天体轨道长期数值积分的误差估计方法

数值积分方法是进行天体力学研究的重要工具, 尤其对于行星历表的研究工作而言. 由于在使用数值方法计算天体轨道时, 最终误差通常是难以预知的, 所以在面对精度要求较高或者积分时间较长的工作时具体积分方案的设计---尤其是当使用定步长方法时的步长选择---需要十分谨慎, 因为这将意味着是否能在时间成本可以被接受的范围内使解的精度达到要求. 因此, 在使用数值方法解决实际问题时如何快速寻找效率与精度之间的最佳平衡点是每一个数值积分方法的设计者与使用者都会面临的难题. 为解决这一问题, 在定步长条件下对数值积分方法的舍入误差概率分布函数以及截断误差积累量对步长的依赖关系和随时间的增长关系进行了深入研究. 基于所得结论, 提出了一种仅需较少的数值实验资料即可对选择任意时间步长积分至任意积分时刻时的舍入误差概率分布函数与截断误差积累量进行准确估计的方法, 并使用Adams-Cowell方法对该误差估计方法在圆周期轨道条件下进行了验证. 该误差估计方法在未来有望用于不同数值算法的性能对比研究, 同时也可以对数值积分方法求解实际轨道问题时的决策工作带来重要帮助.     

半相接双星室女座UW轨道周期变化的物理机制研究

对大陵五型半相接双星室女座UW的轨道周期变化进行了分析.结果表明该星的轨道周期在长期快速增加(dP/dt=+1.37×10-6天/年)的同时也含有周期为62.3年的周期性变化.利用Brancewicz和Dworak在1980年给出的基本物理参量,对引起轨道周期变化的物理机制进行了分析研究.分析表明一个质量为Ms>0.94M⊙的第三天体的光时轨道效应能对轨道周期的周期性变化成份作出解释.由于在观测上没有发现这个第三天体存在的信息,它有可能是一个致密天体(如白矮星等).轨道周期的长期增加成份可解释为由次星到主星的物质交流引起(dM2/dt=1.43×10-7M⊙/年),这与该系统次星充满的半接几何结构是相一致的.但是,根据双星演化理沦,大陵五型半相接双星应该处于以次星的核反应时标进行物质交换的慢速物质交流演化阶段,而分析发现该星的轨道周期变化时标远小于次星的核反应时标,但接近于次星的热力学时标,揭示了(1)这颗双星处于以次星热力学时标进行物质交换的快速物质交流演化阶段;或(2)系统的星周物质要通过角动量交换对轨道周期的快速增加做贡献.     

引力常数随时间变化对双星轨道演变的长期效应(椭圆轨道情形)

给出了以偏近点角为自变量的变引力常数的摄动方程组的解.解包括轨道半长轴的长期和周期变化项,其他轨道根数在一阶解中无长期项,只有周期项.近星点经度和平经度在二阶解中显示长期项变化.给出了由于引力常数变化对双星轨道演变情况的数值估计,对结果做了讨论并给出结论.     

摄动势对第一类周期解稳定性的影响

本文以极坐标为广义坐标,首先讨论了摄动势对第一类周期解存在性的影响,然后利用KAM理论讨论了摄动势对第一类Poincaré周期轨道稳定性的影响。对于在实际情况中常用的形式为(n_1≤n_2为二个正整数)的摄动势,得到当小天体相对旋转坐标系的运动角速度ω≠0,和时由圆轨道所延拓生成的第一类Poincaré周期轨道是轨道稳定的。作为例子,文中还给出了当大主星体具有扁率或者是带有光环时,第一类Poincaré周期轨道稳定的条件。最后对本文所得到的结果进行了讨论。     

在近距双星系统中同步双星达到轨道圆形化时的轨道和自转演化的数值解和演化趋势

从非同步自转双星的轨道和自转的演化方程组推出同步双星的轨道和自转的演化方程组.用数值积分法给出演化方程组的数值解.计算同步自转双星EKCep达到轨道圆形化的时间和那时的轨道半长轴、轨道偏心率和主星自转角速度的演变数值.最后对轨道和自转的演化趋势做了推论.     

利用3次方位观测天基初轨确定新方法

提出了一种适用于天基空间目标光学观测的初始轨道确定新方法. 通过对比地基和天基观测的几何构型, 分析了利用天基光学观测数据进行初轨确定时计算收敛到观测平台自身轨道的原因. 基于轨道半通径方程和改进Gauss方程, 推导出了斜距条件方程组的解析形式, 将天基光学观测的初轨确定问题转换为求解关于观测时刻斜距变量的非线性条件方程组的问题. 利用轨道能量约束减小了解的搜索区域, 消除了方程组的奇点. 最后利用天基实测数据验证并分析了非线性条件方程组根的性质, 利用低轨光学观测平台对低、中、高轨和大椭圆轨道空间目标的仿真观测数据验证了方法的有效性.     

