月球卫星轨道力学综述

月球探测器的运动通常可分为3个阶段，这3个阶段分别对应3种不同类型的轨道：近地停泊轨道、向月飞行的过渡轨道与环月飞行的月球卫星轨道。近地停泊轨道实为一种地球卫星轨道；过渡轨道则涉及不同的过渡方式(大推力或小推力等)；环月飞行的月球卫星轨道则与地球卫星轨道有很多不同之处，它决不是地球卫星轨道的简单克隆。针对这一点，全面阐述月球卫星的轨道力学问题，特别是环月飞行中的一些热点问题，如轨道摄动解的构造、近月点高度的下降及其涉及的卫星轨道寿命、各种特殊卫星(如太阳同步卫星和冻结轨道卫星等)的轨道特征、月球卫星定轨等。     

关于月球低轨卫星运动的两个问题

对月球低轨卫星的轨道寿命特征和冻结轨道晶状态作了详尽的理论分析,给出它们与轨道倾角之间的关系以及它们相互之间的某种联系,并考虑低轨卫星的主要摄动源,在完整力模型下作了相应的模拟计算,不仅证实了理论分析的正确性,而且为环月运行探测器的轨道设计提供了极有参考价值的数值结果．     

从地面发射月球探测器的窗口选择

典型的月球探测器飞行轨道包括地球停泊轨道段、地月转移轨道段、月球卫星轨道段和着月轨道段。首先介绍了设计从地面发射月球探测器轨道典型的约束条件;然后,借助于二体假设,建立解析表达式,分析各种约束对窗口选择的影响,给出了各轨道段概略的飞行时间和粗窗口;最后,利用精确的探测器轨道动力学模型,计算精窗口,并给出了一则算例,所得结论可为月球探测器轨道发射、轨道设计提供依据。     

伽利略卫星的等势面和正常重力场

给出了轨道面接近赤道面的轨旋同步卫星的正常重力场在等势面上分布的展开式，并讨论了潮汐对其正常重力场的影响，利用这一方法，讨论了伽利略卫星正常重力场及其在等势面上的分布，以及木星的潮汐对伽利略卫星的正常重力场的影响，计算表明，潮汐对伽利略卫星的正常重力场影响不大，其径向的影响grt大约是10^-3-10^-5m/s^2的量级，与重力场在经度和纬度方向的分量接近，通过估算，月球的重力场所受到的潮汐影响要比绝大多数伽利略卫星受到的潮汐影响小。     

卫星跟踪卫星模式中轨道参数需求分析

首次基于半解析法利用GRACE(Gravity Recovery and Climate Experiment)双星K波段星间速度误差、GPS接收机轨道误差和加速度计非保守力误差影响累计大地水准面精度的联合模型开展了卫星跟踪卫星模式中轨道参数的需求分析.建议我国将来首颗重力卫星的平均轨道高度设计为400 km和平均星间距离设计为220 km较优.此研究不仅为我国将来卫星重力测量计划中轨道参数的优化选取以及全球重力场精度的有效和快速估计提供了理论基础和计算保证,同时对将来国际GRACE Follow-On地球重力测量计划和GRAIL(Gravity Recovery and Interior Laboratory)月球重力探测计划的发展方向具有一定的指导意义.     

在电离中电感应阻力对带电卫星轨道根数的摄动影响

研究了在高空电离层中运动的带电荷的卫星受电感应阻力后对轨道根数产生的摄动影响。研究结果表明，电感应阻力对带电卫星的轨道半长轴、轨道偏心率、近地点赤经、历元平赤经均有周期摄动影响，但除对半生长轴有长期摄动效应外对其它轨道根数均无长期摄动。轨道倾角和升交点赤经不受摄动影响。文中以飞行在高度1500km的电离层中的导体卫星作为算例。计算结果显示：带电导体卫星在高空电离层中带有一定电量时电感应阻力对轨道半长轴的缩短产生显著效应。     

日本SELENE月球探测计划和卫星间多普勒跟踪的数学模型

日本月球探测计划（SELENE）定于2004年夏季利用HIIa火箭发射一组共3颗绕月人造卫星。他们是主卫星、跟踪中断卫星和空间VLBI电波源。其主要科学目标之一是利用对绕月卫星的多普勒跟踪数据精确测定月球重力场,研究月球的起源与演化。SELENE计划中实现这个科学目标的关键技术是引入中继卫星,目的在于当处于低轨道的主卫星飞行到月球背面地面观测站无法观测时,采用卫星间跟踪方法（SST）,建立地面站与主卫星之间的联系,以得到月球背面重力场的直接测量数据。介绍了几种典型的四程卫星间多普勒跟踪模式和相应的数学模型,并针对SELENE计划中采用的特殊四程多普勒跟踪模式建立了卫星相对观测站速度与跟踪信号多普勒频移之间的转换关系。提出了利用GEODYNⅡ定轨分析软件处理SELENE多普勒跟踪数据的流程。     

