共查询到20条相似文献，搜索用时 31 毫秒
1.
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
2.
A solution of the Uranus-Neptune planetary canonical equations of motion through the Von Zeipel technique is presented. A unique determinging function which depends upon mixed canonical variables, reduces the 12 critical terms of the Hamiltonian to the set of its secular terms. The Poincaré canonical variables are used. We refer to a common fixed plane, and apply the Jacobi-Radau set of origins. In our expansion we neglected terms of power higher than the fourth with respect to the eccentricities and sines of the inclinations. 相似文献
3.
In this part we obtain the expression for –s
by the application of Smart's method, which involves Taylor's theorem for functions of several variables. We neglected terms of power higher than the fourth with respect to eccentricities and tangents of inclinations. 相似文献
4.
R. Dvorak 《Celestial Mechanics and Dynamical Astronomy》1993,56(1-2):71-80
We present numerical results of the so-called Sitnikov-problem, a special case of the three-dimensional elliptic restricted three-body problem. Here the two primaries have equal masses and the third body moves perpendicular to the plane of the primaries' orbit through their barycenter. The circular problem is integrable through elliptic integrals; the elliptic case offers a surprisingly great variety of motions which are until now not very well known. Very interesting work was done by J. Moser in connection with the original Sitnikov-paper itself, but the results are only valid for special types of orbits. As the perturbation approach needs to have small parameters in the system we took in our experiments as initial conditions for the work moderate eccentricities for the primaries' orbit (0.33e
primaries
0.66) and also a range of initial conditions for the distance of the 3
rd
body (= the planet) from very close to the primaries orbital plane of motion up to distance 2 times the semi-major axes of their orbit. To visualize the complexity of motions we present some special orbits and show also the development of Poincaré surfaces of section with the eccentricity as a parameter. Finally a table shows the structure of phase space for these moderately chosen eccentricities. 相似文献
5.
The behaviour of ‘resonances’ in the spin-orbit coupling in celestial mechanics is investigated in a conservative setting.
We consider a Hamiltonian nearly-integrable model describing an approximation of the spin-orbit interaction. The continuous
system is reduced to a mapping by integrating the equations of motion through a symplectic algorithm. We study numerically
the stability of periodic orbits associated to the above mapping by looking at the eigenvalues of the matrix of the linearized
map over the full cycle of the periodic orbit. In particular, the value of the trace of the matrix is related to the stability
character of the periodic orbit. We denote by ε*
(p/q) the value of the perturbing parameter at which a given elliptic periodic orbit with frequency p/q becomes unstable. A plot of the critical function ε*
(p/q) versus the frequency at different orbital eccentricities shows significant peaks at the synchronous resonance (for low eccentricities)
and at the synchronous and 3:2 resonances (at higher eccentricities) in good agreement with astronomical observations.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
6.
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
7.
Ahmed Aly Kamel 《Celestial Mechanics and Dynamical Astronomy》1971,4(3-4):397-405
To develop the perturbation solution of the non-Hamiltonian system of differential equationsy=g(y, t; ), it is sufficient to obtain the perturbation solution of a Hamiltonian system represented by the HamiltonianK=Y·g(y, t; ) which is linear in the adjoint vectorY. This Hamiltonization allows the direct use of the perturbation methods already established for Hamiltonian systems. To demonstrate this fact, a Hamiltonian algorithm developed by this author and based on the Lie-Deprit transform is applied to the Hamiltonized system and is shown to be equivalent to the application of the non-Hamiltonian form of this same algorithm to the original non-Hamiltonian system. 相似文献
8.
C. L. Goudas 《Astrophysics and Space Science》1968,1(1):6-19
It is shown here that the third integral of the galaxy, whenever its constant is conserved, defines the same surface as the Hamiltonian, and thus does not constitute anynew integral, but a function of the already known integral of energy. In particular, the third integral and the Hamiltonian are found to possess collinear gradients, in accordance with Poincaré's theorem concerning the characteristic exponents in systems with multiple integrals. 相似文献
9.
