Saari’s conjecture for the restricted three-body problem     

G. E. Roberts  L. Melanson 《Celestial Mechanics and Dynamical Astronomy》2007,97(3):211-223

Saari’s conjecture adapted to the restricted three-body problem is proven analytically using BKK theory. Specifically, we show that it is not possible for a solution of the planar, circular, restricted three-body problem to travel along a level curve of the amended potential function unless it is fixed at a critical point (one of the five libration points.) Due to the low dimension of the problem, our proof does not rely on the use of a computer.  相似文献   

Existence and linear stability of equilibrium points in the Robe's restricted three body problem when the first primary is an oblate spheroid     

《Planetary and Space Science》2007,55(4):512-516

The existence and linear stability of equilibrium points in the Robe's restricted three body problem have been studied after considering the full buoyancy force as in Plastino and Plastino and by assuming the hydrostatic equilibrium figure of the first primary as an oblate spheroid. The pertinent equations of motion are derived and existence of all equilibrium points is discussed. It is found that there is an equilibrium point near the centre of the first primary. Further there can be one more equilibrium point on the line joining the centre of the first primary and second primary and infinite number of equilibrium points lying on a circle in the orbital plane of the second primary provided the parameters occurring in the problem satisfy certain conditions. So, there can be infinite number of equilibrium points contrary to the classical restricted three body problem.  相似文献   

Secular Solutions at Triangular Equilibrium Point in the Generalized Photogravitational Restricted Three Body Problem     

B. Ishwar  A. Elipe 《Astrophysics and Space Science》2001,277(3):437-446

In this paper we have found secular solutions at the triangular equilibrium point in the generalized photogravitational restricted three body problem. The problem is generalised in the sense that smaller primary is an oblate spheroid and more massive primary as source of radiation. The triangular point has long or short-period retrograde elliptical orbits. The critical mass parameter decreases with the increase in oblateness and radiation pressure. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

Non-linear stability in the generalized restricted three-body problem     

B. Ishwar 《Celestial Mechanics and Dynamical Astronomy》1996,65(3):253-289

The non-linear stability of the triangular equilibrium point L ₄ in the generalized restricted three-body problem has been examined. The problem is generalized in the sense that the infinitesimal body and one of the primaries have been taken as oblate spheroids. It is found that the triangular equilibrium point is stable in the range of linear stability except for three mass ratios.  相似文献   

Motion around L 4 in the perturbed relativistic R3BP     

Jagadish Singh  Nakone Bello 《Astrophysics and Space Science》2014,351(2):491-497

This paper studies the motion of a test particle (infinitesimal mass) in the neighborhood of the triangular point L ₄ in the framework of the perturbed relativistic restricted three-body problem (R3BP). The problem is perturbed in the sense that a small perturbation is given to the centrifugal force. It is found that the position and stability of the triangular point are affected by both the relativistic factor and a small perturbation in the centrifugal force.  相似文献   

A criterion for the linear stability of theequilibrium points in the perturbed restricted three-body problem and its application in Robe''s problem     

Shu Si-hui  Lu Ben-kui  Chen Wu-shen  Liu Fu-yao   《Chinese Astronomy and Astrophysics》2004,28(4):432-440

A criterion for the linear stability of the equilibrium points in the perturbed restricted three-body problem is given. This criterion is related only to the coefficients of the characteristic equation of the tangent map of an equilibrium point, and this is convenient to use. With this criterion, we have discussed the linear stability of the equilibrium points in the Robe problem under the perturbation of a drag force, derived the linearly stable region of the equilibrium point in the perturbed Robe's problem with the drag given by Hallen et al., and improved as well the results obtained by Giordano et al.  相似文献   

Families of periodic orbits in the planar three-body problem     

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   

The scattering map in the planar restricted three body problem     

E. Canalias  A. Delshams  J. J. Masdemont  P. Roldán 《Celestial Mechanics and Dynamical Astronomy》2006,95(1-4):155-171

We study homoclinic transport to Lyapunov orbits around a collinear libration point in the planar restricted three body problem. A method to compute homoclinic orbits is first described. Then we introduce the scattering map for this problem (defined on a suitable normally hyperbolic invariant manifold) and we show how to compute it using the information already obtained for the homoclinic orbits. An example application to Astrodynamics is also proposed.  相似文献   

Numerical Determination of Homoclinic and Heteroclinic Orbits at Collinear Equilibria in the Restricted Three-Body Problem with Oblateness     

V. S. Kalantonis  C. N. Douskos  E. A. Perdios 《Celestial Mechanics and Dynamical Astronomy》2006,94(2):135-153

Asymptotic motion to collinear equilibrium points of the restricted three-body problem with oblateness is considered. In particular, homoclinic and heteroclinic solutions to these points are computed. These solutions depart asymptotically from an equilibrium point and arrive asymptotically at the same or another equilibrium point and are important reference solutions. To compute an asymptotic orbit, we use a fourth order local analysis, numerical integration and standard differential corrections.  相似文献   

Dynamics in the Center Manifold Around L2 in the Quasi-Bicircular Problem     

M. A. Andreu 《Celestial Mechanics and Dynamical Astronomy》2002,84(2):105-133

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Deformation of a line-element in the phase space at the triangular libration point     

