共查询到20条相似文献，搜索用时 31 毫秒
1.
I. A. Robin 《Celestial Mechanics and Dynamical Astronomy》1981,23(1):97-106
We show that the procedure employed in the circular restricted problem, of tracing families of three-dimensional periodic orbits from vertical self-resonant orbits belonging to plane families, can also be applied in the elliptic problem. A method of determining series of vertical bifurcation orbits in the planar elliptic restricted problem is described, and one such series consisting of vertical-critical orbits (a
v=+1) is given for the entire range (0,1/2) of the mass parameter . The initial segments of the families of three-dimensional orbits which bifurcate from two of the orbits belonging to this series are also given. 相似文献
2.
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
3.
John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1992,53(2):151-183
Four 3 : 1 resonant families of periodic orbits of the planar elliptic restricted three-body problem, in the Sun-Jupiter-asteroid system, have been computed. These families bifurcate from known families of the circular problem, which are also presented. Two of them, I
c
, II
c
bifurcate from the unstable region of the family of periodic orbits of the first kind (circular orbits of the asteroid) and are unstable and the other two, I
e
, II
e
, from the stable resonant 3 : 1 family of periodic orbits of the second kind (elliptic orbits of the asteroid). One of them is stable and the other is unstable. All the families of periodic orbits of the circular and the elliptic problem are compared with the corresponding fixed points of the averaged model used by several authors. The coincidence is good for the fixed points of the circular averaged model and the two families of the fixed points of the elliptic model corresponding to the families I
c
, II
c
, but is poor for the families I
e
, II
e
. A simple correction term to the averaged Hamiltonian of the elliptic model is proposed in this latter case, which makes the coincidence good. This, in fact, is equivalent to the construction of a new dynamical system, very close to the original one, which is simple and whose phase space has all the basic features of the elliptic restricted three-body problem. 相似文献
4.
P. G. Kazantzis 《Astrophysics and Space Science》1978,57(2):499-509
The orbits of a family of three-dimensional periodic orbits in the restricted problem of three bodies form a surface. In this paper we determine the equation of this surface in the case of the orbits of double symmetry of the family which emanates from the equilibrium pointL
1. This equation is obtained numerically by a least squares approximation method. 相似文献
5.
K. Katopodis 《Astrophysics and Space Science》1986,123(2):335-349
In this paper several monoparametric families of periodic orbits of the 3-dimensional general 3-body problem are presented. These families are found by numerical continuation with respect to the small massm
3, of some periodic orbits which belong to a family of 3-dimensional periodic orbits of the restricted elliptic problem. 相似文献
6.
Masaya Masayoshi Saito Kiyotaka Tanikawa 《Celestial Mechanics and Dynamical Astronomy》2009,103(3):191-207
We study the change of phase space structure of the rectilinear three-body problem when the mass combination is changed. Generally,
periodic orbits bifurcate from the stable Schubart periodic orbit and move radially outward. Among these periodic orbits there
are dominant periodic orbits having rotation number (n − 2)/n with n ≥ 3. We find that the number of dominant periodic orbits is two when n is odd and four when n is even. Dominant periodic orbits have large stable regions in and out of the stability region of the Schubart orbit (Schubart
region), and so they determine the size of the Schubart region and influence the structure of the Poincaré section out of
the Schubart region. Indeed, with the movement of the dominant periodic orbits, part of complicated structure of the Poincaré
section follows these orbits. We find stable periodic orbits which do not bifurcate from the Schubart orbit. 相似文献
7.
E. A. Belbruno 《Celestial Mechanics and Dynamical Astronomy》1981,25(4):397-415
A new regularizing transformation for the three-dimensional restricted three-body problem is constructed. It is explicitly derived and is equivalent to a simple rational map. Geometrically it is equivalent to a rotation of the 3-sphere. Unlike the KS map it is dimension preserving and is valid inn dimensions. This regularizing map is applied to the restricted problem in order to prove the existence of a family of periodic orbits which continue from a family of collision orbits. 相似文献
8.
We consider the particular case of the planar three body problem obtained when the masses form an isosceles triangle for all time. Various authors [1, 2, 12, 8, 9, 13, 10] have contributed in the knowledge of the triple collision and of several families of periodic orbits in this problem. We study the flow on a fixed level of negative energy. First we obtain a topological representation of the energy manifold including the triple collision and infinity as boundaries of that manifold. The existence of orbits connecting the triple collision and infinity gives some homoclinic and heteroclinic orbits. Using these orbits and the homothetic solutions of the problem we can characterize orbits which pass near triple collision and near infinity by pairs of sequences. One of the sequences describes the regions visited by the orbit, the other refers to the behaviour of the orbit between two consecutive passages by a suitable surface of section. This symbolic dynamics which has a topological character is given in an abstract form and after it is applied to the isosceles problem. We try to keep globality as far as possible. This strongly relies on the fact that the intersection of some invariant manifolds with an equatorial plane (v=0) have nice spiraling properties. This can be proved by analytical means in some local cases. Numerical simulations given in Appendix A make clear that these properties hold globally. 相似文献
9.
Masaya Masayoshi Saito Kiyotaka Tanikawa 《Celestial Mechanics and Dynamical Astronomy》2010,107(4):397-407
In the present paper, in the rectilinear three-body problem, we qualitatively follow the positions of non-Schubart periodic
orbits as the mass parameter changes. This is done by constructing their characteristic curves. In order to construct characteristic
curves, we assume a set of properties on the shape of areas corresponding to symbol sequences. These properties are assured
by our preceding numerical calculations. The main result is that characteristic curves always start at triple collision and
end at triple collision. This may give us some insight into the nature of periodic orbits in the N-body problem. 相似文献
10.
K. E. Papadakis 《Astrophysics and Space Science》2006,302(1-4):67-82
We study numerically the asymptotic homoclinic and heteroclinic orbits around the hyperbolic Lyapunov periodic orbits which
emanate from Euler's critical points L
1 and L
2, in the photogravitational restricted plane circular three-body problem. The invariant stable-unstable manifolds associated
to these Lyapunov orbits, are also presented. Poincaré surface of sections of these manifolds on appropriate planes and several
homoclinic and heteroclinic orbits for the gravitational case as well as for varying radiation factor q
1, are displayed. Homoclinic-homoclinic and homoclinic-heteroclinic-homoclinic chains which link the interior with the exterior
Hill's regions, are illustrated. We adopt the Sun-Jupiter system and assume that only the larger primary radiates. It is found
that for small deviations of its value from the gravitational case (q
1 = 1), the radiation pressure exerts a significant impact on the Hill's regions and on these asymptotic orbits. 相似文献
11.
We explore the effect of oblateness of Saturn (more massive primary) on the periodic orbits and the regions of quasi-periodic motion around both the primaries in the Saturn-Titan system in the framework of planar circular restricted three-body problem. First order interior and exterior mean motion resonances are located. The effect of oblateness is studied on the location, nature and size of periodic and quasi-periodic orbits, using the numerical technique of Poincare surface of sections. Some of the periodic orbits change to quasi-periodic orbits due to the effect of oblateness and vice-versa. The stability of the orbits around Saturn, Titan and both varies with the inclusion of oblateness. The centers of the periodic orbits around Titan move towards Saturn, whereas those around Saturn move towards Titan. For the orbit around Titan at C=2.9992, x=0.959494, the apocenter becomes pericenter. By incorporating oblateness effect, the orbit around Titan at C=2.99345, x=0.924938 is captured by Saturn, remains in various trajectories around Saturn, and as time progresses it spirals away around both the primaries. 相似文献
12.
