# Design of Mega-Constellations for Global Uniform Coverage with Inter-Satellite Links
## 2 Mega-Constellation Design Model
### Constraints on Collision Avoidance  
This paper deduces the minimum distance between the two satellites in the near-circular orbit with the same altitude to avoid collisions between satellites within the mega-constellation. Supposing that the minimum distance is greater than the safe distance between satellites, in that case, it is regarded that the two satellites will not collide. Figure 3 shows the spatial geometric relationship between satellites \(S_{1}\) and \(S_{2}\) in the ECI coordinate system. \(i_{1},\ i_{2},\ i_{1},\ i_{2},\ \Omega_{1}\ \Omega_{2}\) express the inclination, the argument of latitude, and RAAN of satellite \(S_{1}\) and \(S_{2}\), respectively. Furthermore, \(u_{1}=\angle MOS_{1}\), \(u_{2}=\angle POS_{2}\), \(\delta_{1}=\angle NOS_{1}\), \(\delta_{2}=\angle QOS_{2}\), \(\alpha_{1}=\angle MTN\), \(\alpha_{2}=\angle PTQ\), \(\rho\) is the geocentric angle between two satellites, and \(\rho=\angle S_{1}OS_{2}\). According to the sine theorem of the spherical triangle, in \(\Delta MNS_{1}\) and \(\Delta PQS_{2}\), the following relationship can be derived. \[\left\{ \begin{array}{l}\sin\delta_{1}=\sin u_{1}\cdot\sin i_{1}\\ \sin\delta_{2}=\sin u_{2}\cdot\sin i_{2}\end{array} \right. \tag{19}\]  
In \(\Delta MTS_{1}\) and \(\Delta PTS_{2}\), by the sine and cosine theorems, Equations (20) and (21) can be obtained:  
\[\left\{ \begin{array}{l}\sin\alpha_{1}=\sin u_{1}\cdot\frac{\cos i_{1}}{\cos \delta_{1}}\\ \sin\alpha_{2}=\sin u_{2}\cdot\frac{\cos i_{2}}{\cos\delta_{2}}\end{array} \right. \tag{20}\]  
\[\left\{ \begin{array}{l}\cos u_{1}=\sin\left(\frac{\pi}{2}-\delta_{1}\right) \cdot\cos\alpha_{1}=\cos\delta_{1}\cdot\cos\alpha_{1}\\ \cos u_{2}=\sin\left(\frac{\pi}{2}-\delta_{2}\right)\cdot\cos\alpha_{2}=\cos \delta_{2}\cdot\cos\alpha_{2}\end{array} \right. \tag{21}\]  
Set \(\angle S_{1}TS_{2}=A\), easily found Equation (22) from the geometric relationship. \[A=\Omega_{2}+\alpha_{2}-(\Omega_{1}+\alpha_{1})=\Delta\Omega+\Delta\alpha \tag{22}\]  
The expression for the geocentric angle is:  
\[ \begin{array}{l}\cos\rho=\cos(\frac{\pi}{2}-\delta_{1})\cdot\cos(\frac{\pi} {2}-\delta_{2})+\sin(\frac{\pi}{2}-\delta_{1})\cdot\sin(\frac{\pi}{2}-\delta _{2})\cdot\cos A\\ =\sin\delta_{1}\cdot\sin\delta_{2}+\cos\delta_{1}\cdot\cos\delta_{2}\cdot\cos A \end{array}  \tag{23}\]Combining Equations (20)-(22), the expression of \(\cos A\) can be derived. \[ \begin{array}{c}\cos A=\frac{\cos\Delta\Omega}{\cos\delta_{1}\cdot\cos\delta_{2 }}(\cos u_{1}\cdot\cos u_{2}+\cos i_{1}\cdot\cos i_{2}\cdot\sin u_{1}\cdot \sin u_{2})\\ -\frac{\sin\Delta\Omega}{\cos b_{1}\cdot\cos\delta_{2}}(\cos u_{1}\cdot\sin u _{2}\cdot\cos i_{2}+\sin u_{1}\cdot\cos u_{2}\cdot\cos i_{1})\end{array}  \tag{24}\]  
Ultimately, the expression for \(\cos\rho\) is arrived by combining Equations (19) and (24)  
\[ \begin{array}{c}\cos\rho=\sin u_{1}\cdot\sin u_{2}\cdot\sin i_{1}\cdot\sin i _{2}+\cos\Delta\Omega\cdot(\cos u_{1}\cdot\cos u_{2}+\cos i_{1}\cdot\cos i_{2 }\cdot\sin u_{1}\cdot\sin u_{2})\\ -\sin\Delta\Omega\cdot(\cos u_{1}\cdot\sin u_{2}\cdot\cos i_{2}-\sin u_{1} \cdot\cos u_{2}\cdot\cos i_{1})\end{array}  \tag{25}\]  
Since the phase difference between the two satellites is fixed, \(u_{1}+\Delta u\) can be used to replace \(u_{2}\). Substitute \(k_{1},\ k_{2},\ k_{3},\ k_{4}\) for the known constants in Equation (25) to simplify the expression. \[ \begin{array}{c}\cos\rho=k_{1}\sin u_{1}\cdot\sin(u_{1}+\Delta u)+k_{2}\cos u _{1}\cdot\cos(u_{1}+\Delta u)\\ +k_{3}\cos u_{1}\cdot\sin(u_{1}+\Delta u)+k_{4}\sin u_{1}\cdot\cos(u_{1}+ \Delta u)\end{array} \]  
\[\left\{ \begin{array}{c}k_{1}=\sin i_{1}\cdot\sin i_{2}+\cos\Delta\Omega \cdot\cos i_{1}\cdot\cos i_{2}\\ k_{2}=\cos\Delta\Omega\\ k_{3}=-\sin\Delta\Omega\cdot\cos i_{2}\\ k_{4}=\sin\Delta\Omega\cdot\cos i_{1}\end{array} \right. \tag{26}\]  
The formula is further simplified by expanding \(\Delta u\) and replacing the constants in Equation (26) with \(a\), \(b\), \(c\). \[