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1.
Stability of interplay motions 总被引：2，自引：2，他引：0
M. Hénon 《Celestial Mechanics and Dynamical Astronomy》1977,15(2):243-261
A family of rectilinear periodic solutions of the three-body problem, in which the central body collides alternately with each of the two other bodies, is investigated numerically for all values of the three masses. It is found that for every mass combination there exists just one solution of this kind. The linear stability of the orbits with respect to arbitrary three-dimensional perturbations is also investigated. Domains of stability and instability are displayed in a triangular mass diagram. Their boundaries form one-parameter families of critical orbits, which are tabulated. Limiting cases where one or two masses vanish are studied in detail. The domains of stability cover nearly one half of the total area in the mass diagram: this reinforces the conclusion that real triple stars might have motions of a kind entirely different from the usual hierarchical arrangement. 相似文献
2.
V. V. Orlov A. V. Petrova A. I. Martynova 《Monthly notices of the Royal Astronomical Society》2002,333(3):495-500
The general plane isosceles three-body problem is considered for different ratios of the central body mass to the masses of other bodies. The central body goes through the middle of the segment connecting the other bodies along the perpendicular to this segment. The initial conditions are chosen by two parameters: the virial ratio k and the parameter , where r˙ is the relative velocity of the 'outer' bodies, and R˙ is the velocity of the 'central' body with respect to the mass centre of the 'outer' bodies. The equations of motion are numerically integrated until one of three times: the time of escape of the central body, its time of ejection with R >100 d , or 1000 τ (here d is the mean size, and τ is the mean crossing time of the triple system). The regions corresponding to escapes of the central body after different numbers of triple approaches are found at the plane of parameters k ∈(0,1) and μ ∈(-1,1) . The regions of stable motions are revealed. The zones of regular and stochastic orbits are outlined. The fraction of stochastic trajectories increases with the central mass. The fraction of stable orbits is highest for equal masses of the bodies. 相似文献
3.
Giovanni B. Valsecchi Andrea Carusi Archie E. Roy 《Celestial Mechanics and Dynamical Astronomy》1984,32(3):217-230
Hill-type stability surfaces are computed for the general hierarchical three-body problem for non-zero eccentricities of the initial osculating orbits. Significant differences are found between them and the one obtained for initial zero eccentricities. Application is made to the triple subgroups of the Solar System; in particular it is found that no analytical guarantee of Hill-type stability can be given to any of the satellites against solar perturbations. 相似文献
4.
Dionyssia Psychoyos John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》2005,92(1-3):135-156
A complete study is made of the resonant motion of two planets revolving around a star, in the model of the general planar three body problem. The resonant motion corresponds to periodic motion of the two planets, in a rotating frame, and the position and stability properties of the periodic orbits determine the topology of the phase space and consequently play an important role in the evolution of the system. Several families of symmetric periodic orbits are computed numerically, for the 2/1 resonance, and for the masses of some observed extrasolar planetary systems. In this way we obtain a global view of all the possible stable configurations of a system of two planets. These define the regions of the phase space where a resonant extrasolar system could be trapped, if it had followed in the past a migration process.The factors that affect the stability of a resonant system are studied. For the same resonance and the same planetary masses, a large value of the eccentricities may stabilize the system, even in the case where the two planetary orbits intersect. The phase of the two planets (position at perihelion or aphelion when the star and the two planets are aligned) plays an important role, and the change of the phase, other things being the same, may destabilize the system. Also, the ratio of the planetary masses, for the same total mass of the two planets, plays an important role and the system, at some resonances and some phases, is destabilized when this ratio changes.The above results are applied to the observed extrasolar planetary systems HD 82943, Gliese 876 and also to some preliminary results of HD 160691. It is shown that the observed configurations are close to stable periodic motion. 相似文献
5.
Robert S. Harrington 《Celestial Mechanics and Dynamical Astronomy》1972,6(3):322-327
A wide variety of equal-mass stellar triple systems has been numerically integrated in order to establish factors pertinent to stability. The significant parameters appear to be whether the relative revolution is direct or retrograde, and the ratio of the periastron distance in the outer orbit to the semi-major axis of the inner orbit. For stability, this ratio must be at least 3.5 for direct orbits and at least 2.75 for retrograde orbits. 相似文献
6.
John D. Hadjidemetriou 《Astrophysics and Space Science》1976,40(1):201-224
Several families of the planar general three-body problem for fixed values of the three masses are found, in a rotating frame of reference, where the mass of two of the bodies is small compared to the mass of the third body. These families were obtained by the continuation of a degenerate family of periodic orbits of three bodies where two of the bodies have zero masses and describe circular orbits around a third body with finite mass, in the same direction.The above families represent planetary systems with the body with the large mass representing the Sun and the two small bodies representing two planets or comets. One section of a family is shown to represent the Jupiter family of comets and also a model for the Sun-Jupiter-Saturn system is found.The stability analysis revealed that stability exists for small masses and small eccentricities of the two planets. Planetary systems with relatively large masses and eccentricities are proved to be unstable. In particular, the Jupiter family of comets, for small masses of the two small bodies, and the Sun-Jupiter-Saturn system are proved to be stable. Also, it was shown that resonances are not necessarily associated with instabilities. 相似文献
7.
