Orbit's Structure in the Isosceles Rectilinear Restricted Three-Body Problem

The isosceles rectilinear restricted three-body problem can be considered as the Sitnikov's problem with eccentricity one or, as the isosceles problem when the central mass is zero and the primaries move having consecutive elliptic collisions. We compactify the phase space and analyze the flow on their boundary. This allows us to separate the phase space into different regions depending on the kind of orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

Surfaces of zero kinetic energy in a general three-body problem

We generalize the concept of zero-velocity surface and construct zero-kinetic-energy surfaces. In the space of three mutual distances, we determine the regions where motion is possible; these regions are in the shape of an infinitely long tripod. Motions in the three-body problem are shown to be unstable according to Hill.     

Chaotic Scattering in the Restricted Three‐Body Problem II. Small mass parameters

We study the scattering motion of the planar restricted three‐body problem for small mass parameters μ. We consider the symmetric periodic orbits of this system with μ = 0 that collide with the singularity together with the circular and parabolic solutions of the Kepler problem. These divide the parameter space in a natural way and characterize the main features of the scattering problem for small non‐vanishing μ. Indeed, continuation of these orbits yields the primitive periodic orbits of the system for small μ. For different regions of the parameter space, we present scattering functions and discuss the structure of the chaotic saddle. We show that for μ < μc and any Jacobi integral there exist departures from hyperbolicity due to regions of stable motion in phase space. Numerical bounds for μc are given. This revised version was published online in July 2006 with corrections to the Cover Date.     

Zero velocity surfaces for the general planar three-body problem

The angular momentum and the energy integral of the planar three-body problem are used to establish regions of the physical space where motion is allowed to take place. Although forbidden regions exist for both negative and positive values of the energy of the system, the known integrals of the motion always allow for at least one of the three bodies to escape.     

Linear stability of the triangular equilibrium points in the photogravitational elliptic restricted problem,I

The linear stability of the triangular equilibrium points in the photogravitational elliptic restricted problem is examined and the stability regions are determined in the space of the parameters of mass, eccentricity, and radiation pressure. It is found that radiation pressure of the larger body for solar system cases exerts only a small quantitative influence on the stability regions.     

Short Time Lyapunov Indicators in the Restricted Three-Body Problem

We applied the method of the short time Lyapunov indicators to the planar circular and to the planar elliptic restricted three-body problem in order to study the structure of the phase space in some selected regions. In the circular case we computed the short-time averages of the stretching numbers to distinguish between regular and chaotic domains of the phase space. The results obtained in this way are in good agreement with the corresponding Poincaré's surface of sections. Using the short time Lyapunov indicators we determined the detailed structure of the phase space in the semi-major axis-eccentricity plane of the test particle both in the circular and in the elliptic restricted problem (in the latter case for some values of the eccentricities of the primaries) and we studied the main features of the phase space.This revised version was published online in October 2005 with corrections to the Cover Date.     

Dynamics of Two Planets in the 2/1 Mean-Motion Resonance

The dynamics of two planets near a first-order mean-motion resonance is modeled in the domain of the general three-body planar problem. The system studied is the pair Uranus-Neptune (2/1 resonance). The phase space of the resonance and near-resonance regions is studied by means of surfaces of section and spectral analysis techniques. After a thorough investigation of the topology of the phase space, we find that several regimes of motion are possible for the Uranus-Neptune system, and the regions of transition between the regimes of motion are the seats of chaotic motion. This revised version was published online in July 2006 with corrections to the Cover Date.     

Structure of the 2:1 and 3:2 jovian resonances

The chaotic regions of the phase space in the vicinity of the 2:1 and 3:2 jovian resonances are identified by using a mapping technique derived from a second-order expansion of the disturbing function for the planar elliptical restricted three-body problem. It is shown that both resonances have extensive chaotic regions which in some cases can lead to large changes in the eccentricity of asteroid orbits. Although the 3:2 resonance is shown to be more chaotic than the 2:1 resonance, the existence of the Hilda group of asteroids and the Hecuba gap may be explained by distinct differences in the location of the high-eccentricity regions at each resonance. The problem of the convergence of the expansion of the disturbing function in the outer asteroid belt is also discussed.     

