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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.
In the free‐fall three‐body problem, distributions of escape, binary, and triple collision orbits are obtained. Interpretation
of the results leads us to the existence of oscillatory orbits in the planar three‐body problem with equal masses. A scenario
to prove their existence is described.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
3.
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
4.
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
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.
We consider the particular case of the planar three body problem obtained when the masses form an isosceles triangle for all time. Various authors [1, 2, 12, 8, 9, 13, 10] have contributed in the knowledge of the triple collision and of several families of periodic orbits in this problem. We study the flow on a fixed level of negative energy. First we obtain a topological representation of the energy manifold including the triple collision and infinity as boundaries of that manifold. The existence of orbits connecting the triple collision and infinity gives some homoclinic and heteroclinic orbits. Using these orbits and the homothetic solutions of the problem we can characterize orbits which pass near triple collision and near infinity by pairs of sequences. One of the sequences describes the regions visited by the orbit, the other refers to the behaviour of the orbit between two consecutive passages by a suitable surface of section. This symbolic dynamics which has a topological character is given in an abstract form and after it is applied to the isosceles problem. We try to keep globality as far as possible. This strongly relies on the fact that the intersection of some invariant manifolds with an equatorial plane (v=0) have nice spiraling properties. This can be proved by analytical means in some local cases. Numerical simulations given in Appendix A make clear that these properties hold globally. 相似文献
8.
J.R. Donnison 《Planetary and Space Science》2010,58(10):1169-1179
The dynamical stability of a bound triple system composed of a small binary or minor planetary system moving on a orbit inclined to a central third body is discussed in terms of Hill stability for the full three-body problem. The situation arises in the determination of stability of triple star systems against disruption and component exchange and the determination of stability of extrasolar planetary systems and minor planetary systems against disruption, component exchange or capture. The Hill stability criterion is applied to triple star systems and extrasolar planetary systems, the Sun-Earth-Moon system and Kuiper Belt binary systems to determine the critical distances for stable orbits. It is found that increasing the inclination of the third body decreases the Hill regions of stability. Increasing the eccentricity of the binary also produces similar effects.These type of changes make exchange or disruption of the component masses more likely. Increasing the eccentricity of the binary orbit relative to the third body substantially decreases stability regions as the eccentricity reaches higher values. The Kuiper Belt binaries were found to be stable if they move on circular orbits. Taking into account the eccentricity, it is less clear that all the systems are stable. 相似文献
9.
Saurabh Jha Guillermo Torres Robert P. Stefanik David W. Latham Tsevi Mazeh 《Monthly notices of the Royal Astronomical Society》2000,317(2):375-384
The triple-lined spectroscopic triple system HD 109648 has one of the shortest periods known for the outer orbit in a late-type triple, 120.5 d, and the ratio between the periods of the outer and the inner orbits is small, 22:1. With such extreme values, this system should show orbital element variations over a time-scale of about a decade. We have monitored the radial velocities of HD 109648 with the CfA Digital Speedometers for 8 yr, and have found evidence for modulation of some orbital elements. While we see no definite evidence for modulation of the inner binary eccentricity, we clearly observe variations in the inner and the outer longitudes of periastron, and in the radial velocity amplitudes of the three components. The observational results, combined with numerical simulations, allow us to put constraints on the orientation of the orbits. 相似文献
10.
A. E. Perdiou 《Earth, Moon, and Planets》2008,103(3-4):105-118
We study the multiple periodic orbits of Hill’s problem with oblate secondary. In particular, the network of families of double and triple symmetric periodic orbits is determined numerically for an arbitrary value of the oblateness coefficient of the secondary. The stability of the families is computed and critical orbits are determined. Attention is paid to the critical orbits at which families of non-symmetric periodic orbits bifurcate from the families of symmetric periodic orbits. Six such bifurcations are found, one for double-periodic and five for triple-periodic orbits. Critical orbits at which families of sub-multiple symmetric periodic orbits bifurcate are also discussed. Finally, we present the full network of families of multiple periodic orbits (up to multiplicity 12) together with the parts of the space of initial conditions corresponding to escape and collision orbits, obtaining a global view of the orbital behavior of this model problem. 相似文献
11.
12.
The Kozai mechanism often destabilizes high-inclination orbits. It couples changes in the eccentricity and inclination, and drives high inclination, circular orbits to low inclination, eccentric orbits. In a recent study of the dynamics of planetesimals in the quadruple star system HD 98800, there were significant numbers of stable particles in circumbinary polar orbits about the inner binary pair which are apparently able to evade the Kozai instability.
