# Line-of-Sight Initial Pointing Model of Space Dynamic Optical Network and Its Verification
## 3 Multi-Link LOS Pointing Model
### Fou  
Various factors limit the accuracy of the initial pointing. In addition, there are errors in the azimuth axis and elevation axis. The errors of the two axes are independent of each other and follow a Gaussian distribution, with a mean of zero. The error of the initial pointing angle is a Rayleigh distribution, which can be expressed as follows [11]:  
\[f(\theta_{i})=\frac{1}{\sqrt{2\pi\sigma_{i}}}\exp\left(-\frac{\theta_{i}^{2}} {2\sigma_{i}^{2}}\right) \tag{7}\]  
where \(\theta_{i}=\sqrt{\theta_{\alpha_{i}}^{2}+\theta_{\beta_{i}}^{2}}\) is the initial pointing error angle, \(\theta_{\alpha_{i}}\) is the azimuth direction error angle, \(\theta_{\beta_{i}}\) is the elevation direction error angle, and \(\sigma_{i}\) is the standard deviation of the initial pointing error angle, that is, the pointing error. If the Gaussian distribution has the same standard deviation, \(\sigma_{i}=\sigma_{\alpha_{i}}=\sigma_{\beta_{i}}\), in the azimuth direction error angle and the elevation direction error angle, then the path \(i\) capture probability is as follows:  
\[P_{i}=\int\int_{FOU_{i}}\frac{1}{\sqrt{2\pi\sigma_{i}}}\exp\left(-\frac{\theta _{\alpha_{i}}^{2}}{2\sigma_{i}^{2}}\right)\times\frac{1}{\sqrt{2\pi\sigma_{i} }}\exp\left(-\frac{\theta_{\beta_{i}}^{2}}{2\sigma_{i}^{2}}\right)\mathrm{d} \theta_{\alpha_{i}}\mathrm{d}\theta_{\beta_{i}} \tag{8}\]  
Formula (8) is simplified to a uniform distribution with pole angle \(1/2\pi\), and the integral in the polar coordinate system is as follows:  
\[P_{i}=\int_{0}^{FOU_{i}/2}\frac{\theta_{i}}{\sigma_{i}^{2}}\exp\left(-\frac{ \theta_{i}^{2}}{2\sigma_{i}^{2}}\right)\mathrm{d}\theta_{i} \tag{9}\]  
Because of the pointing error, LOSs will not accurately point to each other, but to \(FOU\). For a capture probability greater than 98.89%, \(P_{i}\geq\) 98.89%, \(FOU_{i}\geq 3\sigma_{i}\) is calculated. It can be seen that the size of \(FOU\) is determined by the pointing error. There are many factors affecting the pointing error, including position error, attitude error, dynamic lag, platform vibration, LOS stability error, and alignment error of system instrument installation.  
When the optical terminal between the network nodes is in a state of follow-up pointing, the pointing error can be obtained using the azimuth and elevation follow-up pointing angle, and then the size of \(FOU\) can be obtained. The azimuth error angle \(\theta_{\alpha_{i}}\) can be obtained by determining the difference between the measured azimuth angle \(\alpha_{i}^{ml}\) and the theoretical value \(\alpha_{i}\), calculated using the above pointing model, and \(\sigma_{\alpha_{i}^{2}}\) can be obtained according to the following formula. Similarly, \(\theta_{\beta_{i}}=\beta_{i}^{m}-\beta_{i}\) is calculated according to the following formula to obtain \(\sigma_{\beta_{i}^{2}}\). The calculation formula is as follows:  
\[\sigma_{\chi_{i}^{2}}=\frac{\sum{(\theta_{\chi_{i}}-\mu_{\chi_{i}})^{2}}}{N_{ \chi_{i}}}\]The size of _FOU_ of the \(i\) route can be obtained, and deduced the formula is as follows:  
\[FOU_{i}=3\sigma_{i}=3\sqrt{\sigma_{a_{i}^{2}}+\sigma_{\beta_{i}^{2}}} \tag{11}\]