满足约束条件的月球卫星飞行轨道的初步设计

讨论满足约束条件的月球卫星飞行轨道的设计问题，将约束条件分类为只与太阳，月球，地球，飞行器和观测站之间的相对位置有关的运行学约束条件以及涉及到飞行器轨道运行的动力学约束条件，在考虑月球卫星轨道的受力情况后，给出一种准确快速地计算和设计满足约束条件的标准飞行轨道的方法，并应用于不同约束条件下月球卫星的轨道预设计，初步讨论了轨道设计的误差分析，轨道跟踪及实时精密定轨等正在进行的其它相关工作。     

在电离中电感应阻力对带电卫星轨道根数的摄动影响

研究了在高空电离层中运动的带电荷的卫星受电感应阻力后对轨道根数产生的摄动影响。研究结果表明 ,电感应阻力对带电卫星的轨道半长轴、轨道偏心率、近地点赤经、历元平赤经均有周期摄动影响 ,但除对半长轴有长期摄动效应外对其它轨道根数均无长期摄动。轨道倾角和升交点赤经不受摄动影响。文中以飞行在高度 1 50 0km的电离层中的导体卫星作为算例。计算结果显示 :带电导体卫星在高空电离层中带有一定电量时电感应阻力对轨道半长轴的缩短产生显著效应     

制约卫星轨道寿命的另一种机制

近点共振会导致太阳系小天体(小行星,自然卫星以及大行星和月球的人造卫星)的轨道偏心率出现变幅较大的长周期变化,特别是以月球和大行星为中心天体的大倾角轨道(确切地说是倾角接近90°的极轨道)卫星,由于类似的原因,偏心率的增大而导致近星距rp=a(1-e)≤ae(ae是中心天体的赤道半径),使其落到中心天体上,结束轨道寿命,这与耗散机制大不相同,因此将对其作理论分析,并以计算实例加以证实.     

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．     

Analytical investigations of quasi-circular frozen orbits in the Martian gravity field

Frozen orbits are always important foci of orbit design because of their valuable characteristics that their eccentricity and argument of pericentre remain constant on average. This study investigates quasi-circular frozen orbits and examines their basic nature analytically using two different methods. First, an analytical method based on Lagrangian formulations is applied to obtain constraint conditions for Martian frozen orbits. Second, Lie transforms are employed to locate these orbits accurately, and draw the contours of the Hamiltonian to show evolutions of the equilibria. Both methods are verified by numerical integrations in an 80 × 80 Mars gravity field. The simulations demonstrate that these two analytical methods can provide accurate enough results. By comparison, the two methods are found well consistent with each other, and both discover four families of Martian frozen orbits: three families with small eccentricities and one family near the critical inclination. The results also show some valuable conclusions: for the majority of Martian frozen orbits, argument of pericentre is kept at 270° because J ₃ has the same sign as J ₂; while for a minority of ones with low altitude and low inclination, argument of pericentre can be kept at 90° because of the effect of the higher degree odd zonals; for the critical inclination cases, argument of pericentre can also be kept at 90°. It is worthwhile to note that there exist some special frozen orbits with extremely small eccentricity, which could provide much convenience for reconnaissance. Finally, the stability of Martian frozen orbits is estimated based on the trace of the monodromy matrix. The analytical investigations can provide good initial conditions for numerical correction methods in the more complex models.     

A Proposed lunar orbiting gravity gradiometer experiment

Analysis of the gravity gradiometer developed by R. L. Forward and C. C. Bell at the Hughes Research Laboratories suggest than an accuracy, in the range 0.1 to 0.5 EU can be expected in a lunar orbiter application. This accuracy will allow gradient anomalies associated with mascons to be mapped with 1% accuracy and should reveal a great deal of new information about the lunar gravity field.The proposed experiment calls for putting such a gradiometer into a closely circular polar orbit at an average height of about 30 km above the lunar surface. This orbit allows the entire lunar surface to be covered in fourteen days, the gradiometer to be checked twice per revolution and results in successive passes above the lunar surface being spaced at about the resolution limit of about 30 km set both by the satellite altitude and instrumental integration time. Doppler tracking will be employed and the spacecraft will carry an electromagnetic altimeter. Gradient and altitude data from the far side of the Moon can be stored for replay when communication is re-established.     

Earth–Moon Weak Stability Boundaries in the restricted three and four body problem

This work deals with the structure of the lunar Weak Stability Boundaries (WSB) in the framework of the restricted three and four body problem. Geometry and properties of the escape trajectories have been studied by changing the spacecraft orbital parameters around the Moon. Results obtained using the algorithm definition of the WSB have been compared with an analytical approximation based on the value of the Jacobi constant. Planar and three-dimensional cases have been studied in both three and four body models and the effects on the WSB structure, due to the presence of the gravitational force of the Sun and the Moon orbital eccentricity, have been investigated. The study of the dynamical evolution of the spacecraft after lunar capture allowed us to find regions of the WSB corresponding to stable and safe orbits, that is orbits that will not impact onto lunar surface after capture. By using a bicircular four body model, then, it has been possible to study low-energy transfer trajectories and results are given in terms of eccentricity, pericenter altitude and inclination of the capture orbit. Equatorial and polar capture orbits have been compared and differences in terms of energy between these two kinds of orbits are shown. Finally, the knowledge of the WSB geometry permitted us to modify the design of the low-energy capture trajectories in order to reach stable capture, which allows orbit circularization using low-thrust propulsion systems.     