A perturbation method for problems with two critical arguments

We develop a semi-numerical perturbation method for problems with two critical arguments. We apply it to a truncated model of the restricted, elliptic three body problem in case of a resonance 2/1 We identify regions of the phase space where chaotic motion is expected because of the presence of homoclinic orbits. One of these regions, the largest one, sits at the entrance to the resonance zone and is associated with a 2/1 resonance between the two critical arguments. The results are compared with numerical results due to Murray (1986)     

Using FLI maps for preliminary spacecraft trajectory design in multi-body environments

Fast Lyapunov Indicator (FLI) maps are presented as a tool for solving spacecraft preliminary trajectory design problems in multi-body environments with long-term stability requirements. In particular, the FLI maps are shown to provide a global overview of the dynamics in the restricted three-body problem that can guide mission designers in selecting long-term stable regions of phase space which are inherently more robust to model parameter perturbations. The FLI is also shown to numerically detect the normally hyperbolic manifolds associated with unstable periodic orbits. These, in turn, provide a global map of the principal heteroclinic connections between the various resonance regions which form the basic backbone of dynamical transfers design. Examples of maps and transfers are provided in the restricted three-body problem modeling the Jupiter–Europa system.     

Some results on the global dynamics of the regularized restricted three-body problem with dissipation

We perform an analysis of the dynamics of the circular, restricted, planar three-body problem under the effect of different kinds of dissipation (linear, Stokes and Poynting–Robertson drags). Since the problem is singular, we implement a regularization technique in the style of Levi–Civita. The effect of the dissipation is often to decrease the semi-major axis; as a consequence the minor body collides with one of the primaries. In general, it is quite difficult to find non-collision orbits using random initial conditions. However, by means of the computation of the Fast Lyapunov Indicators (FLI), we obtain a global view of the dynamics. Precisely, we detect the regions of the phase space potentially belonging to basins of attraction. This investigation provides information on the different regions of the phase space, showing both collision and non-collision trajectories. Moreover, we find periodic orbit attractors for the case of linear and Stokes drags, while in the case of the Poynting–Robertson effect no other attractors are found beside the primaries, unless a fourth body is added to counterbalance the dissipative effect.     

Stability of axial orbits in analytical galactic potentials

We study the stability of axial orbits in analytical galactic potentials as a function of the energy of the orbit and the ellipticity of the potential. The problem is solved by an analytical method, the validity of which is not limited to small amplitudes. The lines of neutral stability divide the parameter space in regions corresponding to different organizations of the main families of orbits in the symmetry planes.     

Mass ratios of three bodies for stable low-inclination three-dimensional periodic motion

The intervals of possible stability, on the -axis, of the basic families of three-dimensional periodie motions of the restricted three-body problem (determined in an earlier paper) are extended into regions of the -m ₃ parameter space of the general three-body problem. Sample three-dimensional periodic motions corresponding to these regions are computed and tested for stability. Six regions, corresponding to the vertical-critical orbitsl1v, m1v,m2v, andilv, survive this preliminary stability test-therefore, emerging as the mass parameters regions allowing the simplest types of stable low inclination three-dimensional motion of three massive bodies.     

Comparative Study of the 2:3 and 3:4 Resonant Motion with Neptune: An Application of Symplectic Mappings and Low Frequency Analysis

The 2:3 and 3:4 exterior mean motion resonances with Neptune are studied by applying symplectic mapping models. The mappings represent efficiently Poincaré maps for the 3D elliptic restricted three body problem in the neighbourhood of the particular resonances. A large number of trajectories is studied showing the coexistence of regular and chaotic orbits. Generally, chaotic motion depletes the small bodies of the effective resonant region in both the 2:3 and 3:4 resonances. Applying a low frequency spectral analysis of trajectories, we determined the phase space regions that correspond to either regular or chaotic motion. It is found that the phase space of the 3:4 resonant motion is more chaotic than the 2:3 one.     

Sundman surfaces and Hill stability in the three-body problem

Using the famous Sundman inequality, we have constructed for the first time the surfaces for the general three-body problem that we suggest calling Sundman surfaces. These surfaces are a generalization of the widely known Hill surfaces in the restricted circular three-body problem. The Sundman surfaces are constructed in a rectangular coordinate system that uses the mutual distances between the bodies as the Cartesian rectangular coordinates. The singular points of the family of these surfaces have been determined. The possible and impossible regions of motion of the bodies have been constructed in the space of mutual distances. We have shown the existence of Hill stable motions and established sufficient criteria for Hill stability of motions. Some of the astronomical applications are considered.     

