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1.
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
2.
We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches.
We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides
the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative
periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with
binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant
families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As
the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between
triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge
to “previously unknown” periodic orbits in the planar problem. 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.
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
5.
The effect of the eccentricity of a planet’s orbit on the stability of the orbits of its satellites is studied. The model
used is the elliptic Hill case of the planar restricted three-body problem. The linear stability of all the known families
of periodic orbits of the problem is computed. No stable orbits are found, the majority of them possessing one or two pairs
of real eigenvalues of the monodromy matrix, while a part of a family with complex instability is found. Two families of periodic
orbits, bifurcating from the Lagrangian points L1, L2 of the corresponding circular case are found analytically. These orbits are very unstable and the determination of their
stability coefficients is not accurate, so we compute the largest Liapunov exponent in their vicinity. In all cases these
exponents are positive, indicating the existence of chaotic motions 相似文献
6.
Applying the method of analytical continuation of periodic orbits, we study quasi-satellite motion in the framework of the three-body problem. In the simplest, yet not trivial model, namely the planar circular restricted problem, it is known that quasi-satellite motion is associated with a family of periodic solutions, called family f, which consists of 1:1 resonant retrograde orbits. In our study, we determine the critical orbits of family f that are continued both in the elliptic and in the spatial models and compute the corresponding families that are generated and consist the backbone of the quasi-satellite regime in the restricted model. Then, we show the continuation of these families in the general three-body problem, we verify and explain previous computations and show the existence of a new family of spatial orbits. The linear stability of periodic orbits is also studied. Stable periodic orbits unravel regimes of regular motion in phase space where 1:1 resonant angles librate. Such regimes, which exist even for high eccentricities and inclinations, may consist dynamical regions where long-lived asteroids or co-orbital exoplanets can be found. 相似文献
7.
The importance of the stability characteristics of the planar elliptic restricted three-body problem is that they offer insight
about the general dynamical mechanisms causing instability in celestial mechanics. To analyze these concerns, elliptic–elliptic
and hyperbolic–elliptic resonance orbits (periodic solutions with lower period) are numerically discovered by use of Newton's
differential correction method. We find indications of stability for the elliptic–elliptic resonance orbits because slightly
perturbed orbits define a corresponding two-dimensional invariant manifold on the Poincaré surface-section. For the resonance
orbit of the hyperbolic–elliptic type, we show numerically that its stable and unstable manifolds intersect transversally
in phase-space to induce instability. Then, we find indications that there are orbits which jump from one resonance zone to
the next before escaping to infinity. This phenomenon is related to the so-called Arnold diffusion.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
8.
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
9.
John D. Hadjidemetriou Th. Christides 《Celestial Mechanics and Dynamical Astronomy》1975,12(2):175-187
A periodic orbit of the restricted circular three-body problem, selected arbitrarily, is used to generate a family of periodic motions in the general three-body problem in a rotating frame of reference, by varying the massm 3 of the third body. This family is continued numerically up to a maximum value of the mass of the originally small body, which corresponds to a mass ratiom 1:m 2:m 3?5:5:3. From that point on the family continues for decreasing massesm 3 until this mass becomes again equal to zero. It turns out that this final orbit of the family is a periodic orbit of the elliptic restricted three body problem. These results indicate clearly that families of periodic motions of the three-body problem exist for fixed values of the three masses, since this continuation can be applied to all members of a family of periodic orbits of the restricted three-body problem. It is also indicated that the periodic orbits of the circular restricted problem can be linked with the periodic orbits of the elliptic three-body problem through periodic orbits of the general three-body problem. 相似文献
10.
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
11.
