## 3 Simulations and Results
### Orbit Determination Simulation  
This section examines the orbit of the Starlink satellite with NORAD ID 58831. Figure 6 shows the reference orbit and corresponding position errors, derived from the SpaceX ephemeris, over a seven-day period starting from the epoch 7 February 2024 19:25:42 UTC. It can be seen that during the first 5 days, the semi-major axis increases almost linearly at a rate of approximately 10.8 km/day. On the sixth day, it remains nearly constant, and  
Figure 4: Statistics of observational gaps obtained by Station 1 during the observation period.  
Figure 5: Distribution of visibility arcs for Starlink satellite with NORAD ID 58836, recorded by Station 1 during the observation period. The red columns indicate durations of the visible arcs, which are measured in minutes. The black columns represent durations of the observational gaps, which are measured in hours.  
from the seventh day onwards, the linear growth pattern resumes. During the period of linear growth of the semi-major axis, the position error can reach approximately 6.5 km. In contrast, when the semi-major axis remains constant, the position error is less than 1 km. The eccentricity and the inclination during the 7 days remain nearly constant. Simulated observations with Station 1 were conducted over a 24 h period beginning at the same epoch, following the procedure described in Section 3.1. In total, 76 sets of observational data were obtained, resulting in seven visible arcs (see Table 2). The time intervals between the consecutive visible arcs were approximately 1.5 h. The first observable moment, i.e., 7 February 2024 20:31:42 UTC, is taken as the selected epoch \(t_{\mathrm{k}}\) for orbit determination. At this epoch, the true position and velocity vectors are \(\mathbf{r}_{0}^{true}\) = [\(-\)5956286.9372 1295514.0729 2914748.1930] m and \(\mathbf{v}_{0}^{true}\) = [1347.5254 \(-\)5509.4750 5184.0691] m/s. The initial estimation of the satellite's position and velocity (\(r_{\mathrm{initial}}\), \(\mathbf{v}_{\mathrm{initial}}\)) at the selected epoch \(t_{\mathrm{k}}\) is computed using [PERSON]'s method, based on the inertial positions corresponding to the first and last timestamps of Arc No. 1, derived from the observational data. The tuning parameter \(\omega_{0}\) for sigma points construction is set to 0.3. The predefined convergence criteria \(\varepsilon\) are taken as \(10^{-2}\). The maximum number of iterations is set to 30.  
As discussed in Section 2, the proposed method removes the requirement for process noise. However, the covariance matrices associated with the initial state vector (\(\textbf{P}_{\mathrm{initial}}\)) and the measurement noise (**R**) must still be empirically specified. Multiple combinations of \(\textbf{P}_{\mathrm{initial}}\) and **R** are tested to evaluate their influence on OD performance. Once the augmented state vector at the chosen epoch is determined, they are propagated through the dynamics equations to generate orbit ephemerides over the following 24 h period. To evaluate the accuracy of OD and OP, RMS errors between the generated orbit and the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Arc ID** & **Starting Time** & **Ending Time** \\ \hline
1 & 7 February 2024 20:31:42 & 7 February 2024 20:35:42 \\
2 & 7 February 2024 22:05:42 & 7 February 2024 22:12:42 \\
3 & 7 February 2024 23:42:42 & 7 February 2024 23:48:12 \\
4 & 8 February 2024 01:21:42 & 8 February 2024 01:23:42 \\
5 & 8 February 2024 02:57:42 & 8 February 2024 03:02:12 \\
6 & 8 February 2024 04:32:42 & 8 February 2024 04:39:42 \\
7 & 8 February 2024 06:09:42 & 8 February 2024 07:14:12 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation results observed from Station 1 for satellite 58831 since 7 February 2024 19:25:42 UTC over a 24 h period.  
Figure 6: Reference orbit and position errors of Starlink satellite with the NORAD ID 58831 since epoch 7 February 2024 19:25:42 UTC.  
true orbit are calculated over the next 24 h in the radial, along-track, cross-track, and total position components.