# Self-Organizing Control of Mega Constellations for Continuous Earth Observation
## Appendix A  
According to Equations (11) and (12),  
\[\cos\angle\text{BC}\leq\cos\left(\frac{2\pi}{P}\right) \tag{120}\]  
\[\cos^{2}\!2\alpha_{\text{max}}\text{sin}^{2}\theta\leq\cos^{2}\left(\frac{2\pi} {P}\right)\left(\text{sin}^{2}\theta\text{cos}^{2}\theta-\text{sin}^{2}\delta \Omega\text{cos}^{2}\theta+\text{cos}^{2}\!2\alpha_{\text{max}}\text{sin}^{2} \delta\Omega\right) \tag{121}\]  
Let \(\tau=\text{sin}^{2}\theta\), \(1-\tau=\cos^{2}\theta\),  
\[ \begin{split}&-\cos\left(\frac{2\pi}{P}\right)\tau^{2}+\left[ \cos^{2}\left(\frac{2\pi}{P}\right)\left(1+\text{sin}^{2}\delta\Omega\right)- \cos^{2}\!2\alpha_{\text{max}}\right]\tau\\ &-\text{cos}^{2}\left(\frac{2\pi}{P}\right)\text{sin}^{2}\!2 \alpha_{\text{max}}\text{sin}^{2}\delta\Omega\geq 0\end{split}  \tag{122}\]  
Let  
\[ \begin{split}& A_{1}=-\text{cos}^{2}\left(\frac{2\pi}{P}\right)\\ & B_{1}=-\text{cos}^{2}\left(\frac{2\pi}{P}\right)\left(1+\text{ sin}^{2}\delta\Omega\right)+\text{cos}^{2}\!2\alpha_{\text{max}}\\ & C_{1}=\text{cos}^{2}\left(\frac{2\pi}{P}\right)\text{sin}^{2} \!2\alpha_{\text{max}}\text{sin}^{2}\delta\Omega\end{split}  \tag{123}\]  
According to Equation (6), it can be proven that  
\[\cos\left(\frac{2\pi}{P}\right)\geq\cos(2\alpha_{\text{max}}) \tag{124}\]  
Therefore,  
\[ \begin{split}& A_{1}\geq 0\\ & B_{1}\leq-\text{cos}^{2}2\alpha_{\text{max}}\text{sin}^{2}\delta \Omega\leq 0\\ & C_{1}\geq 0\end{split}  \tag{125}\]  
The range of \(\tau\) can be obtained by Equation (122)  
\[\frac{-B_{1}-\sqrt{B_{1}^{2}-4A_{1}C_{1}}}{2A_{1}}\leq\tau\leq\frac{-B_{1}+ \sqrt{B_{1}^{2}-4A_{1}C_{1}}}{2A_{1}} \tag{126}\]  
According to Equation (125), it is obvious that  
\[\frac{-B_{1}+\sqrt{B_{1}^{2}-4A_{1}C_{1}}}{2A_{1}}\geq\frac{-B_{1}-\sqrt{B_{1 }^{2}-4A_{1}C_{1}}}{2A_{1}}\geq 0 \tag{127}\]  
Therefore, \(\tau=\text{sin}^{2}\theta\geq 0\) can be proven, yet \(B_{1}^{2}-4A_{1}C_{1}\geq 0\) depends on the \(\delta\Omega\) value. The range of \(\sin\theta\) is  
\[\sqrt{\frac{-B_{1}-\sqrt{B_{1}^{2}-4A_{1}C_{1}}}{2A_{1}}}\leq\sin\theta\leq \sqrt{\frac{-B_{1}+\sqrt{B_{1}^{2}-4A_{1}C_{1}}}{2A_{1}}} \tag{128}\]  
The range of \(\cos\theta\) is  
\[\sqrt{\frac{2A_{1}+B_{1}+\sqrt{B_{1}^{2}-4A_{1}C_{1}}}{2A_{1}}}\leq\cos\theta \leq\sqrt{\frac{2A_{1}+B_{1}-\sqrt{B_{1}^{2}-4A_{1}C_{1}}}{2A_{1}}} \tag{129}\]According to Equations (10) and (13),  
\[\left(\cos\angle DF\right)_{\min}\leq\cos\biggl{(}\frac{2\pi}{S}\biggr{)} \tag{11}\]  
\[\left(\cos\theta\right)_{\min}\leq\cos\biggl{(}\frac{2\pi}{S}\biggr{)}\cos \delta\Omega \tag{12}\]  
\[\sqrt{B_{1}^{2}-4A_{1}C_{1}}\geq-2A_{1}\sin^{2}\biggl{(}\frac{2\pi}{S} \biggr{)}-2A_{1}\cos^{2}\biggl{(}\frac{2\pi}{S}\biggr{)}\sin^{2}\delta\Omega- B_{1} \tag{13}\]  
\[B_{1}^{2}-4A_{1}C_{1}\geq\biggl{[}2A_{1}\sin^{2}\biggl{(}\frac{2\pi}{S} \biggr{)}+2A_{1}\cos^{2}\biggl{(}\frac{2\pi}{S}\biggr{)}\sin^{2}\delta\Omega+ B_{1}\biggr{]}^{2} \tag{14}\]  
Equation (14) ensures that \(\theta\) is a real value, given in Equations (9) and (10). Equation (14) can be simplified as  
\[A_{2}\sin^{2}\delta\Omega-B_{2}\geq 0 \tag{15}\]  
where  
\[ \begin{split}& A_{2}=-2\sin^{2}\biggl{(}\frac{2\pi}{S}\biggr{)} \cos^{2}\biggl{(}\frac{2\pi}{S}\biggr{)}\cos^{2}\biggl{(}\frac{2\pi}{P}\biggr{)} -\cos^{2}\biggl{(}\frac{2\pi}{S}\biggr{)}\cos^{2}2\alpha_{\max}+\cos^{2} \biggl{(}\frac{2\pi}{P}\biggr{)}\cos^{2}2\alpha_{\max}\\ & B_{2}=-\sin^{2}\biggl{(}\frac{2\pi}{S}\biggr{)}\biggl{[}\cos^{2 }\biggl{(}\frac{2\pi}{S}\biggr{)}\cos^{2}\biggl{(}\frac{2\pi}{P}\biggr{)}-\cos ^{2}2\alpha_{\max}\biggr{]}\end{split}  \tag{16}\]  
According to Equation (6),  
\[