The restricted problem of a tri-axial rigid body and two spherical bodies with variable masses

The restricted problem of a tri-axial rigid body and two spherical bodies with variable masses be considered. The general solution of the equations of motion of the tri-axial body be obtained in which the motion of the spherical bodies is determined by the classic nonsteady Gyldén-Meshcherskii problem.     

Existence of particular solutions in the generalised restricted problem of three bodies with variable masses

The restricted problem of three bodies with variable masses is considered. It is assumed that the infinitesimal body is axisymmetric with constant mass and the finite bodies are spherical with variable masses such that the ratio of their masses remains constant. The motion of the finite bodies are determined by the Gyldén-Meshcherskii problem. It is seen that the collinear, triangular, and coplanar solutions not exist, but these solutions exist when the infinitesimal body be a spherical.     

Solar atmospheric heating

The solar atmosphere is heated by a flux of mechanical waves propagating in one or more of the modes: acoustic, Alfvén and gravitational.The acoustic theory is compared with observational data and found inadequate. First, the theory is based quantitatively on the Böhm-Vitense convection zone model, and large-scale convective motions (supergranulation) and magnetic fields (unipolar regions) show that convection has another form. On the other hand, when granular motions are invoked the energy flux is too small. Second, atmospheric heating is localized in faculae, and enhanced acoustic flux beneath these regions is no longer explicable. Finally, the short periods of 10–30 s invoked recently appear inexplicable. Objections to the gravitational wave heating process are given briefly.Previous objections to Alfvén waves as an energy source followed from the belief that fields were generally uniform and of strength 50 G, now known to be incorrect. Models of Alfvén wave generation are based on (i) granule eddy motions, (ii) overstable oscillations in subsurface flux tubes and sunspot flux ropes, and (iii) supergranule motions, both horizontal and vertical.The first provides waves which propagate along thin flux tubes oscillating as taut wires in a compressible fluid; they may explain mottles, fibrils and other small emission features. The second may explain the enormous dissipation in spot groups, including flares. The third has been invoked earlier to explain spicules, and may have effects in the solar wind.     

Sur les petits diviseurs

On montre la convergence d'une modification des séries de Zeipel aux fréquences fixées d'avance qui tiennent compte de perturbations à courte période dans les seconds membres des équations différentielles de la méchanique céleste. À l'aide de ces séries on établit l'existence des solutions quasipériodiques résonantes n'exigeant ancunes paramètres complémentaires.The convergence of a modification of the Zeipel's series of fixed frequences taking into account the short period perturbations of the right hand membres of the differential equations of celestial mechanics has been shown. By means of these series the existence of quasi-periodic solutions depending on the degenerate spectre of frequences without any complementary parametres has been established.     

Evolution d'attitude d'un satellite dans le champ magnetique terrestre

Résumé On donne une représentation possible de l'attitude d'un satellite aimanté dans le champ magnétique terrestre. Le moment magnétique du satellite est de l'ordre de 10 amp m² et la rotation du satellite est voisine de 1 tour/mn. Dans ces conditions, le couple magnétique agissant sur le satellite ne peut plus être considéré comme un couple perturbateur.Dans la mesure où l'on peut admettre que l'axe de rotation du satellite, est fixe dans un repère lié au corps, on est conduit à résoudre un système d'équations différentielles linéaires à coefficients périodiques. Les coefficients contenant le temps ont en facteur une quantité 0<<1/3. On peut développer la solution suivant les puissances de les coefficients de ces développements sont donnés par des formules de récurrence. Les séries convergent en général; au voisinage des points singuliers le rayon de convergence peut être étudié.L'hypothése ci-dessus conduit à une représentation convenable de l'attitude du satellite lorsque la rotationr ₀ n'est pas trop faible. Avec la valeur adoptée pour le moment magnétique, la valeur minimum de la rotationr ₀ est de l'ordre de 0.8 t/mn.

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A possible representation of the motion of a satellite about its centre of mass is investigated. The satellite's magnetic moment is of the order of 10 Amp m² and its spin is about 1 rpm. Under these conditions, the magnetic torque acting on the, satellite can no longer be treated as simple perturbation.In the case where the axis of the satellite's rotation is assumed to be a constant in a system of axes fixed to the satellite, a system, of linear differential equations with periodic coefficients has to be solved. The time dependant coefficients have the small parameter 0<<1/3 as factor. The solution is expanded in power series of the parameter . The coefficients of these, series are given by recurrent formulas. The series generally do converge; in the vicinity of the singular points, the radius of convergence can be studied. The given hypothesis leads to a reasonable representation of the satellite's motion if its rotationr ₀ is not too slow. With the adopted value for the magnetic moment, the minimum value ofr ₀ is of the order of 0.8 rpm.

