Satellite prediction formulae for Vinti's model

For Vinti's dynamical problem, there is proposed a new form of solution wherein all three coordinates are expressed in terms of one independent variable. The formulae for the three co-ordinates are clear generalizations of the corresponding formulae for the Kepler problem while the independent variable corresponds to the true anomaly. The solution is completed by the relation connecting the independent variable with time: the latter is a generalization of the well known Kepler time-angle relationship. From the form and method of solution the main qualitative features of the motion are readily derived.     

A theory for the gravitational potentials of spheroidal stellar systems and its application to the Galaxy

A new formula for the gravitational potential for spheroidal systems is proposed, and is applied to the galactic system. The applied model consists of a disk, a nucleus and a massive halo. Although this model is not quite so accurate as those produced by the superposition of large numbers of simple spheroids, it has the important advantage of retaining analytic simplicity for its acceleration formulae, thereby producing significant economies for orbit computations. As better observational data become available, the method can be readily expanded.     

Integration of the elliptic restricted three-body problem with Lie series

The method of Lie series is used to construct a solution for the elliptic restricted three body problem. In a synodic pulsating coordinate system, the Lie operator for the motion of the third infinitesimal body is derived as function of coordinates, velocities and true anomaly of the primaries. The terms of the Lie series for the solution are then calculated with recurrence formulae which enable a rapid successive calculation of any desired number of terms. This procedure gives a very useful analytical form for the series and allows a quick calculation of the orbit.The project is supported by the Austrian Fonds zur Förderung der wissénschaftlichen Forschung under Project No. 4471.     

Photographic observations of fireballs in Tajikistan

Since 2006, systematic double-station photographic observations of fireballs using all-sky cameras equipped with Zeiss Distagon “fisheye” objectives (f/3.5, f = 30 mm) with a 180° field of view have been carried out at two observatories, Gissar (GisAO) and Sanglokh (IAOS), of the Institute of Astrophysics of the Tajik Academy of Sciences. In the method of astrometric reduction of fireball photographs, the empirical formulae for converting the measured coordinates to horizontal celestial coordinates are used. These formulae contain 12 unknown constants to be determined by the least-squares method and the iteration method. Such an approach enables the determination of the coordinates of an object at any point of the celestial hemisphere with a precision close to the theoretical limit whose value is quite comparable with the measurement errors. In the photometric reduction, the dependence of the measured width of the diurnal star trails on their magnitudes was used. As a result of astrometric and photometric reduction of the double-station photographs of five fireballs, the data on atmospheric trajectories, the coordinates of radiants, orbits in interplanetary space, light curves, and photometric masses of meteoroids which produced fireballs were obtained, and the belonging of fireballs to the known meteor showers was determined as well.     

Solving linearized equations of the N-body problem using the Lie-integration method

Several integration schemes exist to solve the equations of motion of the N -body problem. The Lie-integration method is based on the idea to solve ordinary differential equations with Lie-series. In the 1980s, this method was applied to solve the equations of motion of the N -body problem by giving the recurrence formulae for the calculation of the Lie-terms. The aim of this work is to present the recurrence formulae for the linearized equations of motion of N -body systems. We prove a lemma which greatly simplifies the derivation of the recurrence formulae for the linearized equations if the recurrence formulae for the equations of motions are known. The Lie-integrator is compared with other well-known methods. The optimal step-size and order of the Lie-integrator are calculated. It is shown that a fine-tuned Lie-integrator can be 30–40 per cent faster than other integration methods.     

Construction of a general planetary theory of the first order

All the necessary formulae for constructing a general solution for the motion of a planet, in rectangular coordinates, at the first order of the disturbing masses, in purely literal form in eccentricities and inclinations, are given. The authors present the transformation formulae in the two-body problem which give the correspondence between the constants of integration introduced in the theory and the classical keplerian elements. The practical elaboration of the algorithm and some partial results for the couple of planets Jupiter and Saturn are described.     

A study of the force-field equation for the propagation of galactic cosmic rays

A new development is given of the solution of the equation of the force-field approximation for the propagation of galactic cosmic rays in the interplanetary region. It leads to simpler methods for determining the force-field parameters. A method is given for determining the separable diffusion coefficient from observations of galactic electron spectrum and near-Earth electron spectra; it is shown that this diffusion coefficient is not unique but may have a periodic-like dependence upon rigidity; and the method is used to obtain diffusion coefficients for 1965 and 1968. Approximate formulae relating small changes in intensity and diffusion coefficient are developed and some applications of these noted; in one it is shown that the form of, and changes in, diffusion coefficient deduced previously for a neutron monitor event during June–September 1969 are unnecessarily constrained and therefore probably not correct.     

