Relative motion of near orbiting satellites

The relative motion of two particles on adjacent orbits about the same primary has been investigated under the condition that both motions have the same period. The geometrical properties of the relative displacement and velocity traces, on representative planes, are studied. A complete state of the motion is given; and, the range and range-rate variations, over one or more orbits, are described.It has been found that cusps appear on some of the traces provided that a proper relationship exists between the eccentricity and inclination. (Here, one particle moves on a circular path while the second moves on an ellipse.) The conditions for which cusps appear are given, and typical traces are shown.     

Celestial-mechanical peculiarities of Uranus’s satellite system

We consider the structural peculiarities of Uranus’s satellite system associated with its separation into two groups: inner equatorial satellites moving in nearly circular orbits and distant irregular satellites with retrograde motion in highly elliptical orbits. The intermediate region is free from satellites in a wide range of semimajor axes. By analyzing the evolution of satellite orbits under the combined effect of solar attraction and Uranus’s oblateness, we offer a celestial-mechanical explanation for the absence of equatorial satellites in this region. M.L. Lidov’s studies during 1961–1963 have served as a basis for our analysis.     

The effect of projection on the observed gas velocity fields in barred galaxies

The problem of determining the pattern of gas motions in the central regions of disk spiral galaxies is considered. Two fundamentally different cases—noncircular motions in the triaxial bar potential and motions in circular orbits but with orientation parameters different from those of the main disk—are shown to have similar observational manifestations in the line-of-sight velocity field of the gas. A reliable criterion is needed for the observational data to be properly interpreted. To find such a criterion, we analyze two-dimensional nonlinear hydrodynamic models of gas motions in barred disk galaxies. The gas line-of-sight velocity and surface brightness distributions in the plane of the sky are constructed for various inclinations of the galactic plane to the line of sight and bar orientation angles. We show that using models of circular motions for inclinations i>60° to analyze the velocity field can lead to the erroneous conclusions of a “tilted (polar) disk” at the galaxy center. However, it is possible to distinguish bars from tilted disks by comparing the mutual orientations of the photometric and dynamical axes. As an example, we consider the velocity field of the ionized gas in the galaxy NGC 972.     

Numerical periodic orbits of charged grains around magnetic planets

We apply a numerical searching method to investigate three-dimensional periodic orbits of charged dust particles in planetary magnetospheres. A classic generalized Stormer model of magnetic planets along with the parameters of Saturn is employed. More periodic orbits are found, besides the already known circular periodic orbits in or parallel to the equatorial plane. We divide all these orbits into six categories based on their appearances. By calculating the characteristic multipliers of the orbits, we investigate the stabilities of these periodic orbits.     

The stability of our solar system

Most investigations of the stability of the solar system have been concerned with the question as to whether the very long term effect of the gravitational attractions of the planets on each other will be to alter the nearly coplanar, nearly circular nature of the orbits in which they move. Analytical investigations in the traditions of Laplace, Lagrange, Poisson and Poincaré strongly indicate stability, though rely on asymptotic expansions with difficult analytical properties. The question is related to the existence of invariant tori, which have been proved to exist in certain motions. Numerical integration experiments have thrown considerable light on possible types of motions, especially in fictitious solar systems in which the planetary masses have been increased to enhance the perturbations, and in testing how critical are stability boundary estimates given by Hill surface type methods.     

Bifurcation of Relative Equilibria in the Main Problem of Artificial Satellite Theory for a Prolate Body

In this paper, we study circular orbits of the J ₂ problem that are confined to constant-z planes. They correspond to fixed points of the dynamics in a meridian plane. It turns out that, in the case of a prolate body, such orbits can exist that are not equatorial and branch from the equatorial one through a saddle-center bifurcation. A closed-form parametrization of these branching solutions is given and the bifurcation is studied in detail. We show both theoretically and numerically that, close to the bifurcation point, quasi-periodic orbits are created, along with two families of reversible orbits that are homoclinic to each one of them.     

