Les equations aux variations et la 11e integrale du probleme desN corps

LetH=T(p)?U(q) the Hamiltonian of theN-body problem in cartesian coordinates. In his ‘Nouvelles Méthodes de la Mécanique Céleste’ (tome 1, p. 172) Poincaré observes that the equations leave invariant the differential form $$\alpha = \Sigma (q \delta p + \tfrac{1}{2}p \delta q) - \tfrac{3}{2}t \delta H.$$ This property was rediscovered by Elie Cartan in the framework of his ‘Leçons sur les Invariants Intégraux’ (p. 89, last line). We propose to study some applications of this fact from the point of view of Poincaré, that is: the point of view of the equations of variation.     

Multiple expansion of the tidal potential

The Earth tidal deformation causes an additional gravitational potential. Its effect on the Moon orbital motion has been studied by several authors.In this contribution, we develop this additional potential without specifying the inertial frame chosen.For this purpose, we use the properties of the representation of rotation groups in 3 dimensions space. We finally obtain the interaction potential between the distorted Earth and the Moon which is a necessary preliminary to the study of the evolution of the Earth-Moon system.Nomenclature T.R.O Tide raising object - (, , ) Spherical coordinates of the T.R.O. - (J, _E ) Earth spin axis orientation. _E is the longitude of the ascending node of Earth's equator on thexy-plane - (a ,I ,e , , ,M ) Elliptics elements of the T.R.O     

Emden-chandrasekhar axisymmetric,rigidly rotating polytropes

An approximate analytical method of solving the polytropic equilibrium equations, first developed by Seidov and Kuzakhmedov (1978), has been extended and generalized to equilibrium configurations of axisymmetric systems in rigid rotation, with polytropic index,n =n _p + _n , nearn _p =0, 1, and 5. Though the details of the method depend on the value ofn _p , acceptable results are obtained for | _n | 0.5 to describe slowly rotating configurations in the range 0n1.5, 4.5n5. In the limit of rotational equilibrium configurations, when the distorsion may be large enough, a satisfactory approximation holds only in the range 0n, 1n1.5, 4.5n5.     

Interpretation of type I- and IV mB-bursts and noise storms by mode coupling in the warm plasma

The circular polarization of radiation emitted from the solar type I- and IV mB-bursts and noise storms is not understood very well. For an attempted new explanation the dispersion equations for the ordinary (left-handed) and the extraordinary (right-handed) wave are derived from the well-known tensor conductivity of warm plasma proposing a very small angle between magnetic field and propagation direction, and the plasma parametersX1,Y1. Taking into account a plasma temperature different from zero, conditions of a coupling point (Budden, 1961) are attained very nearly even if the very small collision frequency is neglected. It is shown, that the observed ordinary (left-handed) polarization may be explained by a process of mode-coupling between the originally emitted extraordinary (right-handed) wave and the resulting ordinary wave. The source of the right-handed radiation may be gyro-radiation or erenkov-radiation. The proposed mechanism is in accordance with the main observational facts. It remains open, whether the known magnetic asymmetry of active regions accounts for the prevailing left-handed polarization.     

On stationary solutions of some canonical systems of differential equations

In our article (Zhuravlev, 1979) a formal method of constructing conditionally periodic solutions of canonical systems of differential equations with a quick-rotating phase in the case of sharp commensurability was presented. The existence of stationary (or periodic) solutions of an averaged system of differential equations corresponding to the initial system of differential equations is necessary for an effective application of the method for different problems.Evidently, the stationary solutions do not always exist but in numerous papers on stationary solutions (oscillations or motions), the conditions of existence of such solutions are very often not considered at all. Usually a simple assumption is used that the stationary solutions do exist.Otherwise it is well known that Poincaré's theory of periodic solutions (Poincaré, 1892) let one set up conditions of existence of periodic solutions in different systems of differential equations. Particularly, in papers,Mah (1949, 1956), see alsoexmah (1971), the necessary and sufficient conditions of the existence of periodic solutions of (non-canonical) systems of differential equations which are close to arbitrary non-linear systems are given. For canonical autonomous systems of differential equations the conditions of existence of periodic solutions and a method of calculation are presented in the paperMepmah (1952).In our paper another approach is given and the conditions of existence of stationary solutions of canonical systems of differential equations with a quick-rotating phase are proved. For this purpose Delaunay-Zeipel's transformation and Poincaré's small parameter method are used.     

