Two-line Element Research Articles

ANALYSIS OF THE RE–ENTRY PHASE OF STARLINK–1353 SATELLITE

Eighty-three percent of all operational satellites are deployed in LEO, where Starlink is the largest mega-constellation with 5,233 satellites in working phase (data available for June 2024) (Planet4589, 2023). To date, SpaceX has launched the first 6,505 Starlink satellites out of 11,908 units planned (with a possible supplement of 22,488 units). One of the failed satellites was the Starlink–1353 (2020–025G) unit, for which the controlled decay above Hokkaido Island, on February 21, 2023, was in the field-of-view of one telescope from Kitasubaru Astronomical Observatory. Luckily, in that moment the telescope was performing observations on the Orion Nebula region, in a live streaming session (Kitasubaru, 2024). The major break-up was captured by the video camera of the telescope and based on the frame–by–frame image analysis, W identified 15 fragments after the atmospheric re–entry. According to our analysis based on Debris Assessment Software (DAS) (NASA), the major breakup occurred at 64.4 km above the sea level and some fragments were spread over an area of 15.13 m2. The paper presents hypothesis about the composition of the fragments that crossed the sky above Japan and if these can reach the sea level. In order to study the dynamical behaviour of Starlink–1353, before the moment of planned re–entry maneuvers, the observation recorded by Berthelot Observatory (IAU Code L54) were used. Orbital evolution of the satellite is presented using numerical integration for the available Two Line Elements (TLE) values. Using the last TLE, published five hours before re–entry, an estimated time 10:46 UTC, February 21, 2023 was computed.

Closing the gap between SGP4 and high-precision propagation via differentiable programming

The simplified general perturbations 4 (SGP4) orbital propagation model is one of the most widely used methods for rapidly and reliably predicting the positions and velocities of objects orbiting Earth. Over time, SGP models have undergone refinement to enhance their efficiency and accuracy. Nevertheless, they still do not match the precision offered by high-precision numerical propagators, which can predict the positions and velocities of space objects in low-Earth orbit with significantly smaller errors.In this study, we introduce a novel differentiable version of SGP4, named ∂SGP4. By porting the source code of SGP4 into a differentiable program based on PyTorch, we unlock a whole new class of techniques enabled by differentiable orbit propagation, including spacecraft orbit determination, state conversion, covariance similarity transformation, state transition matrix computation, and covariance propagation. Besides differentiability, our ∂SGP4 supports parallel propagation of a batch of two-line elements (TLEs) in a single execution and it can harness modern hardware accelerators like GPUs or XLA devices (e.g. TPUs) thanks to running on the PyTorch backend.Furthermore, the design of ∂SGP4 makes it possible to use it as a differentiable component in larger machine learning (ML) pipelines, where the propagator can be an element of a larger neural network that is trained or fine-tuned with data. Consequently, we propose a novel orbital propagation paradigm, ML-∂SGP4. In this paradigm, the orbital propagator is enhanced with neural networks attached to its input and output. Through gradient-based optimization, the parameters of this combined model can be iteratively refined to achieve precision surpassing that of SGP4. Fundamentally, the neural networks function as identity operators when the propagator adheres to its default behavior as defined by SGP4. However, owing to the differentiability ingrained within ∂SGP4, the model can be fine-tuned with ephemeris data to learn corrections to both inputs and outputs of SGP4. This augmentation enhances precision while maintaining the same computational speed of ∂SGP4 at inference time. This paradigm empowers satellite operators and researchers, equipping them with the ability to train the model using their specific ephemeris or high-precision numerical propagation data.

Satellite impact on astronomical observations based on the elliptical orbit model

Space-based and ground-based telescopes have extensively documented the impact of satellites on astronomical observations. With the proliferation of satellite mega-constellation programmes, their influence on astronomical observations has become undeniable. Quantifying the impact of satellites on telescopes is crucial. To this end, we enhanced the circular orbit model for satellites and introduced a methodology based on two-line element orbit data. This involves constructing a satellite probability distribution model to evaluate the impact of satellites on telescopes. Using our method, we assessed the satellite impact on global grounded observatories. Our results indicate that the regions most severely affected by satellite interference at present are those near the equator, with latitudes of around ±50 and ±80 degrees experiencing the most significant impact from low-Earth-orbit satellites. Furthermore, we validated the reliability of our method using imaging data obtained from the focal surface acquisition camera of the LAMOST telescope.

