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1.
Intersections of families of three-dimensional periodic orbits which define bifurcation points are studied. The existence conditions for bifurcation points are discussed and an algorithm for the numerical continuation of such points is developed. Two sequences of bifurcation points are given concerning the family of periodic orbits which starts and terminates at the triangular equilibrium pointsL
4,L
5. On these sequences two trifurcation points are identified forµ = 0.124214 andµ = 0.399335. The caseµ = 0.5 is studied in particular and it is found that the space families originating at the equilibrium pointsL
2,L
3,L
4,L
5 terminate on the same planar orbitm
1v of the familym. 相似文献
2.
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
3.
4.
Xi-Yun Hou 《中国天文和天体物理学报》2009,9(4)
In a previous paper, we proposed another special critical value concerning the evolution of the long period family around the equilateral equilibrium points, besides the two values given by Henrard. Are there any other special critical values? After studying the stability curves of the long period family carefully, we gave a negative answer. During the study, we found an interesting family of periodic orbits which we called the homo family. We studied the evolution of this family following the increase of μ. With these findings, we were able to explain the origin of the four branches of periodic families emanating from L4 and the stability results of the equilateral equilibrium points. 相似文献
5.
Antonis D. Pinotsis 《Celestial Mechanics and Dynamical Astronomy》2010,108(2):187-202
We studied systematically cases of the families of non-symmetric periodic orbits in the planar restricted three-body problem.
We took interesting information about the evolution, stability and termination of bifurcating families of various multiplicities.
We found that the main families of simple non-symmetric periodic orbits present a similar dynamical structure and bifurcation
pattern. As the Jacobi constant changes each branch of the characteristic of a main family spirals around a focal point-terminating
point in x- at which the Jacobi constant is C
∞ = 3 and their periodic orbits terminate at the corotation (at the Lagrangian point L4 or L5). As the family approaches asymptotically its termination point infinite changes of stability to instability and vice versa
occur along its characteristic. Thus, infinite bifurcation points appear and each one of them produces infinite inverse Feigenbaum
sequences. That is, every bifurcating family of a Feigenbaum sequence produces the same phenomenon and so on. Therefore, infinite
spiral characteristics appear and each one of them generates infinite new inner spirals and so on. Each member of these infinite
sets of the spirals reproduces a basic bifurcation pattern. Therefore, we have in general large unstable regions that generate
large chaotic regions near the corotation points L4, L5, which are unstable. As C varies along the spiral characteristic of every bifurcating family, which approaches its focal
point, infinite loops, one inside the other, surrounding the unstable triangular points L4 or L5 are formed on their orbits. So, each terminating point corresponds to an asymptotic non-symmetric periodic orbit that spirals
into the corotation points L4, L5 with infinite period. This is a new mechanism that produces very large degree of stochasticity. These conclusions help us
to comprehend better the motions around the points L4 and L5 of Lagrange. 相似文献
6.
The planar restricted three-body problem has an infinite number of families of symmetric periodic solutions (SPSs). The natural SPS families include certain families which are self-closed with respect to small variations in a parameter. These families remain closed for any admissible variations in the mass parameter μ. However, there are closed SRS families of another type, which exist only in bounded intervals of μ and are formed via self-bifurcations of some SPS families. This type of SPS families is poorly understude. This work describes the initial stage (4 bifurcations) of a bifurcation cascade of the natural family i and points out other closed SPS families known to date. 相似文献
7.
An enlarged averaged Hamiltonian is introduced to compute several families of periodic orbits of the planar elliptic 3-body
problem, in the Sun–Jupiter–Asteroid system, near the 4:1 resonance. Four resonant critical point families are found and their
stability is studied. The families of symmetric periodic orbits of the elliptic problem appear near the corresponding fixed
points computed in this model. There is a good agreement for moderate eccentricity of the asteroid for three of these families,
whereas the remaining family cannot be considered as a family of periodic orbits of the real model.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
8.
Bifurcating families around collinear libration points 总被引：1，自引：0，他引：1
The planar and the vertical Lyapunov families are two basic periodic families around the collinear libration points. The stability curves of these two families are given first, and then periodic families bifurcating from them are explored in detail. Several properties of these bifurcating families are found. This study follows a series of the authors’ publications on periodic families around the libration points in the restricted three-body problem. 相似文献
9.
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
10.
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
11.
This paper focuses on some aspects of the motion of a small particle moving near the Lagrangian points of the Earth–Moon system.
