# 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 5: Satellite C39 clock fitting residuals for the ECOM1 (green), ECOM2 (blue) and ECOM1 + BW (red) solutions on DOY 172–173, 2020 (**top**), \(\beta\approx 78.4^{\circ}\)) and DOY 254–255, 2020 (**bottom**), \(\beta\approx 9.5^{\circ}\)). satellite C39 deteriorated most severely, with an increase of 0.7 cm (26.9%) in the fitting residual RMS. In order to illustrate the clock behaviors of eclipsing BeiDou-3 IGSOs in detail, Figure 6 gives the time series of clock fitting residual RMS for satellite C39 based on the ECOM1 + BW model during one eclipse season (days 58-90 of the year 2021). With the decrease of \(|\beta|\), the clock residual RMS is significantly elevated. This clearly indicates the presence of orbit modeling deficiencies during eclipse seasons. As mentioned earlier, there are generally two reasons for the orbit solution degradation of eclipsing satellites. One is the inaccurate attitude modeling [20], and the other is the unexplained non-conservative forces, such as thermal effects from spacecraft's body [21; 22; 23]. Based on reverse kinematic PPP (RTPPP), which is proposed by [PERSON] et al. [36] to monitor and model the yaw behaviors of eclipsing satellites, we perform the first analysis for the yaw-attitude maneuvers of eclipsing BeiDou-3 IGSOs. RKPPP estimates, nominal values [27] and predicted values using the BeiDou continuous yaw-attitude (CYS) model [13; 33] for the yaw angle of satellite C39 in the vicinity of the midnight and noon points are illustrated in Figures 7 and 8. The BeiDou CYS model, developed by [13; 33] for the new BeiDou-2 I06 and later proved to be also applicable to the BeiDou-3 CAST (China Academy of Space Technology, Beijing, China) MEOs, is expressed as follows:  
\[\Psi(U_{s})\ =\ a\tan 2(-\tan\beta,\sin U_{s}) \tag{12}\]  
\[\Psi(U)=\ 90^{\circ}\cdot SIGN(1,\Psi(U_{s}))+[\Psi(U_{s})-90^{\circ}\cdot SIGN (1,\Psi(U_{s}))]\ \cdot\ \cos(\frac{2\pi}{t_{\max}}\cdot\frac{U-U_{s}}{\dot{U}}) \tag{13}\]  
where \(U_{s}\) is the orbit angle at the start of the yaw maneuver. The best value of \(U_{s}\) is \(-6^{\circ}\) or \(174^{\circ}\). The term \(t_{\max}\) is a constant, which represents the duration of the yaw maneuver. Figure 6: Daily RMS of clock fitting residuals for satellite C39 based on the ECOM1 + BW (blue) and ECOM1D + BW (red) models over DOY 58–90, 2021. The area shaded in gray indicates eclipse seasons. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**SRP Model**} & \multicolumn{2}{c}{**C38**} & \multicolumn{2}{c}{**C39**} & \multicolumn{2}{c}{**C40**} \\ \cline{2-7}  & **NE** & **E** & **NE** & **E** & **NE** & **E** \\ \hline ECOM1 + BW & 3.9 & 4.2 & 2.6 & 3.3 & 3.1 & 3.8 \\ ECOM1D + BW & - & 4.0 & - & 2.3 & - & 3.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Mean of daily fitting residual RMS of BeiDou-3 IGSO clocks for the solutions with the ECOM1 + BW and ECOM1D + BW models insides and outsides eclipse seasons in cm. “NE” and “E” denotes the non-eclipse and eclipse seasons, respectively. The values are 3090 s and 5740 s for MEO and IGSO satellites, respectively. \(\Psi(U)\) is the modelled yaw angle at the orbit angle \(U\). From Figures 7 and 8, the yaw maneuvers of BeiDou-3 IGSOs occur when the \(\beta\) -angle is in the range of \([-3^{\circ},3^{\circ}]\) and the \(U\) angle is in the range of approximately \([-6^{\circ},6^{\circ}]\) or \([174^{\circ},186^{\circ}]\). The yaw behaviors of eclipsing BeiDou-3 IGSOs can be well reproduced by the BeiDou continuous yaw-attitude model used in this study [13; 33]. This rules out the first assumption that attitude errors may be the reason for the orbit deterioration and suggests the presence of unmodeled non-conservative orbit perturbations during eclipse seasons. Figure 8: Nominal (green), estimated (blue) and modeled (red) yaw angles of IGSO C39 passing the Earth’s shadow (gray area) under different \(\beta\) -angles. Figure 7: Nominal (green), estimated (blue) and modeled (red) yaw angles of IGSO C39 crossing the noon point at different \(\beta\) -angles.