Orbit Accuracy Research Articles

Analytical expressions for orbital perturbations due to Lorentz force

The selection of orbit propagation models strongly affects various space situational awareness applications like future object population prediction, collision avoidance maneuvers, object detection, object tracking, and catalog maintenance. Two key considerations in the selection of orbit propagation models are accuracy and computational cost. While one can use accurate numerical orbit propagation techniques for short propagation periods, it can be computationally taxing when the propagation period is long and/or when a large number of objects are to be propagated. This can be remedied by using analytical orbit propagation techniques, which not only facilitate fast orbit propagation but also permit meaningful theoretical insight into the structure of the perturbations. Most previous analytical object propagation in the near-Earth realm has been carried out in the presence of zonal and tesseral harmonics of Earth’s gravity, Sun and Moon gravities, atmospheric drag, and solar radiation pressure. However, for high-energy plasma weather and under eclipse where objects can get highly charged, Lorentz force should be included for better accuracy in orbital propagation. Lorentz force should also not be ignored for high-area-to-mass ratio objects. Among the work that has taken Lorentz force into consideration, most have made the simplifying assumption of non-tilted Earth magnetic dipole or have developed analytical formulas valid only for low Earth orbit region. In this paper, new analytical formulas for Lorentz perturbations are developed using a tilted dipole Earth magnetic field and a constant charge-to-mass ratio. The derived analytical formulas are valid for all near-Earth altitudes. Two sets of formulas are derived, one for small eccentricities and one for large eccentricities. The analytical results are validated using numerical simulations. A comparative study is also carried out using different charging levels and area-to-mass ratios for the low Earth orbit and geosynchronous Earth orbit regions.

Multiconstrained Ascent Trajectory Optimization Using an Improved Particle Swarm Optimization Method

This paper researches the ascent trajectory optimization problem in view of multiple constraints that effect on the launch vehicle. First, a series of common constraints that effect on the ascent trajectory are formulated for the trajectory optimization problem. Then, in order to reduce the computational burden on the optimal solution, the restrictions on the angular momentum and the eccentricity of the target orbit are converted into constraints on the terminal altitude, velocity, and flight path angle. In this way, the requirement on accurate orbit insertion can be easily realized by solving a three-parameter optimization problem. Next, an improved particle swarm optimization algorithm is developed based on the Gaussian perturbation method to generate the optimal trajectory. Finally, the algorithm is verified by numerical simulation.

Galileo real-time orbit determination with multi-frequency raw observations

Real-time GNSS-based applications require corresponding real-time orbit products. While traditional GNSS orbits are generated with the dual-frequency IF (Ionosphere-Free) model, the increase of multi-frequency signal satellites brings new challenges for the data processing. Therefore, real-time orbit determination with the multi-frequency UC (Uncombined) model is introduced in this study considering its flexibility. With the derived mathematical model conforming to IGS (International GNSS Service) dual-frequency clock definition and one-week triple-frequency Galileo observation data from 90 IGS network stations, the convergence and accuracy of real-time orbits is assessed and the characteristics of satellite IFCB (Inter-Frequency Clock Bias) are analyzed. Results indicate that the model differences, including dual-frequency IF model, dual-frequency UC model and triple-frequency UC model, contribute to only cm-level differences with CODE (Center for Orbit Determination in Europe) final orbits after a convergence time of around 12 h. The constellation-mean RMS (Root Mean Square) differences of the converged real-time orbits with the CODE final orbits reaches about 5.0 cm, 7.0 cm and 5.0 cm for the radial, tangential and normal directions. The convergence of satellite IFCB is much faster than that of satellite orbit, which reflects a loose correlation between these two parameters. While the Galileo satellite IFCB are temporally stable, the modeling of satellite IFCB may be unreliable when over constrained and becomes even more unstable with commonly encountered datum changes. In summary, real-time GNSS orbit determination with multi-frequency raw observations is feasible and extendable with proper treatment of IFCB.

