GeneralizedF andG series and convergence of the power series solution of theN-body problem

The existence of power series, analogous to the familiarf andg series of the two-body problem, is demonstrated in the case of then-body problem, and recursive formulae are deduced for the derivation of the coefficients of these series. In addition a proof of the convergence of the power series solution of then-body problem is given, based on the developed series.     

A note on an attempt at more efficient Poisson series evaluation

Eliminating many of the trigonometric function calls by a suitable series transformation has resulted in a substantial reduction of on-line computing time while very long Poisson series are being evaluated. A further reduction has been realized by applying a short SNOBOL processor to the FORTRAN coding of the transformed series which eliminates many of the multiplication operations during the course of series evaluation.     

Optimizing expansions of the Earth’s gravity potential and its derivatives over ellipsoidal harmonics

A set of spherical harmonics is the most widely used representation of the Earth’s gravity potential. This series converges outside and on the surface of a reference sphere enveloping the Earth. However, the Earth’s surface is better approximated by the reference ellipsoid—a compressed ellipsoid of revolution that covers the entire Earth. The gravity potential can be expanded in a series over ellipsoidal harmonics on the surface of the reference ellipsoid and on the surface of other external confocal ellipsoids of revolution. In contrast to spherical harmonics, depending on the associated Legendre functions of the first kind, ellipsoidal harmonics depend also on the associated Legendre functions of the second kind. The latter contain the very slowly converging hypergeometric Gauss series. The number of series increases with increasing the order of their derivatives. In this work, we derived new series for the gravitational potential of the Earth and its derivatives over ellipsoidal harmonics. Starting from the first order derivative, all the series corresponding to higher order derivatives depend on the same two hypergeometric Gauss series. The latter converges considerably faster than that for the hypergeometric series previously used when computing the gravity potential and its derivatives.     

Power series representation of partial derivatives required in orbit determination

A program to integrate the equations of motion by series in powers of the time step can be easily modified to furnish the elements of the matrizant in power series of the time step. In particular, if the series representing the motion are obtained recursively, differentiation of the recurrence relations will provide immediately a recursive scheme for computing the coefficient in the power series for the elements of the matrizant.     

时间序列数字滤波后自由度的确定--应用于日长变化与南方涛动指数的相关分析

原始观测时间序列进行数字滤波后，其自由度将大大降低，对各种数字滤波器都可以通过Monte Carlo方法模拟获得对应置信水平的相关系数临界值，从而得到时间序列滤波后的自由度，如果滤波后的频段宽度与Nyquist频率之比为Z，那么对于理想的单边和双边滤波器，时间序列滤波后的自由度为原始值的Z／2和Z倍，非理想滤波器一般不符合此规律。     

Poincaré's méthode nouvelle by skew composition

Poincaré designed the méthode nouvelle in order to build approximate integrals of Hamiltonians developed as series of a small parameter. Due to several critical deficiencies, however, the method has fallen into disuse in favor of techniques based on Lie transformations. The paper shows how to repair these shortcomings in order to give Poincaré’s méthode nouvelle the same functionality as the Lie transformations. This is done notably with two new operations over power series: a skew composition to expand series whose coefficients are themselves series, and a skew reversion to solve implicit vector equations involving power series. These operations generalize both Arbogast’s technique and Lagrange’s inversion formula to the fullest extent possible. This revised version was published online in July 2006 with corrections to the Cover Date.     

