Orbit Prediction Research Articles

Observing the variation of asteroid thermal inertia with heliocentric distance

Thermal inertia is a useful property to characterise a planetary surface since it can be used as a qualitative measure of the regolith grain size. It is expected to vary with heliocentric distance because of its dependence on temperature. However, no previous investigation has conclusively observed a change in thermal inertia for any given planetary body. We have addressed this by using NEOWISE data and the Advanced Thermophysical Model to study the thermophysical properties of the near-Earth asteroids (1036) Ganymed, (1580) Betulia, and (276049) 2002 CE26 as they moved around their highly eccentric orbits. We confirm that the thermal inertia values of Ganymed and 2002 CE26 do vary with heliocentric distance, although the degree of variation observed depends on the spectral emissivity assumed in the thermophysical modelling. We also confirm that the thermal inertia of Betulia did not change for three different observations obtained at the same heliocentric distance. Depending on the spectral emissivity, the variations for Ganymed and 2002 CE26 are potentially more extreme than that implied by theoretical models of heat transfer within asteroidal regoliths, which might be explained by asteroids having thermal properties that also vary with depth. Accounting for this variation reduces a previously observed trend of decreasing asteroid thermal inertia with increasing size, and suggests that the surfaces of small and large asteroids could be much more similar than previously thought. Furthermore, this variation can affect Yarkovsky orbital drift predictions by a few tens of per cent.

Improving orbit prediction accuracy through supervised machine learning

Due to the lack of information such as the space environment condition and resident space objects’ (RSOs’) body characteristics, current orbit predictions that are solely grounded on physics-based models may fail to achieve required accuracy for collision avoidance and have led to satellite collisions already. This paper presents a methodology to predict RSOs’ trajectories with higher accuracy than that of the current methods. Inspired by the machine learning (ML) theory through which the models are learned based on large amounts of observed data and the prediction is conducted without explicitly modeling space objects and space environment, the proposed ML approach integrates physics-based orbit prediction algorithms with a learning-based process that focuses on reducing the prediction errors. Using a simulation-based space catalog environment as the test bed, the paper demonstrates three types of generalization capability for the proposed ML approach: (1) the ML model can be used to improve the same RSO’s orbit information that is not available during the learning process but shares the same time interval as the training data; (2) the ML model can be used to improve predictions of the same RSO at future epochs; and (3) the ML model based on a RSO can be applied to other RSOs that share some common features.

Atmospheric interaction with nanosatellites from observed orbital decay

Nanosatellites have gained considerable presence in low Earth orbits wherein the atmospheric interaction with exposed surfaces plays a fundamental role in the evolution of motion. These aspects become relevant with the increasing applicability of nanosatellites to a broader range of missions objectives. This investigation sets out to determine distinctive drag coefficient development and attributes of atmospheric gas-surface interactions in nanosatellites in the common form of standard 3U CubeSats from observed orbital decay. As orbital decay can be measured with relative accuracy, and its mechanism broken down into its constituent sources, the value of drag-related coefficients can be inferred by fitting modelled orbit predictions to observed data wherein the coefficient of interest is the adjusted parameter. The analysis uses the data of ten historical missions with documented passive attitude stabilisation strategies to reduce uncertainties. Findings indicate that it is possible to estimate fitted drag coefficients in CubeSats with physical representativeness. Assessment of atomic oxygen surface coverage derived from the fitted drag coefficients is broadly consistent with theoretical trends. The proposed methodology opens the possibility to assess atmospheric interaction characteristics by using the unprecedented opportunity arising from the numerous observed orbital decay of nanosatellites.

SPOOK - A comprehensive Space Surveillance and Tracking analysis tool

Space Surveillance and Tracking (SST) is the ability to detect and predict the movement of space debris in orbit around the Earth. SPOOK (SPace Object Observations and Kalman filtering) is a versatile and highly configurable software tool developed by Airbus, for the detection, cataloguing and orbit prediction analysis of Earth orbiting objects. It provides an integrated framework for all the activities related with SST sensor architecture simulation. Throughout this paper, the functionalities of the tool are described and examples of different simulation scenarios are provided.