Higher order ionospheric error correction in BDS precise orbit determination

The ionospheric error affects the accuracy of the Global Navigation Satellite Systems observation and precise orbit determination. Usually, only the first order ionospheric error is considered, which can be eliminated by the ionospheric-free linear combination observation. But the remaining higher order ionospheric error will affect the accuracy of observations and their applications. In this paper, the influence of the higher order ionospheric error have been studied by using the International Geomagnetic Reference Field 13 and the Global Ionosphere Maps model produced by the Center for Orbit Determination in Europe. Focus on ionospheric error, the experiment of paper at doy 302 in 2019, which show that the second order ionospheric error impacting BeiDou Navigation Satellite System (BDS) B1I and B3I observation is 6.3569 mm and 11.8484 mm, respectively. Whereas, the third order ionospheric error impacting BDS B1I and B3I observation is 0.1734 mm and 0.3977 mm, respectively. Due to the current measurement accuracy of BDS carrier-phase observation can reach 2 mm, the influence of high order ionospheric error on observation should be considered. For BDS precise orbit determination, the orbit overlapping results are indicated that its orbit accuracy can be improved approximately 5 mm with the higher order ionospheric error correction, which is also in agreement with the results of Satellite Laser Ranging in this work.

Assessment of single-difference and track-to-track ambiguity resolution in LEO precise orbit determination

Single-difference (SD) ambiguity resolution (AR) and track-to-track (T2T) AR are two typical AR methods in precise orbit determination (POD) for Low Earth Orbit (LEO) satellites, which could improve the accuracy of orbits greatly. In this study, SD AR and T2T AR methods are introduced and analyzed. The performance of these two methods is assessed by three months of GPS observations from the Gravity Recovery and Climate Experiment Follow On (GRACE-FO) twin satellites. Results show that T2T AR is highly dependent on the stability of receiver hardware delays, while SD AR requires Fractional Cycle Bias (FCB) or Integer Recovery Clock (IRC) products. We find that these two methods have comparable performance in Reduced Dynamic Precise Orbit Determination (RDPOD), while SD AR slightly outperforms T2T AR in Kinematic Precise Orbit Determination (KPOD). We also find that SD AR has a higher AR success rate than T2T AR. Therefore, we recommend SD AR as the top choice in LEO orbit determination, and T2T AR can be a good alternative when FCB or IRC products are not available.

Precise Orbit Determination of MEX Flyby Phobos Using Simulated Radiometric and Image Data.

A navigation camera or topography camera is a standard payload for deep space missions and the image data are normally used for auto-navigation. In this work, we study the potential contribution of image data in precise orbit determination for deep space spacecraft. The Mars Express (MEX) spacecraft has generated extensive Phobos image data during flybys of Phobos, but these data have not been used in precise orbit determination because of the difficulty in employing these image data. Therefore, we did an experiment using simulated image data as the first step for exploring how to use real image data in precise orbit determination of spacecraft. Our results demonstrate that image data can provide stronger constraints on orbit in the tangential and normal directions than Doppler data. When the image data were used in the MEX orbit determination during the MEX Phobos flyby, the orbit determination accuracies in the tangential and normal directions were significantly improved. This work will provide a reference for real image data processing during MEX Phobos flyby to improve MEX orbit accuracy as well as Phobos ephemeris accuracy.

The influence of tesseral harmonic $J_{22}$ term on the TLE orbit error of low polar orbit satellite

By expanding the perturbation function of tesseral harmonic $J_{22}$ term of the earth’s non spherical gravitation, it is found that there is a term $\cos 2\Omega _{e}$ related to the rotation of the earth. The period of the term is half a day, and the coefficient includes the factor $\sin ^{2}i$ related to the orbital inclination angle. For the low polar orbit satellite with orbit inclination close to $90^{\circ }$ , $J_{22}$ term has a great impact on the orbit accuracy, which will reach the order of kilometers. Based on the measured orbit data (historical TLEs and GPS precise ephemeris) of the polar satellite GRACE-A/B, the influence of the tesseral harmonic $J_{22}$ term on TLE orbit error is further verified. It is found that the system bias caused by ignoring $J_{22}$ term exists in TLE orbit data for LEO, which shows a periodic change of 180 degrees with the geographic longitude in the in-track direction. In addition, the accuracy of TLE orbit in the northern hemisphere is significantly higher than that in the southern hemisphere, which is related to the distribution of SSN observation network. For GRACE-A satellite, the maximum system bias of TLE orbit can reach close to 1.5 km near the equator of the earth, and 8 km after a 7-day prediction period.

