## 3 Star Tracker Observation Model  
In the process of observing space debris, assuming that the vector of debris observed by the star tracker in the geocentric coordinate system is \(\overset{\rightarrow}{\rho}\), the basic relationship between the instantaneous geocentric position vector of debris \(\overset{\rightarrow}{r}_{deb}\) and the position vector of the satellite in which the star tracker is located is as follows [25]:  
\[\overset{\rightarrow}{\rho}~{}=~{}\overset{\rightarrow}{r}_{deb}-\overset{ \rightarrow}{r}_{sat}~{}=~{}\left\lfloor \begin{array}{l}x_{deb}-x_{sat}\\ y_{deb}-y_{sat}\\ z_{deb}-z_{sat}\end{array} \right\rfloor \tag{4}\]  
where \(\overset{\rightarrow}{r}_{deb}=\left[~{}x_{deb}~{}~{}y_{deb}~{}~{}z_{deb}~{} \right]^{T}\) and \(\overset{\rightarrow}{r}_{sat}~{}=~{}\left[~{}x_{sat}~{}~{}y_{sat}~{}~{}z_{ sat}~{}\right]^{T}\) are the coordinates in the inertial system of the debris and star tracker, respectively, and the spatial polar coordinates of the observed debris \(\overset{\rightarrow}{\rho}\) are expressed as follows:  
\[\overset{\rightarrow}{\rho}~{}=~{}\rho\left[ \begin{array}{l}\cos\delta_{ star}\cos\alpha_{star}\\ \cos\delta_{star}\sin\alpha_{star}\\ \sin\delta_{star}\end{array} \right] \tag{5}\]  
where \(\rho\) is the distance from the debris to the observation star tracker, \(\alpha\) and \(\delta\) the recast-cension and declination of the debris. The angle subscript \"_star_\" indicates the observation vector of the rectascension and declination by the star tracker. \[\overset{\rightarrow}{r}_{deb}~{}=~{}r\left[ \begin{array}{l}\cos\delta_{ earth}\cos\alpha_{earth}\\ \cos\delta_{earth}\sin\alpha_{earth}\\ \sin\delta_{earth}\end{array} \right] \tag{6}\]  
The angle subscript \"_earth_\" indicates the rectascension and declination of the debris vector observed under the center of the celestial sphere, that is, the center of the earth. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Target Diameter D** & \begin{tabular}{c} **Diffuse Target** \\ **Distance R = 200 km** \\ \end{tabular} & \
\begin{tabular}{c} **Diffuse Target** \\ **Distance R = 100 km** \\ \end{tabular} &
\begin{tabular}{c} **Diffuse Target** \\ **Distance R = 50 km** \\ \end{tabular} \\ \hline
1 m & +3.8 Mv & +2.3 Mv & +0.8 Mv \\
0.5 m & +5.3 Mv & +3.81 Mv & +2.31 Mv \\
0.3 m & +6.63 Mv & +4.92 Mv & +3.42 Mv \\
0.2 m & +7.3 Mv & +5.8 Mv & +4.3 Mv \\
0.1 m & +8.8 Mv & +7.3 Mv & +5.8 Mv \\
0.05 m & +9.5 Mv & +8.0 Mv & +6.5 Mv \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Analysis of brightness characteristics of typical spherical targets. After the coordinate transformation,  
\[\left[ \begin{array}{c}\cos\delta_{star}\cos\alpha_{star}\\ \cos\delta_{star}\sin\alpha_{star}\\ \sin\delta_{star}\end{array} \right]\ =\ KA_{1}\left[ \begin{array}{c}\cos\delta_{ earth}\cos\alpha_{earth}\\ \cos\delta_{earth}\sin\alpha_{earth}\\ \sin\delta_{earth}\end{array} \right]+r_{sat} \tag{7}\]  
The relationship between the direction vector of the debris under the celestial sphere, that is, under the center of the earth, and the direction vector of the debris by satellite observation is the coordinate transformation from the coordinate system of the star tracker to the coordinate system of the geocentric coordinate system, where \(k\) is the scaling factor, \(A_{1}\) is the coordinate system rotation matrix, \(r_{sat}\) is the vector of the observation satellite (quasi-satellite optical observation platform) in the geocentric coordinate system, that is, the translation vector from the star tracker coordinate system to the geocentric coordinate system [5]. Since translation and scaling do not change the directionality of the vector, it can be derived  
\[\left[ \begin{array}{c}\cos\delta_{star}\cos\alpha_{star}\\ \cos\delta_{star}\sin\alpha_{star}\\ \sin\delta_{star}\end{array} \right]\ =\ A_{1}\left[ \begin{array}{c}\cos\delta_{ earth}\cos\alpha_{earth}\\ \cos\delta_{earth}\sin\alpha_{earth}\\ \sin\delta_{earth}\end{array}