Box-wing Model Research Articles

Comparison of the SRP spherical model between LEO and GEO satellites

Solar Radiation Pressure (SRP) is a phenomenon caused by the pressure exerted by solar photons on a satellite’s surface when it is exposed to sunlight. It is a form of radiation force and can significantly impact the motion and behaviour of satellites in space. SRP influences a satellite’s orbit by causing changes in its semimajor axis, eccentricity, inclination, argument of perigee, right ascension of the ascending node, and mean anomaly. SRP models are used to simulate the effects of solar radiation pressure on satellites. These models are essential for accurately predicting satellite trajectories and orbital behaviour. There are several types of SRP models, such as spherical model, flat model, box-wing model, faceted model, and analytical SRP models. This research focuses on Telkom 1 and LAPAN A1 satellites, both belonging to Indonesia and positioned in Geostationary Earth Orbit (GEO) and Low Earth Orbit (LEO) orbits, respectively. The study aims to find a comparison of the effects of SRP spherical model on LEO and GEO satellites. Our modelling shows that the semimajor axis and eccentricity are sensitive to SRP, while the inclination and right ascension of the ascending node are not significantly affected. Comparing the effects of SRP on LEO and GEO satellites, we concluded that both LEO and GEO orbit experience the most significant fluctuations in January (perihelion), likely due to the influence of solar radiation pressure.

Optimization of orbit prediction strategies for GNSS satellites

The ability to provide precise orbit prediction of Global Navigation Satellite System (GNSS) satellites is essential in a wide range of applications in space geodesy and Earth sciences, including near real-time and real-time GNSS applications, forming broadcast ephemerides, or supporting Satellite Laser Ranging stations in tracking. In this study, we evaluate the impact of orbit modeling strategies on the accuracy of orbit predictions on short (one-day), and long (multi-day) time scales, focusing on the selection of the optimal approach for handling solar radiation pressure (SRP), the effects of including pseudo-stochastic parameters, and the impact of the arc length used for the initial orbit fit. The analysis includes 300 days of predictions in 2021 and the satellites belonging to the GPS, Galileo, GLONASS, BeiDou-3, (BDS-3), and QZSS constellations. Fitting an initial 2-day orbit arc is the optimal solution for all analyzed navigation satellite constellations. When official satellite construction metadata are available, e.g. for Galileo/QZSS, the hybrid strategy of combining both empirical and physical models, i.e. the extended Empirical CODE Orbit Model (ECOM2) with box-wing models, leads to the best results. Otherwise, using only the ECOM2 is a better choice. Finally, the results indicate that for all navigation satellites, the introduction of pseudo-stochastic parameters deteriorates the prediction quality. When using the optimal prediction strategy, the 95th percentiles of the position errors after the 1st/4th/9th day of prediction are equal to 0.09/0.93/4.52, 0.22/1.71/9.69, 0.20/2.19/11.30, 0.23/1.80/9.39, 0.23/2.28/7.78m for GPS-IIF, Galileo FOC, GLONASS-M, BDS-3 CAST, and QZSS, respectively.

Estimating surface optical properties and thermal thrust for Galileo satellite body and solar panels

Precise orbit determination of GNSS (Global Navigation Satellite System) satellites requires accurate models of perturbing forces acting on the spacecraft, i.e., solar radiation pressure (SRP) and thermal radiation forces. With the officially published satellite metadata, the analytical box-wing model is usually used to describe most of the SRP accelerations and the rest is assumed to be compensated by estimating ECOM/ECOM2 (Empirical CODE Orbit Model) parameters. However, we find that the precision of Galileo satellite orbits shows notable degradation inside eclipse seasons for 3-day-arc solutions and 24-h predictions. For instance, the RMS (root-mean-square) of orbit misclosures increases by about a factor of two in the eclipse season when using the box-wing model as the a priori and the 5-parameter ECOM model on top. The reason is proven to be mostly due to ignoring imbalanced thermal radiation forces (i.e., radiator emission and thermal radiation of solar panels) as satellite thermal properties are unknown. These imbalanced thermal effects cannot be fully absorbed by the ECOM/ECOM2 parameters inside eclipse seasons because the earth’s shadowing of a satellite in orbit causes periodic changes of the thermal environment. To cope with this problem, we first estimate satellite optical and thermal parameters as part of orbit determination based on Galileo tracking data covering 1 year. Then, we add physical thermal radiation models for radiators and solar panels as part of the a priori model and evaluate the performance of different ECOM models in Galileo satellite orbit determination. As shown by orbit misclosures, 24-h orbit predictions and SLR (Satellite Laser Ranging) residuals, the 7-parameter ECOM2 model performs better than the 5-parameter ECOM and the 9-parameter ECOM2 model for Galileo satellites. When using the 7-parameter ECOM2 model on top, the impact of the radiator emission and the thermal radiation of solar panels on Galileo satellite orbits is about 1 and 2 cm, respectively, inside eclipse seasons for 3-day-arc solutions.