Martín Lara André Deprit Antonio Elipe 《Celestial Mechanics and Dynamical Astronomy》1995,62(2):167-181
In the zonal problem of a satellite around the Earth, we continue numerically natural families of periodic orbits with the polar component of the angular momentum as the parameter. We found three families; two of them are made of orbits with linear stability while the third one is made of unstable orbits. Except in a neighborhood of the critical inclination, the stable periodic (or frozen) orbits have very small eccentricities even for large inclinations. 相似文献
8.
John D. Hadjidemetriou Dionyssia Psychoyos George Voyatzis 《Celestial Mechanics and Dynamical Astronomy》2009,104(1-2):23-38
We study orbits of planetary systems with two planets, for planar motion, at the 1/1 resonance. This means that the semimajor axes of the two planets are almost equal, but the eccentricities and the position of each planet on its orbit, at a certain epoch, take different values. We consider the general case of different planetary masses and, as a special case, we consider equal planetary masses. We start with the exact resonance, which we define as the 1/1 resonant periodic motion, in a rotating frame, and study the topology of the phase space and the long term evolution of the system in the vicinity of the exact resonance, by rotating the orbit of the outer planet, which implies that the resonance and the eccentricities are not affected, but the symmetry is destroyed. There exist, for each mass ratio of the planets, two families of symmetric periodic orbits, which differ in phase only. One is stable and the other is unstable. In the stable family the planetary orbits are in antialignment and in the unstable family the planetary orbits are in alignment. Along the stable resonant family there is a smooth transition from planetary orbits of the two planets, revolving around the Sun in eccentric orbits, to a close binary of the two planets, whose center of mass revolves around the Sun. Along the unstable family we start with a collinear Euler–Moulton central configuration solution and end to a planetary system where one planet has a circular orbit and the other a Keplerian rectilinear orbit, with unit eccentricity. It is conjectured that due to a migration process it could be possible to start with a 1/1 resonant periodic orbit of the planetary type and end up to a satellite-type orbit, or vice versa, moving along the stable family of periodic orbits. 相似文献
9.
Krzysztof Go&#;dziewski S&#;awomir Breiter Wojciech Borczyk 《Monthly notices of the Royal Astronomical Society》2008,383(3):989-999
We describe numerical tools for the stability analysis of extrasolar planetary systems. In particular, we consider the relative Poincaré variables and symplectic integration of the equations of motion. We apply the tangent map to derive a numerically efficient algorithm of the fast indicator Mean Exponential Growth factor of Nearby Orbits (MEGNO), a measure of the maximal Lyapunov exponent, that helps to distinguish chaotic and regular configurations. The results concerning the three-planet extrasolar system HD 37124 are presented and discussed. The best-fitting solutions found in earlier works are studied more closely. The system involves Jovian planets with similar masses. The orbits have moderate eccentricities, nevertheless the best-fitting solutions are found in dynamically active region of the phase space. The long-term stability of the system is determined by a net of low-order two-body and three-body mean motion resonances. In particular, the three-body resonances may induce strong chaos that leads to self-destruction of the system after Myr of apparently stable and bounded evolution. In such a case, numerically efficient dynamical maps are useful to resolve the fine structure of the phase space and to identify the sources of unstable behaviour. 相似文献
10.
Winston L. Sweatman 《Celestial Mechanics and Dynamical Astronomy》2006,94(1):37-65
We locate members of an important category of periodic orbits in the Newtonian four-body problem. These systems perform an interplay motion similar to that of the periodic three-body orbit discovered by Schubart. Such orbits, when stable, have been shown to be a key feature and influence on the dynamics of few-body systems. We consider the restricted case where the masses are collinear and are distributed symmetrically about their centre of mass. A family of orbits is generated from the known (three-dimensionally) unstable equal masses case by varying the mass ratio, whilst maintaining the symmetry. The stability of these orbits to perturbation is studied using linear stability analysis, analytical approximation of limiting cases and nonlinear simulation. We answer the natural question: are there any stable periodic orbits of this kind? Three ranges of the mass ratio are found to have stable orbits and three ranges have unstable orbits for three-dimensional motion. The systems closely resemble their three-body counterparts. Here the family of interplay orbits is simpler requiring just one parameter to characterise the mass ratio. Our results provide a further insight into three-body orbits studied previously. 相似文献
11.
Peter J. Shelus 《Celestial Mechanics and Dynamical Astronomy》1972,5(4):483-489
Within the context of the restricted problem of three bodies, we wish to show the effects, caused by varying the mass ratio of the primaries and the eccentricity of their orbits, upon periodic orbits of the infinitesimal mass that are numerical continuations of circular orbits in the ordinary problem of two bodies. A recursive-power-series technique is used to integrate numerically the equations of motion as well as the first variational equations to generate a two-parameter family of periodic orbits and to identify the linear stability characteristics thereof. Seven such families (comprised of a total of more than 2000 orbits) with equally spaced mass ratios from 0.0 to 1.0 and eccentricities of the orbits of the primaries in a range 0.0 to 0.6 are investigated. Stable orbits are associated with large distances of the infinitesimal mass from the perturbing primary, with nearly circular motion of the primaries, and, to a slightly lesser extent, with small mass ratios of the primaries.Conversely, unstable orbits for the infinitesimal mass are associated with small distances from the perturbing primary, with highly elliptic orbits of the primaries, and with large mass ratios. 相似文献
12.
