[ \begin{split}& A_{1}=-\cos^{2}\biggl{(}\frac{2\pi}{P}\biggr{)}\\ & B_{1}=-\cos^{2}\biggl{(}\frac{2\pi}{P}\biggr{)}\bigl{(}1+\sin^{ 2}\!\delta\Omega\bigr{)}+\cos^{2}\!2\alpha_{\mathrm{max}}\\ & C_{1}=\cos^{2}\biggl{(}\frac{2\pi}{P}\biggr{)}\sin^{2}\!2 \alpha_{\mathrm{max}}\sin^{2}\!\delta\Omega\\ &\theta_{\mathrm{min}}=\arcsin\left(\sqrt{\frac{-B_{1}-\sqrt{B _{1}^{2}-4A_{1}C_{1}}}{2A_{1}}}\right)\\ &\theta_{\mathrm{max}}=\arcsin\left(\sqrt{\frac{-B_{1}+\sqrt{B _{1}^{2}-4A_{1}C_{1}}}{2A_{1}}}\right)\end{split}  \tag{15}\]  
According to Equations (10), (13) and (14), the drift range of \(\delta\Omega\) of every two adjacent satellites can be obtained as  
\[0\leq\delta\Omega\leq\arcsin\left(\sqrt{\frac{B_{2}}{A_{2}}}\right) \tag{16}\]where  
\[