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.     

The Lagrange-Jacobi equation in the finite-size many-body problem

We determine the values of the barycentric energy constant that necessarily result in collisions between bodies. The standard Hill stability regions in the problem of four or more bodies are shown to be located inside the regions where collisions are inevitable. Only in the problem of three finite bodies is part of the Hill stability region preserved where the bodies can move without colliding with one another. We point out possible astronomical applications of our results.     

The Photogravitational Hill Problem: Numerical Exploration

We consider the recently introduced version of the classical Lunar Hill problem, the photogravitational Hill problem, and study it's equilibrium points and zero-velocity curves. The full network of families of periodic orbits is numerically explored, their stability is computed and critical orbits are determined. Non-periodic orbits are also computed as points on a surface of section, providing an outlook of the stability regions, chaotic motions and escape.     

Stability in the Full Two-Body Problem

Stability conditions are established in the problem of two gravitationally interacting rigid bodies, designated here as the full two-body problem. The stability conditions are derived using basic principles from the N-body problem which can be carried over to the full two-body problem. Sufficient conditions for Hill stability and instability, and for stability against impact are derived. The analysis is applicable to binary small-body systems such as have been found recently for asteroids and Kuiper belt objects.     

A Review on Co-orbital Motion in Restricted and Planetary Three-body Problems

The 1:1 mean motion resonance may be referred to as the lowest order mean motion resonance in restricted or planetary three-body problems. The five well-known libration points of the circular restricted three-body problem are five equilibriums of the 1:1 resonance. Coorbital motion may take different shapes of trajectory. In case of small orbital eccentricities and inclinations, tadpole-shape and horseshoe-shape orbits are well-known. Other 1:1 libration modes different from the elementary ones can exist at moderate or large eccentricities and inclinations. Coorbital objects are not rare in our solar system, for example the Trojans asteroids and the coorbital satellite systems of Saturn. Recently, dozens of coorbital bodies have been identified among the near-Earth asteroids. These coorbital asteroids are believed to transit recurrently between different 1:1 libration modes mainly due to orbital precessions, planetary perturbations, and other possible effects. The Hamiltonian system and the Hill’s three-body problem are two effective approaches to study coorbital motions. To apply the perturbation theory to the Hamiltonian system, standard procedures involve the development of the disturbing function, averaging and normalization, theory of ideal resonance model, secular perturbation theory, etc. Global dynamics of coorbital motion can be revealed by the Hamiltonian approach with a suitable expansion. The Hill’s problem is particularly suitable for the studies on the relative motion of two coorbital bodies during their close encounter. The Hill’s equation derived from the circular restricted three-body problem is well known. However, the general Hill’s problem whose equation of motion takes exactly the same form applies to the non-restricted case where the mass of each body is non-negligible, namely the planetary case. The Hill’s problem can be transformed into a “canonical shape” so that the averaging principle can be applied to construct a secular perturbation theory. Besides the two analytical theories, numerical methods may be consulted, for example the approach of periodic orbit, the surface of section, and the computation of invariant manifolds carried by equilibriums or periodic orbits.     

The Restricted Hill Four-Body Problem with Applications to the Earth–Moon–Sun System

Starting from the four-body problem a generalization of both the restricted three-body problem and the Hill three-body problem is derived. The model is time periodic and contains two parameters: the mass ratio ν of the restricted three-body problem and the period parameter m of the Hill Variation orbit. In the proper coordinate frames the restricted three-body problem is recovered as m → 0 and the classical Hill three-body problem is recovered as ν → 0. This model also predicts motions described by earlier researchers using specific models of the Earth–Moon–Sun system. An application of the current model to the motion of a spacecraft in the Sun perturbed Earth–Moon system is made using Hill's Variation orbit for the motion of the Earth–Moon system. The model is general enough to apply to the motion of an infinitesimal mass under the influence of any two primaries which orbit a larger mass. Using the model, numerical investigations of the structure of motions around the geometric position of the triangular Lagrange points are performed. Values of the parameter ν range in the neighborhood of the Earth–Moon value as the parameter m increases from 0 to 0.195 at which point the Hill Variation orbit becomes unstable. Two families of planar periodic orbits are studied in detail as the parameters m and ν vary. These families contain stable and unstable members in the plane and all have the out-of-plane stability. The stable and unstable manifolds of the unstable periodic orbits are computed and found to be trapped in a geometric area of phase space over long periods of time for ranges of the parameter values including the Earth–Moon–Sun system. This model is derived from the general four-body problem by rigorous application of the Hill and restricted approximations. The validity of the Hill approximation is discussed in light of the actual geometry of the Earth–Moon–Sun system. This revised version was published online in July 2006 with corrections to the Cover Date.     

