一类具非零角动量的平面三体系统的数值分析

对一类具非零角动量的平面三体系统研究其三体构形对系统演化的影响．根据Agekian和Anosova提出的构形图(homology map)，三体系统按其构形特点分属于4个不同的区域．通过数值计算，考察了初始位置位于不同区域中的构形颗粒(homolgydrop)的演化，并就有关性质与Heinamaki等人研究的角动量为零的三体系统作了比较指出，构形颗粒的组成系统全部发生解体的时间在L区域最早，H区域最晚，这与零角动量系统不同．还对4个区域的三体系统的寿命进行了统计分析，得到了各区域中末解体的系统数随时间指数衰减的函数关系．     

倾角函数的四精度计算试验

倾角函数是天体力学分析理论中一种常用的函数.当把摄动方程展开成时间和根数的形式时需要用到.历史上提出了很多经典的倾角函数递推算法,并在双精度平台下开发了Fortran程序.进行了1次四精度计算倾角函数的试验,结果表明:L平面递推方法的四精度计算精度可达10-22,计算速度比双精度Jacobi方法快6倍.     

计算倾角函数的改进Gooding方法

将Gooding方法中的Vklm(I)改写为便于计算的形式,并利用标准递推过程计算Aklm(I).在规定k和l同奇偶的前提下改写了Gooding的计算程序,使程序缩短了一半,提高了程序计算效率和可读性,计算时间缩短了41％,计算精度和稳定性也略有提高.     

利用Jacobi多项式计算倾角函数

研究了一种利用Jacobi多项式计算倾角函数的方法,该方法表达式非常简单,不存在k和l是否同奇偶和计算非整数阶乘的问题,也不存在k＜0和k≥0的转换问题,递推公式可使用标准的Jacobi多项式的递推公式.而且,计算精度和适应阶数可与Gooding方法相当,计算时间比Gooding方法省9％.     

倾角函数及其导数的定积分计算方法

给出了一种倾角函数及其导数的定积分计算方法,表达式十分简单,其计算精度:倾角函数可达10^-15,导数可达10^-13,可与Gooding方法相媲美.该方法的稳定性和适用倾角范围均较好,可供倾角函数的最高阶数L_max≤50时使用.     

两种利用d-函数计算倾角函数方法的比较

利用d-函数的Blanco递推(d-funl)和Risbo递推(d-fun2),可以得到两种计算倾角函数及其导数的方法.这两种方法均有较高的精度和稳定性.对于小倾角,d-fun2的精度优于d-fun1,而对于大多数其他倾角,d-fun1的精度优于d-fun2;但d-fun2的稳定性明显优于d-funl;计算速度d-funl比d-fun2约快7倍.但是这两种方法均有sin I=0的奇点.另外,d-函数方法直接计算出来的就是正规化的倾角函数,不能实现倾角函数的无奇点计算,因此不适合在小倾角卫星动力学中应用.     

用Hamilton方法计算弹性地球的章动序列

本文利用Hamilton方法研究弹性地球自转运动，采用地球模型PREM参数，给出了形状轴的章动序列．结果表明我们的方法是可行的，计算是可靠的．弹性地幔对地球章动的影响仅在毫角秒量级上，它相对液核对地球竟动的影响要小得多．     

用Hamilton方法计算滞弹地球自转的潮汐变化

在日月引潮力势作用下地球产生弹性形变．地幔粘滞性子致这个形变对于引潮力滞后,成为引起地球自转长期减慢的原因之一．地幔滞弹性也使有效洛夫数k增加,并使自转变化的周期项位相滞后,即产生反常位相项．本文首先用Hamilton方法计算了地球的形变．然后考虑到地幔的滞弹性,计算了在日月引潮力作用下的地球自转长期减慢和滞弹性对周期（带谐）变化的影响．     

