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1.
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
2.
John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1977,16(1):61-76
It is proved that monoparametric families of periodic orbits of theN-body problem in the plane, for fixed values of all masses, exist in a rotating frame of reference whosex axis contains always two of the bodiesP
1 andP
2. A periodic motion of theN-body problem is obtained as a continuation ofN–2 symmetric periodic orbits of the circular restricted three-body problem whose periods are in integer dependence, by increasing the masses of the originallyN–2 massless bodiesP
3, ...,P
k. The analytic continuation, for sufficiently small values of theN–2 bodiesP
3
...P
k
and finite values for the masses ofP
1 andP
2 has been proved by the continuation method and the solution itself has been found explicitly to a linear approximation in the small masses. Also, the possible application of the above periodic orbits to the study of the Solar system and of stellar systems is mentioned. 相似文献
3.
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
4.
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
5.
D. Viswanath 《Celestial Mechanics and Dynamical Astronomy》2006,94(2):213-235
The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses
1− μ and μ that circle each other with period equal to 2π. For small μ, a resonant periodic motion of the massless particle
in the rotating frame can be described by relatively prime integers p and q, if its period around the heavier primary is approximately 2π p/q, and by its approximate eccentricity e. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions.
We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study
of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem
are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations
to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with q/p equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances.
We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3,
and 1/2 resonances for any μ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist
for μ small enough in the unaveraged restricted three-body problem. 相似文献
6.
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
7.
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
8.
This paper investigates the stability of equilibrium points in the restricted three-body problem, in which the masses of the
luminous primaries vary isotropically in accordance with the unified Meshcherskii law, and their motion takes place within
the framework of the Gylden–Meshcherskii problem. For the autonomized system, it is found that collinear and coplanar points
are unstable, while the triangular points are conditionally stable. It is also observed that, in the triangular case, the
presence of a constant κ, of a particular integral of the Gylden–Meshcherskii problem, makes the destabilizing tendency of the radiation pressures
strong. The stability of equilibrium points varying with time is tested using the Lyapunov Characteristic Numbers (LCN). It
is seen that the range of stability or instability depends on the parameter κ. The motion around the equilibrium points L
i
(i=1,2,…,7) for the restricted three-body problem with variable masses is in general unstable. 相似文献
9.
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
10.
Alexei V. Tsygvintsev 《Celestial Mechanics and Dynamical Astronomy》2007,99(1):23-29
We consider the Newtonian planar three-body problem with positive masses m
1, m
2, m
3. We prove that it does not have an additional first integral meromorphic in the complex neighborhood of the parabolic Lagrangian
orbit besides three exceptional cases ∑m
i
m
j
/(∑m
k
)2 = 1/3, 23/33, 2/32 where the linearized equations are shown to be partially integrable. This result completes the non-integrability analysis
of the three-body problem started in papers [Tsygvintsev, A.: Journal für die reine und angewandte Mathematik N 537, 127–149
(2001a); Celest. Mech. Dyn. Astron. 86(3), 237–247 (2003)] and based on the Morales–Ramis–Ziglin approach. 相似文献
11.
The effect of small perturbation in the Coriolis and centrifugal forces on the location of libration point in the ‘Robe (1977)
restricted problem of three bodies’ has been studied. In this problem one body,m
1, is a rigid spherical shell filled with an homogeneous incompressible fluid of densityϱ
1. The second one,m
2, is a mass point outside the shell andm
3 is a small solid sphere of densityϱ
3 supposed to be moving inside the shell subject to the attraction ofm
2 and buoyancy force due to fluidϱ
1. Here we assumem
3 to be an infinitesimal mass and the orbit of the massm
2 to be circular, and we also suppose the densitiesϱ
1, andϱ
3 to be equal. Then there exists an equilibrium point (−μ + (ɛ′μ)/(1 + 2μ), 0, 0). 相似文献
12.
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
13.
The restricted three-body problem in Schwarzschild's gravitational field is analyzed. The existen- ce of the equilibrium points in the orbital plane is discussed and the corresponding positions are established. There are three collinear libration points, and, if they exist, two triangular libration points (situated in the orbital plane of the primaries). If triangular points exist, they may not form equilateral triangles; the triangles are isosceles for equal masses of the primaries, and scalene else. 相似文献
14.
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
15.
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
16.
In a recent paper, published in Astrophys. Space Sci. (337:107, 2012) (hereafter paper ZZX) and entitled “On the triangular libration points in photogravitational restricted three-body problem
with variable mass”, the authors study the location and stability of the generalized Lagrange libration points L
4 and L
5. However their study is flawed in two aspects. First they fail to write correctly the equations of motion of the variable
mass problem. Second they attribute a variable mass to the third body of the restricted three-body model, a fact that is not
compatible with the assumptions used in deriving the mathematical formulation of this model. 相似文献
17.
Martha Alvarez-Ramírez Joaquín Delgado 《Celestial Mechanics and Dynamical Astronomy》2003,87(4):371-381
We consider the symmetric planar (3 + 1)-body problem with finite masses m
1 = m
2 = 1, m
3 = µ and one small mass m
4 = . We count the number of central configurations of the restricted case = 0, where the finite masses remain in an equilateral triangle configuration, by means of the bifurcation diagram with as the parameter. The diagram shows a folding bifurcation at a value consistent with that found numerically by Meyer [9] and it is shown that for small > 0 the bifurcation diagram persists, thus leading to an exact count of central configurations and a folding bifurcation for small m
4 > 0. 相似文献
18.
Previously, we have considered the equations of motion of the three-body problem in a Lagrange form (which means a consideration of relative motions of 3-bodies in regard to each other). Analysing such a system of equations, we considered the case of small-body motion of negligible mass m 3 around the second of two giant-bodies m 1, m 2 (which are rotating around their common centre of masses on Kepler’s trajectories), the mass of which is assumed to be less than the mass of central body. In the current development, we have derived a key parameter η that determines the character of quasi-circular motion of the small third body m 3 relative to the second body m 2 (planet). Namely, by making several approximations in the equations of motion of the three-body problem, such the system could be reduced to the key governing Riccati-type ordinary differential equations. Under assumptions of R3BP (restricted three-body problem), we additionally note that Riccati-type ODEs above should have the invariant form if the key governing (dimensionless) parameter η remains in the range 10?2 Open image in new window 10?3. Such an amazing fact let us evaluate the forbidden zones for Moon’s orbits in the inner solar system or the zones of distances (between Moon and Planet) for which the motion of small body could be predicted to be unstable according to basic features of the solutions of Riccati-type. 相似文献
19.
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) μR (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
20.
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