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1.
The main problem in the orbit determination of the space debris population orbiting our planet is identifying which separate sets of data belong to the same physical object. The observations of a given object during a passage above an observing station are collectively called a Too Short Arc (TSA): data from a TSA cannot allow for a complete determination of an orbit. Therefore, we have to solve first the identification problem, finding two or more TSAs belonging to the same physical object and an orbit fitting all the observations. This problem is well known for the determination of orbits of asteroids: we shall show how to apply the methods developed for preliminary orbit determination of heliocentric objects to geocentric objects. We shall focus on the definition of an admissible region for space debris, both in the case of optical observations and radar observations; then we shall outline a strategy to perform a full orbit determination.  相似文献   

2.
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.  相似文献   

3.
We study the possible degeneracies for the normal matrix of the observations from the Earth of the motion of a satellite around a planet, and give the possible solutions to the loss of precision in the orbit determination caused by the rank deficiency. Finally we discuss the methods available to control the instability in the orbit determination resulting from the degeneracy.  相似文献   

4.
A modified Laplacian technique is described for initial orbit determination of asteroids from CCD observations and its applications for orbit determination of the main belt asteroids and near Earth asteroids. The proposed modification is based on a simultaneous improvement of both the orbital elements and the derivatives of spherical coordinates in frames of Laplace's method. It provides an orbit which represents the used observations with the residuals comparable with errors of these observations. The improved values of the derivatives might be used as ephemeris parameters for identification of newly discovered objects.  相似文献   

5.
Initial asteriod orbits are determined by a least squares adjustment of an arbitrary number (N) of optical and radar observations. The usual separation, into an orbit determination by three observations and a subsequent differential orbit improvement, is combined into a single algorithm. A priori information is used for very small arcs. Ephemerides very suitable for linking are obtained by strictly linear computations.  相似文献   

6.
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.  相似文献   

7.
A method to deal with uncertainties in initial orbit determination (IOD) is presented. This is based on the use of Taylor differential algebra (DA) to nonlinearly map uncertainties from the observation space to the state space. When a minimum set of observations is available, DA is used to expand the solution of the IOD problem in Taylor series with respect to measurement errors. When more observations are available, high order inversion tools are exploited to obtain full state pseudo-observations at a common epoch. The mean and covariance of these pseudo-observations are nonlinearly computed by evaluating the expectation of high order Taylor polynomials. Finally, a linear scheme is employed to update the current knowledge of the orbit. Angles-only observations are considered and simplified Keplerian dynamics adopted to ease the explanation. Three test cases of orbit determination of artificial satellites in different orbital regimes are presented to discuss the feature and performances of the proposed methodology.  相似文献   

8.
This paper presents a comprehensive analysis of the Mars orbital phase of the Mariner 9 trajectory as determined from Earth based radio data. Both the method and accuracy of the orbit determination process are reviewed. Analysis is presented to show the effects of Mars gravity model and node in the plane of the sky errors on the accuracy of orbit determination. In addition the long term evolution of the orbit from insertion through the first 500 revolutions is presented, and decomposed into effects from the Mars garvity field,n-body perturbations, and solar radiation pressure. Since the orbit period is nearly commensurable with the Mars rotational period, the orbit experiences significant resonance perturbations. The primary perturbation is in-track with a maximum amplitude of 1000 km and a wavelength of 39 spacecraft revolutions.This paper was presented at the AIAA/AAS Astrodynamics Conference, Palo Alto, California, September 11 and 12, 1972. At this time Mariner 9 operations were still underway. The operational life of Mariner 9 ended October 27, 1972, when the supply of nitrogen gas, used for attitude stabilization, was depleted. This paper represents one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, under NASA Contract No. NAS 7-100.  相似文献   

9.
It has long been recognized and demonstrated in the astrodynamic literature that three observations of angular position are not always sufficient to determine a preliminary orbit. One reason for this is due to the fact that as the plane of the observer's motion approaches the plane of the orbit of the observed object, the determination of the orbit of the object becomes indeterminant. Merely changing the coordinate system will not eliminate the inherent indeterminacy or singularity. When the observed object is moving in the same plane as the observer, their relative motion is described in two dimensions rather than three. The problem reduces to defining two components of position and two of velocity given only three angular measures and no solution is possible. Although this singularity is a rather old, albeit infrequently arising problem in celestial mechanics, it has received renewed interest due to the advent of satellite observatories that observe other spacecraft. In this new circumstance the plane of the observer's motion is rather frequently near the plane of the object (12% to 35% of the time) and the co-planar singularity becomes a subject that deserves additional attention.It is the purpose of this paper to develop a practical and simple method of orbit determination using four observations. This method also allows one to avoid the problem of multiple orbit-determination solution roots, and provides numerical indices that are useful in assessing the degree of indeterminacy in any given observer/object geometry. This paper does not dwell at length on the theory of orbital singularities, since they have been already treated in celestial mechanics literature. Instead, the emphasis is on the details of a new computational technique, which has been found to be computationally more efficient than previous four-observation methods, and which is unique in being formulated in the geocentric system and involves only one scalar quantity in the correction process.The equations for the new method are developed and a numerical example is presented that demonstrates the efficiency of the method.  相似文献   