Advancing the Solar Radiation Pressure Model for BeiDou-3 IGSO Satellites

In the absence of detailed surface information, empirical solar radiation pressure (SRP) models, such as the five-parameter Empirical CODE Orbit Model (ECOM1) and its extended version-ECOM2, are widely used for modeling SRP forces acting on GNSS satellites. This study shows that the orbits of BeiDou-3 Inclined Geosynchronous Orbit satellites (IGSOs) determined with the ECOM1 model suffer from systematic once-per-revolution radial orbit errors, which can be partly reduced by the ECOM2 model. To eliminate such orbit errors, the BeiDou-3 IGSO optical coefficients are solved by using an adjustable box-wing (ABW) model and then introduced into an a priori box-wing SRP model to enhance the ECOM1 model (ECOM1 + BW). In the ABW solution, in addition to satellite body and solar panels, the contributions of the communication payloads installed on BeiDou-3 IGSO ±X panels on the SRP are also considered, which markedly improves the stability of the optical coefficient estimates. The efficiency of the developed a priori box-wing model is demonstrated through eliminated once-per-revolution radial orbit errors and decreased day boundary discontinuities. However, the orbit solutions still show significant degradations during eclipse seasons. The results of the first yaw-attitude analysis for eclipsing BeiDou-3 IGSOs show that their yaw behaviors are the same as those of BeiDou-3 CAST (China Academy of Space Technology) MEOs (Medium Earth Orbit satellites), and have been well considered in the study. This rules out the possibility that attitude errors are the potential reason for the orbit deterioration. By introducing a once-per-revolution sine term in the Sun direction (Ds term) and keeping Ds active during the Earth’s shadow transitions to the ECOM1 + BW model, the orbit performance inside the eclipse seasons is significantly improved and can be comparable to that outside the eclipse seasons.

Improving the Orbits of the BDS-2 IGSO and MEO Satellites with Compensating Thermal Radiation Pressure Parameters

The orbit accuracy of the navigation satellites relies on the accurate knowledge of the forces on the spacecraft, in particular the non-conservative perturbations. This study focuses on the Inclined Geosynchronous Orbit (IGSO) and Medium Earth Orbit (MEO) satellites of the regional Chinese BeiDou Navigation Satellite System (BDS-2), for which apparent deficiencies of non-conservative models are identified and evidenced in the Satellite Laser Ranging (SLR) residuals. The orbit errors derived from the empirical 5-parameter Extended CODE Orbit Model (ECOM) as well as a semi-analytical adjustable box-wing model show prominent dependency on the Sun elongation angle, even in the yaw-steering attitude mode. Hence, a periodic acceleration in the normal direction of the +X surface, presumably generated by the mismodeled thermal radiation pressure, is introduced. The SLR validations reveal that the Sun elongation angle-dependent systematic errors were significantly reduced, and the orbit accuracy was improved by 10–30% to approximately 4.5 cm and 3.0 cm for the BDS-2 IGSO and MEO satellites, respectively.

