Data and POD estimation strategy
This paper used the Multi-GNSS EXperiment (MGEX) network data (Montenbruck et al. 2017) in 2019 for the POD of BeiDou satellites. During this period, 50–60 stations received BDS data. The orbit determination adopted non-difference Pseudo-range Combination (PC) and Carrier-phase Combination (LC) observations with a 1-day arc length. All data are processed using the GNSS analysis software package developed by Shanghai Astronomical Observatory (SHAO). This software uses a least-squares estimator to process GNSS data to generate orbits, clocks, Earth Orientation Parameter (EOP), Solution Independent EXchange Format (SINEX) and other products. It also could implement the SLR orbit validation. The software applies the Box-Wing Solar Radiation Pressure Model (SRPM) and Box-Wing Earth Radiation Pressure Model (ERPM).
The SRPM uses the Box-wing model to calculate the initial value and adds five parameters of the ECOM model (D0, Y0, B0, BC, and BS) for the BDS satellite (Arnold et al. 2015; Duan and Hugentobler 2021). The ERPM is a numerical model with the measured Earth radiation (albedo and infrared) as the input. The radiation data is calculated according to the formula to obtain the Earth grid’s reflection coefficient and thermal infrared radiation coefficient. The radiation data comes from the Clouds and the Earth’s Radiant Energy System (CERES) data of NASA. The thermal radiation is considered according to the thermal conditions of each satellite’s surface and substituted into the actual thermal environment; the finite element analysis and the average temperature analysis of the surface heat balance method are carried out to calculate the perturbation force of the thermal radiation force. The antenna thrust is also considered. BDS satellite L-band transmit power is taken from the IGS metadata SINEX file (IGSMAIL-8015, from Peter Steigenberger) (Duan et al. 2022).
The other models follow the IERS conventions 2010 or recommended by IGS. The observation types, error correction models, and estimated parameters are listed in Table 2. To facilitate the orbit check, we refer to the satellites with laser observations published by the International Laser Ranging Service (ILRS) and select satellites C01, C08, C10, C11, C13, C19, C20, C21, C29, and C30 to calculate the occurrence of eclipse throughout the year. Figure 5 illustrates the eclipse statistics for these satellites in 2019. The horizontal axis represents the Day of Year (DOY), and the vertical axis represents the eclipse occurrence.
Eclipse occurrence of BeiDou satellites in 2019
According to the statistics, the Earth’s shadowing days for the satellites C01, C08, C10, C11, C13, C20, C21, C29, and C30 were 101 days, 47 days, 58 days, 89 days, 48 days, 56 days, 56 days, 57 days, 93 days and 93 days, respectively. The SRP model was SRPM which was a priori box-wing model. The same attitude equation was used during the eclipse in which C13 and C16 kept yaw-steering (YS), other satellites of BDS-2 IGSO/MEO used yaw-steering orbit-normal (YS-ON), and the switches between YS and ON take place when the sun elevation angle is approximately ± 4 deg (Dai et al. 2015). BDS-3 Shanghai Engineering Center for Microsatellites (SECM) satellites use YS-ON, and BDS-3 China Academy of Space Technology (CAST) satellites keep YS (Li et al. 2020). The yaw control model of CAST satellites was established by Dilssner (2017). The formula for the yaw angle was described as below:
$${psi }_{mathrm{cast}}=mathrm{ATAN}2(-mathrm{tan}{beta }_{d},mathrm{sineta })$$
(3.1)
where ({beta }_{d}) is modified Sun elevation angle:
$$left{begin{array}{l}{beta }_{d}=beta +ftimes left(mathrm{SIGN}left({beta }_{0},beta right)-beta right)\ f=left{begin{array}{c}frac{1}{1+{d}_{m}times {mathrm{sin}}^{4}eta },{beta }_{0}le beta \ 0,{beta }_{0}>beta end{array}right.end{array}right.$$
(3.2)
This modified Sun elevation angle ensures a minimum angular distance of ({beta }_{0}) between the Sun’s vector and the spacecraft’s z-axis. (beta ) is the Sun elevation angle, (upeta ) is the geocentric orbit angle between the satellite and orbit midnight. (mathrm{SIGN} left({beta }_{0},beta right)) is a FORTRAN function returning the value of ({beta }_{0}) with the sign of (beta ). ({d}_{m}) is a dimensionless constant, and equals to 80,000. (f) is a bell-shaped smoothing function of the orbit angle (upeta ).