Two-Line Element Outlier and Space Event Detection Method Based on Multi-Strategy Genetic Algorithm

The detection of two-line element (TLE) outliers and space events play a crucial role in enhancing spatial situational awareness. Therefore, this paper addresses the issue of TLE outlier detection methods that often overlook the mutual influence of multiple factors. Hence, a Multivariate Gaussian Mixture Model (MGMM) is introduced to consider the interdependencies among various indicators. Additionally, a Multi-strategy Genetic Algorithm (MGA) is employed to adjust the complexity of the MGMM, allowing it to accurately learn the actual distribution of TLE data. Initially, the proposed method applies probabilistic fits to the predicted error rate changes for both the TLE semi-major axis and the orbital inclination. Chaos initialization, a posterior probability penalty, and local optimization iterations are subsequently integrated into the genetic algorithm. These enhancements aim to estimate the MGMM parameters, addressing issues related to poor robustness and the susceptibility of the MGMM to converge to local optima. The algorithm’s effectiveness is validated using TLE data from typical space targets. The results demonstrate that the optimized algorithm can efficiently detect outliers and maneuver events within complex TLE data. Notably, the comprehensive detection performance index, measured, using the F1 score, improved by 15.9% compared to the Gaussian mixture model. This significant improvement underscores the importance of the proposed method in bolstering the security of complex space environments.

Space Domain Awareness Observations Using the Buckland Park VHF Radar

There is increasing interest in space domain awareness worldwide, motivating investigation of the use of non-traditional sensors for space surveillance. One such class of sensor is VHF wind profiling radars, which have a low cost relative to other radars typically applied to this task. These radars are ubiquitous throughout the world and may potentially offer complementary space surveillance capabilities to the Space Surveillance Network. This paper updates an initial investigation on the use of Buckland Park VHF wind profiling radars for observing resident space objects in low Earth orbit to further investigate the space surveillance capabilities of the sensor class. The radar was operated during the Australian Defence “SpaceFest” 2019 activity, incorporating new beam scheduling and signal processing functionality that extend upon the capabilities described in the initial investigation. The beam scheduling capability used two-line element propagations to determine the appropriate beam direction to use to observe transiting satellites. The signal processing capabilities used a technique based on the Keystone transform to correct for range migration, allowing the development of new signal processing modes that allow the coherent integration time to be increased to improve the SNR of the observed targets, thereby increasing the detection rate. The results reveal that 5874 objects were detected over 10 days, with 2202 unique objects detected, representing a three-fold increase in detection rate over previous single-beam direction observations. The maximum detection height was 2975.4 km, indicating a capability to detect objects in medium Earth orbit. A minimum detectable RCS at 1000 km of −10.97 dBm2 (0.09 m2) was observed. The effects of Faraday rotation resulting from the use of linearly polarised antennae are demonstrated. The radar’s utility for providing total electron content (TEC) measurements is investigated using a high-range resolution mode and high-precision ephemeris data. The short-term Fourier transform is applied to demonstrate the radar’s ability to investigate satellite rotation characteristics and monitor ionospheric plasma waves and instabilities.

Non-Cooperative LEO Satellite Orbit Determination Using Single Station for Space-Based Opportunistic Positioning

Space-based opportunistic positioning is a crucial component of resilient positioning, navigation, and timing (PNT) systems, and it requires the acquisition of orbit information for non-cooperative low Earth orbit (LEO) satellites. Traditional methods for orbit determination (OD) of non-cooperative LEO satellites have difficulty in achieving a balance between reliability, hardware costs, and availability duration. To address these challenges, this study proposes a framework for single-station orbit determination of non-cooperative LEO satellites. By utilizing signals of opportunity (SOPs) captured by a single ground station, the system performs initial orbit determination (IOD), precise orbit determination (POD), and orbit prediction (OP), enabling the long-term determination of satellite positions and velocities. Under the proposed framework, the reliability and real-time performance are dependent on the initial orbit determination and the orbit calculation based on the dynamical model. To achieve initial orbit determination, a three-step algorithm is designed. (1) An improved search method is employed to estimate a coarse orbit using single-pass Doppler measurements. (2) Data association is conducted to obtain multi-pass Doppler observations. (3) The least squares (LS) is implemented to determine the initial orbit using the associated multi-pass Doppler measurements and the coarse orbit. Additionally, to enhance computational efficiency, two fast orbit calculation algorithms are devised. These algorithms leverage the numerical stability of the Runge–Kutta integrator to reduce computations and exploit the strong correlation among nearby time intervals of orbits with small eccentricities to minimize redundant calculations, thereby achieving orbit calculation efficiently. Finally, through positioning experiments, the determined orbits are demonstrated to have accuracy comparable to that of two-line elements (TLE) updated by the North American Aerospace Defense Command (NORAD).

