Dynamical evolution of high area-to-mass ratio objects in Molniya orbits

For special demands, some notable orbit types have been developed by human, including the Molniya orbits, which have a relatively high eccentricity up to about 0.7, and a period of 12 h. Considering that space debris with high area-to-mass ratio (A/M) has been discovered, such objects may also exist in Molniya orbits due to spacecraft and upper stages fragmentation events. However, there are not sufficient studies of the complex dynamical phenomena of such orbits. These studies can enrich the knowledge about the long-term evolution of these orbits, be helpful to propose uncatalogued objects observation and identification, and also set the protected region as well as active debris removal. In this paper, the characteristics of 2:1 resonance of Molniya satellite orbits are studied. A large set of numerical simulations, including all the relevant perturbations, is carried out to further investigate the main characteristics, and special attention is payed to the dynamical evolution of objects with high A/M, particularly affected by the direct solar radiation pressure. The long-term dynamical evolution of orbital elements, as well as the dependency of lifetime on the A/M value, is discussed.     

Apparent rotation properties of space debris extracted from photometric measurements

Knowledge about the rotation properties of space debris objects is essential for the active debris removal missions, accurate re-entry predictions and to investigate the long-term effects of the space environment on the attitude motion change. Different orbital regions and object’s physical properties lead to different attitude states and their change over time.Since 2007 the Astronomical Institute of the University of Bern (AIUB) performs photometric measurements of space debris objects. To June 2016 almost 2000 light curves of more than 400 individual objects have been acquired and processed. These objects are situated in all orbital regions, from low Earth orbit (LEO), via global navigation systems orbits and high eccentricity orbit (HEO), to geosynchronous Earth orbit (GEO). All types of objects were observed including the non-functional spacecraft, rocket bodies, fragmentation debris and uncorrelated objects discovered during dedicated surveys. For data acquisition, we used the 1-meter Zimmerwald Laser and Astrometry Telescope (ZIMLAT) at the Swiss Optical Ground Station and Geodynamics Observatory Zimmerwald, Switzerland. We applied our own method of phase-diagram reconstruction to extract the apparent rotation period from the light curve. Presented is the AIUB’s light curve database and the obtained rotation properties of space debris as a function of object type and orbit.     

Optimising filtering of two-line element sets to increase re-entry prediction accuracy for GTO objects

Predicting re-entry epoch of space objects enables managing the risk to ground population. Predictions are particularly difficult for objects in highly-elliptical orbits, and important for objects with components that can survive re-entry, e.g. rocket bodies (R/Bs). This paper presents a methodology to filter two-line element sets (TLEs) to facilitate accurate re-entry prediction of such objects. Difficulties in using TLEs for precise analyses are highlighted and a set of filters that identifies erroneous element sets is developed. The filter settings are optimised using an artificially generated TLE time series. Optimisation results are verified on real TLEs by analysing the automatically found outliers for exemplar R/Bs. Based on a study of 96 historical re-entries, it is shown that TLE filtering is necessary on all orbital elements that are being used in a given analysis in order to avoid considerably inaccurate results.     

关于HEO空间飞行体的轨道寿命问题

低轨地球卫星的轨道寿命主要取决于大气的耗散作用,其轨道在不断变小(即高度降低)变圆的状态下进入地球稠密大气层中陨落.但HEO(Highly Eccentric Orbit)类型的空间飞行体的运行轨道是一个近地点高度很低,远地点高度却很高的大偏心率椭圆轨道,其轨道寿命主要由第三体(日、月)引力摄动所决定,而且还与其轨道的初始状态有密切关系,特别是慢变量Ω(轨道升交点经度)和ω(轨道近地点幅角),决定了偏心率e的长周期变化状态,从而制约了HEO类型空间飞行体的轨道寿命.本文将根据地球卫星轨道变化规律进行理论分析,阐明这一力学机制,并给出相应的数值验证.      

