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Comparison of shadow models and their impact on precise orbit determination of BeiDou satellites during eclipsing phases

Abstract

Solar radiation pressure (SRP) is an extremely critical perturbative force that affects the GNSS satellites’ precise orbit determination (POD). Its imperfect modelling is one of the main error sources of POD, whose magnitude is even to10−9 m/s2. The shadow factor (i.e., eclipse factor) is one crucial parameter of SRP, generally estimated by the cylindrical model, the conical model, or shadow models considering the Earth’s oblateness and the atmospheric effect, such as the Perspective Projection Method atmosphere (PPMatm) model and Solar radiation pressure with Oblateness and Lower Atmospheric Absorption, Refraction, and Scattering Curve Fit (SOLAARS-CF) model. This paper applies the former four shadow models to determine the corresponding precise orbit using BeiDou satellites’ ground-based observation, and then compared and assessed the orbit accuracy through Satellite Laser Ranging (SLR) validation and Inter-Satellite Link (ISL) check. The results show that the PPMatm model’s accuracy is equivalent to the SOLAARS-CF model. Compared with the conical shadow model, SLR validations show the orbit accuracy from the PPMatm and SOLAARS-CF model can be generally improved by 2–10 mm; ISL range check shows that the Root Mean Square (RMS) can be decreased by 2–7 mm. These results show that the shadow model in GNSS POD should fully consider the Earth’s oblateness and the atmospheric effect, especially for the perturbative acceleration higher than 10–10 m/s2.

Graphical Abstract

Introduction

Solar radiation pressure is a primary error source in orbit determination of navigation satellites, and it varies with the irradiated area, mass, satellite attitude, the surface optical property of satellites, solar irradiance, etc. (Musen 1960; Rodriguez-Solano et al. 2012; Zhang et al. 2019; Villiger and Dach 2021). The shadow factor (i.e., eclipse factor) is one important parameter related to the irradiated area and is an essential factor of SRP. An eclipse occurs when the Earth or the Moon blocks the sunlight to the satellites. A satellite can confront three phases depending on its relative positions to the Earth/Moon and the Sun. If the satellite is fully exposed to sunlight without any block area, it is in the full phase with the shadow factor 1. When the satellite moves partly behind the Earth, it enters the penumbra phase with the shadow factor between 0 and 1. Alternatively, the shadow factor is set zero when the satellite is completely occluded by the Earth or Moon and cannot receive any sunlight which is termed the umbra phase.

The shadow factor is calculated based on the type of the Earth’s shadow towards the Sun. Kozai (1963) first discovered and discussed the shadow function by proposing a discontinuous step function to depict the penumbra transition and evaluate the influence on the satellite orbit determination. Ferraz-Mello (1972) put forward the cylinder shadow model, a discontinuous shadow function with only two phases: full and umbra phases. The cylinder model ignored the penumbra, resulting in a significant deviation of orbit determination. Hubaux et al. (2012) proposed a cone shadow model that defined penumbra regions and used a hyperbolic tangent sigmoidal function with scaled parameters to ensure the width of the 0–1 S-shaped curve equals to the time that satellites spent across the penumbra. Adhya et al. (2004) proposed a relatively simple method to add the Earth’s oblateness modelling. They estimated the phases by the number of line intersection points between the satellite to the Sun-edge points and the Earth. When there was no intersection point, the satellite was in the full phase; when there was an intersection point, the satellite was in the penumbra; when there were two intersection points, the satellite was in the umbra. Vokrouhlicky et al. (1993) redefined the position and radius of the Earth by the positional relationship between Sunlight, Earth, and satellite, thereby determined the osculating spherical surface of the Earth, which matches the radius of curvature of the circular Earth at the grazing point. Robertson and Shoemaker (2014) corrected the derivation of the formula of Vokrouhlicky et al. (1996) and compensated the Earth’s oblateness based on Adhya et al. (2004) and Vokrouhlicky et al. (1994). Robertson and Shoemaker (2014) presented a shadow model (SOLAARS-CF) accounting for Earth’s oblateness and atmospheric effect by curve fitting based on a physics-based model (SOLAARS). The SOLAARS model fully integrated the atmospheric extinction modelling (including Rayleigh scattering, aerosol extinction, molecular absorption, and cloud extinction) and the Earth’s oblateness, but it was complex and computational. The SOLAARS-CF model significantly reduced the data and computation cost. Inspired by the perspective projection algorithm proposed by Oswald et al. (1982), Li et al. (2019) built PPM and PPMatm shadow models. The difference of PPM and PPMatm is that the latter adds modelling of atmospheric effects.