A systematic approach to generate periodic orbits in the elliptic restricted problem of three bodies in introduced. The approach is based on (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to periodic orbits in the elliptic restricted problem. Two families of periodic orbits of the elliptic restricted problem are found by this approach. The mass ratio of the primaries of these orbits is equal to that of the Sun-Jupiter system. The sidereal mean motions between the infinitesimal body and the smaller primary are in a 2:5 resonance, so as to approximate the Sun-Jupiter-Saturn system. The linear stability of these periodic orbits are studied as functions of the eccentricities of the primaries and of the infinitesimal body. The results show that both stable and unstable periodic orbits exist in the elliptic restricted problem that are close to the actual Sun-Jupiter-Saturn system. However, the periodic orbit closest to the actual Sun-Jupiter-Saturn system is (linearly) stable. 相似文献
13.
The dynamical evolution of triple systems with equal and unequal-mass components and different initial velocities is studied. It is shown that, in general, the statistical results for the planar and three-dimensional triple systems do not differ significantly. Most (about 85%) of the systems disrupt; the escape of one component occurs after a triple approach of the components. In a system with unequal masses, the escaping body usually has the smallest mass. A small fraction (about 15%) of stable or long-lived systems is formed if the angular momentum is non-zero. Averages, distributions and coefficients of correlations of evolutionary characteristics are presented: the life-time, angular momentum, numbers of wide and close triple approaches of bodies, relative energy of escapers, minimum perimeter during the last triple approach resulting in escape, elements of orbits of the final binary and escaper. 相似文献
14.
In our work, the method that can help to predict the existence of distant objects in the Solar system is demonstrated. This
method is connected with statistical properties of a heliocentric orbital complex of meteoroids with high eccentricities.
Heliocentric meteoroid orbits with high eccentricities are escape routes for dust material from distant parental objects with
near-circular orbits to Earth-crossing orbits. Ground-based meteor observations yield trajectory information from which we
can derive their place of possible origin: comets, asteroids, and other objects (e.g. Kuiper Objects) in the Solar system
or even interstellar space. Statistical distributions of radius vectors of nodes, and other parameters of orbits of meteoroids
contain key information about position of greater bodies. We analyze meteor orbits with high eccentricities that were registered
in 1975–1976 in Kharkiv (Ukraine). The orbital data of the Kharkiv electronic catalogue are received from observations of
radiometeors with masses 10−6−10−3 g. 相似文献
15.
John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1981,23(3):277-286
Recent results on periodic orbits are presented and it is shown that the periodic orbits can be used in the study of planetary systems and triple or multiple stellar systems. Triple stellar systems are stable even for close approaches of the three components. Also stable triple systems exist with nearly zero angular momentum. For the planetary systems a global view is obtained from which it is clear which configurations are stable or unstable and also what factors affect the stability. Also, the relation between resonance and instability is studied by making use of periodic orbits. 相似文献
16.
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
17.
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
18.
P. C. Kammeyer 《Celestial Mechanics and Dynamical Astronomy》1983,30(4):329-342
In this paper the existence of families of symmetric periodic orbits in the rectilinear three body problem with the middle mass much larger than the masses on the outside is rigorously established. A number of these families are continued numerically and their stability properties as orbits of the planar general problem of three bodies are studied. 相似文献
19.
In this paper we consider a restricted equilateral four-body problem where a particle of negligible mass is moving under the Newtonian gravitational attraction of three masses (called primaries) which move on circular orbits around their center of masses such that their configuration is always an equilateral triangle (Lagrangian configuration). We consider the case of two bodies of equal masses, which in adimensional units is the parameter of the problem. We study numerically the existence of families of unstable periodic orbits, whose invariant stable and unstable manifolds are responsible for the existence of homoclinic and heteroclinic connections, as well as of transit orbits traveling from and to different regions. We explore, for three different values of the mass parameter, what kind of transits and energy levels exist for which there are orbits with prescribed itineraries visiting the neighborhood of different primaries. 相似文献
20.
Nikolaos Georgakarakos 《Monthly notices of the Royal Astronomical Society》2006,366(2):566-574
In previous papers, we developed a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular. We considered systems with well-separated components and different initial setups (e.g., coplanar and non-coplanar orbits). However, the systems we examined had comparable masses. In the present paper, the validity of some of the formulae derived previously is tested by numerically integrating the full equations of motion for systems with smaller mass ratios (from 10−3 to 103 , i.e. systems with Jupiter-sized bodies). There is also discussion about HD 217107 and its planetary companions. 相似文献