The restricted two-body problem in constant curvature spaces

We perform the bifurcation analysis of the Kepler problem on and . An analog of the Delaunay variables is introduced. We investigate the motion of a point mass in the field of a Newtonian center moving along a geodesic on and (the restricted two-body problem). For the case of a small curvature, the pericenter shift is computed using the perturbation theory. We also present the results of numerical analysis based on an analogy with the motion of a rigid body.     

The Schwarzschild Problem in Astrophysics

The Schwarzschild problem (the two-body problem associated to apotential of the form A/r + B/r³ has been qualitativelyinvestigated in an astrophysical framework, exemplified by two likelysituations: motion of a particle in the photogravitational field ofan oblate, rotating star, or in that of a star which generates aSchwarzschild field. Using McGehee-type transformations, regularizedequations of motion are obtained, and the collision singularity isblown up and replaced by the collision manifold (a torus)pasted on the phase space. The flow on is fullycharacterized. Then, reducing the 4D phase space to dimension 2, theglobal flow in the phase plane is depicted for all possible values ofthe energy and for all combinations of nonzero A and B. Eachphase trajectory is interpreted in terms of physical motion,obtaining in this way a telling geometric and physical picture of themodel.     

Kepler's problem in constant curvature spaces

In this article the generalization of the motion of a particle in a central field to the case of a constant curvature space is investigated. We found out that orbits on a constant curvature surface are closed in two cases: when the potential satisfies Iaplace-Beltrami equation and can be regarded as an analogue of the potential of the gravitational interaction, and in the case when the potential is the generalization of the potential of an elastic spring. Also the full integrability of the generalized two-centre problem on a constant curvature surface is discovered and it is shown that integrability remains even if elastic forces are added.     

Special Solutions in a Simplified Restricted Three-Body Problem with Gravitational Radiation Taken into Account

The distinctive feature of the relativistic restricted three-body problem within the c ^–5 order of accuracy (2 post-Newtonian approximation) is the presence of the gravitational radiation. To simplify the problem the motion of the massive binary components is assumed to be quasi-circular. In terms of time these orbits have linearly changing radii and quadratically changing phase angles. By substituting this motion into the Newtonian-like equations of motion one gets the quasi-Newtonian restricted quasi-circular three-body problem sufficient to take into account the main indirect perturbations caused by the binary radiation terms. Such problem admits the Lagrange-like quasi-libration solutions and rather simple quasi-circular orbits lying at large distance from the binary.     

The main mean motion commensurabilities in the planar circular and elliptic problem

We study the motion of asteroids in the main mean motion commensurabilities in the frame of the planar restricted three-body problem. No assumption is made about the size of the eccentricity of the asteroid. At small to moderate eccentricity, we recover existing results (shape of the phase space and location of secondary resonances). We also provide global pictures of the dynamics in the region of secondary resonances. At high eccentricity, the phase space portraits of the integrable part of the Hamiltonian show new families of stable orbits for the 3:2 and 2:1 cases and the secular resonances ₅ and ₆ are located.     

Resonant motion in the restricted three body problem

The resonant structure of the restricted three body problem for the Sun- Jupiter asteroid system in the plane is studied, both for a circular and an elliptic orbit of Jupiter. Three typical resonances are studied, the 2 : 1, 3 : 1 and 4 : 1 mean motion resonance of the asteroid with Jupiter. The structure of the phase space is topologically different in these cases. These are typical for all other resonances in the asteroid problem. In each case we start with the unperturbed two-body system Sun-asteroid and we study the continuation of the periodic orbits when the perturbation due to a circular orbit of Jupiter is introduced. Families of periodic orbits of the first and of the second kind are presented. The structure of the phase space on a surface of section is also given. Next, we study the families of periodic orbits of the asteroid in the elliptic restricted problem with the eccentricity of Jupiter as a parameter. These orbits bifurcate from the families of the circular problem. Finally, we compare the above families of periodic orbits with the corresponding families of fixed points of the averaged problem. Different averaged Hamiltonians are considered in each resonance and the range of validity of each model is discussed.     

Non—integrability of Hill's lunar problem

We prove that Hill's lunar problem does not possess a second analytic integral of motion, independent of the Hamiltonian. In order to obtain this result, we avoid the usual normalization in which the angular velocity of the rotating reference frame is put equal to unit. We construct an artificial Hamiltonian that includes an arbitrary parameter b and show that this Hamiltonian does not possess an analytic integral of motion for in an open interval around zero. Then, by selecting suitable values of , b and using the invariance of the Hamiltonian under scaling in the units of length and time, we show that the Hamiltonian of Hill's problem does not possess an integral of motion, analytically continued from the integrable two–body problem in a rotating frame.     