太阳系P型逆行小天体运动的稳定区域问题

本文讨论了太阳系Ｐ型逆行小天体运动的稳定区域问题，首先，指出了Ｐ型逆行小天体顺行小天体的Ｈｉｌｌ稳定区域及数值稳定区域的差别，然后，讨论了大行星间Ｐ型逆行小天体存在的可能性问题。     

Planar motions of a charged particle in a magnetic-binary system with spherical or oblate primaries

In this paper we present for the first time simple symmetric motions in the planar magnetic-binary problem where both primaries are spherical bodies or oblate spheroids. From the study of this case it follows that there is a dense and complicated distribution of the families of such motions in the phase space. Our results also show that the orbital characteristics of the particle and the configuration of the phase space are appreciably affected by the oblateness of the primaries only if these parameters become sufficiently large.     

On the effect of the eccentricity of a planetary orbit on the stability of satellite orbits

The effect of the eccentricity of a planet’s orbit on the stability of the orbits of its satellites is studied. The model used is the elliptic Hill case of the planar restricted three-body problem. The linear stability of all the known families of periodic orbits of the problem is computed. No stable orbits are found, the majority of them possessing one or two pairs of real eigenvalues of the monodromy matrix, while a part of a family with complex instability is found. Two families of periodic orbits, bifurcating from the Lagrangian points L₁, L₂ of the corresponding circular case are found analytically. These orbits are very unstable and the determination of their stability coefficients is not accurate, so we compute the largest Liapunov exponent in their vicinity. In all cases these exponents are positive, indicating the existence of chaotic motions     

Refined model for the evolution of distant satellite orbits

We consider a model that describes the evolution of distant satellite orbits and that refines the solution of the doubly averaged Hill problem. Generally speaking, such a refinement was performed previously by J. Kovalevsky and A.A. Orlov in terms of Zeipel’s method by constructing a solution of the third order with respect to the small parameter m, the ratio of the mean motions of the planet and the satellite. The analytical solution suggested here differs from the solutions obtained by these authors and is closest in form to the general solution of the doubly averaged problem (∼m ²). We have performed a qualitative analysis of the evolutionary equations and conditions for the intersection of satellite orbits with the surface of a spherical planet with a finite radius. Using the suggested solution, we have obtained improved analytical time dependences of the elements of evolving orbits for a number of distant satellites of giant planets compared to the solution of the doubly averaged Hill problem and, thus, achieved their better agreement with the results of our numerical integration of the rigorous equations of perturbed motion for satellites.     

Axisymmetric field generation within an ambient axial field

The generation of magnetic field in a homogeneous, electrically conducting fluid – as required for the dynamo generation of the fields of many astrophysical bodies – is normally a threshold process; the dynamo mechanism, applicable to such bodies in unmagnetised environments, requires motions of sufficient strength to overcome the innate magnetic diffusion. In the presence of an ambient field, however, the critical nature of the field generation process is relaxed. Motions can distort and amplify the ambient field for all amplitudes of flow. For motions with appropriate geometries, an internal ‘dynamo‐like’ field of appreciable strength can be generated, even for relatively weak flows. At least a minority of planets, moons and other bodies exist within significant external astrophysical fields. For these bodies, the ambient field problem is more relevant than the classical dynamo problem, yet it remains relatively little studied. In this paper we consider the effect of an axial ambient field on a spherical mean‐field α ²ω dynamo model, through nonlinear calculations with α ‐quenching feedback. Ambient fields of varying strengths, and both stationary and oscillatory in time, are imposed. Particular focus is placed on the effects of these fields on the equatorial symmetry and the time dependence of the preferred solutions. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The stability of periodic orbits in the three-body problem

A method is developed to study the stability of periodic motions of the three-body problem in a rotating frame of reference, based on the notion of surface of section. The method is linear and involves the computation of a 4×4 variational matrix by integrating numerically the differential equations for time intervals of the order of a period. Several properties of this matrix are proved and also it is shown that for a symmetric periodic motion it can be computed by integrating for half the period only.This linear stability analysis is used to study the stability of a family of periodic motions of three bodies with equal masses, in a rotating frame of reference. This family represents motion such that two bodies revolve around each other and the third body revolves around this binary system in the same direction to a distance which varies along the members of the family. It was found that a large part of the family, corresponding to the case where the distance of the third body from the binary system is larger than the dimensions of the binary system, represents stable motion. The nonlinear effects to the linear stability analysis are studied by computing the intersections of several perturbed orbits with the surface of sectiony ₃=0. In some cases more than 1000 intersections are computed. These numerical results indicate that linear stability implies stability to all orders, and this is true for quite large perturbations.     

