OpenOrb: Open‐source asteroid orbit computation software including statistical ranging

Abstract— We are making an open‐source asteroid orbit computation software package called OpenOrb publicly available. OpenOrb is built on a well‐established Bayesian inversion theory, which means that it is to a large part complementary to orbit‐computation packages currently available. In particular, OpenOrb is the first package that contains tools for rigorously estimating the uncertainties resulting from the inverse problem of computing orbital elements using scarce astrometry. In addition to the well‐known least‐squares method, OpenOrb also contains both Monte‐Carlo (MC) and Markov‐Chain MC (MCMC; Oszkiewicz et al. [2009]) versions of the statistical ranging method. Ranging allows the user to obtain sampled, non‐Gaussian orbital‐element probability‐density functions and is therefore optimized for cases where the amount of astrometry is scarce or spans a relatively short time interval. Ranging‐based methods have successfully been applied to a variety of different problems such as rigorous ephemeris prediction, orbital element distribution studies for transneptunian objects, the computation of invariant collision probabilities between near‐Earth objects and the Earth, detection of linkages between astrometric asteroid observations within an apparition as well as between apparitions, and in the rigorous analysis of the impact of orbital arc length and/or astrometric uncertainty on the uncertainty of the resulting orbits. Tools for making ephemeris predictions and for classifying objects based on their orbits are also available in OpenOrb. As an example, we use OpenOrb in the search for candidate retrograde and/or high‐inclination objects similar to 2008 KV₄₂ in the known population of transneptunian objects that have an observational time span shorter than 30 days.     

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.     

Orbit computation for transneptunian objects

Using statistical orbital ranging, we systematically study the orbit computation problem for transneptunian objects (TNOs). We have automated orbit computation for large numbers of objects, and, more importantly, we are able to obtain orbits even for the most sparsely observed objects (observational arcs of a few days). For such objects, the resulting orbit distributions include a large number of high-eccentricity orbits, in which TNOs can be perturbed by close encounters with Neptune. The stability of bodies on the computed orbits has therefore been ascertained by performing a study of close encounters with the major planets. We classify TNO orbit distributions statistically, and we study the evolution of their ephemeris uncertainties. We find that the orbital element distributions for the most numerous single-apparition TNOs do not support the existence of a postulated sharp edge to the belt beyond 50 AU. The technique of statistical ranging provides ephemeris predictions more generally than previously possible also for poorly observed TNOs.     

Asteroid orbits using phase-space volumes of variation

We present a statistical orbit computation technique for asteroids with transitional observational data, that is, a moderate number of data points spanning a moderate observational time interval. With the help of local least-squares solutions in the phase space of the orbital elements, we map the volume of variation as a function of one or more of the elements. We sample the resulting volume using a Monte Carlo technique and, with proper weights for the sample orbital elements, characterize the six-dimensional orbital-element probability density function. The volume-of-variation (VOV) technique complements the statistical ranging technique for asteroids with exiguous observational data (short time intervals and/or small numbers of observations) and the least-squares technique for extensive observational data. We show that, asymptotically, results using the new technique agree closely with those from ranging and least squares. We apply the technique to the near-Earth object 2004 HA₃₉, the main-belt object 2004 QR and the transneptunian object 2002 CX₂₂₄ recently observed at the Nordic Optical Telescope on La Palma, illustrating the potential of the technique in ephemeris prediction. The VOV technique helps us assess the phase transition in orbital-element probability densities, that is, the non-linear collapse of wide orbital-element distributions to narrow localized ones. For the three objects above, the transition takes place for observational time intervals of the order of 10 h, 5 d and 10 months, respectively, emphasizing the significance of the orbital-arc fraction covered by the observations.     

Detection of Earth-mass and super-Earth Trojan planets using transit timing variation method

We have carried out an extensive study of the possibility of the detection of Earth-mass and super-Earth Trojan planets using transit timing variation method with the Kepler space telescope. We have considered a system consisting of a transiting Jovian-type planet in a short period orbit, and determined the induced variations in its transit timing due to an Earth-mass/super-Earth Trojan planet. We mapped a large section of the phase space around the 1:1 mean-motion resonance and identified regions corresponding to several other mean-motion resonances where the orbit of the planet would be stable. We calculated transit timing variations (TTVs) for different values of the mass and orbital elements of the transiting and perturbing bodies as well as the mass of central star, and identified orbital configurations of these objects (ranges of their orbital elements and masses) for which the resulted TTVs would be within the range of the variations of the transit timing of Kepler’s planetary candidates. Results of our study indicate that in general, the amplitudes of the TTVs fall within the detectable range of timing precision obtained from the Kepler’s long-cadence data, and depending on the parameters of the system, their magnitudes may become as large as a few hours. The probability of detection is higher for super-Earth Trojans with slightly eccentric orbits around short-period Jovian-type planets with masses slightly smaller than Jupiter. We present the details of our study and discuss the implications of its results.     

Can GJ 876 host four planets in resonance?

