The exponential growth of mega-constellation satellites, typified by SpaceX’s Starlink, poses unprecedented challenges for existing space surveillance, particularly when tracking uncooperative spacecrafts executing continuous orbit-raising/deorbiting maneuvers. This situation makes the conventional orbit determination (OD) and prediction (OP) struggle with three critical issues: insufficient observational data, unknown maneuvering parameters, and the cumulative effects of unmodeled thrust. To address these issues, this study proposes a piecewise estimation-based OD method by developing a semi-analytical thrust acceleration (TA) model. The TA model employs time-explicit polynomial expansions with state-dependent coefficients to characterize the continuous low-thrust effect. The OD system integrates a piecewise least-squares estimation algorithm with dynamic compensation, enabling accurate TA resolution within one-day observation windows. Specifically, the OP system incorporates the latest TA estimate to account for the future continuous low-thrust effect. Experiments with sparse radar observations of Starlink satellites demonstrate the effectiveness of the proposed method. The TA estimation errors remain below 0.5% relative to the reference obtained from precise ephemerides. The OP capabilities maintain one-day and two-day position accuracy below 2 and 4 km, respectively, improving by more than 60% compared to the unified OD method. More importantly, the approach exhibits operational robustness, achieving OD convergence with initial TA errors up to 35%. These advantages make the proposed approach a practicable solution for autonomous catalog maintenance of maneuvering spacecraft. • A piecewise estimation-based orbit determination (OD) framework is developed for uncooperative spacecrafts with continuous low-thrust maneuver. • The continuous low-thrust effect is modeled by a semi-analytical expansion comprised of constant, linear, and nonlinear terms related to the spacecraft’s state, with the relative estimation errors of thrust acceleration (TA) below 0.5% under sparse tracking data condition, compared to those from the precise ephemerides. • Accurate orbit prediction (OP) is achievable by incorporating the latest TA estimate into the orbit dynamics system, with the one-/two-day position prediction error better than 2 km/4 km, respectively, for the maneuvering Starlink satellites. • The developed piecewise estimation-based OD framework demonstrates high robustness, achieving convergence with the initial TA errors up to 35%.

Integrated precise orbit determination of GPS and LEO satellites with an optimized integer ambiguity resolution strategy: multiple baselines with soft constraints

Autonomous optical flow-based initial orbit determination

Abstract This study presents a comprehensive evaluation of five widely used Solar Radiation Pressure (SRP) models applied to precise orbit determination (POD) using L-band ground tracking data and joint L-band with Ka-band Inter-Satellite Link (ISL) observations. The evaluated models are ECOM1, ECOM2, the Adjustable Box-Wing (ABW) model, and ECOM1/ECOM2 augmented with a box-wing a priori model (BW+ECOM1, BW+ECOM2). Through rigorous validation via Satellite Laser Ranging (SLR) residuals, SRP acceleration analysis, orbit prediction, and clock offset fitting residuals, key limitations and strengths of each model are identified. Results demonstrate that empirical models like ECOM1 and ECOM2 alone exhibit significant deficiencies, particularly for satellites with complex structures (e.g., C45/C46). These deficiencies manifest as degraded radial accuracy and large SLR residuals (up to 30 cm) correlated with Sun elevation angle (β) due to unmodeled 1-cpr signals in the Sun direction during high β angles. Augmenting ECOM1 with a box-wing model (BW+ECOM1) yields optimal performance while requiring the fewest estimated parameters across all metrics in L-band POD solutions. Specifically, this scheme achieves SLR residuals Root Mean Square (RMS) of 4.1 cm for China Academy of Space Technology (CAST) satellites and 2.7 cm for Shanghai Engineering Center for Microsatellites (SECM) satellites, 1-day 3D orbit prediction accuracy better than 25 cm, and clock residuals RMS of 0.11 ns. The integration of Ka-band ISL observations significantly improves the SLR validation, particularly for models with numerous parameters (e.g., ABW), with SLR residuals RMS reduction up to 35%. The L/Ka-band POD achieves optimal 3D prediction accuracies of 13.9 cm for the BW+ECOM2 scheme. The L/Ka-band POD reduces clock residuals to an RMS of 0.1 ns, regardless of SRP model selection. This work provides critical insights and practical references for selecting optimal SRP modeling strategies in BDS-3 POD.

