On the number of isolating integrals in resonant systems with 3 degrees of freedom

The 221-resonance case for a potential problem with three degrees of freedom is characterized by the existence of two isolating approximate integrals apart from the energy. This result completes a statement by Gustavson concerning the number of formal integrals in resonant Hamiltonian systems.     

On the Relationship Between Fast Lyapunov Indicator and Periodic Orbits for Continuous Flows

It is already known (Froeschlé et al., 1997a) that the fast Lyapunov indicator (hereafter FLI), i.e. the computation on a relatively short time of a quantity related to the largest Lyapunov indicator, allows us to discriminate between ordered and weak chaotic motion. Using the FLI many results have been obtained on the standard map taken as a model problem. On this model we are not only able to discriminate between a short time weak chaotic motion and an ordered one, but also among regular motion between non resonant and resonant orbits. Moreover, periodic orbits are characterised by constant FLI values which appear to be related to the order of periodic orbits (Lega and Froeschlé, 2001). In the present paper we extend all these results to the case of continuous dynamical systems (the Hénon and Heiles system and the restricted three-body problem). Especially for the periodic orbits we need to introduce a new value: the orthogonal FLI in order to fully recover the results obtained for mappings.     

Generalized Lyapunov exponents indicators in Hamiltonian dynamics: An application to a double star system

The Lyapunov characteristic numbers (LCNs) which are defined as the mean value of the distribution of the local variations of the tangent vectors to the flow (=ln _k ⁱ ) (see Froeschlé, 1984) have been found to be sensitive indicators of stochasticity. So we computed the distribution of these local variations and determined the moments of higher order for the integrable and stochastic regions in a binary star system with =0.5.     

The peculiarities of diffusion of the galactic cosmic rays at low energies

I give an interpretation of a result of Simpsonet al. (1988) on the variation with kinetic energyT _i of the mean pathlengthX _m (T _i ) of the galactic cosmic rays (CGRs) in the range 0.1T _i 10.0 GeV nucl^–1. I argue that the data onX _m (T _i ) may be interpreted in terms of a model of GCR diffusion on the one-dimensional Alfvén-wave turbulence, having a cutoff in the spectrum at frequencies _h , where _h is the proton gyrofrequency. The cutoff results in changing of the character of variation of the GCR diffusion coefficientD(T _i )T ^a in the rangeT _i 1 GeV nucl^–1 towards some more complicated variation at 0.1T _i 1.0 GeV nucl^–1 due to the peculiarities of the pitch-angle scattering at 90⁰.     

On the number of isolating integrals in systems with three degrees of freedom

Dynamical systems with three degrees of freedom can be reduced to the study of a fourdimensional mapping. We consider here, as a model problem, the mapping given by the following equations: $$\left\{ \begin{gathered} x_1 = x_0 + a_1 {\text{ sin (}}x_0 {\text{ + }}y_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{)}} \hfill \\ y_1 = x_0 {\text{ + }}y_0 \hfill \\ z_1 = z_0 + a_2 {\text{ sin (}}z_0 {\text{ + }}t_0 {\text{)}} + b{\text{ sin (}}x_0 {\text{ + }}y_0 {\text{ + }}z_{\text{0}} {\text{ + }}t_{\text{0}} {\text{) (mod 2}}\pi {\text{)}} \hfill \\ t_1 = z_0 {\text{ + }}t_0 \hfill \\ \end{gathered} \right.$$ We have found that as soon asb≠0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).     

Rational approximations for a Hamiltonian System with three degrees of freedom

We develop a new method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. The idea of the method is to express the solution of the nonlinear ODE in the formx=N/D ⁿ , whereN andD are Fourier series andn is an appropriate constant. We apply this method to a galactic potential with three degrees of freedom.Paper presented at the 11th European Regional Astronomical Meetings of the IAU on New Windows to the Universe, held 3–8 July, 1989, Tenerife, Canary Islands, Spain.     

Fourier analysis of local Lyapunov characteristic exponents for satellite-type motions

For a binary star system with equal masses the transition from regular to stochastic motion has been determined by means of the Lyapunov Characteristic Numbers (LCN) in a paper of Gonczi and Froeschlé (1981), where the initial distance of the massless body to the second primary was used as parameter of integration. This time we analyse the Fourier spectra of the local LCNs (ln _k ⁱ ) of this system for different intervals of time.     

Mutual Potential of Homogeneous Polyhedra

The mutual gravitational potential between a pair of homogeneous polyhedra is expressed using an infinite series. The nested volume integrals are evaluated analytically and result in simple tensor expressions containing no special functions. However, complexity increases as O(6 ⁿ ), where n is the term degree. An alternate formulation due to Liebenthal is also presented.     