Integration theory for the restricted problem of three axisymmetric bodies

In this paper the circular planar restricted problem of three axisymmetric ellipsoids S _i (i = 1, 2, 3), such that their equatorial planes coincide with the orbital plane of the three centres of masses, be considered. The equations of motion of infinitesimal body S ₃ be obtained in the polar coordinates. Using iteration approach we have given an approximation for another integral, which independent of the Jacobian integral, in the case of P-type orbits (near circular orbits surrounding both primaries).     

Particle motion in the field of a five-dimensional charged black hole

In this paper, we have investigated the geodesics of neutral particles near a five-dimensional charged black hole using a comparative approach. The effective potential method is used to determine the location of the horizons and to study radial and circular trajectories. This also helps us to analyze the stability of radial and circular orbits. The radius of the innermost stable circular orbits have also been determined. Contrary to the case of massive particles for which, the circular orbits may have up to eight possible values of specific radius, we find that the photons will only have two distinct values for the specific radii of circular trajectories. Finally we have used the dynamical systems analysis to determine the critical points and the nature of the trajectories for the timelike and null geodesics.     

Secular evolution of the orbits of hypothetical satellites of Uranus

The problem of the secular perturbations of the orbit of a test satellite with a negligible mass caused by the joint influence of the oblateness of the central planet and the attraction by its most massive (or main) satellites and the Sun is considered. In contrast to the previous studies of this problem, an analytical expression for the full averaged perturbing function has been derived for an arbitrary orbital inclination of the test satellite. A numerical method has been used to solve the evolution system at arbitrary values of the constant parameters and initial elements. The behavior of some set of orbits in the region of an approximately equal influence of the perturbing factors under consideration has been studied for the satellite system of Uranus on time scales of the order of tens of thousands of years. The key role of the Lidov–Kozai effect for a qualitative explanation of the absence of small bodies in nearly circular equatorial orbits with semimajor axes exceeding ~1.8 million km has been revealed.     

Disturbance compensating control of orbit radially aligned two-craft Coulomb formation

This paper investigates the orbit radial stabilization of a two-craft virtual Coulomb structure about circular orbits and at Earth–Moon libration points. A generic Lyapunov feedback controller is designed for asymptotically stabilizing an orbit radial configuration about circular orbits and collinear libration points. The new feedback controller at the libration points is provided as a generic control law in which circular Earth orbit control form a special case. This control law can withstand differential solar perturbation effects on the two-craft formation. Electrostatic Coulomb forces acting in the longitudinal direction control the relative distance between the two satellites and inertial electric propulsion thrusting acting in the transverse directions control the in-plane and out-of-plane attitude motions. The electrostatic virtual tether between the two craft is capable of both tensile and compressive forces. Using the Lyapunov’s second method the feedback control law guarantees closed loop stability. Numerical simulations using the non-linear control law are presented for circular orbits and at an Earth–Moon collinear libration point.     

Investigation of the Galactic bar based on photometry and stellar proper motions

A new method for selecting stars in the Galactic bar based on 2MASS infrared photometry in combination with stellar proper motions from the Kharkiv XPM catalogue has been implemented. In accordance with this method, red clump and red giant branch stars are preselected on the color-magnitude diagram and their photometric distances are calculated. Since the stellar proper motions are indicators of a larger velocity dispersion toward the bar and the spiral arms compared to the stars with circular orbits, applying the constraints on the proper motions of the preselected stars that take into account the Galactic rotation has allowed the background stars to be eliminated. Based on a joint analysis of the velocities of the selected stars and their distribution on the Galactic plane, we have confidently identified the segment of the Galactic bar nearest to the Sun with an orientation of 20°–25° with respect to the Galactic center-Sun direction and a semimajor axis of no more than 3 kpc.     

