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1.
The Sitnikov problem is one of the most simple cases of the elliptic restricted three body system. A massless body oscillates
along a line (z) perpendicular to a plane (x,y) in which two equally massive bodies, called primary masses, perform Keplerian orbits around their common barycentre with
a given eccentricity e. The crossing point of the line of motion of the third mass with the plane is equal to the centre of gravity of the entire
system. In spite of its simple geometrical structure, the system is nonlinear and explicitly time dependent. It is globally
non integrable and therefore represents an interesting application for advanced perturbative methods. In the present work
a high order perturbation approach to the problem was performed, by using symbolic algorithms written in Mathematica. Floquet
theory was used to derive solutions of the linearized equation up to 17th order in e. In this way precise analytical expressions for the stability of the system were obtained. Then, applying the Courant and
Snyder transformation to the nonlinear equation, algebraic solutions of seventh order in z and e were derived using the method of Poincaré–Lindstedt. The enormous amount of necessary computations were performed by extensive
use of symbolic programming. We developed automated and highly modularized algorithms in order to master the problem of ordering
an increasing number of algebraic terms originating from high order perturbation theory. 相似文献
2.
We study the stability of motion in the 3-body Sitnikov problem, with the two equal mass primaries (m
1 = m
2 = 0.5) rotating in the x, y plane and vary the mass of the third particle, 0 ≤ m
3 < 10−3, placed initially on the z-axis. We begin by finding for the restricted problem (with m
3 = 0) an apparently infinite sequence of stability intervals on the z-axis, whose width grows and tends to a fixed non-zero value, as we move away from z = 0. We then estimate the extent of “islands” of bounded motion in x, y, z space about these intervals and show that it also increases as |z| grows. Turning to the so-called extended Sitnikov problem, where the third particle moves only along the z-axis, we find that, as m
3 increases, the domain of allowed motion grows significantly and chaotic regions in phase space appear through a series of
saddle-node bifurcations. Finally, we concentrate on the general 3-body problem and demonstrate that, for very small masses, m
3 ≈ 10−6, the “islands” of bounded motion about the z-axis stability intervals are larger than the ones for m
3 = 0. Furthermore, as m
3 increases, it is the regions of bounded motion closest to z = 0 that disappear first, while the ones further away “disperse” at larger m
3 values, thus providing further evidence of an increasing stability of the motion away from the plane of the two primaries,
as observed in the m
3 = 0 case. 相似文献
3.
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
4.
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
5.
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
6.
This article deals with the region of motion in the Sitnikov four-body problem where three bodies (called primaries) of equal masses fixed at the vertices of an equilateral triangle. Fourth mass which is finite confined to moves only along a line perpendicular to the instantaneous plane of the motions of the primaries. Contrary to the Sitnikov problem with one massless body the primaries are moving in non-Keplerian orbits about their centre of mass. It is investigated that for very small range of energy h the motion is possible only in small region of phase space. Condition of bounded motions has been derived. We have explored the structure of phase space with the help of properly chosen surfaces of section. Poincarè surfaces of section for the energy range ?0.480≤h≤?0.345 have been computed. We have chosen the plane (q 1,p 1) as surface of section, with q 1 is the distance of a primary from the centre of mass. We plot the respective points when the fourth body crosses the plane q 2=0. For low energy the central fixed point is stable but for higher value of energy splits in to an unstable and two stable fixed points. The central unstable fixed point once again splits for higher energy into a stable and three unstable fixed points. It is found that at h=?0.345 the whole phase space is filled with chaotic orbits. 相似文献
7.
In the problem of 2+2 bodies in the Robe’s setup, one of the primaries of mass m*1m^{*}_{1} is a rigid spherical shell filled with a homogeneous incompressible fluid of density ρ
1. The second primary is a mass point m
2 outside the shell. The third and the fourth bodies (of mass m
3 and m
4 respectively) are small solid spheres of density ρ
3 and ρ
4 respectively inside the shell, with the assumption that the mass and the radius of third and fourth body are infinitesimal.
We assume m
2 is describing a circle around m*1m^{*}_{1}. The masses m
3 and m
4 mutually attract each other, do not influence the motion of m*1m^{*}_{1} and m
2 but are influenced by them. We also assume masses m
3 and m
4 are moving in the plane of motion of mass m
2. In the paper, the equations of motion, equilibrium solutions, linear stability of m
3 and m
4 are analyzed. There are four collinear equilibrium solutions for the given system. The collinear equilibrium solutions are
unstable for all values of the mass parameters μ,μ
3,μ
4. There exist an infinite number of non collinear equilibrium solutions each for m
3 and m
4, lying on circles of radii λ,λ′ respectively (if the densities of m
3 and m
4 are different) and the centre at the second primary. These solutions are also unstable for all values of the parameters μ,μ
3,μ
4, φ, φ′. Such a model may be useful to study the motion of submarines due to the attraction of earth and moon. 相似文献
8.