Robert L. Devaney 《Celestial Mechanics and Dynamical Astronomy》1979,19(4):391-404
We investigate specific homothetic solutions of then-body problem which both begin and end in a simultaneous collision of all of the particles. Under a suitable change of variables, these solutions become heteroclinic orbits, i.e., they lie in the intersection of the stable and unstable manifolds of distinct equilibrium points. Our main result is that these manifolds intersect transversely along these orbits. This proves that the homothetic solutions are structurally stable.Partially supported by NSF Grant MCS 77-00430. 相似文献
13.
We apply the numerical technique of Poincare surface of section to investigate the dichotomy present in the Earth-Moon system, considering the framework of planar, circular, restricted three-body problem. A study on the transition of quasi-periodic orbits (oscillatory type dichotomy) present at the Jacobi constant C=2.85 shows that the dichotomy discussed here exist not at a particular value of the mass ratio and the Jacobi constant. It is observed that as C increases, the range of mass ratio at which the dichotomy pertains increases, even though the mass ratio at which the transition of orbits takes place decreases. 相似文献
14.
Joaquín Delgado Fernández 《Celestial Mechanics and Dynamical Astronomy》1988,44(3):299-307
In a recent paper [3], Lacomba and Llibre showed numerically the existence of two transversal ejection-collision orbits in Hill's problem for a valueC=5 of the Jacobian constant. This result can be used to prove the non-existence ofC
1-extendable regular integrals for Hill's problem. Here we give an analytic proof of the existence of four ejection-collision orbits which are transversal for large enough values ofC. 相似文献
15.
M. Hénon 《Celestial Mechanics and Dynamical Astronomy》1974,10(3):375-388
We show by a general argument that periodic solutions of the planar problem of three bodies (with given masses) form one-parameter families. This result is confirmed by numerical investigations: two orbits found earlier by Standish and Szebehely are shown to belong to continuous one-parameter families of periodic orbits. In general these orbits have a non-zero angular momentum, and the configuration after one period is rotated with respect to the initial configuration. Similar general arguments whow that in the three-dimensional problem, periodic orbits form also one-parameter families; in the one-dimensional problem, periodic orbits are isolated. 相似文献
16.
Global stability regions are found for classi orbits of the circular restricted 3-body problem for primary masses equal and Jacobi constantK>15.5. As this constant decreases, the stability, region shrinks extremely rapidly. 相似文献
17.
V. V. Markellos 《Celestial Mechanics and Dynamical Astronomy》1975,12(2):215-224
The critical orbits, corresponding to bifurcations of the generating family and its branches, are considered more closely and the part off is investigated that has branches of very high order only. Three families of periodic solutions of the elliptic problem are also determined in an effort to follow the evolution of the stability region aroundf when the eccentricity of the primaries is increased to non-zero values. 相似文献
18.
M. Michalodimitrakis 《Astrophysics and Space Science》1981,75(2):289-305
By generalizing the restricted three-body problem, we introduce the restricted four-body problem. We present a numerical study of this problem which includes a study of equilibrium points, regions of possible motion and periodic orbits. Our main motivation for introducing this problem is that it can be used as an intermediate step for a systematic exploration of the genral four-body problem. In an analogous way, one may introduce the restrictedN-body problem. 相似文献
19.
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
20.
In this paper, families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem are presented.
Three bodies of masses m
1, m
2 and m
3 (primaries) lie always at the apices of an equilateral triangle, while each moves in circle about the center of mass of the
system fixed at the origin of the coordinate system. A massless fourth body is moving under the Newtonian gravitational attraction
of the primaries. The fourth body does not affect the motion of the three bodies. We investigate the evolution of these families
and we study their linear stability in three cases, i.e. when the three primary bodies are equal, when two primaries are equal
and finally when we have three unequal masses. Series, with respect to the mass m
3, of critical periodic orbits as well as horizontal and vertical-critical periodic orbits of each family and in any case of
the mass parameters are also calculated. 相似文献