Dynamical causes of asymmetry in the arrangement of gaps in the asteroid belt

The centers of the gaps observed in the asteroid belt are displaced toward Jupiter from their positions that correspond to the exact commensurability between the mean motions of an asteroid and Jupiter. Using the current theory of stability and nonlinear oscillations of Hamiltonian systems, we point out the dynamical causes of this asymmetry. Our analysis is performed in terms of the plane circular restricted three-body problem. The orbits that correspond to Poincaré periodic solutions of the first kind are taken as unperturbed asteroid orbits.     

Equatorial orbits in superposed dipole and uniform magnetic fields

In the present work we study the equatorial motions of charged par ticles that are performed within a field consisting of the superposition of a dipole field—that could represent the magnetic field of a planet — and of a uniform magnetic field normal to the dipole's equator. We use a non-dimensional coordinate system in which the velocity of the charged particle is unit. The model depends on two parameters: the constant of the generalized momentum and the parameter of the interplanetary magnetic field. It is proved that the motion is always bounded. The regions of the motion and the corresponding orbits are studied with respect to the constant of the generalized momentum. We also, investigate numerically conditional periodic and asymptotic orbits.     

Families of periodic orbits in the planar three-body problem

A periodic orbit of the restricted circular three-body problem, selected arbitrarily, is used to generate a family of periodic motions in the general three-body problem in a rotating frame of reference, by varying the massm ₃ of the third body. This family is continued numerically up to a maximum value of the mass of the originally small body, which corresponds to a mass ratiom ₁:m ₂:m ₃?5:5:3. From that point on the family continues for decreasing massesm ₃ until this mass becomes again equal to zero. It turns out that this final orbit of the family is a periodic orbit of the elliptic restricted three body problem. These results indicate clearly that families of periodic motions of the three-body problem exist for fixed values of the three masses, since this continuation can be applied to all members of a family of periodic orbits of the restricted three-body problem. It is also indicated that the periodic orbits of the circular restricted problem can be linked with the periodic orbits of the elliptic three-body problem through periodic orbits of the general three-body problem.     

Mars: Satellite origin and angular momentum

The origin of Phobos and Deimos is considered with a view to accounting for the existence of very small satellites with circular orbits in the Martian equatorial plane, and simultaneously for the suspected angular momentum deficiency of the Mars system. All models considered failed to satisfy at least one requirement, and the problem is considered more puzzling than is at first apparent. The Martian angular momentum deficiency, if physically significant, may be unrelated to the present satellites' origin, but might relate to a large ancient satellite, long ago destroyed. Accretion onto Mars of large amounts of asteroidal dust brought in by Poynting-Robertson drag may have some bearing on the angular momentum problem.     

Numerical models for the study of motion of lunar satellites

The present study deals with numerical modeling of the elliptic restricted three-body problem as well as of the perturbed elliptic restricted three-body (Earth-Moon-Satellite) problem by a fourth body (Sun). Two numerical algorithms are established and investigated. The first is based on the method of the series solution of the differential equations and the second is based on a 5th-order Runge-Kutta method. The applications concern the solution of the equations and integrals of motion of the circular and elliptical restricted three-body problem as well as the search for periodic orbits of the natural satellites of the Moon in the Earth-Moon system in both cases in which the Moon describes circular or elliptical orbit around the Earth before the perturbations induced by the Sun. After the introduction of the perturbations in the Earth-Moon-Satellite system the motions of the Moon and the Satellite are studied with the same initial conditions which give periodic orbits for the unperturbed elliptic problem.     

First-order theory of weakly eccentric orbital motion

Starting from the analytical theory of perturbed circular motions presented in Celestial Mechanics (Bois, 1994), this paper presents an extended resolution valid also for small eccentricity orbits. The solution is of the first order of a small parameter characterizing the magnitude of disturbing forces. The solution has the form of Fourier series with the coefficients given by iterative formation laws. The solution is free from singularities due to small eccentricity or inclination. As an example of numerical application the equatorial artificial satellite orbits are analyzed. For some high satellite orbits with small eccentricity the difference between the numerical integration and the analytical model does not exceed few centimeters per one revolution.On leave from Astronomical Observatory of A. Mickiewicz University, Soneczna 36, PL60-286 Pozna, Poland.