     

Alfvén waves in the solar atmosphere

The nonlinear propagation of Alfvén waves on open solar magnetic flux tubes is considered. The flux tubes are taken to be vertical and axisymmetric, and they are initially untwisted. The Alfvén waves are time-dependent axisymmetric twists. Their propagation into the chromosphere and corona is investigated by solving numerically a set of nonlinear time-dependent equations, which couple the Alfvén waves into motions parallel to the initial magnetic field (motion in the third coordinate direction is artificially suppressed). The principal conclusions are: (1) Alfvén waves can steepen into fast shocks in the chromosphere. These shocks can pass through the transition region into the corona, and heat the corona. (2) As the fast shocks pass through the transition region, they produce large-velocity pulses in the direction transverse to B _o. The pulses typically have amplitudes of 60 km s^–1 or so and durations of a few tens of seconds. Such features may have been observed, suggesting that the corona is in fact heated by fast shocks. (3) Alfvén waves exhibit a strong tendency to drive upward flows, with many of the properties of spicules. Spicules, and the observed corrugated nature of the transition region, may therefore be by-products of magnetic heating of the corona. (4) It is qualitatively suggested that Alfvén waves may heat the upper chromosphere indirectly by exerting time-dependent forces on the plasma, rather than by directly depositing heat into the plasma.     

Sur les petits diviseurs I. (complement). Les series de V. Zeipel [on the small divisors. I. (complement). the V. Zeipel's series]

Résumé On démontre ici la convergence des séries de v. Zeipel excluant les termes à courte période de la fonction perturbatrice.

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The convergence of the v. Zeipel's series excluding the short periodic terms of the perturbation function has been established.

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Cette remarque est due à M.N.N. Vassiliéff, à qui l'auteur expresse ici sa reconnaissance.     

Contribution to an analytical lunar theory

During the last year, the project of an analytical lunar theory was revived. All programs for computations on literal trigonometric series with rational coefficients were rewritten for an IBM 360-65 with new principles as far as the internal structure of series is concerned. The computational loops have been programmed and results of the calculations for the first loops will be presented. The method is essentially based on the theory derived by J. Chapront and L. Mangeney.To prepare the successive approximations, all the expressions are computed by incremental formulae. A special device has been prepared in order to keep the denominators only in those terms where their development in power series affects the convergence of the coefficients.

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Résumé Nous avons repris l'année dernière le projet d'une thèorie analytique de la Lune. Tous les programmes de calculs sur des séries trigonométriques littérales à coefficients rationnels ont été réécrits pour un IBM 360-65, avec de nouveaux principes quant à la structure interne des séries. Les boucles de calcul ont été programmées et nous donnons les premiers résultats. La méthode repose sur la théorie due à J. Chapront et L. Mangeney.Pour les approximations suivantes les expressions sont calculées par accroissements. Chaque fois que le développement en série du dénominateur d'un coefficient affecte la convergence de celui-ci, il est conservé.

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Communication presented at the conference on Lunar Dynamics and Observational Coordinate Systems held January 15-17, 1973 at the Lunar Science Institute, Houston, Tex., U.S.A.     

Theorie analytique programmee de la libration physique de la lune

Résumé La rotation de la Lune autour de son centre de gravité est traitée par une méthode analytique, en tenant compte de son mouvement orbital. On développe une théorie Hamiltonienne en utilisant les variables d'Andoyer et l'on démontre que les écarts, purement périodiques, à trois relations de résonances similaires aux lois de Cassini, sont les variables canoniques du problème. Le potentiel est exprimé dans ces nouvelles coordonnées et l'Hamiltonien est développé jusqu'au deuxième degré en les petites variables. Un système d'équations donne le vrai centre de libration qu'on trouve proche du centre défini par les lois de Cassini. Un second système, résolu par un processus d'iterations, donne les séries de la libration, analytiques par rapport aux constantes du potentiel de la Lune et trigonométriques en les arguments de Delaunay. La question de convergence est brièvement abordée, mais sans démonstration.

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The rotation of the Moon about its center of mass, taking into account the orbital motion, is treated analytically. A Hamiltonian theory is developed in terms of the Andoyer variables. The periodic parts of departures from three resonances, equivalent to Cassini's laws, are found to be the canonical variables of the problem. The potential is expressed as a function of these new coordinates and the whole Hamiltonian is developed to the second degree in these small variables. One system of equations gives the real center of libration which is found to be near the center defined by Cassini's laws. A second system solved by iterations, gives the libration as analytical series in the constants of the Moon's potential, and trigonometric series in Delaunay arguments. The question of convergence is briefly exposed without any demonstration.

