Parallel initial orbit determination using angles-only observation pairs

We propose a type of admissible-region analysis for track initiation in multi-satellite problems when angles are the primary observable. Pairs of optical observations are used to calculate candidate orbits via a Lambert solver by hypothesizing range values. The method is attractive because it allows multiple levels of parallelization of the track-initiation process. Orbital element partitions are introduced to divide the admissible region into smaller search spaces to be processed on individual computer nodes. For a specified rectangular partition in the space of orbital elements, constraints are developed to bound the values of range that will lead to initial orbit hypotheses (data association hypotheses) associated with that partition. These bounds allow us to parallelize the generation of candidate orbits, because each element-space partition can be handled independently of the others. Several constraints are developed and shown to limit the range pair hypotheses effectively to the constrained admissible region based on the orbital element partitions. Examples are provided to highlight the topology of the proposed constraints.     

Dealing with uncertainties in angles-only initial orbit determination

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.     

A numerical comparison with the Ceplecha analytical meteoroid orbit determination method

Abstract– Analytic methods by Ceplecha have long been used for the determination of meteoroid heliocentric orbits. These methods include both the derivation of an initial atmospheric contact position and velocity state, and the calculation of an orbit at infinity based on zenithal attraction assumptions. Herein, we describe a numerical integration‐based verification for a portion of the Ceplecha methods, a verification driven by the need for an accurate meteoroid ephemeris in the hours before atmospheric contact. We show a close correspondence in analytic and numerical results, with a previously undocumented minor correction to a meteoroid’s longitude of the ascending node.     

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.     

A new method for the determination of an intermediate orbit using three positions of a small body on the celestial sphere

We propose a new method for the determination of the preliminary orbit of a small celestial body using three pairs of its angular coordinates in three moments of time. The method is based on the use of the intermediate orbit we constructed earlier using three position vectors and the corresponding time moments. This intermediate orbit accounts for the main part of the perturbations of the motion of the body under study. We compare the results obtained by the classical Lagrange-Gauss method, Herrick-Gibbs method, generalized Herrick-Gibbs method, and the new method by the examples of the determination of the orbit of the small planet 1566 Icarus. The comparison showed that the new method is a highly efficient tool for the study of perturbed motion. It is especially efficient when applied to high-precision observational data covering short arcs of the orbit.     

A semi-analytical method of orbit computation

It is efficient to compute the long-period and secular perturbations by numerical intergration and to use the classical analytical solution for the short-period perturbations.     

Errors in orbit determination

The gaussian noise which affects tracking measurements causes an error in the computed value of the orbit parameters. This study provides a method for evaluating: (a) the length of the arc over which the satellite must be tracked; (b) the number of measurements to be made along this arc; (c) the position of the arc with respect to the orbit, necessary to reach the desired accurary of the calculated orbit parameters for a given pointing error of the tracking antenna. It has been assumed that the errors of the tracking measurements have a known gaussian probability distribution, which may differ for each measurement. The equations relating the orbit parameters to the measurements performed have been linearized. It has been shown that the orbit parameters are gaussian random variables and their variance has been calculated as a function of (a), (b) and (c).This paper was presented at the AIAA/AAS meeting, Princeton University, August 1969.     

A multi-arc approach for chaotic orbit determination problems

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.     

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.     

A new determination of the orbit and masses of the Be binary system δ Scorpii

The binary star δ Sco (HD143275) underwent remarkable brightening in the visible in 2000, and continues to be irregularly variable. The system was observed with the Sydney University Stellar Interferometer (SUSI) in 1999, 2000, 2001, 2006 and 2007. The 1999 observations were consistent with predictions based on the previously published orbital elements. The subsequent observations can only be explained by assuming that an optically bright emission region with an angular size of  ≳2 ± 1 mas  formed around the primary in 2000. By 2006/2007 the size of this region grew to an estimated ≳4 mas.
We have determined a consistent set of orbital elements by simultaneously fitting all the published interferometric and spectroscopic data as well as the SUSI data reported here. The resulting elements and the brightness ratio for the system measured prior to the outburst in 2000 have been used to estimate the masses of the components. We find   M_A = 15 ± 7 M_⊙  and   M_B = 8.0 ± 3.6 M_⊙  . The dynamical parallax is estimated to be  7.03 ± 0.15 mas  , which is in good agreement with the revised Hipparcos parallax.     

