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1.
The dynamics of circumbinary planetary systems (the systems in which the planets orbit a central binary) with a small binary mass ratio discovered to date is considered. The domains of chaotic motion have been revealed in the “pericentric distance–eccentricity” plane of initial conditions for the planetary orbits through numerical experiments. Based on an analytical criterion for the chaoticity of planetary orbits in binary star systems, we have constructed theoretical curves that describe the global boundary of the chaotic zone around the central binary for each of the systems. In addition, based on Mardling’s theory describing the separate resonance “teeth” (corresponding to integer resonances between the orbital periods of a planet and the binary), we have constructed the local boundaries of chaos. Both theoretical models are shown to describe adequately the boundaries of chaos on the numerically constructed stability diagrams, suggesting that these theories are efficient in providing analytical criteria for the chaoticity of planetary orbits. 相似文献
2.
Possible configurations of the planetary systems of the binary stars α Cen A–BandEZAqr A–C are analyzed. The P-type orbits—circumbinary ones, i.e., the orbits around both stars of the binary, are studied. The choice of these systems is dictated by the fact that α Cen is closest to us in the Galaxy, while EZ Aqr is the closest system whose circumbinary planets, as it turns out, may reside in the “habitability zone.” The analysis has been performed within the framework of the planar restricted three-body problem. The stability diagrams of circumbinary motion have been constructed: on representative sets of initial data (in the pericentric distance–eccentricity plane), we have computed the Lyapunov spectra of planetary motion and identified the domains of regular and chaotic motion through their statistical analysis. Based on present views of the dynamics and architecture of circumbinary planetary systems, we have determined the most probable planetary orbits to be at the centers of the main resonance cells, at the boundary of the dynamical chaos domain around the parent binary star, which allows the semimajor axes of the orbits to be predicted. In the case of EZ Aqr, the orbit of the circumbinary planet is near the habitability zone and, given that the boundary of this zone is uncertain, may belong to it. 相似文献
3.
T. Aikawa 《Astrophysics and Space Science》1993,210(1-2):269-280
Phenomena of bifurcation in hydrodynamic stellar models of radial pulsation are reviewed. By changing control parameters of models, we can see qualitatively different pulsation behaviors in hydrodynamic models with transitions due to various types of bifurcation.In weakly dissipative models (classical Cepheids), the bifurcation is induced by modal resonances. Two types of the modal resonances found in models are discussed: The higherharmonic resonances of the second overtone mode in the fundamental mode pulsator and of the fourth overtone mode in the first overtone pulsator are relevant to observations. The subharmonic resonance between the fundamental and first overtone modes is confirmed in classical Cepheid models.In strongly dissipative models (less-massive supergiant stars), the bifurcation of nonlinear pulsation is induced by the hydrodynamics of ionization zones as well as modal resonances. The sequence of the bifurcation sometimes leads to chaotic behaviors in nonlinear pulsation. The transition routes from regular to the chaotic pulsations found in models are discussed with respect to the theory of chaos in simple dynamical systems: The cascade of period-doubling bifurcation is confirmed to cause chaotic pulsation in W Virginis models. For models of higher luminosity, the tangent bifurcation is found to lead intermittent chaos.Finally, hydrodynamic models for chaotic pulsation with small amplitudes observed in the post-AGB stars are briefly discussed. 相似文献
4.
Cosmogonical theories as well as recent observations allow us to expect the existence of planets around many stars other than the Sun. On an other hand, double and multiple star systems are established to be more numerous than single stars (such as the Sun), at least in the solar neighborhood. We are then faced to the following dynamical problem: assuming that planets can form in a binary early environment (I do not deal here with), does long-term stability for planetary orbits exist in double star systems.Although preliminary studies were rather pessimistic about the possibility of existence of stable planetary orbits in double or multiple star systems, modern computation have shown that many such stable orbits do exist (but possible chaotic behavior), either around the binary as a whole (P-type) or around one component of the binary (S-type), this latter being explored here.The dynamical model is the elliptic plane restricted three-body problem; the phase space of initial conditions is systematically explored, and limits for stability have been established. Stable S-type planetary orbits are found up to distance of their "sun" of the order of half the periastron distance of the binary; moreover, among these stable orbits, nearly-circular ones exist up to distance of their "sun" of the order of one quarter the periastron distance of the binary; finally, among the nearly-circular stable orbits, several stay inside the "habitable zone", at least for two nearby binaries which components are nearly of solar type.Nevertheless, we know that chaos may destroy this stability after a long time (sometimes several millions years). It is therefore important to compute indicators of chaos for these stable planetary orbits to investigate their actual very long-term stability. Here we give an example of such a computation for more than a billion years. 相似文献
5.