Sur une generalisation des equations lagrangiennes permettant une extension de la transformation KS

Résumé Cet arricle est composé de deux parties, l'une établie par M. Langlois, l'autre par Mme Losco. La première partie est consacrée à l'étude des équations de Poincaré qui sont les équations de Lagrange du mouvement lorsqu'on introduit des pseudo-paramètres. Une application de ces équations est intéressante à envisager lorsque l'on mélange coordonnées et pseudo-paramètres et que les coordonnées sont ignorables dans le lagrangien. On établit alors un théorème de réduction des équations du mouvement par des relations invariantes. La transformation KS entre dans ce cadre. La seconde partie concerne la construction de matrices généralisant KS. Ce sont des matrices dont les premières lignes définissent des variablesQ , les dernières lignes des pseudo-paramètres et pour lesquelles on peut appliquer le théoreme de réduction établi précédemment.Le mouvement général du corps solide dansR ⁿ permet une construction de telles matrices, de même que KS est associée à une rotation deR ⁿ.

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This paper is composed of two parts, the first one established by M. Langlois, the other one by L. Losco. First a study of Poincaré's equations is made, which are Lagrangian equations where use is made of some quasi-coordinates. One application of these equations is very interesting when some coordinates are ignorable in the Lagrangian. A theorem of reduction is obtained with invariant relations. KS is of this kind. Then matrices are constructed which generalize KS. There are matrices of coordinates and quasi-coordinates, which allow application of the theorem of reduction previously obtained. The general motion, helicoidal motion, of a rigid body inR ⁿ-space allows to obtain such matrices, just as KS corresponds to a rotation inR ⁴.Some results have been briefly published in two notes mentioned at the end.

     

Les equations aux variations du probleme de Störmer

Résumé Une théorie a déjà été établie [3] concernant tout système lagrangienL(q, ,t) qui possède des intégrales premières ou plus généralement des formes invariantes, provenant par, exemple d'invariances géométriques. Cet article est une application concrète et directe aux équations aux variations du problème de Störmer qui intéressent actuellement des chercheurs en Mécanique [4].

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The variational equations of Störmer's problem

A theory has already been established [3] concerning all lagrangiansL(q, ,t) which possess the integrals or more generally invariant forms, originating for example from geometric invariances. This paper is a direct application to the variational equations of Störmer's problem that has captured the interest of many researchers in celestial mechanics [4].

     

The photoelectric elements of AS Camelopardalis

The classical situation of dynamical systems with two degrees of freedom is completely modified by the introduction of a stochastic element which provides a slow dissolution of the zone of stability. Using a Fokker-Planck method it has been shown that the Gambler's ruin problem is a rather good approximation of the diffusion process. Furthermore the orbits exhibit aC-system character even during the diffusion process.     

The paschen discontinuity in B stars

Measurements of the Paschen discontinuity in stars withT _eff10⁴K leads to the conclusion that theD _P/D_B ratio increases with temperature faster than expected. The increase ofD _P/D_B with (logg)^–1 is also steep.

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Résumé Les mesures de la discontinuité de Paschen dans les étoiles deT _eff10⁴K nous montrent que le rapportD _P/D_B augmente avec la température plus rapidement que prevu. L'augment deD _P/D_B avec (logg)^–1 est aussi remarquable.

     

