共查询到20条相似文献，搜索用时 22 毫秒
1.
The spatial restricted rhomboidal five-body problem, or shortly, SRRFBP, is a five body problem in which four positive masses, called the primaries, move two by two in coplanar circular motions with the center of mass fixed at the origin such that their configuration is always a rhombus, the fifth mass being negligible and not influencing the motion of the four primaries. The Hamiltonian function that governs the motion of the fifth mass is derived and has three degrees of freedom depending periodically on time. Using a synodical system of coordinates, we fix the primaries in order to eliminate the time dependence. With the help of the Hamiltonian structure, we characterize the regions of possible motion. The vertical $z$ axis is invariant and we study what we call the rhomboidal Sitnikov problem. Unlike the classical Sitnikov problem, no chaos exists and the behavior of the fifth mass is quite predictable, periodic solutions of arbitrary long periods are shown to exist and we study numerically their linear horizontal stability. 相似文献
2.
V. A. Shefer 《Solar System Research》2005,39(3):239-246
The motion of minor Solar System bodies having close encounters with major planets is described using the model of motion within the framework of the perturbed restricted three-body problem. The actual motion of a minor body is represented as a combination of two motions, namely, the motion of a fictitious attracting center with a variable mass and the motion with respect to the fictitious center. The position and mass of the fictitious center are chosen so that, when the minor body collides with any of the primaries, the fictitious center carries into the center of inertia of the colliding body and the mass of the fictitious center becomes identical to the mass of this body. The regularizing KS-transformation and Sundman’s time transformation were applied to coordinates and velocities. As a result, a system of differential equations of motion that are quasilinear within the nearest vicinity of each of the primary attracting bodies was obtained. These equations are characterized by a numerical behavior during the encounters of the minor body with the primaries that is essentially better than that of the initial equations of motion. The motion of comets Brooks 2 and Gehrels 3, which have fairly close encounters with Jupiter, is simulated.__________Translated from Astronomicheskii Vestnik, Vol. 39, No. 3, 2005, pp. 272–280.Original Russian Text Copyright © 2005 by Shefer. 相似文献
3.
The restricted three-body problem is reconsidered by replacing the point-like primaries of the classical problem by a pair of axisymmetric rigid bodies which have a plane of symmetry perpendicular to their axes, and the infinitesimal mass by a gyrostat. The conditions for the circular motion of the primaries around their center of mass are stated and they yield the classification of all possible orientations of these bodies into four groups according to the value of their angular velocity. Then the equations of motion of the gyrostat are derived and solved for the equilibrium configurations of the system. 相似文献
4.
《New Astronomy》2020
We consider the square configuration of photo-gravitational elliptic restricted five-body problem and study the Sitnikov motions. The four radiating primaries are of equal mass placed at the vertices of square and the fifth body having negligible mass performs oscillations along a straight line perpendicular to the orbital plane of the primaries. The motion of the fifth body is called vertical periodic motion and the main aim of this paper is to study the effect of radiation pressure on these periodic motions in the linear approximation. Moreover, the effects of radiation pressure on the motion of fifth body have been examined with the help of Poincare surfaces of section. By escalating the radiation pressure, surrounding periodic tubes and islands disappear and chaotic motion occurs near the hyperbolic points. Further, by escalating the radiation pressure, the main stochastic region joins the escaping one. 相似文献
5.
Peter J. Shelus 《Celestial Mechanics and Dynamical Astronomy》1972,5(4):483-489
Within the context of the restricted problem of three bodies, we wish to show the effects, caused by varying the mass ratio of the primaries and the eccentricity of their orbits, upon periodic orbits of the infinitesimal mass that are numerical continuations of circular orbits in the ordinary problem of two bodies. A recursive-power-series technique is used to integrate numerically the equations of motion as well as the first variational equations to generate a two-parameter family of periodic orbits and to identify the linear stability characteristics thereof. Seven such families (comprised of a total of more than 2000 orbits) with equally spaced mass ratios from 0.0 to 1.0 and eccentricities of the orbits of the primaries in a range 0.0 to 0.6 are investigated. Stable orbits are associated with large distances of the infinitesimal mass from the perturbing primary, with nearly circular motion of the primaries, and, to a slightly lesser extent, with small mass ratios of the primaries.Conversely, unstable orbits for the infinitesimal mass are associated with small distances from the perturbing primary, with highly elliptic orbits of the primaries, and with large mass ratios. 相似文献
6.
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
7.
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
8.
The motion around the libration points is studied with Charlier's first order theory assuming the isotropic mass variation of the primaries does not exceed the order of the small primary and the derivatives of the masses with respect to time are negligible second order quantities. The results are analogous to those obtained with constant primary masses. 相似文献
9.
《New Astronomy》2021
This paper examines the effects of triaxiality of both the primaries on the position and stability of the oblate infinitesimal mass in the neighborhood of triangular equilibrium points in the framework of Elliptical restricted three body problem. We have found the solutions for the locations of triangular equilibrium points. We have investigated the stability of infinitesimal mass around the triangular equilibrium points.It is observed that the infinitesimal motion around triangular equilibrium points are stable under certain condition with respect to triaxiality of primaries. We have applied the method of averaging used by Grebenivok, throughout the analysis of the stability of the infinitesimal mass around the triangular equilibrium points. We have exploited simulation technique using MATLAB 15 to analyze the stability of the system. The critical mass ratio depends on the triaxiality, oblateness, semi- major axis and eccentricity of the elliptical orbits. 相似文献
10.