IGS′92联测期间地极坐标的高频波动

本文分析了ＩＧＳ＇９２联测期间七个ＧＰＳ数据处理中心提供的极坐标序列。通过谱分析、最小二乘拟合和Ｆ检验，表明在这些序列中存在一些共同的高频波动：在Ｘ方向上具有２７．０，１６．５，１３．４和１０．４天的周期，在Ｙ方向上的波动周期约为２０．５，１５．８和１０．０天。并且每个序列与ＥＯＰ（ＩＥＲＳ）９２Ｃ０４之间都存在一个系统差。计算与分析表明，这些系统偏离的主要原因是由于在用ＧＰＳ资料解算Ｘ、Ｙ时，不同分析中心采用了不同系统的台站坐标（或者说只有部分台站采用了固定的台站坐标），从而造成这些序列所在的参考架与ＩＴＲＦ９１之间存在一个平移和旋转。最后，计算了该期间的大气角动量激发函数，可部分地解释该期间的Ｘ、Ｙ高频波动的原因。     

On the expansions of intermediate orbit for general planetary theory

Intermediate orbit for general planetary theory is constructed in the form of multivariate Fourier series with numerical coefficients. The structure and efficiency of the derived series are illustrated by giving various statistical properties of the coefficients.The ability of the recently proposed elliptic function approach to compress the Fourier series representing the intermediate orbit is investigated. Our results confirm that when mutual perturbations of a pair of planets are considered the elliptic function approach is quite efficient and allows one to compress the series substantially. However, when perturbations of three or more planets are under study the elliptic function approach does not give any advantages.     

Studies in the application of recurrence relations to special perturbation methods

The numerical integration of the differential equations describing dynamical systems has been shown in previous papers of this series to be most effectively accomplished by an explicit Taylor series method.In this paper we show that one explicit Taylor series method, developed earlier in this series and which appears to possess a high degree of versatility, yields considerable gains in efficiency over classical single-step and multi-step methods. (In this context efficiency is a measure of the time taken to carry out a calculation of a specific accuracy).For a given accuracy criterion governing the local truncation error (LTE) it is found that the Taylor series method is generallytwice as fast as the classical multi-step method and up totwenty times faster than the classical single-step method.     

Radius of the Roche equipotential surfaces

In literature, there is no exact analytical solution available for determining the radius of Roche equipotential surfaces of distorted close binary systems in synchronous rotation. However, Kopal (Roche Model and Its Application to Close Binary Systems, Advances in Astronomy and Astrophysics, Academic Press, New York 1972) and Morris (Publ. Astron. Soc. Pac. 106:154, 1994) have provided the approximate analytical solutions in the form of infinite mathematical series. These series expressions have been commonly used by various authors to determine the radius of the Roche equipotential surfaces, and hence the equilibrium structures of rotating stars and stars in the binary systems. However, numerical results obtained from these approximating series expressions are not very accurate. In the present paper, we have expanded these series expressions to higher orders so as to improve their accuracy. The objective of this paper is to check, whether, there is any effect on the accuracy of these series expressions when the terms of higher orders are considered. Our results show that in most of the cases these expanded series give better results than the earlier series. We have further used these expanded series to find numerically the volume radius of the Roche equipotential surfaces. The obtained results are in good agreement with the results available in literature. We have also presented simple and accurate approximating formulas to calculate the radius of the primary component in a close binary system. These formulas give very accurate results in a specified range of mass ratio.     

Evaluation of the light changes for distorted eclipsing binaries

This paper discusses a method for improving on the numerical evaluation of the light changes exhibited by a distorted eclipsing binary system.In the theory formulated by Kopal (1959), certain boundary integrals due to the distortion of both components have been calculated in terms of the Appell hypergeometric series of the first kind. The values of the four parameters appearing in these series differ according as to whether one is dealing with a partial or an annular eclipse.To accelerate the numerical evaluation of the light changes one should avoid recomputing such infinite series for contiguous values of the parameters. This can be achieved by making use of certain recursion formulae which hold for the foregoing series.We have provided here a procedure that yields forty-eight recursion formulae for the Appell hypergeometric series and have specifically calculated four new independent recursion formulae relevant to the astrophysical problem.     

