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1.
Kiyotaka Tanikawa 《Celestial Mechanics and Dynamical Astronomy》2000,76(3):157-185
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. 相似文献
2.
Masaya Masayoshi Saito Kiyotaka Tanikawa 《Celestial Mechanics and Dynamical Astronomy》2010,107(4):397-407
In the present paper, in the rectilinear three-body problem, we qualitatively follow the positions of non-Schubart periodic
orbits as the mass parameter changes. This is done by constructing their characteristic curves. In order to construct characteristic
curves, we assume a set of properties on the shape of areas corresponding to symbol sequences. These properties are assured
by our preceding numerical calculations. The main result is that characteristic curves always start at triple collision and
end at triple collision. This may give us some insight into the nature of periodic orbits in the N-body problem. 相似文献
3.
The existence of escape and nonescape orbits arbitrarily close to the homothetic equilateral triplecollision orbit is considered analytically in the threebody problem with zero initial velocities and equal masses. It is proved that escape orbits in the initial condition space are distributed around three kinds of isosceles orbits. It is also proved that nonescape orbits are distributed in between the escape orbits where different particles escape. In order to show this, it is proved that the homotheticequilateral orbit is isolated from other triplecollision orbits as far as the collision at the first triple encounter is concerned. Moreover, the escape criterion is formulated in the planarisosceles problem and translated into the words of regularizing variables. The result obtained by us explains the orbital structure numerically. 相似文献
4.
The planar isosceles three-body problem where the two symmetric bodies have small masses is considered as a perturbation of
the Kepler problem. We prove that the circular orbits can be continued to saddle orbits of the Isosceles problem. This continuation
is not possible in the elliptic case. Their perturbed orbits tend to a continued circular one or approach a triple collision.
The basic tool used is the study of the Poincaré maps associated with the periodic solutions.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
5.
We consider the motions of particles in the one-dimensional Newtonian three-body problem as a function of initial values.
Using a mapping of orbits to symbol sequences we locate the initial values leading to triple collisions. These turn out to
form curves which give clear structure to the region in which the motions depend sensitively on initial conditions. In addition
to finding the triple collision orbits we also locate orbits which end up to a triple collision in both directions of time,
that is, orbits which are finite both in space and time.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
6.
P. Delibaltas 《Celestial Mechanics and Dynamical Astronomy》1983,29(2):191-204
In the general three-body problem, in a rotating frame of reference, a symmetric periodic solution with a binary collision is determined by the abscissa of one body and the energy of the system. For different values of the masses of the three bodies, the symmetric periodic collision orbits form a two-parametric family. In the case of equal masses of the two bodies and small mass of the third body, we found several symmetric periodic collision orbits similar to the corresponding orbits in the restricted three-body problem. Starting with one symmetric periodic collision orbit we obtained two families of such orbits. Also starting with one collision orbit in the Sun-Jupiter-Saturn system we obtained, for a constant value of the mass ratio of two bodies, a family of symmetric periodic collision orbits. 相似文献
7.
Marco Giancotti Mauro Pontani Paolo Teofilatto 《Celestial Mechanics and Dynamical Astronomy》2012,114(1-2):55-76
Analysis and design of low-energy transfers to the Moon has been a subject of great interest for many decades. This paper is concerned with a topological study of such transfers, with emphasis to trajectories that allow performing lunar capture and those that exhibit homoclinic connections, in the context of the circular restricted three-body problem. A fundamental theorem stated by Conley locates capture trajectories in the phase space and can be condensed in a sentence: “if a crossing asymptotic orbit exists then near any such there is a capture orbit”. In this work this fundamental theoretical assertion is used together with an original cylindrical isomorphic mapping of the phase space associated with the third body dynamics. For a given energy level, the stable and unstable invariant manifolds of the periodic Lyapunov orbit around the collinear interior Lagrange point are computed and represented in cylindrical coordinates as tubes that emanate from the transformed periodic orbit. These tubes exhibit complex geometrical features. Their intersections correspond to homoclinic orbits and determine the topological separation of long-term lunar capture orbits from short-duration capture trajectories. The isomorphic mapping is proven to allow a deep insight on the chaotic motion that characterizes the dynamics of the circular restricted three-body, and suggests an interesting interpretation, and together corroboration, of Conley’s assertion on the topological location of lunar capture orbits. Moreover, an alternative three-dimensional representation of the phase space is profitably employed to identify convenient lunar periodic orbits that can be entered with modest propellant consumption, starting from the Lyapunov orbit. 相似文献
8.
