On the Centre Manifold of Collinear Points in the Planar Three-Body Problem

We study the existence of invariant tori in a neighbourhood of the collinear equilibrium points of the planar three-body problem. To this end some properties of the normal form of the Hamiltonian reduced to the 4D central manifold are proved. Using this normal form, we show that the nondegeneracy conditions of KAM theorem are satisfied for all positive masses, including the 2:1 resonance case. The evaluation of the conditions is done numerically.     

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.     

The Inverse Problem for Collinear Central Configurations

We consider the problem: given a collinear configuration of n bodies, find the masses which make it central. We prove that for n ≤ 6, each configuration determines a one-parameter family of masses (after normalization of the total mass). The parameter is the center of mass when n is even and the square of the angular velocity of the corresponding circular periodic orbit when n is odd. The result is expected to be true for any n. This revised version was published online in July 2006 with corrections to the Cover Date.     

Collinear Equilibrium Solutions of Four-body Problem

We discuss the equilibrium solutions of four different types of collinear four-body problems having two pairs of equal masses. Two of these four-body models are symmetric about the center-of-mass while the other two are non-symmetric. We define two mass ratios as μ ₁ = m ₁/M _T and μ ₂ = m ₂/M _T, where m ₁ and m ₂ are the two unequal masses and M _T is the total mass of the system. We discuss the existence of continuous family of equilibrium solutions for all the four types of four-body problems.     

Collinear Central Configuration in Four-Body Problem

In the n-body problem a central configuration is formed if the position vector of each particle with respect to the center of mass is a common scalar multiple of its acceleration vector. We consider the problem: given a collinear configuration of four bodies, under what conditions is it possible to choose positive masses which make it central. We know it is always possible to choose three positive masses such that the given three positions with the masses form a central configuration. However for an arbitrary configuration of four bodies, it is not always possible to find positive masses forming a central configuration. In this paper, we establish an expression of four masses depending on the position x and the center of mass u, which gives a central configuration in the collinear four body problem. Specifically we show that there is a compact region in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of the compact region, it is always possible to choose positive masses to make the configuration central.     

Periodic Orbits of a Collinear Restricted Three-Body Problem

In this paper we study symmetric periodic orbits of a collinear restricted three-body problem, when the middle mass is the largest one. These symmetric periodic orbits are obtained from analytic continuation of symmetric periodic orbits of two collinear two-body problems.     

The Evolution of the Stable and Unstable Manifold of an Equilibrium Point

We consider the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter,ν. The eigenvalues of the linearized system are complex for ν < 0 and purely imaginary for ν > 0. Thus for ν < 0 the equilibrium has a two‐dimensional stable manifold and a two‐dimensional unstable manifold, but for ν > 0 these stable and unstable manifolds are gone. We study the system defined by the truncated generic normal form in this situation. One of two things happens depending on the sign of a certain quantity in the normal form expansion. In one case the two families detach as a single invariant manifold and recedes from the equilibrium as ν tends away from 0 through positive values. In the other case the stable and unstable manifold are globally connected for ν < 0 and the whole structure of these manifolds shrinks to the equilibrium as ν → 0 and disappears. These considerations have interesting implications about Strömgren's conjecture in celestial mechanics and the blue sky catastrophe of Devaney.     

An Addendum to the Damour and Deruelle's Solution of the Post-Newtonian Two-Body Problem

A complement to the the analytical solution given by Damour and Deruelle (D&D) in order to solve the differential equations that describe the conic-like and rectilinear post-Newtonian two-body problem is presented. We show the geometric relation between the eccentric and true anomalies for the elliptic-like and hyperbolic-like cases and expressions for their velocities are obtained. A post-Newtonian version of the Lambert's theorem is shown. This revised version was published online in July 2006 with corrections to the Cover Date.     