Properties of the high area-to-mass ratio space debris population at high altitudes

In the framework of its space debris research activities ESA established an optical survey program to study the space debris environment at high altitudes, in particular in the geostationary ring and in the geostationary transfer orbit region. The Astronomical Institute of the University of Bern (AIUB) performs these surveys on behalf of ESA using ESA’s 1-m telescope in Tenerife. Regular observations were started in 1999 and are continued during about 120–140 nights per year. Results from these surveys revealed a substantial amount of space debris at high altitudes in the size range from 0.1 to 1 m. Several space debris populations with different dynamical properties were identified in the geostationary ring. During the searches for debris in the geostationary transfer orbit region a new population of objects in unexpected orbits, where no potential progenitors exist, was found. The orbital periods of these objects are clustered around one revolution per day; the eccentricities, however, are scattered between 0 and 0.6. By following-up some of these objects using the ESA telescope and AIUB’s 1-m telescope in Zimmerwald, Switzerland, it was possible to study the properties of this new population. One spectacular finding from monitoring the orbits over time spans of days to months is the fact that these objects must have extreme area-to-mass ratios, which are by several orders of magnitudes higher than for ‘normal-type’ debris. This in turn supports the hypothesis that the new population actually is debris generated in or near the geostationary ring and which is in orbits with periodically varying eccentricity and inclination due to perturbations by solar radiation pressure. In order to further study the nature of these debris, multi-color and temporal photometry (light curves) were acquired with the Zimmerwald telescope. The light curves show strong variations over short time intervals, including signals typical for specular reflections. Some objects exhibit distinct periodic variations with periods ranging from 10 to several 100 s. All this is indicative for objects with complicated shapes and some highly reflective surfaces.     

Dynamical evolution of high area-to-mass ratio debris released into GPS orbits

A large set of simulations, including all the relevant perturbations, was carried out to investigate the long-term dynamical evolution of fictitious high area-to-mass ratio (A/M) objects released, with a negligible velocity variation, in each of the six orbital planes used by Global Positioning System (GPS) satellites. As with similar objects discovered in near synchronous trajectories, long lifetime orbits, with mean motions of about 2 revolutions per day, were found possible for debris characterized by extremely high area-to-mass ratios. Often the lifetime exceeds 100 years up to A/M ∼ 45 m²/kg, decreasing rapidly to a few months above such a threshold. However, the details of the evolution, which are conditioned by the complex interplay of solar radiation pressure and geopotential plus luni-solar resonances, depend on the initial conditions. Different behaviors are thus possible. In any case, objects like those discovered in synchronous orbits, with A/M as high as 20–40 m²/kg, could also survive in this orbital regime, with semi-major axes close to the semi-synchronous values, with maximum eccentricities between 0.3 and 0.7, and with significant orbit pole precessions (faster and wider for increasing values of A/M), leading to inclinations between 30° and more than 90°.     

Collision probability and spacecraft disposition in the geostationary orbit

Since 1963 approximately 300 satellites have been launched into the geostationary orbit, followed possibly by another additional 200 satellites up to the year 2000. Ground surveillance with radar and optical sensors able to detect objects of 1 m minimum size in the geostationary ring indicates a total population of several hundred which includes active and defunct satellites and spent upper stages. In addition, a population of untrackable objects is conjectured, whose size can only be estimated, possibly several thousand of smaller objects.

The purpose of this paper is to review the long-term evolution of orbits in the geostationary ring and at higher altitude, the collision probabilities and disposition options.

The major perturbations are considered including attitude-orbit cross-coupling effects which could cause secular orbit perturbations.

Collision probabilities for current and projected populations are reviewed considering different approaches, such as a deterministic treatment of the uncontrolled population and a stochastic modeling for the controlled satellites. Also, colocation, that is sharing of the same longitude slot by several operational satellites, is a potential source for collision, if no preventive measures are taken.

As regards spacecraft disposition options, the conclusion is that reorbiting is currently the only practical measure to safeguard the geostationary orbit. In this recommended procedure the defunct satellites are inserted into a so-called graveyard orbit, located suffieciently high above the geostationary orbit.     

Resonant quasi-periodic near-rectilinear Halo orbits in the Elliptic-Circular Earth-Moon-Sun Problem

Motivated by the near-future re-exploration of the cislunar space, this paper investigates dynamical substitutes of the Earth-Moon’s resonant Near-Rectilinear Halo Orbits (NRHOs) under the Elliptic-Circular Restricted Four-Body Problem formulation of the Earth-Moon-Sun system. This model considers that the Earth and Moon move in elliptical orbits about each other and that a third body, the Sun, moves in a circular orbit about the Earth-Moon barycenter. By making use of this higher-fidelity dynamical model, we are able to incorporate the Sun’s influence and the Moon’s eccentricity, two of the most significant perturbations of the cislunar environment. As a result of these perturbations, resonant periodic NRHOs of the Earth-Moon Circular Restricted Three-Body Problem (CR3BP) are hereby replaced by two-dimensional quasi-periodic tori that better represent the dynamical evolution of satellites near the vicinity of the Moon. We present the steps and algorithms needed to compute these dynamical structures in the Elliptic-Circular model and subsequently assess their utility for spacecraft missions. We focus on the planned orbit for the NASA-led Lunar Gateway mission, a 9:2 synodic resonant L₂ southern NRHO, as well as on the 4:1 synodic and 4:1 sidereal resonances, due to the proximity to the nominal orbit and their advantageous dynamical properties. We verify that the dynamical equivalents of these orbits preserve key dynamical attributes such as eclipse avoidance and near-linear stability. Furthermore, we find that the higher dimensionality of quasi-periodic solutions offers interesting alternatives to mission designers in terms of phasing maneuvers and low-altitude scientific observations.     