Efficient and automatic image reduction framework for space debris detection based on GPU technology

In the last years, the increasing number of space debris has triggered the need of a distributed monitoring system for the prevention of possible space collisions. Space surveillance based on ground telescope allows the monitoring of the traffic of the Resident Space Objects (RSOs) in the Earth orbit. This space debris surveillance has several applications such as orbit prediction and conjunction assessment. In this paper is proposed an optimized and performance-oriented pipeline for sources extraction intended to the automatic detection of space debris in optical data. The detection method is based on the morphological operations and Hough Transform for lines. Near real-time detection is obtained using General Purpose computing on Graphics Processing Units (GPGPU). The high degree of processing parallelism provided by GPGPU allows to split data analysis over thousands of threads in order to process big datasets with a limited computational time. The implementation has been tested on a large and heterogeneous images data set, containing both imaging satellites from different orbit ranges and multiple observation modes (i.e. sidereal and object tracking). These images were taken during an observation campaign performed from the EQUO (EQUatorial Observatory) observatory settled at the Broglio Space Center (BSC) in Kenya, which is part of the ASI-Sapienza Agreement.

Comparison of Ultra-Rapid Orbit Prediction Strategies for GPS, GLONASS, Galileo and BeiDou.

Currently, ultra-rapid orbits play an important role in the high-speed development of global navigation satellite system (GNSS) real-time applications. This contribution focuses on the impact of the fitting arc length of observed orbits and solar radiation pressure (SRP) on the orbit prediction performance for GPS, GLONASS, Galileo and BeiDou. One full year’s precise ephemerides during 2015 were used as fitted observed orbits and then as references to be compared with predicted orbits, together with known earth rotation parameters. The full nine-parameter Empirical Center for Orbit Determination in Europe (CODE) Orbit Model (ECOM) and its reduced version were chosen in our study. The arc lengths of observed fitted orbits that showed the smallest weighted root mean squares (WRMSs) and medians of the orbit differences after a Helmert transformation fell between 40 and 45 h for GPS and GLONASS and between 42 and 48 h for Galileo, while the WRMS values and medians become flat after a 42 h arc length for BeiDou. The stability of the Helmert transformation and SRP parameters also confirmed the similar optimal arc lengths. The range around 42–45 h is suggested to be the optimal arc length interval of the fitted observed orbits for the multi-GNSS joint solution of ultra-rapid orbits.

Approximate analytic solution of motion for spacecraft around an oblate planet

The oblateness of a planet has an important effect on the surrounding spacecraft and is usually regarded as a significant perturbation factor to spacecraft’s motion. In this paper, an approximate analytic solution of motion for spacecraft which considering this perturbation factor is investigated. The perturbed dynamic model which taking true anomaly as the independent variable is firstly built in non-inertial coordinate system moving with a Kepler two-body orbit, then the six-dimensional state transition matrix about the state deviation vector is introduced. After that, the expression of J2 term’s gravitational potential in this non-inertial coordinate system is obtained based on spherical trigonometry, and the analytic solution for every variable of the state deviation vector is then derived by complicated mathematic transformation. Numerical simulation results done by the presented method have been compared with that done by numerical integration as well as an existing first-order analytic solution. The comparison results show that the presented method has high accuracy, and is more accurate than the first-order analytic solution within one or few orbit revolutions, which makes it more suitable for short-term orbit prediction of the satellites, especially for those suborbital flight vehicles.

A fast cold-start method of GPS receiver based on satellite orbit prediction

Time to first fix (TTFF) is one of the crucial indicators to evaluate the performance of a GPS receiver. In this paper, an orbit prediction algorithm to reduce the TTFF of GPS receivers without a network connection is presented. Satellite orbit is predicted by numerically integrating the satellite’s equation of motion. Satellite’s initial position and velocity value, as well as the Earth’s polar motion parameters, used in prediction correspond to the locally collected previous broadcast ephemeris. As the solar radiation pressure (SRP) is one of the most critical factors causing orbit prediction error, an empirical SRP model with two parameters that change with the movement of satellite and Earth is also described. The presented algorithm is verified by GPS satellites using the initial conditions divided from three groups of broadcast ephemeris data. Simulation results show that, with the SRP model described in this paper the satellite’s position error limited to 30 meters within 7 days prediction.