Parallel/Vector Integration Methods for Dynamical Astronomy

This paper reviews three recent works on the numerical methods to integrate ordinary differential equations (ODE), which are specially designed for parallel, vector, and/or multi-processor-unit(PU) computers. The first is the Picard-Chebyshev method (Fukushima, 1997a). It obtains a global solution of ODE in the form of Chebyshev polynomial of large (> 1000) degree by applying the Picard iteration repeatedly. The iteration converges for smooth problems and/or perturbed dynamics. The method runs around 100-1000 times faster in the vector mode than in the scalar mode of a certain computer with vector processors (Fukushima, 1997b). The second is a parallelization of a symplectic integrator (Saha et al., 1997). It regards the implicit midpoint rules covering thousands of timesteps as large-scale nonlinear equations and solves them by the fixed-point iteration. The method is applicable to Hamiltonian systems and is expected to lead an acceleration factor of around 50 in parallel computers with more than 1000 PUs. The last is a parallelization of the extrapolation method (Ito and Fukushima, 1997). It performs trial integrations in parallel. Also the trial integrations are further accelerated by balancing computational load among PUs by the technique of folding. The method is all-purpose and achieves an acceleration factor of around 3.5 by using several PUs. Finally, we give a perspective on the parallelization of some implicit integrators which require multiple corrections in solving implicit formulas like the implicit Hermitian integrators (Makino and Aarseth, 1992), (Hut et al., 1995) or the implicit symmetric multistep methods (Fukushima, 1998), (Fukushima, 1999). This revised version was published online in July 2006 with corrections to the Cover Date.     

几类辛方法的数值稳定性研究

主要对一阶隐式Euler辛方法M1、二阶隐式Euler中点辛方法M2、一阶显辛Euler方法M3和二阶leapfrog显辛积分器M4共4种辛方法及一些组合算法进行了通常意义下的线性稳定性分析．针对线性哈密顿系统,理论上找到每个数值方法的稳定区,然后用数值方法检验其正确性．对于哈密顿函数为实对称二次型的情况,为了理论推导便利,特推荐采用相似变换将二次型的矩阵对角化来研究辛方法的线性稳定性．当哈密顿分解为一个主要部分和一个小摄动次要部分且二者皆可积时,无论是线性系统还是非线性系统,这种主次分解与哈密顿具有动势能分解相比,明显扩大了辛方法的稳定步长范围．     

天文动力学方程数值积分中的一种有效变步法

利用积分曲线的曲率控制步长的技巧，使天文动力学方程数值解法的精度和速度有较大提高，这种方法适用于天体精密定轨以及一些精度要求高的常微分方程初值问题的数值积分。     

Processing Symplectic Methods for Near-Integrable Hamiltonian Systems

Processing techniques are used to approximate the exact flow of near-integrable Hamiltonian systems depending on a small perturbation parameter. We study the reduction of the number of conditions for the kernel for this type of Hamiltonians and we build third, fourth and fifth order methods which are shown to be more efficient than previous algorithms for the same class of problems. This revised version was published online in July 2006 with corrections to the Cover Date.     

A numerical-analytical method for studying the orbital evolution of distant planetary satellites

We describe an approximate numerical-analytical method for calculating the perturbations of the elements of distant satellite orbits. The model for the motion of a distant satellite includes the solar attraction and the eccentricity and ecliptic inclination of the orbit of the central planet. In addition, we take into account the variations in planetary orbital elements with time due to secular perturbations. Our work is based on Zeipel’s method for constructing the canonical transformations that relate osculating satellite orbital elements to the mean ones. The corresponding transformation of the Hamiltonian is used to construct an evolution system of equations for mean elements. The numerical solution of this system free from rapidly oscillating functions and the inverse transformation from the mean to osculating elements allows the evolution of distant satellite orbits to be studied on long time scales on the order of several hundred or thousand satellite orbital periods.     

Dynamic portrait of the planetary 2/1 mean-motion resonance – I. Systems with a more massive outer planet

We analyse the global structure of the phase space of the planar planetary 2/1 mean-motion resonance in cases where the outer planet is more massive than its inner companion. Inside the resonant domain, we show the existence of two families of periodic orbits, one associated to the librational motion of resonant angle (σ-family) and the other related to the circulatory motion of the difference in longitudes of pericentre (  Δϖ  -family). The well-known apsidal corotation resonances (ACR) appear as intersections between both families. A complex web of secondary resonances is also detected for low eccentricities, whose strengths and positions are dependent on the individual masses and spatial scale of the system.
The construction of dynamical maps for various values of the total angular momentum shows the evolution of the families of stable motion with the eccentricities, identifying possible configurations suitable for exoplanetary systems. For low–moderate eccentricities, several different stable modes exist outside the ACR. For larger eccentricities, however, all stable solutions are associated to oscillations around the stationary solutions.
Finally, we present a possible link between these stable families and the process of resonance capture, identifying the most probable routes from the secular region to the resonant domain, and discussing how the final resonant configuration may be affected by the extension of the chaotic layer around the resonance region.     