Efi Meletlidou Simos Ichtiaroglou Fabian Josef Winterberg 《Celestial Mechanics and Dynamical Astronomy》2001,80(2):145-156
We prove that Hill's lunar problem does not possess a second analytic integral of motion, independent of the Hamiltonian. In order to obtain this result, we avoid the usual normalization in which the angular velocity of the rotating reference frame is put equal to unit. We construct an artificial Hamiltonian that includes an arbitrary parameter b and show that this Hamiltonian does not possess an analytic integral of motion for in an open interval around zero. Then, by selecting suitable values of , b and using the invariance of the Hamiltonian under scaling in the units of length and time, we show that the Hamiltonian of Hill's problem does not possess an integral of motion, analytically continued from the integrable two–body problem in a rotating frame. 相似文献
10.
Using a 12th order expansion of the perturbative potential in powers of the eccentricities and the inclinations, we study
the secular effects of two non-coplanar planets which are not in mean–motion resonance. By means of Lie transformations (which
introduce an action–angle formulation of the Hamiltonian), we find the four fundamental frequencies of the 3-D secular three-body
problem and compute the long-term time evolutions of the Keplerian elements. To find the relations between these elements,
the main combinations of the fundamental frequencies common to these evolutions are identified by frequency analysis. This
study is performed for two different reference frames: a general one and the Laplace plane. We underline the known limitations
of the linear Laplace–Lagrange theory and point out the great sensitivity of the 3-D secular three-body problem to its initial
values. This analytical approach is applied to the exoplanetary system Andromedae in order to search whether the eccentricities evolutions and the apsidal configuration (libration of ) observed in the coplanar case are maintained for increasing initial values of the mutual inclination of the two orbital
planes.
Anne-Sophie Libert is FNRS Research Fellow. 相似文献
11.
A problem of stability of odd 2-periodic oscillations of a satellite in the plane of an elliptic orbit of arbitrary eccentricity is considered. The motion is supposed to be only under the influence of gravitational torques.Stability of plane oscillations was investigated earlier (Zlatoustovet al., 1964) in linear approximation. In the present paper a problem of stability is solved in the non-linear mode. Terms up to the forth order inclusive are taken into consideration in expansion of Hamiltonian in a series.It is shown that necessary conditions of stability obtained in linear approximation coincide with sufficient conditions for almost all values of parameters ande (inertial characteristics of the satellite and eccentricity of the orbit). Exceptions represent either values of the parameters ,e when a problem of stability cannot be solved in a strict manner by non-linear approximation under consideration, or values of the parameters which correspond to resonances of the third and fourth orders. At the resonance of the third order oscillations are unstable, but at the resonance of the fourth order both unstability and stability of the satellite's oscillations take place depending on the values of the parameters ,e. 相似文献
12.
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
13.
For a given family of orbits f(x,y) = c
* which can be traced by a material point of unit in an inertial frame it is known that all potentials V(x,y) giving rise to this family satisfy a homogeneous, linear in V(x,y), second order partial differential equation (Bozis,1984). The present paper offers an analogous equation in a synodic system Oxy, rotating with angular velocity . The new equation, which relates the synodic potential function (x,y), = –V(x, y) + % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSqaaSqaai% aaigdaaeaacaaIYaaaaaaa!3780!\[\tfrac{1}{2}\]2(x
2 + y
2) to the given family f(x,y) = c
*, is again of the second order in (x,y) but nonlinear.As an application, some simple compatible pairs of functions (x,y) and f(x, y) are found, for appropriate values of , by adequately determining coefficients both in and f. 相似文献
14.
W. E. Wiesel 《Celestial Mechanics and Dynamical Astronomy》1976,13(1):3-37
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
15.
Ian W. Walker A. Gordon Emslie Archie E. Roy 《Celestial Mechanics and Dynamical Astronomy》1980,22(4):371-402
An expansion of the force function ofn-body dynamical systems, where the equations of motion are expressed in the Jacobian coordinate system, is shown to give rise naturally to a set of (n–1) (n–2) dimensionless parameters
ki
li
{i = 2,...,n;k = 2,...,i – 1 (i 3);l =i + 1,...,n (i n – 1)}, representative of the size of the disturbances on the Keplerian orbits of the various bodies. The expansion is particularized to the casen=3 which involves the consideration of only two parameters
23 and
32. Further, the work of Szebehely and Zare (1977) is reviewed briefly with reference to a sufficient condition for the stability of corotational coplanar three-body systems, in which two of the bodies form a binary system. This condition is sufficient in the sense that it precludes any possibility of an exchange of bodies, i.e. Hill type stability, however, it is not a necessary condition. These two approaches are then combined to yield regions of stability or instability in terms of the parameters
23 and
32 for any system of given masses and orbital characteristics (neglecting eccentricities and inclinations) with the following result: that there is a readily applicable rule to assess the likelihood of stability or instability of any given triple system in terms of
23 and
32.Treating a system ofn bodies as a set of disturbed three-body systems we use existing data from the solar system, known triple systems and numerical experiments in the many-body problem to plot a large number of triple systems in the
23,
32 plane and show the results agree well with the
23,
32 analysis above (eccentricities and inclinations as appropriate to most real systems being negligible). We further deal briefly with the extension of the criteria to many-body systems wheren>4, and discuss several interesting cases of dynamical systems. 相似文献
16.