A new family of periodic orbits for the restricted problem 总被引：1，自引：0，他引：1
E. A. Belbruno 《Celestial Mechanics and Dynamical Astronomy》1981,25(2):195-217
A new family of periodic orbits of the three-dimensional restricted three-body problem which continue off from a consecutive collision orbit are numerically studied. Their behavior for varying energy is unexpected. In particular, associated with our system is a countable set of resonant energy values and each time the energy passes through one of them the periodic orbit forms a loop by self-intersection. Any number of loops can form by this process and the resulting orbits take on an interesting appearance. 相似文献
9.
In this paper, we study the invariant manifold and its application in transfer trajectory problem from a low Earth parking orbit to the Sun-Earth \(L_{1}\) and \(L_{2}\)-halo orbits with the inclusion of radiation pressure and oblateness. Invariant manifold of the halo orbit provides a natural entrance to travel the spacecraft in the solar system along some specific paths due to its strong hyperbolic character. In this regard, the halo orbits near both collinear Lagrangian points are computed first. The manifold’s approximation near the nominal halo orbit is computed using the eigenvectors of the monodromy matrix. The obtained local approximation provides globalization of the manifold by applying backward time propagation to the governing equations of motion. The desired transfer trajectory well suited for the transfer is explored by looking at a possible intersection between the Earth’s parking orbit of the spacecraft and the manifold. 相似文献
10.
We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches.
We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides
the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative
periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with
binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant
families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As
the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between
triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge
to “previously unknown” periodic orbits in the planar problem. 相似文献
11.
12.
Victor Vladimirovich Orlov Anna V. Petrova Kiyotaka Tanikawa Masaya M. Saito Alija I. Martynova 《Celestial Mechanics and Dynamical Astronomy》2008,100(2):93-120
The rectilinear equal-mass and unequal-mass three-body problems are considered. The first part of the paper is a review that
covers the following items: regularization of the equations of motion, integrable cases, triple collisions and their vicinities,
escapes, periodic orbits and their stability, chaos and regularity of motions. The second part contains the results of our
numerical simulations in this problem. A classification of orbits in correspondence with the following evolution scenarios
is suggested: ejections, escapes, conditional escapes (long ejections), periodic orbits, quasi-stable long-lived systems in
the vicinity of stable periodic orbits, and triple collisions. Homothetic solutions ending by triple collisions and their
dependence on initial parameters are found. We study how the ejection length changes in response to the variation of the triple
approach parameters. Regions of initial conditions are outlined in which escapes occur after a definite number of triple approaches
or a definite time. In the vicinity of a stable Schubart periodic orbit, we reveal a region of initial parameters that corresponds
to trajectories with finite motions. The regular and chaotic structure of the manifold of orbits is mostly defined by this
periodic orbit. We have studied the phase space structure via Poincaré sections. Using these sections and symbolic dynamics,
we study the fine structure of the region of initial conditions, in particular the chaotic scattering region. 相似文献
13.
The importance of the stability characteristics of the planar elliptic restricted three-body problem is that they offer insight
about the general dynamical mechanisms causing instability in celestial mechanics. To analyze these concerns, elliptic–elliptic
and hyperbolic–elliptic resonance orbits (periodic solutions with lower period) are numerically discovered by use of Newton's
differential correction method. We find indications of stability for the elliptic–elliptic resonance orbits because slightly
perturbed orbits define a corresponding two-dimensional invariant manifold on the Poincaré surface-section. For the resonance
orbit of the hyperbolic–elliptic type, we show numerically that its stable and unstable manifolds intersect transversally
in phase-space to induce instability. Then, we find indications that there are orbits which jump from one resonance zone to
the next before escaping to infinity. This phenomenon is related to the so-called Arnold diffusion.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
14.