Here, we isolate this feature and investigate the dynamics through numerical and analytical models. The results show that the Kozai mechanism of the outer star is disrupted by a nodal libration induced by the inner binary pair on a shorter time-scale. By empirically modelling the period of the libration, a criteria for determining the high-inclination stability limits in general triple systems is derived. The nodal libration feature is interesting and, although affecting inclination and node only, shows many parallels to the Kozai mechanism. This raises the possibility that high-inclination planets and asteroids may be able to survive in multistellar systems. 相似文献
Here, we isolate this feature and investigate the dynamics through numerical and analytical models. The results show that the Kozai mechanism of the outer star is disrupted by a nodal libration induced by the inner binary pair on a shorter time-scale. By empirically modelling the period of the libration, a criteria for determining the high-inclination stability limits in general triple systems is derived. The nodal libration feature is interesting and, although affecting inclination and node only, shows many parallels to the Kozai mechanism. This raises the possibility that high-inclination planets and asteroids may be able to survive in multistellar systems. 相似文献
13.
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
14.
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
15.
Lennard F. Bakker Tiancheng Ouyang Duokui Yan Skyler Simmons Gareth E. Roberts 《Celestial Mechanics and Dynamical Astronomy》2010,108(2):147-164
We apply the analytic-numerical method of Roberts to determine the linear stability of time-reversible periodic simultaneous
binary collision orbits in the symmetric collinear four-body problem with masses 1, m, m, 1, and also in a symmetric planar
four-body problem with equal masses. In both problems, the assumed symmetries reduce the determination of linear stability
to the numerical computation of a single real number. For the collinear problem, this verifies the earlier numerical results
of Sweatman for linear stability with respect to collinear and symmetric perturbations. 相似文献
16.
Rudolf Dvorak 《Celestial Mechanics and Dynamical Astronomy》1997,68(1):63-73
We report on the different results concerning the stability of the hierarchical triple systems where a close binary is accompanied
by a third star. There are different possible approaches to answer the question of the stability limits for such triple stars:
the most direct investigations can be undertaken in integrating numerically the respective equations of motion for many different
initial conditions. It is then difficult to take into account all the important parameters like eccentricities, inclination,
phases and masses. Analytical approaches and qualitative methods are more approriate to deal with this problem; the respective
results confirm the numerically found results that:
1. for prograde orbits the ratio semimajor axis of the inner orbits to the periastron position of the outer orbit is approximately
3.2
2. for retrograde orbits this ratio is just some 10 percents smaller
3. the results are not sensitive in what concerns the masses involved
4. There is a tendency that the inclinations and eccentricities change slightly the stability limits mentioned above.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
17.
Lennard F. Bakker Tiancheng Ouyang Duokui Yan Skyler Simmons 《Celestial Mechanics and Dynamical Astronomy》2011,110(3):271-290
We analytically prove the existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise
symmetric equal mass four-body problem. This is an extension of our previous proof of the analytic existence of a symmetric
periodic simultaneous binary collision orbit in a regularized planar fully symmetric equal mass four-body problem. We then
use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar
pairwise symmetric 1, m, 1, m four-body problem for m between 0 and 1. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability
when 0.54 ≤ m ≤ 1, and are linearly unstable when 0 < m ≤ 0.53. 相似文献
18.
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
19.
We present a numerical study of the set of orbits of the planar circular restricted three body problem which undergo consecutive
close encounters with the small primary, or orbits of second species. The value of the Jacobi constant is fixed, and we restrict
the study to consecutive close encounters which occur within a maximal time interval. With these restrictions, the full set
of orbits of second species is found numerically from the intersections of the stable and unstable manifolds of the collision
singularity on the surface of section that corresponds to passage through the pericentre. A ‘skeleton’ of this set of curves
can be computed from the solutions of the two-body problem. The set of intersection points found in this limit corresponds
to the S-arcs and T-arcs of Hénon’s classification which verify the energy and time constraints, and can be used to construct
an alphabet to describe the orbits of second species. We give numerical evidence for the existence of a shift on this alphabet
that describes all the orbits with infinitely many close encounters with the small primary, and sketch a proof of the symbolic
dynamics. In particular, we find periodic orbits that combine S-type and T-type quasi-homoclinic arcs. 相似文献
20.
It is usually believed that we know everything to be known for any separable Hamiltonian system, i.e. an integrable system
in which we can separate the variables in some coordinate system (e.g. see Lichtenberg and Lieberman 1992, Regular and Chaotic Dynamics, Springer). However this is not always true, since through the separation the solutions may be found only up to quadratures,
a form that might not be particularly useful. A good example is the two-fixed-centers problem. Although its integrability
was discovered by Euler in the 18th century, the problem was far from being considered as completely understood. This apparent
contradiction stems from the fact that the solutions of the equations of motion in the confocal ellipsoidal coordinates, in
which the variables separate, are written in terms of elliptic integrals, so that their properties are not obvious at first
sight. In this paper we classify the trajectories according to an exhaustive scheme, comprising both periodic and quasi-periodic
ones. We identify the collision orbits (both direct and asymptotic) and find that collision orbits are of complete measure
in a 3-D submanifold of the phase space while asymptotically collision orbits are of complete measure in the 4-D phase space.
We use a transformation, which regularizes the close approaches and, therefore, enables the numerical integration of collision
trajectories (both direct and asymptotic). Finally we give the ratio of oscillation period along the two axes (the ‘rotation
number’) as a function of the two integrals of motion.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献