# Impacts of Arc Length and ECOM Solar Radiation Pressure Models on BDS-3 Orbit Prediction
## 3 Results and Analysis
### Impacts of the Arc Length on Orbit Prediction  
The user range error (URE) is used to verify the accuracy of the predicted orbit. The satellite URE is associated with the maximum opening angle of the satellite relative to Earth [15]. Without considering the satellite clock error, the MEO satellites' URE is computed as follows [21]:  
\[{\rm URE}_{\rm MEO}=\sqrt{(0.99\Delta{\rm R})^{2}+(0.14\Delta{\rm T})^{2}+(0.14 \Delta{\rm N})^{2}} \tag{5}\] \begin{table}
\begin{tabular}{l l} \hline \hline
**Models** & **Reference/Source** \\ \hline Geopotential & EGM 2008, 12 \(\times\) 12 degree [24] \\ N-body gravitation & JPL DE405 ephemeris [25] \\ EOP & Fixed to IERS EOP 14 C04 dataset \\ SRP model & ECOM5 model, ECOM7 model, ECOM9 model \\ Solid earth tides, pole tides & IERS conventions 2010 [26] \\ Ocean tides & FES2004 [27] \\ Relativity effect & IERS2010 \\ Integration & Collocation Integration method \\ Integration step size & 900 s \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dynamical models and integration method. where \(\Delta\)R, \(\Delta\)T, \(\Delta\)N are the radial, along-track, and cross-track components of the satellite orbit, respectively. For IGSO and GEO satellites, URE is computed as follows. \[\text{URE}_{\text{GEO/IGSO}}=\sqrt{\left(\Delta\text{R}\right)^{2}+\left(0.99 \Delta\text{T}\right)^{2}+\left(0.99\Delta\text{N}\right)^{2}} \tag{6}\]  
It can be seen from the Formulas (5) and (6) that the most significant contribution to URE is the radial component of the satellite orbit. The orbit prediction accuracy of BDS-3 MEO satellites is first calculated in the experiment. The 24-72 h satellite-orbit arc lengths are used as initial observation data. IGS's final precise ephemeris is used as the reference. We use the root mean square (RMS) in the radial (R), along-track (T) and cross-track (N) directions and URE to evaluate the accuracy of BDS-3-predicted orbits. The accuracy of the first 6 h and 24 h of predicted orbits is used to evaluate the impact of the arc length on BDS-3 MEO orbit prediction. As shown in Figure 1, the smallest orbital URE is in the fitted arc length of 42-48 h for both 6 h and 24 h orbit prediction. It can be observed that the accuracy of the radial, along-track, and cross-track are similar to that of the URE, and the best fitting arc length is 42-48 h. The results also show that when the prediction intervals are 6 h, the URE is the smallest at the arc length of 42 h, and the URE, radial, cross-track, and along-track are 0.029 m, 0.019 m, 0.022 m, and 0.054 m, respectively. When the prediction intervals are 24 h, the URE is also the smallest at the arc length of 42 h, and the URE, radial, cross-track, and along-track are 0.051 m, 0.027 m, 0.034 m, and 0.111 m, respectively. Meanwhile, the orbit prediction accuracy of BDS-3 IGSO satellites is analyzed. BDS-3 has three IGSO satellites (C38, C39, and C40). In this contribution, these BDS-3 IGSO satellites are taken as an example to explore the best-fitting arc length. As is shown in Figure 2, the results of BDS-3 IGSO are consistent with those of BDS-3 MEO satellites. The smallest orbital URE is in the fitted arc length of 42-48 h for both 6 h or 24 h length orbit prediction. The results show that when the prediction intervals are 6 h, the URE is the smallest at the arc length of 42 h, and the URE, radial, cross-track, and along-track are 0.193 m, 0.125 m, 0.080 m, and 0.125 m, respectively. When the prediction intervals are 24 h, the URE is also the smallest at the arc length of 42 h, and the URE, radial, cross-track, and along-track are 0.231 m, 0.108 m, 0.080 m, and 0.189 m, respectively. It is noteworthy that, compared to BDS-3 MEO satellites, the accuracy of BDS-3 IGSO satellite orbit prediction is significantly reduced when the fitted arc length is less than 36 h. The reason may be that the BDS-3 IGSO satellite has a longer orbital period than the MEO  
Figure 1: The accuracy of the first 6 h (**top**) and 24 h (**bottom**) of predicted BDS-3 MEO orbits with different arc lengths. satellite.