Change Detection Using Time and Look Angle Series of Geocoded Tandem-X Staring Spotlight Images

This article describes the analysis of changes in time and look angle series of TanDEM-X (TDX) and TerraSAR-X (TSX) staring spotlight images over the regions around Kjeller near Oslo in Norway and the town Nikel in Russia near the Norwegian border. It is shown that change detection may be done with TDX/TSX with shorter intervals than the repeat cycle using time series of geocoded images with varying look angles. Using look angle series of geocoded images can help to identify changes due to look angle variation. Accurate science orbit data and very good overlap of scenes both in time and look angle make the TDX/TSX data very attractive for both coherent and incoherent change detection.

Acquisition of Positional Accuracy with Comparative Analysis of GPS and EGNOS in Urban Constituency

Over the years, precise positioning has been the ultimate goal for Satellite Navigation Systems. The American Global Navigation Satellite System deliver the position and time information intended for various sectors such as vehicle tracking, oil exploration, atmospheric studies, astronomical telescope pointing, airport and harbor security tracking etc. Corresponding technological competitors such as Russian Global Navigation Satellite System (GLONASS), European Union’s GALILEO, China’s BeiDou and Japanese Quasi Zenith Satellite System (QZSS) are few other versions of Satellite based Augmentation Systems. Nevertheless, stern security measures, geographical statistics and stimulation of diverse Electronic Gadgets at indoor/outdoor surroundings make it critical to acquire data about any vicinity with seamless accessibility, accuracy and integrity with satellite links. In this paper, positional accuracy has been tested with analysis of EGNOS, EDAS and simple GPS receiver models at Rome City, Italy. To support results, various real time experiments/tests has been performed with GPS Receiver SIRF Demo software. The test was conducted on-board a car by installing a laptop equipped with GPS Receiver plus supportive SBAS (EGNOS particularly) through three diverse bus routes of locality and outcomes of few tested samples inside the Rome City center are specified to check the availability of desired satellite signals. Subsequently, comparative analysis has been executed between the simple GPS data received and GPS + EGNOS data collected during daytime traffic. The strength of test signals reveals accuracy of EGNOS in open terrain area with less congestion. Furthermore, Asian and European Advanced GPS systems are compared in terms of performance as well as feasibility of authentic, accurate and swift satellite navigation systems.

Discussion of different Remote sensing satellite possibilities for scientifical Earth observations

More than 1,000 satellites are launched into space, and they differ in their functions, rotation orbits, resolution, and other properties. Scientists divide the satellites into low-resolution, medium-resolution, high-resolution, and very high-resolution satellites by their properties. Now, the biggest challenge facing scientists is to use some of these different resolution images in their field. To get the expected result, it is very important to analyze the image that needs an which gives more accurate results. Therefore, the main attention of this article is aimed to find the answer to these problems. In this article 3 satellite images which have different resolution are analyzed. The possibility of middle-resolution images of MODIS, high-resolution images of Landsat, and very high-resolution images of WorldView-2 (WV-2) satellites using GIS are analyzed. A research area was the Syrdarya region, and downloaded different images of satellites of this area and compared with using e Cognition. According to the results, a more accurate satellite image for irrigation sets information is WorldView-2 images. In comparison analysis, it shows more accurate properties than other satellite images. As irrigation sets are small objects for the analysis, very high spatial resolution satellite images are important. Water discharge and surface change happen very fast; thus, it requires daily monitoring of the condition. And in this case, the temporal resolution of the MODIS and Landsat is 16 day, and it is a too long period.

Sounding the Atmospheric Density at the Altitude of LARES and Ajisai during Solar Cycle 24

During Solar Cycle 24, the passive spherical satellites LARES and Ajisai, placed in nearly circular orbits with mean geodetic altitudes between 1450 and 1500 km, were used as powerful tools to probe the neutral atmosphere density and the performances of six thermospheric models in orbital regimes for which the role of dominant atomic species is contended by hydrogen and helium, and accurate satellite measurements are scarce. The starting point of the analysis was the accurate determination of the secular semi-major axis decay rate and the corresponding neutral drag acceleration in a satellite- centered orbital system. Then, for each satellite, thermospheric model and solar activity level, the drag coefficients capable of reproducing the orbital decay observed were found. These coefficients were finally compared with the physical drag coefficients computed for both satellites in order to assess the biases affecting the thermospheric density models. None of them could be considered unconditionally the best; the specific outcome depending on solar activity and the regions of the atmosphere crossed by the satellites. During solar maximum conditions, an additional density bias linked to the satellite orbit inclination was detected.