T.J. Kalvouridis 《Astrophysics and Space Science》1998,260(3):309-325
Our intention in this article is to present a new model for the investigation of the motion of a particle of negligible mass
in a multibody surrounding. The proposed general planar configuration consists of ν = n-1 primaries arranged in equal arcs
on an ideal ring and a central body of different mass located at the centre of mass of the system. We formulate the general
equations of motion and we study the stationary solutions and the zero-velocity contours for various values of ν.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
11.
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
12.
This paper studies the motion and orbital stability of the infinitesimal mass in the vicinity of the equilateral (triangular) Lagrangian points of the elliptic restricted three body problem, considering photo gravitational effects of both the primaries. The stability of the triangular points is studied under the effects of radiating primaries around the binary system (Achird, Luyten, αCen AB, Kruger-60, Xi-Bootis); using simulation technique by drawing different curves of zero velocity. 相似文献
13.
An analytical approach to small amplitude solutions of the extended nearly circular Sitnikov problem
J. Hagel 《Celestial Mechanics and Dynamical Astronomy》2009,103(3):251-266
The model of extended Sitnikov Problem contains two equally heavy bodies of mass m moving on two symmetrical orbits w.r.t the centre of gravity. A third body of equal mass m moves along a line z perpendicular to the primaries plane, intersecting it at the centre of gravity. For sufficiently small distance from the
primaries plane the third body describes an oscillatory motion around it. The motion of the three bodies is described by a
coupled system of second order differential equations for the radial distance of the primaries r and the third mass oscillation z. This problem which is dealt with for zero initial eccentricity of the primaries motion, is generally non integrable and
therefore represents an interesting dynamical system for advanced perturbative methods. In the present paper we use an original
method of rewriting the coupled system of equations as a function iteration in such a way as to decouple the two equations
at any iteration step. The decoupled equations are then solved by classical perturbation methods. A prove of local convergence
of the function iteration method is given and the iterations are carried out to order 1 in r and to order 2 in z. For small values of the initial oscillation amplitude of the third mass we obtain results in very good agreement to numerically
obtained solutions. 相似文献
14.
G P. Horedt 《Celestial Mechanics and Dynamical Astronomy》1974,10(3):319-326
The delimitations of the librational motion around the Lagrangian triangular pointsL 4,L 5 are investigated within the framework of the restricted circular three body problem according to Brown's and Thüring's theory. The isotropic mass variation of the primaries does not exceed the order of the small primary and the derivatives of the masses with respect to the time are negligible second order quantities. The amplitude of the maximum elongations with respect to the small primary remains unchanged. The expression for the maximum variation of the distance of the particle from the large mass has the same form as in the classical problem with constant masses. 相似文献
15.
We deal with the problem of a zero mass body oscillating perpendicular to a plane in which two heavy bodies of equal mass orbit each other on Keplerian ellipses. The zero mass body intersects the primaries plane at the systems barycenter. This problem is commonly known as theSitnikov Problem. In this work we are looking for a first integral related to the oscillatory motion of the zero mass body. This is done by first expressing the equation of motion by a second order polynomial differential equation using a Chebyshev approximation techniques. Next we search for an autonomous mapping of the canonical variables over one period of the primaries. For that we discretize the time dependent coefficient functions in a certain number of Dirac Delta Functions and we concatenate the elementary mappings related to the single Delta Function Pulses. Finally for the so obtained polynomial mapping we look for an integral also in polynomial form. The invariant curves in the two dimensional phase space of the canonical variables are investigated as function of the primaries eccentricity and their initial phase. In addition we present a detailed analysis of the linearized Sitnikov Problem which is valid for infinitesimally small oscillation amplitudes of the zero mass body. All computations are performed automatically by the FORTRAN program SALOME which has been designed for stability considerations in high energy particle accelerators. 相似文献
16.
This paper studies the stability of Triangular Lagrangian points in the model of elliptical restricted three body problem, under the assumption that both the primaries are radiating. The model proposed is applicable to the well known binary systems Achird, Luyten, αCen AB, Kruger-60, Xi-Bootis. Conditional stability of the motion around the triangular points exists for 0≤μ≤μ ?, where μ is the mass ratio. The method of averaging due to Grebenikov has been exploited throughout the analysis of stability of the system. The critical mass ratio depends on the combined effects of radiation of both the primaries and eccentricity of this orbit. It is found by adopting the simulation technique that the range of stability decreases as the radiation pressure parameter increases. 相似文献
17.
In this paper we consider a restricted equilateral four-body problem where a particle of negligible mass is moving under the Newtonian gravitational attraction of three masses (called primaries) which move on circular orbits around their center of masses such that their configuration is always an equilateral triangle (Lagrangian configuration). We consider the case of two bodies of equal masses, which in adimensional units is the parameter of the problem. We study numerically the existence of families of unstable periodic orbits, whose invariant stable and unstable manifolds are responsible for the existence of homoclinic and heteroclinic connections, as well as of transit orbits traveling from and to different regions. We explore, for three different values of the mass parameter, what kind of transits and energy levels exist for which there are orbits with prescribed itineraries visiting the neighborhood of different primaries. 相似文献
18.
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
19.
Vladislav V. Sidorenko 《Celestial Mechanics and Dynamical Astronomy》2011,109(4):367-384
This paper is devoted to the special case of the restricted circular three-body problem, when the two primaries are of equal
mass, while the third body of negligible mass performs oscillations along a straight line perpendicular to the plane of the
primaries (so called periodic vertical motions). The main goal of the paper is to study the stability of these periodic motions
in the linear approximation. A special attention is given to the alternation of stability and instability within the family
of periodic vertical motions, whenever their amplitude is varied in a continuous monotone manner. 相似文献
20.
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