# The Availability of Space Service for Inter-Satellite Links in Navigation Constellations
## 3 Analysis of Availability  
The ideal goal for navigation satellites is to ensure that all users in the service space can observe four or more satellites for real-time positioning, and the suboptimal goal is at least one satellite for time synchronization. The GPS service volume, which includes the TSV, MEO SSV, and HEO/GEO SSV, is shown in Figure 2. Numerical results reveal that 97% of the users in the MEO SSV can obtain four-fold availability using the main lobe of L1, while only 1% of the users can obtain four-fold availability, and 80% of the users can obtain single-fold availability at an altitude 36,000 km [5]. The availability may be enhanced if the side lobes are considered. However, there are two problems hidden in these seemingly satisfactory results: (1) the availability needs to be improved, especially in the HEO/GEO SSV; and (2) the navigation signal received at the GEO is approximately 25 dBW weaker than that at an altitude of 3000 km. Use of the ISL signal is expected to solve these problems because its natural advantage is that it is designed for users that are in space.  
Figure 2 shows two additional parameters of the ISL signal: \(\theta_{beam}\) is the beamwidth and is related to the gain of the antenna, and \(\theta_{scan}\) is the scanning angle and is related to the field of view.  
The availability of the ISL is defined as the number of visible navigation satellites at an arbitrary time. This assumes that the signal can be received as long as users are in the LOS of the antenna. In order to eliminate the influence of the ionosphere and troposphere, altitudes under 1000 km are considered to be invisible, as shown in Figure 3.  
Figure 1: Characteristics of the ISL signal: (**a**) range to MEO11; and (**b**) rangerate to MEO11.  
Figure 2: Coverage characteristics of navigation signal and ISL signal.  
The off-axis angle \(\theta\) is the angle between a line extending from the navigation satellite to the space user, and a line extending from the navigation satellite to the center of the earth. A navigation satellite that is visible to the space user must meet Equation (1):  
\[\left\{\begin{array}{l}\theta_{obs}<\theta<\theta_{scan},\ \ \ \text{or}\\ \theta_{obs}>\theta\ \ \ \text{and}\ \ \ \left\|\overrightarrow{R_{s}}- \overrightarrow{R_{u}}\right\|<(R_{E}+H_{s})\cos(\theta_{obs})\end{array}\right. \tag{1}\]  
where \(\theta_{obs}=\arcsin((R_{E}+1000)/(R_{E}+H_{s}))\) is the Earth obstructing angle, \(\theta_{scan}\) is the maximum scanning angle, \(R_{E}\) (6378.137 km) is the radius of the Earth, \(H_{s}\) is the altitude of the navigation satellite, \(\overrightarrow{R_{s}}=(x_{s},y_{s},z_{s})\) is the position of the satellite in ECEF (Earth-Centered, Earth-Fixed) coordinates, and \(\overrightarrow{R_{u}}=(x_{u},y_{u},z_{u})\) is the position of the user in the same coordinate system. The upper condition means the user outside the maximum scanning angle is unavailable. The lower means the visible user and satellite should be in the same side of Earth if \(\theta<\theta_{obs}\). The cosine of \(\theta\) can be computed as  
\[\cos(\theta)=\frac{<\overrightarrow{R_{s}},\overrightarrow{R_{s}}- \overrightarrow{R_{u}}>}{\left\|\overrightarrow{R_{s}}\right\|\left\| \overrightarrow{R_{s}}-\overrightarrow{R_{u}}\right\|} \tag{2}\]  
Hence, we get  
\[\theta=\arccos(\frac{x_{s}\left(x_{s}-x_{u}\right)+y_{s}\left(y_{s}-y_{u} \right)+z_{s}\left(z_{s}-z_{u}\right)}{\sqrt{x_{s}^{2}+y_{s}^{2}+z_{s}^{2}} \sqrt{\left(x_{s}-x_{u}\right)^{2}+\left(y_{s}-y_{u}\right)^{2}+\left(z_{s}-z _{u}\right)^{2}}}) \tag{3}\]  
The visibility can be determined by Equations (1) and (3).