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1.
In this article we treat the 'Extended Sitnikov Problem' where three bodies of equal masses stay always in the Sitnikov configuration.
One of the bodies is confined to a motion perpendicular to the instantaneous plane of motion of the two other bodies (called
the primaries), which are always equally far away from the barycenter of the system (and from the third body). In contrary
to the Sitnikov Problem with one mass less body the primaries are not moving on Keplerian orbits. After a qualitative analysis
of possible motions in the 'Extended Sitnikov Problem' we explore the structure of phase space with the aid of properly chosen
surfaces of section. It turns out that for very small energies H the motion is possible only in small region of phase space and only thin layers of chaos appear in this region of mostly
regular motion. We have chosen the plane (
) as surface of section, where r is the distance between the primaries; we plot the respective points when the three bodies are 'aligned'. The fixed point
which corresponds to the 1 : 2 resonant orbit between the primaries' period and the period of motion of the third mass is
in the middle of the region of motion. For low energies this fixed point is stable, then for an increased value of the energy
splits into an unstable and two stable fixed points. The unstable fixed point splits again for larger energies into a stable
and two unstable ones. For energies close toH = 0 the stable center splits one more time into an unstable and two stable ones. With increasing energy more and more of
the phase space is filled with chaotic orbits with very long intermediate time intervals in between two crossings of the surface
of section. We also checked the rotation numbers for some specific orbits.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
2.
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
3.
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
4.
The Sitnikov configuration is a special case of the restricted three-body problem where the two primaries are of equal masses
and the third body of a negligible mass moves along a straight line perpendicular to the orbital plane of the primaries and
passes through their center of mass. It may serve as a toy model in dynamical astronomy, and can be used to study the three-dimensional
orbits in more applicable cases of the classical three-body problem. The present paper concerns the straight-line oscillations
of the Sitnikov family of the photogravitational circular restricted three-body problem as well as the associated families
of three-dimensional periodic orbits. From the stability analysis of the Sitnikov family and by using appropriate correctors
we have computed accurately 49 critical orbits at which families of 3D periodic orbits of the same period bifurcate. All these
families have been computed in both cases of equal and non-equal primaries, and consist entirely of unstable orbits. They
all terminate with coplanar periodic orbits. We have also found 35 critical orbits at which period doubling bifurcations occur.
Several families of 3D periodic orbits bifurcating at these critical Sitnikov orbits have also been given. These families
contain stable parts and close upon themselves containing no coplanar orbits. 相似文献
5.
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
6.
We present results about the stability of vertical motion and its bifurcations into families of 3-dimensional (3D) periodic orbits in the Sitnikov restricted N-body problem. In particular, we consider ν = N ? 1 equal mass primary bodies which rotate on a circle, while the Nth body (of negligible mass) moves perpendicularly to the plane of the primaries. Thus, we extend previous work on the 4-body Sitnikov problem to the N-body case, with N = 5, 9, 15, 25 and beyond. We find, for all cases we have considered with N ≥ 4, that the Sitnikov family has only one stability interval (on the z-axis), unlike the N = 3 case where there is an infinity of such intervals. We also show that for N = 5, 9, 15, 25 there are, respectively, 14, 16, 18, 20 critical Sitnikov periodic orbits from which 3D families (no longer rectilinear) bifurcate. We have also studied the physically interesting question of the extent of bounded dynamics away from the z-axis, taking initial conditions on x, y planes, at constant z(0) = z 0 values, where z 0 lies within the interval of stable rectilinear motions. We performed a similar study of the dynamics near some members of 3D families of periodic solutions and found, on suitably chosen Poincaré surfaces of section, “islands” of ordered motion, while away from them most orbits become chaotic and eventually escape to infinity. Finally, we solve the equations of motion of a small mass in the presence of a uniform rotating ring. Studying the stability of the vertical orbits in that case, we again discover a single stability interval, which, as N grows, tends to coincide with the stability interval of the N-body problem, when the values of the density and radius of the ring equal those of the corresponding system of N ? 1 primary masses. 相似文献
7.
We study numerically the photogravitational version of the problem of four bodies, where an infinitesimal particle is moving under the Newtonian gravitational attraction of three bodies which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle. The fourth body does not affect the motion of the three bodies (primaries). We consider that the primary body m 1 is dominant and is a source of radiation while the other two small primaries m 2 and m 3 are equal. In this case (photogravitational) we examine the linear stability of the Lagrange triangle solution. The allowed regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions of the equilibrium points on the orbital plane are given. The existence and the number of the collinear and the non-collinear equilibrium points of the problem depends on the mass parameters of the primaries and the radiation factor q 1. Critical masses m 3 and radiation q 1 associated with the existence and the number of the equilibrium points are given. The stability of the relative equilibrium solutions in all cases are also studied. In the last section we investigate the existence and location of the out of orbital plane equilibrium points of the problem. We found that such critical points exist. These points lie in the (x,z) plane in symmetrical positions with respect to (x,y) plane. The stability of these points are also examined. 相似文献
8.