John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》2006,95(1-4):225-244
We present a global view of the resonant structure of the phase space of a planetary system with two planets, moving in the same plane, as obtained from the set of the families of periodic orbits. An important tool to understand the topology of the phase space is to determine the position and the stability character of the families of periodic orbits. The region of the phase space close to a stable periodic orbit corresponds to stable, quasi periodic librations. In these regions it is possible for an extrasolar planetary system to exist, or to be trapped following a migration process due to dissipative forces. The mean motion resonances are associated with periodic orbits in a rotating frame, which means that the relative configuration is repeated in space. We start the study with the family of symmetric periodic orbits with nearly circular orbits of the two planets. Along this family the ratio of the periods of the two planets varies, and passes through rational values, which correspond to resonances. At these resonant points we have bifurcations of families of resonant elliptic periodic orbits. There are three topologically different resonances: (1) the resonances (n + 1):n, (2:1, 3:2, ...), (2) the resonances (2n + 1):(2n-1), (3:1, 5:3, ...) and (3) all other resonances. The topology at each one of the above three types of resonances is studied, for different values of the sum and of the ratio of the planetary masses. Both symmetric and asymmetric resonant elliptic periodic orbits exist. In general, the symmetric elliptic families bifurcate from the circular family, and the asymmetric elliptic families bifurcate from the symmetric elliptic families. The results are compared with the position of some observed extrasolar planetary systems. In some cases (e.g., Gliese 876) the observed system lies, with a very good accuracy, on the stable part of a family of resonant periodic orbits. 相似文献
12.
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
13.
G. A. Tsirogiannis E. A. Perdios V. V. Markellos 《Celestial Mechanics and Dynamical Astronomy》2009,103(1):49-78
We present an improved grid search method for the global computation of periodic orbits in model problems of Dynamics, and
the classification of these orbits into families. The method concerns symmetric periodic orbits in problems of two degrees
of freedom with a conserved quantity, and is applied here to problems of Celestial Mechanics. It consists of two main phases;
a global sampling technique in a two-dimensional space of initial conditions and a data processing procedure for the classification
(clustering) of the periodic orbits into families characterized by continuous evolution of the orbital parameters of member
orbits. The method is tested by using it to recompute known results. It is then applied with advantage to the determination
of the branch families of the family f of retrograde satellites in Hill’s Lunar problem, and to the determination of irregular families of periodic orbits in a
perturbed Hill problem, a species of families which are difficult to find by continuation methods.
相似文献
14.
Kyriaki I. Antoniadou Anne-Sophie Libert 《Celestial Mechanics and Dynamical Astronomy》2018,130(6):41
We consider a planetary system consisting of two primaries, namely a star and a giant planet, and a massless secondary, say a terrestrial planet or an asteroid, which moves under their gravitational attraction. We study the dynamics of this system in the framework of the circular and elliptic restricted three-body problem, when the motion of the giant planet describes circular and elliptic orbits, respectively. Originating from the circular family, families of symmetric periodic orbits in the 3/2, 5/2, 3/1, 4/1 and 5/1 mean-motion resonances are continued in the circular and the elliptic problems. New bifurcation points from the circular to the elliptic problem are found for each of the above resonances, and thus, new families continued from these points are herein presented. Stable segments of periodic orbits were found at high eccentricity values of the already known families considered as whole unstable previously. Moreover, new isolated (not continued from bifurcation points) families are computed in the elliptic restricted problem. The majority of the new families mainly consists of stable periodic orbits at high eccentricities. The families of the 5/1 resonance are investigated for the first time in the restricted three-body problems. We highlight the effect of stable periodic orbits on the formation of stable regions in their vicinity and unveil the boundaries of such domains in phase space by computing maps of dynamical stability. The long-term stable evolution of the terrestrial planets or asteroids is dependent on the existence of regular domains in their dynamical neighbourhood in phase space, which could host them for long-time spans. This study, besides other celestial architectures that can be efficiently modelled by the circular and elliptic restricted problems, is particularly appropriate for the discovery of terrestrial companions among the single-giant planet systems discovered so far. 相似文献
15.
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
16.
A systematic approach to generate periodic orbits in the elliptic restricted problem of three bodies in introduced. The approach is based on (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to periodic orbits in the elliptic restricted problem. Two families of periodic orbits of the elliptic restricted problem are found by this approach. The mass ratio of the primaries of these orbits is equal to that of the Sun-Jupiter system. The sidereal mean motions between the infinitesimal body and the smaller primary are in a 2:5 resonance, so as to approximate the Sun-Jupiter-Saturn system. The linear stability of these periodic orbits are studied as functions of the eccentricities of the primaries and of the infinitesimal body. The results show that both stable and unstable periodic orbits exist in the elliptic restricted problem that are close to the actual Sun-Jupiter-Saturn system. However, the periodic orbit closest to the actual Sun-Jupiter-Saturn system is (linearly) stable. 相似文献
17.
