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.     

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.     

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.     

The Recovery as an Important Part of NEA Astrometric Follow-up

The number of known near-Earth asteroids (NEAs) has increased rapidly in recent years due to large surveys. This discovery process has to be followed by follow-up observations to obtain a sufficient number of precise astrometric data needed for an accurate orbit determination of newly discovered bodies.Accurate orbit determination requires observations from at least two oppositions. If asteroids are not found in the next apparition, different from the discovery apparition, then they can be considered lost. This is particularly embarrassing for NEAs. If data for different apparitions are not found in the course of precovery surveys or in other archive data, then it is necessary to prepare targeted observations of a particular NEA in the second convenient apparition. Therefore NEA recovery is a very important part of NEA follow-up.We discuss here methods, techniques, and results of planned recoveries at the Klet' Observatory using a 0.57-m telescope equipped with a CCD detector. The Klet' NEA recovery subprogram has brought 21 planned NEA recoveries since 1997, including seven NEAs belonging to the potentially hazardous asteroid category.We briefly mention the overall work on NEA recoveries provided by several NEO follow-up programs as well as the need for communication resources supporting astrometric observers. Finally we present here a planned extension of the Klet' NEA recovery subprogram to fainter objects by means of a new 1.06-m reflector.     

Orbit Determination of Binary Asteroids

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.     

地月DRO星载光学测量近地小行星轨道确定

针对地基光学监测系统对近地小行星在近太阳方向的监测存在盲区的问题,提出了远距离逆行轨道(Distant Retrograde Orbit,DRO)天基光学平台对近地小行星进行跟踪定轨的方法.通过可视性分析,筛选仿真观测数据,利用美国宇航局喷气推进实验室(Jet Propulsion Laboratory,JPL)公布的小行星初始轨道信息对不同轨道类型的目标天体进行轨道确定,将计算结果与参考轨道对比分析.仿真结果表明:在测量精度2角秒,定轨弧长3年的情况下,DRO平台对仿真算例中所选择的近地小行星的定轨精度可以达到几十公里量级,其中Atira型轨道精度可达10公里以内.由此可见,DRO天基平台对近地小行星具有较好的监测能力,定轨精度能实现对目标小行星的精确跟踪,并对其进行轨道预报.     

Determination of asteroid masses from their close encounters with Mars

An obstacle to the asteroid mass determination lies in the difficulty in isolating the gravitational perturbation exerted by a single asteroid on the planets, being strongly correlated and mixed up with those of many other asteroids. This hindrance may be avoided by the method of analysis presented here: an asteroid mass is estimated in correspondence with its close encounters with Mars where the acceleration it induces on the planet can be sufficiently disentangled from those generated by the remaining asteroid masses to calculate. We test this technique in the analysis of range observations to Mars Global Surveyor and Mars Express performed from 1999 to 2007. For this purpose, we adopt the dynamical model of the planetary ephemeris INPOP06 (Fienga et al., 2008), which includes the gravitational influences of the 300 most perturbing asteroids of the Martian orbit. We obtain the solutions of 10 asteroid masses that have the largest effects on this orbit over the period examined: they are generally in good agreement with determinations recently published.     

Die Entdeckung eines neuen Asteroiden (1988BJ) vom “Hungaria”-Typ

During an observational program of asteroids at the observatory of Haute Provence (France) an asteroid of 17^th magnitude was discovered on 22^nd January 1988 (Elst 1988). Further observations at the European Southern Observatory (ESO) and a preliminary determination of an orbit revealed that it belongs to the “Hungaria” family.     

Theoretical meteor radiants of Apollo,Amor, and Aten asteroids

A compilation of theoretical meteor radiants is presented for all numbered (through 2525) asteroids which approach the Earth's orbit to within 0.20 AU. On the basis of orbital similarity, asteroids associated with current meteor streams and Prairie Network fireballs are listed; plausible associations with medieval fireball radiants are also given. The best defunct comet candidates in terms of meteoric evidence appear to be 2101 Adonis and 2201 1947XC. Asteroids which may be either extinct comets or perturbed main belt asteroids accompanied by collisional debris (represented by fireballs) are 1917 Cuyo, 2202 Pele, 2061 Anza, and 2340 Hathor. 1566 Icarus and 1981 Midas are the only asteroids whose orbits approach to less than 0.07 AU of the Earth's orbit, have a northern radiant, and still show no certain meteoric activity. The majority of Atens, Apollos, and Amors do not pass sufficiently close (<0.07 AU) to the Earth's orbit for a reasonable expectation of meteoric activity, or have radiants south of ?20° declination, requiring southern hemisphere observations.     

天体力学的几个热点问题

阐述当今天体力学前沿课题中的几个热点问题，近地小行星与地球的交会及其动力演化，航天器定轨新手段中的得一星跟踪自主定轨方法以及星际探测中的轨道力学问题。     

The asteroid ephemerides program at the U.S. Naval Observatory

The U.S. Naval Observatory has begun a program of ephemeris improvement and reference frame determination from the main belt asteroids. The program is, currently, starting out with a limited set of observations of the larger asteroids to determine the equator and equinox corrections for the USNO W1J00 transit circle observations catalog, and, if possible, improve the orbits of these asteroids based on this limited set of observations. For this project, transit circle observations of the Sun and the planets Mercury through Jupiter, are also being used to determine the equator, equinox, and ephemeris corrections, the next goal is to improve the orbits of the larger asteroids in the optical reference frame using observation series that cover a much longer period of time. This will allow the exploration of the differences between the dynamical reference frame based on radar observations of main belt asteroids and its relation with the optical reference frame. Other goals include the exploration of the mass distribution in the main asteroid belt from high precision observations, and the effect of this mass on the ephemerides of the major planets.     