GLONASS precise orbit determination with identification of malfunctioning spacecraft

Due to the continued development of the GLONASS satellites, precise orbit determination (POD) still poses a series of challenges. This study examines the impact of introducing the analytical tube-wing model for GLONASS-M and the box-wing model for GLONASS-K in a series of hybrid POD strategies that consider both the analytical model and a series of empirical parameters. We assess the perturbing accelerations acting on GLONASS spacecraft based on the analytical model. All GLONASS satellites are equipped with laser retroreflectors for satellite laser ranging (SLR). We apply the SLR observations for the GLONASS POD in a series of GNSS + SLR combined solutions. The application of the box-wing model significantly improves GLONASS orbits, especially for GLONASS-K, reducing the STD of SLR residuals from 92.6 to 27.6 mm. Although the metadata for all GLONASS-M satellites reveal similar construction characteristics, we found differences in empirical accelerations and SLR offsets not only between GLONASS-M and GLONASS-M+ but also within the GLONASS-M+ series. Moreover, we identify satellites with inferior orbit solutions and check if we can improve them using the analytical model and SLR observations. For GLONASS-M SVN730, the STD of the SLR residuals for orbits determined using the empirical solution is 48.7 mm. The STD diminishes to 41.2 and 37.8 mm when introducing the tube-wing model and SLR observations, respectively. As a result, both the application of the SLR observations and the analytical model significantly improve the orbit solution as well as reduce systematic errors affecting orbits of GLONASS satellites.

EXPERIMENTAL ANALYSIS OF DOUBLE BOX WING UAV.

In an attempt to reduce the induced drag on a wing, Prandtl found that induced drag reduced significantly by highly increasing the number of vertically offset wings. The same result could be obtained by joining the wingtips of two vertically offset wings. This helped increase payload capacity and also reduced fuel consumption and emissions. Such a wing configuration came to be known as Prandtl’s box wing. In this work, the design and analysis of a box wing aircraft model has been carried out. The preliminary analysis is performed using XFLR5, and the computational analysis is done with the help of ANSYS 18.2. The values of experiments are computed with the help of MATLab R2017. The box wing model has shown a nearly 53.74% reduction in drag as compared with conventional wing models. The computational results of drag have been compared and validated with the results of analytical and the experimental results from the wind tunnel and found to be within 10% of the computational result. Since the drag of the box wing is significantly lesser than the conventional wings the box wing is a feasible configuration which can be used to design various aircrafts including Unmanned Aerial Vehicles and Commercial Planes.

Validation of Multi-Year Galileo Orbits Using Satellite Laser Ranging

Satellite laser ranging (SLR) observations provide an independent validation of the global navigation satellite system (GNSS) orbits derived using microwave measurements. SLR residuals have also proven to be an important indicator of orbit radial accuracy. In this study, SLR validation is conducted for the precise orbits of eight Galileo satellites covering four to eight years (the current longest span), provided by multiple analysis centers (ACs) participating in the multi-GNSS experiment (MGEX). The purpose of this long-term analysis (the longest such study to date), is to provide a comprehensive evaluation of orbit product quality, its influencing factors, and the effect of perturbation model updates on precise orbit determination (POD) processing. A conventional ECOM solar radiation pressure (SRP) model was used for POD. The results showed distinct periodic variations with angular arguments in the SRP model, implying certain defects in the ECOM system. Updated SRP descriptions, such as ECOM2 or the Box-Wing model, led to significant improvements in SLR residuals for orbital products from multiple ACs. The standard deviation of these residuals decreased from 8–10 cm, before the SRP update, to about 3 cm afterward. The systematic bias of the residuals was also reduced by 2–4 cm and the apparent variability decreased significantly. In addition, the effects of gradual SRP model updates in the POD were evident in orbit comparisons. Orbital differences between ACs in the radial direction were reduced from the initial 10 cm to better than 3 cm, which is consistent with the results of SLR residual analysis. These results suggest SLR validation to be a powerful technique for evaluating the quality of POD strategies in GNSS orbits. Furthermore, this study has demonstrated that perturbation models, such as SRP, provide a better orbit modeling for the Galileo satellites.

