Symmetry,reduction and relative equilibria of a rigid body in the J2 problem

The J₂ problem is an important problem in celestial mechanics, orbital dynamics and orbital design of spacecraft, as non-spherical mass distribution of the celestial body is taken into account. In this paper, the J₂ problem is generalized to the motion of a rigid body in a J₂ gravitational field. The relative equilibria are studied by using geometric mechanics. A Poisson reduction process is carried out by means of the symmetry. Non-canonical Hamiltonian structure and equations of motion of the reduced system are obtained. The basic geometrical properties of the relative equilibria are given through some analyses on the equilibrium conditions. Then we restrict to the zeroth and second-order approximations of the gravitational potential. Under these approximations, the existence and detailed properties of the relative equilibria are investigated. The orbit–rotation coupling of the rigid body is discussed. It is found that under the second-order approximation, there exists a classical type of relative equilibria except when the rigid body is near the surface of the central body and the central body is very elongated. Another non-classical type of relative equilibria can exist when the central body is elongated enough and has a low average density. The non-classical type of relative equilibria in our paper is distinct from the non-Lagrangian relative equilibria in the spherically-simplified Full Two Body Problem, which cannot exist under the second-order approximation. Our results also extend the previous results on the classical type of relative equilibria in the spherically-simplified Full Two Body Problem by taking into account the oblateness of the primary body. The results on relative equilibria are useful for studies on the motion of many natural satellites, whose motion are close to the relative equilibria.     

Stability of relative equilibria of the full spacecraft dynamics around an asteroid with orbit–attitude coupling

The full dynamics of spacecraft around an asteroid, in which the spacecraft is considered as a rigid body and the gravitational orbit–attitude coupling is taken into account, is of great value and interest in the precise theories of the motion. The spectral stability of the classical relative equilibria of the full spacecraft dynamics around an asteroid is studied with the method of geometric mechanics. The stability conditions are given explicitly based on the characteristic equation of the linear system matrix. It is found that the linearized system decouples into two entirely independent subsystems, which correspond to the motions within and outside the equatorial plane of the asteroid respectively. The system parameters are divided into three groups that describe the traditional stationary orbit stability, the significance of the orbit–attitude coupling and the mass distribution of the spacecraft respectively. The spectral stability of the relative equilibria is investigated numerically with respect to the three groups of system parameters. The relations between the full spacecraft dynamics and the traditional spacecraft dynamics, as well as the effect of the orbit–attitude coupling, are assessed. We find that when the orbit–attitude coupling is strong, the mass distribution of the spacecraft dominates the stability of the relative equilibria; whereas when the orbit–attitude coupling is weak, both the mass distribution and the traditional stationary orbit stability have significant effects on the stability. We also give a criterion to determine whether the orbit–attitude coupling needs to be considered.     

Roto-orbital dynamics of a triaxial rigid body around a sphere. Relative equilibria and stability

We study the roto-orbital motion of a triaxial rigid body around a sphere, which is assumed to be much more massive than the triaxial body. The associated dynamics of this system, which consists of a normalized Hamiltonian with respect to the fast angles (partial averaging), is investigated making use of variables referred to the total angular momentum. The first order approximation of this model is integrable. We carry out the analysis of the relative equilibria, which hinges principally in the dihedral angle between the orbital and rotational planes and the ratio among the moments of inertia $\rho =(B-A)/(2C-B-A)$. In particular, the dynamics of the body frame, though formally given by the classical Euler equations, experiences changes of stability in the principal directions related to the roto-orbital coupling. When $\rho =1/3$, we find a family of relative equilibria connected to the unstable equilibria of the free rigid body.     

