On the stability of equilibria of Hamiltonian systems near the main resonances

The equilibrium point O of an autonomous Hamiltonian system of two degrees of freedom is considered for small-oscillation frequencies related as ₂=2₁+. If under the precise resonance (=0) the equilibrium is unstable, the inner diameter () of the domain of stability containing the point O is estimated. It is shown that for the normalized variables ()/b where b is the corresponding resonance coefficient. The estimates () for other main resonances are reported.     

The three principal secular resonances ν5, ν6, and ν16 in the asteroidal belt

We review theoretical and numerical results obtained for secular resonant motion in the asteroidal belt. William's theory (1969) yields the locations of the principal secular resonances ₅, ₆, and ₁₆ in the asteroidal belt. Theories by Nakai and Kinoshita (1985) and by Yoshikawa (1987) allow us to model the basic features of orbital evolution at the secular resonances ₁₆ and ₆, respectively. No theory is available for the secular resonance v₅. Numerical experiments by Froeschlé and Scholl yield quantitative and new qualitative results for orbital evolutions at the three principal secular resonances ₅, ₆, and ₁₆. These experiments indicate possible chaotic motion due to overlapping resonances. A secular resonance may overlap with another secular resonance or with a mean motion resonance. The role of the secular resonances as possible sources of meteorites is discussed.     

The critical inclination problem — 30 years of progress

The critical inclination problem in artificial satellite theory, first discovered 30 years ago, has aroused great interest and provoked much discussion and controversy in the intervening years. It was this problem which essentially provided the seed corn for the development of the theory of the Ideal Resonance Problem (IRP). The latter theory provides good first-approximation solutions to a number of important resonance problems in celestial mechanics. It is not applicable, however, to certain other interesting resonant systems within the solar system. For these resonances a new fundamental mathematical model of resonance, in the spirit of the IRP, has recently been formulated and successfully applied.This paper reviews the history of the critical inclination problem and highlights the controversies it has generated over the years. The Problem's strong connection with the IRP is outlined with both thenormal andabnormal forms featuring. Finally, with reference to the critical inclination problem, the essential properties of the newer fundamental model are described and compared with the IRP. A strong correspondence is established between recent independent investigations of a variety of resonance problems and earlier work of Andoyer.     

The virial-based solution for the velocity of gaseous sphere gravitational contraction

It is found that the relationship between the potential energy and the moment of the inertia independent of the radial mass distribution obtained earlier for the sphere also holds in the case of the ellipsoidal mass distribution for the ellipsoid of rotation.The possibility of application of the energy virial relations for solution of the evolutionary problems of the gravitating gaseous sphere, with the help of the relationship found earlier, is demonstrated. The physical conditions on the gaseous sphere boundary are introduced. The existence of two branches of evolution, the proton one and the electron one, is established. The problem of the gravitational contraction velocity during sphere evolution is solved. The relationship between the boundary temperature and the gaseous sphere radius as well as between the luminosity and the body mass is obtained. Some limiting relations for the final stage of the gaseous sphere evolution are found.

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Перенос резонансного излучения И случайные ълуждания Фотонов

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Transfer of resonance radiation in infinite medium is considered as a process of random walks of photons. Close relation is shown to exist between the problems of transfer of line radiation and the stable distributions of the probability theory. This relation is used as a basis of a new method for the investigation of the asymptotic properties of the radiation field far from the sources.

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On the coupling of lunisolar resonances for Earth satellite orbits

The resonance C1 occurs when the longitude of the perigee measured from the equinox becomes a slow angle in the doubly averaged equations of motion. This resonance is one of the critical inclination family with I 46°. For prograde Earth satellite orbits, up to five critical points can be identified. Only simple pitchfork bifurcations occur for the single resonance C1. A two degrees of freedom system is studied to check how a coupling of two lunisolar resonances affects the results furnished by the analysis of an isolated resonance case. In the system with two critical angles (g+h and h,+2 , seven types of critical points have been identified. The critical points arise and change their stability through 11 bifurcations. If the initial conditions are selected close to the critical points, the system becomes chaotic as shown in Poincaré maps.     

О лагранжевых и зйлеровых решеиях обощенной задачи трех тел

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This paper is a continuation and a generalization of one published earlier (Duboshin, 1969): it discusses the problem whether there exist the Lagrangian and the Eulerian solutions of the generalized three-body (material points) problem. Every point in this generalized problem acts on another, one with a force (attractive or repulsive) directed along the straight line passing through these points, and in an arbitrary manner depending on time, mutual distance and its derivatives, the first and the second. Here, generally speaking, the third axiom of dynamics (law of action and reaction) is not presupposed as fulfilled, that is, it is supposed that every two material points interact in a different way.This most general assumption being made, we establish the conditions which must dictate the laws of the interactions, so that the three points can always remain at the apexes of the equilateral triangle (Langrangian solution), or remain always on a straight line (Eulerian solution).The author believes that such general treatment of the three-body problem can be useful for theoretical studies in celestial mechanics and also for practical applications in the study of isolated stellar systems.

