## 5 Capacity Trends Analysis
### Dynamics of Nodes  
We now study the impact of dynamic node positions on availability in the network capacity model. The IWSSN is a high dynamic network. Satellites are located in orbits above the Earth and are usually mobile relative to the ground. New satellites and sensors may inject into the IWSSN, and invalid nodes may also be ruled out from the network. Although satellites in different orbits may be mobile relative to each other, in most practical scenarios, inter-satellite links (ISLs) are designed to be stable (ISLs are generally designed to connect satellites in the same orbital plane or satellites in adjacent orbital planes with slow relative movement in most practical space scenarios, e.g., ISLs in Iridium [38, 39], advanced extremely high frequency (AEHF) [43] and tracking and data relay satellite (TDRS) [44] systems; there may be high dynamic ISLs, such as in multi-layered satellite network (MLSN) [45]; the special cases are not considered in our current work). Hence, the main factors affecting the availability are the line-of-sights between satellite nodes and sensor nodes. In order to study the impact of dynamic node positions on the network capacity model. We focus on the interaction of nodes' dynamics and availability in Equation (3), _i.e._, we study capacity uniquely as a function of availability time. Hence, we only consider \(\varphi\left(d,p,t\right)\) and assume that the other constraints \(r_{i}\left(d,p,t\right)\), \(s_{i}\left(d,p,t\right)\) and \(\eta_{i}\left(d,p,t\right)\) in Equation (3) are constant. Then, from Equations (1) and (3), capacity is directly proportional to availability time and can be expressed as:  
\[C_{G}^{\times}\propto\frac{1}{T}\sum\
olimits_{i=1}^{N}\sum\limits_{d\in D_{i }}\sum\limits_{p\in P_{i}^{d}}\int_{t_{0}}^{T+t_{0}}\varphi_{idp}\left(t\right) \mathrm{d}t \tag{4}\]  
where \(\varphi_{idp}\left(t\right)\) is equal to \(\varphi_{i}\left(d,p,t\right)\). We are interested in the major contributors to the path availability \(\varphi_{idp}\left(t\right)\), _i.e._, the dynamically-varying duration of each satellite node in the orbit pass over visible remote sensor nodes. Hence, we should determine if there is visibility between satellite nodes and sensor nodes; that is, if the sensor node is in the coverage area of a satellite during the orbit. The percentage of passes that results in visibility between a satellite node and remote sensor node can be evaluated using simple geometric model as a function of satellite inclination, altitude and the sensor latitude [46]. We verify this model in our simulation, as shown in Figure 2. The altitudes of the satellitesare 550 km, 880 km, 1450 km and 8050 km, respectively. We select these altitudes from low to high, and the recursive orbit period of these altitudes are all two days. Statistical coverage analysis is done for satellite inclinations \(i_{s}\) and the sensor latitudes \(l_{g}\) in the Northern Hemisphere (_i.e._, \(0^{\circ}\leq i_{s}\leq 90^{\circ}\) and \(0^{\circ}\leq l_{g}\leq 90^{\circ}\)). The minimum elevation angle \(e_{\min}\) at which the remote sensor can establish a link to an orbiting satellite in the simulation is \(10^{\circ}\). From Figure 2, the general trend is that visibility duration increases as the satellite inclination approaches the sensor's latitude. Obviously, satellites with low inclinations only cover equatorial and near-equatorial areas, while satellites with polar orbits cover remote sensors with all latitudes. The percent coverage of low latitude areas is reduced compared to high latitude areas when satellite inclination increases. Useful information about the visibility duration trends can be extracted from Figure 2 as a function of satellite and sensor node parameters. For instance, under the network topology constraint, a satellite with inclination \(i_{s}=30^{\circ}\) cannot cover any sensor nodes with latitudes \(l_{g}>51^{\circ}\) for an 880-km orbit and \(l_{g}>58^{\circ}\) for a 1450 km orbit.