10.
连线干涉测量(Connected Element Interferometry, CEI)是一种全天时全天候的被动测角技术, 已用于空间目标的跟踪监视. 地球静止轨道(Geostationary Earth Orbit, GEO)卫星需要频繁机动以保持轨位或完成其他任务, 其机动后的快速轨道恢复能力对于监视预警极为重要. 针对基于CEI的GEO短弧定轨和预报, 分析了定轨算法的形亏和数亏, 在附加先验轨道约束的短弧定轨基础上, 提出了轨道半长轴初值的自适应优化方法. 利用亚太七号卫星的CEI仿真和实测数据进行了短弧定轨和预报, 实验结果表明, 采用优化后的半长轴初值, 30min短弧定轨和10min预报的卫星位置分量精度均优于4km, 能够满足非合作GEO目标机动后快速轨道恢复的需求.  相似文献   

11.
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.  相似文献   

12.
Summary Attention is called to the method(s) of orbit determination as mentioned in the heading, deriving the values of the coefficients to be applied, by adjustment by least squares, of the power series in either rectangular coordinate, to the values observed, of this, -as initiated in the past by several authors, among others by Silva and De Caro in the 30's and 40's, and critically studied by the present author in the 50ies, with merely a short note published (Lund Ann. 12, C1–C7 (1953)). The avail of modern computers for overcoming the numerical trouble by taking into account the terms of higher order, than those until the fourth necessary for the orbit computation — an account unevitable for a seccessful accomplishment, as shown by the author — may justify a resumed interest in those methods, as the only ones allowing to extract just the information contained in the observations of such a short arc — without any hypothesis in advance.  相似文献   

13.
Chaotic dynamical systems are characterized by the existence of a predictability horizon, connected to the notion of Lyapunov time, beyond which predictions of the state of the system are meaningless. In order to study the main features of orbit determination in the presence of chaos, Spoto and Milani (Celest Mech Dyn Astron 124:295–309, 2016) applied the classical least-squares fit and differential correction algorithm to determine a chaotic orbit and a dynamical parameter of a simple discrete system—Chirikov standard map (cf. Chirikov in Phys Rep 52:263, 1979)—with observations distributed beyond the predictability horizon. They found a time limit beyond which numerical calculations are affected by numerical instability: the computability horizon. In this article, we aim at pushing forward such inherent obstacle to numerical calculations in chaotic orbit determination by applying the classical and the constrained multi-arc method (cf. Alessi et al. in Mon Not R Astron Soc 423:2270–2278, 2012) to the same dynamical system. These strategies entail the determination of an orbit when observations are grouped in separate observed arcs. For each arc, a set of initial conditions is determined and, in the case of the constrained multi-arc method, all subsequent arcs are constrained to belong to the same trajectory. We show that the use of these techniques in place of the standard least-squares method has significant advantages: Not only can we perform accurate numerical calculations well beyond the computability horizon, in particular, the constrained multi-arc strategy improves considerably the determination of the dynamical parameter.  相似文献   

14.
The traditional least square estimation (LSE) method for orbit determination will not be optimal if the error of observational data does not obey the Gaussian distribution. In order to solve this problem, the least p-norm (Lp) estimation method is presented in this paper to deal with the non-Gaussian distribution cases. We show that a suitable selection of parameter p may guarantee a reasonable orbit determination result. The character of Lp estimation is analyzed. It is shown that the traditional Lp estimation method is not a robust method. And a stable Lp estimating based on data depth weighting is put forward to deal with the model error and outlier. In the orbit determination process, the outlier of observational data and coarse model error can be quantitatively described by their weights. The farther is the data from the data center, the smaller is the value of data depth and the smaller is the weighted value accordingly. The result of the new Lp method is stabler than that of the traditional Lp estimation and the breakdown point could be up to 1/2. In addition, the orbit parameter is adaptively estimated by residual analysis and matrix estimation method, and the estimation efficiency is enhanced. Finally, by taking the Space-based Space Surveillance System as an example and performing simulation experiments, we show that if there are system error or abnormal value in the observational data or system error in satellite dynamical model and space-based observation platform, LSE will not be optimal even though the observational data obeys the Gaussian distribution, and the orbit determination precision by the self-adaptive robust Lp estimation method is much better than that by the traditional LSE method.  相似文献   