An analysis of BDS-3 real-time PPP: Time transfer, positioning, and tropospheric delay retrieval

With the development of China’s BDS-3, more studies focused on its performance in precise point positioning (PPP), mainly in post-processing mode. This study conducts a comprehensive investigation of BDS-3 real-time PPP with Centre National d’Etudes Spatiales (CNES) real-time products (SSRA00CNE0) from the perspective of time transfer, positioning, and tropospheric delay retrieval. First, the availability and accuracy of BDS-3 real-time products were evaluated over 41 days. The product availability was about 80%. The accuracy of BDS-3 satellite orbit were 0.060, 0.115 and 0.065 m in the radial, along-track, and cross-track component. In addition, the standard deviation (STD) of satellite clock difference was 0.455 ns. Afterwards, precise time transfer, positioning, and Zenith troposphere delay (ZTD) estimation derived from BDS-3 real-time PPP were assessed with 7-day processing. The root mean square (RMS) of smoothed residuals is 0.020 ns, while it is 0.001 ns for BDS-3 PPP with WUM final products for time transfer. Furthermore, the RMS values of positioning errors derived from real-time static BDS-3 PPP positioning were 0.021, 0.010, and 0.03 m in the east (E), north (N), and up (U) component, respectively. Moreover, the RMS of ZTD estimation is 9.9 mm, which is comparable to GPS real-time PPP solutions.

Long-term orbit dynamics of decommissioned geostationary satellites

In nominal mission scenarios, geostationary satellites perform end-of-life orbit maneuvers to reach suitable disposal orbits, where they do not interfere with operational satellites. This research investigates the long-term orbit evolution of decommissioned geostationary satellite under the assumption that the disposal maneuver does not occur and the orbit evolves with no control. The dynamical model accounts for all the relevant harmonics of the terrestrial gravity field at the typical altitude of geostationary orbits, as well as solar radiation pressure and third-body perturbations caused by the Moon and the Sun. Orbit propagations are performed using two algorithms based on different equations of motion and numerical integration methods: (i) Gauss planetary equations for modified equinoctial elements with a Runge-Kutta numerical integration scheme based on 8–7th-order Dorman and Prince formulas; (ii) Cartesian state equations of motion in an Earth-fixed frame with a Runge-Kutta Fehlberg 7/8 integration scheme. The numerical results exhibit excellent agreement over integration times of decades. Some well-known phenomena emerge, such as the longitudinal drift due to the resonance between the orbital motion and Earth's rotation, attributable to the J22 term of the geopotential. In addition, the third-body perturbation due to Sun and Moon causes two major effects: (a) a precession of the orbital plane, and (b) complex longitudinal dynamics. This study proposes an analytical approach for the prediction of the precessional motion and show its agreement with the (more accurate) orbit evolution obtained numerically. Moreover, long-term orbit propagations show that the above mentioned complex longitudinal dynamics persists over time scales of several decades. Frequent and unpredictable migrations toward different longitude regions occur, in contrast with the known effects due only to the perturbative action of J22.

Energy behavior of Boris algorithm

Boris numerical scheme due to its long-time stability, accuracy and conservative properties has been widely applied in many studies of magnetized plasmas. Such algorithms conserve the phase space volume and hence provide accurate charge particle orbits. However, this algorithm does not conserve the energy in some special electromagnetic configurations, particularly for long simulation times. Here, we empirically analyze the energy behavior of Boris algorithm by applying it to a 2D autonomous Hamiltonian. The energy behavior of the Boris method is found to be strongly related to the integrability of our Hamiltonian system. We find that if the invariant tori is preserved under Boris discretization, the energy error can be bounded for an exponentially long time, otherwise the said error will show a linear growth. On the contrary, for a non-integrable Hamiltonian system, a random walk pattern has been observed in the energy error.

