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。     

A velocity scaling method with least-squares correction of several constraints

A new scale transformation to the integrated velocity vector is designed to monitor the accumulation of numerical errors in several integrals of motion. The scale factor is derived from the least-squares correction that minimizes the sum of the squares of the errors of these integrals. In order to preserve an invariant, we employ the velocity scaling method for rigorously satisfying the constraint. When adjusting many constants, the new scheme like other existing methods is valid to typically reduce the integration errors below those of an uncorrected integrator. Via integral invariant relations, the new method is also able to treat slowly-varying quantities, such as the Keplerian energy and the Laplace vector, for a perturbed Keplerian problem or each of multiple bodies in the solar system dynamics. Consequently it does nearly agree with the rigorous dual scaling method in the sense of drastically improving the integration accuracy. As one of its advantages, the implementation of the new method is significantly easier than that of other methods. In particular, the method can be simply applied to a complicated dynamical system with some constraints.     

稳定化方法与后稳定化方法的比较

对BaumgaLrte的稳定化和Chin的后稳定化进行了详尽讨论与数值比较．用经典数值方法并结合这两种稳定化方式都能提高数值精度和改善数值稳定性．在最佳稳定参数下稳定化精度一般不等价于后稳定化．两者精度优劣并无常定．考虑到Baumgarte的稳定化使得数值积分的右函数更复杂和增加计算耗费,尤其是存在稳定参数最佳选取的麻烦,故推荐后稳定化投入实算．但值得注意的是用后稳定化与没有经过稳定化处理的经典积分器来比不宜扩大积分步长．     

Numerical calculations in the orbital determination of an artificial satellite for a long arc

Numerical methods are usually used for the computation of ephemerides with perturbations for the precise orbital determination of an artificial satellite. But their numerical stability will be encountered in a long arc. In this case the use the improved Encke special perturbation methods has been suggested. The results of this paper show that Encke's method does indeed have a certain effectiveness, but cannot yet completely resolve the numerical stability, and the more efficient method is to use the energy integral or its variational relation to control the growth of the along-track error in general numerical calculations so that the aim of stabilization can be achieved.     

New adaptive time step symplectic integrator:an application to the elliptic restricted three-body problem

The time-transformed leapfrog scheme of Mikkola Aarseth was specifically designed for a second-order differential equation with two individually separable forms of positions and velocities.It can have good numerical accuracy for extremely close two-body encounters in gravitating few-body systems with large mass ratios,but the non-time-transformed one does not work well.Following this idea,we develop a new explicit symplectic integrator with an adaptive time step that can be applied to a time-dependent Hamiltonian.Our method relies on a time step function having two distinct but equivalent forms and on the inclusion of two pairs of new canonical conjugate variables in the extended phase space.In addition,the Hamiltonian must be modified to be a new time-transformed Hamiltonian with three integrable parts.When this method is applied to the elliptic restricted three-body problem,its numerical precision is explicitly higher by several orders of magnitude than the nonadaptive one's,and its numerical stability is also better.In particular,it can eliminate the overestimation of Lyapunov exponents and suppress the spurious rapid growth of fast Lyapunov indicators for high-eccentricity orbits of a massless third body.The present technique will be useful for conservative systems including N-body problems in the Jacobian coordinates in the the field of solar system dynamics,and nonconservative systems such as a time-dependent barred galaxy model in a rotating coordinate system.     

Time dependent monochromatic scattering of radiation in one dimensional media: Analytical and numerical solutions

In order to choose a numerical method for solving the time dependent equations of radiative transport, we obtain an exact solution for the time dependent radiation field in a one dimensional infinite medium with monochromatic, isotropic scattering for sources with an arbitrary spatial distribution and an arbitrary time variation of their power. The Lax-Wendroff method seems to be the most suitable. Because it is assumed that radiation delay is caused by the finite speed of light, the following difficulty arises when the numerical method is used: the region of variation of the variables (position τ and time t) is triangular (the inequality τ ≤ t). This difficulty is overcome by expanding the unknown functions in series in terms of small values of the time and position. By comparing the numerical and exact solutions for a point source with a given time dependence for its power and with pure scattering, the number of steps in the variables required to obtain a desired accuracy is estimated. This numerical method can be used to calculate the intensity and polarization of the radiation from sources in the early universe during epochs close to the recombination epoch. __________ Translated from Astrofizika, Vol. 51, No. 1, pp. 109–123 (February 2008).     