V. Szebehely  R. McKenzie 《Celestial Mechanics and Dynamical Astronomy》1981,23(2):131-137

The flow in the projection of the phase space into the configuration space is presented in the neighborhood of a neutrally (or critically) stable equilibrium point in the restricted problem of three bodies. The projection is a line-element every point of which has zero initial velocity. After the elapse of various times the mapping (the rotations and elongations) of the line-element is described showing chaotic behavior.  相似文献   

The stability of the Lagrangian point L4     

D. S. Schmidt 《Celestial Mechanics and Dynamical Astronomy》1988,45(1-3):201-206

The Hamiltonian function of the restricted problem of three bodies near the triangular Lagrangian point is normalized through sixth order terms with the help of MACSYMA. The same calculations were done previously with an algebraic processor in order to establish the stability at a critical value of the mass ratio.  相似文献   

The restricted problem of three bodies with rigid dumb-bell satellite     

W. J. Robinson 《Celestial Mechanics and Dynamical Astronomy》1973,8(2):323-330

This paper is concerned with an extension of the classical restricted problem of three bodies in three dimensions. Usually, the satellite is considered to be a point mass. Here, the satellite is assumed to have a simple structure. The equations of motion are obtained and some of their consequences are discussed.  相似文献   

Asymptotic solutions of the restricted problem near the equilateral Lagrangian points     

R. Cid  S. Ferrer  J. A. Caballero 《Celestial Mechanics and Dynamical Astronomy》1985,35(2):189-200

The restricted problem in the vicinity of the Lagrangian point L₄ is studied by finding a convergent binomial expansion of the disturbing function. Using a Hamiltonian formulation in Delaunay variables and removing the short-period terms a resonance problem (already considered by Giacaglia (1970) in an attempt of enlarging the Ideal Resonance) is obtained. It is shown that this extension is reducible to Garfinkel's ideal resonance in the libration region.  相似文献   

About stability of libration points in the restricted photogravitational three body problem     

Krzysztof Goździewski  Andrzej J. Maciejewski  Zuzanna Niedzielska 《Celestial Mechanics and Dynamical Astronomy》1991,52(2):195-201

Nonlinear stability of the triangular libration point in the photogravitational restricted three body problem was investigated in the whole range of the parameters. Some results obtained earlier are corrected. The method for proper determination of cases when stability cannot be determined by four order terms of the hamiltonian was proposed.  相似文献   

Normal forms for real linear Hamiltonian systems with purely imaginary eigenvalues     

N. Burgoyne  R. Cushman 《Celestial Mechanics and Dynamical Astronomy》1974,8(4):435-443

This note gives a concise algorithm for computing a normal form for a real linear Hamiltonian differential equatin which has purely imaginary eigenvalues. This algorithm is then applied to the differential equation which comes from the quadratic terms of the Hamiltonian of the restricted three body problem at a Lagrange equilateral triangle equilibrium point.  相似文献   

On the stability of the lagrangian points in the spatial restricted problem of three bodies     

Alessandra Celletti  Antonio Giorgilli 《Celestial Mechanics and Dynamical Astronomy》1990,50(1):31-58

The problem of stability of the Lagrangian equilibrium point of the circular restricted problem of three bodies is investigated in the light of Nekhoroshev-like theory. Looking for stability over a time interval of the order of the estimated age of the universe, we find a physically relevant stability region. An application of the method to the Sun-Jupiter and the Earth-Moon systems is made. Moreover, we try to compare the size of our stability region with that of the region where the Trojan asteroids are actually found; the result in such case is negative, thus leaving open the problem of the stability of these asteroids.  相似文献   

Periodic solutions of the singly averaged hill problem     

M. A. Vashkov’yak 《Astronomy Letters》2006,32(10):707-715

Based on the ideas of Lyapunov’s method, we construct a family of symmetric periodic solutions of the Hill problem averaged over the motion of a zero-mass point (a satellite). The low eccentricity of the satellite orbit and the sine of its inclination to the plane of motion of the perturbing body are parameters of the family. We compare the analytical solution with numerical solutions of the averaged evolutionary system and the rigorous (nonaveraged) equations of the restricted circular three-body problem.  相似文献   

Application of two special orbits in the orbit determination of lunar satellites     

Peng Liu  Xi-Yun Hou  Jing-Shi Tang  Lin Liu 《中国天文和天体物理学报》2014,(10)

Using inter-satellite range data,the combined autonomous orbit determination problem of a lunar satellite and a probe on some special orbits is studied in this paper.The problem is firstly studied in the circular restricted three-body problem,and then generalized to the real force model of the Earth-Moon system.Two kinds of special orbits are discussed:collinear libration point orbits and distant retrograde orbits.Studies show that the orbit determination accuracy in both cases can reach that of the observations.Some important properties of the system are carefully studied.These findings should be useful in the future engineering implementation of this conceptual study.  相似文献   

On some applications of invariant manifolds     

Xi-Yun Hou  Lin Liu  Yu-Hui Zhao 《中国天文和天体物理学报》2011,11(1)

Taking transfer orbits of a collinear libration point probe, a lunar probe and an interplanetary probe as examples, some applications of stable and unstable invariant manifolds of the restricted three-body problem are discussed. Research shows that transfer energy is not necessarily conserved when invariant manifolds are used. For the cases in which the transfer energy is conserved, the cost is a much longer transfer time.  相似文献