A Comparative Study on the Solar Radiation Pressure Modeling in GPS Precise Orbit Determination

For Global Positioning System (GPS) precise orbit determination (POD), the solar radiation pressure (SRP) is the dominant nongravitational perturbation force. Among the current SRP models, the ECOM and box-wing models are widely used in the International GNSS Service (IGS) community. However, the performance of different models varies over different GPS satellites. In this study, we investigate the performances of different SRP models, including the box-wing and adjustable box-wing as a priori models, and ECOM1 and ECOM2 as parameterization models, in the GPS POD solution from 2017 to 2019. Moreover, we pay special attention to the handling of the shadow factor in the SRP modeling for eclipsing satellites, which is critical to achieve high-precision POD solutions but has not yet been fully investigated. We demonstrate that, as an a priori SRP model, the adjustable box-wing has better performance than the box-wing model by up to 5 mm in the orbit day boundary discontinuity (DBD) statistics, with the largest improvement observed on the BLOCK IIR satellites using the ECOM1 as a parameterization SRP model. The box-wing model shows an insignificant orbit improvement serving as the a priori SRP model. For the eclipsing satellites, the three-dimensional (3D) root mean square (RMS) values of orbit DBD are improved when the shadow factor is applied only in the D direction (pointing toward to Sun) than that in the three directions (D, Y, and B) in the satellite frame. Different SRP models have comparable performance in terms of the Earth rotation parameter (ERP) agreement with the IERS EOP 14C04 product, whereas the magnitude of the length of day (LoD) annual signal is reduced when the shadow factor is applied in the D direction than in the three directions. This study clarifies how the shadow factor should be applied in the GPS POD solution and demonstrates that the a priori adjustable box-wing model combined with ECOM1 is more suitable for high-precision GPS POD solutions, which is useful for the further GNSS data analysis.

An analysis of a priori and empirical solar radiation pressure models for GPS satellites

Abstract. Among the different non-conservative forces acting on GPS satellites, solar radiation pressure (SRP) has the greatest influence and inappropriate modeling of it can introduce an acceleration with the order of 1 × 10−7 m s−2. There are a variety of empirical, analytical, and hybrid empirical-physical models to describe the SRP effect. Among them, the empirical model developed at the Center for Orbit Determination in Europe (CODE) and analytical models based on a box-wing prototype, namely box-shape bus with solar panels, are widely used in the International GNSS Service (IGS) community. To investigate the effects of different a priori SRP models on top of empirical parameterization, two sets of parameters based on the Empirical CODE Orbit Model (ECOM) and two a priori models including the analytical box-wing model and the empirical GPS Solar Pressure Model (GSPM) are tested for the different GPS satellites. Orbit comparison of different SRP scenarios shows that: (1) the two parameterizations of ECOM perform differently for Block IIA and IIR/IIR-M satellites but lead to fewer differences for Block IIF satellites in terms of orbit difference pattern. The 3D RMS of orbit difference of two parameterizations are 25, 30 and 21 mm for each block type. (2) Adoption of a priori model or change of the ECOM parameterization mainly lead to orbit differences varying with both elevation of the Sun w.r.t. the orbit plane and the satellites' argument of latitude w.r.t. the noon point, which is supposed to be related to the special geometry and attitude of every block type. These differences are especially obvious in radial direction. Analysis of estimated parameters of ECOM indicates that (3) the GSPM.04 performs better than box-wing model to describe the main constant solar radiation. It is found (4) that the asymmetry of estimated ECOM parameters in B direction (i.e., the direction completing the orthogonal system with D direction and satellite's solar panel axes), observed for three Block IIR satellites, causes corresponding asymmetrical orbit differences in radial direction when reduced ECOM parameters are used. This does not apply to the extended ECOM parameterization tested in this study, which indicates the insufficiency of reduced ECOM to parameterize asymmetrical satellites.

