Bifurcations of Plane to 3d Periodic Orbits in the Restricted Three-Body Problem with Oblateness     

E.A. Perdios  S.S. Kanavos  V.V. Markellos 《Astrophysics and Space Science》1998,262(1):75-87

We consider the bifurcation of 3D periodic orbits from the plane of motion of the primaries in the restricted three-body problem with oblateness. The simplest 3D periodic orbits branch-off at the plane periodic orbits of indifferent vertical stability. We describe briefly suitable numerical techniques and apply them to produce the first few such vertical-critical orbits of the basic families of periodic orbits of the problem, for varying mass parameter and fixed oblateness coefficent A₁ = 0.005, as well as for varying A₁ and fixed = 1/2. The horizontal stability of these orbits is also determined leading to predictions about the stability of the branching 3D orbits.  相似文献   

Bifurcations of plane with three-dimensional periodic orbits in the elliptic restricted problem     

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   

Bifurcations of planar to three-dimensional periodic orbits in the general three-body problem     

V. V. Markellos 《Celestial Mechanics and Dynamical Astronomy》1981,25(1):3-31

We study the generation of three-dimensional periodic orbits of the general three-body problem from special generating plane orbits, the vertical-critical orbits. The bifurcation process is examined analytically and geometrically. A method of obtaining numerically continuous sets of vertical-critical orbits is outlined, and applied for the determination of 16 monoparametric sets including all possible types of such orbits corresponding to all possible types of symmetry of the bifurcating three-dimensional orbits. The stability of all bifurcation orbits is assessed. Examples of three-dimensional periodic orbits generated from the bifurcation orbits are given.  相似文献   

Families of periodic orbits in the restricted four-body problem     

A. N. Baltagiannis  K. E. Papadakis 《Astrophysics and Space Science》2011,336(2):357-367

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   

Three-dimensional periodic motion of three finite masses around collinear equilibrium configurations     

V. V. Markellos 《Celestial Mechanics and Dynamical Astronomy》1981,25(4):319-344

Three-dimensional periodic motions of three bodies are shown to exist in the infinitesimal neighbourhood of their collinear equilibrium configurations. These configurations and some characteristic quantities of the emanating three-dimensional periodic orbits are given for many values of the two mass parameters, =m ₂/(m ₁+m ₂) andm ₃, of the general three-body problem, under the assumption that the straight line containing the bodies at equilibrium rotates with unit angular velocity. The analysis of the small periodic orbits near the equilibrium configurations is carried out to second-order terms in the small quantities describing the deviation from plane motion but the analytical solution obtained for the horizontal components of the state vector is valid to third-order terms in those quantities. The families of three-dimensional periodic orbits emanating from two of the collinear equilibrium configurations are continued numerically to large orbits. These families are found to terminate at large vertical-critical orbits of the familym of retrograde periodic orbits ofm ₃ around the primariesm ₁ andm ₂. The series of these termination orbits, formed when the value ofm ₃ varies, are also given. The three-dimensional orbits are computed form ₃=0.1.  相似文献   

A Note on Parameter Values for Stable Low-Inclination Periodic Motion in the Restricted Three-Body Problem with Radiation     

E.A. Perdios  K.E. Papadakis  V.V. Markellos 《Astrophysics and Space Science》1997,254(1):151-153

The intervals of the mass parameter (μ) values for possible stability of the basic families of 3D periodic orbits in the restricted three-body problem determined elsewhere are now extended into regions of theμ - q ₁ parameter space of the photogravitational restricted three-body problem, where q ₁ is the radiation factor of m ₁ and it is assumed that m ₂ does not radiate. Several 3D periodic orbits corresponding to these regions are computed and tested for stability and seven regions, corresponding to the vertical-critical orbits l1v, l'1v, l6v, m1v, m2v and i1v, survive this stability test, emerging as the regions allowing the simplest types of stable low inclination 3D motion of the infinitesimal particle. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

Examples of predominantly two-body orbits in the Sun-Jupiter asteroid system     

C. G. Zagouras  P. E. Moran 《Earth, Moon, and Planets》1981,25(1):67-94

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   

Highly perturbed periodic oscillations around a small primary     

V. V. Markellos  P. E. Moran  W. Black 《Astrophysics and Space Science》1975,33(2):385-419

A global picture of the families of simple periodic orbits in terms of their characteristics is given in the part of the plane (γ, ξ₁) of representation near the singularity corresponding to the small primary, for the case of the restricted problem with a small value of the mass parameter μ. The value used for μ is smaller than the critical value of Routh. The picture is found to be qualitatively different from the one corresponding to μ larger than the critical value. By means of the stability parameters four new families, consisting of asymmetric periodic orbits, are shown to exist as bifurcations of families of symmetric periodic orbits.  相似文献   

The Structure of Periodic Solutions in the Restricted Problem of the Three Bodies II     

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   

Periodic motions in the spatial Chermnykh restricted three-body problem     

A. E. Perdiou  A. A. Nikaki  E. A. Perdios 《Astrophysics and Space Science》2013,345(1):57-66

Three-dimensional motions in the Chermnykh restricted three-body problem are studied. Specifically, families of three-dimensional periodic orbits are determined through bifurcations of the family of straight-line periodic oscillations of the problem which exists for equal masses of the primaries. These rectilinear oscillations are perpendicular to the plane of the primaries and give rise to an infinite number of families consisting entirely of periodic orbits which belong to the three-dimensional space except their respective one-dimensional bifurcations as well as their planar terminations. Many of the computed branch families are continued in all mass range that they exist.  相似文献   