Hydrodynamics of Underwater «Physical Explosion»

The modern scientific and technological revolution has led to profound changes in the technique and technology of marine seismic research, ensuring an increase in labor productivity, a significant increase in the volume of work, as well as an increase in their efficiency. At the same time, along with the use of floating piezoelectric seismograph, digital recording equipment, modern electronic computing equipment and more accurate satellite navigation systems, one of the important factors that contributed to improving the efficiency of seismic exploration was the introduction of a new generation of seismic signal excitation devices – non-explosive sources – into the practice of marine seismic research. Of these, the most widely used in seismic exploration in water areas around the world are pneumatic sources, in which elastic waves are excited by underwater exhaust of compressed air.

An Improved Multi-Satellite Method for Evaluating Real-Time BDS Satellite Clock Offset Products

Two methods are widely used for evaluating the precision of satellite clock products, namely the single-satellite method (SSM) and the multi-satellite method (MSM). In the satellite clock product evaluation, an important issue is how to eliminate the timescale difference. The SSM selects a reference satellite to eliminate the timescale difference by between-satellite differencing, but its evaluation results are susceptible to the gross errors in the referenced satellite clock offsets. In the MSM, the timescale difference is first estimated and then removed. Unlike the GPS, the BeiDou Navigation Satellite System (BDS) consists of three types of satellites, namely geosynchronous earth orbit (GEO), inclined geosynchronous orbit (IGSO), and medium earth orbit (MEO) satellites. The three types of satellites have uneven orbital accuracy. In the generation of satellite clock products, the orbital errors are partly assimilated into the clock offsets. If neglecting the orbital accuracy difference of the three types of BeiDou satellites, the MSM will obtain biased estimates of the timescale difference and finally affect the clock product evaluation. In this study, an improved multi-satellite method (IMSM) is proposed for evaluating the real-time BDS clock products by removing the assimilated orbital errors of the three types of BDS satellites when estimating the timescale difference. Three real-time BDS clock products disseminated by three different International GNSS Service (IGS) analysis centers, namely CLK16, CLK20, and CLK93, over a period of two months are used to validate this method. The results indicate that the assimilated orbital errors have a significant impact on the estimation of the timescale difference. Subsequently, the IMSM is compared with the SSM in which the referenced satellite is rigorously chosen, and their RMS difference is only 0.08 ns, which suggests that the evaluation results obtained by the IMSM are accurate. Compared with the traditional MSM, the IMSM improves the RMS by 0.16, 0.11, and 0.07 ns for CLK16, CLK20, and CLK93, respectively. Finally, three real-time BDS clock products are evaluated using the proposed method, and results reveal a significant precision difference among them.

On the correctness of orbital solutions obtained from a small set of points. Orbit of HIP 53731

HIP 53731 is a binary consisting of stars of the spectral types K0 and K9. The orbit of this object was constructed previously by Cvetković et al. and improved by Tokovinin. It should be noted that there is a 180° ambiguity in the position angles of some published measurements. Speckle interferometric observations were obtained in 2007–2020 (21 measurements) at the 6-m telescope of the SAO RAS (BTA) by the authors of this article. The analysis of new data together with previously published ones made it possible to construct an accurate orbit of HIP 53731 and to halve the already known values of the orbital period of the system. As a result of the study, the mass sum, the masses of each component and their spectral types were determined by two independent methods. According to the qualitative classification of orbits, the orbital solution has grade 2 – “good” (observations cover more than half of the orbital period and correspond to different phases).

On GPS data quality of GRACE-FO and GRACE satellites: effects of phase center variation and satellite attitude on precise orbit determination

As the gravity formation flying spacecraft jointly developed by National Aeronautics and Space Administration and German Research Centre for Geosciences, the Gravity Recovery and Climate Experiment (GRACE) and GRACE follow-on (GRACE-FO) satellites have adopted the same K-band ranging system and orbit design in order to detect the Earth’s gravity field information. Different from the BlackJack receiver onboard GRACE, a TriG GNSS receiver is loaded on the GRACE-FO satellites. The orbit determination accuracy of GRACE is better than 2.5 cm, but for GRACE-FO there is no comprehensive assessment of orbit accuracy. We discuss the quality of Global Positioning System data, as well as effects of phase center variation (PCV) model and attitude for GRACE and GRACE-FO. The results show that there is no significant difference in terms of rate of change in ionospheric delay (IOD) and multipath effect, which suggests that the performance of TriG receiver is as excellent as that of BlackJack receiver. After using PCV corrections, the root mean square (RMS) errors of kinematic and reduced-dynamic (RD) orbit residuals decrease by 0.4–0.5 and 0.6–0.9 mm, respectively. Satellite laser ranging RMS errors for RD orbit solutions are lower than 2.59 cm whether PCV corrections are used or not. The effect of attitude data on kinematic and RD orbits indicates that nominal attitude data can reliably replace measured attitude data in GRACE-FO orbit determination.