Photoelectric line profiles in umbral spectra

Umbral line profiles of H, Na D₂ and Hei 10830 have been observed photoelectrically with a pressure scanning spectrometer, URSIES (Ultravariable Resolution Single Interferometer Echelle Scanner). Intermittent pulse counting techniques are applied with integration times as short as 0.8 s which permits selection of moments of good seeing and guiding even in poor climates.The H line profile shows in a medium-sized spot (A _u = 100 × 10_–6 SH) a pronounced weakening in the wings but a persistence of the line core in agreement with the results of Fricke and Elsässer (1965).The profile of the Hei 10830 line shows an unexpected strengthening over the umbra of the two spots which have been observed. The strengthening of the line in the umbral spectrum of a third spot has been confirmed by Dr. Mallia at the Oxford Solar Station (Gornergrat, Switzerland).Na D₂ line profile together with continuum observations at 4260, 5890, and 6530 have been accounted for with an empirical (T, ) relation by one of us (Yun, 1971). The wavelength dependence and the absolute values from 4000 Å to 14000 Å of the intensity curve generated by this relation shows very good agreement with the intensities observed by Wöhl et al. (1970) in a spot of similar diameter.Precision observations of the continuum at 5890 indicate umbra-to-disk continuum ratios that are higher (0.104) than the Mercury calibrated value by Mattig (0.086) even when a maximal correction for scattered light is applied. Only Zwaan's (1965) cool umbral model atmosphere approaches Mattig's continuum value, but this model renders the absorption wings in Na D₂ much too deep. It is suggested that future definitive observations should be carried out with coronagraphs under coronal skies or in satellites.Work supported by NASA Grant NGR-39-005-066.     

Hamiltonian Stability of Spin–Orbit Resonances in Celestial Mechanics

The behaviour of ‘resonances’ in the spin-orbit coupling in celestial mechanics is investigated in a conservative setting. We consider a Hamiltonian nearly-integrable model describing an approximation of the spin-orbit interaction. The continuous system is reduced to a mapping by integrating the equations of motion through a symplectic algorithm. We study numerically the stability of periodic orbits associated to the above mapping by looking at the eigenvalues of the matrix of the linearized map over the full cycle of the periodic orbit. In particular, the value of the trace of the matrix is related to the stability character of the periodic orbit. We denote by ε_* (p/q) the value of the perturbing parameter at which a given elliptic periodic orbit with frequency p/q becomes unstable. A plot of the critical function ε_* (p/q) versus the frequency at different orbital eccentricities shows significant peaks at the synchronous resonance (for low eccentricities) and at the synchronous and 3:2 resonances (at higher eccentricities) in good agreement with astronomical observations. This revised version was published online in July 2006 with corrections to the Cover Date.     

The local invariants under rotations about an axis and their applications

The three possible scalar products formed from the position-speed state vector lead to Stumpff's local invariants. The assumption of a distinguished axis (001) gives rise to the axial local invariantsz and under the rotations about this axis.-If a satellite moves in the field of a rotationally symmetric central body, then there will be an appropriate set of invariants for which a regular, non-linear system of differential equations holds. It can be used to derive recurrence relations for time series expansions and special perturbation methods, which are characterised by the occurrence of the scalar products of the perturbational acceleration with the position and speed vectors and (001).Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969.     

Magnetic fields and flares in the region CMP 20 September 1963

The position of bright knots of 30 flares at their very beginning relative to the high-resolution isogauss maps of the longitudinal component (H ) and maps of the transverse component (H ) of magnetic field are considered for seven days during the passage of the active and large spot group in Sept. 1963 (see Table I and maps on Figures 1–8).The flare bright knots occur simultaneously in regions of opposite magnetic polarity, and the majority of these knots are adjacent to neutral line H = 0, although not coinciding precisely with this line (Figure 9). Lenticular form of flare knots and the motions of bright material of flares is restrained by transversal field H . Also flares are closely associated (83%) with so-called bifurcated regions, where specific crossing of transverse components takes place (Figures 4–5). There is well-expressed (80%) coincidence of flare knots with the strongest (positive or negative) electric currents as determined from the relation j = c/4 rot H. The relation of results obtained to some existing theories of flares is briefly discussed.U.S. Nat. Acad. of Science - U.S.S.R. Acad. Nauk. Exchange Scientist Program; now at CSIRO Division of Physics, Australia.     

