Uncertainty-aware Cube algorithm for medium-term collision risk assessment

The number of Earth orbiting objects is constantly growing, and some orbital regions are becoming risky environments for space assets of interest, which are increasingly threatened by accidental collisions with other objects, especially in Low-Earth Orbit (LEO). Collision risk assessment is performed by various methods, both covariance and non-covariance based. The Cube algorithm is a non-covariance-based method used to estimate the collision rates between space objects, whose concept consists in dividing the space in cubes of fixed dimension and, at each time instant, checking if two or more objects share the same cube. Up to now its application has been limited to the long-term scenarios of orbital debris evolutionary models, where considering the uncertainties is not necessary and impractical. Within operative contexts, instead, medium-term collision risk analysis may be an important task, in which the propagation-related uncertainties play a prominent role, but the timescale poses challenges for the application of standard covariance-based conjunction analysis techniques. In this framework, this paper presents an approach for the evaluation of the medium-term collision frequency for objects in LEO, called Uncertainty-aware Cube method. It is a modified version of the Cube, able to take the possible errors in the space objects’ position into account for the detection of the conjunctions. As an object’s orbit is propagated, the along-track position error grows more and more, and each object could potentially be in a different position with respect to the one determined by numerical propagation and, thus, in a different cube. Considering the uncertainties, at each time instant the algorithm associates more than one cube to each object and checks if they share at least one cube. If so, a conjunction is detected and a degree of confidence is evaluated. The performance of the method is assessed in different LEO scenarios and compared to the original Cube method.     

An analytical model for the prediction of a micro-dosimeter response function

A rapid analytical procedure for the prediction of a micro-dosimeter response function in low Earth orbit (LEO), correlated with the Space Transportation System (STS, shuttle) Tissue Equivalent Proportional Counter (TEPC) measurements is presented. The analytical model takes into consideration the energy loss straggling and chord length distribution of the detector, and is capable of predicting energy deposition fluctuations in a cylindrical micro-volume of arbitrary aspect ratio (height/diameter) by incoming ions through both direct and indirect (δ ray) events. At any designated (ray traced) target point within the vehicle, the model accepts the differential flux spectrum of Galactic Cosmic Rays (GCRs) and/or trapped protons at LEO as input. On a desktop PC, the response function of TEPC for each ion in the GCR/trapped field is computed at the average rate of 30 s/ion. The ionizing radiation environment at LEO is represented by O’Neill’s GCR model (2004), covering charged particles in the 1 ? Z ? 28 range. O’Neill’s free space GCR model is coupled with the Langley Research Center (LaRC) angular dependent geomagnetic cutoff model to compute the transmission coefficient in LEO. The trapped proton environment is represented by a LaRC developed time dependent procedure which couples the AP8MIN/AP8MAX, Deep River Neutron Monitor (DRNM) and F10.7 solar radio frequency measurements. The albedo neutron environment is represented by the extrapolation of the Atmospheric Ionizing Radiation (AIR) measurements. The charged particle transport calculations correlated with STS 51 and 114 flights are accomplished by using the most recent version (2005) of the LaRC deterministic High charge (Z) and Energy TRaNsport (HZETRN) code. We present the correlations between the TEPC model predictions (response function) and TEPC measured differential/integral spectra in the lineal energy (y) domain for both GCR and trapped protons, with the conclusion that the model correctly accounts for the increase in flux at low y values where energetic ions are the primary contributor. We further discuss that, even with the incorporation of angular dependency in the cutoffs, comparison of the GCR differential/integral flux between STS 51 and 114 TEPC measured data and current calculations indicates that there still exists an underestimation by the simulations at low to mid range y values. This underestimation is partly related the exclusion of the secondary pion particle production from the current version of HZETRN.     

Onboard orbit determination using GPS observations based on the unscented Kalman filter

Spaceborne GPS receivers are used for real-time navigation by most low Earth orbit (LEO) satellites. In general, the position and velocity accuracy of GPS navigation solutions without a dynamic filter are 25 m (1σ) and 0.5 m/s (1σ), respectively. However, GPS navigation solutions, which consist of position, velocity, and GPS receiver clock bias, have many abnormal excursions from the normal error range for space operation. These excursions lessen the accuracy of attitude control and onboard time synchronization. In this research, a new onboard orbit determination algorithm designed with the unscented Kalman filter (UKF) was developed to improve the performance. Because the UKF is able to obtain the posterior mean and covariance accurately by using the second-order Taylor series expansion through the sampled sigma points that are propagated by using the true nonlinear system, its performance can be better than that of the extended Kalman filter (EKF), which uses the linearized state transition matrix to predict the covariance. The dynamic models for orbit propagation applied perturbations due to the 40 × 40 geo-potential, the gravity of the Sun and Moon, solar radiation pressure, and atmospheric drag. The 7(8)th-order Runge–Kutta numerical integration was applied for orbit propagation. Two types of observations, navigation solutions and C/A code pseudorange, can be used at the user’s discretion. The performances of the onboard orbit determination were verified using real GPS data of the CHAMP and KOMPSAT-2 satellites. The results of the orbit determination were compared with the precision orbit ephemeris (POE) of the CHAMP and KOMPSAT-2 satellites.     