MESSENGER observations of the plasma environment near Mercury

The MESSENGER Fast Imaging Plasma Spectrometer (FIPS) measured the bulk plasma characteristics of Mercury's magnetosphere and solar wind environment during the spacecraft's first two flybys of the planet on 14 January 2008 (M1) and 6 October 2008 (M2), producing the first measurements of thermal ions in Mercury's magnetosphere. In this work, we identify major features of the Mercury magnetosphere in the FIPS proton data and describe the data analysis process used for recovery of proton density (n_p) and temperature (T_p) with a forward modeling technique, required because of limitations in measurement geometry. We focus on three regions where the magnetospheric flow speed is likely to be low and meets our criteria for the recovery process: the M1 plasma sheet and the M1 and M2 dayside and nightside boundary-layer regions. Interplanetary magnetic field (IMF) conditions were substantially different between the two flybys, with intense reconnection signatures observed by the Magnetometer during M2 versus a relatively quiet magnetosphere during M1. The recovered ion density and temperature values for the M1 quiet-time plasma sheet yielded n_p∼1–10 cm⁻³, T_p∼2×10⁶ K, and plasma β∼2. The nightside boundary-layer proton densities during M1 and M2 were similar, at n_p∼4–5 cm⁻³, but the temperature during M1 (T_p∼4–8×10⁶ K) was 50% less than during M2 (T_p∼8×10⁶ K), presumably due to reconnection in the tail. The dayside boundary layer observed during M1 had a density of ∼16 cm⁻³ and temperature of 2×10⁶ K, whereas during M2 this region was less dense and hotter (n_p∼8 cm⁻³ and T_p∼10×10⁶ K), again, most likely due to magnetopause reconnection. Overall, the southward interplanetary magnetic field during M2 clearly produced higher T_p in the dayside and nightside magnetosphere, as well as higher plasma β in the nightside boundary, ∼20 during M2 compared with ∼2 during M1. The proton plasma pressure accounts for only a fraction (24% for M1 and 64% for M2) of the drop in magnetic pressure upon entry into the dayside boundary layer. This result suggests that heavy ions of planetary origin, not considered in this analysis, may provide the “missing” pressure. If these planetary ions were hot due to “pickup” in the magnetosheath, the required density for pressure balance would be an ion density of ∼1 cm⁻³ for an ion temperature of ∼10⁸ K.     

The behaviour of certain helium lines in the P Cygni spectrum

Sixty-two spectrograms of the supergiant P Cyg, obtained with the Coudé spectrograph of the 2 m RCC telescope at NAO (National Astronomical Observatory, Bulgaria) were used in order to study the behaviour (we mean the displacements of the absorption components only) of the Hei spectrum of this star. Twelve lines from four series (2³ P ⁰–n³S, 2³P⁰ –n ³ D, 2 ¹ P ⁰ –n ¹ D, and 2¹ S–n ¹ P ⁰) were examined. It was established that all observed cases can be reduced to a few (three or four) which are typical. These cases are stated in the form of the comparison between the hydrogen and helium lines. All profiles were rectified and their complex structure was taken into consideration. The fixed peculiarities were explained by an optical depth effect.     

Sur l'existence de certaines configurations d'equilibre relatif dans le probleme desN corps

Résumé On sait que les positions d'équilibre relatif dans le problème des trois corps, où les corps se trouvent aux sommets d'un triangle équilatéral, existent lorsque les masses sont quelconques; tandis que pourn=4 (voir [3]) et pourn=5 (voir [4]), les positions d'équilibre relatifs, où les corps se trouvent aux sommets d'un polygone régulier de n cotés, existent seulement si les masses sont égales. L'objet de cet article est de montrer que ce dernier résultat est vrai pour toutn4.

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It is known that in the three body problem, the equilateral configuration of relative equilibrium exists for all values of masses, while in then-body problem, forn=4 (see [3]) andn=5 (see [4]), the position of relative equilibrium where the bodies are at the vertices of a regular polygon withn sides, exists only if the masses are equal.We prove in this paper, that this last result is true for alln4.

     

Wave reflection and wave disorder in the solar transition zone and corona

The reflection coefficient for sound or Alfvén waves reaching the transition zone is evaluated. A family of temperature profiles, including T ^5/2 dT/dz = constant, permits analytical solutions for the velocity and yields the reflection coefficient as a function of both the wavelength and the temperature jump across the zone. When the temperature jump is large, even waves appreciably shorter than the zone thickness are reflected efficiently.Wave reflection disorders the waves in and below the transition zone, because rising waves there interact with reflected waves in a manner more similar to turbulence than to shock steepening.The distribution in directions of hydromagnetic waves is determined by the non-uniformity of their sources. Most inhomogeneities in the wave source cause the waves to resemble isotropic fastmode waves more than Alfvén waves. This places severe restrictions on possible sources of Alfvén waves.     

On Szebehely's problem for holonomic systems involving generalized potential functions

We consider the problem of finding the generalized potential function V = U _i(q ¹, q ²,..., q ⁿ)q ⁱ + U(q ¹, q ²,...;q ⁿ) compatible with prescribed dynamical trajectories of a holonomic system. We obtain conditions necessary for the existence of solutions to the problem: these can be cast into a system of n – 1 first order nonlinear partial differential equations in the unknown functions U ₁, U ₂,...;, U _n, U. In particular we study dynamical systems with two degrees of freedom. Using adapted coordinates on the configuration manifold M ₂ we obtain, for potential function U(q ¹, q ²), a classic first kind of Abel ordinary differential equation. Moreover, we show that, in special cases of dynamical interest, such an equation can be solved by quadrature. In particular we establish, for ordinary potential functions, a classical formula obtained in different way by Joukowsky for a particle moving on a surface.Work performed with the support of the Gruppo Nazionale di Fisica Matematica (G.N.F.M.) of the Italian National Research Council.     