Equations are presented for the computation of tangent maps for use in nearly Keplerian motion, approximated by use of a symplectic
leapfrog map. The resulting algorithms constitute more accurate and efficient methods to obtain the Liapunov exponents and
the state transition matrix, and can be used to study chaos in planetary motions, as well as in orbit determination procedures
from observations. Applications include planetary systems, satellite motions and hierarchical, nearly Keplerian systems in
general.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
6.
John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》2008,102(1-3):69-82
We study the evolution of an extrasolar planetary system with two planets, for planar motion, starting from an exact resonant periodic motion and increasing the deviation from the equilibrium solution. We keep the semimajor axes and the eccentricities of the two planets fixed and we change the initial conditions by rotating the orbit of the outer planet by Δω. In this way the resonance is preserved, but we deviate from the exact periodicity and there is a transition from order to chaos as the deviation increases. There are three different routes to chaos, as far as the evolution of (ω 2 ? ω 1) is concerned: (a) Libration → rotation → chaos, with intermittent transition from libration to rotation in between, (b) libration → chaos and (c) libration → intermittent interchange between libration and rotation → chaos. This indicates that resonant planetary systems where the angle (ω 2 ? ω 1) librates or rotates are not different, but are closely connected to the exact periodic motion. 相似文献
7.
Massimiliano Guzzo 《Icarus》2006,181(2):475-485
The motion of the giant planets from Jupiter to Neptune is chaotic with Lyapunov time of approximately 10 Myr. A recent theory explains the presence of this chaos with three-planet mean-motion resonances, i.e. resonances among the orbital periods of at least three planets. We find that the distribution of these resonances with respect to the semi-major axes of all the planets is compatible with orbital instability. In particular, they overlap in a region of 10−3 AU with respect to the variation of the semi-major axes of Uranus and Neptune. Fictitious planetary systems with initial conditions in this region can undergo systematic variations of semi-major axes. The true Solar System is marginally in this region, and Uranus and Neptune undergo very slow systematic variations of semi-major axes with speed of order 10−4 AU/Gyr. 相似文献
8.
Tassos Bountis Thanos Manos Chris Antonopoulos 《Celestial Mechanics and Dynamical Astronomy》2012,113(1):63-80
We use probability density functions (pdfs) of sums of orbit coordinates, over time intervals of the order of one Hubble time, to distinguish weakly from strongly chaotic orbits in a barred galaxy model. We find that, in the weakly chaotic case, quasi-stationary states arise, whose pdfs are well approximated by q-Gaussian functions (with 1 <?q < 3), while strong chaos is identified by pdfs which quickly tend to Gaussians (q =?1). Typical examples of weakly chaotic orbits are those that ??stick?? to islands of ordered motion. Their presence in rotating galaxy models has been investigated thoroughly in recent years due to their ability to support galaxy structures for relatively long time scales. In this paper, we demonstrate, on specific orbits of 2 and 3 degree of freedom barred galaxy models, that the proposed statistical approach can distinguish weakly from strongly chaotic motion accurately and efficiently, especially in cases where Lyapunov exponents and other local dynamic indicators appear to be inconclusive. 相似文献
9.
R. Brasser 《Monthly notices of the Royal Astronomical Society》2001,324(4):1109-1116
In this paper the effect of the Galactic tidal field on a Sun–comet pair will be considered when the comet is situated in the Oort cloud and planetary perturbations can be neglected. First, two averaged models were created, one of which can be solved analytically in terms of Jacobi elliptic functions. In the latter system, switching between libration and circulation of the argument of perihelion is prohibited. The non-averaged equations of motion are integrated numerically in order to determine the regions of the ( e , i ) phase space in which chaotic orbits can be found, and an effort is made to explain why the chaotic orbits manifest in these regions only. It is evident that for moderate values of semimajor axis, a ∼50 000 au , chaotic orbits are found for ( e <0.15 , 40°≤ i ≤140°) as determined by integrating the evolution of the comet over a period of 104 orbits. These regions of chaos increase in size with increasing semimajor axis. The typical e-folding times for these orbits range from around 600 Myr to 1 Gyr and thus are of little practical interest, as the time-scales for chaos arising from passing stars are much shorter. As a result of Galactic rotation, the chaotic regions in ( e , i ) phase space are not symmetric for prograde and retrograde orbits. 相似文献
10.
We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of symmetric periodic orbits by applying two schemes of analytical continuation. In the first scheme, we start from the 2/1 (or 1/2) resonant periodic orbits of the restricted problem and in the second scheme, we start from vertical critical periodic orbits of the general planar problem. Most of the periodic orbits are unstable, but many stable periodic orbits have been, also, found with mutual inclination up to 50?–60?, which may be related with the existence of real planetary systems. 相似文献
11.