Downlink Analysis of a Low-Earth Orbit Satellite Considering an Airborne Interference Source Moving on Various Trajectory

This paper analyzes low-Earth-orbit (LEO) satellite downlinks when an airborne interference source moves parallel to the satellite trajectory by considering the relative angle differences between the satellites and the interference sources. To make the experimental interference situations more like actual environments, the LEO trajectories are obtained from two-line element set (TLE) data. Airborne interference sources with various altitudes move parallel to the LEO trajectories, and a jamming to signal (J/S) ratio is calculated based on the relative angle differences between the ground station, the LEO satellite, and the interference source. To accurately calculate the J/S ratio, we should apply the sidelobe gain from which the interference signal enters to the ground station antenna. In order to calculate the relative angle difference ψ, the coordinates of the satellite and the interference source are converted from the World Geodetic System 1984 (WGS84) to the ground station-centered east–north-up (ENU) system. The resulting J/S ratio demonstrates that the distance and the relative angle difference ψ between the ground stations, LEO satellite, and airborne interference source appear to be important factors causing changes in the J/S ratio. Among them, the relative angle difference ψ, which determines the sidelobe gain of the ground station antenna, is the most significant factor affecting the J/S ratio variation.

The effect of baseline on the uncertainties of range determination performed by two-site astrometric observations: A sample case for the triangulation method: TURKSAT 3A and TURKSAT 4A

The uncertainties in range determination depending on two-site locations for GEO satellites were calculated and shown that the uncertainty in the range depends not only on linear baseline distance between the locations but also both latitude and longitude of geographic coordinates of the sites. By taking the first observing site as fixed and changing the location of the second site, the triangulated range calculations were performed by using SGP4 orbital model and published Two-Line Elements (TLE). Additionally, as a sample of triangulated observation, the results of nearly synchronous two-site astrometric measurements of two GEO satellites, TURKSAT 3A and TURKSAT 4A, were presented. These astrometric observations were carried out in TÜBITAK National Observatory (TUG), Antalya and TURKSAT Company’s site, Ankara. Although the baseline between the observatory sites corresponds to a relatively short distance of about 400 km, measured range determinations well stays under the error level expected by Choi et al. (2010).

Data-driven satellite orbit prediction using two-line elements

Orbit prediction is crucial for space situational awareness operations. Low earth orbit satellites are subjected to external forces such as atmospheric drag, radiation, and gravity. However, the well-known Kepler propagation model ignores these external forces. The simplified perturbation model included only the main external forces. In this study, a nonlinear programming model for orbit prediction using public two-line elements (TLE) is proposed. It has been proven that our models exhibit better performance than the standard Kepler and SPG4 models in terms of orbit prediction accuracy. Moreover, the proposed models were simple, computationally effective, and robust to disturbances. The sensitivity analysis indicates that a right ascension of the ascending node, a perigee argument, and a mean anomaly of the orbital elements are the most sensitive parameters in our models. The results also revealed that our method can be generalized to any low-earth orbit satellite with adequate data.

Radar observation and recontruction of Cosmos 1408 fragmentation

The population of objects in space has increased dramatically over recent decades. Space debris now represents the majority of objects in space resulting from inactive satellites, breakups, collisions and fragmentations. It has become a concern for institutions all over the world and, as such, it has led to the fostering of several programmes to counter the issues. Among these, the use of ground-based sensors for Space Surveillance Tracking (SST) activities and services and tools for analysing fragmentations play a crucial role.This work presents the activities carried out by Politecnico di Milano, Italian Space Agency and Italian National Institute of Astrophysics in this framework, using data from SST networks and the observation measurements from Bistatic Radar for LEo Survey (BIRALES), an Italian bistatic radar belonging to the EUropean Space Surveillance and Tracking (EUSST), which contributed most to the monitoring of the cloud of fragments. Exploiting Two-Line Elements (TLEs) of observed fragments, a reverse engineering approach is used to reconstruct a fragmentation in orbit through the use of the software suite PUZZLE developed at Politecnico di Milano. The analyses focus on studying the fragmentation of the Cosmos 1408 satellite, which occurred on November 15th 2021 following an Anti-SATellite (ASAT) missile test. More than 1000 trackable pieces and millions of smaller debris (estimated from numerical analysis) were produced by this event, increasing the population of inactive objects around the Earth, and threatening nearby orbiting objects.First, the processing method adopted from BIRALES in observing Cosmos debris is presented and discussed and a critical analysis about the derivable information is conducted. Then, these data and those from SST network observations are used to identify the epoch and the location of the fragmentation. In this procedure, the software toolkit PUZZLE, developed by Politecnico di Milano within a project funded by the Italian Space Agency and extended through the European Research Council, is used.