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Ce travail a été soutenu par une bourse du Centre National d'Etudes Spatiales.     

Alfvén waves in sunspots

The propagation of Alfvén waves in a simple model of a sunspot is considered. The vertical structure near the center of the umbra is modelled realistically, but the horizontal structure is not considered. The full wave equation is solved, without recourse to the WKB approximation. Only wave propagation in the vicinity of the central field line in an axially symmetric spot is examined, and it is assumed that this field line is open. By taking wave reflections into account, we find that the observations of non-thermal motions near the temperature minimum (Beckers, 1976) and in the corona (Beckers and Schneeberger, 1977) are both consistent with an upward-propagating Alfvénic energy flux density of a few times 10⁷ erg cm^–2 s^–1. This flux density is too small to cool the sunspot, but it is large enough to supply the energy requirements of the transition region and corona above a sunspot. This conclusion depends on the assumptions that the observed motions are indeed Alfvénic with periods near 180 s.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

The nature of the sunspot phenomenon

The properties of Alfvén waves in a vertical column of field are pointed out as a guide in treating the complicated problem of overstability. There are internal Alfvén waves of arbitrary form propagating along the magnetic field, without disturbing the fluid outside the column. There are also surface waves which involve the fluid both inside and outside the column of field. The surface waves propagate at a speed less than the Alfvén speed.Convective forces couple the internal and external fluid motions. If the forces are not too strong, the identity of the modes, as internal waves or surface waves, is maintained. The surface waves are unstable and, we suggest, may contribute to some of the activity of a sunspot. We suggest that the internal Alfvén wave modes are of more central interest for producing the basic sunspot phenomena. They represent the degenerate case, and their form is worked out in some detail. The overstable Alfvén wave modes peak sharply near the outer edge of the field, and do not strongly disturb the fluid outside. We suggest that this effect contributes to the sharp edge of the sunspot umbra.Recent observations by Giovanelli show intense wave activity originating inside the edge of the umbra. We tentatively identify the activity with the peak in the overstable modes within the umbra.This work was supported in part by the National Aeronautics and Space Administration under Grant NGL 14-001-001.     

Periodic solutions in Hill's relativistic problem

In this article the existence of periodic solutions in Hill's relativistic problem is demonstrated using Poincaré's small parameter method. This method guarantees the convergence of the series representing the periodic solutions.     

Nonthermal Velocities in the Solar Transition Zone and Corona

Nonthermal velocities are presented for spectral lines covering the temperature range 10 4–10 6 K, measured from high-spectral-resolution data for several solar features observed at the limb by the high resolution telescope and spectrograph (HRTS), including a coronal hole, quiescent regions and several small-scale active regions. These results are compared with predictions based on acoustic waves and heating via Alfvén waves. It is likely that more than one mechanism is operating simultaneously, in particular, resonant Alfvén wave heating, which is very sensitive to background plasma motions.     

On Coronal Loop Heating by Torsional Alfvén Waves

Heating of coronal loops by linear resonant Alfvén waves, excited by the footpoints motions in the photosphere, has been studied. The analysis of single-layer heating is extended to multilayer heating, in semiempirical treatment. Heating and nonthermal velocities in different layers of loops in X-ray bright points, active regions, and large-scale structures are estimated. The average value of velocity is found to be in agreement with the observations.     

Particle acceleration and the decay of soft X-ray non-thermal line broadening in solar flares

The relationship between the production of -ray emitting particles and non-thermal soft X-ray line broadening is investigated. A model of particle acceleration via the stochastic interaction with MHD turbulence is assumed and the time development of the wave energy density derived under the condition of energy conservation between waves and particles. The inferred numbers and energy distribution of accelerated protons for four -ray flares are used to define the wave energy density and its temporal development. The presence of Alfvén wave turbulence is considered as the source of the non-thermal motions in the ambient plasma. These motions are observed as excess widths in the soft X-ray line emission from these events. The decay of the waves via the particle acceleration process is compared with the observed decays of this non-thermal line broadening. Our results show that both the -ray emission and excess soft X-ray line widths in these flares can be explained by the single physical phenomenon of Alfvén wave turbulence.     

The 3/2 resonance

The perturbation method for two critical arguments is applied to the 3/2 resonance between Jupiter's and the Asteroid's mean motions. The radius of convergence of the series in eccentricity is smaller than in the 2/1 case and the chaotic regions in the plane (a,e) are less important.     