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
12.
B. Funk E. Pilat-Lohinger R. Dvorak F. Freistetter B. Èrdi 《Celestial Mechanics and Dynamical Astronomy》2004,90(1-2):43-50
Since the first extrasolar planet was discovered about 10 years ago, a major point of dynamical investigations was the determination
of stable regions in extrasolar planetary systems where additional planets may exist. Using numerical methods, we investigate
the dynamical stability in known multiple planetary systems (HD74156, HD38529, HD168443, HD169830) with special interest on
the region between the two known planets and on the mean motion resonances inside this region. As a dynamical model we take
the restricted 4-body problem containing the host star, the two planets and massless test-planets. For our numerical integrations,
we used the Lie-integrator and additionally the Fast Lyapunov Indicators as a tool for detecting chaotic motion. We also investigated
the inner resonances with the outer planet and the outer resonances with the inner planet with test-planets located inside
the resonances. 相似文献
13.
《天文和天体物理学研究(英文版)》2017,(1)
One of the most puzzling problems in astrophysics is to understand the anomalous resistivity in collisionless magnetic reconnection that is believed extensively to be responsible for the energy release in various eruptive phenomena. The magnetic null point in the reconnecting current sheet, acting as a scattering center, can lead to chaotic motions of particles in the current sheet, which is one of the possible mechanisms for anomalous resistivity and is called chaos-induced resistivity. In many interesting cases, however, instead of the magnetic null point, there is a nonzero magnetic field perpendicular to the merging field lines, usually called the guide field, whose effect on chaos-induced resistivity has been an open problem. By use of the test particle simulation method and statistical analysis, we investigate chaos-induced resistivity in the presence of a constant guide field. The characteristics of particle motion in the reconnecting region, in particular, the chaotic behavior of particle orbits and evolving statistical features, are analyzed. The results show that as the guide field increases, the radius of the chaos region increases and the Lyapunov index decreases. However, the effective collision frequency, and hence the chaos-induced resistivity, reach their peak values when the guide field approaches half of the characteristic strength of the reconnection magnetic field. The presence of a guide field can significantly influence the chaos of the particle orbits and hence the chaos-induced resistivity in the reconnection sheet, which decides the collisionless reconnection rate. The present result is helpful for us to understand the microphysics of anomalous resistivity in collisionless reconnection with a guide field. 相似文献
14.
The stability of the motion of a hypothetical planet in the binary system ?? Cen A?CB has been investigated. The analysis has been performed within the framework of a planar (restricted and full) three-body problem for the case of prograde orbits. Based on a representative set of initial data, we have obtained the Lyapunov spectra of the motion of a triple system with a single planet. Chaotic domains have been identified in the pericenter distance-eccentricity plane of initial conditions for the planet through a statistical analysis of the data obtained. We have studied the correspondence of these chaotic domains to the domains of initial conditions that lead to the planet??s encounter with one of the binary??s stars or to the escape of the planet from the system. We show that the stability criterion based on the maximum Lyapunov exponent gives a more clear-cut boundary of the instability domains than does the encounterescape criterion at the same integration time. The typical Lyapunov time of chaotic motion is ??500 yr for unstable outer orbits and ??60 yr for unstable inner ones. The domain of chaos expands significantly as the initial orbital eccentricity of the planet increases. The chaos-order boundary has a fractal structure due to the presence of orbital resonances. 相似文献
15.
The planetary dynamics of 4/3, 3/2, 5/2, 3/1 and 4/1 mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance. Both planar and spatial cases are examined. In the spatial problem, families of periodic orbits are obtained after analytical continuation of vertical critical orbits. The linear stability of orbits is also examined. Concerning initial conditions nearby stable periodic orbits, we obtain long-term planetary stability, while unstable orbits are associated with chaotic evolution that destabilizes the planetary system. Stable periodic orbits are of particular importance in planetary dynamics, since they can host real planetary systems. We found stable orbits up to 60° of mutual planetary inclination, but in most families, the stability does not exceed 20°–30°, depending on the planetary mass ratio. Most of these orbits are very eccentric. Stable inclined circular orbits or orbits of low eccentricity were found in the 4/3 and 5/2 resonance, respectively. 相似文献
16.
