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We study two and three-dimensional resonant periodic orbits, usingthe model of the restricted three-body problem with the Sun andNeptune as primaries. The position and the stability character ofthe periodic orbits determine the structure of the phase space andthis will provide useful information on the stability and longterm evolution of trans-Neptunian objects. The circular planarmodel is used as the starting point. Families of periodic orbitsare computed at the exterior resonances 1/2, 2/3 and 3/4 withNeptune and these are used as a guide to select the energy levelsfor the computation of the Poincaré maps, so that all basicresonances are included in the study. Using the circular planarmodel as the basic model, we extend our study to more realisticmodels by considering an elliptic orbit of Neptune and introducingthe inclination of the orbit. Families of symmetric periodicorbits of the planar elliptic restricted three-body problem andthe three-dimensional problem are found. All these orbitsbifurcate from the families of periodic orbits of the planarcircular problem. The stability of all orbits is studied. Althoughthe resonant structure in the circular problem is similar for allresonances, the situation changes if the eccentricity of Neptuneor the inclination of the orbit is taken into account. All theseresults are combined to explain why in some resonances there aremany bodies and other resonances are empty. 相似文献
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John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1993,56(4):563-599
A mapping model is constructed to describe asteroid motion near the 3 : 1 mean motion resonance with Jupiter, in the plane. The topology of the phase space of this mapping coincides with that of the real system, which is considered to be the elliptic restricted three body problem with the Sun and Jupiter as primaries. This model is valid for all values of the eccentricity. This is achieved by the introduction of a correcting term to the averaged Hamiltonian which is valid for small values of the ecentricity.We start with a two dimensional mapping which represents the circular restricted three body problem. This provides the basic framework for the complete model, but cannot explain the generation of a gap in the distribution of the asteroids at this resonance. The next approximation is a four dimensional mapping, corresponding to the elliptic restricted problem. It is found that chaotic regions exist near the 3 : 1 resonance, due to the interaction between the two degrees of freedom, for initial conditions close to a critical curve of the circular model. As a consequence of the chaotic motion, the eccentricity of the asteroid jumps to high values and close encounters with Mars and even Earth may occur, thus generating a gap. It is found that the generation of chaos depends also on the phase (i.e. the angles andv) and as a consequence, there exist islands of ordered motion inside the sea of chaotic motion near the 3 : 1 resonance. Thus, the model of the elliptic restricted three body problem cannot explain completely the generation of a gap, although the density in the distribution of the asteroids will be much less than far from the resonance. Finally, we take into account the effect of the gravitational attraction of Saturn on Jupiter's orbit, and in particular the variation of the eccentricity and the argument of perihelion. This generates a mixing of the phases and as a consequence the whole phase space near the 3 : 1 resonance becomes chaotic. This chaotic zone is in good agreement with the observations. 相似文献
3.
John D. Hadjidemetriou 《Astrophysics and Space Science》1969,3(1):31-45
The effects of non-isotropic ejection of mass from either component of a binary system on the orbital elements are studied, for the case of a small initial eccentricity of the relative orbit, when all the ejected mass falls on the other component. The problem is transformed to an equivalent two-body problem with isotropic variation of mass, plus a perturbing force which is a function of the intial conditions of ejection of the particles and their final, positions and velocities when they fall on the surface of the other star. The variation of the orbital elements are derived. It is shown that, to first-order terms in the eccentricity, the secular change of the semimajor axis is equal to the one corresponding to the case of zero initial eccentricity. On the contrary, the secular change of the eccentricity is smaller and it depends on the variations of mass ejection due to the finite eccentricity. 相似文献
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John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1987,43(1-4):371-390
A review is presented of periodic orbits of the planetary type in the general three-body problem and fourbody problem and the restricted circular and elliptic tnreebody problem. These correspond to planetary systems with one Sun and two or three planets (or a planet and its satellites), the motion of asteoids and also planetary systems with two Suns. The factors which affect the stability of the above configurations are studied in connection with resonance or additional perturbations. Finally, the correspondence of the periodic orbits in the restricted three-body problem with the fixed points obtained by the method of averaging or the method of surface of section is indicated. 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.
Victor A. Brumberg Eugene V. Brumberg: John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1999,75(4):301-301
The aim of this book is to present techniques for the study of motion of solar system objects in highly eccentric orbits. Instead of using the usual anomalies (mean, true, eccentric), the authors define and use a new kind of anomaly, the elliptic anomaly.In this way, it is possible, in a theory using perturbation series expansions, to make the ratio: (accuracy)/(number of needed terms), higher than in the classical techniques. The book consists of six chapters. The first chapter deals with the elliptic anomaly in the two-body problem and the second chapter presents the general technique to construct first-order perturbation theory in elliptic function expansions. The next three chapters deal with applications of the new technique to artificial satellites and asteroids, in highly eccentric orbits. The last chapter describes the basic algorithms of the theory.The tools developed in the book demand the use of computer algebra, which is implemented by means of Mathematica 3.0.The book is well written and the new technique is clearly presented and related to the existing techniques, making it useful to all those who use analytical or semi-analytical methods for the study of highly eccentric motion.
Celestial Mechanics at High Eccentricities, Gordon and Breach Publishers, US$95, GBP 59, EUR 79, ISBN 90-5699-212-0 相似文献
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John D. Hadjidemetriou 《Astrophysics and Space Science》1969,3(2):330-344
The problem of mass exchange in a close binary system is studied from the point of view of the evolution of the orbital elements. It is assumed that the original orbit is nearly circular and one of the components has expanded and fills the area inside the equipotential surface passing through the inner Lagrangian pointL
1, losing mass to the other component. The mass is assumed to be ejected along the tangent to the equipotential surface passing throughL
1 in retrograde orbits.It is proved that the eccentricity remains very small. The semimajor axis increases in almost all cases where the mass is being transferred from the less massive to the more massive component, and decreases when the mass is being transferred from the more massive to the less massive component. It is also shown that if the more massive star evolves first and loses mass to its companion, the process of mass exchange continues automatically until the originally more massive component becomes the less massive one and the binary system remains in an almost static condition for long intervals of time, the less massive component occupying the area inside the equipotential surface passing throughL
1 and surrounding the star. 相似文献
10.
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