Variation of ionospheric total electron content in Taiwan region of the equatorial anomaly from 1994 to 2003

The ionospheric total electron content (TEC) in the northern hemispheric equatorial ionospheric anomaly (EIA) region is studied by analyzing dual-frequency signals of the Global Position System (GPS) acquired from a chain of nine observational sites clustered around Taiwan (21.9–26.2°N, 118.4–112.6°E). In this study, we present results from a statistical study of seasonal and geomagnetic effects on the EIA during solar cycle 23: 1994–2003. It is found that TEC at equatorial anomaly crests yield their maximum values during the vernal and autumnal months and their minimum values during the summer (except 1998). Using monthly averaged I_c (magnitude of TEC at the northern anomaly crest), semi-annual variations is seen clearly with two maxima occurring in both spring and autumn. In addition, I_c is found to be greater in winter than in summer. Statistically monthly values of I_c were poorly correlated with the monthly Dst index (r = −0.22) but were well correlated with the solar emission F_10.7 index (r = 0.87) for the entire database for the period during 1994–2003. In contrast, monthly values of I_c were correlated better with Dst (r ? 0.72) than with F_10.7 (r ? 0.56) in every year during the low solar activity period (1994–1997). It suggests that the effect of solar activity on I_c is a longer term (years), whereas the effect of geomagnetic activity on I_c is a shorter term (months).     

Accurate estimation of relative atmospheric density error on the example of uncertain geomagnetic activity information

Errors in neutral atmospheric density are the dominant contributor to unrealistic orbital state-vector covariances in low Earth orbits (LEO). Density uncertainty is caused by model-uncertainty at spatial scales below and within the model resolution, as well as input-uncertainty of the environmental parameters supplied to the semi-empirical atmospheric model.The paper at hand provides multiple contributions. First, analytic equations are derived to estimate the relative density error due to an input parameter uncertainty in any of the environmental parameters supplied to the model. Second, it is shown on the example of uncertain geomagnetic activity information, how to compute the required inputs to facilitate the accurate estimation of the relative density error.A clamped cubic splining approach for the conversion from geomagnetic amplitude (ap) to the kp index is postulated to perform this uncertainty propagation, as other algorithms were found unsuitable for this task. Results of numerical simulations with three popular semi-empirical models are provided to validate the set of derived equations. It is found that geomagnetic input uncertainty is especially important to consider in case of low global geomagnetic activity. The findings seamlessly integrate with prior work by the authors to perform density-uncertainty considering orbit estimation.     

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.     

Shape and motion estimate of LEO debris using light curves

We have derived a tri-axial ellipsoidal model of an LEO object, a Cosmos 2082 rocket body, including its rotational axis direction, rotation period, precession, and a compositional parameter, using only light curve data from an optical telescope. The brightness of the object was monitored for two days and least-squares fitting was used to determine these values. The derived axial ratios are 100:18:18, the coordinates of the rotational axis direction on the celestial sphere are R.A. = 305.8° and Dec. = 2.6°, and its observed average rotation period is 41 s. When precession is considered, its amplitude and precession period are 30.5° and 29.4 min. These results show that optical light curve data are sufficient to determine the shapes and the motions of LEO objects.     

Two-manoeuvres transfers between LEOs and Lissajous orbits in the Earth–Moon system

The purpose of this work is to compute transfer trajectories from a given Low Earth Orbit (LEO) to a nominal Lissajous quasi-periodic orbit either around the point L₁ or the point L₂ in the Earth–Moon system. This is achieved by adopting the Circular Restricted Three-Body Problem (CR3BP) as force model and applying the tools of Dynamical Systems Theory.     

Orbital debris–debris collision avoidance

We focus on preventing collisions between debris and debris, for which there is no current, effective mitigation strategy. We investigate the feasibility of using a medium-powered (5 kW) ground-based laser combined with a ground-based telescope to prevent collisions between debris objects in low-Earth orbit (LEO). The scheme utilizes photon pressure alone as a means to perturb the orbit of a debris object. Applied over multiple engagements, this alters the debris orbit sufficiently to reduce the risk of an upcoming conjunction. We employ standard assumptions for atmospheric conditions and the resulting beam propagation. Using case studies designed to represent the properties (e.g. area and mass) of the current debris population, we show that one could significantly reduce the risk of nearly half of all catastrophic collisions involving debris using only one such laser/telescope facility. We speculate on whether this could mitigate the debris fragmentation rate such that it falls below the natural debris re-entry rate due to atmospheric drag, and thus whether continuous long-term operation could entirely mitigate the Kessler syndrome in LEO, without need for relatively expensive active debris removal.     

A high order method for orbital conjunctions analysis: Sensitivity to initial uncertainties

A high order method to quickly assess the effect that uncertainties produce on orbital conjunctions through a numerical high-fidelity propagator is presented. In particular, the dependency of time and distance of closest approach to initial uncertainties on position and velocity of both objects involved in a conjunction is studied. The approach relies on a numerical integration based on differential algebraic techniques and a high-order algorithm that expands the time and distance of closest approach in Taylor series with respect to relevant uncertainties. The modeled perturbations are atmospheric drag, using NRLMSISE-00 air density model, solar radiation pressure with shadow, third body perturbation using JPL’s DE405 ephemeris, and EGM2008 gravity model. The polynomial approximation of the final position is used as an input to compute analytically the expansion of time and distance of closest approach. As a result, the analysis of a close encounter can be performed through fast, multiple evaluations of Taylor polynomials. Test cases with objects ranging from LEO to GEO regimes are considered to assess the performances and the accuracy of the proposed method.     

空间目标碰撞预警的协方差计算与应用

碰撞概率是碰撞预警工程中空间目标危险交会的重要判据之一,其计算精度会受到预报协方差计算精度的影响. 本文统计计算了两种形式的预报协方差. 一是利用精密数值预报模型对卫星精密根数进行预报,将预报根数与精密根数的差作为样本,统计得出1～7天不同预报期的预报协方差;二是采用 SGP4/SDP4预报模型对TLE数据进行预报,将预报根数与TLE根数的差作为样本,统计得出1～7d不同预报期的预报协方差. 分别分析两种方法中包含变轨过程和无变轨情况下的轨道预报精度. 结合2012年某低轨道卫星的危险交会,分析了采用不同协方差时,协方差精度对碰撞预警精度的影响. 协方差计算可为实际碰撞预警工程提供参考.      

Feasibility analysis of LEO and GEO large space debris de/re-orbiting taking into account launch mass of spacecraft-collector and its configuration layout

Analysis of the efficiency of two basic strategies for de/re-orbiting large space debris objects to disposal orbits (DO) is given. Large objects in LEO are classified into groups with similar orbital inclinations and comprise primarily last stages of launch vehicles, in GEO vicinity the paper studies upper stages. Under the first de/re-orbiting variant, it is assumed a spacecraft-collector is equipped with several thruster de/re-orbiting kits (TDKs); one of them can be fixed on an object and is capable of de/re-orbiting an object to a DO independently of the collector. In the second variant, a collector operates as a space tug: transfers objects to a DO and then returns to the next objects in line. The authors study possible configuration layouts of collectors in LEO and near GEO. The available analogous projects are analyzed. The efficiency of both de/re-orbiting variants can be properly compared using the estimations of collector's dry mass and having at one's disposal the parameters of the maneuvers required for transfers between all objects in the group. As reasonable criteria of effectiveness, one can consider (separately or jointly) the launch mass of an equipped collector, its ΔV budget, and the required number of such active spacecraft. Two de/re-orbiting variants are compared in terms of these criteria via mass-energy diagrams constructed for each group of objects in both altitude regions. Analysis of these diagrams shows that low Earth orbits can be more efficiently cleaned under the first de-orbiting variant by using a two-stage space system consisting of an active spacecraft carrying TDKs. For GEO, it is expedient to choose the second re-orbiting variant using a single-stage spacecraft. Our analysis shows that LEO cleaning is an order of magnitude more expensive than that for GEO, hence the problem of LEO population should be given increased attention.     

Evolution of the orbital elements for objects with high area-to-mass ratios in geostationary transfer orbits

A new population of uncatalogued objects in geosynchronous Earth orbits (GEO), with a mean motion of about 1 rev/day and eccentricities up to 0.6, has been identified recently. The first observations of this new type of objects were acquired in the framework of the European Space Agency’s (ESA) search for space debris in GEO and the geostationary transfer orbit (GTO) using the ESA 1-m telescope on Tenerife. Earlier studies have postulated that the perturbations due to the solar radiation pressure can lead to such large eccentricities for GEO objects with a high area-to-mass ratio (A/M). The simulations showed that the eccentricities of GEO objects with large A/M exhibit periodic variations with periods of about one year and amplitudes depending on the value of A/M. The findings of these studies could be confirmed by observations from the ESA 1-m telescope on Tenerife.     

