TLE-Aided Orbit Determination Using Single-Station SLR Data

It is difficult to use the single-station satellite laser ranging (SLR) data for orbit determination, due to the singular geometrical distribution of the observations. The single-station data produced by performing the diffuse- reflection SLR on the earth-orbiting space debris are therefore ineffective for orbit improvement. To solve this problem, we propose an orbit determination method by using single-station SLR data in aid of the two-line element set (TLE). For verifying its feasibility, this method is implemented and applied to the orbit determination of the satellite Ajisai, using the single-station SLR data of five passes in one day and the corresponding TLE. And on this basis, the five-day orbit prediction is generated, the result indicates that the errors of predicted positions are less than 40 m. In addition, the potential application of this method in the orbit improvement of space debris is discussed.     

Single lunation <Emphasis Type="Italic">N</Emphasis>-observation orbits

Initial orbit determination by least squares of N observations is essentially a linear problem if the coordinates x ₀ and x ₁ at two standard epochs are used as elements. The orbit of a main belt object is approximated within the observational errors by a third degree polynomial during a month. A 4-observation orbit is useful for the initial linking between two nights. Parallax is treated rigorously and future simultaneous space and Earth based observations determine the critical distance directly. The N-observation method is a great simplification of the classical 3-observation orbit followed by a differential correction by N observations.     

两行根数辅助的SLR单站定轨

单站测距资料定轨的困难限制了漫反射SLR(Satellite Laser Ranging)测距资料的应用.为此,提出利用两行根数模拟多站SLR测距资料作为辅助,实现单站SLR测距资料定轨的方法.该方法对卫星Ajisai单站SLR测距资料定轨并生成5 d预报轨道,误差小于40 m,实现利用单站测距资料的轨道改进,验证了方法的可行性.     

The space debris environment of the Earth

37 years of space activities have led to a large number of anthropogenic objects orbiting the Earth. Ground-based observations with radar and optical facilities reveal the existence of about 7500 objects in space, which do not represent an immediate excessive danger. However, adequate actions are required to keep the long-term debris hazard for manned and unmanned missions within acceptable tolerances. In this paper the space debris environment of the Earth and its future evolution are described. New developments which could have a major impact on the space environment, are the the planned multi-satellite constellations for communications purposes or solar power stations in Earth orbit. Finally, methods for debris reduction are outlined. Space debris is a global problem which can only be effectively solved by international cooperation.     

Innovative methods of correlation and orbit determination for space debris

We propose two algorithms to provide a full preliminary orbit of an Earth-orbiting object with a number of observations lower than the classical methods, such as those by Laplace and Gauss. The first one is the Virtual debris algorithm, based upon the admissible region, that is the set of the unknown quantities corresponding to possible orbits for a given observation for objects in Earth orbit (as opposed to both interplanetary orbits and ballistic ones). A similar method has already been successfully used in recent years for the asteroidal case. The second algorithm uses the integrals of the geocentric 2-body motion, which must have the same values at the times of the different observations for a common orbit to exist. We also discuss how to account for the perturbations of the 2-body motion, e.g., the J ₂ effect.     

From Astrometry to Celestial Mechanics: Orbit Determination with Very Short Arcs

Contemporary surveys provide a huge number of detections of small solar system bodies, mostly asteroids. Typically, the reported astrometry is not enough to compute an orbit and/or perform an identification with an already discovered object. The classical methods for preliminary orbit determination fail in such cases: a new approach is necessary. When the observations are not enough to compute an orbit we represent the data with an attributable (two angles and their time derivatives). The undetermined variables range and range rate span an admissible region of solar system orbits, which can be sampled by a set of Virtual Asteroids (VAs) selected by an optimal triangulation. The attributable results from a fit and has an uncertainty represented by a covariance matrix, thus the predictions of future observations can be described by a quasi-product structure (admissible region times confidence ellipsoid), which can be approximated by a triangulation with each node surrounded by a confidence ellipsoid. The problem of identifying two independent short arcs of observations has been solved. For each VA in the admissible region of the first arc we consider prediction at the time of the second arc and the corresponding covariance matrix, and we compare them with the attributable of the second arc with its own covariance. By using the penalty (increase in the sum of squares, as in the algorithms for identification) we select the VAs which can fit together both arcs and compute a preliminary orbit. Even two attributables may not be enough to compute an orbit with a convergent differential corrections algorithm. The preliminary orbits are used as first guess for constrained differential corrections, providing solutions along the Line Of Variations (LOV) which can be used as second generation VAs to further predict the observations at the time of a third arc. In general the identification with a third arc will ensure a least squares orbit, with uncertainty described by the covariance matrix.     