On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries     

C. Douskos  V. Kalantonis  P. Markellos  E. Perdios 《Astrophysics and Space Science》2012,337(1):99-106

The existence of new equilibrium points is established in the restricted three-body problem with equal prolate primaries. These are located on the Z-axis above and below the inner Eulerian equilibrium point L ₁ and give rise to a new type of straight-line periodic oscillations, different from the well known Sitnikov motions. Using the stability properties of these oscillations, bifurcation points are found at which new types of families of 3D periodic orbits branch out of the Z-axis consisting of orbits located entirely above or below the orbital plane of the primaries. Several of the bifurcating families are continued numerically and typical member orbits are illustrated.  相似文献   

The flattening of three-dimensional periodic orbits in the restricted problem     

C. Zagouras 《Astrophysics and Space Science》1977,50(2):383-407

A number of partly known families of symmetric three-dimensional periodic orbits of the restricted three-body (=0.4) problem are numerically continued in both ends until they terminate with orbits in the plane of motion of the primaries. The families of plane symmetric periodic orbits from which they bifurcate are identified and many orbit illustrations are given.  相似文献   

Multiple Periodic Orbits in the Hill Problem with Oblate Secondary     

A. E. Perdiou 《Earth, Moon, and Planets》2008,103(3-4):105-118

We study the multiple periodic orbits of Hill’s problem with oblate secondary. In particular, the network of families of double and triple symmetric periodic orbits is determined numerically for an arbitrary value of the oblateness coefficient of the secondary. The stability of the families is computed and critical orbits are determined. Attention is paid to the critical orbits at which families of non-symmetric periodic orbits bifurcate from the families of symmetric periodic orbits. Six such bifurcations are found, one for double-periodic and five for triple-periodic orbits. Critical orbits at which families of sub-multiple symmetric periodic orbits bifurcate are also discussed. Finally, we present the full network of families of multiple periodic orbits (up to multiplicity 12) together with the parts of the space of initial conditions corresponding to escape and collision orbits, obtaining a global view of the orbital behavior of this model problem.  相似文献   

Periodic orbits and bifurcations in the Sitnikov four-body problem when all primaries are oblate     

L. P. Pandey  I. Ahmad 《Astrophysics and Space Science》2013,345(1):73-83

We study the motions of an infinitesimal mass in the Sitnikov four-body problem in which three equal oblate spheroids (called primaries) symmetrical in all respect, are placed at the vertices of an equilateral triangle. These primaries are moving in circular orbits around their common center of mass. The fourth infinitesimal mass is moving along a line perpendicular to the plane of motion of the primaries and passing through the center of mass of the primaries. A relation between the oblateness-parameter ‘A’ and the increased sides ‘ε’ of the equilateral triangle during the motion is established. We confine our attention to one particular value of oblateness-parameter A=0.003. Only one stability region and 12 critical periodic orbits are found from which new three-dimensional families of symmetric periodic orbits bifurcate. 3-D families of symmetric periodic orbits, bifurcating from the 12 corresponding critical periodic orbits are determined. For A=0.005, observation shows that the stability region is wider than for A=0.003.  相似文献   

The manifolds of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem     

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   

The genealogy of periodic orbits in a plane rotating galaxy     

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   

Families of three-dimensional plane-symmetric periodic orbits in the restricted three-body problem: Sun-Jupiter case     

P. G. Kazantzis 《Astrophysics and Space Science》1979,61(2):357-367

Two new families of three-dimensional simple-symmetric periodic orbits are determined numerically in the Sun-Jupiter case of the restricted three-body problem. These families emanate from the vertical-critical orbits (_v = 1,c _v = 0)of the familiesi andl of plane symmetric simpleperiodic orbits direct around the Sun and the Sun-Jupiter respectively. Further, the numerical technique employed in the determination of these families has been described and interesting results have been pointed out. Also, computer plots of the orbits of these families have been shown in conical projections.  相似文献   

The stability parameters of three-dimensional periodic three-body orbits     

E. Perdios  K. Papadakis  V. V. Markellos 《Astrophysics and Space Science》1988,146(1):19-26

Formulae containing the elements of the variational matrix are obtained which determine the linear isoenergetic stability parameters of three-dimensional periodic orbits of the general three-boy problem. This requires the numerical integration of the variational equations but produces the stability parameters with the effective accuracy of the numerical integration. The conditions for stability, criticality, and bifurcations are briefly examined and the stability determination procedure is tested in the determination of some three-dimensional periodic orbits of low inclination bifurcating from vertical-critical coplanar orbits.  相似文献   

Periodic orbits around an oblate spheroid     

I. Stellmacher 《Celestial Mechanics and Dynamical Astronomy》1981,23(2):145-158

The general properties of certain differential systems are used to prove the existence of periodic orbits for a particle around an oblate spheroid.In a fixed frame, there are periodic orbits only fori=0 andi near /2. Furthermore, the generating orbits are circles.In a rotating frame, there are three families of orbits: first a family of periodic orbits in the vicinity of the critical inclination; secondly a family of periodic orbits in the equatorial plane with 0<e<1; thirdly a family of periodic orbits for any value of the inclination ife=0.  相似文献   

Numerical continuation of periodic orbits from the restricted to the general 3-dimensional 3-body problem     

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