The structure of periodic solutions in the restricted problem of three bodies

We consider the basic families of plane-symmetric simply-periodic orbits in the Sun-Jupiter case of the plane restricted three-body problem and we study their horizontal and vertical stabilities. We give the critical orbits of these families, corresponding to the vertical stability parameter = 1 and in future communications we shall give the three-dimensional families which emanate from these plane bifurcations.     

Поле излучения в полубесконечной атмосфере, содержащей источники энергии

, , , , . , , . , , - - ( , ..). - .     

Planar orbits through second variations

The development and application of a predictor-corrector method for the computation of families of periodic motions as well as of singular periodic solutions from which branchings or change in the stability character occur, based on the use of second order variations is presented. Numerical results obtained by means of this method are also given. It is found that this algorithm consumes somewhat less computer time in determining orbits-members of a family of periodic orbits, while it represents a unique tool for the determination of branchings of various orders, as well as of the precise members of each family at which the orbits change from unstable to stable and viceversa.     

New kinds of asymmetric periodic orbits in the restricted three-body problem

The procedure of numerical ascent from families of planar to three-dimensional periodic orbits and the subsequent descent to the plane is proved efficient in determining new families of planar asymmetric periodic orbits in the restricted three-body problem. Two such families are computed and described for values of the mass parameter for which it has been found that they exist. Two new families of three-dimensional asymmetric periodic orbits are also presented in this paper.     

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.

---

. , . , . . .

     

About the stability of the libration points of a rotating triaxial ellipsoid in a degenerate case

The problem of stability of the equilibrium points in the problem of motion of a mass point in the neighbourhood of a rotating triaxial ellipsoid in a degenerate case (the Gaussian of the Hamiltonian is equal to zero) is investigated.It is proved that the equilibrium points in the degenerate case are stable in the strict sence.

---

Résumé On étudie le problème de la stabilité des positions d'équilibre relatif dans le problème du mouvement du point matériel au voisinage de l'ellipsoid à trois axes tournant dans un cas dégénéré (le gaussian du Hamiltonian égal zéro).On a démontré que des positions d'équilibre relatif dans le dégénéré sont stables dans le sens rigoureux.

---

( )., .

     

Families of three-dimensional axisymmetric periodic orbits in the restricted three-body problem: Sun-Jupiter case

Families of three-dimensional axisymmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families bifurcate from the vertical-critical orbits (_v = 1,b _v = 0) of the basic plane familiesi andI. Further the predictor-corrector procedure employed to reveal these families has been described and interesting numerical results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections.     

О лагранжевых и зйлеровых решеиях обощенной задачи трех тел

(, 1969). ( ), ( ), , , . , (=), , , .. , . , , - ( ), ( ). , .

---

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.

     

Effect of perturbations on the stability of triangular points. In the restricted problem of three bodies with variable mass

The effect of small perturbations and in the coriolis and the centrifugal forces respectively on the stability of the triangular points in the restricted problem of three bodies with variable mass has been studied. It is found that the range of stability of triangular points increases or decreases depending upon whether the perturbation point (, ) lies in one or the other of the two parts in which the (, ) plane is divided by the line J₈–J₉=0 where J₈ and J₉ depend upon , the constant due to the variation in mass governed by Jeans' law.     

On the time variation of rotation parameters of pulsar NP 0532

Calculations of the rotation parameters at initial stage of evolution on the basis of the model suggested by Jones and published observational data of NP 0532 were made. It is shown that the observed dependence of the angular velocity of NP 0532 and its first derivative with respect to time can be explained by an increase of the angle between the magnetic moment and the rotation axis. The time variation of rotation parameters essentially depends on the value and time variation of the viscosity of the star. In spite of the fact that the observational data are not extensive enough, it is possible to conclude that initial rotation frequency of NP 0532 did not exceed essentially 50 Hz.

---

, , NP 0532 . , . , . , , NP 0532 50 , .

