Encke's special perturbation technique associated with the KS regularized variables

In this paper of the series, a special perturbation technique of Encke-type associated with the KS regularized variables will be developed for satellite motions in the Earth's gravitational field with axial symmetry. Its computational algorithm is of recursive nature and could be applied for any perturbed conic motion whatever the number of the zonal harmonic coefficients may be. Applications of the algorithm are also included.Now at the Department of Mathematics, Girls College of Education, Jeddah, Saudi Arabia.     

On the numerical transformation of variables in perturbation theory

Once the generating function of a Lie-type transformation is known, canonical variables can be transformed numerically by application of a Runge-Kutta type integration method or any other appropriate numerical integration algorithm. The proposed approach avails itself of the fact, that the transformation is defined by a system of differential equations with a small parameter as the independent variable. The integration of such systems arising in the perturbation theories of Hori and Deprit is discussed. The method allows to compute numerical values of periodic perturbations without deriving explicitly the perturbation series. This saving of an algebraic work is achieved at the expense of multiple evaluations of the generator's derivatives.     

On the choice of reference orbit,canonical variables,and perturbation method in satellite theory

The author's second-order artificial satellite theory (Aksnes, 1970) is reviewed and compared with that of Kozai (1962). These theories differ in that the former makes use of: (1) an intermediate orbit, being a rotating ellipse instead of a fixed ellipse, (2) Hill variables instead of Delaunay variables, and (3) Hori's perturbation method in Lie series rather than Von Zeipel's method in Taylor series.It is demonstrated that because of these differences, the former theory enjoys a greater simplicity and compactness, it is non-singular at zero eccentricity, and the process of deriving the perturbations is considerably simplified (Aksnes, 1972). For example, the number of second-order short-period terms due to the planet's oblateness (J ₂) is reduced by a factor of about three (Hori, 1970). The intermediate orbit and Hori's perturbation method contribute about equally to this reduction.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

A note on the correspondence between Mathieu's transformations and redundant variables in lagrangian mechanics

It is shown that, to any change of variables:q _i=q_i(r, t) (i=1,..., n; =1,...,n+m; mn) increasing the number of variables, it is possible to associate a Mathieu's transformation and conversely. The results are applied to the theory of the osculating plane of motion.

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Resumé On montre qu'à toute transformation:q _i=q_i(r, t)(i=1,..., n; =1,...,n+m; mn) augmentant le nombre de variables, on peut associer une transformation de Mathieu et réciproquement. Les résultats sont appliqués à la théorie du plan osculateur du mouvement.

     

Albedo perturbation models: General formalism and applications to LAGEOS

The force due to radiation pressure on a satellite of arbitrary shape is written in a general form within a formalism similar to that used in the theory of radiative transfer in atmospheres. Then the corresponding integrals are evaluated for the simple case of a spherically symmetric satellite, and applied to model the perturbation due to the Earth-reflected radiation flux on LAGEOS. For this purpose, the optical behaviour of the Earth's surface and atmosphere is described as a combination of Lambertian diffusion (continents), partial specular reflection consistent with Fresnel law (oceans) and anisotropic diffusion according to Chandrasekhar's radiative transfer theory (clouds). The in-plane Gauss componentsT andS vs. mean anomaly are computed for a simple orbital geometry and for different models of the Earth's optical properties. A sensitive dependence is found on the assumed cloud distribution, with significant perturbations possibly arising from oceanic specular reflection when the satellite is close to the Earth's shadow boundaries.On leave from Astronomical Institute, Charles University, védská 8, 15000 Prague 5, Czechoslovakia     

On the use of the autonomous Birkhoff equations in Lie series perturbation theory

In this article, we present the Lie transformation algorithm for autonomous Birkhoff systems. Here, we are referring to Hamiltonian systems that obey a symplectic structure of the general form. The Birkhoff equations are derived from the linear first-order Pfaff–Birkhoff variational principle, which is more general than the Hamilton principle. The use of 1-form in formulating the equations of motion in dynamics makes the Birkhoff method more universal and flexible. Birkhoff’s equations have a tensorial character, so their form is independent of the coordinate system used. Two examples of normalization in the restricted three-body problem are given to illustrate the application of the algorithm in perturbation theory. The efficiency of this algorithm for problems of asymptotic integration in dynamics is discussed for the case where there is a need to use non-canonical variables in phase space.     

