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.     

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. (Mon Not R Astron Soc 451(2):1883–1891, 2015b), 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 four 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 velocity, which is the rate of change of the line of sight. In the same spirit of Gronchi et al. (Mon Not R Astron Soc 451(2):1883–1891, 2015b), 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.     

Orbit determination with the two-body integrals

We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results from the observation error statistics. The performance of this method has been investigated by using a large set of simulated observations of the Pan-STARRS project.     

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.     

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%.     

Multiple solutions in preliminary orbit determination from three observations

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.     

Determination of the orbital parameters of binary pulsars

We present a simple method for determination of the orbital parameters of binary pulsars, using data on the pulsar period at multiple observing epochs. This method uses the circular nature of the velocity space orbit of Keplerian motion and produces preliminary values based on two one-dimensional searches. Preliminary orbital parameter values are then refined using a computationally efficient linear least-squares fit. This method works for random and sparse sampling of the binary orbit. We demonstrate the technique on (i) the highly eccentric binary pulsar PSR J0514−4002 (the first known pulsar in the globular cluster NGC 1851) and (ii) 47 Tuc T, a binary pulsar with a nearly circular orbit.     

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.     

Orbit determination of space debris: admissible regions

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.     

Asteroid close encounters characterization using differential algebra: the case of Apophis

A method for the nonlinear propagation of uncertainties in Celestial Mechanics based on differential algebra is presented. The arbitrary order Taylor expansion of the flow of ordinary differential equations with respect to the initial condition delivered by differential algebra is exploited to implement an accurate and computationally efficient Monte Carlo algorithm, in which thousands of pointwise integrations are substituted by polynomial evaluations. The algorithm is applied to study the close encounter of asteroid Apophis with our planet in 2029. To this aim, we first compute the high order Taylor expansion of Apophis’ close encounter distance from the Earth by means of map inversion and composition; then we run the proposed Monte Carlo algorithm to perform the statistical analysis.     

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.     

Asteroid identification over apparitions

We present a new method for the linking of scarce asteroid astrometry over apparitions, and apply it both to simulated and real data to prove its feasibility. Up to date, there has not been a robust method available to search for linkages between the approximately 50,000 provisionally designated sets of asteroid astrometry spanning less than two days. Unless such a scarce set of astrometry is linked to another set of astrometry, the underlying object can be considered lost as the ephemeris uncertainties are substantial. The new method, which can tackle the challenges, is based on Ranging, which is a fully nonlinear, statistical orbital inversion method. Ranging properly treats astrometric uncertainties and propagates the uncertainty to the resulting orbital-element probability density, which is sampled by a set of orbits. The new orbital-element-space multiple-address-comparison (oMAC) method uses dimensionality-reduction techniques and tree structures to efficiently search for overlapping probability densities in the orbital-element phase space. Overlapping probability densities indicate a candidate linkage between astrometric observation sets. To accept a candidate linkage, we have to find a many-body orbital solution which reproduces the observed positions within the observational uncertainties. To find the linking orbit, we use a multi-step approach starting from a Monte-Carlo generation of possible orbits in a reduced volume of the orbital-element phase space and ending with a least-squares orbital solution, which, in addition to the Sun's gravitation, also takes into account the gravitational influence of the relevant planets. The new multiple-address-comparison method has a loglinear computational complexity, that is, it scales as O(nlogn), where n is the number of included observation sets. It has recently also been implemented for the ephemeris-space multiple-address-comparison (eMAC) method, which is optimized for the short-term linking of scarce astrometry.     

Laplacian Orbit Determination and Differential Corrections

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.     

Linking large-parallax Spitzer–CFHT–VLT astrometry of asteroids

We show that the new ephemeris-space multiple-address-comparison (eMAC) method solves asteroid linking problems despite large parallaxes by applying the method to astrometric asteroid observation sets obtained nearly simultaneously with the Spitzer space telescope, the Canada–France–Hawaii Telescope (CFHT), and European Southern Observatory's Very Large Telescope (VLT). For main-belt asteroids, the parallax between Spitzer and the Earth-based telescopes is approximately one degree which is large as compared to a typical parallax for solely Earth-based telescopes in the arcseconds regime. In the eMAC method, we reduce the initially huge amount of possible linkages between observation sets by comparing samples of ephemerides that have been computed separately for all sets at, say, three common dates. If the non-zero ephemeris probability densities overlap at all common dates, we try to find an orbit solution for these so-called trial linkages. If there exists an orbit which reproduces all the astrometric observations assuming predefined observational errors, we call it a linkage. Known asteroids are independently identified among Spitzer, CFHT, and VLT astrometry, and comparing the identified observations to the linkages found shows that the method found all known correct linkages present in the data. In addition, we also found five previously unpublished linkages between Spitzer astrometry and Earth-based astrometry. Based on our simulations, we found virtually all Spitzer-related linkages between two single-night observation sets, and more than 99.4% of linkages between two single-night observation sets obtained by Earth-based observatories. Virtually all correct linkages consisting of at least three single-night sets were also detected. The results show that large-parallax discovery observations made from a spacecraft can be linked to Earth-based follow-up observations to ensure that the objects are not lost. Furthermore, we compute the heliocentric and Spitzer-centric distances as well as the corresponding solar phase angles at the dates of Spitzer observations. Based on comparisons to simulated geocentric observations, we also show that, for typical nearly-simultaneous observations, the parallax reduces the distance uncertainties by several orders of magnitude.     

Strategy and Scenario for Radioastron Space VLBI Observations

Space VLBI's highly dynamic geometry, ability to access the space radio telescope (SRT) only via distant communication links, very expensive mission cost, and scientific goals define the basic strategy and scenario for mission control and radio source observations. These are very different from those for ground-based VLBI. Space VLBI strategy must be based on the limitation of SRT repointings, periodic orbit determination before astronomical observations, preliminary preparation and checking of space and ground facilities, and recommended observing sequences and modes. A control and observation scenario is considered for an in-orbit-checkout period, and also for short (1-orbit – 1 week) and long (1 week and more) observation sessions. Examples and illustrations are given for the Radioastron space VLBI Project.     

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.     

Circular and zero-inclination solutions for optical observations of Earth-orbiting objects

Situational awareness of Earth-orbiting particles is important for human extraterrestrial activities. Given an optical observation, an admissible region can be defined over the topocentric range/range-rate space, with each point representing a possible orbit for the object. However, based on our understanding of Earth orbiting objects, we expect that certain orbits in that distribution, such as circular or zero-inclination orbits, would be more likely than others. In this research, we present an analytical approach for describing the existence of such special orbits for a given observation pass, and investigate topological features of the range/range-rate space by means of singularities in orbital elements.     

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.     

Gravity assist space pruning based on differential algebra

In this paper a differential algebra version of the gravity assist space pruning algorithm is presented. The use of differential algebraic techniques is proposed to overcome the two main drawbacks of the existing algorithm, i.e., the steep increase of the number of function evaluations with the number of planets involved in the transfer, and the use of a bounding procedure that relies on Lipschitzian tolerances. Differential algebra allows us to process boxes in place of grid points, and to substitute pointwise evaluations of the constraint functions with their Taylor expansions. Thanks to the particular instance of multi-gravity assist problems dealt with, all the planet-to-planet legs can be treated independently, and forward and backward constraining can be applied. The proposed method is applied to preprocess the search space of sample interplanetary transfers and it also serves as a stepping stone towards a fully rigorous treatment of the pruning process based on Taylor models.     

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.