Preliminary Orbit Determination Research Articles

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.

Optimal Control Approach to Lambert’s Problem and Gibbs’ Method

This paper presents the optimal control approach to solve both Lambert’s problem and Gibbs’ method, which are commonly used for preliminary orbit determination. Lambert’s problem is reinterpreted with Hamilton’s principle and is converted to an optimal control problem. Various extended Lambert’s problems are formulated by modifying the weighting and constraint settings within the optimal control framework. Furthermore, Gibbs’ method is also converted to an extended Lambert’s problem with two position vectors and one orbit energy with the help of the proposed orbital energy computation algorithm. The proposed extended Lambert’s problem and Gibbs’ method are numerically solved with the Lobatto pseudospectral method, and their accuracies are verified by numerical simulations.

Preliminary orbits with line-of-sight correction for LEO satellites observed with radar

We propose a method to account for the Earth oblateness effect in preliminary orbit determination of satellites in low orbits with radar observations. This method is an improvement of the one described in (Gronchi et al 2015), which uses a pure Keplerian dynamical model. Since the effect of the Earth oblateness is strong at low altitudes, its inclusion in the model can sensibly improve the initial orbit, giving a better starting guess for differential corrections and increasing the chances to obtain their convergence. The input set consists of two tracks of radar observations, each one composed of at least 4 observations taken during the same pass of the satellite. A single observation gives the topocentric position of the satellite, where the range is very accurate, while the line of sight direction is poorly determined. From these data we can compute by a polynomial fit the values of the range and range rate at the mean epochs of the two tracks. In order to obtain a preliminary orbit we wish to compute the angular velocities, that is the rate of change of the line of sight. In the same spirit of (Gronchi et al 2015), we also wish to correct the values of the angular measurements, so that they fit the selected dynamical model if the same holds for the radial distance and velocity. The selected model is a perturbed Keplerian dynamics, where the only perturbation included is the secular effect of the $J_2$ term of the geopotential. The proposed algorithm models this problem with 8 equations in 8 unknowns.

Improved Laplace Earth Satellite Orbit Determination Using Line-of-Sight Projective Coordinates

This study aimed to improve the accuracy, efficiency, range of applicability, and numerical stability of the classical Laplace method for preliminary orbit determination when the observing sensor and the observed object are both in idealized Keplerian geocentric elliptical orbits. The improvements that were made flow largely from representing the line of sight by stabilized projective rather than angular coordinates, such as declination and right ascension or local azimuth and elevation. The Laplace classical method determines the orbital position and velocity of an object in space from three line-of-sight unit vectors pointing from the observing sensor to the observed object and the corresponding sensor inertial positions and velocities at the three observation times. The performance improvement of the modernized algorithm on a set of two orbit test cases is shown, in which the observing electro-optical sensor is carried aboard a geostationary satellite. Moreover, this paper shows how to solve the problem of choosing the right orbit from the two feasible candidates that correspond to the two roots of the classical deflated seventh degree polynomial with a single additional line-of-sight observation.

Parabolic orbit determination. Comparison of the Olbers method and algebraic equations

In this paper, the Olbers method for the preliminary parabolic orbit determination (in the Lagrange–Subbotin modification) and the method based on systems of algebraic equations for two or three variables proposed by the author are compared. The maximum number of possible solutions is estimated. The problem of selection of the true solution from the set of solutions obtained both using additional equations and by the problem reduction to finding the objective function minimum is considered. The results of orbit determination of the comets 153P/Ikeya-Zhang and 2007 N3 Lulin are cited as examples.

