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1.
An analysis of ordered and chaotic regions of motion in the outer asteroid belt has shown that once the eccentricity of Jupiter is introduced the chaotic regions of the circular model are quite easily depleted. This suggests that also objects in neighbouring regions must be strongly perturbed. Therefore it is not surprising that many outer belt asteroids have been reported in the literature as resonant or anyway dynamically protected. By using the planar elliptic restricted 3-body model we have investigated the motion of outer belt asteroids which had not been suspected to librate. We find 3 cases of libration and 11 cases of e, coupling that can be explained within the theory of secular resonances. It is thus established that in the outer belt only resonant and dynamically protected asteroids can have lifetimes of the same order as the age of the Solar System.  相似文献   

2.
The resonant structure of the restricted three body problem for the Sun- Jupiter asteroid system in the plane is studied, both for a circular and an elliptic orbit of Jupiter. Three typical resonances are studied, the 2 : 1, 3 : 1 and 4 : 1 mean motion resonance of the asteroid with Jupiter. The structure of the phase space is topologically different in these cases. These are typical for all other resonances in the asteroid problem. In each case we start with the unperturbed two-body system Sun-asteroid and we study the continuation of the periodic orbits when the perturbation due to a circular orbit of Jupiter is introduced. Families of periodic orbits of the first and of the second kind are presented. The structure of the phase space on a surface of section is also given. Next, we study the families of periodic orbits of the asteroid in the elliptic restricted problem with the eccentricity of Jupiter as a parameter. These orbits bifurcate from the families of the circular problem. Finally, we compare the above families of periodic orbits with the corresponding families of fixed points of the averaged problem. Different averaged Hamiltonians are considered in each resonance and the range of validity of each model is discussed.  相似文献   

3.
Motions of asteroids in mean motion resonances with Jupiter are studied in three-dimensional space. Orbital changes of fictitious asteroids in the Kirkwood gaps are calculated by numerical integrations for 105 – 106 years. The main results are as follows: (1) There are various motions of resonant asteroids, and some of them are very complicated and chaotic and others are regular. (2) The eccentricity of some asteroids becomes very large, and the variation of the inclination is large while the eccentricity is large. (3) In the 3:1 resonance, there is a long periodic change in the variation of the inclination, when (7 : ) is a simple ratio (7: longitude of perihelion, : longitude of node). (4) In the 7:3 resonance, the variation of the inclination of some resonant asteroids is so large that prograde motion becomes retrograde. Some asteroids in the 7:3 resonance can collide with the Sun as well as with the inner planets.  相似文献   

4.
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   

5.
We study the motion of asteroids in the main mean motion commensurabilities in the frame of the planar restricted three-body problem. No assumption is made about the size of the eccentricity of the asteroid. At small to moderate eccentricity, we recover existing results (shape of the phase space and location of secondary resonances). We also provide global pictures of the dynamics in the region of secondary resonances. At high eccentricity, the phase space portraits of the integrable part of the Hamiltonian show new families of stable orbits for the 3:2 and 2:1 cases and the secular resonances 5 and 6 are located.  相似文献   

6.
Schubart's model of a planar, elliptic restricted three-body problem is used to study the orbital motion of the Hilda asteroids from thePalomar-Leiden Survey. The 3:2 resonant coupling to Jupiter of some of these small asteroids are found to be stable. However, some of the small asteroids with absolute magnitudeg>15 have large amplitude of variation in their orbital elements in one libration period. Since the lifetime scales against catastrophic collision of the Hilda asteroids are estimated to be several times larger than those of the main belt objects, a significant portion of these resonant asteroids could be the original members of the Hilda group. From this point of view, it is suggested that such size-dependence of resonant orbital motions might be the result of the cosmogonic effects ofjet stream accretion.  相似文献   

7.
The planar motion of a Trojan asteroid is considered within the framework of the elliptic restricted three-body problem. The solution is derived asymptotically to second order taking the square root of the Jupiter-Sun mass ratio and the orbital eccentricity of Jupiter as first order quantities. The results are given in explicit form for the coordinates as functions of the true anomaly of Jupiter including both short and long periodic terms resulting from the orbital accentricity of Jupiter.  相似文献   

8.
Jack Wisdom 《Icarus》1983,56(1):51-74
The sudden eccentricity increases discovered by J. Wisdom (Astron J.87, 577–593, 1982) are reproduced in numerical integrations of the planar-elliptic restricted three-body problem, verifying that this phenomenon is real. Maximum Lyapunov characteristic exponents for trajectories near the 31 commensurability are computed both with the mappings presented in Wisdom (1982) and by numerical integration of the planar-elliptic problem. In all cases the agreement is excellent, indicating that the mappings accurately reflect whether trajectories are chaotic or quasiperiodic. The mappings are used to trace out the chaotic zone near the 31 commensurability, both in the planar-elliptic problem and to a more limited extent in the three-dimensional elliptic problem. The outer boundary of the chaotic zone coincides with the boundary of the 31 Kirkwood gap in the actual distribution of asteroids within the errors of the asteroid orbital elements.  相似文献   

