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1.
Michel Hénon 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):87-100
We present five families of periodic solutions of Hill’s problem which are asymmetric with respect to the horizontal ξ axis. In one of these families, the orbits are symmetric with respect to the vertical η axis; in the four others, the orbits are without any symmetry. Each family consists of two branches, which are mirror images
of each other with respect to the ξ axis. These two branches are joined at a maximum of Γ, where the family of asymmetric periodic solutions intersects a family
of symmetric (with respect to the ξ axis) periodic solutions. Both branches can be continued into second species families for Γ → − ∞. 相似文献
2.
The stability evolution of family f of the planar circular restricted three-body problem in the Earth–Moon case is explored numerically using the Poincaré surface
of section. It is shown that third order resonances are the main cause of the reduction of the stability region of retrograde
satellites. Several branches of family f are also computed and these are seen by the configuration of their family characteristics to roughly determine the stability
region. Previous results on smaller mass ratios of primaries are thus extended to the Earth–Moon system. 相似文献
3.
The vertical stability character of the families of short and long period solutions around the triangular equilibrium points
of the restricted three-body problem is examined. For three values of the mass parameter less than equal to the critical value
of Routh (μ
R
) i.e. for μ = 0.000953875 (Sun-Jupiter), μ = 0.01215 (Earth-Moon) and μ = μ
R
= 0.038521, it is found that all such solutions are vertically stable. For μ > (μ
R
) vertical stability is studied for a number of ‘limiting’ orbits extended to μ = 0.45. The last limiting orbit computed by
Deprit for μ = 0.044 is continued to a family of periodic orbits into which the well known families of long and short period
solutions merge. The stability characteristics of this family are also studied. 相似文献
4.
5.
P. S. Soulis K. E. Papadakis T. Bountis 《Celestial Mechanics and Dynamical Astronomy》2008,100(4):251-266
We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits
in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the
fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted
three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” ż0 varies within a finite interval (while z
0 tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve
corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D
branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these
orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the
z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion
near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of
ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity. 相似文献
6.
The work presented in paper I (Papadakis, K.E., Goudas, C.L.: Astrophys. Space Sci. (2006)) is expanded here to cover the evolution of the approximate general solution of the restricted problem covering symmetric and escape solutions for values of μ in the interval [0, 0.5]. The work is purely numerical, although the available rich theoretical background permits the assertions that most of the theoretical issues related to the numerical treatment of the problem are known. The prime objective of this work is to apply the ‘Last Geometric Theorem of Poincaré’ (Birkhoff, G.D.: Trans. Amer. Math. Soc. 14, 14 (1913); Poincaré, H.: Rend. Cir. Mat. Palermo 33, 375 (1912)) and compute dense sets of axisymmetric periodic family curves covering the initial conditions space of bounded motions for a discrete set of values of the basic parameter μ spread along the entire interval of permissible values. The results obtained for each value of μ, tested for completeness, constitute an approximation of the general solution of the problem related to symmetric motions. The approximate general solution of the same problem related to asymmetric solutions, also computable by application of the same theorem (Poincaré-Birkhoff) is left for a future paper. A secondary objective is identification-computation of the compact space of escape motions of the problem also for selected values of the mass parameter μ. We first present the approximate general solution for the integrable case μ = 0 and then the approximate solution for the nonintegrable case μ = 10−3. We then proceed to presenting the approximate general solutions for the cases μ = 0.1, 0.2, 0.3, 0.4, and 0.5, in all cases building them in four phases, namely, presenting for each value of μ, first all family curves of symmetric periodic solutions that re-enter after 1 oscillation, then adding to it successively, the family curves that re-enter after 2 to 10 oscillations, after 11 to 30 oscillations, after 31 to 50 oscillations and, finally, after 51 to 100 oscillations. We identify in these solutions, considered as functions of the mass parameter μ, and at μ = 0 two failures of continuity, namely: 1. Integrals of motion, exempting the energy one, cease to exist for any infinitesimal positive value of μ. 2. Appearance of a split into two separate sub-domains in the originally (for μ = 0) unique space of bounded motions. The computed approximations of the general solution for all values of μ appear to fulfill the ‘completeness’ criterion inside properly selected sub-domains of the domain of bounded motions in the (x, C) plane, which means that these sub-domains are filled countably densely by periodic family curves, which form a laminar flow-line pattern. The family curves in this pattern may, or may not, be intersected by a ‘basic’ family curve segment of order from 1 up to 3. The isolated points generating asymptotic solutions resemble ‘sink’ points toward which dense sets of periodic family curves spiral. The points in the compact domain in the (x, C) plane resting outside the domain of bounded motions (μ = 0), including the gap between the two large sub-domains (μ > 0) created by the aforementioned split, generate escape motions. The gap between the two large sub-domains of bounded motions grows wider for growing μ. Also, a number of compact gaps that generate escape motions exist within the body of the two sub-domains of bounded motions. The approximate general solutions computed include symmetric, heteroclinic, asymptotic, collision and escape solutions, thus constituting one component of the full approximate general solution of the problem, the second and final component being that of asymmetric solutions. 相似文献
7.
