As mentioned earlier, due to BeiDou-3 IGSO satellites with a cuboid shape, the orbits determined with the ECOM1 model may have systematic radial orbit errors. Therefore, the performance of ECOM1, ECOM2 and ECOM1 + BW models for BeiDou-3 IGSO orbit determination is analyzed. Although all BeiDou-3 IGSOs are equipped with laser retroreflectors for Satellite Laser Ranging that allows for external accuracy assessment of orbit radial component, none of BeiDou-3 IGSOs have been tracked by the International Laser Ranging Service so far. Other than SLR, the clock estimates of high-stable satellite clocks can also be used as a quality indicator for radial orbit modeling issues because radial orbit errors are mapped to the estimated satellite clocks [
9,
12,
15,
23]. Due to the PHM clocks equipped on BeiDou-3 IGSOs, the residuals of the daily estimated clocks after a second-order polynomial fitting are used as a measure of radial orbit accuracy. Of course, the orbit boundary discontinuity (DBD) [
35] is also a very important index for assessing orbit quality and will be discussed in detail.
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
-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
is greater than
. Once the ECOM2 model is used, this
-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.
A significant part of the systematic
-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
and DOY 254–255, 2020
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
-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
-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
-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 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
, 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 Dilssner 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
Figure 7 and
Figure 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:
where
is the orbit angle at the start of the yaw maneuver. The best value of
is
or
. The term
is a constant, which represents the duration of the yaw maneuver. The values are 3090 s and 5740 s for MEO and IGSO satellites, respectively.
is the modelled yaw angle at the orbit angle
.
From
Figure 7 and
Figure 8, the yaw maneuvers of BeiDou-3 IGSOs occur when the
-angle is in the range of
and the
angle is in the range of approximately
or
. 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.
The signal presented in the BeiDou-3 IGSO clock fitting residuals (
Figure 6) can be also observed in the Galileo FOC (Full Operational Capability) satellites during eclipse seasons [
37]. Previous studies confirmed that the deterioration of Galileo FOC clocks estimates was attributed to the unaccounted thermal radiations from the spacecraft body, and this modeling deficiency can be compensated by the estimation of the
parameter during eclipse seasons and also keeping
active during Earth’s shadow transitions [
22,
23,
37].
We found that adding the
term in the ECOM1 + BW model and keeping it active in Earth’s shadows., i.e., the ECOM1D + BW model, is very effective for eclipsing BeiDou-3 IGSOs. The mean of daily clock residual RMS of eclipsing BeiDou-3 IGSOs for the ECOM1D + BW solution is given in
Table 5. Compared with the ECOM1 + BW solution, the clock residual RMS of eclipsing BeiDou-3 IGSOs decreases by 0.5 cm (13.3%) on average. The elevated clock residual RMS is well reduced, and the RMS statistics inside and outside the eclipse season are on the same level (
Figure 6). To further highlight that the clock estimates benefit from the consideration of the
term, the clock residuals of satellite C39 on DOY 74–75, 2021
for the ECOM1 + BW and ECOM1D + BW solutions are shown in
Figure 9. It can be seen that the stability of clock estimates based on the ECOM1 + BW model does not conform to the characteristics of PHM clocks on BeiDou-3 IGSOs. When switching the ECOM1 + BW model to the ECOM1D + BW model, the clock residuals become very stable. The above results clearly indicate that the orbit modeling defects of eclipsing BeiDou-3 IGSOs can be covered by the introduction of the
term also active in Earth’s shadows.
3.2.2. Orbit Assessment Based on Orbit Boundary Discontinuities
The orbit DBD in the radial, along- and cross-track, and as well as three-dimension (3D) directions are defined as [
35]:
where
and
refer to days.
and
are the geocentric satellite positions of the last epoch of day
and the first epoch of day
, respectively.
is the rotation matrix converted from a geocentric terrestrial reference frame to an orbital frame. The terms
,
,
and
are orbit DBDs in the radial, along- and cross-track as well as three-dimension (3D) directions, respectively.
The mean values of 3D orbit DBDs of satellites C38, C39 and C40 for the solutions with ECOM1, ECOM2 and ECOM1 + BW models inside and outside eclipse seasons are listed in
Table 6. For the non-eclipse periods, the ECOM1 + BW solution provides the best orbit quality in the three-dimension direction, followed by the ECOM2 and ECOM1 solutions.
Figure 10 shows the DBD
statistics of BeiDou-3 IGSOs in the radial, along- and cross-track directions outside eclipse seasons. The ECOM1 + BW model shows the best performance, except that the satellite C39 cross-track orbit consistency for the ECOM1 + BW model is slightly worse than that of the ECOM2 model. The radial and 3D orbit consistency of BeiDou-3 IGSOs for the ECOM1 + BW solution improves by about 2.2 (32.7%) and 5.1 cm (29.5%), 1.2 (21.1%) and 2.6 cm (17.6%) relative to the solutions with the ECOM1 and ECOM2 models on average, respectively.
To investigate whether orbit DBDs show
-dependent variations like daily clock residuals (
Figure 4), the orbit DBD time series of BDS-3 IGSO C39 based on different SRP models in along-track, cross-track, and radial directions are given in
Figure 11. It can be clearly seen that there are no
-dependent systematic variations in orbit DBDs. This should be due to the fact that the
-dependent systematic orbit errors are highly correlated from one day to the next and cannot be reflected in the overlap statistics.
From
Table 6 and
Figure 11, the BeiDou-3 IGSO orbit consistency shows a significantly lower performance during eclipse seasons. The orbit 3D DBDs during eclipse seasons are 4–5 times larger than those in the non-eclipse seasons, except that C40 orbit 3D DBDs inside its first (1–23 January 2020) and second (28 June–22 July 2020) eclipse seasons during our experiment only increase by 15–20%. Similar to the clock estimates presented in the previous section, the degraded orbits during eclipse seasons can be improved by using the ECOM1D + BW model.
Figure 12 plots the BeiDou-3 IGSO orbit 3D DBDs for the ECOM1 + BW and ECOM1D + BW solutions as a function of
-angle. It can be clearly seen that the orbit 3D DBDs of eclipsing C38 and C39, and C40 inside its third eclipse season (27 December 2020–17 January 2021) increase significantly with the decrease of
. Once the ECOM1D + BW model is employed, the elevated orbit errors are basically removed. The mean 3D orbit DBDs of eclipsing C38, C39 and C40 for the ECOM1D + BW solution are given in
Table 6. Compared with the ECOM1 + BW solution, the 3D orbit DBDs of satellite C40 inside its third eclipse season, and eclipsing C38 and C39 decreased by 54.7 cm (79.8%) on average, while those of C40 inside its first and second eclipse seasons are reduced by 3.6 cm (21.6%). The orbit consistency inside eclipse seasons can be comparable to that outside eclipse seasons.