Orbit determination with very short arcs: II. Identifications

When the observational data are not enough to compute a meaningful orbit for an asteroid/comet we can represent the data with an attributable, i.e., 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 means of an optimal triangulation [Milani, A., Gronchi, G.F., de' Michieli Vitturi, M., Kne?evi?, Z., 2004. Celest. Mech. Dyn. Astron. 90, 59-87]. The attributable 4 coordinates are the result of a fit and they have an uncertainty, represented by a covariance matrix. Two short arcs of observations, represented by two attributables, can be linked by considering for each VA (in the admissible region of the first arc) the covariance matrix for the prediction at the time of the second arc, and by comparing it with the attributable of the second arc with its own covariance. By defining an identification penalty we can select the VAs allowing to fit together both arcs and compute a preliminary orbit. Two attributables may not be enough to compute an orbit with convergent differential corrections. Thus the preliminary orbit is used in a constrained differential correction, providing solutions along the Line Of Variation 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 well determined orbit, to which additional sets of observations can be attributed. To test these algorithms we use a large scale simulation and measure the completeness, the reliability and the efficiency of the overall procedure to build up orbits by accumulating identifications. Under the conditions expected for the next generation asteroid surveys, the methods developed in this and in the preceding papers are efficient enough to be used as primary identification methods, with very good results. One important property is that the completeness in finding the possible identifications is as good for comparatively rare orbits, such as the ones of Near-Earth Objects, as for main belt orbits.     

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.     

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%.     

<Emphasis Type="Italic">N</Emphasis>-observations and radar orbits

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.     

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.     

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.     

Development of an observational error model

In calculating the orbit of a minor planet with a least-squares algorithm, current practice is to assume that all observations of a given era have the same uncertainty, and that the errors in these observations are uncorrelated. These assumptions are unrealistic; and they lead to sub-optimal orbits.Our objective is to develop and validate an observational error model that provides realistic estimates of the uncertainties and correlations in asteroid observations. When used to populate the covariance matrix of the least-squares algorithm, the resulting orbits are shown to more accurately and precisely represent asteroid trajectories.     

Three tenuous rings/arcs for three tiny moons

Using Cassini images, we examine the faint material along the orbits of Methone, Anthe and Pallene, three small moons that reside between the orbits of Mimas and Enceladus. A continuous ring of material covers the orbit of Pallene; it is visible at extremely high phase angles and appears to be localized vertically to within ±25 km of Pallene's inclined orbit. By contrast, the material associated with Anthe and Methone appears to lie in longitudinally confined arcs. The Methone arc extends over ∼10° in longitude around the satellite's position, while the Anthe arc reaches ∼20° in length. The extents of these arcs are consistent with their confinement by nearby corotation eccentricity resonances with Mimas. Anthe has even been observed to shift in longitude relative to its arc in the expected manner given the predicted librations of the moon.     

基于智能优化算法的小天体初轨确定

经典的初轨确定方法包括Laplace方法和Gauss方法以及它们的各种变化形式. 除这些经典方法之外, 基于当今光学观测数据的特点, 学者们也陆续提出了一些其他的初轨确定方法, 包括双r (目标距离观测者的距离)方法和可行域方法. 双r方法的一种实现方式是通过猜测某两个时刻(通常是定轨弧段的首、末时刻)目标离观测者的距离, 结合观测者在空间中的位置矢量, 即可求解相应的Lambert弧段作为目标轨道的初始猜测. 进一步, 以其他观测时刻的RMS (Root Mean Square)为优化变量可以改进初始猜测从而确定初轨. 可行域方法则是针对一组初始观测参数(包括赤经、赤纬及其变率), 根据一些初始假设将目标(离观测者的)距离及其变率约束在可行域内, 并通过三角划分逐步逼近的方式寻找到使观测RMS最小的猜测解. 针对一系列模拟观测数据以及实测数据, 将智能优化算法(粒子群算法)应用于这两种初轨方法, 并将结果与改进的Laplace算法的结果进行比较. 由于双r方法不仅可以用于短弧定轨还可用于长弧关联, 所以进一步给出了针对长弧段数据的关联结果.     

Orbit determination with the two-body integrals. II

The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of a celestial body, which may be obtained either by optical or by radar observations. We write polynomial equations for this problem, which can be solved using the powerful tools of computational Algebra. An algorithm to decide if the linkage of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. This paper continues the research started in Gronchi et al. (Celest. Mech. Dyn. Astron. 107(3):299–318, 2010), where the angular momentum and the energy integrals were used. The use of a suitable component of the Laplace–Lenz vector in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half.     

Ephemeris of a highly eccentric orbit: Explorer 28

An ephemeris has been obtained for Explorer 28 (IMP 3) which agrees well with 2 years of radio observations and with SAO observations a year later. This ephemeris is generated over the 3 year lifetime by a numerical integration method utilizing a set of initial conditions, at launch and without requiring further differential correction. Because highly eccentric orbits are difficult to compute with acceptable accuracy and because a long continuous arc has been obtained which compares with actual data to a known precision, this ephemeris may be used as a standard for computing highly eccentric orbits in the Earth-Moon system.Orbit improvement was used to obtain the initial conditions which generated the ephemeris. This improvement was based on correcting the energy by adjusting the semimajor axis to match computed times of perigee passage with the observed. This procedure may generate errors in semimajor axis to compensate for model errors in the energy; however this compensation error is also implicit in orbit determination itself.     