Spin Axis Determination of Defunct GLONASS Satellites Using Photometry Observation

GLONASS, a satellite navigation system developed in Russia since 1976, is defunct and orbits in an unstable attitude. The satellites in these problems are not managed and there is no precise information, which can increase the risk of collisions with other space objects. In this study, detailed attitude dynamic have to be analyzed through photometry data, which requires spin period and spin axis. The light curve data is obtained by observing through the photometer at the Graz station and the power spectrum is calculated to obtain the cycle of the satellite. The geometric relationship between observer and sun is analyzed for GLONASS-50 satellite. The box-wing model is applied to obtain the phase reflection of the satellite and obtain the Irradiation of the satellite through this information. In Light Curve and Irradiation, the spin axis is calculated for each peak points with the distance square minimum technique. The spin axis of the GLONASS-50 satellite is RA = 116°, Dec = 92°.

Enhanced solar radiation pressure model for GPS satellites considering various physical effects

Solar radiation pressure (SRP) is the dominant non-gravitational perturbation for GPS satellites. In the IGS (International GNSS Service), this perturbation is modeled differently by individual analysis centers (ACs). The two most widely used methods are the Empirical CODE orbit Model (ECOM, ECOM2) and the JPL GSPM model. When using ECOM models, a box-wing model or other a priori models, as well as stochastic pulses at noon or midnight, are optionally adopted by some ACs to compensate for the deficiencies of the ECOM or ECOM2 model. However, both box-wing and GSPM parameters were published many years ago. There could be an aging effect going with time. Also, optical properties and GSPM parameters of GPS Block IIF satellites are currently not yet published. In this contribution, we first determine Block-specific optical parameters of GPS satellites using GPS code and phase measurements of 6 years. Various physical effects, such as yaw bias, radiator emission in the satellite body-fixed − X and Y directions and the thermal radiation of solar panels, are considered as additional constant parameters in the optical parameter adjustment. With all the adjusted parameters, we form an enhanced box-wing model adding all the modeled physical effects. In addition, we determine Block-specific GSPM parameters by using the same GPS measurements. The enhanced box-wing model and the GSPM model are then taken as a priori model and are jointly used with ECOM and ECOM2 model, respectively. We find that the enhanced box-wing model performs similarly to the GSPM model outside eclipse seasons. RMSs of all the ECOM and ECOM2 parameters are reduced by 30% compared to results without the a priori model. Orbit misclosures and orbit predictions are improved by combining the enhanced box-wing model with ECOM and ECOM2 models. In particular, the improvement in orbit misclosures for the eclipsing Block IIR and IIF satellites, as well as the non-eclipsing IIA satellites, is about 25%, 10% and 10%, respectively, for the ECOM model. Therefore, the enhanced box-wing model is recommended as an a priori model in GPS satellite orbit determination.

Geocenter coordinates derived from multi-GNSS: a look into the role of solar radiation pressure modeling

The Global Navigational Satellite System (GNSS) technique is naturally sensitive to the geocenter motion, similar to all satellite techniques. However, the GNSS-based estimates of the geocenter used to contain more orbital artifacts than the geophysical signals, especially for the Z component of the geocenter coordinates. This contribution conveys a discussion on the impact of solar radiation pressure (SRP) modeling on the geocenter motion estimates. To that end, we process 3 years of GPS, GLONASS, and Galileo observations (2017–2019), collected by a globally distributed network of the ground stations. All possible individual system-specific solutions, as well as combinations of the available constellations, are tested in search of characteristic patterns in geocenter coordinates. We show that the addition of a priori information about the SRP-based forces acting on the satellites using a box-wing model mitigates a great majority of the spurious signals in the spectra of the geocenter coordinates. The amplitude of the 3 cpy (about 121 days) signal for GLONASS has been reduced by a factor of 8.5. Moreover, the amplitude of the spurious 7 cpy (about 52 days) signal has been reduced by a factor of 5.8 and 3.1 for Galileo and GPS, respectively. Conversely, the box-wing solutions indicate increased amplitudes of the annual variations in the geocenter signal. The latter reaches the level of 10–11 mm compared to 4.4 and 6.0 mm from the satellite laser ranging observations of LAGEOS satellites and the corresponding GNSS series applying extended empirical CODE orbit model (ECOM2), respectively. Despite the possible improvement in the GLONASS-based Z component of the geocenter coordinates, we show that some significant power can still be found at periods other than annual. The GPS- and Galileo-based estimates are less affected; thus, a combination of GPS and Galileo leads to the best geocenter estimates.

