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1.
In this work we reveal for the first time that in the three dipole problem only asymmetric periodic orbits exist.For these periodic orbits — planar and three dimensional — of a charged particle moving under the influence of the electromagnetic field of the three dipoles we give their symplectic relations using the Hamiltonian formulation which is related to the symplectic matrix. Also we study the properties of the symplectic matrix and we give the relations there are among the variations of a periodic solution. These relations have been used to check the accuracy of numerical integration of equations of first order variations. 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.
In the three dipole problem where enormous electromagnetic forces obstruct the three dimensional movement of the charged particle we determined for the first time families of three dimensional asymmetric periodic orbits. We study how these families appear, branching from the planar motion and we develop the procedures we have followed to determine them numerically. Also we give their characteristics and the conical projections and plottings of some orbits. 相似文献
4.
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
5.
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
6.
A systematic and detailed discussion of planar periodic orbits, of a charged particle moving under the influence of an electromagnetic field of three celestial bodies, is given for the first time. In this problem the periodic orbits are all asymmetric. Numerical procedures are applied to find the families of these orbits and to study their stability. Moreover, the bifurcations of these families with families of three dimensional asymmetric periodic orbits are given. 相似文献
7.
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
8.
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
9.
We apply a numerical searching method to investigate three-dimensional periodic orbits of charged dust particles in planetary magnetospheres. A classic generalized Stormer model of magnetic planets along with the parameters of Saturn is employed. More periodic orbits are found, besides the already known circular periodic orbits in or parallel to the equatorial plane. We divide all these orbits into six categories based on their appearances. By calculating the characteristic multipliers of the orbits, we investigate the stabilities of these periodic orbits. 相似文献
10.
George Voyatzis John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》2006,95(1-4):259-271
We study the dynamics of 3:1 resonant motion for planetary systems with two planets, based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating frame) and the basic families of symmetric and asymmetric periodic orbits are computed. Four symmetric families bifurcate from the family of circular orbits of the two planets. Asymmetric families bifurcate from the symmetric families, at the critical points, where the stability character changes. There exist also asymmetric families that are independent of the above mentioned families. Bounded librations exist close to the stable periodic orbits. Therefore, such periodic orbits (symmetric or asymmetric) determine the possible stable configurations of a 3:1 resonant planetary system, even if the orbits of the two planets intersect. For the masses of the system 55Cnc most of the periodic orbits are unstable and they are associated with chaotic motion. There exist however stable symmetric and asymmetric orbits, corresponding to regular trajectories along which the critical angles librate. The 55Cnc extra-solar system is located in a stable domain of the phase space, centered at an asymmetric periodic orbit. 相似文献
11.
G. Voyatzis T. Kotoulas J.D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》2005,91(1-2):191-202
We present families of periodic orbits and their stability for the exterior mean motion resonances 1:2, 1:3 and 1:4 with Neptune
in the framework of the planar circular restricted three-body problem. We found that in each resonance there exist two branches
of symmetric elliptic periodic orbits with stable and unstable segments. Asymmetric periodic orbits bifurcate from the corresponding
symmetric ones. Asymmetric periodic orbits are stable and the motion in their neighbourhood is a libration with respect to
the resonant angle variable. In all the families of asymmetric periodic orbits the eccentricity extends to high values. Poincaré
sections reveal the changes of the topology in phase space. 相似文献
12.
Accurate numerical continuation of families of plane symmetric direct periodic orbits around the large primary in the Sun-Jupiter case of the restricted problem of three bodies allows the determination of the vertical branching points where families of three-dimensional symmetric periodic orbits bifurcate from the planar ones. Three families of plane periodic orbits, and the initial segments of ten bifurcating families of three-dimensional ones are determined. The stability of these families is examined and examples of their orbits are illustrated. 相似文献
13.
K.E. Papadakis 《Astrophysics and Space Science》2004,293(3):271-287
The regularized equations of motion of the planar Hill problem which includes the effect of the oblateness of the larger primary
body, is presented. Using the Levi-Civita coordinate transformation as well as the corresponding time transformation, we obtain
a simple regularized polynomial Hamiltonian of the dynamical system that corresponds to that of two uncoupled harmonic oscillators
perturbed by polynomial terms. The relations between the synodic and regularized variables are also given. The convenient
numerical computations of the regularized equations of motion, allow derivation of a map of the group of families of simple-periodic
orbits, free of collision cases, of both the classical and the Hill problem with oblateness. The horizontal stability of the
families is calculated and we determine series of horizontally critical symmetric periodic orbits of the basic families g and g'.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
14.