     

О колебаниях хвоста магнитосферы в приБлижении квазигидро динамики

-- () ., , , , , . , , . , .     

Движение малых частиц в окрестностях протосолнца переменной светимости

, , , , . , . , () (), , , . — .     

Метагалактические протоны сверхвысоких энергий

, . 2.7°- . . . E3×10¹⁹ (1.6) .     

Bifurcations in systems of three degrees of freedom

We study the bifurcations of families of double and quadruple period orbits in a simple Hamiltonian system of three degrees of freedom. The bifurcations are either simple or double, depending on whether a stability curve crosses or is tangent to the axis b=–2. We have also generation of a new family whenever a given family has a maximum or minimum or .The double period families bifurcate from simple families of periodic orbits. We construct existence diagrams to show where any given family exists in the control space (, ) and where it is stable (S), simply unstable (U), doubly unstable (DU), or complex unstable (), We construct also stability diagrams that give the stability parameters b₁ and b₂ as functions of (for constant ), or of (for constant ).The quadruple period orbits are generated either from double period orbits, or directly from simple period orbits (at double bifurcations). We derive several rules about the various types of bifurcations. The most important phenomenon is the collision of bifurcations. At any such collision of bifurcations the interconnections between the various families change and the general character of the dynamical system changes.     

On the restrited three-body problem when the mass parameter is small

We study some aspects of the restricted three-body problem when the mass parameter is sufficiently small. First, we describe the global flow of the two-body rotating problem, =0, and we use it for the analysis of the collision and parabolic orbits when 0. Also we show that for any fixed value of the Jacobian constant and for any >0, there exists a ₀>0 such that if the mass parameter [0,₀], then the set of bounded orbits which are not contained in the closure of the set of symmetric periodic orbits has Lebesgue measure less than .Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics.     

The asymptotic limit of the form-factors α and β product for celestial bodies

The applicability of the properties of central configurations proceeding from the many-body problem to study of gaseous sphere cloud evolution during its gravitational contraction is justified. It is shown that the product runs to a constant value in the asymptotic time limit of simultaneous collision of all the particles of the cloud where is a form-factor of the potential energy and is a form-factor of the moment of inertia.The spherical bodies as well as ellipsoids of rotation and general ellipsoids with a one-dimensional mass distribution (k),k[0, 1] are found to possess the property =const.

---

. , - , , ., , - =const., , (k),k[0, 1].

     

К теории кривых Блеска сверхновых звезд

, . . . (10R ) ( 10 000R ). NGC 5457, NGC 6946 NGC 5236. , , II I . . . , . . 10⁵⁰÷10⁵² , .     

Examples of predominantly two-body orbits in the Sun-Jupiter asteroid system

Accurate numerical continuation of families of plane symmetric direct periodic orbits around the large primary in the Sun-Jupiter case of the restricted problem of three bodies allows the determination of the vertical branching points where families of three-dimensional symmetric periodic orbits bifurcate from the planar ones. Three families of plane periodic orbits, and the initial segments of ten bifurcating families of three-dimensional ones are determined. The stability of these families is examined and examples of their orbits are illustrated.     

Numerical determination of families of three-dimensional double-symmetric periodic orbits in the restricted three-body problem,II

New families of three-dimensional double-symmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families bifurcate from the vertical-critical orbits ( _v = – 1, b _v – 0) of the basic plane familiesi, g ₁, g₂, c andI. Further, the predictor-corrector procedure employed to reveal these families has been described and interesting numerical results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections.     

Numerical determination of families of three-dimensional double-symmetric periodic orbits in the restricted three-body problem. I.

New families of three-dimensional double-symmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families bifurcate from the vertical-critical orbits ( _v = –1,c _v ),c _v=0) of the basic plane familiesi,g ₁,g ₂,h,a,m andl. Further the numerical procedure employed in the determination of these families has been described and interesting results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections.