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1.
Message and Taylor (1978) have given values of the mean eccentricities and commen-surabilities which correspond to bifurcation orbits of families of symmetric periodic orbits with families of asymmetric periodic orbits in the limit as the mass ratio tends to zero. These bifurcations have been given in a way that they seem to be isolated and unrelated from the whole structure of the periodic orbits of the system.In this paper a numerical investigation of the horizontal stability of the family I and its branches reveals the above bifurcations orbits in the Sun-Jupiter case of the restricted three-body problem and associates these orbits with the whole structure of the system, giving extensive information on them. 相似文献
2.
V. V. Markellos 《Celestial Mechanics and Dynamical Astronomy》1980,21(3):291-309
The three-dimensional general three-body problem is formulated suitably for the numerical determination of periodic orbits either directly or by continuation from the three-dimensional periodic orbits of the restricted problem. The symmetry properties of the equations of motion are established and the algorithms for the numerical determination of families of periodic orbits are outlined. A normalization scheme based on the concept of the invariable plane is introduced to simplify the process. All three types of symmetric orbit, as well as the general type of asymmetric orrbit, are considered. Many threedimmensional p periodic orbits are given. 相似文献
3.
The one-dimensional Newtonian three-body problem is known to have stable (quasi-)periodic orbits when the masses are equal. The existence and size of the stable region is discussed here in the case where the three masses are arbitrary. We consider only the stability of the periodic (generalized) Schubart's (1956) orbit. If this orbit is linearly stable it is almost always surrounded by a region of stable quasi-periodic orbits and the size and shape of this stable region depends on the masses. The three-dimensional linear stability of the periodic orbits is also determined. Final results show that the region of stability has a complicated shape and some of the stable regions in the mass-plane are quite narrow. The non-linear three-dimensional stability is studied independently by extensive numerical integrations and the results are found to be in agreement with the linear stability analysis. The boundaries of stable region in the mass-plane are given in terms of polynomial approximations. The results are compared with a similar work by Héenon (1977).We thank the referee for pointing out this reference to us. 相似文献
4.
Alejandro M. Leiva Carlos Bruno Briozzo 《Celestial Mechanics and Dynamical Astronomy》2008,101(3):225-245
Starting from 80 families of low-energy fast periodic transfer orbits in the Earth–Moon planar circular Restricted Three Body
Problem (RTBP), we obtain by analytical continuation 11 periodic orbits and 25 periodic arcs with similar properties in the
Sun–Earth–Moon Quasi-Bicircular Problem (QBCP). A novel and very simple procedure is introduced giving the solar phases at
which to attempt continuation. Detailed numerical results for each periodic orbit and arc found are given, including their
stability parameters and minimal distances to the Earth and Moon. The periods of these orbits are between 2.5 and 5 synodic
months, their energies are among the lowest possible to achieve an Earth–Moon transfer, and they show a diversity of circumlunar
trajectories, making them good candidates for missions requiring repeated passages around the Earth and the Moon with close
approaches to the last. 相似文献
5.
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
6.
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
7.
M. Hénon 《Celestial Mechanics and Dynamical Astronomy》1974,10(3):375-388
We show by a general argument that periodic solutions of the planar problem of three bodies (with given masses) form one-parameter families. This result is confirmed by numerical investigations: two orbits found earlier by Standish and Szebehely are shown to belong to continuous one-parameter families of periodic orbits. In general these orbits have a non-zero angular momentum, and the configuration after one period is rotated with respect to the initial configuration. Similar general arguments whow that in the three-dimensional problem, periodic orbits form also one-parameter families; in the one-dimensional problem, periodic orbits are isolated. 相似文献
8.
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
9.
The mechanism by which ‘vertical’ branches consisting of symmetric, three-dimensional periodic orbits bifurcate from families of plane orbits at ‘veertical self-resonant’ orbits is discussed, with emphasis on the relationship between symmetry properties and multiplicity, and methods for the numerical determination of such branches are described. As examples, eight new families of all symmetry classes which branch vertically from the familyf of retrograde satellite orbits in the Sun-Jupiter case of the restricted problem (μ=0.000 95), are given in their entirety; these branches are found, as expected, to occur in pairs, each pair arising from the same self-resonant orbit, and their symmetry properties following the predicted pattern. The stability and other properties of the branch orbits are discussed. 相似文献
10.
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
11.
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
12.
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
13.
In this paper, the periodic orbits around triangular points in the range of linear stability of the restricted three body problem, when the smaller primary and the test particle have the shape of an oblate spheroid and the larger primary is a radiation emitter with the allowance for the gravitational potential from the belt, is studied. It is observed that the orbits around these points are elliptical and have long and short periodic orbits. The period, orientation, eccentricities, the semi-major and semi-minor axis of the elliptic orbits are found. The study includes some numerical examples in the case of the Sun-Earth and Sun-Jupiter systems. 相似文献
14.
M. Michalodimitrakis 《Celestial Mechanics and Dynamical Astronomy》1979,19(3):263-277
It is shown that the vertical critical orbits of the general planar problem of three bodies can be used as starting points for finding monoparametric families of three-dimensional periodic orbits. Several numerical examples are given. 相似文献
15.
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
16.
Poincaré's continuation method is applied to the elliptic restricted problem for the computation of four families of doubly symmetric three-dimensional periodic orbits emanating from similar orbits corresponding to zero eccentricity. The results are given in four tables and the orbits are characterized in regard to their stability. 相似文献
17.
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
18.
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
19.
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
20.
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