We introduce a three-dimensional version of Hill’s problem with oblate secondary, determine its equilibrium points and their
stability and explore numerically its network of families of simple periodic orbits in the plane, paying special attention
to the evolution of this network for increasing oblateness of the secondary. We obtain some interesting results that differentiate
this from the classical problem. Among these is the eventual disappearance of the basic family g′ of the classical Hill problem and the existence of out-of-plane equilibrium points and a family of simple-periodic plane
orbits non-symmetric with respect to the x-axis. 相似文献
8.
Michael Nauenberg 《Celestial Mechanics and Dynamical Astronomy》2007,97(1):1-15
Numerical solutions are presented for a family of three dimensional periodic orbits with three equal masses which connects
the classical circular orbit of Lagrange with the figure eight orbit discovered by C. Moore [Moore, C.: Phys. Rev. Lett. 70, 3675–3679 (1993); Chenciner, A., Montgomery, R.: Ann. Math. 152, 881–901 (2000)]. Each member of this family is an orbit with finite angular momentum that is periodic in a frame which rotates
with frequency Ω around the horizontal symmetry axis of the figure eight orbit. Numerical solutions for figure eight shaped
orbits with finite angular momentum were first reported in [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and mathematical proofs for the existence of such orbits were given in [Marchal, C.: Celest. Mech. Dyn. Astron.
78, 279–298 (2001)], and more recently in [Chenciner, A. et al.: Nonlinearity 18, 1407–1424 (2005)] where also some numerical solutions have been presented. Numerical evidence is given here that the family
of such orbits is a continuous function of the rotation frequency Ω which varies between Ω = 0, for the planar figure eight
orbit with intrinsic frequency ω, and Ω = ω for the circular Lagrange orbit. Similar numerical solutions are also found for
n > 3 equal masses, where n is an odd integer, and an illustration is given for n = 21. Finite angular momentum orbits were also obtained numerically for rotations along the two other symmetry axis of the
figure eight orbit [Nauenberg, M.: Phys. Lett. 292, 93–99 (2001)], and some new results are given here. A preliminary non-linear stability analysis of these orbits is given
numerically, and some examples are given of nearby stable orbits which bifurcate from these families. 相似文献
9.
E. A. Perdios 《Celestial Mechanics and Dynamical Astronomy》2007,99(2):85-104
This paper deals with the Sitnikov family of straight-line motions of the circular restricted three-body problem, viewed as
generator of families of three-dimensional periodic orbits. We study the linear stability of the family, determine several
new critical orbits at which families of three dimensional periodic orbits of the same or double period bifurcate and present
an extensive numerical exploration of the bifurcating families. In the case of the same period bifurcations, 44 families are
determined. All these families are computed for equal as well as for nearly equal primaries (μ = 0.5, μ = 0.4995). Some of the bifurcating families are determined for all values of the mass parameter μ for which they exist. Examples of families of three dimensional periodic orbits bifurcating from the Sitnikov family at double
period bifurcations are also given. These are the only families of three-dimensional periodic orbits presented in the paper
which do not terminate with coplanar orbits and some of them contain stable parts. By contrast, all families bifurcating at
single-period bifurcations consist entirely of unstable orbits and terminate with coplanar orbits. 相似文献
10.
New doubly-symmetric families of comet-like periodic orbits in the spatial restricted (N + 1)-body problem 总被引:1,自引:0,他引:1
For any positive integer N ≥ 2 we prove the existence of a new family of periodic solutions for the spatial restricted (N +1)-body problem. In these solutions the infinitesimal particle is very far from the primaries. They have large inclinations
and some symmetries. In fact we extend results of Howison and Meyer (J. Diff. Equ. 163:174–197, 2000) from N = 2 to any positive integer N ≥ 2.