On secular stability of rapidly rotating stars of arbitrary structure

The aim of the present paper will be to utilize Poincaré's criteria to investigate secular stability of self-gravitating configurations, of arbitrary structure, in the state of rapid rotation. The potential energy, a knowledge of which is necessary for application of these criteria, will be determined by an extension of Clairaut's method; and its evaluation in terms of suitably chosen generalized coordinates carried out explicitly to quantities of fourth order in superficial oblateness, for configurations of arbitrary internal structure.The method employed can, moreover, clearly be extended to attain accuracy of any order — at the expense of mere manipulative work which lends itself to machine automation; and the angular velocity of axial rotation can be an arbitrary function of position as well as of the time. An application of our results to homogeneous configurations in rigig-body rotation will be undertaken to demonstrate that our method, when applied to a case for which a closed solution exists, leads to results which are consistent with it.     

Clausius’ virial vs. total potential energy in the dynamics of a two-component system

In a gravitational virialized bound system built up of two components, one of which is embedded in the other, the Clausius’ virial energy of one subcomponent is not, in general, equal to its total potential energy, as occurs in a single system without external forces. This is the main reason for the presence, in the case of two non-coinciding concentric spheroidal subsystems, of a minimum (in absolute value) in the Clausius’ virial of the inner component B, when it assumes a special configuration characterized by a value of its semi-major axis we have named tidal radius. The physical meaning, connected with its appearance, is to introduce a scale length on the gravity field of the inner subsystem, which is induced from the outer one. Its relevance in the galaxy dynamics has been stressed by demonstrating that some of the main features of Fundamental Plane may follow as consequence of its existence. More physical insight into the dynamics of a two-component system may be got by looking at the location of this scale length inside the plots of the potential energies of each subsystem and of the whole system and by also taking into account the trend of the anti-symmetric residual-energy, that is the difference between the tidal and the interaction-energy of each component. Some thermodynamical arguments related to the inner component are also added to prove how special is the tidal radius configuration. Moreover, the role of the divergency at the center of the two subsystems in obtaining this scale length is considered. For the sake of simplicity the analysis has been performed in the case of a frozen external component even if this constraint does not appear to be too relevant in order to preserve the main results.     

On the Numerical Stability of Some Symplectic Integrators

In this paper, we analyze the linear stabilities of several symplectic integrators, such as the first-order implicit Euler scheme, the second-order implicit mid-point Euler difference scheme, the first-order explicit Euler scheme, the second-order explicit leapfrog scheme and some of their combinations. For a linear Hamiltonian system, we find the stable regions of each scheme by theoretical analysis and check them by numerical tests. When the Hamiltonian is real symmetric quadratic, a diagonalizing by a similar transformation is suggested so that the theoretical analysis of the linear stability of the numerical method would be simplified. A Hamiltonian may be separated into a main part and a perturbation, or it may be spontaneously separated into kinetic and potential energy parts, but the former separation generally is much more charming because it has a much larger maximum step size for the symplectic being stable, no matter this Hamiltonian is linear or nonlinear.     

A global regularisation for integrating the Caledonian symmetric four-body problem

Several papers in the last decade have studied the Caledonian symmetric four-body problem (CSFBP), a restricted four-body system with a symmetrically reduced phase space. During these studies, difficulties have arisen when the system approaches a close encounter. These are due to collision singularities causing numerical integration algorithms to fail. In this paper, we give the full details of a regularisation approach that now enables us to study these close encounters and collision events. The resulting equations of motion can be efficiently integrated by a high-order integrator. The results from numerical testing of the algorithm verify that the regularisation is advantageous in preserving numerical stability. The effectiveness of the approach is illustrated for a range of CSFBP orbits. Numerical experiments show that the newly developed regularisation algorithm has excellent energy conservation properties.     