Analysis of impulsive maneuvers to keep orbits around the asteroid 2001SN263

The strongly perturbed environment of a small body, such as an asteroid, can complicate the prediction of orbits used for close proximity operations. Inaccurate predictions may make the spacecraft collide with the asteroid or escape to the deep space. The main forces acting in the dynamics come from the solar radiation pressure and from the body’s weak gravity field. This paper investigates the feasibility of using bi-impulsive maneuvers to avoid the aforementioned non-desired phenomena (collisions and escapes) by connecting orbits around the triple system asteroid 2001SN263, which is the target of a proposed Brazilian space mission. In terms of a mathematical formulation, a recently presented rotating dipole model is considered with oblateness in both primaries. In addition, a “two-point boundary value problem” is solved to find a proper transfer trajectory. The results presented here give support to identifying the best strategy to find orbits for close proximity operations, in terms of long orbital lifetimes and low delta- $V$ consumptions. Numerical results have also demonstrated the significant influence of the spacecraft orbital elements (semi-major axis and eccentricity), angular position of the Sun and spacecraft area-to-mass ratio, in the performance of the bi-impulsive maneuver.

A Software Architecture Design for Autonomous Formation Flying Control

Based on currently developed on-board platforms, this paper proposes a software architecture for orbiting spacecraft for autonomously establishing formation, station-keeping, and reconfiguring maneuvers. Components of the software architecture include orbital prediction (OP), orbital control (OC), and relative navigation (RN); these not only meet the constraining requirements of attitude control, thermal control, thrusters, and the ground-based operator, but also have a minimal impact on the on-board data handling (OBDH) system. All of the OP, OC, and RN algorithms are packaged as a subroutine (OPCNS) called by the OBDH system at regular intervals. The in-advance control instruction set (CIS) is temporarily stored in random access memory to be read by OPCNS as a formal argument, rather than in any fixed space of read only memory. The simplified general perturbations 4 method is introduced to perform a specific time interval of OP based on filtered relative measurements of RN. The relative-orbital-element (ROE) control method is employed to create in-advance instructions, depending on whether or not the predicted ROEs trigger the threshold values of any maneuver and then to allocate these instructions into the CIS following the proposed relationship rules between all of the ROEs and the ground control instruction. To tolerate the temporary divergence of the RN filter after any orbital maneuver, an ROE-based unscented Kalman filter is enhanced by using the control instruction in current execution to accelerate the co- nvergence. A 3-craft formation scenario for interferometric synthetic aperture radar measurements is used to validate all of the functions of the proposed software architecture.

A deformation model of flexible, HAMR objects for accurate propagation under perturbations and the self-shadowing effects

A new type of space debris in near geosynchronous orbit (GEO) was recently discovered and later identified as exhibiting unique characteristics associated with high area-to-mass ratio (HAMR) objects, such as high rotation rates and high reflection properties. Observations have shown that this debris type is very sensitive to environmental disturbances, particularly solar radiation pressure, due to the fact that its motion depends on the actual effective area, orientation of that effective area, reflection properties and the area-to-mass ratio of the object is not stable over time. Previous investigations have modelled this type of debris as rigid bodies (constant area-to-mass ratios) or discrete deformed body; however, these simplifications will lead to inaccurate long term orbital predictions. This paper proposes a simple yet reliable model of a thin, deformable membrane based on multibody dynamics. The membrane is modelled as a series of flat plates, connected through joints, representing the flexibility of the membrane itself. The mass of the membrane, albeit low, is taken into account through lump masses at the joints. The attitude and orbital motion of this flexible membrane model is then propagated near GEO to predict its orbital evolution under the perturbations of solar radiation pressure, Earth’s gravity field (J2), third body gravitational fields (the Sun and Moon) and self-shadowing. These results are then compared to those obtained for two rigid body models (cannonball and flat rigid plate). In addition, Monte Carlo simulations of the flexible model by varying initial attitude and deformation angle (different shape) are investigated and compared with the two rigid models (cannonball and flat rigid plate) over a period of 100 days. The numerical results demonstrate that cannonball and rigid flat plate are not appropriate to capture the true dynamical evolution of these objects, at the cost of increased computational time.