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. 相似文献
15.
Joerg Waldvogel 《Celestial Mechanics and Dynamical Astronomy》1973,8(2):189-198
In this paper we discuss some aspects of the isosceles case of the rectilinear restricted problem of three bodies, where two primaries of equal mass move on rectilinear ellipses, and the particle is confined to the symmetry axis of the system. In particular, the behaviour near a collision of the primaries and also near a collision of all three bodies is investigated. It is shown that this latter singularity is a triple collision in the sense of Siegel's theory. Furthermore, asymptotic expansions for the particle's motion during a parabolic and a hyperbolic escape are derived.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972. 相似文献
16.
Robert L. Devaney 《Celestial Mechanics and Dynamical Astronomy》1982,28(1-2):25-36
We describe in detail the qualitative behavior of solutions of the planar isosceles problem which come close to triple collision. Using a method of McGehee, one may describe a neighborhood of triple collision for any mass ratios. For sufficiently small mass ratios, we exhibit infinitely many distinct periodic and collision/ejection solutions of this problem.Partially supported by N.S.F. Grant MCS 81-01855.Paper presented at the 1981 Oberwolfach Conference on Mathematical Methods in Celestial Mechanics. 相似文献
17.
We distinguish between regular orbits, that bifurcate from the main families of periodic orbits (those that exist also in the unperturbed case) and irregular periodic orbits, that are independent of the above. The genuine irregular families cannot be made to join the regular families by changing some parameters. We present evidence that all irregular families appear inside lobes formed by the asymptotic curves of the unstable periodic orbits. We study in particular a dynamical system of two degrees of freedom, that is symmetric with respect to the x-axis, and has also a triple resonance in its unperturbed form. The distribution of the periodic orbits (points on a Poincaré surface of section) shows some conspicuous lines composed of points of different multiplicities. The regular periodic orbits along these lines belong to Farey trees. But there are also lines composed mainly of irregular orbits. These are images of the x-axis in the map defined on the Poincaré surface of section. Higher order iterations of this map , close to the unstable triple periodic orbit, produce lines that are close to the asymptotic curves of this unstable orbit. The homoclinic tangle, formed by these asymptotic curves, contains many regular orbits, that were generated by bifurcation from the central orbit, but were trapped inside the tangle as the perturbation increased. We found some stable periodic orbits inside the homoclinic tangle, both regular and irregular. This proves that the homoclinic tangle is not completely chaotic, but contains gaps (islands of stability) filled with KAM curves. 相似文献
18.
We present some results of a numerical exploration of the rectilinear problem of three bodies, with the two outer masses equal. The equations of motion are first given in relative coordinates and in regularized variables, removing both binary collision singularities in a single coordinate transformation. Among our most important results are seven periodic solutions and three symmetric triple collision solutions. Two of these periodic solutions have been continued into families, the outer massm
3 being the family parameter. One of these families exists for all masses while the second family is a branch of the first at a second-kind critical orbit. This last family ends in a triple collision orbit.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978. 相似文献
19.
Kiyotaka Tanikawa Hiroaki Umehara Hiroshi Abe 《Celestial Mechanics and Dynamical Astronomy》1995,62(4):335-362
A numerical procedure is devised to find binary collision orbits in the free-fall three-body problem. Applying this procedure, families of binary collision orbits are found and a sequence of triple collision orbits are positioned. A property of sets of binary collision orbits which is convenient to search triple collision orbits is found. Important numerical results are formulated and summarized in the final section. 相似文献
20.
We consider a restricted three-body problem where the primaries are moving in an elliptic collision orbit and the infinitesimal
mass moves in a three dimensional space. This paper is devoted to prove analytically the existence of several families of
symmetric periodic solutions as continuation of Keplerian circular orbits. In our approach the perturbing parameter is related
with the energy of the primaries. 相似文献