The model for the motion of the particle is the so-called bicircular problem (BCP), that includes the effect of Earth and
Moon as in the spatial restricted three body problem (RTBP), plus the effect of the Sun as a periodic time-dependent perturbation
of the RTBP. Due to this periodic forcing coming from the Sun, the Lagrangian points are no longer equilibrium solutions for
the BCP. On the other hand, the BCP has three periodic orbits (with the same period as the forcing) that can be seen as the
dynamical equivalent of the Lagrangian points. In this work, we first discuss some numerical methods for the accurate computation
of quasi-periodic solutions, and then we apply them to the BCP to obtain families of 2-D tori in an extended neighbourhood
of the Lagrangian points. These families start on the three periodic orbits mentioned above and they are continued in the
vertical (z and ż) direction up to a high distance. These (Cantor) families can be seen as the continuation, into the BCP, of the Lyapunov
family of periodic orbits of the Lagrangian points that goes in the (z, ż) direction. These results are used in a forthcoming work [9] to find regions where trajectories remain confined for a
very long time. It is remarkable that these regions seem to persist in the real system.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
12.
The main focus of this paper is calculation of the diameters of asteroids belonging to five families (Vesta, Eos, Eunomia, Koronis, and Themis). To do that, we used the HCM algorithm applied for a data set containing 292,003 numbered asteroids, and a numerical procedure for choosing the crucial parameter of the HCM, called “the cutting velocity” vcut. It was established with a precision as high as 1 m s?1. Thereafter, we used the WISE (Wide‐field Infrared Survey Explorer) catalog to set a range of albedo for the largest members of each family considered. The albedo data were supported by the data concerning color classification (SDSS MOC4). The asteroids with albedo out of this range were classified as interlopers and were therefore disqualified as family members. Sizes were calculated for the asteroids with albedo within the acceptable range. For the other asteroids (those chosen by means of the HCM, but with albedo not listed in the WISE), the value of albedo of the largest member of the family was adopted. Results are given in a set of figures showing the families on the planes (a, e), (a, i), (e, i). Diameters and volumes of the asteroids that are the individual members of a family were calculated on the basis of their known or assumed albedo and on their absolute magnitude. Volumes of the parent bodies of the families were found on the basis of the cumulative volume distribution of these families. We also studied the secular resonances of the family members. We have shown that the locations of members of the considered asteroid families are related to the lines of secular resonances z1, z2, and z3 with Saturn. 相似文献
13.
Edward Belbruno Jaume Llibre Mercè Ollé 《Celestial Mechanics and Dynamical Astronomy》1994,60(1):99-129
In this paper we deal with the circular Sitnikov problem as a subsystem of the three-dimensional circular restricted three-body problem. It has a first analytical part where by using elliptic functions we give the analytical expressions for the solutions of the circular Sitnikov problem and for the period function of its family of periodic orbits. We also analyze the qualitative and quantitative behavior of the period function. In the second numerical part, we study the linear stability of the family of periodic orbits of the Sitnikov problem, and of the families of periodic orbits of the three-dimensional circular restricted three-body problem which bifurcate from them; and we follow these bifurcated families until they end in families of periodic orbits of the planar circular restricted three-body problem. We compare our results with the previous ones of other authors on this problem. Finally, the characteristic curves of some bifurcated families obtained for the mass parameter close to 1/2 are also described. 相似文献
14.
For monoparametric familiesf(x,y)=c of planar orbits, created by a planar potentialV(x,y), we introduce the notion of the family boundary curves (FBC). All members of the familyf(x,y)=c are traced in an allowable region of thexy plane, defined by the corresponding FBC, with total energyE=E(c) varying along the family. Family boundary curves are also found for two-parametric familiesf(x,y,b)=c. The relation of equilibrium points and asymptotic orbits, possibly possessed by the potentialV(x,y), to be FBC is studied. 相似文献
15.
When μ is smaller than Routh’s critical value μ
1 = 0.03852 . . . , two planar Lyapunov families around triangular libration points exist, with the names of long and short
period families. There are periodic families which we call bridges connecting these two Lyapunov families. With μ increasing from 0 to 1, how these bridges evolve was studied. The interval (0,1) was divided into six subintervals (0, μ
5), (μ
5, μ
4), (μ
4, μ
3), (μ
3, μ
2), (μ
2, μ
1), (μ
1, 1), and in each subinterval the families B(pL, qS) were studied, along with the families B(qS, qS′). Especially in the interval (μ
2, μ
1), the conclusion that the bridges B(qS, qS′) do not exist was obtained. Connections between the short period family and the bridges B(kS, (k + 1)S) were also studied. With these studies, the structure of the web of periodic families around triangular libration points
was enriched. 相似文献
16.