A robust estimation method in orbit improvement

Orbit improvement is studied via processing simulated optical data of 20 satellites. Using our criterion the fraction of outliers falls to less than 10% in most cases. Our method, which ensures the accuracy of orbit improvement, consists in using the Huber estimator twice, followed by iterating the Hampel estimator until convergence. The parameters C₀ and C₁ in the Hampel estimation may be given the fixed values of 2.2 and 3.6, while the value of C₂ varies in a definite way with the size of the outlier considered.     

A projective approach to orbit determination from three sight-lines

Concepts from projective geometry are used to provide a coherent framework for the determination of orbits from observation data comprising lines of sight at three known times. A novel way of presenting the results in a finite diagram is introduced, The effectiveness of the approach is demonstrated by an example, using a simple spreadsheet. A computer-graphic implementation is recommended.     

Application of the total least squares method to the determination of preliminary orbit

Based on the analysis of the characteristics of the equations of condition in the UVM2 (Unit Vector Method 2), the total least squares method (TLS) is introduced into the orbital determination and the linearization of the vis-viva formula in the original algorithm is thereby avoided. The calculated results from simulation and observation data show that the application of TLS to UVM2 is effective.     

GEO卫星定轨可视化软件设计

介绍了基于Windows系统开发的GEO卫星定轨可视化软件,该软件是采用Microsoft Visual Studio 2005软件平台,利用Visual Basic.NET编程技术开发设计的,具有预处理观测数据资料、解算GEO卫星精密轨道、分析和图形化轨道解算结果等功能。该软件界面友好、可操作性强、方便省时,有效地提高了GEO卫星定轨工作效率。     

A method of intermediate orbit determination based on four positions of a minor body on the celestial sphere

A new method of computing the preliminary orbit of a celestial body based on four pairs of angle measurements has been suggested. The method makes use of preliminary orbit previously constructed by the author based on two position vectors and a corresponding time interval, taking into account the main part of the perturbations in the motion of the body under study. Using the example of constructing the orbit of the minor planet 1383 Limburgia, the results obtained using a four-position procedure of the Gaussian type based on the solution of a two-body problem have been compared with those of the new method. The comparison showed the new method to be highly efficient for perturbed motion studies. It is especially advantageous in the case of high-accuracy observation data on small orbital arcs.     

A new procedure for the solution of the classical problem of minimal orbit determination from three lines of sight

A new method has been developed for obtaining multiple solutions of the classical angles-only initial-orbit-determination problem. The method operates by a higher-order Newton correction of the assumed values for two of the unknown ranges, with the author's universal Lambert algorithm at the heart of the procedure. The observations are permitted to span several revolutions when the orbit is elliptic, and the method is free of artificial singularities in the configuration of observers and sight-lines; thus it has been successfully used with some test problems that would not be solvable by existing methods.Former affiliation: Royal Aerospace Establishment, Farnborough     

Unified iterative methods in orbit determination

The classical problem of Keplerian orbit determination from only three measurements of time and angular coordinates (t _i, _i, _i) has been solved here numerically in two different ways, using Newton's method for non-linear equations in both cases. The first method (Perov, 1989) is based on KS variables, whereas the second emphasizes the fundamental part played by the unified Lambert's equation and the related formulae in that kind of applications. These two methods have been compared and put into practice in various numerical tests based on real asteroid orbits and ficitious Keplerian asteroid, comet and artificial satellite orbits in order to try the stability of these methods for peculiar orbits.     

Initial orbit determination using multiple observations

A novel approach for initial orbit determination based on multiple angles-only observations is presented. The proposed technique is iterative and uses Lagrangian coefficients, f and g. The proposed method does not show singularity for the coplanar cases. In addition, the method is capable of handling multiple observations, providing higher accuracy, whereas the level of the algorithm complexity and processor running time remain almost invariant. The technique presented is compared with the Double r-iteration and Gauss’ methods using data corrupted by noise to simulate true measurements. Results show that the proposed method is a valid alternative to the classical methods of orbit determination.