# A Method to Determine BeiDou GEO/IGSO Orbital Maneuver Time Periods
## 2 Theory of Orbital Maneuver Period Detection
### Analysis of Satellite Orbit Maneuvers  
During normal operation of the satellite, in order to correct the perturbation from the space environment, orbit maneuvering is necessary to adjust the position of the satellite. When the satellite is maneuvering, the working time of the propulsion system is much less than the operating period of the satellite. Therefore, satellite orbits are adjusted by a pulsing method. The pulsing thrust makesthe satellite velocity change suddenly, but the variation of radial distance is small [19]. The common user cannot obtain information of the maneuvering thrust because of secrecy. However, changes in the radial direction can easily be reflected by pseudo-observations of stations. The effects of orbital maneuvers in the radial direction are discussed below.  
In Figure 1, the blue track is the orbit of the satellite before maneuvering and the yellow track is the orbit after maneuvering, and O is the center of the Earth. At time \(t_{0}\), r denote the position vector of the satellite, and \(v_{0}\) is the velocity of the satellite. At that time, \(\Delta v\) denote the orbit maneuvering increasing the instantaneous velocity of the satellite; \(v_{1}\) is the velocity of the satellite after the orbit thrust action; and \(\beta\) is the complementary angle of the angle between r and \(v_{1}\):  
\[\dot{r_{1}}=v_{1}sin\beta=(v_{0}+\Delta v)sin\beta=v_{0}sin\beta+\Delta vsin\beta \tag{1}\]  
\[\Delta\mathrm{d}=\Delta vsin\beta\Delta\mathrm{t} \tag{2}\]  
where \(v_{0}\) is the velocity of the satellite at time \(t_{0}\); \(\Delta v\) is the orbit maneuvering increasing the instantaneous velocity of the satellite; \(v_{1}\) is the velocity of the satellite after the orbit thrust action; \(\beta\) is the complementary angle of the angle between r and \(v_{1}\); and \(\dot{r_{1}}\) is the velocity vector of the satellite in the radial direction after orbit maneuvering, that is accumulating during a time period of \(\Delta\mathrm{t}\) and \(\Delta\mathrm{d}\) denotes the variation of distance due to the orbital maneuvers. The results of analyzing the effect of increased velocity of orbit maneuvering on radial distance are as follows.  
The satellite orbital maneuvers are divided into two situations: one is the orbital adjustment to make the satellite go into the designed orbit, and the other is the orbit maneuvering, which ensures that the satellite runs in the normal orbit. The maneuvering thrust of the latter is less than that of the former, and the increased velocity of orbit maneuvering and the value of \(\beta\) are small. In addition, the efficiency of the orbit maneuvering thrust in the tangential direction is twice the radial direction. Efficiency is estimated using the amount of kinetic energy consumed by the satellite propulsion systems, which depends on the eccentricity of different satellites constellation. The efficiency of the radial orbit maneuvering thrust to correct the mean longitude of the satellite is less, and the increased velocity is 2.66 m per second to adjust 0.1\({}^{\circ}\). It only needs to be 0.28 m per second to achieve it in the tangential direction, the time it takes to obtain the longitude change depends on the orbital semimajor axis (the derivation is not described in this paper, and is proved in [19]). In order to extend the service time of satellites, save fuel used by the propulsion systems, and raise efficiency for orbit maneuvering, the orbit maneuver's increased velocity in the tangential direction is greater than the radial direction, which reduces the values of \(\Delta v\) and \(\beta\). Therefore, the variation of distance in the radial direction and  
Figure 1: Schematic diagram of satellite orbit maneuvering.  
the variation of pseudo-range observations are small due to the orbital maneuvers. It is difficult to determine the start time of orbital maneuvers by using the satellite orbit before maneuvering and, similarly, to detect the end time after maneuvering.