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1.
We present a numerical study of the set of orbits of the planar circular restricted three body problem which undergo consecutive
close encounters with the small primary, or orbits of second species. The value of the Jacobi constant is fixed, and we restrict
the study to consecutive close encounters which occur within a maximal time interval. With these restrictions, the full set
of orbits of second species is found numerically from the intersections of the stable and unstable manifolds of the collision
singularity on the surface of section that corresponds to passage through the pericentre. A ‘skeleton’ of this set of curves
can be computed from the solutions of the two-body problem. The set of intersection points found in this limit corresponds
to the S-arcs and T-arcs of Hénon’s classification which verify the energy and time constraints, and can be used to construct
an alphabet to describe the orbits of second species. We give numerical evidence for the existence of a shift on this alphabet
that describes all the orbits with infinitely many close encounters with the small primary, and sketch a proof of the symbolic
dynamics. In particular, we find periodic orbits that combine S-type and T-type quasi-homoclinic arcs. 相似文献
2.
L. Benet T. H. Seligman D. Trautmann 《Celestial Mechanics and Dynamical Astronomy》1998,71(3):167-189
We study the scattering motion of the planar restricted three‐body problem for small mass parameters μ. We consider the symmetric
periodic orbits of this system with μ = 0 that collide with the singularity together with the circular and parabolic solutions
of the Kepler problem. These divide the parameter space in a natural way and characterize the main features of the scattering
problem for small non‐vanishing μ. Indeed, continuation of these orbits yields the primitive periodic orbits of the system
for small μ. For different regions of the parameter space, we present scattering functions and discuss the structure of the
chaotic saddle. We show that for μ < μc and any Jacobi integral there exist departures from hyperbolicity due to regions of
stable motion in phase space. Numerical bounds for μc are given.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
3.
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
4.
M. Michalodimitrakis 《Astrophysics and Space Science》1978,58(1):125-129
It is proved that the vertical critical orbits of the planar circular restricted three-body problem can be used as starting points for finding periodic orbits of the three-dimensional generalN-body problem. A numerical example is given. 相似文献
5.
Abimael Bengochea Manuel Falconi Ernesto Pérez-Chavela 《Astrophysics and Space Science》2011,333(2):399-408
We present some families of horseshoe periodic orbits in the general planar three-body problem for the case of two equal masses.
The considered system is a symmetric version of the one formed by Saturn, Janus and Epimetheus. We use a mass ratio equal
to 35×10−5, corresponding to 105 times the Saturn-Janus mass parameter of the restricted case; for this mass ratio the satellites have a significantly bigger
influence on the planet than in the classical Saturn, Janus and Epimetheus system. To obtain periodic orbits, we search those
horseshoe orbits passing through two reversible configurations. A particular kind of periodic orbits where the minor bodies
follow the same path is discussed. 相似文献
6.
M. Michalodimitrakis 《Astrophysics and Space Science》1980,68(1):253-268
Starting from the vertical critical periodic orbits of the planar Hill's problem we computed five families of three-dimensional periodic orbits. It is found that their behaviour is qualitatively similar to that of the corresponding families of the restricted problem. 相似文献
7.
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
8.
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
9.
S. Ichtiaroglou K. Katopodis Michalodimitrakis 《Journal of Astrophysics and Astronomy》1989,10(4):367-380
Three-dimensional planetary systems are studied, using the model of the restricted three-body problem for Μ =.001. Families
of three-dimensional periodic orbits of relatively low multiplicity are numerically computed at the resonances 3/1, 5/3, 3/5
and 1/3 and their stability is determined. The three-dimensional orbits are found by continuation to the third dimension of
the vertical critical orbits of the corresponding planar problem 相似文献
10.
G. Voyatzis T. Kotoulas J. D. Hadjidemetriou 《Monthly notices of the Royal Astronomical Society》2009,395(4):2147-2156
The 2/1 resonant dynamics of a two-planet planar system is studied within the framework of the three-body problem by computing families of periodic orbits and their linear stability. The continuation of resonant periodic orbits from the restricted to the general problem is studied in a systematic way. Starting from the Keplerian unperturbed system, we obtain the resonant families of the circular restricted problem. Then, we find all the families of the resonant elliptic restricted three-body problem, which bifurcate from the circular model. All these families are continued to the general three-body problem, and in this way we can obtain a global picture of all the families of periodic orbits of a two-planet resonant system. The parametric continuation, within the framework of the general problem, takes place by varying the planetary mass ratio ρ. We obtain bifurcations which are caused either due to collisions of the families in the space of initial conditions or due to the vanishing of bifurcation points. Our study refers to the whole range of planetary mass ratio values [ρ∈ (0, ∞)] and, therefore we include the passage from external to internal resonances. Thus, we can obtain all possible stable configurations in a systematic way. As an application, we consider the dynamics of four known planetary systems at the 2/1 resonance and we examine if they are associated with a stable periodic orbit. 相似文献
11.