考虑太阳摄动的小行星附近轨道动力学

本文研究了艳后星(216 Kleopatra)和爱神星(433 Eros)附近的周期轨道,在考虑太阳引力摄动的情况下,发现了以往所遗漏的216 Kleopatra轨道族和环绕433 Eros的12族周期轨道,并且给出了它们的特性。研究结果表明,太阳引力对小行星平衡点位置的影响很小,但是对平衡点上航天器运动的影响较大。同族不稳定轨道中,大Jacobi常数轨道更容易在摄动后保持轨道原来特性,这很好地解释了小行星卫星在较远轨道上长期存在的可能性。     

轴对称磁静平衡态Ⅰ.微变换方法

轴对称磁静平衡态被广泛用来描述不同尺度的恒星大气结构。对这类平衡态,可引入所谓磁通量函数来描述磁场,它满足一变系数、非线性椭圆型方程。迄今为止,人们只就线性情况的某些特例给出该方程的解析解;对较为复杂的情况,特别是非线性情况,还未找到有效的办法进行解析处理。本文将应用微变换方法求上述变系数、非线性方程的相似解,并给出普遍结果。有关特解的具体形式及对太阳大气的应用将在本系列的下几篇论文中另作讨论。      

On the physical state in the narrow-line region of the classical Seyfert galaxy NGC 7469

Based on the spectrophotometric data, by decomposition of the observed continuum, the power - law continuum characteristics of the central source are obtained. The behaviour of both electron temperature T_e and density n_e in the Narrow - Line Region (NLR) of NGC 7469 is discussed on the thermal and ionization equilibria calculations. T_e in NLR of the Sy1 galaxies are higher than the Sy2 ones, and the possible explanation is the lack of the dense (n_e 10¹⁰ cm⁻³) zone close to the central source in the Sy2 galaxies.     

二维磁流体动力学平衡日冕(Ⅱ)

基于解析和数值相结合的方法,进一步讨论了非均匀引力场中日冕的二维磁流体动力学平衡。对临界点进行了比较仔细的处理。得到了包含闭场区、中性片和开场区的大尺度日冕磁场位形,闭场区和中性片构成冕流结构。在高纬和低纬地区几个太阳半径之外,等离子体径向流动速度超过了局地声速和局地Alfvén速度。在1AU处,太阳风速度可达到400kms_-1以上      

Cleft formation in pumpkin balloons

NASA’s development of a large payload, high altitude, long duration balloon, the Ultra Long Duration Balloon, centers on a pumpkin shape super-pressure design. Under certain circumstances, it has been observed that a pumpkin balloon may be unable to pressurize into the desired cyclically symmetric equilibrium configuration, settling into a distorted, undesired state instead. Success of the pumpkin balloon for NASA requires a thorough understanding of the phenomenon of multiple stable equilibria and developing of means for the quantitative assessment of design measures that prevent the occurrence of undesired equilibrium. In this paper, we will use the concept of stability to classify cyclically symmetric equilibrium states at full inflation and pressurization. Our mathematical model for a strained equilibrium balloon, when applied to a shape that mimics the Phase IV-A balloon of Flight 517, predicts instability at float. Launched in Spring 2003, this pumpkin balloon failed to deploy properly. Observations on pumpkin shape type super-pressure balloons that date back to the 1980s suggest that within a narrowly defined design class of pumpkin shape super-pressure balloons where individual designs are fully described by the number of gores n_g and by a single measure of the bulging gore shape, the designs tend to become more vulnerable with the growing number of gores and with the diminishing size of the bulge radius r_B Weight efficiency considerations favor a small bulge radius, while robust deployment into the desired cyclically symmetrical configuration becomes more likely with an increased bulge radius. In an effort to quantify this dependency, we will explore the stability of a family of balloon shapes parametrized by (n_g, r_B) which includes a design that is very similar, but not identical, to the balloon of Flight 517. In addition, we carry out a number of simulations that demonstrate other aspects related to multiple equilibria of pumpkin balloons.     