9.
We study the capture and crossing probabilities in the 3:1 mean motion resonance with Jupiter for a small asteroid that migrates from the inner to the middle Main Belt under the action of the Yarkovsky effect. We use an algebraic mapping of the averaged planar restricted three-body problem based on the symplectic mapping of Hadjidemetriou (Celest Mech Dyn Astron 56:563–599, 1993), adding the secular variations of the orbit of Jupiter and non-symplectic terms to simulate the migration. We found that, for fast migration rates, the captures occur at discrete windows of initial eccentricities whose specific locations depend on the initial resonant angles, indicating that the capture phenomenon is not probabilistic. For slow migration rates, these windows become narrower and start to accumulate at low eccentricities, generating a region of mutual overlap where the capture probability tends to 100 %, in agreement with the theoretical predictions for the adiabatic regime. Our simulations allow us to predict the capture probabilities in both the adiabatic and non-adiabatic cases, in good agreement with results of Gomes (Celest Mech Dyn Astron 61:97–113, 1995) and Quillen (Mon Not RAS 365:1367–1382, 2006). We apply our model to the case of the Vesta asteroid family in the same context as Roig et al. (Icarus 194:125–136, 2008), and found results indicating that the high capture probability of Vesta family members into the 3:1 mean motion resonance is basically governed by the eccentricity of Jupiter and its secular variations.  相似文献   

10.
A comparative study is made between the 2/1 and the 3/2 resonant asteroid motion, with the aim to understand their different behaviour (gap in the 2/1 resonance, group in the 3/2 resonance). A symplectic mapping model is used, for each of these two resonances, assuming the asteroid is moving in the three-dimensional space under the gravitational perturbation of Jupiter. It is found that these resonances differ in several points, and although there is, in general, more chaos in the phase space close to the 3/2 resonance, even in the model of circular orbit of Jupiter, there are regions, close to the secondary resonances, which are less chaotic in the 3/2 resonance compared to the 2/1 resonance, and consequently trapping can take place.  相似文献   

11.
Earlier work indicates a comparatively rapid chaotic evolution of the orbits of some Hilda asteroids that move at the border of the domain occupied by the characteristic parameters of the objects at the 3/2 mean motion resonance. A simple Jupiter–Saturn model of the forces leads to numerical results on some of these cases and allows a search for additional resonances that can contribute to the chaotic evolution. In this context the importance of the secondary resonances that depend on the period of revolution of the argument of perihelion is pointed out. Among the studied additional resonances there are three-body resonances with arguments that depend on the mean longitudes of Jupiter, Saturn, and asteroid, but on slowly circulating angular elements of the asteroid as well, and the frequency of these arguments is close to a rational ratio with respect to the frequency of the libration due to the basic resonance.  相似文献   

12.
We consider the application of the statistical method of phase mixing to the approximate Poincaré solution to resonant motion. The two Poincaré integrals of the motion for the restricted problem of three bodies are introduced to first order in the eccentricity. The theory of the phase mixing of an initialad hoc distribution of particles is then developed for this dynamical system, and the absence of significant evolution of the system far from resonance is verified.A selection of results is given for the 21, 31, and 52 resonances, which show in general a peak on the low side of exact resonance and a gap on the high side. The amplitudes of both the peak and the gap decrease, and their relative separation increases as the resonance order increases, or as the initial distribution is shifted to higher eccentricities. Comparison with large numbers of numerically integrated orbits gives good agreement with the model, at least for small eccentricities. However, the model is unable to exhibit the clean gaps shown by the real asteroid belt. Hence, a purely statistical model of the Kirkwood gaps is ruled out, and we must search for an additional mechanism. Some speculation on possible additional mechanisms is offered.  相似文献   

13.
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   

14.
Two families of symmetric periodic orbits of the planar, general, three-body problem are presented. The masses of the three bodies include ratios equal to the Sun-Jupiter-Saturn system and the periods of the orbits of Jupiter and Saturn are in a 25 resonance. The (linear) stability of the orbits are studied in relation to eccentricity and mass variations. The generation of the two families of periodic orbits follows a systematic approach and employs (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to the elliptic restricted problem and from the circular and elliptic problems to the general problem through bifurcation phenomena relating the three dynamical systems. The approach also provides insight into the evolutionary process of periodic orbits continued from the restricted problems to the general problem.  相似文献   

15.
The restricted problem of the motion of a point of negligible mass (asteroid) in anN-planetary system is considered. It is assumed that all the planets move about the central body (Sun) along circular orbits in the same plane and the mean motions of the asteroid and the planets are incommensurable. The asteroid orbit evolution is described as a first approximation by secular equations with the perturbing function averaged by the mean longitudes of the asteroid and the planets. For small values of the asteroid orbit eccentricity an expression for the secular part of the perturbing function has been obtained. This expression holds for the arbitrary values of the asteroid orbit semiaxis which are different from those of the planet orbit radii. The stability of the asteroid circular orbits in a linear approximation with respect to the eccentricity is studied. The critical inclinations for a Solar system model are calculated.  相似文献   