Extension of fast periodic transfer orbits from the Earth–Moon RTBP to the Sun–Earth–Moon Quasi-Bicircular Problem

Starting from 80 families of low-energy fast periodic transfer orbits in the Earth–Moon planar circular Restricted Three Body Problem (RTBP), we obtain by analytical continuation 11 periodic orbits and 25 periodic arcs with similar properties in the Sun–Earth–Moon Quasi-Bicircular Problem (QBCP). A novel and very simple procedure is introduced giving the solar phases at which to attempt continuation. Detailed numerical results for each periodic orbit and arc found are given, including their stability parameters and minimal distances to the Earth and Moon. The periods of these orbits are between 2.5 and 5 synodic months, their energies are among the lowest possible to achieve an Earth–Moon transfer, and they show a diversity of circumlunar trajectories, making them good candidates for missions requiring repeated passages around the Earth and the Moon with close approaches to the last.     

Escape regions of a quartic potential

We investigate the escape regions of a quartic potential and the main types of irregular periodic orbits. Because of the symmetry of the model the zero velocity curve consists of four summetric arcs forming four open channels around the lines y = ± x through which an orbit can escape. Four unstable Lyapunov periodic orbits bridge these openings.We have found an infinite sequence of families of periodic orbits which is the outer boundary of one of the escape regions and several infinite sequences of periodic orbits inside this region that tend to homoclinic and heteroclinic orbits. Some of these sequences of periodic orbits tend to homoclinic orbits starting perpendicularly and ending asymptotically at the x-axis. The other sequences tend to heteroclinic orbits which intersect the x-axis perpendicularly for x > 0 and make infinite oscillations almost parallel to each of the two Lyapunov orbits which correspond to x > 0 or x < 0.     

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.     

Orbital element distributions in the Oort cloud

In this paper we consider orbital element distributions for comets moving on admissible orbits in the Oort cloud and distributions for some functions that depend on the orbital elements. Moreover, we find the probability of an event that an arbitrarily chosen admissible orbit belongs to the set (r) of orbital elements and the distribution of circular velocities in the cloud.     

Time evolution of orbital uncertainties for the impactor candidate 2004 AS1

We evaluate the asteroid impact risk from the discovery night onwards using six-dimensional statistical orbit computation techniques to examine the a posteriori probability density of the orbital elements. Close to the discovery moment the observational data of an object are typically exiguous: the number of observations is very small and/or the covered orbital arc is very short. For such data, the covariance matrices computed in the linear approximation (e.g., with the least-squares technique) are known to fail to describe the uncertainties in the orbital parameters. The technique of statistical ranging gives us rigorous means to assess the orbital uncertainties already on the discovery night. To examine the time evolution of orbital uncertainties, we make use of a new nonlinear Monte Carlo technique of phase-space sampling using volumes of variation, which complements the ranging technique for exiguous data and the least-squares technique for extensive observational data. We apply the statistical techniques to the near-Earth Asteroid 2004 AS₁, which grabbed the attention of asteroid scientists because, for one day, it posed the highest and most immediate impact risk so far recorded. We take this extreme case to illustrate the ambiguities in the impact risk assessment for short arcs. We confirm that the weighted fraction of the collision orbits at discovery was large but conclude that this was mostly due to the discordance of the discovery-night observations. This case study highlights the need to introduce a regularization in terms of an a priori probability density to secure the invariance of the probabilistic analysis especially in the nonlinear orbital inversion for short arcs. We remark that a predominant role of the a priori can give indications of the feasibility of the probabilistic interpretation, that is, how reliable the results derived from the a posteriori probability density are. Nevertheless, the strict mathematical definition of, e.g., the collision probability remains valid, and our nonlinear statistical techniques give us the means to always deduce, at the very least, order-of-magnitude-estimates for the collision probability.     

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.     

Error analyses of resonant orbits for geodesy

An error analysis of resonant orbits for geodesy indicates that attempts to use resonance to recover high order geopotential coefficients may be seriously hampered by errors in the geopotential. This effect, plus the very high correlations (up to .99) of the resonant coefficients with each other and the orbital period in single satellite solutions, makesindividual resonant orbits of limited value for geodesy. Multiple-satellite, single-plane solutions are only a slight improvement over the single satellite case. Accurate determination of high order coefficients from low altitude resonant satellites requires multiple orbit planes and small drift-periods to reduce correlations and effects of errors of non-resonant geopotential terms. Also, the effects of gravity model errors on low-altitude resonant satellites make the use of tracking arcs exceeding two to three weeks of doubtful validity. Because high-altitude resonant orbits are less affected by non-resonant terms in the geopotential, much longer tracking arcs can be used for them.     

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.