Improving solar radiation pressure modeling for GLONASS satellites

The Russian Global Navigation Satellite System (GLONASS) satellites have a stretched body shape and take a specific attitude mode inside the eclipse. Based on previous studies, the new Empirical CODE orbit model (ECOM2) performs better than the classical ECOM model if a satellite has elongated shape or does not maintain yaw-steering mode, and the use of an a priori box-wing (BW) model improves the orbits significantly when employing the ECOM model. However, we find that the ECOM model performs better than the ECOM2 model for GLONASS satellites outside eclipse seasons, while it performs two times worse in eclipse seasons. The use of the conventional box-wing model results in very little improvement. By assessing the ECOM Y _{0} estimates, we conclude that there are potential radiators on the -x surface of GLONASS satellites causing orbit perturbations also inside the eclipse. The higher-order Fourier terms of the ECOM2 model can compensate for such effects better than the ECOM model. Based on this finding, we first confirm that GLONASS-K satellites take a similar attitude mode as GLONASS-M satellites inside the eclipse. Then, we adjust optical parameters of GLONASS satellites as part of precise orbit determination (POD) considering the potential radiator and thermal radiation effects. Finally, the adjusted parameters are introduced into a new box-wing model and jointly used with the ECOM and ECOM2 model, respectively. Results show that the amplitude and the dependency of the empirical parameters on the beta angle are greatly reduced for both ECOM and ECOM2 models. Rather than the conventional box-wing model, the new box-wing model reduces the orbit misclosure between two consecutive arcs for both GLONASS-M and GLONASS-K satellites. In particular, the improvement in GLONASS-M satellites is more than 30% for the ECOM model during eclipse seasons. Further evaluation from 24-h predicted orbits demonstrates that the improvement during eclipse seasons is mainly in along- and cross-track directions. Finally, we validate GLONASS satellite orbits using Satellite Laser Ranging (SLR) observations. The use of the new box-wing model reduces the spurious pattern of the SLR residuals as a function of beta and Delta u significantly, and the linear dependency of the SLR residuals on the elongation drops from as large as -0.760 mm/deg to almost zero for both ECOM and ECOM2 models. In general, GLONASS-M satellites benefit more from the new a priori box-wing model and the BW+ECOM model results in the best SLR residuals, with an improvement of about 50% and 20%, respectively, for the mean and standard deviation (STD) values with respect to the orbit products without a priori model.

GPS III Vespucci: Results of half a year in orbit

The spacecraft nicknamed Vespucci is the first GPS III satellite launched in December 2018. Numerous receivers of the global International GNSS Service network tracked the signals of this new generation of GPS spacecraft from January to July 2019. This data set serves as the basis for analysis of broadcast ephemeris performance and clock stability, as well as solar radiation pressure and satellite antenna phase center models. Different empirical orbit models are tested, and box-wing models are developed based on approximate dimensions, mass, and assumed optical properties. The box-wing models show in general a better performance than the empirical models. Compared to Block IIF satellites, the stability of the GPS III rubidium clock is higher for integration times up to 10 s. At longer integration times, the stability of both clocks is similar but the GPS III clock does not suffer from frequency-specific line bias variations that affect the apparent IIF clock. Estimated antenna phase center offsets agree on the centimeter level with the calibrations published by the manufacturer. Measurements with a 30-m high-gain antenna revealed that the satellite did not stop signal transmission in July 2019 but switched to non-standard codes and codes beyond the tracking capabilities of commercial GNSS receivers. Prior to this hibernation mode, a 70 days test without broadcast ephemeris update was conducted during which the user range error increased to roughly 1 km.