SL9彗星撞击木星时S碎片的CCD观测

用上海天文台１．５６ｍ望远镜和ＣＣＤ照相机，观测ｓ＝５碎片时，发现了类似彗尾的奇异现象。     

Collision Probability for Earth-Crossing Asteroids Using Orbital Ranging

We introduce new techniques for the computation of the collision probability for Earth-crossing asteroids in the case of short observational arcs and/or small numbers of observations. The techniques rely on the orbital element probability density computed using statistical orbital ranging. We apply the techniques to the Earth-crossing asteroid 1998 OX₄with non-vanishing collision probability in numerous close approaches after the year 2012 (inclusive). We study the invariance of the collision probability in transformations between different orbital element sets, and develop a Spearman rank correlation measure for the validity of the linear approximation. We introduce an optimized, fast version of the statistical ranging method.     

Homotopy Continuation Method of Arbitrary Order of Convergence for the Two-Body Universal Initial Value Problem

In this paper, an efficient iterative method of arbitrary positive integer order of convergence 2 will be established for the two-body universal initial value problem. The method is of dynamic nature in the sense that, on going from one iterative scheme to the subsequent one, only additional instruction is needed. Moreover, which is the most important, the method does not need any a priori knowledge of the initial guess. A property which avoids the critical situations between divergent to very slow convergent solutions, that may exist in other numerical methods which depend on initial guess. Some applications of the method are also given.     

卫星与空间碎片碰撞预警的快速算法

提出了一种空间目标与空间碎片碰撞预警的快速计算方法，该算法不仅计算速度快，而且减缓了沿迹误差的影响，延长了预警的有效期，利用该方法，检测一个卫星与5800个空间碎片在一天内是否碰撞，计算时间小于3秒；用2天前的资料，预报计算法国CERISE卫星和空间碎片的碰撞时间误差约为0．02秒，空间位置误差约为500—600米．     

Motion close to the Hopf Bifurcation Of the vertical family of periodic orbits of L4

The paper deals with different kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that takes place close to the Lyapunov vertical family of periodic orbits of the triangular equilibrium point L₄ in the 3D restricted three-body problem (RTBP) for the mass parameter, μ greater than (and close to) μ_R (Routh’s mass parameter). Consequences of such bifurcation, concerning the confinement of the motion close to the hyperbolic orbits and the 3D nearby tori are also described.     

A Search for Collision Orbits in the Free-Fall Three-Body Problem II

A numerical procedure to systematically find collision orbits in the planar three-body problem has been developed in the preceding paper (Tanikawa et al., 1995). Using this procedure, a search for binary and triple collision orbits has been carried out in the free-fall three-body problem. Some detailed structures of a part of the initial value space are discussed. Various interesting orbits have been found. Examples are oscillatory orbits in which ejected particles change from ejection to ejection, and orbits which are not isosceles initially but nearly isosceles after escape. Some results of isosceles problems (Simó and Martínez, 1988) are extended to non-isosceles problems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical Determination of Homoclinic and Heteroclinic Orbits at Collinear Equilibria in the Restricted Three-Body Problem with Oblateness

Asymptotic motion to collinear equilibrium points of the restricted three-body problem with oblateness is considered. In particular, homoclinic and heteroclinic solutions to these points are computed. These solutions depart asymptotically from an equilibrium point and arrive asymptotically at the same or another equilibrium point and are important reference solutions. To compute an asymptotic orbit, we use a fourth order local analysis, numerical integration and standard differential corrections.     

The Existence and Stability of Families of Displacement Two-Body Orbits

The two-body problem is considered with an additional thrust induced acceleration. Stationary solutions are obtained to this problem in a rotating frame of reference which generates families of displaced circular orbits when viewed from an internal frame of reference. The existence and stability of these orbits is considered along with applications such as in-situ observations of Saturn's ring system and spacecraft proximity operations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Second Species Periodic Orbits of the Elliptic 3 Body Problem

We consider the plane restricted elliptic 3 body problem with small mass ratio and small eccentricity and prove the existence of many periodic orbits shadowing chains of collision orbits of the Kepler problem. Such periodic orbits were first studied by Poincaré for the non-restricted 3 body problem. Poincaré called them second species solutions.     

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.