Expansion of the Hamiltonian for the planetary problem into a Poisson series in the heliocentric reference frame

An expansion of the Hamiltonian for the N-planet problem into a Poisson series using a system of modified (complex) Poincare´ canonical elements in the heliocentric coordinate system is constructed. The Lagrangian and Hamiltonian formalisms are used. The first terms in the expansions of the principal and complementary parts of the disturbing function are presented. Estimates of the number of terms in the presented expansions have been obtained through numerical experiments. A comparison with the results of other authors is made.     

The Expansion of the Distance Between Two Arbitrary Gravitating Points as Applied to Specific Models of Celestial Bodies

For the complete system of biharmonic functions a suitable representation in spheroidal coordinates is found. It is used for expanding the distance between two arbitrary gravitating points and its inverse quantity in appropriate series. Such expansions can be of interest and useful in a number of problems of celestial mechanics and stellar dynamics.     

Simultaneous creation of electron-positron pairs and photons due to their interaction in Robertson-Walker universes

The influence of the electromagnetic interaction upon particle creation in expanding Robertson-Walker universes is investigated applying an S-matrix scheme. We discuss all processes contributing to the number density of created particles up to second order in the coupling constant. One process is of particular importance being the only source of photons. It is considered in more detail and evaluated explicitly for a time-symmetrically expanding radiation-dominated Friedman universe as well as statically bounded expansions.     

Book Review

The aim of this book is to present techniques for the study of motion of solar system objects in highly eccentric orbits. Instead of using the usual anomalies (mean, true, eccentric), the authors define and use a new kind of anomaly, the elliptic anomaly.In this way, it is possible, in a theory using perturbation series expansions, to make the ratio: (accuracy)/(number of needed terms), higher than in the classical techniques. The book consists of six chapters. The first chapter deals with the elliptic anomaly in the two-body problem and the second chapter presents the general technique to construct first-order perturbation theory in elliptic function expansions. The next three chapters deal with applications of the new technique to artificial satellites and asteroids, in highly eccentric orbits. The last chapter describes the basic algorithms of the theory.The tools developed in the book demand the use of computer algebra, which is implemented by means of Mathematica 3.0.The book is well written and the new technique is clearly presented and related to the existing techniques, making it useful to all those who use analytical or semi-analytical methods for the study of highly eccentric motion. Celestial Mechanics at High Eccentricities, Gordon and Breach Publishers, US$95, GBP 59, EUR 79, ISBN 90-5699-212-0     

Expansions in power series by computer

The paper describes a technique of expanding power series solutions to sets of equations that can be applied formally to a set of equations by a compiler type program. A program suitable for an algebra system is generated that when run will produce the power series expansions by recurrence relations. Examples are given from Celestial mechanics.     

Fourier techniques for an analysis of eclipsing binary light curves: IV

The aim of the present paper is to establish two further series expansions (alternative to those given in Demircan, 1979a, b, c), for the observed light changes of eclipsing binary system. The coefficients of these expansions have also been expressed in the form of general series expansions in terms of the eclipse elementsr _1.2,i andL ₁ of the spherical model on which all other distorted models may be based (Kopal, 1975, 1976) in an analysis in the frequency-domain.     

A FORTRAN-based Poisson series processor and its applications in celestial mechanics

The present article describes the design and applications of our Poisson series subroutine package developed and maintained since 1968. The programs are written in standard FORTRAN-77 and are almost independent of the wordlength of the particular computer. The system has no restriction on the number of polynomial and angular variables and the storage allocations are completely automatic, invisible to the user.The nucleus of the system consists of about 20 basic traffic subroutines that handle the terms of the different series. Besides these subroutines, we have a number of I/O routines as well as arithmetic subroutines and a large number of Celestial Mechanics applications such as the classical expansions of the Kepler Problem and several expansions of Disturbing Functions. A preprocessor has also been built, allowing the user to write code in a high-level language, such as Jefferys' Trigman, and then translate it in our call-statements.The system was developed on several different computers: first on the Univac 1108 at the Jet Propulsion Laboratory in 1968, then the IBM 360-91 at UCLA, Los Angeles, California and finally the CDC 6600 and CYBER 170/750 at the University of Texas, Austin, Texas. The latest version is entirely geared towards the IBM-PC and compatibles.     

