# A Method to Determine BeiDou GEO/IGSO Orbital Maneuver Time Periods
## 2 Theory of Orbital Maneuver Period Detection
### Detection Factor of Orbital Maneuver Time Period  
In order to find a method to determine the period of orbit maneuvering and combine with the above analysis results, a schematic scheme of orbital maneuvering was used (Figure 2). In Figure 2, the solid blue line denotes the orbit of the satellite before orbit maneuvering; the solid yellow line denotes the orbit of the satellite after orbit maneuvering; the dotted blue line denotes the forward predicted orbit after orbit maneuvering; the dotted yellow line denotes the backward predicted orbit before orbit maneuvering; and the solid red line denotes the orbit during orbit maneuvering. The dotted green line denotes the measured distance between station and satellite from the receiver; the dotted red line is the calculated distance using the station coordinates and the predicted orbits by precise products on the day after orbital maneuvers; and the dotted black line is calculated distance using the station coordinates and predicted orbits by precise products on the day before orbital maneuvers. The intersection point of the solid blue and red lines is the start time and the intersection point of the solid red and yellow lines is the end time for orbit maneuvering:  
\[\widehat{D}_{obs}=D_{obs}+c\delta t^{j}-c\delta t_{i}+\delta I+\delta T \tag{3}\]  
where \(D_{obs}\) is the pseudo-range observation; \(c\) is the velocity of light; \(\delta t^{j}\) is the clock offset of the satellite; \(\delta t_{i}\) is the clock offset of the receiver; \(\delta I\) is the ionospheric delay correction; \(\delta T\) is the tropospheric delay correction; and \(\widehat{D}_{obs}\) is the value of pseudo-range observation reduced the clock offset of the satellite and receiver, the ionospheric delay correction and the tropospheric delay correction. It needs to be emphasized that in order to correctly calculate the clock offset of the receiver, it needs to be estimated through the least squares and robust methods [17; 18]:  
\[S_{for}=\sqrt{\left(X_{for}^{j}-X_{i}\right)^{2}+\left(Y_{for}^{j}-Y_{i} \right)^{2}+\left(Z_{for}^{j}-Z_{i}\right)^{2}} \tag{4}\]  
Figure 2: Schematic scheme of orbital maneuver detection. where \(X_{i},Y_{i},Z_{i}\) are the station coordinates; \(X^{j}_{for^{\prime}},Y^{j}_{for^{\prime}},Z^{j}_{for}\) are the spatial coordinates of satellite \(j\), which is calculated by the forward predicted orbit by the precise products on the day after orbital maneuvers; and \(S_{for}\) is the distance calculated between the satellite and the station. \[S_{back}=\sqrt{\left(X^{j}_{back}-X_{i}\right)^{2}+\left(Y^{j}_{back}-Y_{i} \right)^{2}+\left(Z^{j}_{back}-Z_{i}\right)^{2}} \tag{5}\]  
where \(X^{j}_{back},Y^{j}_{back},Z^{j}_{back}\) are the coordinates of the satellite, calculated by the backward predicted orbit by the precise products on the day before orbital maneuvers; and \(S_{back}\) is the distance calculated between the satellite and the station. Whether or not orbital maneuvers occur, \(\widehat{D}_{obs}\) is closer to the real distance between the satellite and the station, which cannot be influenced by orbit maneuvering. However, from Figure 2 we know that \(S_{back}\) is inaccurate, with increasing errors before the start time of orbital maneuvers, because of the incorrect backward predicted orbit (dotted yellow line in Figure 2). Thus, the start time of orbit maneuvering can be detected by using the difference between \(S_{back}\) and \(\widehat{D}_{obs}\). Similarly, the end time can be determined by using the difference between \(S_{for}\) and \(\widehat{D}_{obs}\):  
\[V_{back}=\left|S_{back}-\widehat{D}_{obs}\right| \tag{6}\]  
where \(V_{back}\) is the absolute value of the difference between \(S_{back}\) and \(\widehat{D}_{obs}\):  
\[V_{for}=\left|S_{for}-\widehat{D}_{obs}\right| \tag{7}\]  
where \(V_{for}\) is the absolute value of the difference between \(S_{for}\) and \(\widehat{D}_{obs}\).