Un formulaire pour le calcul des perturbations d'ordres élevés dans les problèmes planétaires

The authors present formulas in compact form for constructing high order planetary perturbations with respect to the disturbing masses. They have been built by an iterative process and give the variations of osculating elements. Singularities due to vanishing eccentricities and inclinations are not present in the differential equations. All elementary operations are manipulations of Fourier series with numerical coefficients, and great care has been taken to economize algebraic operations. Results are presented in three forms:

1. vectorial form, with real components which may be useful in numerical integrations;
2. complex form, to put in evidence the symmetries of the system of variables;
3. scalar form, which is the most elaborate. This last form has been used for constructing the first order perturbations for any pair of planets. Two illustrations are given (Jupiter and Saturn, Venus and Earth). Further remarks are made about the practical manipulation of Fourier series, resolution of Kepler's equation in complex form and construction by iteration of the inverse of the distance between two bodies.

     

Astrometric Observations of the Galilean Moons of Jupiter at the Pulkovo Normal Astrograph in 2016−2017

We present the results of observations of the Galilean moons of Jupiter carried out at the Normal Astrograph of the Pulkovo Observatory in 2016?2017. We obtained 761 positions of the Galilean moons of Jupiter in the system of the Gaia DR1 catalog (ICRF, J2000.0) and 854 differential coordinates of the satellites relative to each other. The mean errors in the satellites’ normal places and the corresponding root-mean-square deviations are ε_α = 0.0020′′, ε_δ = 0.0027′′, σ_α = 0.0546′′, and σ_δ = 0.0757′′. The equatorial coordinates of the moons are compared to the motion theories of planets and satellites. On average, the (O–C) residuals in the both coordinates relative to the motion theories are less than 0.031′′. The best agreement with observations is achieved by a combination of the EPM2015 and V. Lainey-V.2.0|V1.1 motion theories, which yields the average (O–C) residuals of approximately 0.02″. Peculiarities in the behavior of the (O–C) residuals and error values in Ganymede have been noticed.     

The Liège Oscillation code

The Liège Oscillation code can be used as a stand-alone program or as a library of subroutines that the user calls from a Fortran main program of his own to compute radial and nonradial adiabatic oscillations of stellar models. We describe the variables and the equations used by the program and the methods used to solve them. A brief account is given of the use and the output of the program.     

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.

---

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.

     

Le problème desn corps et les invariants intégraux

This paper is the text of a course offered at Cortina d'Ampezzo (NATO Advanced Study Institute-Long time prediction in dynamics 1975). The integral invariants were introduced by H. Poincaré and E. Cartan. This mathematical notion is very tractable by the use of exterior algebra of different forms. Work at the University of Besançon introduces again this theory, in Celestial Mechanics. This paper has two parts: mathematical tools, then an application to then-body problem where, by the use of the 11th local integral, I found some new equations and developed them for the collinear triple collision of three bodies.     

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.

---

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.

     

Mesure des Raies de [Sii] dans Trois Nébuleuses planétaires de L'hémisphère Austral

Three southern planetary nebulae (NGC 2818, He 2-130, and NGC 3132) have been observed with the IDS (Image Dessector Scanner) combined with the Boller and Chivens spectrograph mounted at the Cassegrain focus of the 1.52 m telescope of the ESO in Chile. The spectrograph dispersion was 60 Å mm^–1 in the spectral range 6170–7298 Å. The slit aperture was 4×4. Spectra were obtained from an array of positions across each nebula along the E-W direction and/or N-S direction. The data reduction followed the standard IHAP routines for IDS observations. In order to derive electron density, only the [Sii] lines (6617 Å–6731 Å) are given in this paper. The results are in agreement with a shell structure for the observed nebulae.

---

---

Toutes les observations ont été faites à l'Observatoire européen Austral (ESO) au Chili.     