Near-Earth Asteroid orbit determination with Physics-Informed Extreme Learning Machine

Autonomous real-time precise orbit determination for LEO satellite based on multi-GNSS receiver: Galileo, BDS-3, and GPS

Since coming into full operation in 2020, the BeiDou-3 Navigation Satellite System (BDS-3) has provided global users with positioning, navigation and time-synchronization services. Satellite clock bias is a key factor that affects real-time precise point positioning (PPP), precise orbit determination and the optimization of navigation message parameters; high-precision prediction of clock bias is therefore critical for improving the accuracy and reliability of BDS-3. To further enhance the prediction accuracy and stability of satellite clock bias, we propose a hybrid model based on Mamba-LSTM. This combined model leverages the strengths of the Multimodal Adaptive Model Building Algorithm (Mamba) and the Long Short-Term Memory neural network (LSTM) to predict satellite clock bias. Using precise BDS-3 satellite clock bias data from the International GNSS Service (IGS), we carried out prediction experiments. First, we compared the proposed model’s predictive performance with that of the Mamba and LSTM models. In short-term (6 h) and long-term (24 h) prediction scenarios, the average prediction RMSE of Mamba-LSTM improved by approximately 41.7% and 48% relative to Mamba, and by approximately 50.4% and 54.7% relative to the LSTM results, respectively. Next, we ran comparison experiments against traditional neural networks—the BP model and the CNN model. In mid-term (12 h) and long-term (24 h) prediction scenarios, the average prediction RMSE of Mamba-LSTM improved by approximately 59.6% and 63.1% compared with BP, and by approximately 52.4% and 56.2% compared with CNN, respectively. The results indicate that the Mamba-LSTM hybrid model can significantly improve the accuracy and stability of satellite clock bias prediction.

ABSTRACT Precise orbit determination of near-Earth asteroids (NEAs) provides a critical foundation for exploration missions. Considering that cislunar space is a critical domain for future human space activities and the unique positional advantage of the Earth-Moon L1 point, space-based optical observations from the Earth-Moon L1 point’s orbit hold significant potential for improving NEAs orbit determination accuracy. Coordinating such space-based observations with ground-based optical observations is expected to provide essential technical support for high-precision orbit determination. This study focused on two representative NEAs, 2016HO3 and Apophis. We validated the effectiveness of existing optical observations by processing 328 latest ground-based data of 2016HO3. The calculated position true error of the initial state vector was 31.932 km, with a position uncertainty of 24.293 km. The Yarkovsky effect parameter A 2 of 2016HO3 was solved to be (−1.257 ± 0.343)×10−13 AU/day2. Furthermore, we conducted ground-space based orbit determination simulation analyses for 2016HO3 and Apophis to quantitatively evaluate the potential advantages of combining optical observations from the Earth-Moon L1 point’s orbit with ground-based observations in enhancing orbit determination accuracy. The results indicate that within the observation accuracy considered in this study, space-based optical observations reduce orbit determination uncertainty by up to 20% for 2016HO3 and 62% for Apophis. The inclusion of space-based observations reduces the position true error by up to 50% for 2016HO3 and 83% for Apophis, and it can enhance the signal-to-noise ratio (SNR) of A 2. Additionally, we found that the orbit error of the L1 point probe has a notable influence on the position true error of NEAs, reaching several kilometers for 2016HO3 and several hundred meters for Apophis.

Objective: This paper undertakes an analytical study of the shadow of the Schwarzschild black hole. Method: In order to accomplish this objective, it is necessary to emphasize the definition of a black hole, its primary characteristics, and its significance in the field of cosmology. In this case, the shadow of an object is defined as the shape that the object possesses in space, as perceived by an observer when considering solely the properties of spacetime. Results: The analytical solution is derived using the Hamilton-Jacobi equation, the method of separation of variables, and the introduction of a Carter-type conserved quantity. Conclusions: The determination of constant-radius orbits is achieved through the utilization of impact parameters and celestial coordinates.