Exoplanet Predictions Based on Harmonic Orbit Resonances

The current exoplanet database includes 5454 confirmed and candidate planets observed with the Kepler mission. We find 932 planet pairs from which we extract distance and orbital period ratios. While earlier studies used a logarithmic spacing, which lacks a physical model, we employ here the theory of harmonic orbit resonances, which contains quantized ratios instead, to explain the observed planet distance ratios and to predict undetected exoplanets. We find that the most prevailing harmonic ratios are (2:1), (3:2), and (5:3) in 73% of the cases, while alternative harmonic ratios of (5:4), (4:3), (5:2), and (3:1) occur in the other 27% of the cases. Our orbital predictions include 171 exoplanets, 2 Jupiter moons, 1 Saturn moon, 3 Uranus moons, and 4 Neptune moons. The accuracy of the predicted planet distances amounts to a few percent, which fits the data significantly better than the logarithmic spacing. This information may be useful for targeted exoplanet searches with Kepler data and to estimate the number of live-carrying planets in habitable zones.

Impacts of Earth rotation parameters on GNSS ultra-rapid orbit prediction: Derivation and real-time correction

Analysis centers (ACs) for global navigation satellite systems (GNSSs) cannot accurately obtain real-time Earth rotation parameters (ERPs). Thus, the prediction of ultra-rapid orbits in the international terrestrial reference system (ITRS) has to utilize the predicted ERPs issued by the International Earth Rotation and Reference Systems Service (IERS) or the International GNSS Service (IGS). In this study, the accuracy of ERPs predicted by IERS and IGS is analyzed. The error of the ERPs predicted for one day can reach 0.15mas and 0.053ms in polar motion and UT1-UTC direction, respectively. Then, the impact of ERP errors on ultra-rapid orbit prediction by GNSS is studied. The methods for orbit integration and frame transformation in orbit prediction with introduced ERP errors dominate the accuracy of the predicted orbit. Experimental results show that the transformation from the geocentric celestial references system (GCRS) to ITRS exerts the strongest effect on the accuracy of the predicted ultra-rapid orbit. To obtain the most accurate predicted ultra-rapid orbit, a corresponding real-time orbit correction method is developed. First, orbits without ERP-related errors are predicted on the basis of ITRS observed part of ultra-rapid orbit for use as reference. Then, the corresponding predicted orbit is transformed from GCRS to ITRS to adjust for the predicted ERPs. Finally, the corrected ERPs with error slopes are re-introduced to correct the predicted orbit in ITRS. To validate the proposed method, three experimental schemes are designed: function extrapolation, simulation experiments, and experiments with predicted ultra-rapid orbits and international GNSS Monitoring and Assessment System (iGMAS) products. Experimental results show that using the proposed correction method with IERS products considerably improved the accuracy of ultra-rapid orbit prediction (except the geosynchronous BeiDou orbits). The accuracy of orbit prediction is enhanced by at least 50% (error related to ERP) when a highly accurate observed orbit is used with the correction method. For iGMAS-predicted orbits, the accuracy improvement ranges from 8.5% for the inclined BeiDou orbits to 17.99% for the GPS orbits. This demonstrates that the correction method proposed by this study can optimize the ultra-rapid orbit prediction.

Analytic Free-Molecular Aerodynamics for Rapid Propagation of Resident Space Objects

Aerodynamic forces and moments are significant perturbations on low-Earth-orbiting objects, second in magnitude to the nonspherical gravity field. Traditionally, the aerodynamic perturbations are calculated using a direct simulation Monte Carlo method. Under certain assumptions, these forces and moments can be described analytically via free-molecular flow theory. Using symbolic manipulation techniques, exact expressions for the free-molecular aerodynamics of analytic shapes can be derived. In this investigation, analytic expressions for the aerodynamic force and moment coefficients of primitive and composite parametric surfaces are derived, then validated against industry-standard direct simulation Monte Carlo techniques. A framework for the rapid and accurate calculation of free-molecular aerodynamics of composite geometries based on superposition is described. This framework is applied to axisymmetric composite geometries. Results within 6% of direct simulation Monte Carlo calculations are obtained in 0.05% of the time. The analytic aerodynamics models enable rapid trajectory and uncertainty propagation for low-Earth-orbiting objects. A case study on aerodynamic perturbations of a low-Earth-orbit nanosatellite is included to demonstrate application of these analytic models. The case study shows that these derived analytical free-molecular aerodynamics produce results that are applicable to inclusion in rapid trajectory propagation tools for orbit prediction and conceptual mission design.

