## 3 Simulations and Results
### Orbit Determination Simulation
#### 3.2.3 Statistical Analysis with Various Satellites  
A comprehensive validation was conducted for all 22 of the Starlink satellites mentioned in Section 3.1, with NORAD IDs 58826-58847. This validation followed a three-phase procedure: (1) observational data simulation, (2) orbit determination, and (3) 24 h orbit prediction. The observation simulation began on 7 February 2024 and lasted 24 h for every satellite. According to [PERSON] [3], ground-based radar is commonly used to observe LEO satellites. Typical measurement accuracy is several tens of meters for range (\(\rho\)) and several tens of arcseconds for angles (\(Az\), \(El\)). In this subsection, two measurement noise levels  
Figure 11: Iterative histories of position errors and \(a_{\mathrm{T}}\) estimates at the chosen epoch for Case 3 with different initial covariance values of position and velocity: (a) position errors vs. iteration number for Case 3; (b) \(a_{\mathrm{T}}\) estimates vs. iteration number for Case 3. Figure 12: 24 h RMS three-dimensional position errors based on OD results for different initial covariance values of position and velocity. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **RMS Position Error (m)** & **Maximum Position Error (m)** \\ \hline
1-E & 1274 & 2768 \\
1-E\({}^{\prime}\) & 900 & 2057 \\
1-F & 1395 & 3085 \\
1-F\({}^{\prime}\) & 1217 & 2408 \\
3-E & 2080 & 5143 \\
3-E\({}^{\prime}\) & 1060 & 2690 \\
3-F & 2308 & 5725 \\
3-F\({}^{\prime}\) & 1508 & 3424 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: 24 h position errors of OP using different initial covariances of \(a_{\mathrm{T}}\). are used for comparison: [100 m, 100 arcsec, 100 arcsec] and [200 m, 200 arcsec, 200 arcsec]. These conservative, higher noise values are chosen to test the performance of our method. If higher-precision measurements are available, the performance of our approach is expected to improve accordingly. The initial state covariance values, representing the uncertainty about the satellite's Earth-Centered Inertial (ECI) position and velocity vectors as well as the along-track acceleration, were set to [(10,000 m)\({}^{2}\), (10,000 m)\({}^{2}\), (10,000 m)\({}^{2}\), (10 m/s)\({}^{2}\), (10 m/s)\({}^{2}\), (10 m/s)\({}^{2}\), (10\({}^{-4}\) m/s\({}^{2}\))\({}^{2}\)]. The simulation results presented in Sections 3.2.1 and 3.2.2 demonstrate that this configuration of initial state uncertainties achieves stable and favorable orbit determination and prediction performance. A statistical analysis was performed by performing 30 independent observation simulations, OD and OP for each satellite. The 24 h RMS position errors of OP were computed for all trials. Figure 13 presents the resulting mean RMS and maximum position errors for each satellite (except the satellite with NORAD ID 58837). For clarity in the graphical representation, the satellite number series on the x-axis corresponds directly to the NORAD IDs in ascending order. The results show that, for a measurement noise level of [100 m, 100 arcsec, 100 arcsec], the maximum position prediction errors are all less than 6 km. The mean RMS values of the position prediction errors are below 2.5 km. For a noise level of [200 m, 200 arcsec, 200 arcsec], the maximum position prediction errors remain below 7 km, and the mean RMS values are still under 2.5 km. In general, decreased observation accuracy results in greater errors in orbit determination and prediction for most satellites. It should be noted that, during the observation simulation period, the observational data for 14 satellites contained a gap of approximately 15 h. The achieved level of OP accuracy demonstrates that our method remains robust and reliable even in the presence of observational gaps of around 10 h.  
Figure 13: Predicted position errors over 24 h for 21 climbing Starlink satellites: (**a**) with measurement noise level of [100 m, 100 arcsec, 100 arcsec]; (**b**) with measurement noise level of [200 m, 200 arcsec, 200 arcsec].