Branching solutions and Lie series

Branching solutions of algebraic equations are treated using Lie series. A new method is proposed to derive Puiseux expansions. Newton's diagram is considered in the context of Lie series. An application of finding equilibrium points of a Hamiltonian system near resonances is also demonstrated.     

Gravity assist space pruning based on differential algebra

In this paper a differential algebra version of the gravity assist space pruning algorithm is presented. The use of differential algebraic techniques is proposed to overcome the two main drawbacks of the existing algorithm, i.e., the steep increase of the number of function evaluations with the number of planets involved in the transfer, and the use of a bounding procedure that relies on Lipschitzian tolerances. Differential algebra allows us to process boxes in place of grid points, and to substitute pointwise evaluations of the constraint functions with their Taylor expansions. Thanks to the particular instance of multi-gravity assist problems dealt with, all the planet-to-planet legs can be treated independently, and forward and backward constraining can be applied. The proposed method is applied to preprocess the search space of sample interplanetary transfers and it also serves as a stepping stone towards a fully rigorous treatment of the pruning process based on Taylor models.     

Flare-associated coronal expansion phenomena

Two classes of coronal expansion phenomena have been studied in Sacramento Peak coronal movies: Slow, slightly decelerated expansion phenomena (v=∼10− =∼ 2 km/sec) and fast, accelerated, quasi-exploding arches (v =∼ 10 - > 100 km/sec). The various phenomena were found to be associated with flares in different ways: The slow expansions were long lived post-flare phenomena initiated by the flare; the accelerated expanding arches were either (a) arches expanding prior to and apparently exploding at flare onset, or (b) arches apparently emerging from the flare (probably in its initial phase) and rapidly expanding and exploding, or (c) the expansion and disruption of (originally stable) coronal arches during occurrence of a distant flare. These expansions may be considered as evidence for corresponding flare associated changes in the coronal magnetic field. Mitteilung aus dem Fraunhofer Institut Nr. 96.     

Secular variations in the restricted problem: A numerical technique for studying stability of equilibrium solutions

In this paper we develop and implement an explicit numerical technique for studying stability of equilibrium solutions concerning the secular variations in the restricted problem. In our implementation, we use fourth-order expansions for the secular terms, but the method itself is independent of the particular order used as upper limit in the required expansions.     

Analytical methods for the orbits of artificial satellites of the Moon

The motion of a close artificial satellite of the Moon is considered. The principal perturbations taken into account are caused by the nonsphericity of the Moon and the attraction of the Earth and the Sun. To begin with, the expansions of the disturbing functions due to the nonsphericity of the primary body and the action of the disturbing mass-point body have been derived. The second expansion is produced in terms of the Keplerian elements of a satellite and the spherical coordinates of the disturbing body. Both expansions are valid for an arbitrary reference plane. The motion of a satellite of the Moon is studied in the selenocentric coordinate system referred to the Lunar equator and rotating with respect to the fixed ecliptic system. However, the coordinate exes in the equatorial plane are chosen so that the angular speed of rotation of the system is small. The motion of the satellite is described by means of the contact elements which enable one to utilize the conventional Lagrange's planetary equations and may be regarded as the generalization of the notion of the osculating elements to the case of the disturbing function depending not only o the coordinates and the time but on the velocities as well. Two methods are proposed to represent the motion of Lunar satellites over long intervals of time: the von Zeipel method and the Euler method of analytical integration with application of the variation-of-elements technique at every step of integration. The second method is exposed in great detail.Presented at the Meeting of Commission 7 of the IAU on Analytical Methods for the Orbits of Artificial Celestial Objects 14-th General Assembly of the IAU, Brighton, 1970.     

Interpretation of the apparent north-south asymmetry and fluctuations of galactic rotation

A model of the apparent north-south asymmetry of the rotation curve of the Galaxy and of the asymmetry appearing for the rotation of its outskirts has been presented, in terms of interaction redshifts. Fluctuations in the rotation curve for neutral hydrogen are discussed and interpreted as expanding motions in the arms. The expansion of the disk of the Galaxy which has been suggested previously as an explanation for the asymmetry has been shown to meet some serious difficulties. A remarkable north-south symmetry both in the spatial distribution and in the kinematics of neutral hydrogen in the subsolar region follows from the present model.     