With the growing demand for real-time Low Earth Orbit (LEO) satellite clock products for LEO-augmented Positioning, Navigation, and Timing (PNT) services, efficient filter-based LEO satellite clock determination followed by an ultra-short clock prediction is the most thought-after approach. While the accuracy of LEO real-time products is paramount, it is equally important to apply Integrity Monitoring (IM) techniques to ensure their reliability in practical applications. In this study, an IM algorithm for filter-based LEO satellite clock and orbit determination is proposed, taking into consideration the bias propagation. The algorithm is tested using Sentinel-3B onboard GNSS observations together with the National Centre for Space Studies (CNES) real-time products. Key factors influencing the clock (and orbit) Protection Levels (PLs), including various errors in the observation domain like the observation noise, the GNSS real-time product noise and biases, are analyzed. The results demonstrated that bias propagation plays a dominant role in the computed PLs. When assuming an integrity risk of 1 × 10⁻ 5 , increasing the overbounding Standard Deviation (STD) of the zenith-referenced phase noise and GNSS real-time Signal-In-Space Ranging Errors (SISREs) from 0.005 m to 0.05 m only raises the LEO satellite clock PLs from the ns-level to less than 20 ns, and the orbital PLs from the decimeter to meter level. However, when the bias propagation and accumulation effects of the real-time GNSS products are considered, e.g., assuming an overbounding bias of 0.02 m, the clock PLs can increase to beyond 50 ns. This demonstrates the significant contribution of the bias propagation to filter-based LEO satellite clock and orbital PLs.

Kinematic orbit determination for BDS-3 satellites with inter-satellite link data

Improvement of Generalized Laplacian method of initial orbit determination with Maximum Likelihood constraint

Calibration of spaceborne accelerometer through satellite dynamic precise orbit determination: Enhancements and evaluations

The FOCUS mission is an integrative project developed at the Universidad Nacional de San Martín (UNSAM), Argentina, featuring a constellation of small satellites equipped with X-band Synthetic Aperture Radar (SAR) sensors. Designed with autonomous orbit control, the mission enables Interferometric SAR (InSAR) applications for critical infrastructure monitoring, providing scalable and cost-effective global observation capabilities. This paper presents the modeling, design, and numerical evaluation of the Attitude and Orbit Determination and Control System (AODCS) for the FOCUS mission. The analysis incorporates realistic constraints, including actuator saturation, sensor noise, underactuation effects, and hardware limitations—specifically regarding magnetorquer magnetic moments, reaction wheel capacities, and propulsion unit impulse bounds. Utilizing the NASA 42 attitude and orbit simulator, numerical simulations were conducted to assess stability, pointing accuracy, and agile maneuver tracking through specialized guidance laws. The results confirm that the proposed AODCS architecture achieves stable, responsive performance and supports continuous orbit maintenance, ensuring adequate target acquisition per orbit. Additionally, the selection of star trackers allows achieving a secondary objective through the detection of Resident Space Objects.

Abstract LEO satellites will enhance the GNSSs in future Positioning, Navigation, and Timing services. In addition to achieving high orbital accuracy of the LEO satellites, ensuring the reliability and safety of the Near-Real-Time LEO satellite Precise Orbit Determination is equally essential for maintaining integrity in LEO-augmented positioning and timing, yet it is less studied. Compared to the favorable data conditions of scientific LEO satellites in post-processing mode, GNSS observations collected by navigation-oriented LEO satellites may encounter significant discontinuities due to not only tracking problems, but also transmission delays and interruptions within potential data downlinking for NRT ground-based POD, posing challenges to its Integrity Monitoring. In particular, the Protection Level during observation gaps may encounter large increases and strongly harm the IM availability. This study investigates the algorithm calculating the NRT LEO satellite PLs under various observation gaps, followed by different lengths of observation tails based on the reduced-dynamic model. The strategy estimating (RD) and not estimating (CD) piecewise-constant stochastic accelerations in the POD process are both tested. While the former benefits the POD accuracy in complete data conditions, the latter exhibited its advantage in reducing the PLs within large data gaps. Results showed that during observation discontinuities, the RD strategy yields significantly degraded PLs in all three directions, which increase rapidly with gap duration. In the 9-hour gap case, the along-track PL exceeds 20 m, while under nominal conditions, it remains around 0.51 m. In contrast, the CD strategy achieves more stable PLs, i.e., below 2 m even during a 9-h gap, although its convergence after observation recovery is slower. Furthermore, the study derives suitable thresholds of alert limits to bound 99% of the PLs under various gap conditions, providing a reference for LEO integrity assessment under such conditions.