Total sunspot area as a solar activity index

Observational time series of the total sunspot area A in the visible solar hemisphere are analyzed. A technique that allows the instability of the scale of these series to be found and corrected has been developed. An internally homogeneous series of the index A on the Greenwich scale can be obtained from 1875 to the present. A method for the approximate calculation of the yearly mean A from the Wolf sunspot numbers known since 1700 is suggested to extend this series into the past. The resulting series of the index A characterizes the solar activity variations over a period of ～300 years. These data are used to study processes in the Solar System related to the variability of the central star.     

Applications of the DFT/CLEAN Technique to Solar Time Series

A new technique of Fourier analysis, DFT/CLEAN, has been adapted for the study of solar time series. The technique was developed by Roberts and his collaborators (1987, 1994) to address the limitations of other techniques of Fourier analysis and the shortcomings of many astronomical time series. The utility of the technique is illustrated with several applications to solar time series.     

Literal solution for Hill's lunar problem

A computer program for the manipulation of power series in several variables is used to find the first order solution to Hill's lunar problem. The ratiom of the mean motion of the Sun to that of the Moon is kept as a formal parameter. The series inm are known to converge very poorly. It is shown how efficient algorithms among them the Lie transformation allow us to compute the series inm as far as they are needed. When the series are evaluated at Brown's numerical value form the results achieve or exceed his accuracy.     

PARSEC: An interactive Poisson series processor for personal computing systems

PARSEC is a PC-based, interactive algebraic manipulation package designed to manipulate series of the kind frequently found in Celestial Mechanics applications and perturbation procedures. The system is fundamentally an input/command interpreter which allows the user to enter algebraic expressions and procedures and to control the flow of series manipulations interactively. The system is designed to allow easy manipulations of polynomial-trigonometric series within the environment of an electronic scratchpad.     

Sur de nouvelles séries pour le problème de masses critiques de Routh dans le problème restreint plan des trois corps

The planar restricted 3-body problem, linearized in the neighborhood of Lagrangian equilibriaL ₄ andL ₅, has in general two distinct eigenvalues and their opposites. When they are pure imaginary and not multiples of each other, they generate two families of periodic solutions called long and short periodic families. This is essentially a consequence of the famous theorem of Liapunov (Siegel, 1956). We showed (Roels, 1971b) how to solve the problem when the eigenvalues are multiples of each other in building series with negative exponents instead of the integer expansions of Siegel (Roels and Lauterman, 1970). When the eigenvalues are equal, which is the case for the mass ratio of Routh, the problem was solved by Deprit and Henrard (1968) using formal series in ordinary unnormalized variables. That leads to very complicated series because of the use of variables that are not well adapted to the problem. The convergence of the series was proven by Meyer and Schmidt (1971). In this paper we solve the problem by using normalized variables. This brings us to build expansions with fractional exponents. So in summary, normalized variables generate integer series in the non-resonant cases, series with negative exponents in the case of resonancek≥3, and series with fractional exponents when the resonance is 1.     

Time Series Modeling of Sunspot Numbers Using Long-Range Cyclical Dependence

This paper deals with the analysis of the sunspot number time series using a new technique based on cyclical long-range dependence. The results show that sunspot numbers have a periodicity of 130 months but, more importantly, that the series is highly persistent, with an order of cyclical fractional integration slightly above 0.30. That means that the series displays long memory, with a large degree of dependence between the observations that tends to disappear very slowly in time.     

A note on velocity-related series expansions in the two-body problem

The present note describes a few important series expansions in the two-body problem. They are related to the magnitudeV of the velocity vector and they are important for the treatment of atmospheric drag with the method of general perturbations. These series have been obtained with computerized Poisson series Manipulations. The results are given to order seven in the eccentricity, for both the Mean Anomaly and the True Anomaly.     

A comment on some oscillatory models of adiabatic stars

Three different oscillatory models of adiabatic stars are reinvestigated. These are the homogenous model, the inverse square model and the Roche model. The ratio between the amplitude of the oscillations and the distance from the center is developed in a power series. For physical conclusions to be drawn, it turns out to be crucial if the power series is divergent or convergent. Mathematical arguments are given which show that the power series are really divergent for all three models.