Snapshot GNSS receivers for low-effort, high-gain space situational awareness

This paper proposes a novel concept of using highly efficient Snapshot Global Navigation Satellite Systems (GNSS) receivers to provide precise position fixes of single or multiple satellites in Low-Earth Orbit (LEO) to improve upper atmospheric modeling and thus contribute to superior space situational awareness (SSA). While tracking of LEO satellites and the use of onboard GNSS receivers for drag measurements and upper atmosphere modeling are well-established techniques, the expected advent of snapshot GNSS receivers for spaceborne scientific applications will allow massive improvements on the GNSS sensor’s Size, Weight, Power and Cost (SWaP-C). With chip-size dimensions of 4x4 mm2, a mass of less than 5 gr, an average power level below 0.1 mW, snapshot receiver technology is expected to provide position fixes in space with an accuracy of ∼19 m (3D r.m.s.), which will surpass the accuracy of Two-Line Elements (TLE) provided by the US Joint Space Operations Center (JSpOC) by at least two orders of magnitude. Equally important to their SWaP-C benefits, Snapshot GNSS receivers will allow mission and spacecraft designers to trade onboard-processing requirements versus payload downlink requirements, leading to either minimum onboard processing or a minimum amount of downlinked data. In this research, we establish the concept and architectural overview of using snapshot GNSS receivers for SSA, including the role of using them in a Distributed Space System (DSS), and detail their characterization and performance in terms of the required GNSS hardware and the impact of these payload on the power budget, the link budget and the OnBoard Data Handling (OBDH) budget of a satellite. It will be shown that these receivers lend themselves especially to their use on femto-, pico- and nano-satellites, although integrated snapshot modules may be flown as auxiliary payloads on micro- or mini-satellites as well. While this work focuses on the implications of the use of snapshot GNSS receivers on spacecraft design for the use of upper atmosphere modeling and SSA, their use may open up other science applications which avoid the need for expensive high-grade GNSS receivers.

Cavsiopy: a Python package to calculate and visualize spacecraft instrument orientation

Spacecraft attitude plays an important role in the observations of various atmospheric, planetary, and terrestrial parameters and phenomena that are of interest to the scientific community. Precise measurements from imagers, particle sensors, and antennas require accurate knowledge of instrument orientation. cavsiopy is an easy-to-install and use, light-weight open-source Python package for researchers who need to consider instrument pointing direction and observation geometry. cavsiopy contains the coordinate transformation routines and the corresponding rotation matrices from the spacecraft orbital reference frame (ORF) to any of the geocentric equatorial inertial for epoch J2000 (GEI J2K)/International Celestial Reference Frame (ICRF), Earth-centered, Earth-fixed (ECEF), International Terrestrial Reference Frame (ITRF), geodetic north-east-down, and geocentric north-east-center coordinate systems. Additionally, cavsiopy includes routines for importing Swarm-E ephemeris and generic two-line-element (TLE) data files; for the calculation of spacecraft azimuth, elevation, and orbital parameters; as well as for the 2D/3D visualization of the geometry between the instrument and the target. Functionality and utilization of cavsiopy for research problems are demonstrated with examples and visualizations for the Radio Receiver Instrument (RRI) and the Fast Auroral Imager (FAI) of e-POP/Swarm-E.

A station-keeping maneuver detection method of non-cooperative geosynchronous satellites

Orbit maneuver detection is one of the major concerns for the work of Space Situation Awareness of GEO protected region. In this study, we propose a non-cooperative GEO satellites station-keeping maneuver detection approach based on the two-dimensional Convolutional Neural Network (CNN) and the public Two-Line Element (TLE). The proposed method focuses on identifying “station-keeping categories” using the time series segments of TLE data. The time series TLE of the selected objects are converted into several equal-sized segments using a window slicing method. These segments are then used to train a two-dimensional CNN classification model that can divide the input data into three categories of maneuver. The applicability of the proposed method on satellites in the same orbital region were further tested using other objects that had not been included during the modeling process. Experimental results demonstrated the feasibility and robustness of the proposed method, with a classification macro F1-score of 94.45%, macro precision of 94.48%, and macro recall of 94.46%. The results suggest that the proposed method provides a promising solution for detecting GEO station-keeping maneuvers.