John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1993,56(1-2):201-219
The resonant structure of the restricted three body problem for the Sun- Jupiter asteroid system in the plane is studied, both for a circular and an elliptic orbit of Jupiter. Three typical resonances are studied, the 2 : 1, 3 : 1 and 4 : 1 mean motion resonance of the asteroid with Jupiter. The structure of the phase space is topologically different in these cases. These are typical for all other resonances in the asteroid problem. In each case we start with the unperturbed two-body system Sun-asteroid and we study the continuation of the periodic orbits when the perturbation due to a circular orbit of Jupiter is introduced. Families of periodic orbits of the first and of the second kind are presented. The structure of the phase space on a surface of section is also given. Next, we study the families of periodic orbits of the asteroid in the elliptic restricted problem with the eccentricity of Jupiter as a parameter. These orbits bifurcate from the families of the circular problem. Finally, we compare the above families of periodic orbits with the corresponding families of fixed points of the averaged problem. Different averaged Hamiltonians are considered in each resonance and the range of validity of each model is discussed. 相似文献
18.
Two families of symmetric periodic orbits of the planar, general, three-body problem are presented. The masses of the three bodies include ratios equal to the Sun-Jupiter-Saturn system and the periods of the orbits of Jupiter and Saturn are in a 25 resonance. The (linear) stability of the orbits are studied in relation to eccentricity and mass variations. The generation of the two families of periodic orbits follows a systematic approach and employs (numerical) continuation from periodic orbits of the first and second kind in the circular restricted problem to the elliptic restricted problem and from the circular and elliptic problems to the general problem through bifurcation phenomena relating the three dynamical systems. The approach also provides insight into the evolutionary process of periodic orbits continued from the restricted problems to the general problem. 相似文献
19.
D. J. Scheeres 《Celestial Mechanics and Dynamical Astronomy》1998,70(2):75-98
Starting from the four-body problem a generalization of both the restricted three-body problem and the Hill three-body problem
is derived. The model is time periodic and contains two parameters: the mass ratio ν of the restricted three-body problem
and the period parameter m of the Hill Variation orbit. In the proper coordinate frames the restricted three-body problem
is recovered as m → 0 and the classical Hill three-body problem is recovered as ν → 0. This model also predicts motions described
by earlier researchers using specific models of the Earth–Moon–Sun system. An application of the current model to the motion
of a spacecraft in the Sun perturbed Earth–Moon system is made using Hill's Variation orbit for the motion of the Earth–Moon
system. The model is general enough to apply to the motion of an infinitesimal mass under the influence of any two primaries
which orbit a larger mass.
Using the model, numerical investigations of the structure of motions around the geometric position of the triangular Lagrange
points are performed. Values of the parameter ν range in the neighborhood of the Earth–Moon value as the parameter m increases
from 0 to 0.195 at which point the Hill Variation orbit becomes unstable. Two families of planar periodic orbits are studied
in detail as the parameters m and ν vary. These families contain stable and unstable members in the plane and all have the
out-of-plane stability. The stable and unstable manifolds of the unstable periodic orbits are computed and found to be trapped
in a geometric area of phase space over long periods of time for ranges of the parameter values including the Earth–Moon–Sun
system.
This model is derived from the general four-body problem by rigorous application of the Hill and restricted approximations.
The validity of the Hill approximation is discussed in light of the actual geometry of the Earth–Moon–Sun system.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
20.
In a recent paper in this journal series of horizontally critical symmetric periodic orbits of the six basic families of the photogravitational restricted three–body problem were computed (Perdios, 2003). In this paper, such series are determined in the framework of the restricted three–body problem when the more massive primary is an oblate spheroid. The vertical stability of the horizontally critical orbits is also computed. 相似文献