As discussed above, the cylinder and conical shadow models supposed the Earth is spheric and meanwhile neglected the atmospheric effects. Due to the omission of the penumbra, the cylinder shadow model is seldom applied and the cone shadow model has got widely used relatively in POD. However, Global Geodetic Observing System (GGOS) requires the future Terrestrial Reference Frame (TRF) with a space benchmark of 1 mm and stability of 0.1 mm/year. It means the orbit accuracy needs to be improved to the millimetre level. Therefore, this paper would like to consider the Earth’s oblateness and atmospheric effect to refine the solar radiation pressure model using four shadow models (conical, PPM, PPMatm, and SOLAARS-CF) and further study their impact on the SRP and the POD of BeiDou satellites. Finally, their impact and performance are validated not only by SLR orbit validation but also by a new method of ISL for orbit check.

Methodology

The details of four shadow models are described in “Shadow models” section, including the conical shadow model, PPM, PPMatm, and SOLAARS-CF model. The conical shadow model does not contain oblateness and atmospheric effect. PPM lacks atmospheric effect compared with the latter two models. The four shadow models are, respectively, applied in the POD of BDS. Their POD results are compared and validated by SLR orbit validation and the ISL check. The validation method is introduced in section “Validation method”.

Shadow models

Conical shadow model

Compared to the cylindrical shadow model, the conical shadow model has a more precise physical meaning and considers the penumbra phase. Apart from occultations of the Sun by the Earth, the Moon’s shadow is also considered. The conical model is generic and may well adapted to compute the eclipse conditions, generally applied to the Earth or the Moon as occulting bodies. In this paper, this conical model is named “3dishes”. When the occultation has no common part between the Earth and the Moon, the shadow factor can be dealt with the Sun occulted by a spherical body (Montenbruck and Gill 2002). Figure 1 shows the occultation of the Sun by the Earth and the Moon. The occulted area \({A}_{0}\) and the remaining fraction of sunlight \({\widetilde{\gamma }}_{0}\) can be expressed as

Fig. 1
figure 1

Occultation of the Sun by the Earth and the Moon (S: the Sun; E: the Earth; M: the Moon). \({\alpha }_{s}\), \({\alpha }_{e}\) and \({\alpha }_{m}\) stand for the radius of the Sun, the Earth, and the Moon

$${A}_{0}={A}_{F{F}^{^{\prime}}}+{A}_{H{H}^{^{\prime}}}-{A}_{{F}^{^{\prime}}{G}^{^{\prime}}{H}^{^{\prime}}} {\widetilde{\gamma }}_{0}=1-\frac{{A}_{0}}{\pi {\alpha }_{s}^{2}},$$
(2.1)

PPM and PPMatm shadow model

PPM and PPMatm shadow models are built on the basis of the perspective projection method, which establishes the relationship between the satellite, the Earth, and the Sun in eclipse by a 3D projection. The method makes objects appear effectively to the human eye; thus, perspective projection can calculate the overlapping area between the Sun’s and Earth’s images.

A new frame system named Image Space Frame (ISF) was established to describe the perspective projection method, as shown in Fig. 2. The origin of ISF is located at the position, where the centre of the Sun is projected on the projection plane, the Z-axis points from the Sun to the satellite, the X-axis points from the origin of ISF to the projection point of the Earth's mass centre, the Y-axis lies in the image plane and completes the right-hand coordinate system. X_ecef, Y_ecef, and Z_ecef stand for the axis of the Earth Centred Earth Fixed (ECEF) coordinate system, respectively. The Sun’s projection on the image plane is a circle. In the penumbra, the shadow area of the apparent solar disk can be described by the overlapping area, which could be calculated by the sector area \({A}_{1}\) and \({A}_{2}\), shown in Fig. 3. The shape of the conical curve includes an ellipse and hyperbola determined by the condition if the Sun’s center is inside the Earth’s image or not (Li et al. 2019). The intersection of Earth’s projection and Sun’s projection can be computed by combining their projection equations, and the shadow factor is calculated by the ratio of the remaining bright area to the total bright area. Then, the PPM shadow model with perspective projection is completed.