Investigating the GALILEO and BeiDou orbit accuracy derived from rapid products

In recent years, the advances of the new Global Navigation Satellite System (GNSS) constellations including, Galileo and BeiDou (BDS), have undergone dramatic changes. Some analysis centers (ACs) produce precise orbit and clock products of Galileo and BeiDou constellations. Currently, three types of Galileo and BeiDou satellite orbit and clock products are available – namely, precise, rapid and ultra-rapid products –. Ultra-rapid and rapid products are generally used for time-constrained applications. Precise orbit determination (POD) of Galileo and BeiDou is much challenging compared with GPS and GLONASS constellations due to the officially undetermined receiver phase center offset (PCO), variations (PCV) of Galileo and BeiDou constellations and, also some other not well-defined factors such as yaw-attitude models and solar radiation pressure. In this study, GALILEO orbit accuracy is investigated using rapid products produced by Center for Orbit Determination in Europe (CODE) GeoForschungsZentrum (GFZ) and Wuhan University (WUHAN), while GFZ and WUHAN rapid products are used for BeiDou constellation only. One month (January) of data in 2020 is used to compute errors of radial, along-track, and cross-track components of Galileo and BeiDou orbit derived by rapid products compared with the CODE final Multi-GNSS Experiment (MGEX) product which is assumed as the reference product. The results show that no significant differences between the products are found for Galileo orbit. For BeiDou orbit, WUHAN rapid product produced the smaller root mean square errors (RMSEs) of orbit components compared with the GFZ rapid product.

LEO Onboard Real-Time Orbit Determination Using GPS/BDS Data with an Optimal Stochastic Model

The advancements of Earth observations, remote sensing, communications and navigation augmentation based on low Earth orbit (LEO) platforms present strong requirements for accurate, real-time and autonomous navigation of LEO satellites. Precise onboard real-time orbit determination (RTOD) using the space-borne data of multiple global navigation satellite systems (multi-GNSS) becomes practicable along with the availability of multi-GNSS. We study the onboard RTOD algorithm and experiments by using America’s Global Positioning System (GPS) and China’s regional BeiDou Navigation Satellite System (BDS-2) space-borne data of the FengYun-3C satellite. A new pseudo-ambiguity parameter, which combines the constant phase ambiguity, the orbit and clock offset error of the GPS/BDS broadcast ephemeris in the line-of-sight (LOS), is defined and estimated in order to reduce the negative effect of the LOS error on onboard RTOD. The analyses on the variation of the LOS error in the GPS/BDS broadcast ephemeris indicate that the pseudo-ambiguity parameter could be modeled as a random walk, and the setting of the power spectral density in the random walk model decides whether the pseudo-ambiguity can be estimated reasonably and the LOS error could be reduced or not. For different types of GPS/BDS satellites, the LOS errors show different variation characteristics, so the power spectral density should be set separately and differently. A numerical search approach is presented in this paper to find the optimal setting of the power spectral density for each type of GPS/BDS satellite by a series of tests. Based on the optimal stochastic model, a 3-dimensional (3D) real-time orbit accuracy of 0.7–2.0 m for position and 0.7–1.7 mm/s for velocity could be achieved only with dual-frequency BDS measurements and the broadcast ephemeris, while a notably superior orbit accuracy of 0.3–0.5 m for position and 0.3–0.5 mm/s for velocity is achievable using dual-frequency GPS/BDS measurements, due to the absorption effect of the estimated pseudo-ambiguity on the LOS error of the GPS/BDS broadcast ephemeris. Compared to using GPS-alone data, the GPS/BDS fusion only marginally improves the onboard RTOD orbit accuracy by about 1–3 cm, but the inclusion of BDS satellites increases the number of the tracked GNSS satellites and thus speeds up the convergence of the filter. Furthermore, the GPS/BDS fusion could help suppress the local orbit errors, ensure the orbit accuracy and improve the reliability and availability of the onboard RTOD when fewer GPS satellites are tracked in some anomalous arcs.