Scale transformations, tree-level perturbation theory and the cosmological matter bispectrum

Soon after the discovery of radio pulsars in 1967, the pulsars are identified as strongly magnetic (typically 10¹² G) rapidly rotating (∼10²− 0.1 Hz) neutron stars. However, the mechanism of particle acceleration in the pulsar magnetosphere has been a longstanding problem. The central problem is why the rotation power manifests itself in both gamma-ray beams and a highly relativistic wind of electron–positron plasmas, which excites surrounding nebulae observed in X-ray. Here we show with a three-dimensional particle simulation for the global axisymmetric magnetosphere that a steady outflow of electron–positron pairs is formed with associated pair sources, which are the gamma-ray emitting regions within the light cylinder. The magnetic field is assumed to be a dipole, and to be consistent, the pair creation rate is taken to be small, so that the model might be applicable to old pulsars such as Geminga. The pair sources are charge-deficient regions around the null surface, and we identify them as the outer gap. The wind mechanism is the electromagnetic induction which brings about fast azimuthal motion and eventually trans-field drift by radiation drag in the close vicinity of the light cylinder and beyond. The wind causes loss of particles from the system. This maintains charge deficiency in the outer gap and pair creation. The model is thus in a steady state, balancing loss and supply of particles. Our simulation implies how the wind coexists with the gamma-ray emitting regions in the pulsar magnetosphere.     

Prediction of satellite motions in the Earth's gravitational field with axial symmetry by the KS regularized theory

In this paper, economical and stable recurrence formulae for the Earth's zonal potential and its gradient for the KS regularized theory will be established for any numberN of the zonal harmonic coefficient. A general recursive computational algorithm based on these formulae is also established for the initial value problem of the KS theory for the prediction of artificial satellites in the Earth's gravitational field with axial symmetry. Applications of the algorithm for the problem of the final state prediction are illustrated by numerical examples of three test orbits each for two geopotential models corresponding toN=2 andN=36. A final state of any desired accuracy is obtained for each case study, a result which shows the flexibility of the algorithm.     

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.     

On the space-time structure and its relation to quantum mechanics

The Weyl covariant formulation of Dirac's equation by means of a rest mass the Weyl type of which is — 1, implies the introduction of a Riemann structure in addition to the Weyl structure. The Ehlers-Pirani-Schild attempt to specify the space-time structure and the lowest order of the WKB approximation are compatible with each other. Only the requirement that the connection to which the Dirac field couples minimally is identical to the connection the autoparallels of which are the world lines of structure-less test particles, leads to a reduction of the Weyl structure to the Riemannian one.     

Global MHD modeling of Mercury's magnetosphere with applications to the MESSENGER mission and dynamo theory

We use a global magnetohydrodynamic (MHD) model to simulate Mercury's space environment for several solar wind and interplanetary magnetic field (IMF) conditions in anticipation of the magnetic field measurements by the MESSENGER spacecraft. The main goal of our study is to assess what characteristics of the internally generated field of Mercury can be inferred from the MESSENGER observations, and to what extent they will be able to constrain various models of Mercury's magnetic field generation. Based on the results of our simulations, we argue that it should be possible to infer not only the dipole component, but also the quadrupole and possibly even higher harmonics of the Mercury's planetary magnetic field. We furthermore expect that some of the crucial measurements for specifying the Hermean internal field will be acquired during the initial fly-bys of the planet, before MESSENGER goes into orbit around Mercury.     