We obtain thex - p
xPoincare phase plane for a two dimensional, resonant, galactic type Hamiltonian using conventional numerical integration,
a second order symplectic integrator and a map based on the averaged Hamiltonian. It is found that all three methods give
good results, for small values of the perturbation parameter, while the symplectic integrator does a better job than the mapping,
for large perturbations. The dynamical spectra are used to distinguish between regular and chaotic motion. 相似文献
17.
Situations arise in celestial mechanics where orbital eccentricities are large and yet it is desirable to maintain the Darwin-Kaula Fourier decomposition of the perturbing function. Evaluation of the appropriate eccentricity functionsG
lpq
(e) requires a double summation which, for practical purposes, must be truncated. In this note criteria have been established for truncation of the expansion for eccentricities 0.75. 相似文献
18.
We present a symplectic mapping model to study the evolution of a small body at the 3/4 exterior resonance with Neptune, for planar and for three dimensional motion. The mapping is based on the averaged Hamiltonian close to this resonance and is constructed in such a way that the topology of its phase space is similar to that of the Poincaré map of the elliptic restricted three-body problem. Using this model we study the evolution of a small object near the 3/4 resonance. Both chaotic and regular motions are found, and it is shown that the initial phase of the object plays an important role on the appearance of chaos. In the planar case, objects that are phase-protected from close encounters with Neptune have regular orbits even at eccentricities up to 0.44. On the other hand objects that are not phase protected show chaotic behaviour even at low eccentricities. The introduction of the inclination to our model affects the stable areas around the 3/4 mean motion resonance, which now become thinner and thinner and finally at is=10° the whole resonant region becomes chaotic. This may justify the absence of a large population of objects at this resonance. 相似文献
19.
N. D. Caranicolas 《Celestial Mechanics and Dynamical Astronomy》1989,47(1):87-96
We derive an algebraic mapping for an autonomous, two-dimensional galactic type Hamiltonian in the 1/1 resonance case. We use the mapping to study the stability of the periodic orbits. Using the x — p
x
Poincaré surface section, we compare the results of the mapping with those found by the numerical integration of the full equations of motion. For small values of the perturbation the results of the two methods are in very good agreement while satisfactory agreement is obtained for larger perturbations. 相似文献
20.
G. Contopoulos C. Efthymiopoulos N. Voglis 《Celestial Mechanics and Dynamical Astronomy》2000,78(1-4):243-263
We apply the theory of the third integral to a self-consistent galactic model, generated by the collapse of a N-body system. The final configuration after the collapse is a stationary triaxial system, that represents an almost prolate non-rotating elliptical galaxy with its longest axis in the z-direction. This system is represented by an axisymmetric potential V plus a small triaxial perturbation V
1. The orbits in the potential V are of three types: box orbits, tube orbits (corresponding to various resonances), and chaotic orbits.The intersections of the box and tube orbits by a Poincaré surface of section z=0 are closed invariant curves. The main tube orbits are like ellipses and form an island of stability on the (R,R) plane.We calculated the third integral I in the potential V for the general non-resonant case and for various resonant cases. The agreement between the invariant curves of the orbits and the level curves of the third integral is good for the box and tube orbits, if we truncate the third integral at an appropriate level. As expected the third integral fails in the case of chaotic orbits. The most important result is the form of the number density F on the Poincaré surface of section. This function decreases exponentially outwards for the box orbits, like Fexp(–bI), while it is constant, as expected, for the chaotic orbits. In the case of the island of the main tube orbits it has a minimum at the center of the island. This can be explained by the form of the near elliptical orbits that are elongated along R, thus they fail to support a self-consistent galaxy, which is elongated along the z-axis. 相似文献