相似文献
11.
Poincaré surface of section technique is used to study the evolution of a family ‘f’ of simply symmetric retrograde periodic orbits around the smaller primary in the framework of restricted three-body problem for a number of systems, actual and hypothetical, with mass ratio varying from 10−7 to 0.015. It is found that as the mass ratio decreases the region of phase space containing the two separatrices shrinks in size and moves closer to the smaller primary. Also the corresponding value of Jacobi constant tends towards 3. 相似文献
12.
In the present article, a family of static spherical symmetric well behaved interior solutions is derived by considering the
metric potential g
44=B(1−Cr
2)−n
for the various values of n, such that (1+n)/(1−n) is positive integer. The solutions so obtained are utilised to construct the heavenly bodies’ like quasi-black holes such
as white dwarfs, neutron stars, quarks etc., by taking the surface density 2×1014 gm/cm3. The red shifts at the centre and on the surface are also computed for the different star models. Moreover the adiabatic
index is calculated in each case. In this process the authors come across the quarks star only. Least and maximum mass are
fond to be 3.4348M
Θ and 4.410454M
Θ along with the radii 21.0932 km and 23.7245 km respectively. 相似文献
13.
In this paper, following the increase of the mass ratio μ, the vertical stability curves of the long and the short period families were studied, and the vertical bifurcation families
from these two families were computed. It is found that these vertical bifurcation families connect the long and short period
families with the spatial periodic family emanating from the equilateral equilibrium points. The evolution details of these
vertical bifurcation families were carefully studied and they are found to be similar to the planar bifurcation families connecting
the long period family with the short period family in the planar case. 相似文献
14.
Fabian Josef Winterberg Efi Meletlidou 《Celestial Mechanics and Dynamical Astronomy》2004,88(1):37-49
We consider Hill's lunar problem as a perturbation of the integrable two-body problem. For this we avoid the usual normalization in which the angular velocity of the rotating frame of reference is put equal to unity and consider as the perturbation parameter. We first express the Hamiltonian H of Hill's lunar problem in the Delaunay variables. More precisely we deduce the expressions of H along the orbits of the two-body problem. Afterwards with the help of the conserved quantities of the planar two-body problem (energy, angular momentum and Laplace–Runge–Lenz vector) we prove that Hill's lunar problem does not possess a second integral of motion, independent of H, in the sense that there exist no analytic continuation of integrals, which are linear functions of in the rotating two-body problem. In connection with the proof of this main result we give a further restrictive statement to the nonintegrability of Hill's lunar problem. 相似文献
15.
F. Cachucho P. M. Cincotta S. Ferraz-Mello 《Celestial Mechanics and Dynamical Astronomy》2010,108(1):35-58
The theory of diffusion in many-dimensional Hamiltonian system is applied to asteroidal dynamics. The general formulation
developed by Chirikov is applied to the Nesvorny-Morbidelli analytic model of three-body (three-orbit) mean-motion resonances
(Jupiter-Saturn-asteroid). In particular, we investigate the diffusion along and across the separatrices of the (5, −2, −2) resonance of the (490) Veritas asteroidal family and their relationship to diffusion
in semi-major axis and eccentricity. The estimations of diffusion were obtained using the Melnikov integral, a Hadjidemetriou-type
sympletic map and numerical integrations for times up to 108 years. 相似文献
16.
Mercè Ollé Joan R. Pacha Jordi Villanueva 《Celestial Mechanics and Dynamical Astronomy》2004,90(1-2):87-107
The paper deals with different kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that
takes place close to the Lyapunov vertical family of periodic orbits of the triangular equilibrium point L4 in the 3D restricted three-body problem (RTBP) for the mass parameter, μ greater than (and close to) μR (Routh’s mass parameter). Consequences of such bifurcation, concerning the confinement of the motion close to the hyperbolic
orbits and the 3D nearby tori are also described. 相似文献
17.
We investigate straight-line orbits (SLO) in Hamiltonian force fields using both direct and inverse methods. A general theorem
is proven for natural Hamiltonians quadratic in the momenta for arbitrary dimensions and is considered in more detail for
two and three dimensions. Next we specialize to homogeneous potentials and their superpositions, including the familiar Hénon–Heiles
problem. It is shown that SLO’s can exist for arbitrary finite superpositions of N-forms. The results are applied to a family of potentials having discrete rotational symmetry as well as superpositions of
these potentials. 相似文献
18.