Light scattering in a spherical,exponential atmosphere,with applications to Venus

The problem of the reflection of light from an optically thick, spherical atmosphere in which the scatterers are distributed exponentially with a scale height small compared to the radius of the planet is discussed. Exact formal solutions are obtained for the single scattered component. Useful approximate analytic solutions, which also include multiply scattered light, are given. The results are applied to the analysis of the Mariner 10 limb and terminator images of Venus. The altitude of the “detached” haze layer discovered by Mariner 10 is at 79–85 km, but in places the haze exists above 100 km. This layer apparently is a stable, planetwide feature which forms at the top of the Pioneer Venus upper haze layer. It was similar in location, scale height, and thickness at the times of the two missions, in contrast to the lower, high-altitude haze which changed dramatically. We discuss two possibilities for the nature of the limb hazes. (1) The lower haze is probably the sulfuric acid cloud and the “detached” layer may be a separate water-ice haze. (2)The “detached” haze layer may not be separate at all, but part of the sulfuric acid haze, and the apparent “gap” at 75–80 km may be the source region of a broadband absorber. The spatial distribution of the strong near-UV absorber, which may be elemental sulfur as first suggested by B. Hapke and R. Nelson (1975, J. Atmos. Sci.32, 1212–1218), is examined in light of our results. Several arguments indicate that there is no nonabsorbing, overlying haze and that the UV absorber extends to the top of the haze 8layer.     

A unified treatment of some expansion procedures in perturbation theory: Lie series,Faà di Bruno operators,and Arbogast's Rule

The problem of expanding the transform of a function under a near-identity change of coordinates is reviewed. A common derivation is given for Musen's Faà di Bruno operators and Kamel's Lie series, and both are related to Arbogast's Rule for composing power series.     

The theory of canonical perturbations applied to attitude dynamics and to the Earth rotation. Osculating and nonosculating Andoyer variables

In the method of variation of parameters we express the Cartesian coordinates or the Euler angles as functions of the time and six constants. If, under disturbance, we endow the “constants” with time dependence, the perturbed orbital or angular velocity will consist of a partial time derivative and a convective term that includes time derivatives of the “constants”. The Lagrange constraint, often imposed for convenience, nullifies the convective term and thereby guarantees that the functional dependence of the velocity on the time and “constants” stays unaltered under disturbance. “Constants” satisfying this constraint are called osculating elements. Otherwise, they are simply termed orbital or rotational elements. When the equations for the elements are required to be canonical, it is normally the Delaunay variables that are chosen to be the orbital elements, and it is the Andoyer variables that are typically chosen to play the role of rotational elements. (Since some of the Andoyer elements are time-dependent even in the unperturbed setting, the role of “constants” is actually played by their initial values.) The Delaunay and Andoyer sets of variables share a subtle peculiarity: under certain circumstances the standard equations render the elements nonosculating. In the theory of orbits, the planetary equations yield nonosculating elements when perturbations depend on velocities. To keep the elements osculating, the equations must be amended with extra terms that are not parts of the disturbing function [Efroimsky, M., Goldreich, P.: J. Math. Phys. 44, 5958–5977 (2003); Astron. Astrophys. 415, 1187–1199 (2004); Efroimsky, M.: Celest. Mech. Dyn. Astron. 91, 75–108 (2005); Ann. New York Acad. Sci. 1065, 346–374 (2006)]. It complicates both the Lagrange- and Delaunay-type planetary equations and makes the Delaunay equations noncanonical. In attitude dynamics, whenever a perturbation depends upon the angular velocity (like a switch to a noninertial frame), a mere amendment of the Hamiltonian makes the equations yield nonosculating Andoyer elements. To make them osculating, extra terms should be added to the equations (but then the equations will no longer be canonical). Calculations in nonosculating variables are mathematically valid, but their physical interpretation is not easy. Nonosculating orbital elements parameterise instantaneous conics not tangent to the orbit. (A nonosculating i may differ much from the real inclination of the orbit, given by the osculating i.) Nonosculating Andoyer elements correctly describe perturbed attitude, but their interconnection with the angular velocity is a nontrivial issue. The Kinoshita–Souchay theory tacitly employs nonosculating Andoyer elements. For this reason, even though the elements are introduced in a precessing frame, they nevertheless return the inertial velocity, not the velocity relative to the precessing frame. To amend the Kinoshita–Souchay theory, we derive the precessing-frame-related directional angles of the angular velocity relative to the precessing frame. The loss of osculation should not necessarily be considered a flaw of the Kinoshita–Souchay theory, because in some situations it is the inertial, not the relative, angular velocity that is measurable [Schreiber, K. U. et al.: J. Geophys. Res. 109, B06405 (2004); Petrov, L.: Astron. Astrophys. 467, 359–369 (2007)]. Under these circumstances, the Kinoshita–Souchay formulae for the angular velocity should be employed (as long as they are rightly identified as the formulae for the inertial angular velocity).     