Stability of equilibrium solutions in the critical case of even-order resonance in periodic Hamiltonian systems with one degree of freedom

The stability in the Lyapunov sense of an equilibrium position in a periodic Hamiltonian system with one degree of freedom is studied. It is assumed that the equilibrium is stable in the first approximation and that there exists an even resonance of order $k$ k , arbitrary. The critical case is considered, i.e., when the system of parameters is such that, in order to draw rigorous conclusions about the stability of the equilibrium position in the Lypaunov sense, terms or order higher than three in the series expansion of the Hamiltonian function must be taken into account. Sufficient conditions are derived for stability and instability.     

On secular stability of tidally distorted stars of arbitrary structure

The aim of the present paper will be to develop a theory which should make it possible to investigate secular stability of close binary systems, consisting of tidally-distorted components of arbitrary internal structure, by a minimization of the potential energy of the system as a whole. In the second section which follows brief introductory remarks, appropriate expressions for the total potential energy of a close binary will be formulated. Section 3 will be concerned mainly with the nature of the tide-generating potential, and its effects on the shape of each star. In Section 4, the amplitudes of partial tides raised by this potential will be specified, for stars of arbitrary structure, correctly to terms of second order in superficial distortion; and in Section 5 we shall investigate the effects of interaction between rotation and tides to the same degree of approximation. The concluding Section 6 will then contain an explicit formulation of different constituents adding up to the total potential energy of the system, which can be used as a basis for its secular stability by the methods outlined already in our previous investigation (Kopal, 1973).     

Blind Deconvolution of Astronomical Images with a Constraint on Bandwidth Determined by the Parameters of the Optical System

Atmospheric turbulence severely restricts the spatial resolution of astronomical images obtained by a large ground-based telescope. In order to reduce effectively this effect, we propose a method of blind deconvolution, with a bandwidth constraint determined by the parameters of the telescope's optical system based on the principle of maximum likelihood estimation, in which the convolution error function is minimized by using the conjugate gradient algorithm. A relation between the parameters of the telescope optical system and the image's frequency-domain bandwidth is established, and the speed of convergence of the algorithm is improved by using the positivity constraint on the variables and the limited-bandwidth constraint on the point spread function. To avoid the effective Fourier frequencies exceed the cut-off frequency, it is required that each single image element (e.g., the pixel in the CCD imaging) in the sampling focal plane should be smaller than one fourth of the diameter of the diffraction spot. In the algorithm, no object-centered constraint was used, so the proposed method is suitable for the image restoration of a whole field of objects. By the computer simulation and by the restoration of an actually-observed image of α Piscium, the effectiveness of the proposed method is demonstrated.     

月球激光测距的新技术方法研究

月球激光测距是国内外所瞩目的宏伟目标 ,代表着单光子探测技术的高峰。本论文的目的是探索提高激光测月回波光子数的新方法 ,进而增加激光测月成功的几率。其思路是源于一个新的想法 :在激光测月过程中引入大气波前倾斜量实时补偿的技术。首先介绍激光测月的现状和其难度所在 :回波光子数太少 ,基本上属于亚光子探测范畴。在现有技术条件下 ,本文对影响激光测月回波光子数的因素逐一进行分析讨论 ,提出应该把激光束截面能量分布和大气湍流效应包括进去。为此分析讨论了大气湍流和大气中光场的基本统计性质、激光束在大气中传输时所受大气湍流的影响以及大气湍流对激光测距的影响 ,得出大气湍流特别对激光测月有着明显影响的结论。在此基础上对传统的激光测距方程进行了修正 ,使其应用到激光测月时更符合真实的情况 ,从而指导补偿的进行。涉及到在激光测月中对受大气湍流而畸变的激光束进行补偿 ,本文抓住重点 ,通过分析看出对大气倾斜量的实时补偿是提高激光测月回波光子数的重要因素。结合激光测月以及云南天文台现有测距系统的实际情况 ,本文独创性地提出利用激光测月目标近旁一定大小区域的扩展面源探测与计算大气倾斜量 ,然后对激光测月中的激光束进行大气倾斜量实时补偿的新技术方法。在分析比较     

