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.     

Explicit Symplectic Algorithms For Time‐Transformed Hamiltonians

By Hamiltonian manipulation we demonstrate the existence of separable time‐transformed Hamiltonians in the extended phase‐space. Due to separability explicit symplectic methods are available for the solution of the equations of motion. If the simple leapfrog integrator is used, in case of two‐body motion, the method produces an exact Keplerian ellipse in which only the time‐coordinate has an error. Numerical tests show that even the rectilinear N‐body problem is feasible using only the leapfrog integrator. In practical terms the method cannot compete with regularized codes, but may provide new directions for studies of symplectic N‐body integration. This revised version was published online in July 2006 with corrections to the Cover Date.     

Tests and comparisons of gravity models

Optical observations of the GEOS satellites were used to obtain orbital solutions with different sets of geopotential coefficients. The solutions were compared before and after modification to high order terms (necessary because of resonance) and then analyzed by comparing subsequent observations with predicted trajectories. The most important source of error in orbit determination and prediction for the GEOS satellites is the effect of resonance found in most published sets of geopotential coefficients. Modifications to the sets yield greatly improved orbits in most cases.The sets of coefficients analyzed are APL 3.5, NWL5E-6, Köhnlein (1967), Rapp (1967), Kaula (1967), Smithsonian Astrophysical Observatory (SAO)M-1 (1966), SAO AGU (1969), SAO COSPAR (1969) and SAO 1969 Standard Earth. The SAO 1969 models generally give better orbital fits and prediction results than the other models above. However these models can be improved by corrections to resonant coefficients.The results of these comparisons suggest that with the best optical tracking systems and gravity models, satellite position error due to gravity model uncertainty can reach 50–100 m during a heavily observed 5–6 day orbital arc. If resonant coefficients are estimated, the uncertainty is reduced considerably.     

Approximate Analytical Theory of Satellite Orbit Prediction Presented in Spherical Coordinates Frame

A comparative review of analytic theories for the motion of Earth satellites in quasi-circular orbits written in the spherical coordinate frame is presented. The theory of motion is developed for satellites in quasi-circular and quasi-equatorial orbits subjected to geopotential, luni-solar and solar radiation pressure force perturbations. The intermediate orbit is Keplerian and the equations of motion are solved by the Lyapunov–Poincaré small parameter method. Both resonant and non-resonant cases are considered. The results can be useful for the development of a complete theory of weakly eccentric orbits.     

Oscillatory orbits in the planar three‐body problem with equal masses

In the free‐fall three‐body problem, distributions of escape, binary, and triple collision orbits are obtained. Interpretation of the results leads us to the existence of oscillatory orbits in the planar three‐body problem with equal masses. A scenario to prove their existence is described. This revised version was published online in July 2006 with corrections to the Cover Date.     

Unified Symbolic Algorithm of Gauss Method for Near-Parabolic Orbits

In this paper, a unified algorithm of Gauss method for near‐parabolic orbits that is valid for both elliptic and hyperbolic cases is established symbolically and numerically. This revised version was published online in July 2006 with corrections to the Cover Date.     

Detection of a Satellite Orbiting The Nucleus of Comet Hale–Bopp (C/1995 O1)

This paper reports on the detection of a satellite around the principal nucleus of comet Hale-Bopp. As shown elsewhere, a successful morphological model for the comet's dust coma necessitates the postulation of overlapping jet activity from a comet pair. The satellite has been detected digitally on images taken with the Hubble Space Telescope's Wide Field Planetary Camera 2 in the planetary mode on five days in May–October 1996. An average satellite-to-primary signal ratio is 0.21 ± 0.03, which implies that the satellite is ∼30 km in diameter, assuming the main nucleus is ∼70 km across. To avoid collision, the separation distance must exceed 50–60 km at all times. The satellite's projected distances on the images vary from 160 to 210 km, or 0.06 to 0.10 arcsec. The satellite was not detected in October 1995, presumably because of its subpixel separation from the primary. The radius of the gravitational sphere of action of the principal nucleus 70 km in diameter is 370–540 km at perihelion, increasing linearly with the Sun's distance: the satellite appears to be in a fairly stable orbit. Its orbital period at ∼180 km is expected to be ∼2–3 days, much shorter than the intervals between the HST observations. If the main nucleus should be no more than 42 km across, Weaver et al.'s upper limit, the satellite's orbit could become unstable, with the object drifting away from the main nucleus after perihelion. Potentially relevant ground-based detections of close companions are reported. Efforts to determine the satellite's orbit and the total mass of the system will get under way in the near future. This revised version was published online in July 2006 with corrections to the Cover Date.     