Orbit improvement for GTO objects using follow-up observations

The Astronomical Institute of the University of Bern was and is conducting several search campaigns for space debris in the geostationary ring (GEO) and the geostationary transfer orbit (GTO). First tests to build up a catalogue of satellites and space debris were performed recently. As the discovery observations of an object usually cover a time interval of a few minutes only, the first orbits determined are assumed to be circular. For GEO objects a circular orbit is a rather good approximation. This is certainly not the case for GTO objects. Nevertheless, a circular orbit seems to be an acceptable approximation for a short time interval around the observations. The first orbits can be used to acquire follow-up observations, which allow the determination of elliptical orbits. The maximum allowed time span between the detection and the follow-up for a successful recovery is found using simulated GTO orbits and observations. Further follow-up observations are simulated in order to study the orbit improvement process and the required force model. The accuracy of orbits needed to build up a catalogue is studied. The simulations are compared to the results achieved with observations acquired by the 1 m telescope on Tenerife.     

Mapping orbits around the asteroid 2001SN263

The present paper has the goal of mapping orbits, with respect to the perturbations, for a spacecraft traveling around the asteroid 2001SN₂₆₃. This asteroid is a triple system, which center of mass is in an elliptic orbit around the Sun. The perturbations considered in the present model are the ones due to the oblateness of the central body, the gravity field of the two satellite bodies (Beta and Gamma), the Sun, the Moon, the asteroids Vesta, Pallas and Ceres and all the planets of the Solar System. This mapping is important, because it shows the relative importance of each force for a given orbit for the spacecraft, helping to make a decision about which forces need to be included in the model for a given accuracy and nominal orbit. Another important application of this type of mapping is to find orbits that are less perturbed, since it is expected that those orbits have good potential to require a smaller number of station-keeping maneuvers. Simulations under different conditions are made to find those orbits. The main reason to study those trajectories is that, currently, there are several institutions in Brazil studying the possibility to make a mission to send a spacecraft to this asteroid (the so-called ASTER mission), because there are many important scientific studies that can be performed in that system. The results showed that Gamma is the main perturbing body, followed by Beta (10 times smaller) and the group Sun–Mars-oblateness of Alpha, with perturbations 1000 times weaker than the effects of Gamma. The other bodies have perturbations 10⁷ times smaller. The results also showed that circular and polar orbits are less perturbed, when compared to elliptical and equatorial orbits. Regarding the semi-major axis, an internal orbit is the best choice, followed by a larger external orbit. The inclination of the orbit plays an important role, and there are values for the inclination where the perturbations show minimum and maximum values, so it is important to make a good decision on those values.     

Charge exchange interactions on near-Earth proton radiation for orbit perturbation of high area-to-mass ratio objects

This novel concept expels neutral gas in the presence of geomagnetically-trapped protons in near-Earth orbit. The expelled neutral gas acts to induce charge exchange collisions with the geomagnetically-trapped protons and induce drag on objects which pass through it. The charge exchange collisions between the neutral gas and the geomagnetically-trapped protons create neutrals with similar kinetic energy that are not confined by the geomagnetic field. The charge exchange neutrals are able to collide with orbital objects and perturb their orbits. The delta-v applied by the charge exchange neutral flux is greatest on high area-to-mass objects. Numerical simulation shows charge exchange neutral impacts produce a delta-v on objects on the order of 3.8 x 10⁻¹¹ m/s at a distance of 1 km from the center of the expelled gas in a 1,000 km orbit. The impulse imparted by charge exchange neutral impacts is at least six orders of magnitude smaller than that provided by the induced drag caused by gas expulsion. The localized drag increase can force a majority of small objects into the orbit of the expelled gas cloud, even if that orbit is retrograde to the initial orbit of the objects. This new technique can be applied to the remediation of space debris.     