Calibration of atmospheric density model using two-line element data

For satellites in orbits, most perturbations can be well modeled, however the inaccuracy of the atmospheric density model remains the biggest error source in orbit determination and prediction. The commonly used empirical atmospheric density models, such as Jacchia, NRLMSISE, DTM, and Russian GOST, still have a relative error of about 10%–30%. Because of the uncertainty in the atmospheric density distribution, high accuracy estimation of the atmospheric density cannot be achieved using a deterministic model. A better way to improve the accuracy is to calibrate the model with updated measurements. Twoline element (TLE) sets provide accessible orbital data, which can be used in the model calibration. In this paper, an algorithm for calibrating the atmospheric density model is developed. First, the density distribution of the atmosphere is represented by a power series expansion whose coefficients are denoted by the spherical harmonic expansions. Then, the useful historical TLE data are selected. The ballistic coefficients of the objects are estimated using the BSTAR data in TLEs, and the parameterized model is calibrated by solving a nonlinear least squares problem. Simulation results show that the prediction error is reduced using the proposed calibration algorithm.

Modified empirical Solar Radiation Pressure model for IRNSS constellation

Navigation with Indian Constellation (NAVIC) also known as Indian Regional Navigation Satellite System (IRNSS) is India’s regional navigation system designed to provide position accuracy better than 20m over India and the region extending to 1500km around India. The reduced dynamic precise orbit estimation is utilized to determine the orbit broadcast parameters for IRNSS constellation. The estimation is mainly affected by the parameterization of dynamic models especially Solar Radiation Pressure (SRP) model which is a non-gravitational force depending on shape and attitude dynamics of the spacecraft. An empirical nine parameter solar radiation pressure model is developed for IRNSS constellation, using two-way range measurements from IRNSS C-band ranging system. The paper addresses the development of modified SRP empirical model for IRNSS (IRNSS SRP Empirical Model, ISEM). The performance of the ISEM was assessed based on overlap consistency, long term prediction, Satellite Laser Ranging (SLR) residuals and compared with ECOM9, ECOM5 and new-ECOM9 models developed by Center for Orbit Determination in Europe (CODE). For IRNSS Geostationary Earth Orbit (GEO) and Inclined Geosynchronous Orbit (IGSO) satellites, ISEM has shown promising results with overlap RMS error better than 5.3m and 3.5m respectively. Long term orbit prediction using numerical integration has improved with error better than 80%, 26% and 7.8% in comparison to ECOM9, ECOM5 and new-ECOM9 respectively. Further, SLR based orbit determination with ISEM shows 70%, 47% and 39% improvement over 10days orbit prediction in comparison to ECOM9, ECOM5 and new-ECOM9 respectively and also highlights the importance of wide baseline tracking network.

GPS Satellite Orbit Prediction at User End for Real-Time PPP System.

This paper proposed the high-precision satellite orbit prediction process at the user end for the real-time precise point positioning (PPP) system. Firstly, the structure of a new real-time PPP system will be briefly introduced in the paper. Then, the generation of satellite initial parameters (IP) at the sever end will be discussed, which includes the satellite position, velocity, and the solar radiation pressure (SRP) parameters for each satellite. After that, the method for orbit prediction at the user end, with dynamic models including the Earth’s gravitational force, lunar gravitational force, solar gravitational force, and the SRP, are presented. For numerical integration, both the single-step Runge–Kutta and multi-step Adams–Bashforth–Moulton integrator methods are implemented. Then, the comparison between the predicted orbit and the international global navigation satellite system (GNSS) service (IGS) final products are carried out. The results show that the prediction accuracy can be maintained for several hours, and the average prediction error of the 31 satellites are 0.031, 0.032, and 0.033 m for the radial, along-track and cross-track directions over 12 h, respectively. Finally, the PPP in both static and kinematic modes are carried out to verify the accuracy of the predicted satellite orbit. The average root mean square error (RMSE) for the static PPP of the 32 globally distributed IGS stations are 0.012, 0.015, and 0.021 m for the north, east, and vertical directions, respectively; while the RMSE of the kinematic PPP with the predicted orbit are 0.031, 0.069, and 0.167 m in the north, east and vertical directions, respectively.