     

Угловые Флуктуации Реликтового излучения, порождаемые космологической турбу лентностью

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Взаимооействие Излучения Рентгеновскооо Истооника с АтмосФерой Нормальной Звезды в Тесных Двойных Системах

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Non-restricted double-averaged three body problem in Hill's case

A limiting case of the problem of three bodies (m ₀,m ₁,m ₂) is considered. The distance between the bodiesm ₀ andm ₁ is assumed to be much less than that between their barycenter and the bodym ₂ so that one may use Hill's approximation for the potential of interaction between the bodiesm ₁ andm ₂. In the absence of resonant relations the potential, double-averaged by the mean longitudes ofm ₁ andm ₂, describes the secular evolution of the orbits in the first approximation of the perturbation theory.As Harrington has shown, this problem is integrable. In the present paper a qualitative investigation of the evolution of the orbits and comparison with the analogous case in the restricted problem are carried out.The set of initial data is found, for which a collision between the bodiesm ₀ andm ₁ takes place.The region of the parameters of the problem is determined, for which plane retrograde motion is unstable.In a special example the results of approximate analysis are compared with those of numerical integration of the exact equations of the three body problem.

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Some fundamental elements of universe mechanics

An essential part in the mechanics under study is taking into consideration the effect of motions of the Universe objects upon that of an individual one surrounded by them including those infinitely far from it. Only macro-objects of the Universe are meant here.

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Zusammenfassung Ein wesentlicher Bestandteil der Mechanik unter unserer Betrachtung ist die Berechnung des Einflusses auf die Bewegung eines individuellen Objektes von Bewegungen der Universum Objekte die es umringen einschließlich jene Objekte, die unendlich entfernt sind. Nur Makroobjekte des Weltalles sind in der Absicht dabei.

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On the stability of small quasiperiodic motions in the hamiltonian systems

Orbital stability of quasiperiodic motions in the many dimensional autonomic hamiltonian systems is considered. Studied motions are supposed to be not far from equilibrium, the number of their basic frequencies may be not equal to the number of degrees of freedom, and the procedure of their construction is supposed to be converged. The stability problem is solved in the strict nonlinear mode.Obtained results are used in the stability investigation of small plane motions near the lagrangian solutions of the three-dimensional circular restricted three-body problem. The values of parameters for which the plane motions are unstable have been found.

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ОБ ионизации межзвездного магния

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Discovery of a New Class of Predicted Resonance Objects beyond Jupiter

In the regions of mean diurnal motions between the orbits of Jupiter and Saturn, predicted earlier by the authors, five asteroids have been discovered that move in 1:2 and 2:3 Lindblad orbital resonances with Jupiter (external orbital commensurability) and in 2:1 resonance with Saturn (internal version of commensurability). In addition to this, in the precalculated stable resonance zones between the giant planets Saturn and Uranus, three objects have been found that possess third-order (2:5) orbital commensurability with Saturn; nine objects have been discovered between the orbits of Uranus and Neptune, whose mean motions are in 1:3 and 1:4 orbital resonances with Saturn, and more than 200 libration-stable objects, linked by lower-order orbital resonances with Neptune and Uranus have been found in the Kuiper belt.     

Hubble expansion in a euclidean framework

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Результаты и проблемы исследования синхротронной неустойчивости

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Метагалактические протоны сверхвысоких энергий

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εлектроны космических лучей радиогало галактики

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Аккреция вещества с магнитным полем на коллапсирующую звезду. II. Самосогласованная стационарная картина

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Interaction between the interstellar medium and solar wind plasma

The interaction processes governing the penetration of the interstellar gas into the solar neighbourhood are re-examined — as well as photo-ionization and charge-exchange processes, proton elastic collisions and electron ionizations help reduce the nearby gas densities. The total destruction rate varies little during the solar cycle, by perhaps 10%. Particle heating, particularly via the elastic collisions, determines the gas characteristics in the gravitationally focussed tail—enhanced H-density is prevented, while the He-tail is effectively hotter than 10³ K.Termination of the solar wind is rediscussed in the light of both electron heating and the stronger gas/plasma interaction. The spiral interplanetary field is taken to break up and the subsonic plasma flow to be controlled by the pressure of slowly cooling electrons. The terminating collisionless shock is then, if it exists at all, very weak (M ₁<1.4), subcritical, and energetically unimportant. Cosmic rays are little affected by this sonic transition, but at least the electron component should be modulated by plasma turbulence throughout the ionizing flow.

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