Prior to the detection of its outermost Uranus-mass object, it had been suggested that GJ 876 could host an Earth-sized planet in a 15-day orbit. Observation, however, did not support this idea, but instead revealed evidence for the existence of a larger body in a ~125-day orbit, near a three-body resonance with the two giant planets of this system. In this paper, we present a detailed analysis of the dynamics of the four-planet system of GJ 876, and examine the possibility of the existence of other planetary objects interior to its outermost body. We have developed a numerical scheme that enables us to search the orbital parameter-space very effectively and, in a short time, identify regions where an object may be stable. We present details of this integration method and discuss its application to the GJ 876 four-planet system. The results of our initial analysis suggested possible stable orbits at regions exterior to the orbit of the outermost planet and also indicated that an island of stability may exist in and around the 15-day orbit. However, examining the long-term stability of an object in that region by direct integration revealed that the 15-day orbit becomes unstable and that the system of GJ 876 is most likely dynamically full. We present the results of our study and discuss their implications for the formation and final orbital architecture of this system.     

环月飞行器精密定轨的模拟仿真

以中国正在实施的探月计划“嫦娥1号”工程为背景，分析了在中国联合S波段(USB)测控网和甚长基线射电干涉(VLBI)跟踪网的现有空间分布、观测精度水平下的环月飞行器精密定轨．采用的方法是模拟仿真计算，即首先模拟观测数据，然后在计入各误差源的影响后进行求解，并对解算结果进行比较．模拟仿真的工具是美国宇航局哥达德飞行中心的空间数据分析软件系统GEODYN．环月飞行的主要误差源是月球重力场，为此首先讨论了目前精度最高的月球重力场模型JGL165P1的(形式)误差．在模拟了测距、测速以及VLBI的时延、时延率数据后，计入月球重力场的误差进行精密轨道确定．定轨时采用了减缩动力学(reduced dynamic)方法，即选用合适的经验加速度参数吸收重力场误差对定轨的影响．结果表明对于一个不将月球重力场作为主要科学目标的探月计划(如“嫦娥1号”)，减缩动力学方法是一个简单、有效地提高环月飞行器定轨精度的方法．     

Method of unit vectors for the determination of preliminary orbit with single-arc and single-station ranging data

We present a method of unit vectors of calculating preliminary orbit with single-station and single-arc ranging data, assuming known orbital plane parameters i and Ω. Practical calculations show that this method is effective. This method is particularly valuable in cases where the angle data are either unavailable or obviously less accurate than the ranging data.     

The effect of secular resonances on the long-term orbital evolution of uncontrollable objects on satellite radio navigation systems in the MEO region

We present the results of the study of long-term orbital evolution of space debris objects, formed from end-of-life space vehicles (SV) of satellite radio navigation systems in the medium Earth orbit (MEO) region. Dynamical features of the evolution of objects in this region have been studied on the basis of 20-year laser surveillance with the Etalon-1 and Etalon-2 satellites and the results of numerical simulation of the long-term evolution of operating and disposal orbits of uncontrolled GLONASS and GPS SVs. It is shown that perturbations from secular lunisolar resonances produce an eccentricity growth for orbits with inclinations chosen for navigation constellations; this significantly changes the positions of these orbits in space and results in the ingress of end-of-life objects into the area of operating SVs.     

Discovery of an asteroid and quasi‐satellite in an Earth‐like horseshoe orbit

Abstract— The newly discovered asteroid 2002 AA₂₉ moves in a very Earth‐like orbit that relative to Earth has a unique horseshoe shape and allows transitions to a quasi‐satellite state. This is the first body known to be in a simple heliocentric horseshoe orbit, moving along its parent planet's orbit. It is similarly also the first true co‐orbital object of Earth, since other asteroids in 1:1 resonance with Earth have orbits very dissimilar from that of our planet. When a quasi‐satellite, it remains within 0.2 AU of the Earth for several decades. 2002 AA₂₉ is the first asteroid known to exhibit this behavior. 2002 AA₂₉ introduces an important new class of objects offering potential targets for space missions and clues to asteroid orbit transfer evolution.     

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.     

The assessment of the semi-analytical method in the long-term orbit prediction of Earth satellites

To understand the long-term evolution and distribution of the space objects, it is necessary to predict their orbits. Compared with the short-term prediction of a few days, the priority concerns of the long-term orbit prediction are the calculation speed, and the accuracies of major orbital elements, including the semi-major axis and eccentricity which define the shape of the orbit, as well as the orbital inclination and the right ascension of ascending node which define the orientation of the orbit. Given these requirements, it is preferable to adopt the semi-analytical method, which averages the system over the orbital period, and integrates the averaged system using the numerical method. It is not new, however, in the available literature, we can hardly find a quantitative assessment regarding its accuracy and speed when it is applied to various types of orbits. In this paper, we would like to report our implementation and assessment of the semi-analytical method, expecting that it would help to estimate its feasibility in the long-term orbit prediction. The quantitative assessment covers the commonly used orbits for the Earth satellites. In some rare and special cases where the performance of our method appears abnormal, we discuss the reasons and possible solutions. We hope our results can provide some useful reference for the similar applications of the semi-analytical method since our method is a relatively common approach in terms of both accuracy and implementation.     

