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.     

Tridiurnal variations in cosmic-ray intensity

The tridiurnal wave in cosmic-ray intensity expected from a free space anisotropy is theoretically calculated for different cosmic-ray stations which are characterized by different shapes of asymptotic cones of acceptance. The amplitude A and the time of maximum Tmax are given for latitude dependence of the form cosⁿ λ and rigidity dependence of the form $\text{R}{}^{\mathrm{\beta }}\text{exp (?(R?}\text{1}\text{R}{}_0\text{))}$, where λ and R are the latitude and rigidity respectively and n, β, R₀ are constants. The values of A and Tmax, are calculated for different values of n, β and R₀ for each station. The dependence of A and Tmax on the anisotropy parameters is studied for the proper selection of cosmic-ray stations whose data may be used in determining these parameters.Available experimental data were used to find the observed amplitudes of the tridiurnal variations at five stations using power spectrum analysis with hanning applied on the averaged trains. Minimum variance analysis of the theoretical and experimental amplitudes showed that β has a value between 1 and 2, R₀ greater than 100 GV and n smaller than 3.     

A model for thinning of the plasma sheet

A one-dimensional model for thinning of the plasma sheet is developed on the basis of launching a fast mode MHD rarefaction wave propagating in the tailward direction along the plasma sheet. Behind the rarefaction wave the pressure is reduced, leading to thinning of the plasma sheet and also to an Earthward plasma flow with a speed on the order of the sound speed a₀. The plasma sheet thickness is reduced by a factor of 2 if an Earthward plasma flow speed of 0.8a₀ is induced. The predictions of the model are in reasonable agreement with observations.     

VLF emissions and substorm activity

The resonant interaction between the whistler mode waves and the energetic electrons near the plasmapause boundary has been studied in the presence of field aligned currents which seem to exist during substorm activity. It is shown that the electrons which carry the current along the direction of the magnetic field enhance the whistler mode growth considerably if the streaming velocity is small compared to the phase velocity of the wave. It is likely that this is one of the mechanisms explaining the intense VLF emissions observed near the plasmapause during substorm activity.     

Nonlinear generation of ion loss-cone waves by electrostatic turbulence in the magnetosphere

Effects of plasma turbulence on the stability of electrostatic ion loss-cone waves are examined. The turbulence is assumed to be electrostatic with frequencies near 1.5 times the electron gyrofrequency and the frequencies of the generated waves are below the ion plasma frequency ω_pi>. A nonlinear growth rate of the order of 10^?2ω_pi may be obtained, when the amplitude of the turbulence is 20 mV/m. This is comparable to previously found growth rates of the linear ion loss-cone instability, in a plasma with large pitch angle anisotropy. Bounce averaged pitch angle diffusion coefficients are also presented for different models of the ion loss-cone wave spectrum.     

Distribution of minor constituents in a thermal interface

A non-steady, two-dimensional, compressible mathematical model of a fluid with constant viscosity and thermal conductivity is given in order to represent a thin atmospheric layer located at 60 km height, as a function of a known initial vertical distribution of temperature, chemical concentrations of minor components and assumed free boundary conditions.The purpose of this paper is to determine the influence of convective laminar processes in the vertical distribution of minor constituents in a thermal interface. Only three minor components are considered, atomic oxygen, molecular oxygen and ozone. Numerical solutions of the stated timedependent convection problem for different Rayleigh numbers show a marked influence of the established flow pattern upon the distribution of minor constituents. In all cases, an enhancement of mean concentration gradients in relatively small regions adjacent to the warmer and colder boundaries was observed.     

Energetic protons associated with a forward-reverse interplanetary shock pair at 1 A.U.

A forward-reverse interplanetary shock was observed on 25 March 1969 by the magnetometer and plasma detector on the HEOS-1 satellite. This relatively rare event was described by Chao et al (1972) who concluded that the shock pair was formed at a distance 0.10–0.13 A.U. upstream of the Earth as a result of the interaction between a fast and a slow solar wind streams. Simultaneous observations of 1 MeV solar proton fluxes were also performed on HEOS-1. A characteristic intensity peak was observed as the forward shock passed by the spacecraft. The evolution of the proton intensity, together with a detailed analysis of anisotropies and pitch angle distributions show a complex dynamic picture of the effect of the forward shock on the ambient proton population. Significant changes in particle fluxes are seen to be correlated with fluctuations in the magnetic field. It is suggested that simple geometrical models of shock-associated acceleration should be expanded to include the effect of magnetic fluctuations on particle fluxes. The interaction region limited by the forward and reverse shocks contained a large variety of magnetic fluctuations. Following the tangential discontinuity separating the fast solar wind stream from the preceding slow stream, a sunward flow was observed in the proton data, followed by a small but significant drop in intensity prior to the reverse shock.     

The Jovian surface condition and cooling rate

A theory which is almost fully analytic is used to investigate Jupiter's cooling rate. We find that a simple model of contraction with adiabatic interior structure gives a total cooling time to the present which is in good agreement with the age of the solar system. The interplay between the surface condition and the cooling rate is exhibited and discussed. The current rate of change of the effective temperature is calculated to be ?1°K/0.145×10⁹ yr. Discrepancies with fully numerical investigations of the Jovian age and cooling rate are noted.     

The structure of the atmosphere of Uranus

Models of the interior of Uranus (Podolak, 1976) suggest that the abundances of such substances as CH₄ are greatly enhanced with respect to solar abundances of heavy elements. Such enhancement leads to a new type of model atmosphere for Uranus, which agrees with observation if the internal energy flux is small (?10%) compared with the absorbed solar energy. An important feature of the models is the presence of a cloud of CH₄ droplets whose top is at a temperature of ?90°K and a pressure of ?4atm. Above the cloud, the atmosphere is stable because of the rapid decrease of the thermal flux with depth. Being saturated, most of the observable gaseous CH₄ is near the cloud; the CH₄ abundance above the cloud, of the order of 5 km-am, is a very sensitive function of the cloud-top temperature.     

Axel dust on Saturn and Titan

The scattering and absorption properties of Axel dust were investigated by means of Mie theory. We find that a flat distribution of particle radii between 0 and 0.1 μm, and an imaginary part of the index of refraction which varies as λ^?2.5 produce a good fit to the variation of Titan's geometric albedo with wavelength (λ) provided that τ_ext, the extinction optical depth of Titan's atmosphere at 5000 Å, is about 10. The real part of the complex index is taken to be 2.0. The model assumes that the mixing ratio of Axel dust to gas is uniform above the surface of Titan. The same set of physical properties for Axel dust also produces a good fit to Saturn's albedo if τ_ext = 0.7 at 5000 Å. To match the increase in albedo shortward of 3500 Å, a clear layer (containing about 7 km-am H₂) is required above the Axel dust. Such a layer is also required to explain the limb brightening in the ultraviolet. These models can be used to analyze the observed equivalent widths of the visible methane bands. The analysis yields an abundance of the order of 1000 m-am CH₄ in Titan's atmosphere. The derived CH₄/H₂ mixing ratio for Saturn is about 3.5 × 10^?3 or an enhancement of about 5 over the solar ratio.