《New Astronomy》2020
The present paper deals with the periodic orbits generated by Lagrangian solutions of the restricted three-body problem when both the primaries are oblate bodies. We have illustrated the periodic orbits for different values of μ, h, σ1 and σ2 (h is energy constant, μ is mass ratio of the two primaries, σ1 and σ2 are oblateness factors). These orbits have been determined by giving displacements along the tangent and normal to the mobile coordinates as defined by (Karimov and Sokolsky, 1989). We have applied the predictor-corrector algorithm to construct the periodic orbits in an attempt to unveil the effect of oblateness of the primaries by taking the fixed values of parameters μ, h, σ1 and σ2. 相似文献
9.
The linear stability of the triangular equilibrium points in the photogravitational elliptic restricted three-body problem is examined and the stability regions are determined in the space of the parameters of mass, eccentricity, and radiation pressure, in the case of equal radiation factors of the two primaries. The full range of values of the common radiation factor is explored, from the gravitational caseq
1 =q
2 =q = 1 down to the critical value ofq = 1/8 at which the triangular equilibria disappear by coalescing on the rotating axis of the primaries. It is found that radiation pressure exerts a significant influence on the stability regions. For certain intervals of radiation values these regions become qualitatively different from the gravitational case as well as the solar system case considered in Paper I. There exist values of the common radiation factor, in the range considered, for which the triangular equilibrium points are stable for the entire range of mass distribution among the primaries and for large eccentricities of their orbits. 相似文献
10.
This paper investigates the combined effect of small perturbations ε,ε′ in the Coriolis and centrifugal forces, radiation pressure q i , and changing oblateness of the primaries A i (t) (i=1,2) on the stability of equilibrium points in the restricted three body problem in which the primaries is a supergiant eclipsing binary system which consists of a pair of bright oblate stars having the appearance of a giant peanut in space and their masses assumed to vary with time in the absence of reactive forces. The equations of motion are derived and the equilibrium points are obtained. For the autonomized system, it is seen that there are more than a pair of the triangular points as κ→∞; κ being the arbitrary sum of the masses of the primaries. In the case of the collinear points, two additional equilibrium points exist on the line joining the primaries when simultaneously κ+ε′<0 and both primaries are oblate, i.e., 0<α i ?1. So there are five collinear equilibrium points in this case. Two non-planar equilibrium points exist for κ>1. Hence, there are at least nine equilibrium points of the system. The stability of these points is explored analytically and numerically. It is seen that the collinear and triangular points are stable with respect to certain conditions controlled by κ while the non-planar equilibrium points are unstable. 相似文献
11.
We consider the photogravitational restricted three-body problem with oblateness and study the Sitnikov motions. The family of straight line oscillations exists only in the case where the primaries are of equal masses as in the classical Sitnikov problem and have the same oblateness coefficients and radiation factors. A perturbation method based on Floquet theory is applied in order to study the stability of the motion and critical orbits are determined numerically at which families of three-dimensional periodic orbits of the same or double period bifurcate. Many of these families are computed. 相似文献
12.
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
13.
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
14.
We study the regions of finite motions in the vicinity of three simple stable periodic orbits in the general problem of three equal-mass bodies with a zero angular momentum. Their distinctive feature is that one of the moving bodies periodically passes through the center of mass of the triple system. We consider the dynamical evolution of plane nonrotating triple systems for which the initial conditions are specified in such a way that one of the bodies is located at the center of mass of the triple system. The initial conditions can then be specified by three parameters: the virial coefficient k and the two angles, φ1 and φ2, that characterize the orientation of the velocity vectors for the bodies. We scanned the region of variation in these parameters k∈(0, 1); φ1, φ2∈(0, π) at steps of δk=0.01; δφ1=δφ2=1° and identified the regions of finite motions surrounding the periodic orbits. These regions are isolated from one another in the space of parameters (k, φ1, φ2). There are bridges that correspond to unstable orbits with long lifetimes between the regions. During the evolution of these metastable systems, the phase trajectory can “stick” to the vicinity of one of the periodic orbits or move from one vicinity to another. The evolution of metastable systems ends with their breakup. 相似文献
15.