On oscillatory motion in the problem of three bodies

In the case of oscillatory motion in the problem of three bodies it is shown that ast the mutual distances between particles cannot separate faster thanCt ^2/3 whereC is some positive constant. As bounding functions of time exist for the other classifications of motion in the three body problem, it follows in general that the mutual distances between particles is 0(t) ast.     

One to One Resonance at High Inclination

We report results from long term numerical integrations and analytical studies of particular orbits in the circular restricted three-body problem. These are mostly high-inclination trajectories in 1 : 1 resonance starting at or near the triangular Lagrangian L₅ point. In some intervals of inclination these orbits have short Lyapunov times, from 100 to a few hundred periods, yet the osculating semi-major axis shows only relatively small fluctuations and there are no escapes from the 1 : 1 resonance. The eccentricity of these chaotic orbits varies in an erratic manner, some of the orbits becoming temporarily almost rectilinear. Similarly the inclination experiences large variations due to the conservation of the Jacobi constant. We studied such orbits for up to 10⁹ periods in two cases and for 10⁶ periods in all others, for inclinations varying from 0° to 180°. Thus our integrations extend from thousands to 10 million Lyapunov times without escapes of the massless particle. Since there are no zero-velocity curves restricting the motion this opens questions concerning the reason for the persistence of the 1 : 1 resonant motion. In the theory sections we consider the mechanism responsible for the observed behavior. We derive the averaged 1 : 1 resonance disturbing function, to second order in eccentricity, to calculate a critical inclination found in the numerical experiment, and examine motion close to this inclination. The cause of the chaos observed in the numerical experiments appears to be the emergence of saddle points in the averaged disturbing potential. We determine the location of several such saddle points in the (, ) plane, with being the mean longitude difference and the argument of pericentre. Some of the saddle points are illustrated with the aid of contour plots of the disturbing function. Motion close to these saddles is sensitive to initial conditions, thus causing chaos.This revised version was published online in October 2005 with corrections to the Cover Date.     

Generalized Problem of Two and Four Newtonian Centers

We consider integrable spherical analog of the Darboux potential, which appear in the problem (and its generalizations) of the planar motion of a particle in the field of two and four fixed Newtonian centers. The obtained results can be useful when constructing a theory of motion of satellites in the field of an oblate spheroid in constant curvature spaces.     

Classification of Motions for Generalization of the two Centers Problem on a Sphere

The generalization of a test particle motion in a central field of two immovable point-like centers to the case of a constant curvature space, on a three-dimensional sphere, is investigated in the paper. The bifurcation set in the plane of integrals of motion is constructed and the classification of the domains of possible motion is carried out on a two-dimensional sphere. The regularization of the Kepler’s problem on a two-dimensional sphere is carried out. This revised version was published online in July 2006 with corrections to the Cover Date.     

Two-dimensional relativistic stellar wind

The anisotropic structure of the relativistic stellar wind is investigated. Both relativistic fluid velocity and relativistic temperature are taken into account. General analysis is carried out in the curvilinear coordinates and the generalization of the dispersion equation is obtained. The topological structure of the individual field lines is the same as in the spherically-symmetric case, except the fact that the magnetic field dependence on distance cannot be establisheda priori. The interaction between neighbouring field lines brings the dependence on the transverse coordinate, numbering the field lines. This dependence leads to the establishing of a new constraint on the global flow topology. The two-dimensional wind structure is analyzed, with the constraint taken into account, in the large distances limit, using the asymptotic expansion into ther ^–1 power series. In the lowest order approximation the constraint reduces to a new global constant of motion. This constant causes the splitting of the two solution families.     

