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1.
A simple approximate model of the asteroid dynamics near the 3:1 mean–motion resonance with Jupiter can be described by a
Hamiltonian system with two degrees of freedom. The phase variables of this system evolve at different rates and can be subdivided
into the ‘fast’ and ‘slow’ ones. Using the averaging technique, wisdom obtained the evolutionary equations which allow to
study the long-term behavior of the slow variables. The dynamic system described by the averaged equations will be called
the ‘Wisdom system’ below. The investigation of the, wisdom system properties allows us to present detailed classification
of the slow variables’ evolution paths. The validity of the averaged equations is closely connected with the conservation
of the approximate integral (adiabatic invariant) possessed by the original system. Qualitative changes in the behavior of
the fast variables cause the violations of the adiabatic invariance. As a result the adiabatic chaos phenomenon takes place.
Our analysis reveals numerous stable periodic trajectories in the region of the adiabatic chaos. 相似文献
2.
The motion of asteroids near the 3:1 Jovian resonance in the restricted planar case is studied using three numerical methods: (a) integrating the full equations of motion, (b) integrating the averaged equations of motion, and (c) using an algebraic mapping recently developed by Wisdom (1982, Astron. J.87, 577–593). The relative merits of each method are investigated. It is concluded that in the regular regions of the phase space, methods b and c give excellent agreement with each other and that provided the maximum eccentricity emax < 0.4 differences with the exact solution (method a) are <6% in emax and <27% in the period of the oscillations. The additions of higher order terms in the expansion of the averaged Hamiltonian provides marginally better agreement with the full integration. This is probably due to the slow convergence of the expansion of the disturbing function at large eccentricities (e > 0.3). In chaotic regions of the phase there is little agreement between the orbital elements at any given time calculated by each method. However, all methods reflect the qualitative behavior of the chaotic trajectories and give good agreement on the bounds of the motion. Since the map is at least 200 times faster than solving the full equations of motion it is an efficient method of rapidly exploring accessible regions of the chaotic phase space. 相似文献
3.
Ivan I. Shevchenko 《Monthly notices of the Royal Astronomical Society》2008,384(3):1211-1220
The chaotic orbital motion of Prometheus and Pandora, the 16th and 17th satellites of Saturn, is studied. Chaos in their orbital motion, as found by Goldreich & Rappaport and Renner & Sicardy, is due to interaction of resonances in the resonance multiplet corresponding to the 121:118 commensurability of the mean motions of the satellites. It is shown rigorously that the system moves in adiabatic regime. The Lyapunov time (the 'time horizon of predictability' of the motion) is calculated analytically and compared to the available numerical–experimental estimates. For this purpose, a method of analytical estimation of the maximum Lyapunov exponent in the perturbed pendulum model of non-linear resonance is applied. The method is based on the separatrix map theory. An analytical estimate of the width of the chaotic layer is made as well, based on the same theory. The ranges of chaotic diffusion in the mean motion are shown to be almost twice as big compared to previous estimates for both satellites. 相似文献
4.
Yi-Sui Sun Ji-Lin Zhou Tian-Ling Zhang Chong-Qing Cheng 《Celestial Mechanics and Dynamical Astronomy》1994,59(4):313-325
For the 3 : 1 Jovian resonance problem, the time scales of the two degrees of freedom of the resonant Hamiltonian are well-separated [5]. With the adiabatic approximation, the solution for the fast oscillations can be found in terms of the slowly varying variables. Thus the rapidly oscillating terms in the slow oscillation equations can be treated as forced terms. We refer to the resonance between the forcing and intrinsic frequencies as a forced secondary one in this paper. We discuss the forced secondary resonances in asteroidal motion at the 3 : 1 commensurability by using Wisdom's method. The results show that the orbits situated originally near the resonance will leave the neighbourhood of resonance and tend to the separatrices and critical points for different energies, respectively. We have not found any stochastic web as expected in this case. Moreover, we study the problem of validity on the approximation of a system.The Project Supported by the National Natural Science Foundation of China. 相似文献
5.
M. Šidlichovský 《Celestial Mechanics and Dynamical Astronomy》1990,49(2):177-196
A nonlinear theory of secular resonances is developed. Both terms corresponding to secular resonances 5 and 6 are taken into account in the Hamiltonian. The simple overlap criterion is applied and the condition for the overlap of these resonances is found. It is shown that in given approximation the value p = (1 - e2)1/2(1 - cosI) is an integral of motion, where the mean eccentricity e and mean inclination I are obtained by eliminating short-period perturbations as well as the nonresonant terms from the planets. The overlap criterion yields a critical value of parameter p depending on the semi-major axis a of the asteroid. For p greater than the critical value, resonance overlap occurs and chaotic motion has to be expected. A mapping is presented for fast calculation of the trajectories. The results are illustrated by level curves in surfaces of section method. 相似文献
6.
K. G. Hadjifotinou John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》2002,84(2):135-157
A symplectic mapping is constructed for the study of the dynamical evolution of Edgeworth-Kuiper belt objects near the 2:3 mean motion resonance with Neptune. The mapping is six-dimensional and is a good model for the Poincaré map of the real system, that is, the spatial elliptic restricted three-body problem at the 2:3 resonance, with the Sun and Neptune as primaries. The mapping model is based on the averaged Hamiltonian, corrected by a semianalytic method so that it has the basic topological properties of the phase space of the real system both qualitatively and quantitatively. We start with two dimensional motion and then we extend it to three dimensions. Both chaotic and regular motion is observed, depending on the objects' initial inclination and phase. For zero inclination, objects that are phase-protected from close encounters with Neptune show ordered motion even at eccentricities as large as 0.4 and despite being Neptune-crossers. On the other hand, not-phase-protected objects with eccentricities greater than 0.15 follow chaotic motion that leads to sudden jumps in their eccentricity and are removed from the 2:3 resonance, thus becoming short period comets. As inclination increases, chaotic motion becomes more widespread, but phase-protection still exists and, as a result, stable motion appears for eccentricities up to e = 0.3 and inclinations as high as i = 15°, a region where plutinos exist. 相似文献
7.
The averaged spin-orbit resonant motion of Mercury is considered, with e the orbital eccentricity, and i o the orbital inclination introduced as very slow functions of time, given by any secular planetary theory. The basis is our Hamiltonian approach (D’Hoedt, S., Lemaître, A.: Celest. Mech. Dyn. Astron. 89:267–283, 2004) in which Mercury is considered as a rigid body. The model is based on two degrees of freedom; the first one is linked to the 3:2 resonant spin-orbit motion, and the second one to the commensurability of the rotational and orbital nodes. Mercury is assumed to be very close to the Cassini equilibrium of the model. To follow the motion of rotation close to this equilibrium, which varies with respect to time through e and i o , we use the adiabatic invariant theory, extended to two degrees of freedom. We calculate the corrections (remaining functions) introduced by the time dependence of e and i o in the three steps necessary to characterize the frequencies at the equilibrium. The conclusion is that Mercury follows the Cassini equilibrium (stays in the Cassini forced state), in an adiabatic behavior: the area around the equilibrium does not change by more than ${\varepsilon}$ for times smaller than ${\frac{1}{\varepsilon}}$ . The role of the inclination and the eccentricity can be dissociated and measured in each step of the canonical transformation. 相似文献
8.
In order to understand the observed oscillations in sunspots we present a new method for calculating the resonant response of a realistic semi-empirical model of the sunspot umbral atmosphere and subphotosphere to magneto-atmospheric waves in a vertical magnetic field. The depth dependence of both the adiabatic coefficient and the turbulent pressure is taken into account. This requires an extension of the wave equations by Ferraro & Plumpton (1958). We compare the coefficients of wave transmission, re flection, and conversion between fast mode and slow mode waves for different assumptions, compare the results with those from earlier modelling efforts, and point out possible sources of mistakes. The depth dependence of the adiabatic coefficient strongly influences the resulting spectrum of resonance frequencies. The condition of a conservation of wave flux is violated if the depth dependence of the turbulent pressure is not properly considered. 相似文献
9.
A full characterization of a nonintegrable dynamical system requires an investigation into the chaotic properties of that system. One such system, the restricted problem of three bodies, has been studied for over two centuries, yet few studies have examined the chaotic nature of some ot its trajectories. This paper examines and classifies the onset of chaotic motion in the restricted three-body problem through the use of Poincaré surfaces of section, Liapunov characteristic numbers, power spectral density analysis and a newly developed technique called numerical irreversibility. The chaotic motion is found to be intermittent and becomes first evident when the Jacobian constant is slightly higher thanC
2. 相似文献
10.
In the transneptunian classical region (), an unexpected orbital excitation in eccentricity and inclination, dynamically distinct populations and the presence of chaotic regions are observed. For instance, the 7:4 mean motion resonance () appears to have been causing unique dynamical excitation according to observational evidences, namely, an apparent shallow gap in number density and anomalies in the colour distribution, both features enhanced near the 7:4 mean motion resonance location. In order to investigate the resonance dynamics, we present extensive computer simulation results totalizing almost 10,000 test particles under the effect of the four giant planets for the age of the solar system. A chaotic diffusion experiment was also performed to follow tracks in phase space over 4-5 Gyr. The 7:4 mean motion resonance is weakly chaotic causing irregular eccentricity and inclination evolution for billions of years. Most 7:4 resonant particles suffered significant eccentricities and/or inclinations excitation, an outcome shared even by those located in the vicinity of the resonance. Particles in stable resonance locking are rare and usually had 0.25<e<0.3. For other regions, 7:4 resonants had quite large mobility in phase space typically leaving the resonance (and being scattered) after reaching a critical e∼0.2. The escape happened in 108-109 yr time scales. Concerning the inclination dependence for 7:4 resonants, we found strong instability islands for approximately i>10°. Taking into account those particles still locked in the resonance at the end of the simulations, we determined a retainability of 12-15% for real 7:4 resonant transneptunian objects (TNOs). Lastly, our results demonstrate that classical TNOs associated with the 7:4 mean motion resonance have been evolving continuously until present with non-negligible mixing of populations. 相似文献
11.
Some asteroids in Earth‐crossing orbits avoid close approaches by entering in a mean motion resonance whenever the distance between the two orbits is small. These orbits are ‘Toro class’ according to the classification of (Milani et al., 1989). This protection mechanism can be understood by a semi‐averaged model, in which the fast variables are removed and the dynamical variables are the critical argument and the semimajor axis, with dependence upon a slow parameter. The adiabatic invariant theory can be applied to this model and accounts for all the qualitative features of the orbits in this class, including the onset of the libration when the orbit distance is small. Because of the neglected perturbations by the other planets, this theory is approximate and the adiabatic invariant is conserved only with low accuracy moreover, the Toro state can be terminated by a close approach to another planet (typically Venus). “Would you tell me, please, which way I ought to go from here?” “That depends a good deal on where you want to get to,” said the Cat. Alice in Wonderland, L. Carroll 相似文献
12.
Jack Wisdom 《Meteoritics & planetary science》2020,55(4):766-770
In Wisdom (2017), I presented new simulations of meteorite transport from the chaotic zones associated with major resonances in the asteroid belt: the ν6 secular resonance, the 3:1 mean motion resonance with Jupiter, and the 5:2 mean motion resonance with Jupiter. I found that the observed afternoon excess (the fact that approximately twice as many meteorites fall in the afternoon as in the morning) of the ordinary chondrites is consistent with chaotic transport from the 3:1 resonance, contradicting prior reports. Here I report an additional study of the transport of meteorites from ν6 secular resonance and the 3:1 mean motion resonance. I use an improved integration algorithm, and study the evolution of more particles. I confirm that the afternoon excess of the ordinary chondrites is consistent with transport from the 3:1 resonance. 相似文献
13.
Anne Lemaitre Sandrine D’Hoedt Nicolas Rambaux 《Celestial Mechanics and Dynamical Astronomy》2006,95(1-4):213-224
Our purpose is to build a model of rotation for a rigid Mercury, involving the planetary perturbations and the non-spherical shape of the planet. The approach is purely analytical, based on Hamiltonian formalism; we start with a first-order basic averaged resonant potential (including J
2 and C
22, and the first powers of the eccentricity and the inclination of Mercury). With this kernel model, we identify the present equilibrium of Mercury; we introduce local canonical variables, describing the motion around this 3:2 resonance. We perform a canonical untangling transformation, to generate three sets of action-angle variables, and identify the three basic frequencies associated to this motion. We show how to reintroduce the short-periodic terms, lost in the averaging process, thanks to the Lie generator; we also comment about the damping effects and the planetary perturbations. At any point of the development, we use the model SONYR to compare and check our calculations. 相似文献
14.
The global semi-numerical perturbation method proposed by Henrard and Lemaître (1986) for the 2/1 resonance of the planar elliptic restricted three body problem is applied to the 3/1 resonance and is compared with Wisdom's perturbative treatment (1985) of the same problem. It appears that the two methods are comparable in their ability to reproduce the results of numerical integration especially in what concerns the shape and area of chaotic domains. As the global semi-numerical perturbation method is easily adapted to more general types of perturbations, it is hoped that it can serve as the basis for the analysis of more refined models of asteroidal motion. We point out in our analysis that Wisdom's uncertainty zone mechanism for generating chaotic domains (also analysed by Escande 1985 under the name of slow Hamiltonian chaotic layer) is not the only one at work in this problem. The secondary resonance
p
= 0 plays also its role which is qualitatively (if not quantitatively) important as it is closely associated with the random jumps between a high eccentricity mode and a low eccentricity mode. 相似文献
15.
H. Varvoglis 《Earth, Moon, and Planets》1991,54(3):257-267
Many trajectories of the third body are integrated numerically in a modified elliptical restricted three body problem (ERTBP), in which the eccentricity, e, of the orbit of the second primary varies sinusoidally with time. It is found that, in the case of the 2:1 resonance, the introduction of the time variability of e modifies significantly the behaviour of the trajectories of the third body. In particular their osculating eccentricity e, present the following two notable features: (a) In all cases it shows a definite chaotic variation, which appears at significantly shorter time-scales than the one found by Wisdom in the e = constant case. (b) In many cases it shows a significant increase, up and beyond the (critical) value e
crit = 0.52. As a result the third body approaches the first primary at distances smaller than 0.29 (where by we denote the semi-major axis of the trajectory of the second primary around the first), which in the actual Sun-Jupiter-asteroid problem corresponds to the semi-major axis of Mars. Our result might be of interest in the context of explaining the Kirkwood gaps at the resonances where the osculating eccentricity of asteroid trajectories calculated in the classical (e = constant) ERTBP does not reach Mars crosser values. 相似文献
16.
Comparative Study of the 2:3 and 3:4 Resonant Motion with Neptune: An Application of Symplectic Mappings and Low Frequency Analysis 总被引：2，自引：1，他引：1
The 2:3 and 3:4 exterior mean motion resonances with Neptune are studied by applying symplectic mapping models. The mappings
represent efficiently Poincaré maps for the 3D elliptic restricted three body problem in the neighbourhood of the particular
resonances. A large number of trajectories is studied showing the coexistence of regular and chaotic orbits. Generally, chaotic
motion depletes the small bodies of the effective resonant region in both the 2:3 and 3:4 resonances. Applying a low frequency
spectral analysis of trajectories, we determined the phase space regions that correspond to either regular or chaotic motion.
It is found that the phase space of the 3:4 resonant motion is more chaotic than the 2:3 one. 相似文献
17.
N. D. Caranicolas 《Journal of Astrophysics and Astronomy》2001,22(4):309-319
We present a map for the study of resonant motion in a potential made up of two harmonic oscillators with quartic perturbing
terms. This potential can be considered to describe motion in the central parts of non-rotating elliptical galaxies. The map
is based on the averaged Hamiltonian. Adding on a semi-empirical basis suitable terms in the unperturbed averaged Hamiltonian,
corresponding to the 1:1 resonant case, we are able to construct a map describing motion in several resonant cases. The map
is used in order to find thex − p
x
Poincare phase plane for each resonance. Comparing the results of the map, with those obtained by numerical integration of
the equation of motion, we observe, that the map describes satisfactorily the broad features of orbits in all studied cases
for regular motion. There are cases where the map describes satisfactorily the properties of the chaotic orbits as well. 相似文献
18.
The dynamics of space debris with very high A/m near the geostationary orbit is dominated by the gravitational coefficient C
22 and the solar radiation pressure. An analysis of the stability of the orbits by the chaos indicator MEGNO and frequency analysis map FAM shows chaotic layers around the separatrix and reveals a web of sub-structures associated to resonances with the annual period
of the Sun. This succession of stable thin islands and chaotic layers can be reproduced and explained by a quite simple toy
model, based on a pendulum approach, perturbed, through the eccentricity, by the external (Sun) frequency. The use of suitable
action-angle variables in the circulation and libration regions of the pendulum allows to point out new resonances between
the geostationary libration angle and the Sun’s longitude. They correspond very well (positions, shape, width) to the structures
visible on the FAM representations. 相似文献
19.
Á. Bazsó R. Dvorak E. Pilat-Lohinger V. Eybl Ch. Lhotka 《Celestial Mechanics and Dynamical Astronomy》2010,107(1-2):63-76
It is known since the seminal study of Laskar (1989) that the inner planetary system is chaotic with respect to its orbits and even escapes are not impossible, although in time scales of billions of years. The aim of this investigation is to locate the orbits of Venus and Earth in phase space, respectively, to see how close their orbits are to chaotic motion which would lead to unstable orbits for the inner planets on much shorter time scales. Therefore, we did numerical experiments in different dynamical models with different initial conditions—on one hand the couple Venus–Earth was set close to different mean motion resonances (MMR), and on the other hand Venus’ orbital eccentricity (or inclination) was set to values as large as e = 0.36 (i = 40°). The couple Venus–Earth is almost exactly in the 13:8 mean motion resonance. The stronger acting 8:5 MMR inside, and the 5:3 MMR outside the 13:8 resonance are within a small shift in the Earth’s semimajor axis (only 1.5 percent). Especially Mercury is strongly affected by relatively small changes in initial eccentricity and/or inclination of Venus, and even escapes for the innermost planet are possible which may happen quite rapidly. 相似文献
20.
John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1999,73(1-4):65-76
We present a 3-D symplectic mapping model that is valid at the 2:1 mean motion resonance in the asteroid motion, in the Sun-Jupiter-asteroid
model. This model is used to study the dynamics inside this resonance and several features of the system have been made clear.
The introduction of the third dimension, through the inclination of the asteroid orbit, plays an important role in the evolution
of the asteroid and the appearance of chaotic motion. Also, the existence of the secondary resonances is clearly shown and
their role in the appearance of chaotic motion and the slow diffusion of the elements of the orbit is demonstrated.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献