The Trojan problem

There has been a renewed interest in the Trojan problem in recent years. Significant progress has been made in discovering and understanding dynamical features of motion of Jupiter's Trojan asteroids. The dynamics of hypothetical Trojan-type asteroids of other major planets has also been the subject of several recent investigations. This paper offers an overview on the current status of researches on real and hypothetical Trojan asteroids of the major planets. Results of analytical and numerical works are surveyed. Questions of dynamical properties, long-term evolution of orbits, stability regions around the triangular Lagrangian points are discussed among other problems of the Trojans.     

Rigorous estimates for the series expansions of Hamiltonian perturbation theory

In the present paper we prove a theorem giving rigorous estimates in the problem of bringing to normal form a nearly integrable Hamiltonian system, using methods of classical perturbation theory, i.e. series expansions in the small parameter . For any order of normalization, we give a lower bound _* ^r for the convergence radius of the normalized Hamiltonian, and a greater bound for the remainder, i.e. the non normalized part of the Hamiltonian. As an application, we consider the case of weakly coupled harmonic oscillators with highly nonresonant frequencies and show how, by optimizing, for fixed , the orderr of normalization, one gets for the remainder a greater bound of the formAe ^–(*₁/) ^a , with positive constantsA,a and ₁ ^* exponential estimate of Nekhoroshev's type.     

On Broucke's velocity-related series expansions in the two-body problem

This short article supplements a recent paper by Dr R. Broucke on velocity-related series expansions in the two-body problem. The derivations of the Fourier and Legendre expansions of the functionsF(v), \(\sqrt {F(\upsilon )} \) and \(\sqrt {{1 \mathord{\left/ {\vphantom {1 {F(\upsilon )}}} \right. \kern-0em} {F(\upsilon )}}} \) are given, where $$F(\upsilon ) = (1 - e^2 )/(1 + 2e\cos \upsilon + e^2 ), e< 1$$ In the two-body problem,v is identified with the true anomaly,e the eccentricity andF(v) equals (an/V)². Some interesting relations involving Legendre polynomials are also noted.     

Asymptotic expansions in the perturbed two-body problem with application to systems with variable mass

The main theorems of the theory of averaging are formulated for slowly varying standard systems and we show that it is possible to extend the class of perturbation problems where averaging might be used. The application of the averaging method to the perturbed two-body problem is possible but involves many technical difficulties which in the case of the two-body problem with variable mass are avoided by deriving new and more suitable equations for these perturbation problems. Application of the averaging method to these perturbation problems yields asymptotic approximations which are valid on a long time-scale. It is shown by comparison with results obtained earlier that in the case of the two-body problem with slow decrease of mass the averaging method cannot be applied if the initial conditions are nearly parabolic. In studying the two-body problem with quick decrease of mass it is shown that the new formulation of the perturbation problem can be used to obtain matched asymptotic approximations.     

A note on velocity-related series expansions in the two-body problem

The present note describes a few important series expansions in the two-body problem. They are related to the magnitudeV of the velocity vector and they are important for the treatment of atmospheric drag with the method of general perturbations. These series have been obtained with computerized Poisson series Manipulations. The results are given to order seven in the eccentricity, for both the Mean Anomaly and the True Anomaly.     

Series expansions for encounter-type solutions of Hill's problem

Hill's problem is defined as the limiting case of the planar three-body problem when two of the masses are very small. This paper describes analytic developments for encounter-type solutions, in which the two small bodies approach each other from an initially large distance, interact for a while, and separate. It is first pointed out that, contrary to prevalent belief, Hill's problem is not a particular case of the restricted problem, but rather a different problem with the same degree of generality. Then we develop series expansions which allow an accurate representation of the asymptotic motion of the two small bodies in the approach and departure phases. For small impact distances, we show that the whole orbit has an adiabatic invariant, which is explicitly computed in the form of a series. For large impact distances, the motion can be approximately described by a perturbation theory, originally due to Goldreich and Tremaine and rederived here in the context of Hill's problem.     

Fractal structures for the Jacobi Hamiltonian of restricted three-body problem

We study the dynamical chaos and integrable motion in the planar circular restricted three-body problem and determine the fractal dimension of the spiral strange repeller set of non-escaping orbits at different values of mass ratio of binary bodies and of Jacobi integral of motion. We find that the spiral fractal structure of the Poincaré section leads to a spiral density distribution of particles remaining in the system. We also show that the initial exponential drop of survival probability with time is followed by the algebraic decay related to the universal algebraic statistics of Poincaré recurrences in generic symplectic maps.     

New contributions to the problem of capture

We present a treatment of libration-point capture in the restricted three-body problem. Examples of capture are given, and a long-term numerical integration is presented, to illustrate major features of orbits arising from capture. A theory of lifetimes is given, providing order-of-magnitude (though rather conservative) estimates of the time a body remains captured. A general capture criterion, giving bounds on admissible values of the postcapture semimajor axis, for given values of eccentricity and inclination. This criterion is used to demonstrate that, in general, direct postcapture orbits lie outside retrograde ones. We also emphasize the importance of mass-change, of one or both primaries, in producing capture. This phenomenon is shown to give rise to a new type of capture, “pull-down capture,” which produces retrograde orbits. The effects of nebular drag also are noted.These results suggest the improbability of a capture origin for Jupiter's outer satellites within the last 4+ billion years, or since the solar system reached its present dynamical configuration. Computations indicate, however, that either mass-change or nebular drag could have been effective in producing capture. The outer satellite groups are shown to resemble Hirayama families physically, thus supporting a hypothesis of capture followed by collisional fragmentation.     

A note on expansions of functions of velocity in the two-body problem

Fourier expansions of functions of velocity in the two-body problem are obtained in terms of both the true anomaly and the mean anomaly.     