Semi-analytical theory of mean orbital motion for geosynchronous space debris under gravitational influence

This paper provides a hamiltonian formulation of the equations of motion of an artificial satellite or space debris orbiting the geostationary ring. This theory of order 1 has been formulated using canonical and non-singular elements for eccentricity and inclination. The analysis is based on an expansion in powers of the eccentricity and of the inclination. The theory accounts for the influence of the Earth gravity field expanded in spherical harmonics, paying a particular attention to the resonance occurring for geosynchronous objects. The luni-solar perturbations are also taken into account. We present the resonant motion and its main characteristics: equilibria, stability, fundamental frequencies and width of the resonant area by comparison with a basic analytical model. Finally, we show some results concerning the long term dynamics of a typical space debris under the influence of the gravitational field of the Earth and the luni-solar interactions.     

具有食饵选择的食物链模型的定性分析

生态学研究发现,多种食饵选择对捕食者-食饵系统的稳定性、持久性等方面有着重要的影响。处于食物链中间层的种群,既是顶层捕食者的食饵,同时又是捕食者,因此中间捕食者的种群数量变化,对整个生态系统有着不可忽略的影响。假设中间捕食者具有其他可供选择的食饵,建立了一个由顶级捕食者、中间捕食者和食饵所构成的三维食物链系统,并对此系统展开动力学分析,推导出系统各个平衡点的存在性和稳定性条件。对Hopf分支的存在性条件进行了深入的分析,并以食饵选择参数为分支参数,对可能出现的Hopf分支情况进行了数值模拟。结合理论和数值结果,分析食饵选择性对食物链系统的稳定性产生的影响。      

What are the physical mechanisms of eruptions and CMEs?

CMEs are due to physical phenomena that drive both, eruptions and flares in active regions. Eruptions/CMEs must be driven from initially force-free current-carrying magnetic field. Twisted flux ropes, sigmoids, current lanes and pattern in photospheric current maps show a clear evidence of currents parallel to the magnetic field. Eruptions occur starting from equilibria which have reached some instability threshold. Revisiting several data sets of CME observations we identified different mechanisms leading to this unstable state from a force free field. Boundary motions related to magnetic flux emergence and shearing favor the increase of coronal currents leading to the large flares of November 2003. On the other hand, we demonstrated by numerical simulations that magnetic flux emergence is not a sufficient condition for eruptions. Filament eruptions are interpreted either by a torus instability for an event occurring during the minimum of solar activity either by the diffusion of the magnetic flux reducing the tension of the restraining arcade. We concluded that CME models (tether cutting, break out, loss of equilibrium models) are based on these basic mechanisms for the onset of CMEs.     

基于θ-D非线性控制方法的伴随轨道控制

为了实现燃料消耗最少的伴随轨道控制,研究使用θ-D次优非线性控制方法设计伴随轨道的优化控制律。相对运动模型中考虑了地球形状摄动的影响,且适用于在椭圆轨道上运行的目标航天器。采用状态变量增广法,对相对运动模型进行处理,以满足θ-D控制方法的要求。采用相对轨道要素设计了理想的基准伴随轨道。仿真计算结果表明,所设计的控制律是稳定的,能够以较少的燃料消耗实现伴随轨道的控制。     

不完整测量下集群飞行器协同相对定位方法

测距和测向是实现飞行器间相对测量的常用手段。通常而言,飞行器间相互测距精度远高于飞行器绝对位置导航精度。因此,提出了一种基于集群飞行器间测距和对目标测向的协同相对定位方法,利用集群飞行器间相互测距和各飞行器对非合作目标测向来实现集群飞行器及非合作目标的相对定位。方法分为静态相对定位方法和动态相对定位方法两种。基于飞行器间相互测距和对非合作目标测向来实现协同相对定位的方案,不需要对非合作目标进行完整的相对位置测量,简化了对飞行器测量设备配置的要求。同时,相比基于飞行器绝对位置导航和对非合作目标测向的相对定位方案,能获得更高的相对定位精度。     