Sur la stabilite de certains mouvements particuliers d'un satellite artificiel de la terre soumis aux efforts aerodynamiques

L'auteur considère le mouvement d'un satellite artificiel de la terre évoluant suffisamment près de celle-ci pour qu'on ait à tenir compte des efforts aérodynamiques et en admettant que l'air est un fluide parfait incompressible en mouvement irrotationnel.Il met le problème en équations et, dans le cas d'un satellite de révolution, démontre l'existence de mouvements particuliers où le centre de gravité du satellite a un mouvement circulaire uniforme le satellite tournant uniformément autour de son axe perpendiculaire au plan du cercle. Il donne des conditions suffisantes de stabilité et d'instabilité de ces mouvements au moyen de la méthode de Liapounoff.

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The author considers the motion of an artificial satellite of the Earth revolving sufficiently near it so that the aerodynamic forces can be taken into account. It is supposed that air is a perfect incompressible fluid in irrotational motion.The problem is posed in the form of equations and proves the existence of particular motions in the case of a satellite of revolution when the centre of gravity of the satellite has uniform circular motion, the satellite revolving uniformly around its axis, perpendicular to the plan of the circle.Sufficient conditions for stability and instability of particular motions are given by the method of Liapunov.

     

Influence du champ magnétique terrestre sur le mouvement d'un satellite autour de son centre de gravité

Résumé On étudie l'effet du champ magnétique terrestre sur le mouvement d'un satellite autour de son centre de gravité. Le satellite possède une symétrie dynamique et un moment magnétique propre dirigé suivant l'un des axes principaux d'inertie; le champ magnétique terrestre est assimilé au champ d'un dipôle dont les pôles coïncident avec les pôles terrestres. On néglige les perturbations de la trajectoire du satellite qui est supposée circulaire. La position du satellite par rapport à son centre de gravité est repérée dans un système d'axes lié au plan de l'orbite et le mouvement est décrit à l'aide des angles d'Euler , , . La symétrie sphérique et le choix du moment magnétique sur l'un des axes d'inertie permettent d'éliminer l'angle .La solution pour et peut se développer en séries de puissance d'un petit paramètre . Les séries convergent pour ||<1.Lorsque le moment magnétique est faible on la rotation du satellite rapide, est faible. Les développements sont calculés effectivement jusqu'à ².La comparaison des résultats avec l'intégration numérique du système d'équations différentielles est satisfaisante.

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The effect of the Earth's magnetic field on the motion of a satellite around its centre of mass is investigated. The satellite is assumed to be dynamically symmetric and to be magnetized in the same direction as that of a principal axis. The Earth's magnetic field is assumed to be a dipole field whose poles coincide with the rotation poles of the Earth. The satellite's orbit is circular and perturbations are neglected. The position of the satellite with respect to its centre of mass is given with respect to a coordinate system fixed in the orbital plane and the motion is described by Euler's angles , , . The spherical symmetry and the coincidence of the magnetic moment with a principal axis allow one to eliminate the angle .The solution for and , can be expanded in power series for small parameter .The series converge for <1. is small for a small magnetic moment or a high angular velocity of the rotating satellite. The terms of the expansion of the series are calculated up to ².The comparison of the results with those obtained by numerical integration of the differential equation is satisfactory.

     

The periodicity of a charged particle motion in a magnetic dipole field

The author's purpose is to present in this article the evidence of his intuition on the periodic behavior of a charged particle, moving in a magnetic dipole field, that is achieved by adopting the Alfvén's analysis of the motion into the composite of two motions and studying the contribution of these motions into the general behavior of the charged particle, separately.     

Sur la recherche des mouvements stationnaires dans les systèmes ayant des variables cycliques

Résumé Ayant défini la notion de système lié associé à un système mécanique comportant des variables cycliques, on montre que l'ensemble des mouvements stationnaires du système coïncide avec l'ensemble des mouvements stationnaires du système lié. L'étude de la stabilité de ces mouvements montre que si le système lié est stable, il en est de même pour le système initial. La recherche des mouvements stationnaires des gyrostats fournit une application de cette étude.

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A constrained system is associated with a mechanical system having cyclic coordinates. We prove that the set of steady motions of this mechanical system and the set of steady motions of the constrained system are the same. Investigating the stability of these motions, we prove that, if the constrained system is stable, then the mechanical system is also stable. As an example, we consider the problem of the relative equilibrium of gyrostat satellites.