Fig. 2
figure 2

Relationship between ECEF and Image Space Frame (ISF)

Fig. 3
figure 3

Overlapping area of the Sun occulted by the Earth. a Shows the case where the circle and ellipse intersect, and the center of the solar disk is outside the ellipse; b shows the case where the circle intersects the ellipse, but the center of the circle is inside ellipse; c shows the case where the circle and hyperbola intersect, and the center of the circle is inside of the circle; d shows the case where the circle and the hyperbola intersect, but the center of the circle is inside of the overlapping region. In the figure, P1 and P2 are the intersection points between the circle and the conical curve, Pe and Ps is the projection point of the Earth’s center and the Sun’s center, \({A}_{1}\) and \({A}_{2}\) are the area formed by the intersection points and the conical curve

The PPMatm shadow model considers the change of solar radiation in the atmosphere based on the PPM model. It regarded radiation reduction as a linear function defined as

$$f\left(h\right)=\left({\mu }_{2}-{\mu }_{1}\right)\frac{h}{{h}_{0}}+{\mu }_{1}$$
(2.2)

where h is the distance from any point to the solid Earth in the direction of the center of the solid Earth’s projection to the center of the Sun’s projection; \({\mu }_{2}\), \({\mu }_{1}\) represent the radiation reduction coefficients at the boundary of the atmosphere and solid Earth, respectively (\({\mu }_{2}=1\), \({\mu }_{1}=0\))\(;{h}_{0}\) is the height of the Top of Atmosphere (TOA) assumed as an ellipsoid that wraps the solid Earth.

The shadow factor is computed by relative positions between the projections of the Sun, the Earth, and the TOA, which is calculated by applying the perspective projection algorithm to the solid Earth and the TOA ellipsoid separately in Fig. 4. This figure introduces the situation in which the solid Earth and atmosphere partially occult the Sun. Other cases have been described in detail by Li (Li et al. 2019). The shadow factor \({\widetilde{\gamma }}_{0}\) in Fig. 4b can be derived as follows:

Fig. 4
figure 4

a Different situations for the positional relationship between the Sun, the Earth, and the atmosphere; b situation when the solid Earth and the atmosphere partially occult the solar disk. A1 and A2 are the area of the Sun occulted by the atmosphere and solid Earth. Oe is the center of the Earth’s image, and Os is the center of the Sun’s image. H1 and H2 are the upper and lower boundaries for the overlapping part between the Sun’s image and the atmosphere’s image; H2 and H3 are the upper and lower boundaries for the overlapping part between the Sun’s image and the solid Earth’s image

$${\widetilde{\gamma }}_{0}=1-\frac{0.5(\mathrm{f}\left({H}_{1}\right)+\mathrm{f}({H}_{2})){A}_{1}+{0.5(\mathrm{f}\left({H}_{3}\right)+\mathrm{f}({H}_{2}))A}_{2}}{\pi {\alpha }_{s}^{2}}$$
(2.3)

SOLAARS-CF shadow model

SOLAARS-CF is a curve-fitting shadow model coming from a physical model (SOLAARS) (Robertson et al. 2015). It uses the equatorial radius and polar radius of the Earth to recalculate the spherical radius of the Earth and adjusts the positions of the satellite and the Sun. This method obtains the osculating sphere, which is similar to the Earth’s ellipsoid. Thus, the influence of the Earth’s oblateness can be included in the eclipse. In addition, the SOLAARS model still considers atmospheric scattering and dissipation effects, including Rayleigh scattering, aerosol dissipation, molecular absorption, and cloud dissipation. However, the input parameters of SOLAARS are enormous, and thus the calculation is complicated. Robertson (2015) used the hyperbolic tangent Sigmoid function proposed by Hubaux et al. (2012) to describe the satellite performance during the eclipse and rebuilt named SOLAARS-CF shadow model by curve fitting of SOLAARS’s results. SOLAARS-CF model needs only the position of the Sun and satellite. The shadow factor can be expressed as below:

$$\left\{\begin{array}{c}{\widetilde{\gamma }}_{0}=\frac{1+{a}_{1}+{a}_{2}+{a}_{1}\mathit{tan}h\left({a}_{3}\left({r}_{E}^{^{\prime}}-{a}_{4}\right)\right)+{a}_{2}tanh\left({a}_{5}\left({r}_{E}^{^{\prime}}-{a}_{6}\right)\right)+tanh\left({a}_{7}\left({r}_{E}^{^{\prime}}-{a}_{8}\right)\right)}{2+2{a}_{1}+2{a}_{2}}\\ {r}_{E}^{^{\prime}}=\Vert \left({\overrightarrow{r}}_{1}^{^{\prime}}-\frac{{\overrightarrow{r}}_{2}^{^{\prime}}\bullet {\overrightarrow{r}}_{1}^{^{\prime}}}{\Vert {\overrightarrow{r}}_{2}^{^{\prime}}\Vert }*\frac{{\overrightarrow{r}}_{2}^{^{\prime}}}{\Vert {\overrightarrow{r}}_{2}^{^{\prime}}\Vert }\right)\Vert *\frac{p}{{R}_{osc}}\\ d=-\frac{{\overrightarrow{r}}_{2}^{^{\prime}}\bullet {\overrightarrow{r}}_{1}^{^{\prime}}}{\Vert {\overrightarrow{r}}_{2}^{^{\prime}}\Vert }\end{array}\right.$$
(2.4)

\({\widetilde{\gamma }}_{0}\) denote the shadow factor. (\({a}_{1}-{a}_{8}\)) are fit coefficients and shown in Table 1; tanh is the hyperbolic tangent function; \({\overrightarrow{r}}_{1}^{^{\prime}}\), \({\overrightarrow{r}}_{2}^{^{\prime}}\) is the adjusted satellite position and the Sun position, respectively; \({R}_{osc}\) is the adjusted spherical Earth’s radius; p is the Earth’s equatorial radius; \({r}_{E}^{^{\prime}}\) is the distance from the satellite to the center of the Earth after considering the Earth’s oblateness; d is the component of the distance from the satellite to the Earth in the Sun–Earth’s direction and is in \({10}^{6}\)m. Table 1 lists an unique set of coefficients which are generated by the trust-region-reflective curve-fitting algorithm (Robertson 2015).

Table 1 Fit coefficients for the shadow factor

Validation method

SLR is a commonly used mean of external orbital validation (Bury et al. 2019; Hackel et al. 2015). This paper adopts SLR orbit validation and proposes a new mean named ISL orbit check. ISL refers to the interconnection between satellite and satellite through electromagnetic waves (Maine et al. 2003). After the interconnection, information and data can be shared and transmitted, and the distance between satellites can also be measured. For BeiDou satellites, the primary purpose of the ISL is to realize the information transmission and two-way range measurement. Once the ground operation control system is not available, the inter-satellite ranging is used to maintain the satellite navigation message updating independently (Maine et al. 2003). Compared with low-speed and wide-beam GPS UHF band ISLs (Rajan 2002; Rajan et al. 2003) and low-speed and wide-beam GLONASS S-band ISLs (Revnivykh, 2012), Ka-band medium-speed ISLs are used in BeiDou constellation (Tang et al. 2018). The BeiDou ISL’s ranging accuracy can reach to 0.1–0.3 ns (3–10 cm), and the ISL transmission delay and the receiving delay residual are 0.0084 ns and 0.0399 ns, respectively (Tang et al. 2018; Meng et al. 2017). It is higher than the current BDS orbit accuracy in the along/cross (A/C) direction (10 cm) (Li et al. 2020; http://www.igmas.org/). The observed time is ordinarily different from each other by up to 3 s, and the transformation error from one satellite to another is proved to be less than 4 mm. Therefore, it is necessary to transform the dual one‐way measurements at different times into a standard epoch (Tang et al. 2018; Yang et al. 2018). The two-way ISL observation equation is given as follows:

$$\left\{\begin{array}{c}p\left({t}_{0}\right)=\frac{{P}_{AB}+{P}_{BA}}{2}=\frac{{\rho }_{AB}\left({t}_{0}\right){+\rho }_{BA}\left({t}_{0}\right)}{2}-c{(\Delta }_{A}+{\Delta }_{B})-\frac{{\delta }_{AB}{+\delta }_{BA}}{2}-\frac{{\varepsilon }_{AB}{+\varepsilon }_{BA}}{2}\\ {\Delta }_{A}=\frac{{D}_{RA}+{D}_{LA}}{2}\\ {\Delta }_{B}=\frac{{D}_{RB}+{D}_{LB}}{2}\end{array}\right.$$
(2.5)

\({P}_{AB}\) and \({P}_{BA}\) are the distance of satellite A to B and B to A; \({\rho }_{AB}\) and \({\rho }_{BA}\) are the pseudo-range values between satellites; \({t}_{0}\) is the common epoch; \({\Delta }_{A}\) and \({\Delta }_{B}\) are the hardware delay; c is the velocity of light in vacuum; \({D}_{RB}\) and \({D}_{LB}\) are the reception and launch delay of satellite B; \({D}_{RA}\) and \({D}_{LA}\) are the reception and launch delay of satellite A; \({\delta }_{AB}\) and \({\delta }_{BA}\) represent observation corrections such as relativistic effect correction and antenna phase center correction; \({\varepsilon }_{AB}\) and \({\varepsilon }_{BA}\) represent observation noise.

The hardware delay is estimated as a constant parameter in the short term (a specific POD arc). However, the parameters may vary when the satellite-related delays are not necessarily equal to each other during different ISL pairs. Therefore, we can directly estimate the ISL-pair-dependent constant parameter \({\Delta }_{AB}\) within a POD arc as follows (Wang et al. 2019):

$${\Delta }_{AB}=\frac{{D}_{RB}+{D}_{LB}+{D}_{RA}+{D}_{LA}}{2}$$
(2.6)

This paper compares the difference between the precision orbit and the ISL results to evaluate the accuracy of the four shadow models. Combining the results of precision orbit determination, the verification equation can be obtained by:

$$\upomega ({t}_{0})=\left|{\overrightarrow{r}}_{B}({t}_{0})-{\overrightarrow{r}}_{A}({t}_{0})\right|-P({t}_{0})$$
(2.7)

\(\upomega ({t}_{0})\) is the check residual value; \({\overrightarrow{r}}_{B}({t}_{0})\) and \({\overrightarrow{r}}_{A}({t}_{0})\) are satellites’ positions in the inertial coordinate system.

Comparison of shadow models and their impact on POD

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.

Table 2 Correction models and parameter estimation strategy
Fig. 5
figure 5

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:

$${\psi }_{\mathrm{cast}}=\mathrm{ATAN}2(-\mathrm{tan}{\beta }_{d},\mathrm{sin\eta })$$
(3.1)

where \({\beta }_{d}\) is modified Sun elevation angle:

$$\left\{\begin{array}{l}{\beta }_{d}=\beta +f\times \left(\mathrm{SIGN}\left({\beta }_{0},\beta \right)-\beta \right)\\ f=\left\{\begin{array}{c}\frac{1}{1+{d}_{m}\times {\mathrm{sin}}^{4}\eta },{\beta }_{0}\le \beta \\ 0,{\beta }_{0}>\beta \end{array}\right.\end{array}\right.$$
(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:

$${S}_{oy}=\left\{\begin{array}{c}-\mathrm{sin}\left(3 \mathrm{deg}\right),\beta >0\\ \mathrm{sin}\left(3 \mathrm{deg}\right),\beta <0\end{array}\right.$$
(3.3)

The yaw-angle \({\psi }_{\mathrm{secm}}\) is expressed as:

$${\psi }_{\mathrm{secm}}=\mathrm{ATAN}2({S}_{oy},\mathrm{sin\eta cos}\beta )$$
(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.

Fig. 6
figure 6

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

Table 3 Time the satellite enters and exits the shadow (unit: SOD)

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).

Fig. 7
figure 7figure 7

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.