Stars and substars nearest to the Sun: A study review

The structure of the solar neighborhood and its position in the galaxy, including the galactocentric distances, are analyzed. Catalogues are examined of near-sun stars and substars, which are compiled from ground and space based observations in the optical and infrared spectral regions. The problem of classifying celestial bodies in the galaxy by astrophysical and cosmogonic criteria is discussed. The problem of determining the main characteristics of the nearest stars and substars is analyzed. The statistical relationships are investigated between the main characteristics of stars and substars on the basis of their physical evolutionary models. The physical sense of these relationships is discussed. The calculated differential distribution functions of the astrophysical properties of stars and substars are examined.     

Approximate General Solution Of The Two-Dimensional Duffing Dynamical System

This paper presents the approximate general solution of the triple well, double oscillator non-linear dynamical system. This system is non-integrable and the approximate general solution is calculated by application of the Last Geometric Theorem of Poincaré (Birkhoff, 1913, 1925). The original problem, known as the Duffing one, is a 1 degree of freedom system that, besides the conservative force component, includes dumping and external forcing terms (see details in the web site: http://www.uncwil.edu/people/hermanr/chaos/ted/chaos.html). The problem considered here is a 2 degree of freedom, autonomous and conservative one, without dumping, and of axisymmetric potential. The space of permissible motions is scanned for identification of all solutions re-entering after from one to nine oscillations and the precise families of periodic solutions are computed, including their stability parameter, covering all cases with periods T corresponding to 4osc/T. Seven sub-domains of the space of solutions were investigated in detail by zooming, an operation that proved the possibility to advance the accuracy of the approximate general solution to the level permitted by the integration routine. The approximation of the general solution, although impressive, provides clear evidence of the complexity of the problem and the need to proceed to larger period families. Nevertheless, it allows prediction of the areas where chaos and order regions in the Poincaré surfaces of section are to be expected. Examples of such surfaces of sections, as well as of types of closed solutions, are given. Two peculiar points of the space of solutions were identified as crossing, or source points from which infinite families of periodic solutions emanate. The morphology and stability of solutions of the problem are studied and discussed.     

A Note on Parameter Values for Stable Low-Inclination Periodic Motion in the Restricted Three-Body Problem with Radiation

The intervals of the mass parameter (μ) values for possible stability of the basic families of 3D periodic orbits in the restricted three-body problem determined elsewhere are now extended into regions of theμ - q ₁ parameter space of the photogravitational restricted three-body problem, where q ₁ is the radiation factor of m ₁ and it is assumed that m ₂ does not radiate. Several 3D periodic orbits corresponding to these regions are computed and tested for stability and seven regions, corresponding to the vertical-critical orbits l1v, l'1v, l6v, m1v, m2v and i1v, survive this stability test, emerging as the regions allowing the simplest types of stable low inclination 3D motion of the infinitesimal particle. This revised version was published online in July 2006 with corrections to the Cover Date.     

Active solar radio regions at metric frequencies and the interplanetary sector structures

The possible relation between type I noise active regions and the polarity distribution of the interplanetary magnetic field is examined for the period from 13 March to 21 August, 1968 (Solar Rotation Numbers 1842–1847) by using data from ground-based and satellite observations. In general four type I radio regions appeared during each solar rotation period except for Rotation No. 1842. The number of type I regions is the same as the number of sector boundaries. This result suggests that the configuration of the photospheric magnetic field extending into the interplanetary space may be related to the origin of the type I radio regions. Statistically the passage of the sector boundaries is delayed by approximately 5 days after the central meridian passage of the type I noise regions on the solar disk.The position of the source of the sector boundaries and its relation to the type I radio regions are investigated by taking into account the mean bulk velocity of solar winds as observed by space probes. A model of the large-scale structure of type I radio regions and their relation to the sector structure of the magnetic field as observed in the interplanetary space is briefly discussed.NASA Research Associate at the University of Maryland.