# Jamming Analysis between Non-Cooperative Mega-Constellations Based on Satellite Network Capacity
## 3 Satellite Network Capacity Calculation Method
### Transmission Rate  
In the inclined-orbit constellation \(\Psi_{A}\), the intra-plane ISLs and inter-plane ISLs have similarities; therefore, we only give the proof of the intra-plane ISLs' transmission rate here. When \(M\) is odd, for any intra-plane ISL \(L^{(i,j+1)}_{(i,j)}\), \(L^{(i,j+1)}_{(i,j)}\) will be passed through when \(A(i,j)\) sends data to \(A(i,(j+1)\) mod \(M)\), \(A(i,(j+2)\) mod \(M)\), \(\dots\), \(A(i,(j+\frac{M-1}{2})\) mod \(M)\). Therefore, \(L^{(i,j+1)}_{(i,j)}\) is used \(\frac{M-1}{2}\) times. The maximum transmission rate of \(A(i,j)\) is \(\bar{R}^{\times^{\prime}}_{i,j,t}\). Meanwhile, \(L^{(i,j+1)}_{(i,j)}\) will also be passed through when \(A(i,(j-1)\) mod \(M)\) sends data to \(A(i,(j+1)\) mod \(M)\), \(A(i,(j+2)\) mod \(M)\), \(\dots\), \(A(i,(j+\frac{M-1}{2}-1)\) mod \(M)\). \(L^{(i,j+1)}_{(i,j)}\) is passed through \(\frac{M-1}{2}-1\) times. The maximum transmission rate under the jamming condition from \(A(i,j-1)\) to \(A(i,j)\) is \(\bar{R}^{\times^{\prime}}_{i,(j-1)\) mod \(M\). By analogy, \(L^{(i,j+1)}_{(i,j)}\) will still be passed through until \(A(i,(j-(\frac{M-1}{2}-1))\) mod \(M)\) sends data to \(A(i,j+1)\) mod \(M)\). Under this  
Figure 3: +Grid topology of mega-constellation network. condition, \(L_{(i,j)}^{(i,j+1)}\) is used once. The maximum transmission rate is \(\hat{R}_{i,(j-\frac{M-1}{2})\bmod M,t}^{\
earrow}\). Therefore, the actual maximum transmission rate of \(L_{(i,j)}^{(i,j+1)}\) is  
\[ \begin{split} R_{(i,j)}^{(i,j+1)}&=\sum\limits_{k=0 }^{(M-1)/2-1}\left(\frac{M-1}{2}-k\right)\hat{R}_{i,(j-k)\bmod M,t}^{\
earrow} \\ &=\sum\limits_{k=0}^{(M-1)/2}\left(\frac{M-1}{2}-k\right)\hat{R}_ {i,(j-k)\bmod M,t}^{\
earrow}\end{split}  \tag{9}\]  
When \(M\) is even, \(L_{(i,j)}^{(i,j+1)}\) will be passed through when \(A(i,j)\) sends data to \(A(i,(j+1)\bmod M)\), \(A(i,(j+2)\bmod M)\), \(\ldots\), \(A(i,(j+\frac{M-2}{2})\bmod M)\). \(L_{(i,j)}^{(i,j+1)}\) is utilized \(\frac{M-2}{2}\) times. When \(A(i,j)\) sends data to \(A(i,(j+\frac{M}{2})\bmod M)\), both forward and reverse ISLs can be used; therefore, \(L_{(i,j)}^{(i,j+1)}\) is passed through \(\frac{1}{2}\) times. The maximum transmission rate is \(\hat{R}_{i,j,t}^{\
earrow}\). When \(A(i,(j-1)\bmod M)\) sends data to \(A(i,(j+1)\bmod M)\), \(A(i,(j+2)\bmod M)\), \(\ldots\), \(A(i,(j+\frac{M}{2}-1)\bmod M)\), \(L_{(i,j)}^{(i,j+1)}\) will be passed through \(\frac{M-1}{2}-1\) times. The maximum transmission rate is \(\hat{R}_{i,(j-1)\bmod M,t}^{\
earrow}\). By analogy, \(L_{(i,j)}^{(i,j+1)}\) will still be passed through until \(A(i,(j-\frac{M}{2})\bmod M)\) sends data to \(A(i,(j+1)\bmod M)\). Under this condition, \(L_{(i,j)}^{(i,j+1)}\) is used \(\frac{M}{2}-\left(\frac{M}{2}-1\right)-\frac{1}{2}=\frac{1}{2}\) times. The maximum transmission rate is \(\hat{R}_{i,(j-\frac{M-2}{2})\bmod M,t}^{\
earrow}\). Therefore, the actual maximum transmission rate of \(L_{(i,j)}^{(i,j+1)}\) is  
\[ \begin{split} R_{(i,j)}^{(i,j+1)}&=\sum\limits_{k=0 }^{(M-2)/2}\left(\frac{M-2}{2}+\frac{1}{2}-k\right)\hat{R}_{i,(j-k)\bmod M,t }^{\
earrow}\\ &=\sum\limits_{k=0}^{(M-2)/2}\left(\frac{M-1}{2}-k\right)\hat{R} _{i,(j-k)\bmod M,t}^{\
earrow}\end{split}  \tag{10}\]  
In summary, the maximum transmission rate of intra-plane ISL \(L_{(i,j)}^{(i,j+1)}\) is  
\[R_{(i,j)}^{(i,j+1)}=\left\{ \begin{array}{l}\sum\limits_{k=0}^{\frac{M-2}{2 }}\left(\frac{M-1}{2}-k\right)\hat{R}_{i,(j-k)\bmod M,t}^{\
earrow},M=2n\\ \sum\limits_{k=0}^{\frac{M-1}{2}}\left(\frac{M-1}{2}-k\right)\hat{R}_{i,(j-k) \bmod M,t}^{\
earrow},\text{otherwise}\end{array} \right. \tag{11}\]  
Similarly, the maximum transmission rates of intra-plane ISL \(L_{(i,j)}^{(i,j-1)}\), inter-plane ISL \(L_{(i,j)}^{(i+1,j)}\) and \(L_{(i,j)}^{(i-1,j)}\) are \(R_{(i,j)}^{(i,j-1)}\), \(R_{(i,j)}^{(i+1,j)}\) and \(R_{(i,j)}^{(i-1,j)}\), respectively. \[