The SECM attitude control model is given by Xia et al. (2018). When the sun elevation (beta ) is within ± 3 deg, the sun vector component ({S}_{oy}) can be determined as below:
$${S}_{oy}=left{begin{array}{c}-mathrm{sin}left(3 mathrm{deg}right),beta >0\ mathrm{sin}left(3 mathrm{deg}right),beta <0end{array}right.$$
(3.3)
The yaw-angle ({psi }_{mathrm{secm}}) is expressed as:
$${psi }_{mathrm{secm}}=mathrm{ATAN}2({S}_{oy},mathrm{sineta cos}beta )$$
(3.4)
Only the shadow model has been changed for comparison without GNSS data, models, and strategy changes. Moreover, the errors in the force model may be absorbed by other estimated parameters. Thus, we avoid using any estimated empirical parameters.
Primarily, multistep integration methods such as the Adams–Cowell (AC) integrator with fixed step size are widely used in GNSS POD (Bhattarai et al. 2019; Huang and Zhou, 1992). However, the traditional AC integrator used in eclipse duration analysis is inaccurate. Although the deficiencies can be partly mitigated by shortening the step size of the traditional AC integrator, the computation cost and round-off errors are still significantly increased with a small step size (Montenbruck et al. 2017). Therefore, the adjustable-step integration method combining multistep and single-step around the eclipse will solve the problems. The output epochs of the modified integrator are kept the same as the traditional AC integrator, but a single-step integrator is activated when the current output epoch is going to enter, pass and leave an eclipse interval (Ju et al. 2017). This paper also adopted this strategy.
Comparison of shadow factor and SRP acceleration
The shadow factor and SRP acceleration derived from four shadow models were compared. Figure 6 shows the shadow factors of different shadow models for the different types of satellites. To make the entire eclipse process appear on a figure, the x-axis scale of the umbra phase is adjusted, and its unit is Second of Day (SOD). It can be seen from the figure that the satellite’s response to radiation from full phase to umbra and from umbra to full phase is a similar and symmetrical changing process. Due to the consideration of the influence of the Earth’s oblateness and atmospheric effect, compared with the conical shadow model, the satellites enter the shadow earlier and come out of the shadow later when using the latter three models. Table 3 lists the moment that the satellite enters and exits the different phases. ({T}_{mathrm{enter}}) is the moment when the satellite enters the penumbra from full phase to umbra; ({T}_{mathrm{umbra}}) is the time when the satellite enters the umbra; ({T}_{mathrm{penumbra}}) is the time when the satellite enters penumbra from umbra to full phase; ({T}_{mathrm{out}}) is the time when the satellite is back to full phase. Taking C10 as an example, the moments entering shadow estimated from the models 3dishes, PPM, PPMatm, and SOLAARS-CF are 59465 s, 59459 s, 59442 s, and 59421 s, respectively. The latter three models show the satellite enters the shadow 6 s, 23 s, and 56 s earlier than the 3dishes model, respectively. The period of four models in the umbra phase is 3866 s, 3909 s, 3909 s, and 3864 s, respectively. Afterward, the satellite begins to enter the penumbra from the umbra, and the moments of entering the full phase estimated from four models are 63629 s, 63632 s, 63649 s, and 63656 s, respectively. The SOLAARS-CF model enters the eclipse first and leaves the eclipse last. This may be because it is a fitting model and more sensitive to changes of shadow factors.