编队飞行卫星群构型保持及初始化

导出了基于相对轨道要素的编队飞行卫星群轨道相对运动控制的轨道机动方程；提出了编队飞行卫星群轨道相对运动控制的轨道机动控制策略，利用不同方向的脉冲控制相对运动的轨道参数，包括轨道平面内机动和轨道平面外机动控制。根据相对轨道要素的变轨机动控制，进行编队飞行卫星群构型的初始化。这样的构型初始化可以视作一次特殊的变轨机动控制，很容易实现编队飞行构型的初始化机动。     

基于约化相对轨道拟平根数的长期稳定高精度卫星编队导航技术

针对基于星间相对测量的相对导航算法中由测量方程将相对轨道拟平根数转化为相对位置过程导致的模型非线性,提出一种基于约化相对轨道拟平根数的卫星编队导航方法.该方法通过编队卫星之间一段时间的切向漂移估计半长轴偏差,合理处理半长轴偏差对双星相对动力学的影响,克服了模型线性化造成的误差,能够实现长期稳定的高精度卫星编队导航.     

可见光/激光组合的相对位姿测量研究

针对可重构航天器模块超近距离对接场景下特征点消失问题，提出一种可见光/激光的高精度相对位姿测量方法。该方法先通过可见光/激光组合对初始相对横滚角误差进行校零，完成粗校准环节；然后通过激光二维双镜反射法进行精确的相对位姿解算，获得高精度相对位姿数据。仿真结果表明，在现有技术条件下，该方法在模块相对距离在100cm内可不依赖视觉特征点，实现±1.2mm和±0.03°的相对位姿测量精度，为模块航天器对接后续的超近距离高精度位姿控制奠定基础。     

基于自适应滤波的编队卫星实时相对定轨

提出基于自适应滤波的编队卫星实时相对定轨算法,利用2005-12-09—10两颗GRACE（Gravity Recovery and Climate Experiment）卫星的GPS（Global Positioning System）实测数据进行实时相对定轨试验计算,采用JPL（Jet Propulsion Laboratory）轨道对试验结果外部检核,结果表明:①自适应滤波相对定轨通过自适应因子,可以较好地平衡编队卫星的观测信息和相对动力学信息,其相对定轨结果精度优于Kalman滤波相对定轨结果;②自适应滤波相对定轨结果随着星间基线缩短而精度提高;③两颗GRACE卫星采用单频伪距和广播星历进行自适应滤波相对定轨,可以得到精度优于6cm的星间基线。     

基于GPS相对伪距差分的相对导航方法研究

研究基于GPS相对伪距差分技术的RVD相对导航方法。分析了在相对位置零位处对相对伪距差分模型进行线性化后的模型误差,并推导了由模型误差所引起的相对导航参数确定误差;提出了模型误差修正方法和采用强跟踪滤波器的非线性滤波方法等两种改进方法。最后,通过数学仿真验证了两种改进方法的有效性。     

卫星编队飞行的相对轨道自主确定研究

为了研究卫星编队飞行相对轨道的自主确定,基于相对轨道根数建立编队卫星间的相对运动方程,利用测量所得到的星间距离和方位信息作为观测量。不同于目前广泛采用的扩展卡尔曼滤波算法,设计Unscented Kalman Filter(UKF)算法实现卫星编队飞行的相对轨道自主确定。仿真结果表明这种相对轨道自主确定方案能获得较高的定轨精度。     

二维磁流体动力学平衡日冕

应用解析和数值相结合的方法,本文处理了球对称太阳引力场中具有磁场的非粘、可压缩、完全导电等离子体的二维磁流体动力学平衡问题.得到了两种不同的解:(1)一种相应于磁场具有闭合区和开场区的稳定、自洽的等离子体流动,开场区几个太阳半径之外等离子体径向速度达到大值,超过了局地声速和局地Alfén速度;(2)等离子体速度总是小于局地声速和局地Alfvén速度,并当径向距离很大时速度趋于零.