# Link-State Aware Hybrid Routing in the Terrestrial-Satellite Integrated Network +
## 3 System Model
### Problem Formulation
Similarly, the interplane ISL distance increases with the increasing of orbital height and inclination, while it decreases with the satellites number in an orbit plane. The change of interplane ISL lengths are simulated with Iridium and Hongyun project, and as shown in Figure 4, due to the difference in satellite constellation parameters, the ISL distance varies periodically in the time domain, and the change of Iridium ranges from 1680 km to 4370 km, which is more than two times wider than that of Hongyun. Based on the above analysis, the distance of interplane ISL is time-varying, and the orbiting movements of satellites also lead to dynamic distances of TSL. Meanwhile, the achievable data rate of ISL is inversely proportion to the square of distance [38]. In other words, ISLs have time-varying link capacity. If the duration of \(\Delta\tau\) is too large, the transmission delay of the link will be inaccurate, which results in suboptimal path planning, while the short value of \(\Delta\tau\) will waste the computing resources. To this end, based on the dynamic characteristics of ISL distance, the relationship function between time interval and satellite constellation parameters can be derived. Figure 3: The geometry trajectory of satellite. Assuming that the shortest path \(U\) between satellites in satellite network \(G_{\rm S}(V_{s},E_{ss})\) is composed of \(N\) links, the propagation delay of the shortest path \(U\) is is \(\sum\limits_{(s_{i}^{t},s_{j}^{t})\in E_{ss}}^{N}T_{p}(s_{i}^{t},s_{j}^{t})\). Since the ISL distance is continuous over time, the satellite network \(G_{\rm S}(V_{s},E_{ss})\) will be static in a time interval \(\Delta\tau\), i.e., the neighbors of each satellite in \(\Delta\tau\) can not change. To obtain the optimum time interval, the delay corresponding to ISL distance variations are adopted as the reference for the \(\Delta\tau\) setting, so that the path accuracy and computing resources can be guaranteed. Denoting the maximum delay corresponding to ISL length variation as \(\lambda\), all propagation delays after \(\Delta\tau\) need to satisfy:  
\[|\sum\limits_{(s_{i}^{t},s_{j}^{t})\in E_{ss}}^{N}T_{p}(s_{i}^{t+\Delta\tau},s _{j}^{t+\Delta\tau})-\sum\limits_{(s_{i}^{t},s_{j}^{t})\in E_{ss}}^{N}T_{p}(s_{ i}^{t},s_{j}^{t})|\leq\lambda. \tag{16}\]  
Let \(T_{\rm max}(t)\) denotes the maximum interplane ISL propagation delay of the shortest path \(U\), then we have  
\[\sum\limits_{(s_{i}^{t},s_{j}^{t})\in E_{ss}}^{N}T_{p}(s_{i}^{t},s_{j}^{t})<N \times T_{\rm max}. \tag{17}\]  
Denoting \(\Delta d_{\rm max}(s_{i}^{\Delta\tau},s_{j}^{\Delta\tau})\) as the maximum distance change of ISL in a \(\Delta\tau\), (17) can be transformed as  
\[|\sum\limits_{(s_{i}^{t},s_{j}^{t})\in E_{ss}}^{N}T_{p}(s_{i}^{t+\Delta\tau},s _{j}^{t+\Delta\tau})-\sum\limits_{(s_{i}^{t},s_{j}^{t})\in E_{ss}}^{N}T_{p}(s_ {i}^{t},s_{j}^{t})|\leq\frac{N\times\Delta d_{\rm max}(s_{i}^{\Delta\tau},s_{ j}^{\Delta\tau})}{v_{p}}. \tag{18}\]  
Figure 4: Interplane ISL distance vs. time. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Constellation** & **Iridium** & **Hongyun** \\ \hline No. of planes & 6 & 13 \\ No. of satellites per plane & 11 & 12 \\ Orbital inclination & 86\({}^{\circ}\) & 86.5\({}^{\circ}\) \\ Orbital height & 788 km & 1000 km \\ Interplane ISLs length & 4032.9 km & 3790 km \\ Intraplane ISLs propagation delay & 13.5 ms & 12.6 ms \\ Longest interplane ISLs distance & 4370 km & 2366 km \\ Longest interplane ISLs propagation delay & 14.5 ms & 7.89 ms \\ Shortest interplane ISLs distance & 1680 km & 1426 km \\ Shortest interplane ISLs propagation delay & 5.6 ms & 4.75 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The comparison between ISLs.