a Shadow factors for satellite C10 (IGSO) in 2019/007. b Shadow factors for satellite C14 (MEO) in 2019/060. c Shadow factors for satellite C01 (GEO) in 2019/069
Figure 7 shows the SRP acceleration variation status. It can be seen PPMatm and SOLAARS-CF models show the highest degree of conformity, and they are distinguishably different from the conical model. The different types of satellites have a similar trend. Figure 7a shows the acceleration changes of the C10 on behalf of IGSO during the eclipse in the direction of a-direction (along trace), c-direction (orbital surface normal), and r-direction (radial trace), and the scale of the x-axis has also been adjusted in eclipse. The SRP acceleration gradually decreases during the penumbra phase and becomes nearly zero in the umbra phase. Figure 7b, c shows the acceleration changes of the C14 on behalf of MEO and the C01 on behalf of GEO during the eclipse. The right side of the y-axis indicates the SRP acceleration difference: dif1 is the difference between PPM and 3dishes, dif2 is the difference between PPMatm and 3dishes, and dif3 is the difference between SOLAARS-CF and 3dishes. They show similar characteristics. The acceleration difference between 3dishes and other models is about 10−9 m/s2 in the a-direction and c-direction, and it can reach to 10−8 m/s2 in the r-direction. Therefore, the Earth’s oblateness and atmospheric effect must be considered in shadow factor models, especially under present circumstances that perturbation acceleration accuracy stepped toward 10−10 m/s2 (Montenbruck 2020).
a SRP acceleration variation of C10 (IGSO) in eclipse in 2019/007. b SRP acceleration variation of C14 (MEO) in eclipse in 2019/060. c SRP acceleration variation of C01 (GEO) in eclipse in 2019/069. Each figure is composed of SRP acceleration (left) and their differences (right) between other three models with regard to 3dishes. In each side of figure, including along trace direction (top), cross trace direction (middle) and radial trace direction (bottom)
SLR orbit validation
To evaluate the impact of the four shadow models on the POD of BeiDou satellites during the eclipse more reliably, this paper uses the two-way full rate data of SLR (provided by ILRS) to perform an external check. Before SLR checks (Pearlman et al. 2002), the systematic errors of the original SLR data should be corrected, including the distance errors caused by the tide correction, the atmospheric refraction, the relativistic effect, the distance deviation of the laser reflection points on the surface of the satellite from the Center of Mass (CoM) (which is given in Table 4) and the station systematic deviation. The observation correction follows the models recommended by the IERS convention 2010 (Petit and Luzum 2010), and more information about BDS in International Laser Ranging Service (ILRS) can be obtained at https://ilrs.cddis.eosdis.nasa.gov/missions/satellitemissions/currentmissions/bdm2com.html. Only when the original observation data deducts these errors, the SLR data can be efficiently used for calculating the residual value. The processed SLR data accuracy is around 1 cm (Combrinck 2010), which is higher than the orbit determination accuracy of the BDS in the radial (R) direction (3–5 cm). This paper only compared the SLR validation results in eclipse, because the evaluation of the entire orbital arc will obscure the orbital accuracy of the eclipse period and the duration of the eclipse is small relative to the entire orbital arc. Since there is no corresponding laser observation data for C01 and C19 during the eclipse, this paper has calculated the SLR check results of the remaining 8 satellites. All the results of selected satellites are depicted in Table 5. It can be seen the PPM, PPMatm and SOLAARS-CF have smaller residual values than that of 3dishes. The mean value can be decreased by up to 10 mm (C10), and the rest are generally decreased by 2–6 mm; the RMS value can be decreased by up to 10 mm (C10), and the rest are generally 2–5 mm. Comparing the SLR check results of the PPMatm and SOLAARS-CF models, the accuracy of the two models is comparable and better than that of the other two models. The performance of the validation result of C08 and C13 is not well, and this may be due to inadequate data and the low data quality. The SLR calculation results of all satellites are showed in Additional file 1.
ISL check
The ISL measurements of BeiDou Satellites from DOY 032 to DOY 120 in 2019 are chosen to check the precision of the BeiDou satellite orbit during the eclipse in this period. C19, C21, C27, and C30 are selected as the objectives in this paper. Table 6 gives the satellite information of ISL. To better evaluate the shadow models, we keep only transmitting satellites are eclipsing. Table 7 lists the ISL check results. The orbit accuracy of PPM, PPMatm, and SOLAARS-CF is higher than 3dishes, and PPMatm and SOLAARS-CF are better than other two models. The mean value is decreased by up to 8 mm for C27, and generally decreased by 2–6 mm for the others. The RMS value is decreased by up to 7 mm for C27, and generally decreased by 2–5 mm. The ISL calculation results of all links are showed in Additional file 1.
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