Existence of formal integrals of symplectic integrators

A recurrent method of solving the formal integrals of symplectic integrators is given. The special examples show that there are no long-term variations in all integrals of the Hamiltonian system in addition to the energy one when symplectic integrators are used in the numerical studies of the system. As an application of the formal integrals, the relation between them and the linear stability of symplectic integrators is discussed.     

致密星的X射线辐射时延现象

能谱仅反映了高能天体辐射的部分性质，而高能X射线辐射的时变可以反映致密天体的动态性质。因此，在构造高能辐射的理论模型时，既要考虑X辐射的能谱性质，也要考虑其时变性质。特别对高能天体X辐射的快速光变，一般认为其发生在致密天体附近，致密天体快速光变的研究可以揭示出致密天体附近的物理性质，因此，时变可以对X辐射模型给以很强的限制。X射线源不同能量光子到达观测者的时间差称为X射线辐射的时延。介绍了高能X辐射时变中的时延分析方法及其最新发展，综述了致密双星系统中黑洞侯选林和中子星的时延现象，并简单介绍了为解释这些现象所发展的模型，以及时延对模型的限制。     

Stability of the planar full 2-body problem

The stability of the Full Two-Body Problem is studied in the case where both bodies are non-spherical, but are restricted to planar motion. The mutual potential is expanded up to second order in the mass moments, yielding a highly symmetric yet non-trivial dynamical system. For this system we identify all relative equilibria and determine their stability properties, with an emphasis on finding the energetically stable relative equilibria and conditions for Hill stability of the system. The energetically stable relative equilibria always correspond to the classical “gravity gradient” configuration with the long ends of the two bodies pointed at each other, however there always exists a second equilibrium in this configuration at a closer separation that is unstable. For our model system we precisely map out the relations between these different configurations at a given value of angular momentum. This analysis identifies the fundamental physical constraints and limitations that exist on such systems, and has immediate applications to the stability of asteroid systems that are fissioned due to a rapid spin rate. Specifically, we find that all contact binary asteroids which are spun to fission will initially lie in an unstable dynamical state and can always re-impact. If the total system energy is positive, the fissioned system can disrupt directly from this relative equilibrium, while if it is negative the system is bound together.     

Frequency analysis of a dynamical system

Frequency analysis is a new method for analyzing the stability of orbits in a conservative dynamical system. It was first devised in order to study the stability of the solar system (Laskar, Icarus, 88, 1990). It is a powerful method for analyzing weakly chaotic motion in hamiltonian systems or symplectic maps. For regular motions, it yields an analytical representation of the solutions. In cases of 2 degrees of freedom system with monotonous torsion, precise numerical criterions for the destruction of KAM tori can be found. For a 4D symplectic map, plotting the frequency map in the frequency plane provides a clear representation of the global dynamics and describes the actual Arnold web of the system.     

A New Formulation for General Relativistic Force-Free Electrodynamics and Its Applications

We formulate the general relativistic force-free electrodynamics in a new 3 1 language. In this formulation,when we have properly defined electric and magnetic fields,the covariant Maxwell equations could be cast in the traditional form with new vacuum con-stitutive constraint equations. The fundamental equation governing a stationary,axisymmet-ric force-free black hole magnetosphere is derived using this formulation which recasts the Grad-Shafranov equation in a simpler way. Compared to the classic 3 1 system of Thorne and MacDonald,the new system of 3 1 equations is more suitable for numerical use for it keeps the hyperbolic structure of the electrodynamics and avoids the singularity at the event horizon. This formulation could be readily extended to non-relativistic limit and find applica-tions in flat spacetime. We investigate its application to disk wind,black hole magnetosphere and solar physics in both flat and curved spacetime.