面向单脉冲信号分类的集成特征选择与评价

受大量射频干扰信号影响,快速从海量观测数据中准确识别出单脉冲信号已成为天文数据处理的一项重要任务,而设计和提取有效数据特征,是利用机器学习进行单脉冲信号高效识别的决定因素.针对如何选择最优特征,进而提升单脉冲信号的分类精度这一关键问题,设计了面向单脉冲信号分类的集成特征选择方法.方法首先混合单脉冲信号的参数特征、统计特征和抽象特征,然后分别利用5种单一特征选择方法选出各自的最优特征集,最后利用贪心策略对5种单一方法获取的最优特征集进行集成筛选,获取最优集成特征集.实验表明,最优特征集合既包含统计特征也包含抽象特征.在相同特征数量下,利用集成特征选择比单一特征选择能获得更高的模型精度,可使F1值最高提升1.8%.在海量数据背景下,集成特征选择对减少特征数量、提升分类性能和加快数据处理速度具有重要作用.     

Empirical model of motion of space debris in the geostationary region

Regular optical observations of small-sized space debris on a geosynchronous orbit are carried out at the Terskol Peak Observatory (Kabardino-Balkaria). The aim of these observations is to gather data on technogenic pollution of near-Earth space; discover small-sized space debris, determine their parameters, and catalogue them; and maintain an updated catalogue of orbits. The results obtained by processing topocentric measurements of positions of one such object are presented. This object is distinctive in that it is a small-sized fragment of the Fengyun 2D satellite (international designator 2006-053A) and moves about a Lagrangian point at 75° E with a small amplitude. The observational material was accumulated in 12 twoweek sessions within a 6-year interval from 2009 to 2014. The variability of the ratio of the maximum cross section area to the object mass is revealed, and a model of variations of this parameter is proposed. The basic period of variations is 392 days. The validity of the model was verified in a series of independent observations in March, 2015.     

Keplerian integrals,elimination theory and identification of very short arcs in a large database of optical observations

Modern asteroid surveys produce an increasingly large number of observations, which are grouped into very short arcs (VSAs) each containing a few observations of the same object in one single night. To decide whether two VSAs collected in different nights correspond to the same observed object we can attempt to compute an orbit with the observations of both arcs: this is called the linkage problem. Since the number of linkages to be attempted is very large, we need efficient methods of orbit determination. Using the first integrals of Kepler’s motion we can write algebraic equations for the linkage problem, which can be put in polynomial form. In Gronchi et al. (Celest Mech Dyn Astron 123(2):105–122, 2015) these equations are reduced to a polynomial equation of degree 9: the unknown is the topocentric distance of the observed body at the mean epoch of one VSA. Here we derive the same equations in a more concise way, and show that the degree 9 is optimal in a sense that will be specified in Sect. 3.3. We also introduce a procedure to join three VSAs: from the conservation of angular momentum we obtain a polynomial equation of degree 8 in the topocentric distance at the mean epoch of the second VSA. For both identification methods, with two and three VSAs, we discuss how to discard solutions. Finally, we present some numerical tests showing that the new methods give satisfactory results and can be used also when the time separation between the VSAs is large. The low polynomial degree of the new methods makes them well suited to deal with the very large number of asteroid observations collected by the modern surveys.     

Modeling Technogenous Contamination of the Near-Earth Space

In analyzing the technogenous contamination of the near-earth space (NES), the following issues have usually been considered: the estimation of the current level of NES contamination by objects of various size; the modeling of technogenous contamination evolution; the estimation of the probability of spacecraft collisions with space objects (SOs) of various size and the possible implications of hazardous collisions; the determination of characteristics of the flux of SOs of various size through the observation zones of ground-based and onboard means of measurement. The main difficulty in solving the aforementioned problems lies in the deficiency of the experimental data. The available measurement information was obtained in relatively small domains of multidimensional space: the altitude of a point–the latitude of a point–the SO size–the time. As a consequence, additional (a priori) information is invoked for determining the technogenous contamination characteristics at various points of the region mentioned above. The efficient use of experimental data and a priori information constitutes the basic problem of space debris modeling. This paper briefly outlines the data on three space debris (SD) models: ORDEM2000 (Orbital Debris Engineering Model, 2000), MASTER'99 (Meteoroid and Space Debris Terrestrial Environment Reference Model, 2000) and SDPA (Nazarenko, 1997, 2000; Nazarenko and Menshikov, 2001). The features of modeling techniques and the comparative characteristics of technogenous contamination are discussed in the paper.     