In this problem of the restricted (2 + 2) bodies we have considered two magnetic dipoles of masses M
1 and M
2(M
1 > M
2) moving in circular Keplarian orbit about their centre of mass. Two minor bodies of masses m
1, m
2(m
j< M
2) are taken as electric dipoles in the field of rotating magnetic dipoles. These minor bodies interact with each other but do not perturb the primaries.We have found equations of motions which differ from that of Goudas and Petsagouraki's (1985). 相似文献
9.
Marcelo Marchesin Deborah da Paixão Vasconcellos 《Astrophysics and Space Science》2014,352(2):443-460
We study a highly symmetric nine-body problem in which eight positive masses, called the primaries, move four by four, in two concentric circular motions such that their configuration is always a square for each group of four masses. The ninth body being of negligible mass and not influencing the motion of the eight primaries. We assume all the nine masses are in the same plane and that the masses of the primaries are \(m_{1}=m_{2}=m_{3}=m_{4}=\tilde{m}\) and m 5=m 6=m 7=m 8=m and the radii associated to the circular motion of the bodies with mass \(\tilde{m}\) is λ∈[λ 0,1] and for the bodies with mass m is 1. We prove the existence of central configurations which characterize such arrangement of the primaries and we study the influence of the parameter λ, the ratio of the radii of the two circles, on the masses m and \(\tilde{m}\) . We use a synodical system of coordinates to eliminate the time dependence on the equations of motion. We show the existence of equilibria solutions symmetrically distributed on the four quadrants and their dependence on the parameter λ. Finally, we show that there can be 13, 17 or 25 equilibria solutions depending on the size of λ and we investigate their linear stability. 相似文献
10.
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
11.
J. Hagel 《Celestial Mechanics and Dynamical Astronomy》1992,53(3):267-292
A new analytic approach to the solution of the Sitnikov Problem is introduced. It is valid for bounded small amplitude solutions (z
max = 0.20) (in dimensionless variables) and eccentricities of the primary bodies in the interval (–0.4 < e < 0.4). First solutions are searched for the limiting case of very small amplitudes for which it is possible to linearize the problem. The solution for this linear equation with a time dependent periodic coefficient is written up to the third order in the primaries eccentricity. After that the lowest order nonlinear amplitude contribution (being of order z
3) is dealt with as perturbation to the linear solution. We first introduce a transformation which reduces the linear part to a harmonic oscillator type equation. Then two near integrals for the nonlinear problem are derived in action angle notation and an analytic expression for the solution z(t) is derived from them. The so found analytic solution is compared to results obtained from numeric integration of the exact equation of motion and is found to be in very good agreement.
CERN SL/AP 相似文献
12.
S. M. El-Shaboury 《Earth, Moon, and Planets》1990,48(3):243-250
In this paper the circular planar restricted problem of three axisymmetric ellipsoids S
i
(i = 1, 2, 3), such that their equatorial planes coincide with the orbital plane of the three centres of masses, be considered. The equations of motion of infinitesimal body S
3
be obtained in the polar coordinates. Using iteration approach we have given an approximation for another integral, which independent of the Jacobian integral, in the case of P-type orbits (near circular orbits surrounding both primaries). 相似文献
13.
The equations of motion of the 2+2 body problem (two interacting particles in the gravitational field of two much more massive primaries m1 and m2 in circular keplerian orbit) have an integral analogous to the Jacobi integral of the circular 2+1 body problem. We show here that with 2+2 bodies this integral does not give rise to Hill stability, i.e. to confinement for all time in a portion of the configuration space not allowing for some close approaches to occur. This is because all the level manifolds are connected and all exchanges of bodies between the regions surroundingm
1,m
2 and infinity do not contradict the conservation of the integral. However, it is worth stressing that some of these exchanges are physically meaningless, because they involve either unlimited extraction of potential energy from the binary formed by the small bodies (without taking into account their physical size) or significant mutual perturbations between the small masses without close approach, a process requiring, for the Sun-Jupiter-two asteroids system, timescales longer than the age of the Solar System. 相似文献
14.