We have extend Stormer’s problem considering four magnetic dipoles in motion trying to justify the phenomena of extreme “orderlines”
such as the ones observed in the rings of Saturn; the aim is to account the strength of the Lorentz forces estimating that
the Lorentz field, co-acting with the gravity field of the planet, will limit the motion of all charged particles and small
size grains with surface charges inside a layer of about 200 m thickness as that which is observed in the rings of Saturn.
For this purpose our interest feast in the motion of charged particles with neglected mass where only electromagnetic forces
accounted in comparison to the weakness of the Newtonian fields. This study is particularly difficult because in the regions
we investigate these motions there is enormous three dimensional instability. Following the Poincare’s hypothesis that periodic
solutions are ‘dense’ in the set of all solutions in Hamiltonian systems we try to calculate many families of periodic solutions
and to study their stability. In this work we prove that in this environment charged particles can trace planar symmetric
periodic orbits. We discuss these orbits in details and we give their symplectic relations using the Hamiltonian formulation
which is related to the symplectic matrix. We apply numerical procedures to find families of these orbits and to study their
stability. Moreover we give the bifurcations of these families with families of planar asymmetric periodic orbits and families
of three dimensional symmetric periodic orbits. 相似文献
12.
We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of symmetric periodic orbits by applying two schemes of analytical continuation. In the first scheme, we start from the 2/1 (or 1/2) resonant periodic orbits of the restricted problem and in the second scheme, we start from vertical critical periodic orbits of the general planar problem. Most of the periodic orbits are unstable, but many stable periodic orbits have been, also, found with mutual inclination up to 50?–60?, which may be related with the existence of real planetary systems. 相似文献
13.
In the framework of the planar restricted three-body problem we study a considerable number of resonances associated to the basic dynamical features of Kuiper belt and located between 30 and 48 a.u. Our study is based on the computation of resonant periodic orbits and their stability. Stable periodic orbits are surrounded by regular librations in phase space and in such domains the capture of trans-Neptunian object is possible. All the periodic orbits found are symmetric and there is an indication of the existence of asymmetric ones only in a few cases. In the present work first, second and third order resonances are under consideration. In the planar circular case we found that most of the periodic orbits are stable. The families of periodic orbits are temporarily interrupted by collisions but they continue up to relatively large values of the Jacobi constant and highly eccentric regular motion exists for all cases. In the elliptic problem and for a particular eccentricity value of the primary bodies, the periodic orbits are isolated. The corresponding families, where they belong to, bifurcate from specific periodic orbits of the circular problem and seem to continue up to the rectilinear problem. Both stable and unstable orbits are obtained for each case. In the elliptic problem, the unstable orbits found are associated with narrow chaotic domains in phase space. The evolution of the orbits, which are located in such chaotic domains, seems to be practically regular and bounded for long time intervals. 相似文献
14.
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
15.
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
16.
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
17.
The existing explanations for the asteroid distribution in the main belt (between the orbits of Mars and Jupiter) are based
on numerical integration of resonance orbits in models with more than two degrees of freedom. We suggest an approach based
on the investigation of the families of periodic solutions of the planar circular restricted three-body problem, i.e., a model
with two degrees of freedom. This work shows that (a) the distribution of asteroids near the (p + 1)/p resonances and position of the outer boundary of the main asteroid belt can be explained within the planar circular restricted
three-body problem and (b) this problem does not explain the asteroid distribution near other resonances. 相似文献
18.
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
19.
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
20.
Analysis of periodic orbits in the Saturn-Titan system using the method of Poincare section surfaces
We explore 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. The location, nature and size of periodic and quasi-periodic
orbits are studied using the numerical technique of Poincare surface of sections. The maximum amplitude of oscillations about
the periodic orbits is determined and is used as a parameter to measure the degree of stability in the phase space for such
orbits. It is found that the orbits around Saturn remain around it and their stability increases with the increase in the
value of Jacobi constant C. The orbits around Titan move towards it with the increase in C. At C=3.1, the pericenter and apocenter are 358.2 and 358.5 km, respectively. No periodic or quasi-periodic orbits could be found
by the present method around the collinear Lagrangian point L
1 (0.9569373834…). 相似文献