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
15.
A systematic numerical exploration of the families of asymmetric periodic orbits of the restricted three-body problem when
a) the primary bodies are equal and b) for the Earth-Moon mass ratio, is presented. Decades families of asymmetric periodic
solutions were found and three of the simplest ones, in the first case, and ten of the second one are illustrated. All of
these families consist of periodic orbits which are asymmetric with respect to x-axis while are simple symmetric periodic orbits with respect to y-axis (i.e. the orbit has only one perpendicular intersection at half period with y-axis). Many asymmetric periodic orbits, members of these families, are calculated and plotted. We studied the stability of
all the asymmetric periodic orbits we found. These families consist, mainly, of unstable periodic solutions but there exist
very small, with respect to x, intervals where these families have stable periodic orbits. We also found, using appropriate Poincaré surface of sections,
that a relatively large region of phase space extended around all these stable asymmetric periodic orbits shows chaotic motion. 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.
K. E. Papadakis 《Earth, Moon, and Planets》2008,103(1-2):25-42
This paper studies the asymmetric solutions of the restricted planar problem of three bodies, two of which are finite, moving in circular orbits around their center of masses, while the third is infinitesimal. We explore, numerically, the families of asymmetric simple-periodic orbits which bifurcate from the basic families of symmetric periodic solutions f, g, h, i, l and m, as well as the asymmetric ones associated with the families c, a and b which emanate from the collinear equilibrium points L 1, L 2 and L 3 correspondingly. The evolution of these asymmetric families covering the entire range of the mass parameter of the problem is presented. We found that some symmetric families have only one bifurcating asymmetric family, others have infinity number of asymmetric families associated with them and others have not branching asymmetric families at all, as the mass parameter varies. The network of the symmetric families and the branching asymmetric families from them when the primaries are equal, when the left primary body is three times bigger than the right one and for the Earth–Moon case, is presented. Minimum and maximum values of the mass parameter of the series of critical symmetric periodic orbits are given. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, we regularize the equations of motion of the problem using the Levi-Civita transformations. 相似文献
18.
P.G. Kazantzis 《Astrophysics and Space Science》1997,249(1):7-29
In this work we consider four families of plane periodic orbits direct around the Sun which approach Jupiter but they are sufficiently far from it so as to be considered as predominantly two body orbits of the Sun-asteroid system. We study their horizontal and vertical stabilities and we give the exact orbits of bifurcations of these families with three-dimensional families of the same multiplicity or twice the multiplicity of the above families of plane symmetric periodic orbits. Moreover, we give the first segments of the three dimensional families of symmetric periodic orbits which emanate from these plane bifurcations and we study their stability relating it with the stability of the plane bifurcations. 相似文献
19.
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
20.
K. E. Papadakis 《Astrophysics and Space Science》2009,323(3):261-272
In this paper we study the asymptotic solutions of the (N+1)-body ring planar problem, N of which are finite and ν=N−1 are moving in circular orbits around their center of masses, while the Nth+1 body is infinitesimal. ν of the primaries have equal masses m and the Nth most-massive primary, with m
0=β
m, is located at the origin of the system. We found the invariant unstable and stable manifolds around hyperbolic Lyapunov
periodic orbits, which emanate from the collinear equilibrium points L
1 and L
2. We construct numerically, from the intersection points of the appropriate Poincaré cuts, homoclinic symmetric asymptotic
orbits around these Lyapunov periodic orbits. There are families of symmetric simple-periodic orbits which contain as terminal
points asymptotic orbits which intersect the x-axis perpendicularly and tend asymptotically to equilibrium points of the problem spiraling into (and out of) these points.
All these families, for a fixed value of the mass parameter β=2, are found and presented. The eighteen (more geometrically simple) families and the corresponding eighteen terminating
homo- and heteroclinic symmetric asymptotic orbits are illustrated. The stability of these families is computed and also presented. 相似文献