Preliminary Orbit Determination of a Tethered Satellite Using the p-Iteration Method

The possibility of the future deployment of tethered satellites has created a need for a preliminary orbit determination method capable of determining whether or not a satellite is tethered to another satellite. Classical preliminary orbit determination methods, which are used for untethered satellites, typically require two or more position vectors along with their respective observation times in order to determine a preliminary orbital element set. However, these conventional methods can’t distinguish between an untethered satellite and a tethered one, whose motion is modified due to the presence of a tether force. The use of conventional methods on a satellite which is part of a tethered satellite system will result in the calculation of inaccurate orbital elements. Modifications have been made to the p-iteration preliminary orbit determination method in order to allow for the identification of these tethered satellites. The modifications allow for the calculation of a modified gravitational parameter, which can be used to distinguish between a tethered satellite and an untethered one. This paper applies this modified p-iteration method to the problem of the quick identification of a tethered satellite. The performance of this method is evaluated through scenarios of differing tether lengths, levels of observation error, and orbital eccentricities. Due to the need for the preliminary orbit determination to be achieved quickly, only short time intervals between observations were considered. The manner in which this preliminary orbit information can be used to obtain tether parameters for the subsequent differential correction process is also described.

HIGH RESOLUTION IMAGING OF VERY LOW MASS SPECTRAL BINARIES: THREE RESOLVED SYSTEMS AND DETECTION OF ORBITAL MOTION IN AN L/T TRANSITION BINARY

We present high resolution Laser Guide Star Adaptive Optics imaging of 43 late-M, L and T dwarf systems with Keck/NIRC2. These include 17 spectral binary candidates, systems whose spectra suggest the presence of a T dwarf secondary. We resolve three systems: 2MASS J1341$-$3052, SDSS J1511+0607 and SDSS J2052$-$1609; the first two are resolved for the first time. All three have projected separations $<8$ AU and estimated periods of $14-80$ years. We also report a preliminary orbit determination for SDSS J2052$-$1609 based on six epochs of resolved astrometry between 2005$-$2010. Among the 14 unresolved spectral binaries, 5 systems were confirmed binaries but remained unresolved, implying a minimum binary fraction of $47^{+12}_{-11}\%$ for this sample. Our inability to resolve most of the spectral binaries, including the confirmed binaries, supports the hypothesis that a large fraction of very low mass systems have relatively small separations and are missed with direct imaging.

Orbit computation of the TELECOM-2D satellite with a Genetic Algorithm

AbstractIn order to test a preliminary orbit determination method, we fit an orbit of the geostationary satellite TELECOM-2D, as if we did not know any a priori information on its trajectory. The method is based on a genetic algorithm coupled to an analytical propagator of the trajectory, that is used over a couple of days, and that uses a whole set of altazimutal data that are acquired by the tracking network made up of the two TAROT telescopes. The adjusted orbit is then compared to a numerical reference. The method is described, and the results are analyzed, as a step towards an operational method of preliminary orbit determination for uncatalogued objects.

Comparison of different methods to compute a preliminary orbit of Space Debris using radar observations

AbstractWe advertise a new method of preliminary orbit determination for space debris using radar observations, which we call Infang†. We can perform a linkage of two sets of four observations collected at close times. The context is characterized by the accuracy of the range ρ, whereas the right ascension α and the declination δ are much more inaccurate due to observational errors. This method can correct α, δ, assuming the exact knowledge of the range ρ. Considering no perturbations from the J2 effect, but including errors in the observations, we can compare the new method, the classical method of Gibbs, and the more recent Keplerian integrals method. The development of Infang is still on-going and will be further improved and tested.

Two weighted eight-order classes of iterative root-finding methods

In this paper we design, by using the weight function technique, two families of iterative schemes with order of convergence eight. These weight functions depend on one, two and three variables and they are used in the second and third step of the iterative expression. Dynamics on polynomial and non-polynomial functions is analysed and they are applied on the problem of preliminary orbit determination by using a modified Gauss method. Finally, some standard test functions are to check the reliability of the proposed schemes and allow us to compare them with other known methods.