Mean-field approach to spherical dynamo models

The mean-field approach to dynamo theory has proved to be a useful tool in the investigation of cosmical magnetic fields. This paper gives a systematic discussion of this approach for spherical dynamo models as suggested by cosmical bodies. At first some fundamentals of dynamo theory are explained with particular attention to formulations in terms of toroidal and poloidal magnetic fields. Starting from the general ideas of mean-field magnetohydrodynamics the relevant mean-field equations for the models envisaged are derived and discussed. The considerations are not restricted to motions of turbulent nature, motions with more or less regular flow patterns are admitted too. A new representation of the crucial electromotive force caused by the fluctuating motions is given. For an important special case the dependence of this electromotive force on the motions is calculated. The mean-field concept is in particular elaborated under the assumption that the motions show certain symmetry and stationarity properties as to be expected at cosmical bodies. The respective results for the electromotive force caused by the fluctuating motions are discussed in detail. Within this frame the possibilities of dynamo mechanisms are systematically studied. In addition to the well-known α² and αω-mechanisms some others, termed β², βω, and δω-mechanisms, are envisaged, whose relevance for cosmical objects remains to be investigated. In a following paper numerical results are given for dynamo models with mechanisms as envisaged here.     

Search for and study of nearly periodic orbits in the plane problem of three equal-mass bodies

We analyze nearly periodic solutions in the plane problem of three equal-mass bodies by numerically simulating the dynamics of triple systems. We identify families of orbits in which all three points are on one straight line (syzygy) at the initial time. In this case, at fixed total energy of a triple system, the set of initial conditions is a bounded region in four-dimensional parameter space. We scan this region and identify sets of trajectories in which the coordinates and velocities of all bodies are close to their initial values at certain times (which are approximately multiples of the period). We classify the nearly periodic orbits by the structure of trajectory loops over one period. We have found the families of orbits generated by von Schubart’s stable periodic orbit revealed in the rectilinear three-body problem. We have also found families of hierarchical, nearly periodic trajectories with prograde and retrograde motions. In the orbits with prograde motions, the trajectory loops of two close bodies form looplike structures. The trajectories with retrograde motions are characterized by leafed structures. Orbits with central and axial symmetries are identified among the families found.     

Existence of particular solutions in the generalised restricted problem of three bodies with variable masses

The restricted problem of three bodies with variable masses is considered. It is assumed that the infinitesimal body is axisymmetric with constant mass and the finite bodies are spherical with variable masses such that the ratio of their masses remains constant. The motion of the finite bodies are determined by the Gyldén-Meshcherskii problem. It is seen that the collinear, triangular, and coplanar solutions not exist, but these solutions exist when the infinitesimal body be a spherical.     

On the non-linear stability of motions around L5 in the elliptic restricted problem of the three bodies

The non-linear stability of motions around L₅ in the elliptic restricted problem of the three bodies is investigated numerically with emphasis on the effect of the orbital eccentricity of the primaries on the shape of the established stability regions. It is shown that with increasing eccentricity, the width of these regions is decreasing.     

The restricted problem of a tri-axial rigid body and two spherical bodies with variable masses

The restricted problem of a tri-axial rigid body and two spherical bodies with variable masses be considered. The general solution of the equations of motion of the tri-axial body be obtained in which the motion of the spherical bodies is determined by the classic nonsteady Gyldén-Meshcherskii problem.     

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.     

Parametric evolution of periodic orbits in the restricted four-body problem with radiation pressure

The backbone of the analysis in most dynamical systems is the study of periodic motions, since they greatly assist us to understand the structure of all possible motions. In this paper, we deal with the photogravitational version of the rectilinear restricted four-body problem and we investigate the dynamical behaviour of a small particle that is subjected to both the gravitational attraction and the radiation pressure of three bodies much bigger than the particle, the primaries. These bodies are always in syzygy and two of them have equal masses and are located at equal distances from the third primary. We study the effect of radiation on the distribution of the periodic orbits, their stability, as well as the evolution of the families and their main features.