Convergence of the expansions of the disturbing functions in the planar three-body planetary problem

In the framework of the planar three-body planetary problem, conditions are found for the absolute convergence of the expansions of the disturbing functions in powers of the eccentricities, with coefficients represented by trigonometric polynomials with respect to the mean, eccentric, or true anomaly of the inner planet. It is found that using the eccentric or true anomaly as the independent variable instead of the mean anomaly (or time) extends the holomorphy domain of the principal part of the perturbation functions. The expansions of the second parts converge in open bicircles, which admit values of the eccentricity of the inner planet in excess of the Laplace limit.     

An asymptotic solution for the Trojan case of the plane elliptic restricted problem of three bodies

The planar motion of a Trojan asteroid is considered within the framework of the elliptic restricted three-body problem. The solution is derived asymptotically to second order taking the square root of the Jupiter-Sun mass ratio and the orbital eccentricity of Jupiter as first order quantities. The results are given in explicit form for the coordinates as functions of the true anomaly of Jupiter including both short and long periodic terms resulting from the orbital accentricity of Jupiter.     

Polynomial expansions of single-mode motions around equilibrium points in the circular restricted three-body problem

In this work, the single-mode motions around the collinear and triangular libration points in the circular restricted three-body problem are studied. To describe these motions, we adopt an invariant manifold approach, which states that a suitable pair of independent variables are taken as modal coordinates and the remaining state variables are expressed as polynomial series of them. Based on the invariant manifold approach, the general procedure on constructing polynomial expansions up to a certain order is outlined. Taking the Earth–Moon system as the example dynamical model, we construct the polynomial expansions up to the tenth order for the single-mode motions around collinear libration points, and up to order eight and six for the planar and vertical-periodic motions around triangular libration point, respectively. The application of the polynomial expansions constructed lies in that they can be used to determine the initial states for the single-mode motions around equilibrium points. To check the validity, the accuracy of initial states determined by the polynomial expansions is evaluated.     

Theory of the Trojan asteroids

In the previously published Parts I and II of the paper, the author has constructed a formal long-periodic solution for the case of 11 resonance in the restricted problem of three bodies to 0(m ^3/2), wherem is the small mass parameter of the system. The time-dependencet(, ,m), where is the mean synodic longitude and is related to the Jacobi constant, has been expressed by ahyperelliptic integral. It is shown here that with the approximationm=0 in the integrand, the functiont(, , 0) can be expanded in a series involving standardelliptic functions. Then the problem of inversion can be formally solved, yielding the function (t, , 0).Similarly, the normalized period (,m) of the motion can be approximated by theHagihara hyperelliptic integral (, 0), corresponding tom=0. This integral is also expanded into elliptic functions. Asymptotic forms for (, 0) are derived for 0 and for 1, corresponding to the extreme members of thetadpole branch of the family of orbits.     

Three-body planetary problem: study of KAM stability for the secular part of the Hamiltonian

The Hamiltonian representing the average over the mean-motion angles (i.e. the secular part) of the three-body planetary problem is considered. An efficient algorithm constructing invariant tori for the trajectories in phase space is provided. To give a possible practical application, we consider a toy-model including the main terms of the secular part of a hypothetic Sun-Jupiter-Saturn system having eccentricities and inclinations equal to 1/20 of the true ones. The scheme of a KAM proof of the stability of the model is sketched. The proof is “computer assisted”.     

A theory of the Trojan asteroids

The paper constructs a long-periodic solution for the case of 11 resonance in the restricted problem of three bodies. The solution is smoothed by the exclusion of the internal resonant terms arising from the near-commensurability between the long and the short periods of the asteroid.Although the Brown (1911) conjecture regarding the termination of the family of the tadpoles at the Lagrangian pointL ₃ is not supported by our analysis, the conjecture seems to hold in the limit asm0.     

Hamilton系统数值计算的新方法

系统地介绍了近年来对Ｈａｍｉｌｔｏｎ系统数值计算新建立的辛算法和线性对称多步法，并对它们在动力天文中的应用作了一简要回顾。     

Some expansions of the elliptic motion to high eccentricities

Some classic expansions of the elliptic motion — cosmE and sinmE — in powers of the eccentricity are extended to highly eccentric orbits, 0.6627...<e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients are given in terms of the derivatives of Bessel functions with respect to the eccentricity. The expansions have the same radius of convergence (e*) of the extended solution of Kepler's equation, previously derived by the author. Some other simple expansions — (a/r), (r/a), (r/a) sinv, ..., — derived straightforward from the expansions ofE, cosE and sinE are also presented.     

Expansion of the Hamiltonian for the planetary problem into a Poisson series in the heliocentric reference frame

An expansion of the Hamiltonian for the N-planet problem into a Poisson series using a system of modified (complex) Poincare´ canonical elements in the heliocentric coordinate system is constructed. The Lagrangian and Hamiltonian formalisms are used. The first terms in the expansions of the principal and complementary parts of the disturbing function are presented. Estimates of the number of terms in the presented expansions have been obtained through numerical experiments. A comparison with the results of other authors is made.     

Theory of the Trojan asteroids Part II

In the previously published (1977a) Part I of the paper, the author has constructed a formal long-periodic solution for the case of 1:1 resonance in the restricted problem of three bodies. Here the accuracy of the solution is carried fromO(m) toO(m ^3/2), wherem is the mass parameter of the system.Asymptotic approximations for the period of the motion are obtained for the case of small oscillations about the Lagrangian pointL ₄, in agreement with the classical theory, and for the vicinity of a logarithmic singularity on themean separatrix, passing throughL ₃. The regularizing function (), which removes the singularities of the Poincaré type, is extended to all orders, and the result is used to prove the periodicity of the solution.