Paolo Lanzano 《Astrophysics and Space Science》1968,1(1):92-111
The motion of two rotating spheroidal bodies, constituting the components of a binary system in a weak gravitational field, has been considered up to terms of the second order in the small parameterV/c, whereV denotes the velocity of the bodies andc is the velocity of light.The following simplifying assumptions, consistent with a problem of astronomical interest, have been made: (1) the dimensions of the bodies are small compared with their mutual distance; (2) the bodies consist of matter in the fluid state with internal hydrostatic pressure and their oblateness is due to their own rotation; (3) there exist axial symmetry about the axis of rotation and symmetry with respect to the equatorial plane, the same symmetry properties apply to mass densities and stress tensors.The Fock-Papapetrou method was used to ascertain those terms in the equations of motion which are due to the rotation and to the oblateness of each component. Approximate solutions to the Poisson and wave equations were obtained to express the potential and retarded potential at large distances from the bodies generating them. The explicit evaluation of certain integrals has necessitated the use of the Laplace-Clairaut theory for the equibrium configuration of rotating bodies. The final expressions require the knowledge of the mass density as a function of the mean radius of the equipotential surfaces.As an interpretation of the results, the Lagrangian perturbation equations were employed to evaluate the secular motion of the nodal line for the relative orbit of the two components. The results constitute a generalization of Fock's work and furnish the contribution of the mass distribution to the rotation effect of general relativity. 相似文献
15.
K. B. Bhatnagar Usha Gupta Rashmi Bhardwaj 《Celestial Mechanics and Dynamical Astronomy》1994,59(4):345-374
The non-linear stability of the libration pointL
4 in the restricted problem has been studied when there are perturbations in the potentials between the bodies. It is seen that the pointL
4 is stable for all mass ratios in the range of linear stability except for three mass ratios depending upon the perturbing functions. The theory is applied to the following four cases:
相似文献
| (i) | There are no perturbations in the potentials (classical problem). |
| (ii) | Only the bigger primary is an oblate spheroid whose axis of symmetry is perpendicular to the plane of relative motion (circular) of the primaries. |
| (iii) | Both the primaries are oblate spheroids whose axes of symmetry are perpendicular to the plane of relative motion (circular) of the primaries. |
| (iv) | The primaries are spherical in shape and the bigger is a source of radiation. |
16.
We have two mass points of equal masses m
1=m
2 > 0 moving under Newton’s law of attraction in a non-collision parabolic orbit while their center of mass is at rest. We
consider a third mass point, of mass m
3=0, moving on the straight line L perpendicular to the plane of motion of the first two mass points and passing through their
center of mass. Since m
3=0, the motion of m
1 and m
2 is not affected by the third and from the symmetry of the motion it is clear that m
3 will remain on the line L. The parabolic restricted three-body problem describes the motion of m
3. Our main result is the characterization of the global flow of this problem. 相似文献
17.
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
18.
The restricted (equilateral) four-body problem consists of three bodies of masses m 1, m 2 and m 3 (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh’s critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them. 相似文献
19.
József Vanyó Bruno Escribano Julyan H. E. Cartwright Diego L. González Oreste Piro Tamás Tél 《Celestial Mechanics and Dynamical Astronomy》2011,109(2):181-200
We study tidal synchronization and orbit circularization in a minimal model that takes into account only the essential ingredients
of tidal deformation and dissipation in the secondary body. In previous work we introduced the model (Escribano et al. in
Phys. Rev. E, 78:036216, 2008); here we investigate in depth the complex dynamics that can arise from this simplest model of tidal synchronization and
orbit circularization. We model an extended secondary body of mass m by two point masses of mass m/2 connected with a damped spring. This composite body moves in the gravitational field of a primary of mass M ≫ m located at the origin. In this simplest case oscillation and rotation of the secondary are assumed to take place in the plane
of the Keplerian orbit. The gravitational interactions of both point masses with the primary are taken into account, but that
between the point masses is neglected. We perform a Taylor expansion on the exact equations of motion to isolate and identify
the different effects of tidal interactions. We compare both sets of equations and study the applicability of the approximations,
in the presence of chaos. We introduce the resonance function as a resource to identify resonant states. The approximate equations
of motion can account for both synchronization into the 1:1 spin-orbit resonance and the circularization of the orbit as the
only true asymptotic attractors, together with the existence of relatively long-lived metastable orbits with the secondary
in p:q (p and q being co-prime integers) synchronous rotation. 相似文献
20.
O. O. Vasilkova 《Astronomy Letters》2010,36(3):227-230
Dynamical behaviour of a small binary with equal components, each of mass m, is considered under attraction of a heavy body of mass M. Differential equations of the general three-body problem are integrated numerically using the code by S. J. Aarseth (Aarseth,
Zare 1974) for mass ratios m/M within 10−11–10−4 range. The direct and retrograde orbits of light bodies about each other are considered which lie either in the plane of
moving their center of mass or in the plane perpendicular to it. It is shown numerically that the critical separation between
the binary components which leads to disruption of binary is proportional to (m/M)1/3. The criterion can be used for studying (in the first approximation) the motion of double stars and binary asteroids or computing
the parameters of magnetic monopol and antimonopol pairs. 相似文献