15.
This paper presents a non-iterative approach to solve Kepler’s Equation, M = E − e sin E, based on non-rational cubic and rational quadratic Bézier curves. Optimal control point coordinates are first shown to be linear with respect to orbit eccentricity for any eccentric anomaly range. This property yields the development of a piecewise (e.g., 3, 4) solving technique providing accuracies better than 10−13 degree for orbit eccentricity e ≤ 0.99. The proposed method does not require large pre-computed discretization data, but instead solves a cubic/quadratic algebraic equation and uses a single final Halley iteration in only a few lines of code. The method still provides accuracies better than 10−5 degree for the near parabolic worst case (e = 0.9999) with very small mean anomalies (M < 0.0517 deg). The complexity of the proposed algorithm is constant, independent of the parameters e and M. This makes the method suitable for extensive orbit propagations. Presented at the 7th Dynamics and Control of Systems and Structures in Space Conference, July 18–22, 2006, Greenwich, England.  相似文献   

16.
In addition to the detection of an asteroid moon or a binary asteroid, the knowledge of the satellite’s true orbit is of high importance to derive fundamental physical parameters of the binary system such as its mass and to shed light on its possible formation history and dynamical evolution (prograde/retrograde orbit, large/small eccentricity or inclination, etc.). A new methodology for preliminary orbit determination of binary asteroids – and visual binaries in general – is proposed. It is based on Thiele–Innes method combined with a ‘trial and error’ Monte-Carlo technique. This method provides the full set of solutions (bundle of orbits, with the 7 orbital elements) even for a reduced number of observations. The mass is a direct by-product of this orbit determination, from which one can next infer the bulk-density and porosity. In addition to the bundle of orbits, the method provides the marginal probability densities of the foreseen parameters. Such error analysis – since it avoids linear approximation – can be of importance for the prediction of the satellite’s position in the plane-of-sky during future stellar occultations or subsequent observations, but also for the analysis of the orbit’s secular evolution. After briefly describing the method, we present the algorithm and its application to some practical cases, with particular emphasis on asteroids binaries and applications on orbital evolution.  相似文献   

17.
Charlier’s theory (1910) provides a geometric interpretation of the occurrence of multiple solutions in Laplace’s method of preliminary orbit determination, assuming geocentric observations. We introduce a generalization of this theory allowing to take into account topocentric observations, that is observations made from the surface of the rotating Earth. The generalized theory works for both Laplace’s and Gauss’ methods. We also provide a geometric definition of a curve that generalizes Charlier’s limiting curve, separating regions with a different number of solutions. The results are generically different from Charlier’s: they may change according to the value of a parameter that depends on the observations.  相似文献   

18.
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.  相似文献   

19.
Szebehely's renowned equation given in 1974, allowing for potential determination from a given orbit or family of orbits, is proved to be equivalent with an equation deduced in 1963 by Drǎmbǎ. This basic equation in the inverse problem of dynamics, for which the denomination of Drǎmbǎ –Szebehely equation is proposed, is generalized for the motion in the n-dimensional Euclidean space. A method for the determination of the potential function from motion equations is extended to this space.  相似文献   

20.
Laplace’s method is a standard for the calculation of a preliminary orbit. Certain modifications, briefly summarized, enhance its efficacy. At least one differential correction is recommended, and sometimes becomes essential, to increase the accuracy of the computed orbital elements. Difficult problems, lack of convergence of the differential corrections, for example, can be handled by total least squares or ridge regression. The differential corrections represent more than just getting better agreement with the observations, but a means by which a satisfactory orbit can be calculated. The method is applied to three examples of differing difficulty: to calculate a preliminary orbit of Comet 122/P de Vico from 59 observations made during five days in 1995; a more difficult calculation of a possible new object with a poor distribution of observations; Herget’s method fails for this example; and finally a really difficult object, the Amor type minor planet 1982 DV (3288 Seleucus). For this last object use of L1 regression becomes essential to calculate a preliminary orbit. For this orbit Laplace’s method compares favorably with Gauss’s.  相似文献   

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