# Telescopic Network of Zhulong for Orbit Determination and Prediction of Space Objects
## 2 Data and Methods
### Method of Precision OD and OP  
Precise OD can be conducted using continuous multi-day angular arc data. Orbit determination utilizes the least squares estimation method, which calculates an orbit that minimizes the sum of squared residuals between theoretical and actual observations. Set the quantity of the state. \[\mathbf{x}=\{\mathbf{r},\mathbf{v},\mathbf{p}\}^{T} \tag{7}\]  
where \(\mathbf{r}\) and \(\mathbf{v}\) represent the three-dimensional position and velocity vectors of the orbit, respectively. \(\mathbf{p}\) is the parameter vector of the force model to be estimated, typically including drag coefficients, pressure of light, etc. Assuming there is a functional relationship between the measured angular data (\(\mathbf{y}\)) and the initial orbit (\(\mathbf{x_{0}}\)),  
\[\mathbf{y}=f(\mathbf{x_{0}})+\mathbf{n} \tag{8}\]  
where \(\mathbf{n}\) is the measurement noise. The OD process requires minimizing the weighted sum of squares of the difference between the actual observed value \(\mathbf{y}\) and the value calculated using the mathematical and physical model \(f(\mathbf{x_{0}})\), under the conditions of given angular data \(\mathbf{y}\), statistical characteristics of the mathematical model \(f\), and noise \(n\). \[(y-f(\mathbf{x_{0}}))^{T}P(\mathbf{y}-f(\mathbf{x_{0}}))=min \tag{9}\]  
Here, \(P\) is the weight matrix and the estimated value of \(\mathbf{x_{0}}\), denoted as \(\hat{\mathbf{x}}_{0}\), is  
\[\hat{\mathbf{x}}_{0}=\mathbf{x_{0}}+\Delta\hat{\mathbf{x}}_{0} \tag{10}\]  
The equation constructed using the least squares principle is  
\[(\mathbf{A^{T}P\mathbf{A}})\ \Delta\hat{\mathbf{x}}_{0}=\mathbf{A^{T}P}(\mathbf{y}-f(\mathbf{x_{0}})) \tag{11}\]  
where A = \(\left.\left(\frac{\partial f}{\partial x}\right)\right|_{\mathbf{x}=\mathbf{x_{0}}}\) is the partial derivative matrix. The solution of the previous equation is  
\[\Delta\hat{\mathbf{x}}_{0}=(\mathbf{A^{T}P\mathbf{A}})^{-1}\mathbf{A^{T}P}(\mathbf{y}-f(\mathbf{x_{0}})) \tag{12}\]  
The process described above is a continuous iterative process. Initially, \(\hat{\mathbf{x}}_{0}\) is solved and used as an approximation of the initial state quantity. The process is then repeated until the variation of the solved \(\hat{\mathbf{x}}_{0}\) satisfies the condition of convergence, typically when the number of position corrections in the two calculations is less than 1 mm. To obtain an accurate orbit, a set of angular measurements with minimal error is necessary, along with a complete \begin{table}
\begin{tabular}{c c c c} \hline \hline
**NORAD ID** & **Stn6002** & **Stn 5001** & **File Count** \\ \hline \multirow{3}{*}{41240} & \(\overline{z}=\)11 & \(\overline{z}=\)10 & \multirow{3}{*}{16} \\  & \(\overline{z}=\)5.2 & S =3.4 & \\ \multirow{3}{*}{46984} & \(\overline{z}=\)8 & \(\overline{z}=\) & \\  & S =3.4 & \(\overline{z}=\) & \\ \multirow{3}{*}{1328} & \(\overline{z}=\)10 & \multirow{3}{*}{\(\overline{z}=\)10} & \multirow{3}{*}{38} \\  & S =2.6 & & \\ \multirow{3}{*}{All} & \(\overline{z}=\) 9 & \(\overline{z}=\)10 & \\  & S = 3.1 & S =3.4 & \\ \hline \multicolumn{4}{l}{– indicates no data.} \\ \end{tabular}
\end{table} 
Table 2: Error statistics for 2 stations. mathematical model \(f(\mathbf{x_{0}})\). Commonly used mathematical models include the Cowell numerical integration model and the Gauss-Jackson model [34]. All subsequent examples use the Cowell model. The force models used in OD and OP include the gravity field JGM-3 50 \(\times\) 50, third-body gravity, atmospheric mass density model NRLMSISE-00 [35] and JB2008 [36] for atmospheric drag, and solar radiation pressure. The OP can be described as  
\[\left\{ \begin{array}{l}\frac{\mathbf{x}}{dt}=f(\mathbf{x},t)\\ \mathbf{x}(t_{0})=\mathbf{x_{0}}\end{array} \right. \tag{13}\]  
where \(\mathbf{x}_{0}\) and \(f\) have been obtained in the OD.