Numerical integration of the variational equations of satellite orbits

A recurrent power series (RPS) method is constructed for the numerical integration of the equations of motion together with the variational equations of N point masses orbiting around an oblate spheroid. By the term “variational equations” we mean the equations of the partial derivatives of the bodies’ position and velocity components with respect to the initial conditions, the relative masses and the spheroid's oblateness coefficients J₂ and J₄. The construction of recursive relations for the partial derivatives involved in the variational equations is based on partial differentiation of the corresponding recursive relations for the integration of the equations of motion. Since the number of the auxiliary variables needed for this complex system becomes tremendously large when N>1, special care must be taken during computer implementation, so as to minimize the amount of computer memory needed as well as the cost in CPU time. The RPS method constructed in this way is tested for N=1,…,4 using real initial conditions of the Saturnian satellite system. For various sets of satellites, we monitor the behaviour of all the corresponding partial derivatives. The results show a prominent difference in the behaviour of the partial derivatives between resonant and non-resonant orbital systems.     

Behaviour of the SMF method for the numerical integration of satellite orbits

The SMF algorithms were recently developed by the authors as a multistep generalization of the ScheifeleG-functions one-step method. Like the last, the proposed codes integrate harmonic oscillations without truncation error and the perturbing parameter appears as a factor of that error when integrating perturbed oscillations. Therefore they seemed to be convenient for the accurate integration of orbital problems after the application of linearizing transformations, such as KS or BF. In this paper we present several numerical experiments concerning the propagation of Earth satellite orbits, that illustrate the performance of the the SMF method. In general, it provides greater accuracy than the usual standard algorithms for similar computational cost.     

Numerical simulation of time delay interferometry for TAIJI and new LISA

The success of LISA Pathfinder in demonstrating the LISA drag-free requirement paved the way for using space interferometers to detect low-frequency and middle-frequency gravitational waves(GWs). The TAIJI GW mission and the new LISA GW mission propose using an arm length of 3 Gm(1 Gm = 10~6 km) and an arm length of 2.5 Gm respectively. For a space laser-interferometric GW antenna,due to astrodynamical orbit variation, time delay interferometry(TDI) is needed to achieve nearly equivalent equal-arms for suppressing the laser frequency noise below the level of optical path noise, acceleration noise, etc in order to attain the requisite sensitivity. In this paper, we simulate TDI numerically for the TAIJI mission and the new LISA mission. To do this, we work out a set of 2200-day(6-year) optimized science orbits for each mission starting on 2028 March 22 using the CGC 2.7.1 ephemeris framework. Then we use the numerical method to calculate the residual optical path differences of the first-generation TDI configurations and the selected second-generation TDI configurations. The resulting optical path differences of the second-generation TDI configurations calculated for TAIJI, new LISA and eLISA are well below their respective requirements for laser frequency noise cancelation. However, for the first-generation TDI configurations, the original requirements need to be relaxed by 3 to 30 fold to be satisfied. For TAIJI and the new LISA, about one order of magnitude relaxation would be good and recommended; this could be borne on the laser stability requirement in view of recent progress in laser stability, or the GW detection sensitivities of the second-generation TDIs have to be used in the diagnosis of the observed data instead of the commonly used X, Y and Z TDIs.     

Modern Numerical Ephemerides of Planets and the Importance of Ranging Observations for Their Creation

The JPL planetary and lunar ephemerides – DE200/LE200, DE403/LE403, DE405/LE405 and the planetary and lunar ephemerides, EPM87, EPM98, and EPM2000, constructed in the Institute of Applied Astronomy of RAS are described. Common properties and differences of the various ephemerides are given. Graphical comparisons of the DE ephemerides with each other and with the EPM ephemerides are presented. A fairly good agreement of planetary orbits is between DE403, DE405 and EPM98, EPM2000, respectively, over the interval of 120 years (1886–2006) covered by EPM98 and EPM2000. Some differences are explained by a slight disagreement in representing the orbits of Ceres, Pallas, and Vesta as they affect the planets. The accurate radar observations of planets and spacecraft make it possible not only to improve the orbital elements of planets but to determine a broad set of astronomical constants as well: km/AU, parameters of Mars rotation including its precessional rate, the masses of Jupiter, Ceres, Pallas, and Vesta, relativistic parameters of the PPN formalism, the variability of the gravitational constant G. These have been obtained in the fitting process of the DE405 and EPM2000 ephemerides to observational data, including nearly 80000 American and Russian radar observations of planets (1961–1997), ranging and doppler to the Viking and Pathfinder landers, and other miscellaneous measurements from various sources and spacecraft.     

Comparison of Numerical Methods for the Integration of Natural Satellite Systems

We present a new implementation of the recurrent power series (RPS) method which we have developed for the integration of the system of N satellites orbiting a point-mass planet. This implementation is proved to be more efficient than previously developed implementations of the same method. Furthermore, its comparison with two of the most popular numerical integration methods: the 10th-order Gauss–Jackson backward difference method and the Runge–Kutta–NystrRKN12(10)17M shows that the RPS method is more than one order of magnitude better in accuracy than the other two. Various test problems with one up to four satellites are used, with initial conditions obtained from ephemerides of the saturnian satellite system. For each of the three methods we find the values of the user-specified parameters (such as the method's step-size (h or tolerance (TOL)) that minimize the global error in the satellites' coordinates while keeping the computer time within reasonable limits. While the optimal values of the step-sizes for the methods GJ and RKN are all very small (less than T/100, the ones that are suitable for the RPS method are within the range: T/13<h<T/6 (T being the period of the innermost satellite of the problem). Comparing the results obtained by the three methods for these step-sizes and for the various test problems we observe the superiority of the RPS method over GJ in terms of accuracy and over RKN both in accuracy and in speed. This revised version was published online in July 2006 with corrections to the Cover Date.     