短周期彗星表面水冰升华分布研究

彗星是太阳系遗留的原始星子,研究彗星彗核的演化对理解太阳系其他天体的形成和演化历史具有重要意义.在太阳的辐射作用下,彗星携带的挥发性成分会发生升华,并带动尘埃运动,造成彗核物质的损失.因此,彗核的升华活动对其表面形貌甚至整体形状演化都会产生影响.从IAU (International Astronomical Union) MPC (Minor Planet Center)获取轨道数据,并考虑了彗核的自转以及进动,利用MONET (Mass lossdriven shape evolution model)形状演化模型对短周期彗星做数值模拟,计算得到了短周期彗星1P/Halley、9P/Tempel 1、 19P/Borrelly、 67P/C-G (Churyumov-Gerasimenko)、 81P/Wild 2和103P/Hartley 2在一个轨道周期内的太阳辐射能量以及表面侵蚀深度的分布,结合其动力学参数讨论了自转、进动和公转等特性对其表面水冰升华分布的影响以及造成南北侵蚀差异的可能性.     

EPIC 202060577的双星解轨和物理参数研究

EPIC 202060577是一个包含B型子星的食双星系统.根据K2测光数据获得了45个次极小时刻,确定其轨道周期为1.019648 d并给出了历元公式.基于大天区面积多目标光纤光谱望远镜(Large Sky Area Multi-Object Fiber Spectroscopic Telescope, LAMOST1,又称郭守敬望远镜)的光谱数据,使用ROTFIT程序获得了目标系统的主星光谱型、有效温度、金属丰度与表面重力加速度,并计算得到了视向速度曲线.通过PHOEBE (Physics Of Eclipsing Binaries)建模程序和emcee (Affine Invariant Markov chain Monte Carlo Ensemble Sampler)拟合程序,对EPIC 202060577进行了轨道求解并分析了RossiterMcLaughlin效应对视向速度曲线的影响.最终结果表明EPIC 202060577是一个质量比q=0.11的分离结构食双星系统(q=M2/M1, M1、M2分别为主星和伴星质量),主星的光谱型为B2/3,主星和伴星的质量、半径和有效温度分别为M1=12.56M⊙和M2=1.39M⊙,R1=4.58Rsun和R2=1.85Rsun, T1=18979 K和T2=8710 K, M⊙和Rsun分别为太阳质量和半径.根据所获得的物理参数,对EPIC 202060577的演化阶段进行了讨论.     

食双星DS Psc的观测与研究

捅要从2009年到2014年,对食双星DS Psc进行了多色测光和光谱观测,得到了完整覆盖的B、V、R 3色光变曲线以及一批新的光变极小时刻,并首次确定了该双星系统的光谱型.综合已有极小时刻数据,采用O-C方法进行周期分析,得到了新的轨道周期及历元公式.利用Wilson-Devinney方法,对观测所得光变曲线作了拟合分析,首次得到了该双星系统的测光轨道初解.结果表明DS Psc为一个光谱型约为G7V的过相接双星系统.其轨道倾角为66.6°,质量比为2.506.由于其小质量的次子星具有比大质量主子星更高的表面温度,因此DS Psc应属于W次型的W UMa型双星.     

卫星钟差预报模型中周期项的选取方法及性能分析

针对卫星钟差预报(SCB)中周期项选取方法存在的问题,在分析卫星钟周期波动特性的基础上,给出了正确的周期项选取方法,并与现有方法进行了比较.利用IGS(International GNSS Service)的卫星钟差数据,比较分析了二次多项式加周期项模型与传统模型的预报精度.从理论上分析了周期项对传统模型的改善程度及适用条件.结果表明:按照提出方法得到的周期项更符合实际,将其应用于钟差预报时能获得更高的预报精度,大量仿真实验还表明卫星钟周期性波动相对较大时周期项对传统模型有明显改善.     

Numerical models for the study of motion of lunar satellites

The present study deals with numerical modeling of the elliptic restricted three-body problem as well as of the perturbed elliptic restricted three-body (Earth-Moon-Satellite) problem by a fourth body (Sun). Two numerical algorithms are established and investigated. The first is based on the method of the series solution of the differential equations and the second is based on a 5th-order Runge-Kutta method. The applications concern the solution of the equations and integrals of motion of the circular and elliptical restricted three-body problem as well as the search for periodic orbits of the natural satellites of the Moon in the Earth-Moon system in both cases in which the Moon describes circular or elliptical orbit around the Earth before the perturbations induced by the Sun. After the introduction of the perturbations in the Earth-Moon-Satellite system the motions of the Moon and the Satellite are studied with the same initial conditions which give periodic orbits for the unperturbed elliptic problem.     