A model for the infrared brightening of a Nova shell

A model for grain formation in an expanding Nova shell is presented. It is shown that individual grains complete the growth within a very short time interval. In consequence the infrared brightening accompanied with visual darkening observed in several Novae is interpreted in terms of extension of dusty region in the ejected gas shell. Developments of infrared luminosity and visual magnitude are computed on the basis of a model, and compared with the observations of Nova Vul 1976, Nova Ser 1970, and Nova Aql 1975. The constancy and uniformity of grain temperatures for a long period after the infrared maximum observed in Nova Vul 1976 are also discussed.     

The features of auroral bulge expansion

The development of the auroral bulge during substorms is studied using all-sky data from the dense net of stations and also riometer data. A few features seem to be essential for the interpretation of the expansive phase mechanism. The first is the existence of low energy electron precipitation (auroral arcs with the lower border height near 140 km) polewards of the expanding bulge, suggesting that the bulge often arises and develops on closed field lines. Secondly only the localized bulges (with dimensions 2° and 40°Λ) are generated by the continuous deformation of the auroral arc. The greater expansions develop mainly at the expense of the new bright arc formations at the front of the expanding auroral bulge. During each new arc formation impulsive acceleration and precipitation of energetic electrons takes place and brief changes of plasma sheet geometry are sometimes observed at 18 R_E in the magnetotail. This apparently shows a re-distribution of plasma sheet current during the substorm expansive phase.     

On the Possibility of a Finite Model Describing the Universe

In his book “The expanding universe” EDDINGTON (1933) suggested the idea of an astronomical world model containing a finite number of material particles. Starting from this idea the author has speculated about the possibility to create a cosmological model where in addition also the number of geometrical points and time-instants is finite (JÄRNEFELT 1929, 1949, 1951, 1954a, 1954b, 1957, 1964, 1974, JÄRNEFELT and KUSTAANHEIMO 1952, JÄRNEFELT and QVIST 1955). The ultimate goal would be a world model based essentially upon GALOIS fields. In order to explore such a possibility the most simple actually existing physical system, the structure describing a non-relativistic free particle in an one-dimensional Euclidean space, is treated here. A discrete representation of this system is outlined and then a GALOIS -finitization of this platform is discussed. The author suggests that the method presented here for a very special case, could be generalizable and finally extended to a cosmological model.     

Some expansions of the elliptic motion to high eccentricities

Some classic expansions of the elliptic motion — cosmE and sinmE — in powers of the eccentricity are extended to highly eccentric orbits, 0.6627...<e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients are given in terms of the derivatives of Bessel functions with respect to the eccentricity. The expansions have the same radius of convergence (e*) of the extended solution of Kepler's equation, previously derived by the author. Some other simple expansions — (a/r), (r/a), (r/a) sinv, ..., — derived straightforward from the expansions ofE, cosE and sinE are also presented.     

On the expansion of the secular part of the perturbing function of mutual attraction in the satellite systems of planets

We propose a special representation for the secular part of the perturbing function describing the mutual attraction of satellites. In contrast to the known representations, it has a single analytical form for any ratio between the semimajor axes of the perturbed and perturbing satellites. The resulting expression is a partial sum of a power series with respect to the small eccentricities and planet-equatorial inclinations of the satellites’ orbits. This sum includes terms up to and including the fourth degree with respect to these small parameters. The proposed expansion is compared with one of the known expansions for the secular part of the perturbing function.     

Asymptotic theory of the transfer of polarized radiation with resonance scattering in the doppler core of the line

In the first of the series of papers by Ivanov et al. it was shown that the model problem of the transfer of polarized radiation as a result of resonance scattering from two-level atoms in a homogeneous plane atmosphere in the absence of LTE comes down, in the approximation of complete frequency redistribution, to the solution of an integral matrix equation of the Wiener-Hopf type for a (2 × 2) matrix source function S(τ). In the second paper in this series, devoted to the vector Milne problem, complete asymptotic expansions of the matrix I(z) [which is essentially a Laplace transform of the matrix S(τ)] for the case of a Doppler profile of the coefficient of absorption, and the coefficients of asymptotic expansions of S(τ) (τ » 1) are expressed in terms of coefficients of the expansions of I(z). We show that asymptotic expansions of S(τ) can be found directly from an integral matrix equation of the Wiener-Hopf type for S(τ). We give new recursive equations for the coefficients of these expansions, as well as a new derivation of asymptotic expansions of the matrix I, including its second column, which was considered only briefly by Ivanov et al.