New periodic solutions of the three dimensional restricted problem     

Pierre Guillaume 《Celestial Mechanics and Dynamical Astronomy》1974,10(4):475-495

A global review of the symmetric solutions of the restricted problem made in the Introduction opens a window on new symmetric periodic orbits of the two body problem in rotating axes which could be ‘trivially’ continuable to symmetric periodic orbits of the three dimensional restricted problem for small values of μ (see Figure 3). The proof of this possibility of continuation is given in Sections 1, 2, 3 using regularizing variables.  相似文献   

The elliptic restricted problem at the 3 : 1 resonance     

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   

Families of periodic orbits continued in regularizing coordinates     

André Deprit  Julian Palmore 《Celestial Mechanics and Dynamical Astronomy》1969,1(2):150-162

A predictor-corrector algorithm is proposed for continuing analytically families of periodic orbits beyond collision trajectories in the restricted problem of three bodies. It is based on Hill's equation for normal variations in Thiele's regularizing coordinates.  相似文献   

Surface fitting of three-dimensional closed solutions of the restricted three-body problem     

P. G. Kazantzis 《Astrophysics and Space Science》1978,57(2):499-509

The orbits of a family of three-dimensional periodic orbits in the restricted problem of three bodies form a surface. In this paper we determine the equation of this surface in the case of the orbits of double symmetry of the family which emanates from the equilibrium pointL ₁. This equation is obtained numerically by a least squares approximation method.  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   

Periodic orbits and bifurcations in the Sitnikov four-body problem     

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   

The scattering map in the planar restricted three body problem     

E. Canalias  A. Delshams  J. J. Masdemont  P. Roldán 《Celestial Mechanics and Dynamical Astronomy》2006,95(1-4):155-171

We study homoclinic transport to Lyapunov orbits around a collinear libration point in the planar restricted three body problem. A method to compute homoclinic orbits is first described. Then we introduce the scattering map for this problem (defined on a suitable normally hyperbolic invariant manifold) and we show how to compute it using the information already obtained for the homoclinic orbits. An example application to Astrodynamics is also proposed.  相似文献   

Strategies for plane change of Earth orbits using lunar gravity and derived trajectories of family G     

C. F. de Melo  E. E. N. Macau  O. C. Winter 《Celestial Mechanics and Dynamical Astronomy》2009,103(4):281-299

The dynamics of the circular restricted three-body Earth-Moon-particle problem predicts the existence of the retrograde periodic orbits around the Lagrangian equilibrium point L1. Such orbits belong to the so-called family G (Broucke, Periodic orbits in the restricted three-body problem with Earth-Moon masses, JPL Technical Report 32–1168, 1968) and starting from them it is possible to define a set of trajectories that form round trip links between the Earth and the Moon. These links occur even with more complex dynamical systems as the complete Sun-Earth-Moon-particle problem. One of the most remarkable properties of these trajectories, observed for the four-body problem, is a meaningful inclination gain when they penetrate into the lunar sphere of influence and accomplish a swing-by with the Moon. This way, when one of these trajectories returns to the proximities of the Earth, it will be in a different orbital plane from its initial Earth orbit. In this work, we present studies that show the possibility of using this property mainly to accomplish transfer maneuvers between two Earth orbits with different altitudes and inclinations, with low cost, taking into account the dynamics of the four-body problem and of the swing-by as well. The results show that it is possible to design a set of nominal transfer trajectories that require ΔV _Total less than conventional methods like Hohmann, bi-elliptic and bi-parabolic transfer with plane change.  相似文献   

On quasi-satellite periodic motion in asteroid and planetary dynamics     

G. Voyatzis  K. I. Antoniadou 《Celestial Mechanics and Dynamical Astronomy》2018,130(9):59

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   

Families of periodic orbits in the planar three-body problem     

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   

Asymptotic and Periodic Orbits Around L 3 in the Photogravitational Restricted Three-Body Problem     

V. S. Kalantonis  E. A. Perdios  O. Ragos 《Astrophysics and Space Science》2006,301(1-4):157-165

Asymptotic motion near the collinear equilibrium points of the photogravitational restricted three-body problem is considered. In particular, non-symmetric homoclinic solutions are numerically explored. These orbits are connected with periodic ones. We have computed numerically the families containing these orbits and have found that they terminate at both ends by asymptotically approaching simple periodic solutions belonging to the Lyapunov family emanating from L₃.  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   

Periodic collision orbits in the elliptic restricted three-body problem     

R. Broucke 《Celestial Mechanics and Dynamical Astronomy》1971,3(4):461-477

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On the families of periodic orbits which bifurcate from the circular Sitnikov motions     

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   

Motion close to the Hopf Bifurcation Of the vertical family of periodic orbits of L4     

Mercè Ollé  Joan R. Pacha  Jordi Villanueva 《Celestial Mechanics and Dynamical Astronomy》2004,90(1-2):87-107

The paper deals with different kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that takes place close to the Lyapunov vertical family of periodic orbits of the triangular equilibrium point L₄ in the 3D restricted three-body problem (RTBP) for the mass parameter, μ greater than (and close to) μ_R (Routh’s mass parameter). Consequences of such bifurcation, concerning the confinement of the motion close to the hyperbolic orbits and the 3D nearby tori are also described.  相似文献   

The circular restricted four-body problem     

M. Michalodimitrakis 《Astrophysics and Space Science》1981,75(2):289-305

By generalizing the restricted three-body problem, we introduce the restricted four-body problem. We present a numerical study of this problem which includes a study of equilibrium points, regions of possible motion and periodic orbits. Our main motivation for introducing this problem is that it can be used as an intermediate step for a systematic exploration of the genral four-body problem. In an analogous way, one may introduce the restrictedN-body problem.  相似文献   

Computing with certainty individual members of families of periodic orbits of a given period     

V. S. Kalantonis  E. A. Perdios  A. E. Perdiou  M. N. Vrahatis 《Celestial Mechanics and Dynamical Astronomy》2001,80(2):81-96

The accurate computation of families of periodic orbits is very important in the analysis of various celestial mechanics systems. The main difficulty for the computation of a family of periodic orbits of a given period is the determination within a given region of an individual member of this family which corresponds to a periodic orbit. To compute with certainty accurate individual members of a specific family we apply an efficient method using the Poincaré map on a surface of section of the considered problem. This method converges rapidly, within relatively large regions of the initial conditions. It is also independent of the local dynamics near periodic orbits which is especially useful in the case of conservative dynamical systems that possess many periodic orbits, often of the same period, close to each other in phase space. The only computable information required by this method is the signs of various function evaluations carried out during the integration of the equations of motion. This method can be applied to any system of celestial mechanics. In this contribution we apply it to the photogravitational problem.  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   

Numerical investigation of the planar restricted three-body problem     

V. V. Markellos 《Celestial Mechanics and Dynamical Astronomy》1974,9(3):365-380

In this article a method is described for the determination of families of periodic orbits, of the restricted problem of three bodies, as branchings of a given family of stable periodic orbits. Poincaré's method of successive crossings of a surface of section is applied for a value of the mass parameter corresponding to the Sun-Jupiter case of the restricted problem. New families are found, of the type of direct asteroids, having long periods and closing in space after many revolutions of the third body about the Sun. Their stability parameters are also given. The generating family, from which they branch, seems to have special significance for stability considerations.  相似文献   

The bifurcation of periodic orbits of the planar circular restrictedN-body problem to the three-dimensional generalN-body problem     

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