Minima de L'intégrale D'action du Problème Newtoniende 4 Corps de Masses Égales Dans R3: Orbites 'Hip-Hop'

We consider the problem of 4 bodies of equal masses in R ³ for the Newtonian r⁻¹ potential. We address the question of the absolute minima of the action integral among (anti)symmetric loops of class H ¹ whose period is fixed. It is the simplest case for which the results of [4] (corrected in [5]) do not apply: the minima cannot be the relative equilibria whose configuration is an absolute minimum of the potential among the configurations having a given moment of inertia with respect to their center of mass. This is because the regular tetrahedron cannot have a relative equilibrium motion in R ³ (see [2]). We show that the absolute minima of the action are not homographic motions. We also show that if we force the configuration to admit a certain type of symmetry of order 4, the absolute minimum is a collisionless orbit whose configuration ‘hesitates’ between the central configuration of the square and the one of the tetrahedron. We call these orbits ‘hip-hop’. A similar result holds in case of a symmetry of order 3 where the central configuration of the equilateral triangle with a body at the center of mass replaces the square. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Non-Gravitational Perturbations impact on the BepiColombo Radio Science Experiment and the key rôle of the ISA accelerometer: direct solar radiation and albedo effects

The paper is focused on the estimate of the impact of the non-gravitational perturbations on the orbit of the Mercury Planetary Orbiter (MPO), one of the two spacecrafts that will be placed in orbit around the innermost planet of the solar system by the BepiColombo space mission. The key rôle of the Italian Spring Accelerometer (ISA), that has been selected by the European Space Agency (ESA) to fly on-board the MPO, is outlined. In the first part of the paper, through a numerical simulation and analysis we have estimated, over a time span of several years, the long-period behaviours of the disturbing accelerations produced by the incoming direct solar radiation pressure, and the indirect effects produced by Mercury’s albedo. The variations in the orbital parameters of the spacecraft, together with their spectral contents, have been estimated over the analysed period. The direct solar radiation pressure represents the strongest non-gravitational perturbation on the MPO in the very complex radiation environment of Mercury. The order-of-magnitude of this acceleration is quite high, about 10^?6 m/s², because of the proximity to the Sun and the large area-to-mass ratio of the spacecraft. In the second part of the paper, we concentrated upon the short-period effects of direct solar radiation pressure and Mercury’s albedo. In particular, the disturbing accelerations have been compared with the ISA measurement error and the advantages of an on-board accelerometer are highlighted with respect to the best modelling of the non-gravitational perturbations in the strong radiation environment of Mercury. The readings from ISA, with an intrinsic noise level of about $10^{-9}\,m/s^{2}/\sqrt{Hz}The paper is focused on the estimate of the impact of the non-gravitational perturbations on the orbit of the Mercury Planetary Orbiter (MPO), one of the two spacecrafts that will be placed in orbit around the innermost planet of the solar system by the BepiColombo space mission. The key r?le of the Italian Spring Accelerometer (ISA), that has been selected by the European Space Agency (ESA) to fly on-board the MPO, is outlined. In the first part of the paper, through a numerical simulation and analysis we have estimated, over a time span of several years, the long-period behaviours of the disturbing accelerations produced by the incoming direct solar radiation pressure, and the indirect effects produced by Mercury’s albedo. The variations in the orbital parameters of the spacecraft, together with their spectral contents, have been estimated over the analysed period. The direct solar radiation pressure represents the strongest non-gravitational perturbation on the MPO in the very complex radiation environment of Mercury. The order-of-magnitude of this acceleration is quite high, about 10⁻⁶ m/s², because of the proximity to the Sun and the large area-to-mass ratio of the spacecraft. In the second part of the paper, we concentrated upon the short-period effects of direct solar radiation pressure and Mercury’s albedo. In particular, the disturbing accelerations have been compared with the ISA measurement error and the advantages of an on-board accelerometer are highlighted with respect to the best modelling of the non-gravitational perturbations in the strong radiation environment of Mercury. The readings from ISA, with an intrinsic noise level of about in the frequency band of 3·10⁻⁵–10⁻¹ Hz, guarantees a very significant reduction of the non-gravitational accelerations impact on the space mission accuracy, especially of the dominant direct solar radiation pressure.     

Etude photometrique d'un renforcement diffus observé dans la lumière zodiacale, à une distance de 100 R⊙ du soleil

A photograph of the zodiacal light obtained at the Pic du Midi Observatory is studied in order to measure, in absolute units, the brightness of the reinforcement, observed 15° above the ecliptic plan and in a distance of 100R⊙ from the Sun.The obtained brightnesses are compared to the brightness of the zodiacal light given by other authors for the elongations ? ? [23°, 40°]. The calibration of the image was made using the stars in the field of the image and isophotes corrected for extinction were obtained, by the method of isodensities.A discussion of the obtained results is made and the origin of the reinforcement is investigated. The mass evaluation of the interplanetary particles producing this reinforcement has been estimated and permits to us to conclude that it may be due to particles evaporated from the circumsolar region. The mechanism of transfer of momentum to the particles in orbit around the Sun by a convecting ma magnetic field is not elucidated.