Toward the 1-cm Galileo orbits: challenges in modeling of perturbing forces

Precise orbit determination demands knowledge of perturbing forces acting on the satellites of the Global Navigation Satellite Systems (GNSS). The metadata published by the European GNSS Agency for the Galileo satellites allow for the composition of the analytical box-wing model dedicated for coping with the direct solar radiation pressure (SRP), albedo, and infrared radiation (IR). Based on the box-wing model, we evaluated both the magnitude and the characteristic periods of accelerations caused by all the aforementioned forces. We assess which perturbations can be absorbed by the extended Empirical CODE Orbit Model (ECOM2) and what are the consequences of neglecting higher-order ECOM2 coefficients. In order to evaluate the impact of SRP, albedo, IR, and the navigation antenna thrust, we perform a series of precise Galileo orbit determination strategies for Galileo In-Orbit-Validation (IOV), Full Operational Capability (FOC), and two FOC satellites launched into eccentric orbits. The proposed box-wing model is capable of absorbing approximately 97% of the SRP in the Sun-satellite direction, whereas the rest can be mitigated by an additionally estimated small set of empirical parameters. The purely physical box-wing model does not fully handle satellite misorientation and re-radiation effects, such as Y-bias, solar panel rotation lag, that is the misalignment causing a constant acceleration perpendicular to the solar panel axis and the direction to the Sun. However, the box-wing model is especially crucial in terms of the absorption of the higher-order terms of SRP and stabilizes the orbit solutions during the eclipsing periods. Based on the SLR residual analysis, we found a systematic effect at the level up to 50 mm resulting from the omission of the high-order empirical orbit coefficients. We also found that the impact of the albedo, IR, and transmitter antenna thrust on the Galileo orbits reach the level of 5, 14, and 20 mm, respectively. Eventually, we obtain the overall accuracy of the Galileo-FOC orbits at the level of 22.5 mm, even for the eclipsing period for the solution which considers the box-wing model with the estimation of the constant empirical accelerations.

An empirical solar radiation pressure model for satellites moving in the orbit-normal mode

We present a family of empirical solar radiation pressure (SRP) models suited for satellites orbiting the Earth in the orbit normal (ON) mode. The proposed ECOM-TB model describes the SRP accelerations in the so-called terminator coordinate system. The choice of the coordinate system and the SRP parametrization is based on theoretical assumptions and on simulation results with a QZS-1-like box-wing model, where the SRP accelerations acting on the solar panels and on the box are assessed separately. The new SRP model takes into account that in ON-mode the incident angle of the solar radiation on the solar panels is not constant like in the yaw-steering (YS) attitude mode. It depends on the elevation angle of the Sun above the satellite’s orbital plane. The resulting SRP vector acts, therefore, not only in the Sun-satellite direction, but has also a component normal to it. Both components are changing as a function of the incident angle. ECOM-TB has been used for precise orbit determination (POD) for QZS-1 and BeiDou2 (BDS2) satellites in medium (MEO) and inclined geosynchronous Earth orbits (IGSO) based on IGS MGEX data from 2014 and 2015. The resulting orbits have been validated with SLR, long-arc orbit fits, orbit misclosures, and by the satellite clock corrections based on the orbits. The validation results confirm that—compared to ECOM2—ECOM-TB significantly (factor 3–4) improves the POD of QZS-1 in ON-mode for orbits with different arc lengths (one, three, and five days). Moderate orbit improvements are achieved for BDS2 MEO satellites—especially if ECOM-TB is supported by pseudo-stochastic pulses (the model is then called ECOM-TBP). For BDS2 IGSOs, ECOM-TB with its 9 SRP parameters appears to be over-parameterized. For use with BDS2 IGSO spacecraft we therefore developed a minimized model version called ECOM-TBMP, which is based on the same axis decomposition as ECOM-TB, but has only 2 SRP parameters and is supported by pseudo-stochastic parameters, as well. This model shows a similar performance as ECOM-TB with short arcs, but an improved performance with (3-day) long-arcs. The new SRP models have been activated in CODE’s IGS MGEX solution in Summer 2018. Like the other ECOM models the ECOM-TB derivatives might be used together with an a priori model.