Simultaneous measurements of Ca+ K,Ca+ H,Hα, Hβ, and He D3 emission in prominences

Simultaneously measured line intensities of Ca⁺ K and H, H, H and He D₃ are presented and compared with recent model calculations by Heasley and Milkey (1978). The model calculations are compatible with the observations. There is indication that the observed branching of the relation E(Ca⁺ K and H) E(H) depends on the widths of the Ca⁺-lines.     

Effect of perturbations in Coriolis and centrifugal forces on the stability of libration points in the restricted problem

The location and the stability of the libration points in the restricted problem have been studied when small perturbation and are given to the Coriolis and the centrifugal forces respectively. It is seen that the pointsL ₄ andL ₅ form nearly equilateral triangles with the primaries and the pointsL ₁,L ₂,L ₃ remain collinear. It is further observed that for the pointsL ₄ andL ₅, the range of stability increases or decreases depending upon whether the point (, ) lies in one or the other of the two parts in which the (, ) plane is divided by the line 36-19=0 and the stability of the collinear points is not influenced by the perturbations and they remain unstable.     

Radiative heat transfer in surface layers of neutron stars with a magnetic field

Thermal conductivity due to Thomson scattering and free-free absorption of photons is numerically evaluated for a non-relativistic non-degenerate plasma in a magnetic field for a number of values ofb =_B/kT 1000. In the case of pure scattering, simple fitting formulae are derived. Atb6, the magnetic field is shown to decrease (by about one order) the characteristic densities above which heat transfer is mainly determined by free-free transitions.     

Some remarks about cosmology in brans and Dicke's theory

It is established that, in contrast to Einstein's theory, in the cosmological equations of the theory by Brans and Dicke the spatial component of the field equations follows from the remaining equations and the conservation lawT ^ik _jk=0 only if one excludes the power law R ^–2. Some consequences are discussed.     

Mass and orbit estimation of Planet X via a family of comets

The crucial assumption of this paper is, that the observed clustering of aphelion distances of intermediate-period comets in the 70–90 AU range is due to the influence of a tenth planet, called Planet X. We contribute to the search for Planet X a new and extended evaluation of a family of comets assumed to be Planet X's family of comets.By averaging the aphelion distances of comets that belong to a transplutonic family of comets, we get Planet X's semi-major axis a _x = (83.0 ± 5.3) AU. The comets' orbits also yield the upper limit of the planet's orbital eccentricity e _x - 0.019. If this planet played an important part in sending quasi-periodic comet showers to the inner solar system, we can calculate its orbital inclination i _x = 46 .1 ± 3 .6. By distributing all planets' masses into the heliocentric, torus-like zones, in which they were formed, we get the density distribution of the primordial solar nebula. Extrapolating this distribution we find the mass of the planet M _x = (5.1_–2.4 ^+3.6 M _Earth. A few plausible assumptions (e.g. Uranus and Neptune perturbations being caused by Planet X) lead to Planet X's actual location with declination and eccliptic longitude being = 57 ± 17 and = 54 ± 34 , respectively (1989.5 position). In addition, we give Planet X's apparent brightness dependent on its unknown albedo. All those properties and predictions are more or less in agreement with earlier work on Planet X.     

Non-radial oscillations in an axisymmetric MHD incompressible fluid

It is well known from Helioseismology that the Sun exhibits oscillations on a global scale, most of which are non-radial in nature. These oscillations help us to get a clear picture of the internal structure of the Sun as has been demonstrated by the theoretical and observational (such as GONG) studies. In this study we formulate the linearised equations of motion for non-radial oscillations by perturbing the MHD equilibrium solution for an axisymmetric incompressible fluid. The fluid motion and the magnetic field are expressed as scalarsU, V, P andT, respectively. In deriving the exact solution for the equilibrium state, we neglect the contribution due to meridional circulation. The perturbed quantitiesU *, V *, P *, T * are written in terms of orthogonal polynomials. A special case of the above formulation and its stability is discussed.     