Feasibility of performing space surveillance tasks with a proposed space-based optical architecture

Under ESA contract an industrial consortium including Aboa Space Research Oy (ASRO), the Astronomical Institute of the University of Bern (AIUB), and the Dutch National Aerospace Laboratory (NLR), proposed the observation concept, developed a suitable sensor architecture, and assessed the performance of a space-based optical (SBO) telescope in 2005. The goal of the SBO study was to analyse how the existing knowledge gap in the space debris population in the millimetre and centimetre regime may be closed by means of a passive optical instrument. The SBO instrument was requested to provide statistical information on the space debris population in terms of number of objects and size distribution. The SBO instrument was considered to be a cost-efficient with 20 cm aperture and 6° field-of-view and having flexible integration requirements. It should be possible to integrate the SBO instrument easily as a secondary payload on satellites launched into low-Earth orbits (LEO), or into geostationary orbit (GEO). Thus the selected mission concept only allowed for fix-mounted telescopes, and the pointing direction could be requested freely. Since 2007 ESA focuses space surveillance and tracking activities in the Space Situational Awareness (SSA) preparatory program. Ground-based radars and optical telescopes are studied for the build-up and maintenance of a catalogue of objects. In this paper we analyse how the proposed SBO architecture could contribute to the space surveillance tasks survey and tracking. We assume that the SBO instrumentation is placed into a circular sun-synchronous orbit at 800 km altitude. We discuss the observation conditions of objects at higher altitude, and select an orbit close to the terminator plane. A pointing of the sensor orthogonal to the orbital plane with optimal elevation slightly in positive direction (0° and +5°) is found optimal for accessing the entire GEO regime within one day, implying a very good coverage of controlled objects in GEO, too. Simulations using ESA’s Program for Radar and Optical Observation Forecasting (PROOF) in the version 2005 and a GEO reference population extracted from DISCOS revealed that the proposed pointing scenario provides low phase angles together with low angular velocities of the objects crossing the field-of-view. Radiometric simulations show that the optimal exposure time is 1–2 s, and that spherical objects in GEO with a diameter of below 1 m can be detected. The GEO population can be covered under proper illumination nearly completely, but seasonal drops of the coverage are possible. Subsequent observations of objects are on average at least every 1.5 days, not exceeding 3 days at maximum. A single observation arc spans 3° to 5° on average. Using a simulation environment that connects PROOF to AIUB’s program system CelMech we verify the consistency of the initial orbit determination for five selected test objects on subsequent days as a function of realistic astrometric noise levels. The initial orbit determination is possible. We define requirements for a correlator process essential for catalogue build-up and maintenance. Each single observation should provide an astrometric accuracy of at least 1”–1.5” so that the initially determined orbits are consistent within a few hundred kilometres for the semi-major axis, 0.01 for the eccentricity, and 0.1° for the inclination.     

大面质比空间碎片在太阳光压和引力作用下的轨道演化

分析了较高轨道(a > 10000km)大面质比空间碎片的轨道动力学演化问题. 重点讨论了位于地球同步轨道的空间碎片轨道演化问题, 并给出轨道偏心率 随时间演化的表达式. 通过进一步分析得出, 倾角大于63°26'的GTO轨 道空间碎片, 仅在J₂和第三体摄动影响下, 会出现轨道偏心率升高; 而对 于大面质比空间碎片, 在J₂项和太阳光压同时作用下, 当近地点指向的角 变率与太阳平黄经变化率接近时, 会出现长期共振现象, 导致轨道偏心率升 高, 近地点降低. 分析还得出, 轨道演化过程中, 偏心率的最大值与初始轨 道近地点的指向有关.      

The classical Laplace plane as a stable disposal orbit for geostationary satellites

The classical Laplace plane is a frozen orbit, or equilibrium solution for the averaged dynamics arising from Earth oblateness and lunisolar gravitational perturbations. The pole of the orbital plane of uncontrolled GEO satellites regress around the pole of the Laplace plane at nearly constant inclination and rate. In accordance with Friesen et al. (1993), we show how this stable plane can be used as a robust long-term disposal orbit. The current graveyard regions for end-of-life retirement of GEO payloads, which is several hundred kilometers above GEO depending on the spacecraft characteristics, cannot contain the newly discovered high area-to-mass ratio debris population. Such objects are highly susceptible to the effects of solar radiation pressure exhibiting dramatic variations in eccentricity and inclination over short periods of time. The Laplace plane graveyard, on the contrary, would trap this debris and would not allow these objects to rain down through GEO. Since placing a satellite in this inclined orbit can be expensive, we discuss some alternative disposal schemes that have acceptable cost-to-benefit ratios.     

