# A Disruption Tolerant Distributed Routing Algorithm in LEO Satellite Networks
## 2 LEO Satellite Network Model  
In our constellation system, the space router on each satellite communicates with neighboring satellites through four interfaces connected to laser terminals and receives traffic from a ground base router and a space base station through the other two interfaces. In addition to transmitting data packets between itself and the space router, the laser terminal can also inform the space router if the laser link is in a fault state.  
As shown in Figure 1, each satellite is connected to neighboring satellites through inter-plane ISLs and intra-plane ISLs. There are two seams between the two hemispheres, the satellites on both sides of the seam move in opposite directions and do not build links. The four intersatellite link interfaces of the satellites located in the western hemisphere are counted as 0, 1, 2, 3 in the counterclockwise direction from the north direction. As shown in Figure 2, when in the western hemisphere, satellite B communicates with the hetero-orbiting satellite A through its interface 1 and the interface 3 of satellite A. When the two satellites cross the polar region and enter the eastern hemisphere, they do not adjust their movement attitudes, but constantly adjust the positions of interface 1 and interface 3. Finally, in the eastern hemisphere, satellite B still communicates with satellite A through its interface 1 and the interface 3 of satellite A, and the positions of interface 1 and interface 3 are swapped at this time. During the entire process of crossing the polar regions, the inter-plane ISL always maintains link establishment status. From the perspective of routing, the length of the inter-plane ISL is constantly changing, and the link establishment status and the corresponding relationship between the interfaces are always constant.  
For a polar orbit constellation, its constellation configuration is determined by the following parameters: orbit radius \(R\), orbit number \(N\), number of satellites in each orbit \(M\), phase difference between adjacent orbiting satellites \(\varphi\), and right ascension difference of the ascending node of the adjacent orbit \(RI\).  
The length of the intra-plane intersatellite links is fixed, and is calculated by  
\[L_{\varphi}=\sqrt{2}R\sqrt{1-\cos\left(\frac{360^{\circ}}{M}\right)}. \tag{1}\]  
The length of the inter-plane intersatellite links varies with latitude. As shown in Figure 1, if one takes a point \(B^{\prime}\) at the same latitude as star \(B\) on the orbital plane of star \(A\), and take a point A\({}^{\prime}\) at the same latitude as star \(A\) on the orbital plane of star \(B\), then \(AA^{\prime}BB^{\prime}\) is an isosceles trapezoid.  
\[AA^{\prime}=\sqrt{2}R\sqrt{1-\cos(RI)}\cos(latA), \tag{2}\]  
\[BB^{\prime}=\sqrt{2}R\sqrt{1-\cos(RI)}\cos(latB), \tag{3}\]  
\[AB^{\prime}=\sqrt{2}R\sqrt{1-\cos(\varphi)}, \tag{4}\]  
where \(latA\) is the latitude of star \(A\) and \(latB\) is the latitude of star \(B\). If \(A\) and \(B\) are both in the southern or northern hemisphere, \(|latA-latB|=\varphi\), if \(A\) and \(B\) are in different hemispheres, \(|latA|+|latB|=\varphi\).  
Figure 1: Polar orbiting satellite constellation configuration.  
Figure 2: Satellite link establishment status in different hemispheres.  
According to the isosceles trapezoidal waist length formula, the length of the inter-plane intersatellite links is  
\[L_{h}=\sqrt{AB^{\prime 2}+AA^{\prime}bB^{\prime}}=\sqrt{2}R\sqrt{1-\cos(\varphi)+ (1-\cos(RI))\cos(latA)\cos(latB)}. \tag{5}\]