An Analytical Theory with Respect to the Earth’s Zonal Harmonic Term J<sub>2</sub> in Terms of Eccentric Anomaly for Short-Term Orbit Predictions

An Analytical Theory with Respect to the Earth’s Zonal Harmonic Term J<sub>2</sub> in Terms of Eccentric Anomaly for Short-Term Orbit Predictions

Mars spacecraft tracking and analysis of VLBI orbit determination

The Chinese Mars mission will be launched in 2020, which consists of an orbiter and a rover. Mars spacecraft tracking and precise orbit determination are very important for achieving engineering and scientific detection. This paper first analyzes the Mars spacecraft tracking technology, and then introduces VLBI orbit determination results used in the mission of Mars Science Laboratory. Further, for a Mars spacecraft launched in July 2020, we give the angular distance between the spacecraft and the sun from launch to disorbit, the solar plasma delay, and observation elevation, at Shanghai, Beijing, Kunming and Urumqi stations in the arc segments of trajectory correction maneuver, Martian orbit insertion, plane maneuver, and disorbit. Orbit determination and prediction errors in the arc segment of Martian orbit insertion are also presented. Three-dimensional orbit determination error is reduced from 45.7 km to 18.8 km, and the periareon height prediction error (1 sigma) is reduced from 28.2 km to 7.6 km when combined VLBI data with range and Doppler data.

Analysis on the experiment of satellite laser ranging of defunct Tiangong-1 spacecraft

Tiangong-1, which was launched on 29 September 2011, is the first prototype space station of China. During the past two years in its lifetime, Tiangong-1 has been visited by a series of Shenzhou spacecraft. On 21 March 2016, it had formally ended its service and now is replaced by Tiangong-2. The orbit of Tiangong-1 is about 400 km above the ground where Tiangong-2, the international space station and other satellites are located as well. The space debris is now posing great collision risk to these space assets. The satellite laser ranging (SLR) is a space tracking technique with the highest precision. Its precise data lay a solid foundation for reliable conjunction assessment. On October 2002, Greene, Mount Stromlo SLR station of Australia, announced that it was possible to track a piece of space debris of a size of 15 cm at a distance of 1250 km using the SLR technology. In 2011, both Graz SLR station of Austrian and Grasse LLR station of France succeeded in debris laser ranging data. In China, Shanghai Astronomical Observatory obtained debris laser ranging data in 2008, Yunnan Observatory in 2010 and Changchun Observatory in 2014 also obtained debris data using SLR. In 2014, Changchun Observatory observed a total of 244 space debris in altitude about 400 to 1800 km. Tiangong-1 spacecraft was equipped with corner cube reflectors (CCRs), which were designed mainly for orbital rendezvous with Shenzhou spacecraft and now provide convenience for SLR. From March to May of 2016, a coordinated laser ranging experiment on Tiangong-1 was made by Changchun Observatory (ILRS site ID: 7237) of National Astronomical Observatories, Sheshan station (ILRS site ID: 7821) of Shanghai Astronomical Observatory and Purple Mountain Observatory. Tiangong-1 has not been tracked by international laser ranging service (ILRS) and the orbit prediction of consolidated prediction format (CPF) cannot be obtained from ILRS. In general, the two line elements (TLE) of space objects can be downloaded from internet ( https://www.space- track.org) which is released by north American air defense command (NORAD). However, the TLE prediction error of Tiangong-1 (NORAD ID: 37820) is about several kilometers that is not precise enough for SLR. During the experiment, the actual prediction is provided by Purple Mountain Observatory. By the end of the experiment, data of 23 passes was obtained. Precise orbit determination is made by using the SLR data and the root mean squares (RMS) of the SLR observation residuals is about 10 cm. For one pass of scattered data, the orbit determination program is not convergent. Then combined orbit determination is made using one pass of SLR data and optical celestial position data. The relative weight of SLR data to optical data is 100:1 and the RMS of the SLR observation residuals is about 6.9 cm. The new orbit prediction is propagated from the orbit determination of SLR data and its precision is about 200–300 m. The radial precision is better than 100 m that is accurate enough for later laser ranging. In general, the experiment of satellite laser ranging to defunct Tiangong-1 spacecraft will have positive effects on precise orbit determination of both Tiangong-2 and other low earth orbit space objects.