Sur l'instabilite de certaines positions d'equilibre relatif dans le probleme des n corps

Resumé On démontre dans cet article l'instabilité, pour tout n 4, des configurations d'équilibre relatif dans le problème des n corps, oú les n corps soumises aux attractions newtonniennes mutuelles se trouvent aux sommets d'un polygone régulier de n cotés. La preuve consiste à montrer que les équations aux variations, projetées sur le plan P des n corps, possèdent au moins deux exposants caractéristiques complexes connugués dont la parr'e réelle est strictement positive; alors que ces equations projetées sur un axe orthogonal à P possèdent des solutions ayant des termes séculaires.

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We prove in this paper the instability, for all n 4, of the configurations of relative equilibrium in the n-body problem where the n bodies submitted to newtonian mutual attractions are at the vertices of a regular polypon with n sides. For this proof we show that the equations of variations projected to the n bodies plan P have at least two conjugate characteristic exponents with a strictly positive real part; while these equations projected to an orthogonal axis to P have some solutions with secular terms.

     

Lois exponentielles de distance pour les systèmes de satellites

Résumé Une formulation exponentielle de la loi empirique de Titus-Bode a été proposée par Basano et Hugues. Ces auteurs introduisent l'hypothèse de trois planètes manquantes ou trous. Toutes les planètes obéissent à la relation a _n = ⁿ qui donne les demi-grands axes a des planètes pour des valeurs entières de n.Nous proposons une nouvelle méthode qui permet de retrouver la relation de Basano et Hugues pour le système solaire. Nous appliquons cette méthode aux systèmes de satellites de Jupiter, Saturne et Uranus en introduisant des trous pour combler les lacunes dans les séquences de satellites. Nous en tirons trois relations exponentielles de distance, analogues à la relation de Basano et Hugues. Nous constatons que les coefficients sont semblables pour les systèmes solaire, jovien et uranien alors que le coefficient du système de Saturne vaut approximativement la racine carrée des trois autres .Nous expliquons cet espacement exponentiel grâce à un modèle simple d'une nébuleuse gazeuse initiale soumise à de petites perturbations qui engendrent des oscillations dans la distribution de densité. Les minima de la densité perturbée sont donnés par les zéros des fonctions de Bessel décrivant la propagation de la perturbation. Les positions des maxima correspondent aux sites d'accrétion.Tous les trous introduits dans les parties intérieures des systèmes de satellites sont comblés par les anneaux et petits satellites. Dans le système d'Uranus, il reste deux trous vacants qui pourraient être occupés par des petits satellites non encore découverts.

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Exponential distance laws for satellite systems

A revised Titius-Bode law for the Solar system was proposed by Basano and Hugues, by introducing three missing planets. This law can be written a _n = ⁿ (with = 0.2853 AU and = 1.5226), which gives the distances a _n of the nth planet for successive integers n.We propose a new method to find this Basano-Hugues law for the Solar system. Based upon the comparison of the ratios of successive distances, this method can be applied to the satellite systems of the three giants planets Jupiter, Saturn and Uranus by introducing missing satellites to fill the gaps in satellites sequences. We find three exponential distance relations, similar to that of Basano-Hugues. We note that the coefficients for the Solar, Jovian and Uranian systems are almost equal while the Saturnian system's coefficient is nearly the square root of that of the three others.We explain that exponential spacing by a simple model of an initial gaseous nebula subject to small perturbations generating oscillations in the density distribution. The minima of the perturbed density are given by the zeros of Bessel functions describing the perturbation propagation. The maxima positions correspond to accretion sites.All the empty places in the inside parts of satellite systems are occupied by rings and small satellites. In the Uranian system, there are two empty places which could be filled by new undiscovered small satellites.

     

On the number of effective integrals in galactic models

Three different numerical techniques are tested to determine the number of integrals of motion in dynamical systems with three degrees of freedom.

1. The computation of the whole set of Lyapunov Characteristic Exponents (LCE).
2. The triple sections in the configurations space.
3. The Stine-Noid box-counting technique.

These methods are applied to a triple oscillator with coupling terms of the third order. Cases are found for which one effective integral besides the Hamiltonian subsists during a very long time. Such orbits display simultaneously chaotic and quasi-periodic motion, according to which coordinates are considered. As an application, the LCE procedure is applied to a triaxial elliptical galaxy model. Contrary to similar 2-dimensional systems, this 3-dimensional one presents noticeable zones in the phase space without any non-classical integral.