Table 4 Satellite center of mass information (unit: mm)
Table 5 SLR validation results of four shadow models for different satellites (\({N}_{\mathrm{points}}\): Number of observation points; unit: mm)

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.

Table 6 Transmitting and receiving information of ISL
Table 7 Performance of different shadow models by ISL check (\({N}_{\mathrm{points}}\): Number of observation points; unit: mm)

Conclusions

This paper first introduces shadow models' modelling methods, including conical, PPM, PPMatm, and SOLAARS-CF models. PPM is modelled with consideration of the Earth’s oblateness based on the perspective projection algorithm; PPMatm considers the atmospheric effect by linear equations for solar radiation reduction based on PPM; SOLAARS-CF takes the Earth’s oblateness and atmospheric effect into consideration by a curve fitting from a physical-based model (SOLAARS). And then, the shadow factor and SRP acceleration derived by the four shadow models are compared. The satellite enters shadow earlier and comes out of shadow later by PPM, PPMatm, and SOLAARS-CF models than by 3dishes. The SRP acceleration derived from PPMatm and SOLAARS-CF models distinguish with that of 3dishes, and the PPMatm and SOLAARS-CF models have the highest conformity. To further verify the four shadow models, this paper used two validation methods: SLR check and ISL check. It is noteworthy that the ISL check is a new method proposed by this paper. As the result of the SLR check, the shadow models with the Earth’s oblateness and atmospheric effects (PPMatm, SOLAARS-CF) have an improvement of 2–10 mm in RMS compared with the conical model. The ISL check results show that PPMatm and SOLAARS-CF are better than the other two models. Compared with the 3dishes model, the RMS value can be generally decreased by 2–7 mm. The performance of the PPMatm and SOLAARS-CF models is comparable. From the perspective of computational cost, the four models do not have many differences, and the input parameters are the position of the satellite, the Sun, and the Lunar. With the consideration of the precision and reliability of the modelling in eclipse, PPMatm and SOLAARS-CF are recommended for POD processing, and the Earth’s oblateness and atmospheric effect must be taken into consideration in shadow models, especially in the current situation of perturbation acceleration accuracy stepping toward 10–10 m/s2.

Availability of data and materials

The BDS and SLR data sets are available from ftp://cddis.gsfc.nasa.gov/. The ISL measurements are provided by Beijing Satellite Navigation Center.

References

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Acknowledgements

The BDS data set is from the Crustal Dynamics Data Information System (CDDIS). The SLR data set is from the International Laser Ranging Service (ILRS). The ISL data set is provided by Beijing Satellite Navigation Center. Dr. Robbie Robertson helped to build the SOLAARS-CF model. Dr. Chenpan Tang helped to solve the problems in ISL data processing. We express our sincere gratitude to these organizations and individuals. In addition, we also thank the China Scholarship Council (CSC) to support Yan Zhang financially to study in Calgary.

Funding

This work is supported by the National Natural Science Foundation of China (No.11973073), the National Key Research and Development Program of China (No. 2016YFB0501405), the Basic project of Ministry of Science and Technology of China (No.2015FY310200), the Shanghai Key Laboratory of Space Navigation and Position Techniques (No.06DZ22101).

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Contributions

XW proposed the study, supervised the progress of the study, gave the advice for issues, and revised the manuscript; YZ realized the model, analyzed the result, and wrote the manuscript; KX contributed to the refinement of software to process the data; ZL brought suggestions for the model. All authors commented on the manuscript draft and approved the submission. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Xiaoya Wang.

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Supplementary Information

Additional file 1. Figure S1.

(a) SLR check results of all satellites. The y-axis (left) is the omc value; the y-axis (right) is the sun elevation angle; the x-axis is the DOY. Figure S2. (b) ISL results of all links. The y-axis is the check omc value; the x-axis is the DOY.

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Zhang, Y., Wang, X., Xi, K. et al. Comparison of shadow models and their impact on precise orbit determination of BeiDou satellites during eclipsing phases. Earth Planets Space 74, 126 (2022). https://doi.org/10.1186/s40623-022-01684-5

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Keywords

  • Shadow model
  • Earth’s oblateness
  • Atmospheric effect
  • SLR
  • Inter-satellite link