## 5 Capacity Trends Analysis
### Long-Term Variations  
Variations in satellite trajectory can be categorized into two classes: (1) short-term (one or several orbital cycle) variations as described in Section 4.2; and (2) long-term (or seasonal) variations. In high-risk natural disaster areas (e.g., seismic zones), sensors are usually distributed in the long term [1]. The long-term satellite trajectory variations usually have distinct periods and so does the visibility duration dominated by the trajectory. Figure 3 is the simulation result of long-period oscillation of visibility duration variations. In this simulation, satellite orbit parameters are in \(J_{4}\) ( body's fourth dynamic form factor) gravity coefficient [48], as shown in Table 1. Two ground stations are Mohe at (\(53.4^{\circ}\) N, \(122.3^{\circ}\) E), and Sanya at (\(18.2^{\circ}\) N, \(109.5^{\circ}\) E) respectively.  
This oscillation is a very significant trend for network capacity. Space missions are generally designed for the worst case scenario to guarantee their mission objectives will be satisfied and sacrificing the potential advantages in better cases. From the simulation result in Figure 3, it is obvious that if we deal with these naturally-occurring long-term variations well, the network capacity can be improved.  
The long-term visibility duration variations can be largely explained by the perturbations of satellite rotational rates due to Earth's oblateness characterized by the \(J_{4}\) gravity coefficient [48]. The length of each visibility duration is directly related to the maximum elevation angle \(e_{\rm max}\) between the remote sensor and the satellite, where \(e_{\rm max}\) can be calculated using:  
\[e_{\rm max}{=}\frac{\pi}{2}{-}\theta{-}\arcsin\left(\frac{r_{E}\sin\theta}{ \sqrt{(r_{E}^{2}+r^{2}-2r_{E}r\cos\theta)}}\right) \tag{7}\]  
In Equation (7), \(\theta\) is the Earth central angle measured at the center of the Earth from the sub-satellite point to the remote sensor when the longitudes of the sub-satellite point and the sensor are equal. Additionally, the sub-satellite point is the intersection point of a straight line from the satellite to the  
\begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Value** \\ \hline Propagator & \(J_{4}\) Perturbation \\ Start Epoch & 1 January 2012 00:00:00.00 UTCG \\ Stop Epoch & 1 January 2015 00:00:00.00 UTCG \\ Orbit Epoch & 1 January 2012 00:00:00.00 UTCG \\ Apogee Altitude & 800 km \\ Perigee Altitude & 800 km \\ Inclination & 60\({}^{\circ}\) \\ Argument of Perigee & 0\({}^{\circ}\) \\ RAAN & 0\({}^{\circ}\) \\ True Anomaly & 0\({}^{\circ}\) \\ \hline \end{tabular}
\end{table}
Table 1: Satellite orbit parameters.  
Figure 3: Long-term visibility duration variations between the satellite and two ground stations with \(e_{\rm min}{=}0^{\circ}\).  
center of the Earth and the Earth's surface. The maximum elevation angle \(e_{\mathrm{max}}\) dominates the visibility duration of each pass and varies with the successive passes. It is obvious from Equation (7) that the remote sensor always lies in the orbit plane of a satellite that has very little variation in visibility duration (\(\theta\) will be zero, and \(e_{\mathrm{max}}\)=\(\pi/2\)); for example, an equatorial remote sensor and a satellite with zero inclination. Since those satellites orbits and sensors are not in the same plane, we are interested in the amplitude and frequency of visibility duration variability over successive passes. As many of the trends are periodic in nature, models to characterize the variations can be developed. With these models, space missions need not to be designed for the worst case. Additionally, potential advantages in better cases can be utilized for maximizing network capacity.