We discuss the equilibrium solutions of four different types of collinear four-body problems having two pairs of equal masses.
Two of these four-body models are symmetric about the center-of-mass while the other two are non-symmetric. We define two
mass ratios as μ
1 = m
1/M
T and μ
2 = m
2/M
T, where m
1 and m
2 are the two unequal masses and M
T is the total mass of the system. We discuss the existence of continuous family of equilibrium solutions for all the four
types of four-body problems. 相似文献
19.
This paper gives the results of a programme attempting to exploit ‘la seule bréche’ (Poincaré, 1892, p. 82) of non-integrable systems, namely to develop an approximate general solution for the three out of its four component-solutions of the planar restricted three-body problem. This is accomplished by computing a large number of families of ‘solutions précieuses’ (periodic solutions) covering densely the space of initial conditions of this problem. More specifically, we calculated numerically and only for μ = 0.4, all families of symmetric periodic solutions (1st component of the general solution) existing in the domain D:(x
0 ∊ [−2,2],C ∊ [−2,5]) of the (x
0, C) space and consisting of symmetric solutions re-entering after 1 up to 50 revolutions (see graph in Fig. 4). Then we tested the parts of the domain D that is void of such families and established that they belong to the category of escape motions (2nd component of the general solution). The approximation of the 3rd component (asymmetric solutions) we shall present in a future publication. The 4th component of the general solution of the problem, namely the one consisting of the bounded non-periodic solutions, is considered as approximated by those of the 1st or the 2nd component on account of the `Last Geometric Theorem of Poincaré' (Birkhoff, 1913). The results obtained provoked interest to repeat the same work inside the larger closed domain D:(x
0 ∊ [−6,2], C ∊ [−5,5]) and the results are presented in Fig. 15. A test run of the programme developed led to reproduction of the results presented by Hénon (1965) with better accuracy and many additional families not included in the sited paper. Pointer directions construed from the main body of results led to the definition of useful concepts of the basic family of order
n, n = 1, 2,… and the completeness criterion of the solution inside a compact sub-domain of the (x
0, C) space. The same results inspired the ‘partition theorem’, which conjectures the possibility of partitioning an initial conditions domain D into a finite set of sub-domains D
i that fulfill the completeness criterion and allow complete approximation of the general solution of this problem by computing a relatively small number of family curves. The numerical results of this project include a large number of families that were computed in detail covering their natural termination, the morphology, and stability of their member solutions. Zooming into sub-domains of D permitted clear presentation of the families of symmetric solutions contained in them. Such zooming was made for various values of the parameter N, which defines the re-entrance revolutions number, which was selected to be from 50 to 500. The areas generating escape solutions have being investigated. In Appendix A we present families of symmetric solutions terminating at asymptotic solutions, and in Appendix B the morphology of large period symmetric solutions though examples of orbits that re-enter after from 8 to 500 revolutions. The paper concludes that approximations of the general solution of the planar restricted problem is possible and presents such approximations, only for some sub-domains that fulfill the completeness criterion, on the basis of sufficiently large number of families. 相似文献
20.
Roman V. Baluev 《Celestial Mechanics and Dynamical Astronomy》2008,102(4):297-325
The full set of published radial velocity data (52 measurements from Keck + 58 ones from ELODIE + 17 ones from CORALIE) for
the star HD37124 is analysed. Two families of dynamically stable high-eccentricity orbital solutions for the planetary system
are found. In the first one, the outer planets c and d are trapped in the 2/1 mean-motion resonance. The second family of
solutions corresponds to the 5/2 mean-motion resonance between these planets. In both families, the planets are locked in
(or close to) an apsidal corotation resonance. In the case of the 2/1 MMR, it is an asymmetric apsidal corotation (with the
difference between the longitudes of periastra Δω ~ 60°), whereas in the case of the 5/2 MMR it is a symmetric antialigned one (Δω = 180°). It remains also possible that the two outer planets are not trapped in an orbital resonance. Then their orbital
eccentricities should be relatively small (less than, say, 0.15) and the ratio of their orbital periods is unlikely to exceed
2.3 − 2.5. 相似文献