Gaia, an unprecedented observatory for Solar System dynamics

The mission Gaia by European Space Agency (ESA) is expected to fly at the end of 2011 and to perform an all-sky, magnitude-limited survey for 5 years. The probe will not use an input catalogue, and will get high accuracy astrometry and photometry for all sources of magnitude V<20. Low-resolution spectra will also be available. Moving Solar System objects will be observed as well, and their observations will be processed by a specific pipeline in order to retrieve the physical and dynamical characteristics of each object. In this contribution we will mainly focus on the impact of Gaia observations on asteroid dynamics. A dramatic improvement of orbital elements is expected, as well as the measurement of subtle effects such as those related to general relativity (GR). Gaia observations will also be supported by a network of ground-based observation sites, capable of providing follow-up for newly discovered objects that will not receive an adequate coverage from space. Specific strategies for follow-up are being planned and tested. These will need to take into account the peculiar observing geometry (large parallax effect due to the orbit of Gaia around L2) and the time constraints dictated by data processing.     

Reconstruction of objects by direct demodulation

High resolution reconstruction of complicated objects from incomplete and noisy data can be achieved by solving modulation equations iteratively under physical constraints. This direct demodulation method is a powerful technique for dealing with inverse problem in general case. Spectral and image restorations and computerized tomography are only particular cases of general demodulation. It is possible to reconstruct an object in higher dimensional space from observations by a simple lower dimensional instrument through direct demodulation. Our simulations show that wide field and high resolution images of space hard X-rays and soft-rays can be obtained by a collimated non-position-sensitive detector without coded aperture masks.     

Topocentric orbit determination: Algorithms for the next generation surveys

The process of calculating a good orbit from astrometric observations of the same object involves three main steps: preliminary orbit determination, least squares orbit fitting, and quality control assessing the orbit's uncertainty and reliability. For the next generation sky surveys, with much larger number density of observations, new algorithms, or at least substantial revisions of the classical ones, are needed. The classical theory of preliminary orbit algorithms was incomplete in that the consequences of the topocentric correction had not been fully studied. We show that it is possible to rigorously account for topocentric observations and that this correction may increase the number of alternate preliminary orbits without impairing the overall performance. We have developed modified least squares algorithms including the capability of fitting the orbit to a reduced number of parameters. The restricted fitting techniques can be used to improve the reliability of the orbit computing procedure when the observed arcs have small curvature. False identification (where observations of different objects are incorrectly linked together) can be discarded with a quality control on the residuals and a ‘normalization’ procedure removing duplications and contradictions. We have tested our algorithms on two simulations based on the expected performance of Pan-STARRS—one of the next generation all-sky surveys. The results confirm that large sets of discoveries can be handled very efficiently resulting in good quality orbits. In these tests we lost only 0.6 to 1.3% of the possible objects, with a false identification rate in the range 0.02 to 0.06%.     

An Accuracy Analysis of the SGP4/SDP4 Model

Based on the latest release of the SGP4/SDP4 (Simplified General Perturbation Version 4/ Simplified Deep-space Perturbation Version 4) model, in this paper we have designed an orbit determination program. Through calculations for the 1120 objects with various types and orbital elements selected from the space objects database, we have obtained the accuracies of the orbit determination prediction dealt with various types of space objects by the SGP4/SDP4 model. The results show that the accuracies of the near-earth objects are in the order of magnitude of 100 meters; the averages of the orbit determination accuracies of the semi-synchronous and geosynchronous orbits are, respectively, 0.7 and 1.9 km. The orbit determination accuracies of the elliptical orbit objects are related to their eccentricities. Except for few elliptical orbit objects with e > 0.8, the orbit determination errors of the vast majority of the elliptical orbit objects are all less than 10 km. By using the SGP4/SDP4 model to make 3 days predictions for near-earth objects, 30 days for semi-synchronous orbit objects, 15 days for geosynchronous orbit objects and 1 day for elliptical orbit objects, the errors of prediction generally don’t exceed 40 km.     

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.     

Characteristics of small-sized space debris objects using Terskol observatory measurements

The Zeiss-2000 telescope of the International Center for Astronomic and Medico-Ecological Research, National Academy of Sciences of Ukraine (Terskol observatory), with a 2-meter aperture is the largest optical instrument in Europe that is regularly used for investigating space debris in the vicinity of the geostationary orbit. One of the main objectives is to detect and characterize small fragments of space debris that are difficult to approach for other telescopes. During each photometric night, we usually detect four to five unknown fragments of 17th to 20th magnitude. This article provides orbital parameters and physical characteristics of several small-sized fragments of space debris that were detected during observations at Terskol observatory in 2014–2015.     