Gravitational interaction of bodies immersed in fluids

It is shown that Archimedes' principle can be generalized for external gravitational fields due to stationary macroscopic bodies. For instance, a particle immersed in a homogeneous fluid at the centre of spherical symmetry of the fluid, or anywhere in an unbounded homogenous fluid, experiences — in an external field — a force that it would experience in a vacuum if it had an apparent mass less than the actual one by the mass of displaced fluid. Inversely, if one immerses a particle into a symmetrically arranged homogeneous fluid apart from its centre of symmetry, the particle and the fluid produce, at the centre of symmetry of the fluid, a gravitational field that would be produced in vacuo by a particle of the same size and shape but having apparent mass. Simple laboratory experiments, suitable to verify this inverse theorem, are described. On the other hand, the gravitational force between two particles in an infinite homogeneous fluid is reduced by a factor proportional to the product of their apparent masses which can be positive or negative. Two particles with opposite apparent masses repel each other. The results obtained imply corrections to vacuum of the order of (10^–5–10^–4) G of the gravitational constant,G, measured by the common laboratory methods.     

Families of planar orbits generated by homogeneous potentials

The second order partial differential equation which relates the potentialV(x,y) to a family of planar orbitsf(x,y)=c generated by this potential is applied for the case of homogeneousV(x,y) of any degreem. It is shown that, if the functionf(x,y) is also homogeneous, there exists, for eachm, a monoparametric set of homogeneous potentials which are the solutions of an ordinary, linear differential equation of the second order. Iff(x,y) is not homogeneous, in general, there is not a homogeneous potential which can create the given family; only if =f _y /f _x satisfies two conditions, a homogeneous potential does exist and can be determined uniquely, apart from a multiplicative constant. Examples are offered for all cases.     

Compatible potentials and orbits in synodic systems

For a given family of orbits f(x,y) = c ^* which can be traced by a material point of unit in an inertial frame it is known that all potentials V(x,y) giving rise to this family satisfy a homogeneous, linear in V(x,y), second order partial differential equation (Bozis,1984). The present paper offers an analogous equation in a synodic system Oxy, rotating with angular velocity . The new equation, which relates the synodic potential function (x,y), = –V(x, y) + % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSqaaSqaai% aaigdaaeaacaaIYaaaaaaa!3780!\[\tfrac{1}{2}\]²(x ² + y ²) to the given family f(x,y) = c ^*, is again of the second order in (x,y) but nonlinear.As an application, some simple compatible pairs of functions (x,y) and f(x, y) are found, for appropriate values of , by adequately determining coefficients both in and f.     

The velocity field in the atmosphere of δ Cephei

Observations related to the photospheric velocity field of Cephei can be interpreted as follows: during the whole cycle of pulsations the only motion form in the atmosphere is a wave motion with a nearly constant full amplitude of approximately 15 km s^–1, and a wavelength of about 10⁶ km (which are quantities, about equal to the amplitudes of pulsational velocity and radius of the star). There are no significant small-scale turbulent velocity components. The microturbulent and macroturbulent velocities, as derived from spectral line observations, are fully compatible with this picture.     

Radiative transfer in a homogeneous magnetized plasma

On the ground of the proper wave representation the general theory is developed of radiative transfer in a homogeneous plasma with the strong magnetic field ( _B /1). The linear and nonlinear equations are derived which generalize the corresponding equations of scalar radiative transfer theory in isotropic media. The solutions of some problems are given for the cases when the magnetic field is perpendicular to the surface: diffuse reflection of radiation from a semiinfinite medium, provided the sources are placed far from the surface (Milne's problem) and have constant intensity, increase linearly or quadratically with the optical depths, or decrease exponentially from the surface.     

Unsteady magnetohydrodynamic flows in a rotating elasto-viscous fluid

The unsteady flow of an incompressible electrically-conducting and elasto-viscous fluid (Walter's liquidB), filling the semi-infinite space, in contact with an infinite non-conducting plate, in a rotating medium and in the presence of a transverse magnetic field is investigated. An arbitrary time-dependent forcing effect on the motion of the plate is considered and the plate and fluid rotate uniformly as a rigid body. The solution of the problem is obtained with the help of the Laplace transform technique and the analytical expressions for the velocity field as well as for the skin-friction are given.     

Classical Scattering and Block Regularization for the Homogeneous Central Field Problem

The present paper offers an alternative point of view of block regularization for the motion of a particle in a central potential field of the form –x ^–, where x is the distance between the particle and the source and some positive real number.Working in the physical space, we consider the scattering angle determined by the path of the particle as a function of angular momentum. We prove that a particle flow is passing over the collision singularity preserving differentiability with respect to initial data if and only if = 2(1–1/n), n positive integer, n 2.This result coincides with the outcome of block regularization applied by McGehee to the same dynamical problem. We discuss that this identity was to expect since both methods target the same physical constraint over the flow.