GPS定轨中的太阳辐射压模型

对于GPS这样的高轨卫星轨道的确定,最大的误差源为太阳辐射压摄动．近年来IGS各个数据处理中心提供的GPS星历精度越来越高,其中很重要的一个因素就是太阳辐射压摄动模型的不断完善．详细阐述了目前主要的7种太阳辐射压摄动模型后,给出了各种光压摄动模型的计算模型,并利用不同的摄动模型积分卫星轨道,得到不同模型在GPS卫星轨道积分中的精度．结果表明,Bern大学提供的3种模型对太阳辐射压的模拟较为准确,相对于其他4种模型,由其得到的GPS轨道精度有将近一个量级的提高．     

Efficient Symplectic Integration of Satellite Orbits

The use of the extended phase space and time transformations for constructing efficient symplectic methods for computing the long term behavior of perturbed two‐body systems are discussed. Main applications are for artificial satellite orbits. The methods suggested here are efficient also for large eccentricities. This revised version was published online in July 2006 with corrections to the Cover Date.     

Chaotic Scattering in the Restricted Three‐Body Problem II. Small mass parameters

We study the scattering motion of the planar restricted three‐body problem for small mass parameters μ. We consider the symmetric periodic orbits of this system with μ = 0 that collide with the singularity together with the circular and parabolic solutions of the Kepler problem. These divide the parameter space in a natural way and characterize the main features of the scattering problem for small non‐vanishing μ. Indeed, continuation of these orbits yields the primitive periodic orbits of the system for small μ. For different regions of the parameter space, we present scattering functions and discuss the structure of the chaotic saddle. We show that for μ < μc and any Jacobi integral there exist departures from hyperbolicity due to regions of stable motion in phase space. Numerical bounds for μc are given. This revised version was published online in July 2006 with corrections to the Cover Date.     

低轨通信卫星多普勒定位性能分析

为了研究低轨通信卫星多普勒定位性能,首先分析了低轨卫星的对地覆盖特性、信号传输特性以及多普勒频移特性,推导了多普勒定位原理和方法,提出了适用于多普勒定位的精度因子.基于已在轨的铱星和全球星系统,解算了全球范围可见卫星数和定位精度因子,并对相应测站进行了定位仿真实验和误差分析.结果表明:对于铱星和全球星系统,随着纬度降低,卫星可见数减小,多普勒几何精度因子变大;多普勒定位结果精度同时受到频率测量精度、卫星位置误差以及卫星速度误差影响,当卫星位置误差小于10 m、卫星速度误差小于0.1 km·s^-1时,对定位结果影响不大,此时频率测量精度成为影响定位精度的决定性因素,且当频率测量精度为0.01 Hz时,定位精度可达1.18 m.     

小倾角LEO多星天基平台光学跟踪GEO精密定轨

针对地基卫星测控系统(Tracking Telemetry and Command, TT&C)系统对地球静止轨道(Geostation-\lk ary Earth Orbit, GEO)卫星在空间和时间覆盖上的局限性, 提出小倾角低地球轨道(Low Earth Orbit, LEO)多星组网天基平台对GEO卫星进行跟踪定轨的方法. 根据空间环境和光学可视条件对仿真数据进行筛选以模拟真实的观测场景, 利用光学测角数据, 使用数值方法对GEO卫星的轨道进行确定. 结果与参考轨道进行重叠对比, 在平台轨道精度5 m、测量精度5rq\rq、 定轨弧长12 h的情况下, 两颗LEO卫星对GEO卫星进行跟踪定轨的精度可达到千米量级, 4颗LEO卫星对GEO目标进行跟踪定轨的精度可达到百米量级. 随着LEO组网卫星数量的增加, 定轨精度得到了较大的提高.     