Relativity in astrometry,Celestial mechanics and geodesy

Book review

Relativity in astrometry, Celestial mechanics and geodesySoffel, M. H.: 1989, Springer Verlag, DM 98. (Hardcover) (ISBN 3-540-18906-8)     

On singularities in electrodynamics and gravitational theory

The singularity and particle problems in electrodynamics and gravitational theory are discussed. Comparing different modifications of MAXWELL 's electrodynamics and general theory of relativity, arguments are given which show possibilities of solving these problems by a modification that replaces EINSTEIN 's equations by those of the EINSTEIN -BACH -WEYL -type.     

On flares,substorms, and the theory of impulsive flux transfer events

Solar flares and magnetospheric substorms are discussed in the context of a general theory of impulsive flux transfer events (IFTE). IFTE theory, derived from laboratory observations in the Double Inverse Pinch Device (DIPD), provides a quantitative extension of neutral sheet theories to include nonsteady field line reconnection. Current flow along the reconnection line increases with magnetic flux storage. When flux build-up exceeds the level corresponding to a critical limit on the current, instabilities induce a sudden transition in the mode of conduction. The resulting IFTE, indifferent to the specific modes and instabilities involved, is the more energetic, the lower the initial resistivity. It is the more violent, the greater the resulting resistivity increase and the faster its growth. Violent events can develop very large voltage transients along the reconnection line. Persistent build-up promoting conditions produce relaxation oscillations in the quantity of flux and energy stored (build-up-IFTE cycles). It is difficult to avoid the conclusion: flares and substorms are examples of IFTE.     

Tidal disruptions: II. A continuum theory for solid bodies with strength, with applications to the Solar System

We present a comprehensive theory for the breakup conditions for ellipsoidal homogeneous secondary bodies subjected to the tidal forces from a nearby larger primary: for materials ranging from purely fluid ones, to granular rubble-pile gravel-like ones, and to those with either cohesive or granular strength including cohesive rocks and metals. The theory includes but greatly extends the classical analyses given by Roche in 1847, which dealt only with fluids, and also our previous analysis [Holsapple, K.A., Michel, P., 2006. Icarus 183, 331-348], which dealt only with solid but non-cohesive bodies. The results here give the distance inside of which breakup must occur, for both a steadily orbiting satellite and for a passing or impacting object. For the fluid bodies there is a single specific shape (a “Roche Ellipsoid”) that can be in equilibrium at any given distance from a primary, and especially only one shape that can exist at the overall minimum distance (d/R)(ρ/ρp)^1/3=2.455, the classical well-known “Roche limit.” In contrast, solid bodies can exist at a given distance from a primary with a range of shapes. Here we give multiple plots of the minimum distances for various important combinations of body shape, spin, mass density, and the strength parameters characterized by an angle of friction and cohesive strength. Such results can be used in different ways. They can be used to estimate limits on strengths and mass densities for orbiting bodies at a known distance and shape. They can be used to determine breakup distances for passing bodies with an assumed strength and shape. They can be used to constraint physical properties such as bulk density of bodies with a known shape that were known to breakup at a given distance. A collection of approximately 40 satellites of the Solar System is used for comparison to the theory. About half of those bodies are closer than the Roche fluid limit and must have some cohesion and/or friction angle to exist at their present orbital distance. The required solid strength for those states is determined. Finally, we apply the theory to the break up of the SL9 comet at close approach with Jupiter. Our results make clear that the literature estimates of its bulk density depend markedly on unknown parameters such as shape, orientation and spin, and most importantly, material strength characterization.