ESA’s process for the identification and assessment of high-risk conjunction events

ESA’s Space Debris Office provides an operational service for the assessment of collision risks of ESA satellites. Currently, the ENVISAT and ERS-2 missions in low Earth orbits are covered by this service. If an upcoming high-risk conjunction event is predicted based on analysis of Two-Line Element (TLE) data from the US Space Surveillance Network, then independent tracking data of the potential high-risk conjunction object are acquired to improve the knowledge of its orbit. This improved knowledge and the associated small error covariances derived from the orbit determination process scale down the position error ellipsoid at the conjunction epoch. Hence, for the same miss-distance, in most cases an avoidance manoeuvre can be suppressed with an acceptable residual risk.     

On the large-scale structure of the tail current as measured by THEMIS

The magnetic field structure and the spatial characteristics of the large-scale currents in the magnetospheric tail were studied during quiet and moderately disturbed geomagnetic conditions in 2009. The magnetic field of the currents other than the tail current was calculated in terms of a paraboloid model of the Earth’s magnetosphere, A2000, and was subtracted from measurements. It was found on the base of obtained tail current magnetic field radial distribution that the inner edge of the tail current sheet is located in the night side magnetosphere, at distances of about 10 R_E and of about 7 R_E during quiet and disturbed periods respectively. During the disturbance of February 14, 2009 (Dst_min ∼ −35 nT), the B_x and the B_z component of the tail current magnetic field near its inner edge were about 60 nT, and −60 nT that means that strong cross-tail current have been developed. The tail current parameters at different time moments during February 14, 2009 have been estimated. Solar wind conditions during this event were consistent with those during moderate magnetic storms with minimum Dst of about −100 nT. However, the magnetospheric current systems (magnetopause and cross-tail currents) were located at larger geocentric distances than typical during the 2009 extremely quiet epoch and did not provide the expected Dst magnitude. Very small disturbance on the Earth’s surface was detected consistent with an “inflated” magnetosphere.     

Inversion of satellite gradiometry data using statistically modified integral formulas for local gravity field recovery

The satellite gravity gradiometric data can be used directly to recover the gravity anomaly at sea level using inversion of integral formulas. This approach suffers by the spatial truncation errors of the integrals, but these errors can be reduced by modifying the formulas. It allows us to consider smaller coverage of the satellite data over the region of recovery. In this study, we consider the second-order radial derivative (SORD) of disturbing potential (T_rr) and determine the gravity anomaly with a resolution of 1° × 1° at sea level by inverting the statistically modified version of SORD of extended Stokes’ formula. Also we investigate the effect of the spatial truncation error on the quality of inversion considering noise of T_rr. The numerical investigations show satisfactory results when the area of T_rr coverage is the same with that of the gravity anomaly and the integral formula is modified by the biased least-squares modification. The error of recovery will be about 6 mGal after removing the regularization bias in the presence of 1 mE noise in T_rr measured on the orbit.     

轨道姿态误差对TDI-CCD相机行周期及偏流角的影响分析

与其他文献的几何方法不同,本文从运动学角度出发,选定合适的参考系后,按照一系列不同坐标系的坐标转换,完成星下点运动速度矢量在敏感器坐标系的推导,把姿态、轨道、安装偏差作为相关参数,经过一系列坐标变换,表征在最终的偏流角公式及行转移频率动态公式中;在通常的对地稳定指向模式下,姿态参数和安装偏差很小,仅保留一阶小量.根据矢量表达,解析地给出行转移频率及偏流角与轨道、姿态的关系,从而方便分析误差的影响,并以几个典型的低轨为实例作了说明.     

An alternative Shell inversion technique - Analysis and validation based on COSMIC and ionosonde data

Multi-channel Global Positioning System (GPS) carrier phase signals, received by the six low Earth orbiting (LEO) satellites from the Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) program, were used to undertake active limb sounding of the Earth’s atmosphere and ionosphere via radio occultation. In the ionospheric radio occultation (IRO) data processing, the standard Shell inversion technique (SIT), transformed from the traditional Abel inversion technique (AIT), is widely used, and can retrieve good electron density profiles. In this paper, an alternative SIT method is proposed. The comparison between different inversion techniques will be discussed, taking advantage of the availability of COSMIC datasets. Moreover, the occultation results obtained from the SIT and alternative SIT at 500 km and 800 km, are compared with ionosonde measurements. The electron densities from the alternative SIT show excellent consistency to those from the SIT, with strong correlations over 0.996 and 0.999 at altitudes of 500 km and 800 km, respectively, and the peak electron densities (N_mF₂) from the alternative SIT are equivalent to the SIT, with 0.839 vs. 0.844, and 0.907 vs. 0.909 correlation coefficients when comparing to those by the ionosondes. These results show that: (1) the N_mF₂ and h_mF₂ retrieved from the SIT and alternative SIT are highly consistent, and in a good agreement with those measured by ionosondes, (2) no matter which inversion technique is used, the occultation results at the higher orbits (∼800 km) are better than those at the lower orbits (∼500 km).