Estimation of low degree geopotential coefficients using SLR data

Geodetic satellites have been providing the low frequency part of the geopotential models used for precise orbit determination purposes (e.g. JGM3, EGM96, …). Nevertheless they can be used to estimate the temporal variation of selected coefficients, helping to clarify the complex interrelations in the earth-ocean-atmosphere system. In this paper we present the two years long analysis of SLR data from the seven available geodetic satellites (Lageos I–II, Stella, Starlette, Ajisai, Etalon I–II) to recover monthly estimates of low degree geopotential coefficients; the results are obtained analysing the satellites separately and in proper combination. An accurate modelling of the satellite orbits is required in order to separate the geopotential coefficients: we assume as a priori geopotential the JGM3 model together with its associated tides and we take care of non-gravitational effects on the satellites by means of proper empirical estimated accelerations. The time series of the estimated coefficients (J₂, J₃, J₄, J₅) are inspected to detect the sub-annual perturbations related to seasonal variation of mass distribution. Huge residual seasonal signals in the orbit of Stella indicate a strong model deficiency related to the Sun's influence on the environment. The remaining six satellites are homogeneously modelled and build up a three cycles per year oscillation on J₂ and a seasonal oscillation (1 year and six month periods) revealed on the J₄. The origin and possible causes of these signals are further discussed in the text. We also present a preliminary estimate, using twelve years of Lageos-I and Lageos-II observations, that is compared with previous obtained values.     

Comparison of Numerical Methods for the Integration of Natural Satellite Systems

We present a new implementation of the recurrent power series (RPS) method which we have developed for the integration of the system of N satellites orbiting a point-mass planet. This implementation is proved to be more efficient than previously developed implementations of the same method. Furthermore, its comparison with two of the most popular numerical integration methods: the 10th-order Gauss–Jackson backward difference method and the Runge–Kutta–NystrRKN12(10)17M shows that the RPS method is more than one order of magnitude better in accuracy than the other two. Various test problems with one up to four satellites are used, with initial conditions obtained from ephemerides of the saturnian satellite system. For each of the three methods we find the values of the user-specified parameters (such as the method's step-size (h or tolerance (TOL)) that minimize the global error in the satellites' coordinates while keeping the computer time within reasonable limits. While the optimal values of the step-sizes for the methods GJ and RKN are all very small (less than T/100, the ones that are suitable for the RPS method are within the range: T/13<h<T/6 (T being the period of the innermost satellite of the problem). Comparing the results obtained by the three methods for these step-sizes and for the various test problems we observe the superiority of the RPS method over GJ in terms of accuracy and over RKN both in accuracy and in speed. This revised version was published online in July 2006 with corrections to the Cover Date.     

小型光电人卫跟踪仪的摆动叉式ALT-ALT跟踪机架

人卫跟踪仪一般采用地平式跟踪机架 ,由于这种机架固有的天顶盲区 ,致使观测数据不连续而造成卫星精密定轨的困难。讨论了小型光电人卫跟踪仪的ALT ALT机架原理 ,分析证明采用这种机架形式没有天顶盲区、跟踪速度和加速度较小。同时提出了一种新颖的摆动叉式ALT ALT跟踪机架 ,具有全天覆盖无遮挡、体积紧凑小巧等优点 ,其力学性能也十分优良 ,适合小型光电人卫跟踪仪和流动观测仪器使用     