New Trends in Numerical Simulation of the Motion of Small Bodies of the Solar System

A brief survey of the results obtained by the authors in the development and investigation of the algorithms of numerical simulation of the motion of solar system small bodies is given. New approaches to the construction of the algorithms of high-accuracy numerical simulation of the dynamics of small bodies and the methods of the determination of the domain of their possible motions are presented.     

Numerical derivation of forced nutation terms for a rigid earth

We compare the results of a numerical integration of the Euler equations for a rigid Earth model covering a time span of 250 years with Kinoshita's theory for the forced nutations and with a new nutation series by Kinoshita and Souchay. We also present numerical corrections to some of the analytically derived nutation terms.     

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.     

On the efficiency of Runge-Kutta-Nystrom methods with interpolants for solving equations of the form Y″ = F(T,Y, Y′) over short timespans

Runge-Kutta-Nystrom (RKN) codes for the solution of the initial value problem for the general second order differential system have been developed recently, although the methodology on which they are based was known many years ago. In this paper we try to examine the efficiency of several known general Runge-Kutta-Nystrom (GRKN) methods by posing some criteria of cost and accuracy. These methods supplied with the corresponding interpolants, have been applied to some problems of Celestial Dynamics. The results obtained show that these codes have a good response in the approximation of the solution of these problems.Department of Mechanics     

Numerical simulations of the Lyman α forest – a comparison of gadget-2 and enzo

We compare simulations of the Lyman α forest performed with two different hydrodynamical codes, gadget-2 and enzo . A comparison of the dark matter power spectrum for simulations run with identical initial conditions show differences of 1–3 per cent at the scales relevant for quantitative studies of the Lyman α forest. This allows a meaningful comparison of the effect of the different implementations of the hydrodynamic part of the two codes. Using the same cooling and heating algorithm in both codes, the differences in the temperature and the density probability distribution function are of the order of 10 per cent. The differences are comparable to the effects of box size and resolution on these statistics. When self-converged results for each code are taken into account, the differences in the flux power spectrum – the statistics most widely used for estimating the matter power spectrum and cosmological parameters from Lyman α forest data – are about 5 per cent. This is again comparable to the effects of box size and resolution. Numerical uncertainties due to a particular implementation of solving the hydrodynamic or gravitational equations appear therefore to contribute only moderately to the error budget in estimates of the flux power spectrum from numerical simulations. We further find that the differences in the flux power spectrum for enzo simulations run with and without adaptive mesh refinement are also of the order of 5 per cent or smaller. The latter require 10 times less CPU time making the CPU time requirement similar to that of a version of gadget-2 that is optimized for Lyman α forest simulations.     

基于数值模拟的大口径高精度射电望远镜台址风障优化设计研究

随着射电望远镜口径增大、观测频率提高, 对其指向精度的要求也越来越高. 然而, 望远镜服役于野外台站, 台址风扰对天线指向精度的影响在高频段观测时已不能忽略. 由于风扰的时变性, 现有的抗风方法无法保障大口径高指向精度望远镜在高频段的有效观测时长. 因此, 提出了一种基于风障精确布置改善台址风环境的方法. 通过数值模拟构建了风障仿真模型, 并将仿真结果与风洞实测数据比较, 两种孔隙率风障的平均误差分别为3.7%和6.1%, 保证了风障模型的可靠性. 以新疆奇台射电望远镜(QiTai radio Telescope, QTT)台址为例, 基于QTT台址斜坡地形构建了计算域模型, 开展单风障不同高度、不同孔隙率的系列风场仿真试验, 得到了风障参量与下游挡风效果的关系. 基于单风障合理高度和最优孔隙率设置南北风障, 仿真结果表明在确定高度下最优孔隙率可以组合, 孔隙率0.1-0.1组合的风障挡风效果最优, 南方向来风在天线区域可以有效降低75%以上的风速.     

小行星主带上单个行星的质量上限

介绍了N体模拟的Hermite算法,并利用该算法研究了不同质量行星在小行星主带上轨道的演化情况．采用的演化模型是太阳系N体模型(N=7),即把水星、金星、地球的质量加到太阳上,忽略冥王星,同时在小行星主带附近增加一个假想行星,系统演化时间为1亿年．数值模拟显示能够稳定存在于小行星主带上的单个天体的质量上限其量级为10~(25)kg．模拟同时还显示在某些情况下,假想行星与木星之间的低阶共振可以增强系统的稳定性．     

Variations of the IMF Bz component over a solar activity cycle for different regimes of solar wind formation

The behavioural features of the IMF B_z component for different solar wind velocity regimes have been studied. The study revealed a significant difference in variations of the B_z component between high-speed and low-speed regimes. Formation mechanisms for the IMF meridional component as well as the relationship of B_z with dynamical properties of the large-scale magnetic fields on the Sun are discussed.