Orbital correlation of space objects based on orbital elements

Orbital correlation of space objects is one of the most important elements in space object identification. Using the orbital elements, we provide correlation criteria to determine if objects are coplanar,co-orbital or the same. We analyze the prediction error of the correlation parameters for different orbital types and propose an orbital correlation method for space objects. The method is validated using two line elements and multisatellite launching data. The experimental results show that the proposed method is effective, especially for space objects in near-circular orbits.     

On the rotation of co-orbital bodies in eccentric orbits

We investigate the resonant rotation of co-orbital bodies in eccentric and planar orbits. We develop a simple analytical model to study the impact of the eccentricity and orbital perturbations on the spin dynamics. This model is relevant in the entire domain of horseshoe and tadpole orbit, for moderate eccentricities. We show that there are three different families of spin–orbit resonances, one depending on the eccentricity, one depending on the orbital libration frequency, and another depending on the pericenter’s dynamics. We can estimate the width and the location of the different resonant islands in the phase space, predicting which are the more likely to capture the spin of the rotating body. In some regions of the phase space the resonant islands may overlap, giving rise to chaotic rotation.     

Dynamics of rotation of super-Earths

We numerically investigate the dynamics of rotation of several close-in terrestrial exoplanet candidates. In our model, the rotation of the planet is disturbed by the torque of the central star due to the asymmetric equilibrium figure of the planet. We model the shape of the planet by a Jeans spheroid. We use surfaces of section and spectral analysis to explore numerically the rotation phase space of the systems adopting different sets of parameters and initial conditions close to the main spin–orbit resonant states. One of the parameters, the orbital eccentricity, is critically discussed here within the domain of validity of orbital circularization timescales given by tidal models. We show that, depending on some parameters of the system like the radius and mass of the planet, eccentricity etc., the rotation can be strongly perturbed and a chaotic layer around the synchronous state may occupy a significant region of the phase space. 55 Cnc e is an example.     

Astrometric detection of binary companions and planets: Acceleration of proper motion

Apparent acceleration of proper motion is one of the observable manifestations of orbital motion in binary stars. Owing to the increasing accuracy of astrometric measurements, it may also be a method to detect binarity of stars. This paper presents some analytical expressions for the effects of binary motion on proper motions when the orbital period is at least several times the span of observations. We estimate orbit dimensions and distances at which low‐mass companions and planets may be detected around main‐sequence stars, using preliminary estimates of precision for the AMEX, GAIA and SIM space missions.     

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.     

On the Laplacian orbit determination of asteroids

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.     

exofit: orbital parameters of extrasolar planets from radial velocities

Retrieval of orbital parameters of extrasolar planets poses considerable statistical challenges. Due to sparse sampling, measurement errors, parameters degeneracy and modelling limitations, there are no unique values of basic parameters, such as period and eccentricity. Here, we estimate the orbital parameters from radial velocity data in a Bayesian framework by utilizing Markov Chain Monte Carlo (MCMC) simulations with the Metropolis–Hastings algorithm. We follow a methodology recently proposed by Gregory and Ford. Our implementation of MCMC is based on the object-oriented approach outlined by Graves. We make our resulting code, exofit , publicly available with this paper. It can search for either one or two planets as illustrated on mock data. As an example we re-analysed the orbital solution of companions to HD 187085 and HD 159868 from the published radial velocity data. We confirm the degeneracy reported for orbital parameters of the companion to HD 187085, and show that a low-eccentricity orbit is more probable for this planet. For HD 159868, we obtained slightly different orbital solution and a relatively high 'noise' factor indicating the presence of an unaccounted signal in the radial velocity data. exofit is designed in such a way that it can be extended for a variety of probability models, including different Bayesian priors.     

Speckle observations with PISCO in Merate: VIII. Astrometric measurements of visual binaries in 2007 and new orbits of the multiple system Zeta Aqr

We present relative astrometric measurements of visual binaries made during the second semester of 2007, with the speckle camera PISCO at the 102 cm Zeiss telescope of Brera Astronomical Observatory, in Merate. Our sample contains orbital couples as well as binaries whose motion is still uncertain. We obtained 283 new measurements of 279 objects, with angular separations in the range 0″.17–4″.4, and an average accuracy of 0″.014. The mean error on the position angles is 0°.6. Most of the position angles were determined without the usual 180° ambiguity with the application of triple‐correlation techniques and/or by inspection of the long integration files. We also present the new orbit we have computed for Zeta Aqr AB (ADS 15971), for which our measurements lead to large residuals with the previously computed orbit. We were also able to compute the elements of the perturbation orbit Bb‐P caused by an invisible companion, whose mass is estimated at 0.7 M_⊙ (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)