A class of optimal eighth-order derivative-free methods for solving the Danchick–Gauss problem

A derivative-free optimal eighth-order family of iterative methods for solving nonlinear equations is constructed using weight functions approach with divided first order differences. Its performance, along with several other derivative-free methods, is studied on the specific problem of Danchick’s reformulation of Gauss’ method of preliminary orbit determination. Numerical experiments show that such derivative-free, high-order methods offer significant advantages over both, the classical and Danchick’s Newton approach.

A Robust Method of Preliminary Orbit Determination

The preliminary orbit determination with optical angular measure- ments plays an important role in the survey of space objects. The classical method of orbit computation based on the least square error estimation is not robust while outliers occur in the observation. A robust method is proposed by employing the least absolute deviation estimation. The method reduces the problem of orbit determination to a linear programming problem, and gives the variance of the estimation with the bootstrap method. Numerical check shows that the method is effective and robust, and has a high breakdown point.

The Bootstrap Estimation of the Accuracy of Preliminary Orbit Determination

From the point of view of the non-parametric statistics, a general estimation method of the accuracy and confidence interval of preliminary orbit determination is proposed for the occasion without any other information but observational data. Based on the bootstrap method, the estimation relies only on the observational data and does not require the precise orbit determination as a reference, or the assumption of normal distribution of observational errors. Numerical experiments show that this method is very simple in implementa- tion, and may serve as an easy accuracy evaluation for the preliminary orbit determination and for the follow-up employments.

Preliminary Orbit Determination of Artificial Satellites: A Vectorial Sixth-Order Approach

A modified classical method for preliminary orbit determination is presented. In our proposal, the spread of the observations is considerably wider than in the original method, as well as the order of convergence of the iterative scheme involved. The numerical approach is made by using matricial weight functions, which will lead us to a class of iterative methods with a sixth local order of convergence. This is a process widely used in the design of iterative methods for solving nonlinear scalar equations, but rarely employed in vectorial cases. The numerical tests confirm the theoretical results, and the analysis of the dynamics of the problem shows the stability of the proposed schemes.

High-order expansion of the solution of preliminary orbit determination problem

A method for high-order treatment of uncertainties in preliminary orbit determination is presented. The observations consist in three couples of topocentric right ascensions and declinations at three observation epochs. The goal of preliminary orbit determination is to compute a trajectory that fits with the observations in two-body dynamics. The uncertainties of the observations are usually mapped to the phase space only when additional observations are available and a least squares fitting problem is set up. A method based on Taylor differential algebra for the analytical treatment of observation uncertainties is implemented. Taylor differential algebra allows for the efficient computation of the arbitrary order Taylor expansion of a sufficiently continuous multivariate function. This enables the mapping of the uncertainties from the observation space to the phase space as high-order multivariate Taylor polynomials. These maps can then be propagated forward in time to predict the observable set at successive epochs. This method can be suitably used to recover newly discovered objects when a scarce number of measurements is available. Simulated topocentric observations of asteroids on realistic orbits are used to assess the performances of the method.

Artificial satellites preliminary orbit determination by the modified high-order Gauss method

In recent years, high-order methods have shown to be very useful in many practical applications, in which nonlinear systems arise. In this case, a classical method of positional astronomy have been modified in order to hold a nonlinear system in its establishments (that in the classical method is reduced to a single equation). At this point, high-order methods have been introduced in order to estimate the solutions of this system and, then, determine the orbit of the celestial body. We also have implemented a user friendly application, which will allow us to make a numerical and graphical comparison of the different methods with reference orbits, or user defined orbits.

Preliminary Orbit Determination for GEO Objects Using Space-Based Optical Observations

Preliminary orbit determination for geosynchronous Earth orbit (GEO) using two short arcs of space-based optical observations plays a key role in the space-based space surveillance systems. The concept of attributable is introduced to represent the information in a short arc of observations. Using two-body integrals, the polynomial equations for distance and radial velocity between the surveillance satellite and the GEO object at two epochs are constructed. By variables transformation, multiple solutions of these equations are obtained. By covariance analysis, a penalty function is given to select the practical solutions for computing preliminary orbits. The simulation results show this algorithm can successfully obtain preliminary orbits with certain accuracy.