The Steward Observatory LEO Satellite Photometric Survey

The Steward Observatory LEO Satellite Photometric Survey is a comprehensive observational survey to characterize the apparent brightness of the Starlink and OneWeb low Earth orbit satellites and evaluate the potential impact on astronomy. We report the results of over 16,000 independent measurements of nearly 2800 individual satellites. In addition to photometry, we also measured the astrometric position of each satellite and evaluated the accuracy of predicting satellite position with the available two-line element sets. The apparent brightness of a satellite seen in the sky is not constant and depends on the Sun-satellite-observer geometry. To capture this, we designed the survey to create an all-geometries set of measurements to fully characterize the brightness of each population of satellites as seen in the sky. We visualize the data with sky-plots that show the correlation of apparent brightness with on-sky position and relative Sun-satellite-observer geometry. The sky-plots show where in the sky the satellites are brightest. In addition to visual magnitudes, we also present two new metrics: the expected photon flux and the effective albedo. The expected photon flux metric assesses the potential impact on astronomy sensors by predicting the flux for a satellite trail in an image from a theoretical 1 m class telescope and sensor. The effective albedo metric assesses where a satellite is more reflective than baseline, which ties to the physical structure of the satellite and indicates the potential for brightness-reducing design changes. We intend to use this methodology and resulting data to inform the astronomy community about satellite brightness. Observing programs use a variety of telescopes and instruments and look at different parts of the sky. With the expected photon flux metric and a complete all-sky characterization of satellite brightness, observers can evaluate the potential impacts to their projects and possibly avoid the worst effects.

Two‐Line Elements Based Thermospheric Mass Density Specification From Parameter Optimization of a Physics‐Based Model

AbstractAccurately estimating the mass density of the thermosphere can assist satellite operators in planning missions and optimizing satellite orbits, reducing the risk of collisions. The newly developed GOFT (Global Observation‐based Forecast for the Thermosphere) model specifies thermospheric mass density by optimizing uncertainty parameters in a physics‐based model based on model errors. This study extends the model to specify thermospheric mass density based on Two‐Line Elements (TLEs) from numerous space objects. The resulting mass density is validated against accelerometer‐derived mass density from different satellites. The comparison results revealed that assimilating TLEs into the GOFT model significantly improved the accuracy of the thermospheric mass density. In addition, the GOFT model also improved the accuracy of the electron density in the ionosphere, indicating that the assimilation capability of GOFT allows simultaneous specification of coupled ionosphere and thermosphere in future applications.

Demonstration of orbit determination for LEO objects using the Murchison Widefield Array

The rapidly increasing number of satellites in Earth’s orbit motivates the development of Space Domain Awareness (SDA) capabilities using wide field-of-view sensor systems that can perform simultaneous detections. This work demonstrates preliminary orbit determination capability for Low Earth Orbit objects using the 36°×36° field-of-view of the Murchison Widefield Array (MWA) at commercial Frequency Modulated (FM) frequencies (transmitters in 88-108 MHz range). Non-coherent passive radar techniques with the MWA produce spatially smeared detections, due to time averaging in the MWA’s standard signal processing chain. The work develops methods to extract time-stamped measurements of a satellite’s angular coordinates from these data. The developed method was tested on observations of 32 satellite passes and the extracted measurements were used to perform orbit determination for the targets using a least-squares fitting approach. The target satellites span a range in altitude and Radar Cross Section, providing examples of both high and low signal-to-noise detections. The estimated orbital elements for the satellites are validated against the publicly available Two Line Element (TLE) updates provided by the Space Surveillance Network (SSN) and the preliminary estimates are found to be in close agreement. The work also tests for re-acquisition for one target using the orbital elements and finds the trajectory predicted by the method to coincide within 0.2° cross-track and 0.3° in-track for a subsequent pass, reduced to approximately 0.1° cross-track (less than one kilometre) if two passes are used to predict the subsequent pass (using simple two-body propagation). The median uncertainty in the angular position for objects in LEO (range less than 1000 km) is found to be 860 m in the cross-track direction and 780 m in the in-track direction, which are comparable to the typical uncertainty of ∼1000 m in the publicly available TLE information. The techniques, therefore, demonstrate the MWA to be capable of being a valuable contributor to the global SDA community. Based on the understanding of the MWA SDA system, this paper also briefly describes methods to mitigate the impact of FM-reflecting LEO satellites on radio astronomy observations, and how maintaining a catalog of FM-reflecting LEO objects is in the best interests of both SDA and radio astronomy.

