# Advancing the Solar Radiation Pressure Model for BeiDou-3 IGSO Satellites
## 3 Results
### Performance of SRP Models for BeiDou-3 IGSO Satellites
#### 3.2.1 Orbit Assessment Based on Estimated Clocks  
Figure 4 illustrates the fitting residual RMS of daily clock estimates for satellite C39 from the POD solutions with the ECOM1, ECOM2 and ECOM1 + BW models. For each solution, the clock fitting residual RMS of three BeiDou-3 IGSOs, C38, C39 and C40, exhibit similar patterns. From Figure 4, the RMS time series based on the ECOM1 model show pronounced \(\beta\) -dependent variations with peak-to-peak amplitude of about 13.0 cm. The stable and small RMS values only occur for very short time periods when the \(|\beta|\) is greater than 60\({}^{\circ}\). Once the ECOM2 model is used, this \(\beta\) -dependent effect is weakened, but still significant. The solution with the ECOM1 + BW model gives the best results, and the systematic effects are almost eliminated. Table 4 lists the mean of daily clock residual RMS of satellites C38, C39 and C40 for the ECOM1, ECOM2 and ECOM1 + BW solutions. The mean of daily clock residual RMS for the ECOM1 + BW solution decreases by 3.2 cm (47.1%) and 1.4 cm (28.0%) relative to the ECOM1 and ECOM2 solutions on average, respectively. Figure 4: Daily RMS of clock fitting residuals for satellite C39 based on the ECOM1 (green), ECOM2 (blue) and ECOM1 + BW (red) models. The areas shaded in gray indicate eclipse seasons. \begin{table}
\begin{tabular}{c c c c} \hline
**Panel** & **Area** & **\(\alpha\)+\(\delta\)** & \(\rho\) \\ \hline +\(X\) & 8.496 & 0.366 \(\pm\) 0.048 & 0.531 \(\pm\) 0.165 \\ +\(Z\) & 20.871 & 0.589 \(\pm\) 0.169 & 0.001 \(\pm\) 0.172 \\ \(-Z\) & 20.871 & 0.662 \(\pm\) 0.123 & 0.018 \(\pm\) 0.143 \\ \hline \end{tabular}
\end{table} 
Table 3: Estimated optical absorption plus reflectivity (\(\alpha+\delta\)) and diffusion (\(\rho\)) as well as corresponding standard deviations for all BeiDou-3 IGSOs’ satellite bus. The unit of Area is \(m^{2}\). A significant part of the systematic \(\beta\) -dependent pattern in the clock fitting residual RMS for the ECOM1 solution is induced by radial orbit modeling defects, which can be confirmed by the clock residual time series on DOY 172-173, 2020 (\(\beta\approx 78.4^{\circ}\)) and DOY 254-255, 2020 (\(\beta\approx 9.5^{\circ}\)) presented in Figure 5. During DOY 254-255 of the year 2020, there is a once-per-revolution signal with an amplitude of roughly 20 cm in the clock fitting residuals, resulting in a large clock residual RMS. The major reason is that the ECOM1 model cannot fully consider the effect of the rapidly varying cross-section exposed to the Sun on SRP during periods with small \(\beta\) -angles, which introduces systematic once-per-revolution radial orbit errors [9]. This systematic error can be reduced by using the ECOM2 model or almost eliminated when using the developed a priori box-wing model, as shown in Figure 5. For the periods with a large \(\beta\) -angle on DOY 172-173 of the year 2020, the illuminated cross-section of the satellite body does not change much. The ECOM1 model can work as efficiently as the ECOM2 and ECOM1 + BW models in this scenario (Figure 5, top). The SRP modeling defects of the ECOM1 model leads to this systematic \(\beta\) -dependent pattern in the clock fitting residual RMS. Although the clock residual RMS of BeiDou-3 IGSOs can be effectively reduced by using the a priori box-wing model, they are still large in the eclipse seasons. Table 5 gives the mean of daily clock residual RMS of BeiDou-3 IGSOs for the ECOM1 + BW solution inside and outside eclipse seasons. After entering eclipse seasons, the clock residual RMS increases by an average of 0.6 cm (20.9%). In particular, the clock estimates of eclipsing \begin{table}
\begin{tabular}{c c c c} \hline
**Satellite** & **ECOM1** & **ECOM2** & **ECOM1 + BW** \\ \hline C38 & 7.8 & 6.1 & 4.1 \\ C39 & 5.7 & 4.1 & 2.8 \\ C40 & 6.9 & 4.8 & 3.3 \\ \hline \end{tabular}
\end{table} 
Table 4: Mean of daily fitting residual RMS of BeiDou-3 IGSO clocks for the solutions with the ECOM1, ECOM2 and ECOM1 + BW models in cm.