R. Dvorak 《Celestial Mechanics and Dynamical Astronomy》1993,56(1-2):71-80
We present numerical results of the so-called Sitnikov-problem, a special case of the three-dimensional elliptic restricted three-body problem. Here the two primaries have equal masses and the third body moves perpendicular to the plane of the primaries' orbit through their barycenter. The circular problem is integrable through elliptic integrals; the elliptic case offers a surprisingly great variety of motions which are until now not very well known. Very interesting work was done by J. Moser in connection with the original Sitnikov-paper itself, but the results are only valid for special types of orbits. As the perturbation approach needs to have small parameters in the system we took in our experiments as initial conditions for the work moderate eccentricities for the primaries' orbit (0.33e
primaries
0.66) and also a range of initial conditions for the distance of the 3
rd
body (= the planet) from very close to the primaries orbital plane of motion up to distance 2 times the semi-major axes of their orbit. To visualize the complexity of motions we present some special orbits and show also the development of Poincaré surfaces of section with the eccentricity as a parameter. Finally a table shows the structure of phase space for these moderately chosen eccentricities. 相似文献
16.
The linear stability of the inner collinear equilibrium point of the photogravitational elliptic restricted three-body problem is examined and the stability regions are determined in the space of the parameters of mass, eccentricity and radiation pressure. The case of equal radiation factors of the two primaries is considered and the full range of values of the common radiation factor is explored, from the caseq
1 =q
2 =q = 1/8 at which the triangular equilibria disappear by coalescing on the rotating axis of the primaries transferring their stability to the collinear point, down toq = 0 at which value the stability regions in theµ - e plane disappear by shrinking down to zero size. It is found that radiation pressure exerts a significant influence on the stability regions. For certain intervals of radiation values these regions become qualitatively different from the gravitational case as well as the solar system case. They evolve as in the case of the triangular equilibrium point considered in a previous paper. There exist values of the common radiation factor, in the range considered, for which the collinear equilibrium point is stable for the entire range of mass distribution among the primaries and for large eccentricities of their orbits. 相似文献
17.
N. V. Ardeljan G. S. Bisnovatyi-Kogan S. G. Moiseenko 《Astrophysics and Space Science》1996,239(1):1-13
All the families of planar symmetric simple-periodic orbits of the photogravitational restricted plane circular three-body problem, are determined numerically in the case when the primaries are of equal mass and radiate with equal radiation factors (q
1=q2=q). We obtain a global view of the possible patterns of periodic three-body motion while the full range of values of the common radiation factor is explored, from the gravitational case (q=1) down to near the critical value at which the triangular equilibria disappear by coalescing with the inner equilibrium pointL
1 on the rotating axis of the primaries. It is found that for large deviations of its value from the gravitational case the radiation factorq can have a strong effect on the structure of the families. 相似文献
18.
In this paper, we study the existence of libration points and their linear stability when the three participating bodies are axisymmetric and the primaries are radiating, we found that the collinear points remain unstable, it is further seen that the triangular points are stable for 0<μ<μ c , and unstable for where , it is also observed that for these points the range of stability will decrease. In addition to this we have studied periodic orbits around these points in the range 0<μ<μ c , we found that these orbits are elliptical; the frequencies of long and short orbits of the periodic motion are affected by the terms which involve parameters that characterize the oblateness and radiation repulsive forces. The implication is that the period of long periodic orbits adjusts with the change in its frequency while the period of short periodic orbit will decrease. 相似文献
19.
We present special generating plane orbits, the vertical-critical orbits, of the coplanar general three-body problem. These
are determined numerically for various values of m3, for the entire range of the mass ratio of the two primaries. The vertical-critical orbits are necessary in order to specify
the vertically stable segments of the families of plane periodic orbits, and they are also the starting points of the families
of the simplest possible three-dimensional periodic orbits, namely the simple and double periodic. The initial conditions
of the vertical-critical periodic orbits of the basic families l, m, i, h, b and c and their stability parameters are determined.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
20.
Lidia Jiménez-Lara Adolfo Escalona-Buendía 《Celestial Mechanics and Dynamical Astronomy》2001,79(2):97-117
We present some qualitative and numerical results of the Sitnikov problem, a special case of the three-body problem, which offers a great variety of motions as the non-integrable systems typically do. We study the symmetries of the problem and we use them as well as the stroboscopic Poincarée map (at the pericenter of the primaries) to calculate the symmetry lines and their dynamics when the parameter changes, obtaining information about the families of periodic orbits and their bifurcations in four revolutions of the primaries. We introduce the semimap to obtain the fundamental lines l
1. The origin produces new families of periodic orbits, and we show the bifurcation diagrams in a wide interval of the eccentricity (0 0.97). A pattern of bifurcations was found.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献