Procedures for solving Kepler's equation

We review starting formulae and iteration processes for the solution of Kepler's equation, and give details of two complete procedures. The first has been in use for a number of years, but the second is entirely new. The new procedure operates with an iterative process that always gives fourth-order convergence and is taken to only two iterations. The error in the resulting solution then never exceeds 7×10^–15 rad.     

发生在1～2GHz的Ⅲ型爆发

对北京天文台动态频谱仪1996～1999年观测到的68群Ⅲ型爆发作了统计分析,并对这些事件的频率漂移、持续时间、偏振和带宽的基本特性作了定性分析.     

大视场长露光观测中的大气折射影响

利用给出的严格的较差大气折射计算公式,以ＬＡＭＯＳＴ为例,计算了较差大气折射对大视场长露光天文观测的影响．并与其他作者的结果进行了比较．该方法给出的是完整的较差大气折射量,与望远镜和导星方式无关．此外,还讨论了大气色散的影响,并探讨了可能的解决方法     

Fast computation of satellite gravitational gradient

The computation of the Earth's potential function at high order and degree with the method of reference [1], causes overflow most of the time. The normalized method [2–6] can eliminate the overflows, but leads to formulae much more involved than those in reference [1]; besides, the programming is more complex and the computer time required larger. The method presented in this paper has the following features: each component of the satellite gravitational gradient can be computed; the formulae are short and easy to be programme; the method is much quicker than the normalization method and can be carried out with a micro‐computer, without overflow even in the case of Earth's spherical harmonics of order and degree as high as 1025 or higher. This method satisfies the present demand to compute satellite gravitational gradient with high accuracy. Furthermore, we present formulae for the fast computation, without overflow, of the gravitational gradient corresponding to Earth's spherical harmonics up to order and degree of 3170 × 3170 or higher. This revised version was published online in July 2006 with corrections to the Cover Date.     

Lie-series for orbital elements: I. The planar case

Lie-integration is one of the most efficient algorithms for numerical integration of ordinary differential equations if high precision is needed for longer terms. The method is based on the computation of the Taylor coefficients of the solution as a set of recurrence relations. In this paper, we present these recurrence formulae for orbital elements and other integrals of motion for the planar $N$ -body problem. We show that if the reference frame is fixed to one of the bodies—for instance to the Sun in the case of the Solar System—the higher order coefficients for all orbital elements and integrals of motion depend only on the mutual terms corresponding to the orbiting bodies.     

考虑后牛顿效应的二体问题的解和星历计算

本文给出考虑后牛顿(PN)效应的二体问题解所对应的基本关系式,并仿照开普勒(Kepler)运动,给出星历表计算方法和相应的计算公式以及适用于数值研究中的简单形式。     

Efficient numerical routines to find the averaged motion of small dust particles under the Poynting-Robertson force

The orbit-averaged differential equations of motion of dust particles under gravity, radiation pressure and Poynting-Robertson drag were given by Wyatt and Whipple (1950). An integral of motion enables the system of two equations in semi-major axis a and eccentricity e to be reduced to one equation, the solution of which is presented here in terms of analytical formulae. An efficient numerical algorithm to compute the solution is given. Listings of two FORTRAN routines are included.     

Tesseral harmonic perturbations in radial transverse and binormal components

The formulae for the perturbations in radial, transverse and binormal components of the Earth artificial satellite motion have been derived. Perturbations due to the tesseral part of the geopotential are considered. The geopotential expressed in terms of the orbital elements has the form proposed by Wnuk (1988). The formulae for the perturbations have been obtained using the Hori (1966) method. They can be effectively applied in calculation of the perturbations in the components including the coefficients of the high order and degree tesseral harmonics. The derived formulae reveal no singularities at zero eccentricity.     

多点高斯定轨方法

针对光学测角资料给出了一种多点高斯定轨方法,该方法公式简单,容易推广使用。经实测计算表明,观测资料分布较好,该方法对于较大的偏心率和小偏心率轨道计算是可靠和有效的。     

A new tool for the study of periodic orbits

We present the generalization of the method of rational Fourier-series approximations for nonlinear ordinary differential equations in galactic dynamics. In previous papers we presented the formulae which describe simple families of periodic orbits with very good accuracy. In this Letter we generalize our formulae to describe periodic orbits of any resonancem:n.     

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.     

Solution of a Riemann–Hilbert problem for a new expression of Chandrasekhar’s H-function in radiative transfer

The linear singular integral equation derived from the nonlinear integral equation of Chandrasekhar’s H-function in radiative transfer is considered here to develop a new form of H-function as a solution of a Riemann–Hilbert problem using Plemelj and Cauchy integral formulae for complex domain. This new form of H-function is a simple integral of known functions. Forms of H-function both for conservative and nonconservative cases are obtained. Their numerical evaluations are made by Simpson’s one-third rule to arrive at an accuracy to ninth places of decimals.     

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.