G. Contopoulos 《Celestial Mechanics and Dynamical Astronomy》1983,31(2):193-211
We study the various families of periodic orbits in a dynamical system representing a plane rotating barred galaxy. One can have a general view of the main resonant types of orbits by considering the axisymmetric background. The introduction of a bar perturbation produces infinite gaps along the central familyx 1 (the family of circular orbits in the axisymmetric case). It produces also higher order bifurcations, unstable regions along the familyx 1, and long period orbits aroundL 4 andL 5. The evolution of the various types of orbits is described, as the Jacobi constanth, and the bar amplitude, increase. Of special importance are the infinities of period doubling pitchfork bifurcations. The genealogy of the long and short period orbits is described in detail. There are infinite gaps along the long period orbits producing an infinity of families. All of them bifurcate from the short period family. The rules followed by these families are described. Also an infinity of higher order bridges join the short and long period families. The analogies with the restricted three body problem are stressed. 相似文献
17.
Miquel Grau 《Celestial Mechanics and Dynamical Astronomy》1995,61(4):389-401
An enlarged averaged Hamiltonian is introduced to compute some families of periodic orbits of the planar elliptic 3-body problem, in the Sun-Jupiter-Asteroid system, near the 3:1 resonance. Five resonant families are found and their stability is studied, The families of symmetric periodic orbits of the elliptic problem appear near the corresponding fixed points which have been computed in this model and the coincidence is good for moderate values of the eccentricity of the asteroid for two of these families; the other three families do not fulfil the Sundman condition and they cannot be considered as families of periodic orbits of the real model. 相似文献
18.
The association of the Sitnikov family with families of multiple three-dimensional periodic orbits is studied. In particular, the families consisting of three-dimensional periodic orbits bifurcating from self-resonant orbits of the Sitnikov family at double, triple and quadruple period of the bifurcation orbit are considered. The branch families close upon themselves and remain 3D up to their terminations having two common members with the Sitnikov family. By varying the mass parameter we also study the evolution of some of the computed families and find that they become isolas and disappear gradually in three-dimensions by shrinking to point size. 相似文献
19.
P. A. Patsis E. M. Xilouris 《Monthly notices of the Royal Astronomical Society》2006,366(4):1121-1125
The vertical profiles of disc galaxies are built by the material trapped around stable periodic orbits, which form their 'skeletons'. Therefore, knowledge of the stability of the main families of periodic orbits in appropriate 3D models enables one to predict possible morphologies for edge-on disc galaxies. In a pilot survey we compare the orbital structures that lead to the appearance of 'peanut'- and 'X'-like features with the edge-on profiles of three disc galaxies (IC 2531, NGC 4013 and UGC 2048). The subtraction from the images of a model representing the axisymmetric component of the galaxies reveals the contribution of the non-axisymmetric terms. We find a direct correspondence between the orbital profiles of 3D bars in models and the observed main morphological features of the residuals. We also apply a simple unsharp masking technique in order to study the sharpest features of the images. Our basic conclusion is that the morphology of the boxy 'bulges' of these galaxies can be explained by considering disc material trapped around stable 3D periodic orbits. In most models, these building-block periodic orbits are bifurcated from the planar central family of a non-axisymmetric component, usually a bar, at low-order vertical resonances. In such a case, the boxy 'bulges' are parts of bars seen edge-on. For the three galaxies we study, the families associated with the 'peanut' or 'X'-shape morphology are probably bifurcations at the vertical 2/1 or 4/1 resonance. 相似文献
20.
P. Delibaltas 《Astrophysics and Space Science》1976,45(1):207-233
Several families of planar planetary-type periodic orbits in the general three-body problem, in a rotating frame of reference, for the Sun-Jupiter-Saturn mass-ratio are found and their stability is studied. It is found that the configuration in which the orbit of the smaller planet is inside the orbit of the larger planet is, in general, more stable.We also develop a method to study the stability of a planar periodic motion with respect to vertical perturbations. Planetary periodic orbits with the orbits of the two planets not close to each other are found to be vertically stable. There are several periodic orbits that are stable in the plane but vertically unstable and vice versa. It is also shown that a vertical critical orbit in the plane can generate a monoparametric family of three-dimensional periodic orbits. 相似文献