The chaotic behaviour of the motion of the planets in our Solar System is well established. In this work to model a hypothetical extrasolar planetary system our Solar System was modified in such a way that we replaced the Earth by a more massive planet and let the other planets and all the orbital elements unchanged. The major result of former numerical experiments with a modified Solar System was the appearance of a chaotic window at κ E ∈ (4, 6), where the dynamical state of the system was highly chaotic and even the body with the smallest mass escaped in some cases. On the contrary for very large values of the mass of the Earth, even greater than that of Jupiter regular dynamical behaviour was observed. In this paper the investigations are extended to the complete Solar System and showed, that this chaotic window does still exist. Tests in different ‘Solar Systems’ clarified that including only Jupiter and Saturn with their actual masses together with a more ‘massive’ Earth (4 < κ E < 6) perturbs the orbit of Mars so that it can even be ejected from the system. Using the results of the Laplace‐Lagrange secular theory we found secular resonances acting between the motions of the nodes of Mars, Jupiter and Saturn. These secular resonances give rise to strong chaos, which is the cause of the appearance of the instability window. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
17.
Chaotic mixing in noisy Hamiltonian systems 总被引：1，自引：0，他引：1
Henry E. Kandrup Ilya V. Pogorelov Ioannis V. Sideris 《Monthly notices of the Royal Astronomical Society》2000,311(4):719-732
This paper summarizes an investigation of the effects of low-amplitude noise and periodic driving on phase-space transport in three-dimensional Hamiltonian systems, a problem directly applicable to systems like galaxies, where such perturbations reflect internal irregularities and/or a surrounding environment. A new diagnostic tool is exploited to quantify the extent to which, over long times, different segments of the same chaotic orbit evolved in the absence of such perturbations can exhibit very different amounts of chaos. First-passage-time experiments are used to study how small perturbations of an individual orbit can dramatically accelerate phase-space transport, allowing 'sticky' chaotic orbits trapped near regular islands to become unstuck on surprisingly short time‐scales. The effects of small perturbations are also studied in the context of orbit ensembles with the aim of understanding how such irregularities can increase the efficacy of chaotic mixing. For both noise and periodic driving, the effect of the perturbation scales roughly logarithmically in amplitude. For white noise, the details are unimportant: additive and multiplicative noise tend to have similar effects and the presence or absence of friction related to the noise by a fluctuation–dissipation theorem is largely irrelevant. Allowing for coloured noise can significantly decrease the efficacy of the perturbation, but only when the autocorrelation time, which vanishes for white noise, becomes so large that there is little power at frequencies comparable with the natural frequencies of the unperturbed orbit. This suggests strongly that noise-induced extrinsic diffusion, like modulational diffusion associated with periodic driving, is a resonance phenomenon. Potential implications for galaxies are discussed. 相似文献
18.
19.
A. V. Melnikov 《Astronomy Letters》2016,42(2):115-125
The chaotic orbital dynamics of the planet in the wide visual binary star system 16 Cyg is considered. The only planet in this system has a significant orbital eccentricity, e = 0.69. Previously, Holman et al. suggested the possibility of chaos in the orbital dynamics of the planet due to the proximity of 16 Cyg to the separatrix of the Lidov–Kozai resonance. We have calculated the Lyapunov characteristic exponents on the set of possible orbital parameters for the planet. In all cases, the dynamics of 16 Cyg is regular with a Lyapunov time of more than 30 000 yr. The dynamics is considered in detail for several possible models of the planetary orbit; the dependences of Lyapunov exponents on the time of their calculation and the time dependences of osculating orbital elements have been constructed. Phase space sections for the system dynamics near the Lidov–Kozai resonance have been constructed for all models. A chaotic behavior in the orbital motion of the planet in 16 Cyg is shown to be unlikely, because 16 Cyg in phase space is far from the separatrix of the Lidov–Kozai resonance at admissible orbital parameters, with the chaotic layer near the separatrix being very narrow. 相似文献
20.
Practice, i.e., as long as the initial conditions cannot be specified exactlty, the outcome of a chaotic dynamical system can only be specified in statistical terms. Evolution equations (e.g., the Fokker-Planck equation) for a distribution of test particles can then be formulated, and as an alternative to analytical, mostly approximate or idealised solutions one may simulate the problem using Monte Carlo techniques. Such simulations are a well-known tool in the study of completely chaotic many-body systems such as star clusters or planetary rings, where the sample of test particles can indeed be taken to represent a random set of true solutions according to Bowen's shadowing lemma. In this sense the Monte Carlo modelling plays a role analogous to that of averaging or mapping in regular dynamics, i.e.: the exact dynamical system is replaced by a model overlooking the details of the short-term motion but yielding a good approximation to the long-term behaviour. By a further discretization of the problem the stochastic system can be modelled as a Markov chain. Both Monte Carlo simulations and Markov models have been used in cometary dynamics, and we review some examples from this work to illustrate the success as well as limitations of these stochastic modelling techniques. Lyapunov characteristic exponents and Kolmogorov entropy appear to be suitable tools for estimating the underlying stochasticity to which Monte Carlo simulations refer. 相似文献