野值预剔除方法的改进

在初轨有较大误差时，传统的光学资料野值预剔除方法——截断最小二乘L估计法不能有效地剔除野值．从门限和模型两方面改进野值预处理方法，即改用统计中误差定门限．将残差的线性模型改为2次曲线模型，结果为：对于模拟产生的所有卫星弧段，在初轨有10^-4（半长轴α为其5％，以下不再说明）误差情况下，野值比例都能降低到10％以下．     

The accuracy of geopotential models

Extensive tests of two recent geopotential models (GEM 7 and 8) have been made with observations not used in the solutions. Several other recent models are also evaluated. These tests show the accuracy of the satellite derived model (GEM 7, with 400 coefficients) to be about 4.3 m (r.m.s.) with respect to the global geoid surface. The corresponding accuracy of the combined satellite and surface gravimetry model (GEM 8, with 706 coefficients) is found to be 3.9m (r.m.s.). These results include a calibration for the commission errors of the coefficients in the models and an estimate of the errors from omitted coefficients. For GEM 7, the formal precision (commission errors) of the solution gives 0.7 m for the geoid error which after calibration increases to 2.4 m.

Independent observations used in this assessment include: 159 lumped coefficients from 35 resonant orbits of 1 and 9 through 15 revolutions per day, two sets of (8, 8) fields derived from optical-only and laser-only data, sets of zonal and resonant coefficients derived from largely independent sources and geoid undulations measured by satellite altimetry. In addition, the accuracy of GEM 7 has been judged by the gravimetry in GEM 8. The ratio of estimated commission to formal error in GEM 7 and 8 ranges from 2 to 5 in these tests.     

A Study on Orbit Determination by Using the Data of Crossovers from Satellite-borne Altimeter

Starting with the status of the developments of oceanic altimetry satellites, the significance of orbit determination by using altimeter data is introduced. Then the error correction model of the altimeter data and the calculational method of the data of crossovers are analyzed. The modification quantities of the errors and the adopted model concerned in the files of the altimeter data from the aircraft JASON-1 are also introduced in detail. Finally, through the calculations of the simulated data and the data of actual measurements, the highest accuracies of the orbit determination reached by solely using the altimeter data and the data of crossovers are analyzed, respectively. And this work provides a valuable reference to the practical applications in future.     

北斗广域增强系统星钟和星历误差改正算法研究

区域北斗星基增强系统提供等效钟差改正数统一修正星历和钟差误差。随着系统的建设发展,新一代北斗星基增强系统将区分星历和钟差误差改正信息,以提高差分改正精度。由于北斗卫星混合星座设计及区域监测网的局限,星历和钟差误差的高精度分离计算面临着新的挑战。对北斗星基增强系统的星历和钟差改正算法进行了研究,分别采用动力学和运动学模式计算了卫星星历和钟差改正数,并基于北斗实测数据,对两种处理模式的差分改正精度进行了对比研究。试验结果表明,采用动力学和运动学差分方法,得到的双频伪距实时定位精度分别为1.76m和1.78m,定位精度与WAAS及EGNOS相当。利用运动学和动力学差分改正数后均可得到分米级的精密单点定位(precise point position,PPP)结果,其中采用动力学广域差分改正数,收敛后定位精度可达到15cm;采用运动学广域差分改正数,收敛后定位精度可达45cm。     

The Orbital Evolution of Two Sounding Satellites and Analysis of the Accuracy of Orbit Determination

The satellites TC-1 and TC-2 are the two Chinese satellites with great elliptical orbits which are still in orbit around the earth at present. Since the launch the orbits of the two satellites have continuously evolved, which has a certain effect on the orbit determination and prediction precision. The regularities of the orbital evolution of the two sounding satellites are qualitatively and quantitatively analyzed. Under the current tracking mode the corresponding prediction precision of orbit determination is analyzed based on the different stages of the orbital evolution, thereby providing the basis for the adjustment of planning mode by the satellite application departments and the guarantee of normal satellite payload. Finally, the orbital lifetimes of the two satellites are predicted through the trend of the orbital evolution.