16.
The existing explanations for the asteroid distribution in the main belt (between the orbits of Mars and Jupiter) are based on numerical integration of resonance orbits in models with more than two degrees of freedom. We suggest an approach based on the investigation of the families of periodic solutions of the planar circular restricted three-body problem, i.e., a model with two degrees of freedom. This work shows that (a) the distribution of asteroids near the (p + 1)/p resonances and position of the outer boundary of the main asteroid belt can be explained within the planar circular restricted three-body problem and (b) this problem does not explain the asteroid distribution near other resonances.  相似文献   

17.
Many asteroids with a semimajor axis close to that of Mars have been discovered in the last several years. Potentially some of these could be in 1:1 resonance with Mars, much as are the classic Trojan asteroids with Jupiter, and its lesser-known horseshoe companions with Earth. In the 1990s, two Trojan companions of Mars, 5261 Eureka and 1998 VF31, were discovered, librating about the L5 Lagrange point, 60° behind Mars in its orbit. Although several other potential Mars Trojans have been identified, our orbital calculations show only one other known asteroid, 1999 UJ7, to be a Trojan, associated with the L4 Lagrange point, 60° ahead of Mars in its orbit. We further find that asteroid 36017 (1999 ND43) is a horseshoe librator, alternating with periods of Trojan motion. This asteroid makes repeated close approaches to Earth and has a chaotic orbit whose behavior can be confidently predicted for less than 3000 years. We identify two objects, 2001 HW15 and 2000 TG2, within the resonant region capable of undergoing what we designate “circulation transition”, in which objects can pass between circulation outside the orbit of Mars and circulation inside it, or vice versa. The eccentricity of the orbit of Mars appears to play an important role in circulation transition and in horseshoe motion. Based on the orbits and on spectroscopic data, the Trojan asteroids of Mars may be primordial bodies, while some co-orbital bodies may be in a temporary state of motion.  相似文献   

18.
Our investigation is motivated by the recent discovery of asteroids orbiting the Sun and simultaneously staying near one of the Solar System planets for a long time. This regime of motion is usually called the quasi-satellite regime, since even at the times of the closest approaches the distance between the asteroid and the planet is significantly larger than the region of space (the Hill’s sphere) in which the planet can hold its satellites. We explore the properties of the quasi-satellite regimes in the context of the spatial restricted circular three-body problem “Sun–planet–asteroid”. Via double numerical averaging, we construct evolutionary equations which describe the long-term behaviour of the orbital elements of an asteroid. Special attention is paid to possible transitions between the motion in a quasi-satellite orbit and the one in another type of orbits available in the 1:1 resonance. A rough classification of the corresponding evolutionary paths is given for an asteroid’s motion with a sufficiently small eccentricity and inclination.  相似文献   

19.
When asteroids are in the secular resonance 6, the variation of the eccentricity becomes very large. In this paper, the dynamics of this secular resonance 6 is investigated by a simple analytical model, in which the third degree terms of the eccentricity and inclination are taken into account. The eccentricity variations of asteroids located near this resonance are represented clearly by the diagrams of equi-Hamiltonian curves on the plane of versuse ( the longitude of perihelion of asteroids and Saturn,e: the eccentricity of asteroids). These diagrams predict that the eccentricity of these asteroids suffers a large increase or decrease, and that the secular resonance argument librates about 0° and 180°. In order to confirm these predictions, numerical integrations are carried out over one million years. By these integrations, it is found that the eccentricity of secular resonant asteroids becomes more than 0.8, and that the libration about 0° also exists, as well as the libration about 180°. The strongly depopulated region in the asteroidal belt, which corresponds to the position of the secular resonance 6, is also explained well by this analytical model.  相似文献   

20.
Orbital resonances tend to force bodies into noncircular orbits. If a body is also under the influence of an eccentricity-reducing medium, it will experience a secular change in semimajor axis which may be positive or negative depending on whether its orbit is exterior or interior to that of the perturbing body. Thus a dissipative medium can promote either a loss or a gain in orbital energy. This process may explain the resonant structure of the asteroid belt and of Saturn's rings. For reasonable early solar system parameters, it would clear a gap near the 2:1 resonance with Jupiter on a time scale of a few thousand years; the gap width would be comparable to the Kirkwood gap presently at the location in the asteroid belt. Similarly, a gap comparable in width to Cassini's division would be cleared in Saturn's rings at the 2:1 resonance with Mimas in ～106 yr. Most of the material from the gap would be deposited at the outer edge of ring B. The process would also affect the radial distribution of preplanetary material. Moreover, it provides an explanation for the large amplitude of the Titan-Hyperion libration. Consideration of the effects of dissipation on orbits near the stable L4 and L5 points of the restricted three-body problem indicates that energy loss causes particles to move away from these points. This results explains the large amplitude of Trojan asteroids about these points and the possible capture of Trojan into orbit about Jupiter.  相似文献   

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