A Priori Solar Radiation Pressure Model for BeiDou-3 MEO Satellites

Due to the cuboid satellite body of BeiDou-3 satellites, the accuracy of their orbit showed a trend of systematic variation with the sun-satellite-earth angle (ε) using the Extend CODE Orbit Model (ECOM1). Therefore, an a priori cuboid box-wing model (named the cuboid model) is necessary to compensate ECOM1. Considering that the body-dimensions and optical properties of the BeiDou-3 satellites used to construct the box-wing model have not yet been fully released, the adjustable box-wing model (ABW) was used for precise orbit determination (POD). The a priori cuboid box-wing model was directly estimated by the precision radiation accelerations, obtained from ABW POD. When using ECOM1 model, for 14 < β < 40°, a linear systematic variation of D0 related to the elevation of the sun above the orbital plane (β-angle) with a slope of 0.048 nm/s2/°, was found for C30. After adding the cuboid model to assist ECOM1 (named Cuboid + ECOM1), the slope was reduced to 0.005 nm/s2/°, and for C20 satellite, the standard deviation (STD) of D0 was improved, from 1.28 to 0.85 nm/s2 (34%). For satellite laser ranging (SLR) validation, when using the ECOM1 model, the systematic variation with the ε angle was about 14 cm for C20 and C30. After using the Cuboid + ECOM1 model, the variation was significantly reduced to about 5 cm. For C20 and C21, compared with the ECOM1 model, the root mean square (RMS) of the ECOM2 and Cuboid + ECOM1 model was improved by about 0.54 (10.3%) and 0.43 cm (8.7%). For C29 and C30, the RMS of ECOM2 and Cuboid + ECOM1 model was improved for about 0.7 (10.9%) and 1.6 cm (25.6%). Finally, the RMS of the SLR residuals of 4.37 to 4.88 cm was achieved for BeiDou-3 POD.

Demonstrating developments in high-fidelity analytical radiation force modelling methods for spacecraft with a new model for GPS IIR/IIR-M

This paper presents recently developed strategies for high-fidelity, analytical radiation force modelling for spacecraft. The performance of these modelling strategies is assessed using a new model for the Global Positioning System Block IIR and IIR-M spacecraft. The statistics of various orbit model parameters in a full orbit estimation process that uses tracking data from 100 stations are examined. Over the full year of 2016, considering all Block IIR and IIR-M satellites on orbit, introducing University College London’s grid-based model into the orbit determination process reduces mean 3-d orbit overlap values by 9% and the noise about the mean orbit overlap value by 4%, when comparing against orbits estimated using a simpler box-wing model of the spacecraft. Comparing with orbits produced using the extended Empirical CODE Orbit Model, we see decreases of 4% and 3% in the mean and the noise about the mean of the 3-d orbit overlap statistics, respectively. In orbit predictions over 14-day intervals, over the first day, we see smaller root-mean-square errors in the along-track and cross-track directions, but slightly larger errors in the radial direction. Over the 14th day, we see smaller errors in the radial and cross-track directions, but slightly larger errors in the along-track direction.

Accounting for perturbing forces acting on Galileo using a box-wing model

In November 2017, the European Global Navigation Satellite System Agency (GSA) released geometrical and optical information for the Galileo satellites which allowed for the composition of a box-wing model whose main goal is absorption of the direct solar radiation pressure, earth’s albedo, and infrared radiation. In order to evaluate the efficiency of the box-wing model, we test solutions based solely on the empirical models, the pure analytical box-wing model, and a series of hybrid models including the box-wing with different sets of additionally estimated empirical parameters. The hybrid solution, which is based on the box-wing model and on a reduced number of estimated empirical parameters, substantially reduces variabilities of the satellite laser ranging (SLR) residuals, especially for the low elevation angles of the sun above the orbital plane (β), i.e., for eclipsing Galileo satellites. The standard deviation of SLR residuals for |β| < 12.3° decreases from 37 to 25 mm between the solution based on the ECOM2 and the hybrid solution, respectively. We found significant mitigation of the spurious geocenter signal in the Z component and its formal errors, when reducing the number of estimated empirical parameters, and a substantial reduction of the dependency between geocenter coordinates, the geometry of Galileo orbital planes, and the position of the sun. The hybrid box-wing solution with a reduced set of empirical parameters provides thus the best solution for precise orbit determination, orbit predictions, and estimation of geodetic parameters.