Preliminary Orbit Determination for GEO Objects Using Space-Based Optical Observations

Preliminary orbit determination for geosynchronous Earth orbit (GEO) using two short arcs of space-based optical observations plays a key role in the space-based space surveillance systems. The concept of attributable is introduced to represent the information in a short arc of observations. Using two-body integrals, the polynomial equations for distance and radial velocity between the surveillance satellite and the GEO object at two epochs are constructed. By variables transformation, multiple solutions of these equations are obtained. By covariance analysis, a penalty function is given to select the practical solutions for computing preliminary orbits. The simulation results show this algorithm can successfully obtain preliminary orbits with certain accuracy.

Derivative-free high-order methods applied to preliminary orbit determination

From position and velocity coordinates for several given instants, it is possible to determine the orbital elements of the preliminary orbit, taking only into account mutual gravitational attraction forces between the Earth and the satellite. Nevertheless, it should be refined with later observations from ground stations, whose geographic coordinates are previously known. Different methods developed for this purpose need to find a solution of a nonlinear function. In some classical methods it is usual to employ fixed point or secant methods. The second iterative scheme is often used when it is not possible to obtain the derivative of the nonlinear function. Nowadays, there exist efficient numerical methods that are able to highly improve the results obtained by the classical schemes. We will focus our attention on the method of iteration of the true anomaly, in which the secant method is replaced by more efficient methods, such as the second-order Steffensen’s method, as well as other high-order derivative-free methods.

First astrometric observations of space debris with the MéO telescope

The MéO (for Métrologie Optique) telescope is the Satellite and Lunar Laser Ranging (SLR) dedicated telescope of Observatoire de la Côte d’Azur (France) located at plateau de Calern. The telescope uses an altazimuth mount. The motorization of the mount has a capability of 6 deg/s allowing the follow up of Low Earth Orbits (LEO) satellites, as well as Medium Earth Orbits (MEO) and geostationary (GEO) satellites, and the Moon. The telescope has a primary mirror of 1.54 m. It uses a Nasmyth focus equipped with an EMCCD camera. The telescope field of view, defined by the equivalent focal length and the size of the camera, is currently 3.4 arcmin × 3.4 arcmin. Space debris observation with an optical telescope ideally requires a large field of view, accurate pointing, a fast slew rate, a high sensitivity, accurate astrometric positions, and a precise method for orbit propagation. The challenge is to obtain accurate orbits for all debris without compromising the field of view. The MéO telescope has a larger diameter than the ones habitually used for space debris tracking. It should improve the current accuracy of observations in the GEO region. For LEO, such sensitivity should allow observations of small pieces of debris at low altitudes. This paper presents the preliminary experiments carried out to benefit from the high astrometric quality of the instrument, namely the method developed to extract and to compute the astrometric positions of LEO and MEO satellites, as a test of the capabilities of such an instrument (very small field of view, but large aperture) for space debris tracking. Furthermore, we analyse the ability of MéO to keep track of an object for which only a preliminary orbit (computed by the Laplace method from previous observations) is known, so that high precision measurements can be obtained and the object can be catalogued with an updated orbit. The feasibility of our astrometric methods was tested throughout 2010. This paper presents the methods used and the difficulties encountered. The originality of our paper does not strictly reside in the methods used but in their adaptation to a telescope presenting such a small field of view. We discuss our first results obtained for LEO and MEO satellites. In particular, we give an estimate of the astrometric errors for the geodetic satellite Lageos-1 by comparing its positions deduced from our reduction strategy with those obtained from a reference orbit. We also present research to use the Laplace method to produce preliminary orbital elements. In this context, we discuss a novel improvement to the Laplace method. To put it in a nutshell, this article provides original results covering all fields of orbit estimation, from satellites tracking to preliminary orbit determination.