SGP4/SDP4模型精度分析

本文基于最新发布的SGP4/SDP4(Simplified General Perturbation Version 4/Simplified Deep-space Perturbation Version 4)模型设计了一套定轨方案,从空间目标库中挑选出不同类型和轨道参数的1120个目标进行计算,定量给出了SGP4/SDP4模型处理不同类型空间目标的定轨预报精度.结果表明:近地目标定轨精度为百米量级;半同步和同步轨道定轨精度平均为0.7和1.9km.椭圆轨道目标的定轨精度与偏心率有关,除少数e>0.8的椭圆轨道目标,绝大多数椭圆轨道目标定轨误差均小于10km.用SGP4/SDP4模型对近地目标预报3天,半同步轨道预报30天,同步轨道预报15天,椭圆轨道预报1天,预报误差一般不超过40km.     

Formation of the extreme Kuiper-belt binary 2001 QW<Subscript>322</Subscript> through adiabatic switching of orbital elements

Binaries in the Kuiper-belt are unlike all other known binaries in the Solar System. Both their physical and orbital properties are highly unusual and, because these objects are thought to be relics dating back to the earliest days of the Solar System, understanding how they formed may provide valuable insight into the conditions which then prevailed. A number of different mechanisms for the formation of Kuiper-belt binaries (KBBs) have been proposed including; two-body collisions inside the Hill sphere of a larger body; strong dynamical friction; exchange reactions; and chaos assisted capture. So far, no clear consensus has emerged as to which of these mechanisms (if any) can best explain the observed population of KBBs. Indeed, the recent characterization of the mutual orbit of the symmetric (i.e., roughly equal mass) KBB 2001 QW₃₂₂ has only served to complicate the picture because its orbit does not seem readily explicable by any of the available models. The binary 2001 QW₃₂₂ stands out even among the already unusual population of KBBs for the following reasons: its mutual orbit is extremely large (≈10⁵ km or about 30% of the Hill sphere radius), retrograde, it is inclined ≈120^° from the ecliptic and has very low eccentricity, i.e., e ≤ 0.4 (and possibly e ≤ 0.05). Here we propose a hybrid formation mechanism for this object which combines aspects of several of the mechanisms already proposed. Initially two objects are temporarily trapped in a long-living chaotic orbit that lies close to a retrograde periodic orbit in the three-dimensional Hill problem. This is followed by capture through gravitational scattering with a small intruder object. Finally, weak dynamical friction gradually switches the original orbit “adiabatically” into a large, almost circular, retrograde orbit similar to that actually observed.     

Orbit of an Astrometric Binary System

We present a new method to solve the problem of initial orbit determination of any binary system. This method is mainly based on the material available for an observer, for example relative positions at a given time of the couple in the “plane of sky”, namely the tangent plane to the celestial sphere at the position of the primary component. The problem of orbit determination is solved by splitting in successive stages in order to decorrelate the parameters of each other as much as possible. On one hand, the geometric problem is solved using the first Kepler’s law from a single observing run and, on the other hand, dynamical parameters are then inferred from the fit of the Kepler’s equation. At last, the final stage consists in determining the main physical parameters involved in the secular evolution of the system, that is the spin axis and the J₂ parameter of the primary if we assume that it is a quasi-spherical body. As a matter of fact there is no need to make too restrictive initial assumptions (such as circular orbit or zero eccentricity) and initial guesses of parameters required by a non-linear least-squares Levenberg–Marquardt algorithm are finally obtained after each stage. Such a protocol is very useful to study systems like binary asteroids for which all of the parameters should be considered a priori as unknowns. As an example of application, we used our method to estimate the set of the Pluto–Charon system parameters from observations collected in the literature since 1980.     

Analytic characterization of measurement uncertainty and initial orbit determination on orbital element representations

This paper presents an approach to characterize the uncertainty associated with the state vector obtained from the Herrick-Gibbs orbit determination approach using transformation of variables. The approach is applied to estimate the state vector and its probability density function for objects in low Earth orbit using sparse observations. The state vector and associated uncertainty estimates are computed in Cartesian coordinates and Keplerian elements. The approach is then extended to accommodate the $J_2$ perturbation where the state vector is written in terms of mean orbital elements. The results obtained from the analytical approach presented in this paper are validated using Monte Carlo simulations and compared with the often utilized similarity transformation for Kepler, mean, and nonsingular elements. The measurement uncertainty characterization obtained is used to initialize conventional nonlinear filters as well as operate a Bayesian approach for orbit determination and object tracking.