An investigation on space debris of unknown origin using proper elements and neural networks

Proper elements represent a dynamical fingerprint of an object’s inherent state and have been used by small-body taxonomists in characterizing asteroid families. Being linked to the underlying dynamical structure of orbits, Celletti, Pucacco, and Vartolomei have recently adopted these innate orbital parameters for the association of debris from breakup or collision into its parent satellite. Building from this rich astronomical heritage and recent foundations, we introduce an unsupervised learning method—density-based spatial clustering of applications with noise (DBSCAN)—to determine clusters of orbital debris in the space of proper elements. Data is taken from the space-object catalog of trackable Earth-orbiting objects in the form of two-line element sets. Proper elements for debris fragments in low-Earth orbit are computed using an ad hoc numerical scheme, akin to the state-of-the-art Fourier-series-based synthetic method for the asteroid domain. Given the heuristic nature of classical DBSCAN, we investigate the use of neural networks, trained on known families, to augment DBSCAN into a classification problem and apply it to analyst objects of unknown origin.

De-Orbit Maneuver Demonstration Results of Micro-Satellite ALE-1 with a Separable Drag Sail

ALE-1, a micro-satellite created for the demonstration of artificial shooting stars, required orbital descent before mission execution due to safety aspects in orbit. ALE-1 utilized a drag sail called SDOM (Separable De-Orbit Mechanism) for a passive de-orbit maneuver, which was successfully completed, lowering the orbit from about 500 km down to about 400 km. This paper summarizes the detailed history of satellite operation and the results of the de-orbit maneuver demonstration during the past three years. Although the SDOM sail faced difficulty in keeping the desired deployed shape of the drag sail due to mechanical troubles, by letting the sail be a drag flag instead, it could still deliver a meaningful de-orbit performance to allow the satellite to successfully lower the orbit as planned. The de-orbit effect of the drag flag was evaluated using comparisons between orbit propagation simulations and the actual orbit transition flight data provided in the form of TLE (Two-Line Element) sets. Through this study, it is demonstrated that the SDOM can provide orbit transfer capabilities for satellites. Furthermore, the de-orbit performance of the drag flag can be evaluated, which could be an important reference for the future implementation of de-orbit devices to solve space debris problems.

Long-Term Orbital Lifetime Prediction of Highly Eccentric Orbits: A Statistical Approach

This paper concentrates on the long-term orbital lifetime prediction of highly eccentric orbits (HEOs) based on two-line element sets via a statistical approach. Due to the significantly different evolution characteristics of low- and high-inclination HEOs induced by different orbital resonances, two area-to-mass estimation methods are proposed, respectively. The resonance phenomena encountered in the HEO region strongly affect the orbital evolution and cause high sensitivity. Therefore, a statistical approach is adopted to handle these effects and correctly estimate low-inclination HEO lifetime. We use the Monte Carlo method and kernel-density estimation to calculate the probability distribution of the orbital lifetime. Finally, the performance of the method is assessed by the actual orbital lifetimes of space objects that reentered from HEOs in the past 50 years. The results indicate that our statistical approach can improve the orbital lifetime prediction accuracy to a large extent, especially for the low-inclination HEOs. If a relative error of 15% is adopted as the error tolerance, compared with the traditional method based on a single orbital propagation, our statistical method can increase the success rate from 40% to more than 70%. For the high-inclination HEOs, the objects with a relative error below 15% account for more than 90%.

TLE orbit determination using simplex method

Two-Line Element (TLE) datasets are the only orbital data source of Earth-orbiting space objects for many civil users for their research and applications. The datasets have uneven qualities that may affect the reliability of the propagated positions of space objects using a single TLE. The least squares approach to use multiple TLEs also suffers from the poor quality of some TLEs, and reliable error information cannot be available. This paper proposes a simplex algorithm to estimate an optimal TLE from multiple TLEs and obtain the uncertainty of each element. It is a derivative-free technique that can deal with various orbit types. Experiments have demonstrated that using the TLE estimated from the simplex method is more reliable, stable, and effective than those from the batch least squares method. As an application example, the optimal TLE and its uncertainty are used for predicting the fallen area, keeping the actual fallen site in the prediction areas.