低轨目标短弧关联的稳健方法

未来大量低轨星座部署一旦完成,将会对传统的空间目标监视提出更高的要求.广角光学望远镜系统具有广域监视效果,可以同时观测大量目标.但是广角望远镜的观测数据大部分属于“短弧”观测数据,单次观测无法进行空间目标的初始轨道确定.目前有效的解决方案是对光学观测弧段进行准确关联,融合多组观测数据进行初定轨.本文以容许域方法为基础,通过优化拟合两段观测短弧的轨道,结合卡方检验,确定不同观测弧段之间的关联性.其次详细描述了对于低轨目标,用仅测角观测数据进行关联的时候,存在的病态性问题.最后,针对低轨目标短弧关联错误率高的问题,提出了用角度误差特性规律进行误关联识别的方法.误关联的准确识别有效地提高了对于低轨空间目标仅光学观测序列的关联成功率,为后续的空间目标编目提供了有效的数据支撑.     

Heliotropic dust rings for Earth climate engineering

This paper examines the concept of a Sun-pointing elliptical Earth ring comprised of dust grains to offset global warming. A new family of non-Keplerian periodic orbits, under the effects of solar radiation pressure and the Earth’s J₂ oblateness perturbation, is used to increase the lifetime of the passive cloud of particles and, thus, increase the efficiency of this geoengineering strategy. An analytical model is used to predict the orbit evolution of the dust ring due to solar-radiation pressure and the J₂ effect. The attenuation of the solar radiation can then be calculated from the ring model. In comparison to circular orbits, eccentric orbits yield a more stable environment for small grain sizes and therefore achieve higher efficiencies when the orbit decay of the material is considered. Moreover, the novel orbital dynamics experienced by high area-to-mass ratio objects, influenced by solar radiation pressure and the J₂ effect, ensure the ring will maintain a permanent heliotropic shape, with dust spending the largest portion of time on the Sun facing side of the orbit. It is envisaged that small dust grains can be released from a circular generator orbit with an initial impulse to enter an eccentric orbit with Sun-facing apogee. Finally, a lowest estimate of 1 × 10¹² kg of material is computed as the total mass required to offset the effects of global warming.     

Propagation errors analysis of TLE data

Orbit position uncertainty is an important factor for collision avoidance issues. For a single object with high frequency historical data, we can attain its position uncertainty easily. But sometimes data is not enough for errors analysis, orbits need to be classified. In this paper error analysis is made from two-line element sets data (TLEs). The Simplified General Perturbations-4 (SGP4) propagator was used. Statistical errors of debris and R/B are given for lower-altitude orbits which are classified by perigee altitude and eccentricity. The errors results and analysis for SSO (the Sun synchronous orbit) typical orbits are obtained. At last atmospheric drag as a main cause of downrange errors in lower-altitude orbit is analyzed. BSTAR in TLEs is modified to improve prediction precision.     

基于X射线脉冲星的航天器自主导航数值分析研究木

基于X射线脉冲星的天文自主导航方法稳定、可靠、精度高.是航天器自主导航领域重要的发展方向.论文介绍了基于X射线脉冲星导航的基本原理,给出了脉冲星自主定位的滤波模型;基于扩展卡尔曼滤波导航算法,以地球同步轨道为例,分析了不同初始误差、轨道倾角、轨道偏心率以及不同脉冲星方位误差下的导航精度.仿真算例表明,在脉冲星方位误差为0.001"、脉冲到达时间测量误差为0.1μs的情况下,导航定位精度优于1 km,且对初始误差不敏感;脉冲星方位精度、脉冲计时精度、轨道面位置和轨道偏心率是影响导航精度的主要因素.     

基于X射线脉冲星的航天器自主导航数值分析研究

基于X射线脉冲星的天文自主导航方法稳定、可靠、精度高,是航天器自主导航领域重要的发展方向.论文介绍了基于X射线脉冲星导航的基本原理,给出了脉冲星自主定位的滤波模型;基于扩展卡尔曼滤波导航算法,以地球同步轨道为例,分析了不同初始误差、轨道倾角、轨道偏心率以及不同脉冲星方位误差下的导航精度.仿真算例表明,在脉冲星方位误差为0.001″、脉冲到达时间测量误差为0.1μs的情况下,导航定位精度优于1km,且对初始误差不敏感;脉冲星方位精度、脉冲计时精度、轨道面位置和轨道偏心率是影响导航精度的主要因素.