A criterion for the linear stability of theequilibrium points in the perturbed restricted three-body problem and its application in Robe''s problem

A criterion for the linear stability of the equilibrium points in the perturbed restricted three-body problem is given. This criterion is related only to the coefficients of the characteristic equation of the tangent map of an equilibrium point, and this is convenient to use. With this criterion, we have discussed the linear stability of the equilibrium points in the Robe problem under the perturbation of a drag force, derived the linearly stable region of the equilibrium point in the perturbed Robe's problem with the drag given by Hallen et al., and improved as well the results obtained by Giordano et al.     

保持Runge-Lenz向量的数值方法

对孤立积分和能够保持Runge-Lenz向量的梯形公式进行详尽讨论．孤立积分就是限制粒子运动区域的不变量，具有n个自由度的自治可积哈密顿系统且只有n个互相对合的独立孤立积分，并且其他孤立积分的存在对粒子的运动是有意义的，Kepler二体系统存在能量积分、角动量积分和Runge-Lenz向量．对于平面运动情况，这三类积分中只有3个独立孤立积分；而对于三维空间情形，该三类积分仅有5个是独立的．就前者而言，Kepler二体平面运动积分构成该系统中的对称群SO（3），经过Levi-Civita变换，它可以转化为二维各向同性谐振子系统中的对称群，而该对称群能够被梯形公式准确保持，另一方面，对于后者梯形公式对这三类积分的严格保持还可以在5个Kepler轨道根数n、e、i、Ω和w上得到体现。     

Simulation of precise orbit determination of lunar orbiters

Based on the ongoing Chinese lunar exploration mission, i.e. the “Chang'e 1” project, precise orbit determination of lunar orbiters is analyzed for the actual geographical distribution and observational accuracy of the Chinese united S-band (USB) observation and control network as well as the very long baseline interferometry (VLBI) tracking network. The observed data are first simulated, then solutions are found after including the effects of various error sources and finally compared. We use the space data analysis software package, GEODYN, developed at Goddard Space Flight Center, NASA, USA. The primary error source of the flight orbiting the moon is the lunar gravity field. Therefore, the (formal) error of JGL165P1, i.e. the model of the lunar gravity field with the highest accuracy at present, is first discussed. After simulating the data of ranging and velocity measurement as well as the VLBI data of the time delay and time delay rate, precise orbit determination is carried out when the error of the lunar gravity field is added in. When the orbit is determined, the method of reduced dynamics is adopted with the selection of appropriate empirical acceleration parameters to absorb the effect of errors in the lunar gravity field on the orbit determination. The results show that for lunar missions like the “Chang'e 1” project, that do not take the lunar gravity field as their main scientific objective, the method of reduced dynamics is a simple and effective means of improving the accuracy of the orbit determination of the lunar orbiters.     

约束条件和数值积分

自治的哈密顿系统存在约束条件，例如能量积分或广义相对论中的4速度大小为常数，它能否在数值积分过程中始终满足将直接影响数值稳定性．在牛顿力学中哈密顿系统的动能一般为椭圆型，直接运用约束条件对方程进行降阶存在开平方判断正负号的困难，导致应用高精度的经典数值积分器时能量存在耗散．然而相对论力学的度规为双曲型，利用约束条件有可能实行方程降阶．在时空具有一定对称性的情况下，能够找到整个时空的一个全局变换使变换后的度规的主对角线某一元素为零，于是从约束方程中不需开平方能够解出某一动量，顺利实现运动方程的降阶．相对论力学中另一个可以降阶的模型是Mixmaster宇宙模型．数值实验表明将经典算法用于降阶后的运动方程能够严格地满足约束，但不一定能保持辛结构。