Abstract In the space-based gravitational wave detection of the TianQin mission, gravitational perturbations from the Earth–Moon system induce relative motion between the satellites ( ± 4 m s −1 ), leading to a Doppler effect in the inter-satellite laser links. In heterodyne laser interferometry, the Doppler shift causes the beatnote frequencies to drift slowly over time, potentially exceeding the phase read-out bandwidth and resulting in loss of phase locking. Therefore, various frequency plans have been proposed to lock all lasers to a primary laser, which ensures that the beatnote frequencies remain within the phase read-out bandwidth throughout missions. Frequency plans for TianQin must take into account multiple clock sidebands, prevent beatnote signal crosstalk, and maintain frequency separations of 0.5 MHz. On top of the straightforward fixed-offset scheme, this paper proposes a Doppler-compensated scheme, where the frequency offsets are dynamically adjusted according to the Doppler shift, utilizing the precision orbit determination and prediction capability of TianQin. In contrast to the fixed-offset scheme, the Doppler-compensated scheme can halve the variation range of the beatnote frequencies and provides sufficient margin for the phase read-out bandwidth for all 36 laser locking configurations considered in the paper.

<p indent="0mm">This dataset is derived from the next-generation VLBI Global Observing System (VGOS) using its intensive session mode VGOS-INT-A, which provides high-precision time-series data for determining the Earth rotation parameter dUT1. The dataset consists of 204 standardized files in the vgosDb format, with dUT1 solutions exhibiting high timeliness and accuracy. Data acquisition was conducted through the global VGOS station network, utilizing inter-hemispheric ultra-long baselines, broadband signal reception technologies, high-speed sampling, and coordinated multi-band observation strategies. After systematic error correction and radio frequency interference suppression, data were processed using the DiFX software correlator to generate the final dUT1 solution dataset for VGOS-INT-A intensive sessions. External validation against the IERS C04 reference series shows a standard deviation (STD) of <sc>0.048 μs</sc> for the dUT1 values in this dataset, representing a nearly threefold improvement in precision compared to traditional S/X-band VLBI data. This dataset supports near-real-time monitoring of Earth rotation parameters, deep-space spacecraft orbit determination, and crustal deformation analysis, providing critical data for the establishment of high-precision spatiotemporal reference frames and deep-space navigation.

Optimal Cislunar Optical Observation Constellations Design Using Orbit Determination Accuracy Estimate

Abstract The ESA GOCE satellite carried a gravity gradiometer consisting of three pairs of accelerometers on mutually orthogonal axes. For each accelerometer, bias and scale factors have been re-estimated by a dynamic precise orbit determination (POD) using improved gravity field modeling and standards. The kinematic orbit solution included in GPS-based Precise Science Orbit (PSO) product served as the baseline observables for 1210 daily arcs, covering the period from 1 November 2009 to 20 October 2013. Implementing improved force models almost completely resolved the deviations of the Y -axis scale factor obtained in earlier work (Visser and Ijssel 2016). A novel aspect is the verification by comparison with dynamic POD solutions based on SLR observations using 51 two-day orbital arcs. A high level of consistency was obtained between the kinematic PSO- and SLR-based accelerometer calibration parameters, e.g. within 0.01 nm/s $$^{ 2}$$ 2 for the X -axis pointing predominantly in the flight direction in terms of bias. One set of accelerometer scale factors was estimated for the entire mission. These were found to be consistent to within 0.005 for all accelerometer axes. The three-dimensional consistency between the dynamic orbits and the PSO reduced-dynamic orbit solutions has a mean Root-Mean-Square (RMS) of 4.5 and 10 cm, respectively, for the PSO reduced-dynamic and SLR-based dynamic orbit solutions. In addition, the one-dimensional RMS-of-fit of the PSO kinematic orbit solution improved significantly from 6.9 in Visser and Ijssel (2016) to 2.6 cm.

Evaluation of real-time orbit determination performance for LEO satellites using different empirical acceleration models