## 5 Simulation and Mathematical Analysis
### Performance of Orbit Extrapolation  
Figure 7 shows the distance error and velocity error of orbit extrapolation from \(t=1\) to \(t=30\) h when using the SGP4 algorithm in the X, Y, and Z coordinates. Here, due to the SGP4 algorithm employing the TEME coordinate system and the accurate ephemeris adopting the mean equator and mean equinox (MEME) coordinate system, the values from the TEME coordinate system are first transformed to the MEME coordinate system. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **IRIDIUM-143** & **STARLINK-2569** \\ \hline Mean orbital altitude (km) & 789.86 & 554.4 \\ Orbital inclination (degrees) & 86.4 & 53.056 \\ Period (minutes) & 102.86 & 95.593 \\ Eccentricity & 0.00011 & 0.00015 \\ \hline \end{tabular}
\end{table} 
Table 2: Satellite parameters. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Gravitational constant \(G/(N\cdot\text{m}^{2}/\text{kg}^{2})\) & \(6.67\times 10^{-11}\) \\ Earth’s mass \(\text{M}/(\text{kg})\) & \(5.965\times 10^{24}\) \\ Second-order Earth gravitational harmonic \(J_{2}\) & \(1.0826158\times 10^{-3}\) \\ Second-order Earth gravitational harmonic \(J_{3}\) & \(-2.53881\times 10^{-6}\) \\ Second-order Earth gravitational harmonic \(J_{4}\) & \(-1.65597\times 10^{-6}\) \\ Earth’s equatorial radius \(\text{KMPER}/(\text{km})\) & 6378.135 \\ Iterative tolerance \(\Delta(E+\omega)\) & \(1\times 10^{-6}\) \\ Earth’s flattening \(\in\) & \(3.3528\times 10^{-3}\) \\ \hline \end{tabular}
\end{table} 
Table 3: The second line of TLE data. forces, the magnitudes and directions of which are complex and constantly changing. In most cases, the ephemeris extrapolation error tends to increase. However, in some specific situations, the error may fluctuate within a certain range. Nonetheless, it still shows an increasing trend overall. These fluctuations may be caused by the characteristics of the extrapolation algorithm itself, the periodic variations in the satellite's orbit, or other external factors. The position error of the orbit extrapolation in the X coordinate is smaller compared to that in the Y and Z directions. Both the velocity TE and position TE of orbit extrapolation are similar in the X, Y, and Z coordinates. The position errors are larger than the velocity errors. Here, we noted that the observed start time is \(-2.22\) h because the TLE reference time is within the accurate ephemeris' given period. The further statistical results of the SGP4 algorithm are shown in Tables 4 and 5. As these tables show, the velocity errors of orbit extrapolation are smaller, with RMS values of around 0.001, whereas the position errors are larger. The extrapolation errors in the X, Y, and Z coordinates increase significantly after 10 h. Therefore, the orbit extrapolation algorithm based on the MS-FARIMA model will exploit ephemeris data in the X, Y, and Z coordinates from the first 10 h as input training data to then extrapolate the satellite orbit beyond 10 h. Since the ephemeris data in the X, Y, and Z coordinates are similar, we only present the performance of the extrapolation of the X coordinate of the ephemeris data. Figure 8 shows the performance of TE extraction and extrapolation. As Figure 8 shows, based on the original data \(S^{\prime}_{orbit}(t)\), the intrinsic function \(G_{i}(t)\) is obtained through multiple iterations, as depicted in Figure 7. Finally, the TE is derived according to \(S^{\prime}_{orbit}(t)=\sum_{i=1  
Figure 9 shows the values of the PE obtained through the FFT and spectral analysis. It can be observed that PE \(S(t)\) has a period of 1.517 h and an amplitude of 0.02. This indicates that while a PE exists in \(S^{\prime}_{orbit}(t)\), its overall impact on the values is minimal. Figure 10 shows the extrapolation results when using the MS-FARIMA model. As Figure 10 shows, before \(t=10\) h, the extrapolation error exhibited by the proposed MS-FARIMA is close to that of the original SGP4.