Nonlinear treatment for solar radio spikes

We emphasize that a nonlinear treatment is required to realize the diagnostic potentiality of solar spiky emission. The observational constraints including the latest data on the harmonic structure, degree, and sense of polarization are discussed. A set of coupled equations for energy density of high-frequency normal modes of a magneto-active plasma involving the most important nonlinear effects within the three-wave approximation is deduced. The equations include both previously known and new effects. The qualitative evaluations of the equations obtained have provided a few new findings: (i) quasi-linear relaxation of fast electrons on quasi-potential waves (_) occurs in a characteristic time scale of the order of 10 ms if the frequency,f, is about 1 GHz; (ii) the stimulated scattering of the transverse waves on the background plasma particles is shown to be important if the brightness temperature of the spiky emission exceeds 10¹⁵–10¹⁶ K; (iii) the Raman scattering of the transverse waves on background plasma density inhomogeneities may suppress the electron cyclotron maser instability if n _e ² /n _e ² 3 × (10^–4–10^–5).     

Motions in a loop prominence

Motions of loop prominence knots have been studied on H-line filtergrams. By making use of contours of equal photographic density for entire cinegram it has been possible to significantly decrease the error in determining the locations of the knots. The method of mean velocities has been developed, which has permitted for the first time accurate determination of the laws of knot motion with sufficient accuracy. Two types of falling knots are distinguished: (1) those with a constant acceleration that is always below the gravitational acceleration, and (2) those with a constant velocity. The initial velocity of the falling knots is always different from zero. The gasdynamic theory has been developed to explain the deceleration of the two types of knots due to: the work done against the pressure force; pileup plasma raking; and shock-wave generation. The shock mechanism imparts a constant velocity to the motion. The temperature along the knot trajectories has been estimated. The ratio of the densities in the knots to the surrounding medium has been found.The aim of the present work is to gain new insight into the known observations of the motions in active prominences, in particular in the loop prominences, and to understand the reasons for the observed motions.A number of studies of prominence knot motions have been made using filter photographs (cinefilms). It seems to us, however, that the velocities (and especially acceleration) of individual knots have been determined within insufficient accuracy. It will be noted first of all that values of V(t) have been determined with a low accuracy by some authors even for high velocities (V 100 km s^–1). In fact, to achieve at least 10% accuracy in determining V by comparing two photographs obtained with a 30 s interval, it is necessary to measure the location of knots to 0.5 accuracy. The problem is the more complicated as the size of the knots can often reach 5 × 10. This is why the temporal dependence of the velocity of prominence knot motion is represented with a complicated curve.     

Equilibrium structure of stars obeying a generalized differential rotation law

In the present paper we have considered the problem of determining the equilibrium structure of differentially rotating stars in which the angular velocity of rotation varies both along the axis of rotation and in directions perpendicular to it. For this purpose, a generalized law of differential rotation of the type ² =b ₀+b ₁ s ²+b ₂ s ⁴+b ₃ z ²+b ₄ z ⁴+b ₅ z ² s ² (here is a nondimensional measure of the angular velocity of a fluid element distants from the axis of rotation andz from the plane through the centre of the star perpendicular to the axis of rotation, andb's are suitably chosen parameters) has been used. Whereas Kippenhahn and Thomas averaging approach has been used to incorporate the rotational effects in the stellar structure equations, Kopal's results on Roche equipotentials have been used to obtain the explicit form of the stellar structure equations, which incorporate the rotational effects up to second order of smallness in the distortion parameters. The method has been used to compute the equilibrium structure of certain differentially rotating polytropes. Certain differentially rotating polytropes. Certain differentially rotating models of the Sun have also been computed by using this approach.     

Damping constant and turbulence in the solar atmosphere

In the region of the formation of weak and medium-strong lines, the microturbulence increases with height (V _ver=0.7–0.9 km s^-1, V _hor= 1.1–1.5 km s^-1), the macroturbulence decreases (V _ver=1.6–1.4 km s^-1, V _hor= 2.4–1.5 km s^-1), and the total velocity field (vertical component) is depth-independent (1.7 km s^-1). The empirical damping constants for Fe, Ti, Cr, Ni lines are equal 1.3₆, 1.7₆, 1.6₆, 1.6₆, respectively. The correlation length (the Kubo-Anderson process has been used) in the solar photosphere is 520–550 km.