Determining the masses of large asteroids by the dynamical method

The masses of 21 main-belt asteroids were determined by the dynamical method. The masses of 13 asteroids have relative errors of less than 50 percent. When earlier positional observations are excluded and recent, highly accurate ones are used, the error of asteroid mass determination is reduced.     

Some results from the National Observatory of Turkey, Kazan State University, and Nikolaev Astronomical Observatory on small bodies of the solar system

A scientific collaboration between TÜB?TAK National Observatory (Turkey), Kazan State University (Russia) and Nikolaev Astronomical Observatory (Ukraine) involves observations of minor planets and near-Earth asteroids (NEAs) with the 1.5 m Russian-Turkish telescope (RTT150). Regular observations of selected asteroids in the range of 11-18 magnitudes began in 2004 with the view of determining masses of selected asteroids, improving the orbits of the NEAs, and studying physical characteristics of selected asteroids from photometric observations. More than 3000 positions of 53 selected asteroids and 11 NEAs have been obtained with an internal error in the range of 30-300 mas for a single determination. Photometric reductions of more than 4000 CCD frames are in progress. Masses of 21 asteroids were estimated through dynamical method using the ground-based optical observations, mainly from the RTT150 and Minor Planet Center. A comparison of the observational results from the RTT150 in 2004-2005 with observations of the same objects at other observatories allows us to conclude that RTT150 can be used for ground-based support in astrometry for the space mission GAIA.     

Comparing analytical and numerical approaches to meteoroid orbit determination using Hayabusa telemetry

Fireball networks establish the trajectories of meteoritic material passing through Earth's atmosphere, from which they can derive pre‐entry orbits. Triangulated atmospheric trajectory data require different orbit determination methods to those applied to observational data beyond the Earth's sphere of influence, such as telescopic observations of asteroids. Currently, the vast majority of fireball networks determine and publish orbital data using an analytical approach, with little flexibility to include orbital perturbations. Here, we present a novel numerical technique for determining meteoroid orbits from fireball network data and compare it to previously established methods. The re‐entry of the Hayabusa spacecraft, with its known pre‐Earth orbit, provides a unique opportunity to perform this comparison as it was observed by fireball network cameras. As initial sightings of the Hayabusa spacecraft and capsule were made at different altitudes, we are able to quantify the atmosphere's influence on the determined pre‐Earth orbit. Considering these trajectories independently, we found the orbits determined by the novel numerical approach to align closer to JAXA's telemetry in both cases. Using simulations, we determine the atmospheric perturbation to become significant at ~90 km—higher than the first observations of typical meteorite dropping events. Using further simulations, we find the most substantial differences between techniques to occur at both low entry velocities and Moon passing trajectories. These regions of comparative divergence demonstrate the need for perturbation inclusion within the chosen orbit determination algorithm.     

天球参考架零点的测定（Ⅱ）－小行星可观测弧段长度

全轨道均匀分布的小行星观测对天球参考架零点的测定，以及其它一些相关课题的研究非常有利。本文就低纬子午环的观测能力，计算分析了小行星在全轨道观测的星等范围和弧段分布，给出了可观测弧段长度的计算公式，并进行了模拟计算。     

Problems of CCD Photometry of Fast-Moving Asteroids

The problems associated with the photometry of fast-moving asteroids are discussed. The effect of noise in CCD observations on the photometric accuracy is analyzed. A photometric accuracy limitation is shown to exist for observations of asteroids, which is determined by the angular rate of the object and the ratio of the flux from the object and noise due to sky background and dark current. The effective exposure for observing a moving object is determined. The method of overlapping areas is analyzed, which is used for obtaining the lightcurves of fast-moving asteroids. This method includes the determination of the mutual magnitude differences for the entire ensemble of comparison stars, the reduction of the magnitudes of all these stars to the magnitude of one of them adopted as the primary comparison star, the determination of the magnitude of the average star on each frame of the entire series of CCD observations, and the computation of the lightcurve as the difference between the magnitude of the asteroid and that of the average star.     

小行星的搜寻和定轨

本文简要介绍了与小行星有关的一些基本知识和在小行星搜寻方面的国际进展情况 ,侧重与ASTROD项目有关的内容。并介绍了与小行星定轨有关的网络资源     

Yarkovsky detection opportunities: II. Binary systems

We consider the possibility of detecting the Yarkovsky orbital perturbation acting on binary systems among the near-Earth asteroids. This task is significantly more difficult than for solitary asteroids because the Yarkovsky force affects both the heliocentric orbit of the system's center of mass and the relative orbit of the two components. Nevertheless, we argue these are sufficiently well decoupled so that the major Yarkovsky perturbation is in the simpler heliocentric motion and is observable with the current means of radar astrometry. Over the long term, the Yarkovsky perturbation in the relative motion of the two components is also detectable for the best observed systems. However, here we consider a simplified version of the problem by ignoring mutual non-spherical gravitational perturbations between the two asteroids. With the orbital plane constant in space and the components' rotation poles fixed (and assumed perpendicular to the orbital plane), we do not examine the coupling between Yarkovsky and gravitational effects. While radar observations remain an essential element of Yarkovsky detections, lightcurve observations, with their ability to track occultation and eclipse phenomena, are also very important in the case of binaries. The nearest possible future detection of the Yarkovsky effect for a binary system occurs for (66063) 1998 RO₁ in September 2006. Farther out, even more statistically significant detections are possible for several other systems including 2000 DP₁₀₇, (66391) 1999 KW₄ and 1996 FG₃.