Simulation of the Chang’E-5 mission contribution in lunar long wavelength gravity field improvement

The precision of lunar gravity field estimation has improved by means of three to five orders of magnitude since the successful GRAIL lunar mission. There are still discrepancies however, in the low degree coefficients and long wavelength components of the solutions developed by two space research centers (JPL and GSFC). These discrepancies hint at the possibilities for improving the accuracy in the long wavelength part of the lunar gravity field. In the near future, China will launch the Chang’E-5 lunar mission. In this sample-return mission, there will be a chance to do KBRR measurements between an ascending module and an orbiting module. These two modules will fly around lunar at an inclination of ～49 degrees, with an orbital height of 100 km and an inter-satellite distance of 200 km. In our research, we simulated the contribution of the KBRR tracking mode for different GRAIL orbital geometries. This analysis indicated possible deficiencies in the low degree coefficient solutions for the polar satellite-to-satellite tracking mode at various orbital heights. We also investigated the potential contributions of the KBRR to the Chang’E-5 mission goal of lunar gravity field recovery, especially in the long wavelength component. Potential improvements were assessed using various power spectrums of the lunar gravity field models. In addition, we also investigated possible improvements in solving lunar tidal Love number K2. These results may assist the implementation of the Chang’E-5 mission.     

Study of the transfer between libration point orbits and lunar orbits in Earth–Moon system

This paper is devoted to the study of the transfer problem from a libration point orbit of the Earth–Moon system to an orbit around the Moon. The transfer procedure analysed has two legs: the first one is an orbit of the unstable manifold of the libration orbit and the second one is a transfer orbit between a certain point on the manifold and the final lunar orbit. There are only two manoeuvres involved in the method and they are applied at the beginning and at the end of the second leg. Although the numerical results given in this paper correspond to transfers between halo orbits around the \(L_1\) point (of several amplitudes) and lunar polar orbits with altitudes varying between 100 and 500 km, the procedure we develop can be applied to any kind of lunar orbits, libration orbits around the \(L_1\) or \(L_2\) points of the Earth–Moon system, or to other similar cases with different values of the mass ratio.     

Averaged model to study long-term dynamics of a probe about Mercury

This paper provides a method for finding initial conditions of frozen orbits for a probe around Mercury. Frozen orbits are those whose orbital elements remain constant on average. Thus, at the same point in each orbit, the satellite always passes at the same altitude. This is very interesting for scientific missions that require close inspection of any celestial body. The orbital dynamics of an artificial satellite about Mercury is governed by the potential attraction of the main body. Besides the Keplerian attraction, we consider the inhomogeneities of the potential of the central body. We include secondary terms of Mercury gravity field from \(J_2\) up to \(J_6\), and the tesseral harmonics \(\overline{C}_{22}\) that is of the same magnitude than zonal \(J_2\). In the case of science missions about Mercury, it is also important to consider third-body perturbation (Sun). Circular restricted three body problem can not be applied to Mercury–Sun system due to its non-negligible orbital eccentricity. Besides the harmonics coefficients of Mercury’s gravitational potential, and the Sun gravitational perturbation, our average model also includes Solar acceleration pressure. This simplified model captures the majority of the dynamics of low and high orbits about Mercury. In order to capture the dominant characteristics of the dynamics, short-period terms of the system are removed applying a double-averaging technique. This algorithm is a two-fold process which firstly averages over the period of the satellite, and secondly averages with respect to the period of the third body. This simplified Hamiltonian model is introduced in the Lagrange Planetary equations. Thus, frozen orbits are characterized by a surface depending on three variables: the orbital semimajor axis, eccentricity and inclination. We find frozen orbits for an average altitude of 400 and 1000 km, which are the predicted values for the BepiColombo mission. Finally, the paper delves into the orbital stability of frozen orbits and the temporal evolution of the eccentricity of these orbits.     

倾角函数展开及其在分析法轨道预报中的应用

田谐项摄动是分析法轨道预报中的重要部分,其中包含大量倾角函数及其偏导数的计算.由于具有精度更高、速度更快的优点,倾角函数一般通过递推方法计算.以文献中提出的改进Gooding方法为基础,将其给出的程序稍加改进,在计算2–50阶倾角函数时缩短了约24%的计算时间.考虑到分析法预报过程中轨道平倾角变化很小,以泰勒展开式计算倾角函数,可极大提高计算速度,较大程度地减小分析法预报耗时,且引力场阶次越高,减小幅度越大,取50阶时预报耗时缩短了48%.另一方面,以2阶展开式计算倾角函数时,与改进Gooding法相比,分析法预报星历偏差很小.对于500 km高度的低轨卫星,分别以改进Gooding法和2阶泰勒展开式计算倾角函数,预报3天,当地球引力场阶次不高于50时,二者预报星历偏差RMS (Root Mean Square)低于1 mm,且随着轨道高度的增加,预报星历偏差RMS逐渐减小.