Periodic solutions of the singly averaged hill problem

Based on the ideas of Lyapunov’s method, we construct a family of symmetric periodic solutions of the Hill problem averaged over the motion of a zero-mass point (a satellite). The low eccentricity of the satellite orbit and the sine of its inclination to the plane of motion of the perturbing body are parameters of the family. We compare the analytical solution with numerical solutions of the averaged evolutionary system and the rigorous (nonaveraged) equations of the restricted circular three-body problem.     

On the periodically evolving orbits in the singly averaged Hill problem

We continue to analyze the periodic solutions of the singly averaged Hill problem. We have numerically constructed the families of solutions that correspond to periodically evolving satellite orbits for arbitrary initial values of their eccentricities and inclinations to the plane of motion of the perturbing body. The solutions obtained are compared with the numerical solutions of the rigorous (nonaveraged) equations of the restricted circular three-body problem. In particular, we have constructed a periodically evolving orbit for which the well-known Lidov-Kozai mechanism manifests itself, just as in the doubly averaged problem.     

A perturbation method and some applications

For differential equations with one fast variable, a perturbation method is introduced that transforms a solution valid over only a short time interval to a new solution composed of averaged variables plus a periodic function of the averaged variables. The averaged variables are governed by a set of differential equations where the fast variable has been removed and thus can be numerically integrated quickly or solved directly. This method is applied to a perturbed harmonic oscillator with a cubic perturbation, van der Pol's equation, coorbital motion in the restricted three-body problem, and to nearly circular motion of a particle near one of the primaries in the restricted three-body problem.     

Continuation of periodic orbits: Three-dimensional circular restricted to the general three-body problem

It is proved that a periodic orbit of the three-dimensional circular restricted three-body problem can be continued analytically, when the mass of the third body is sufficiently small, to a periodic orbit of the three dimensional general three-body problem in a rotating frame. The above method is not applicable when the period of the periodic orbit of the restricted problem is equal to 2k (k any integer), in the usual normalized units. Several numerical examples are given.     

On the families of periodic orbits which bifurcate from the circular Sitnikov motions

In this paper we deal with the circular Sitnikov problem as a subsystem of the three-dimensional circular restricted three-body problem. It has a first analytical part where by using elliptic functions we give the analytical expressions for the solutions of the circular Sitnikov problem and for the period function of its family of periodic orbits. We also analyze the qualitative and quantitative behavior of the period function. In the second numerical part, we study the linear stability of the family of periodic orbits of the Sitnikov problem, and of the families of periodic orbits of the three-dimensional circular restricted three-body problem which bifurcate from them; and we follow these bifurcated families until they end in families of periodic orbits of the planar circular restricted three-body problem. We compare our results with the previous ones of other authors on this problem. Finally, the characteristic curves of some bifurcated families obtained for the mass parameter close to 1/2 are also described.     

The family I3υ of the three-dimensional general three-body problem (Parts I and II)

We present the biparametric family I_3υ of symmetric periodic orbits of the three-dimensional general three-body problem, found by numerical continuation of the vertical critical orbit I_3υ of the circular restricted three-body problem. The periodic orbits refer to a suitably chosen rotating frame of reference.     

Boundary-value problem formulations for computing invariant manifolds and connecting orbits in the circular restricted three body problem

We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordinary differential equations. Specifically, we consider the circular restricted three-body problem (CR3BP), which models the motion of a satellite in an Earth–Moon-like system. The CR3BP has many well-known families of periodic orbits, such as the planar Lyapunov orbits and the non-planar vertical and halo orbits. We compute the unstable manifolds of selected vertical and halo orbits, which in several cases leads to the detection of heteroclinic connections from such a periodic orbit to invariant tori. Subsequent continuation of these connecting orbits with a suitable end point condition and allowing the energy level to vary leads to the further detection of apparent homoclinic connections from the base periodic orbit to itself, or the detection of heteroclinic connections from the base periodic orbit to other periodic orbits. Some of these connecting orbits are of potential interest in space mission design.     

有摄圆型限制性三体问题平动点稳定的充要条件及应用

摘要给出了一个判断有摄圆型限制性三体问题平动点稳定性的充要条件．该条件只依赖于平动点变分方程的特征方程系数的一个简单关系，使用很方便．用所得到的条件，讨论了任意外力摄动对经典圆型限制性三体问题三角平动点稳定性的影响和惯性阻力摄动对Robe圆型限制性三体问题主要平动点的稳定性的影响．