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1.
Krzysztof Goździewski 《Celestial Mechanics and Dynamical Astronomy》2003,85(1):79-103
In papers (Godziewski and Maciejewski, 1998a, b, 1999), we investigate unrestricted, planar problem of a dynamically symmetric rigid body and a sphere. Following the original statement of the problem by Kokoriev and Kirpichnikov (1988), we assume that the potential of the rigid body is approximated by the gravitational field of a dumb-bell. The model is described in terms of a 2D Hamiltonian depending on three parameters.In this paper, we investigate the stability of triangular equilibria permissible by the dynamics of the model, under the assumption of low-order resonances. We analyze all resonances of order smaller than four, and we examine the stability with application of theorems by Markeev and Sokolsky. These are the possible following cases: the non-diagonal resonance of the first order with two null characteristic frequencies (unstable); resonances of the first order with one nonzero frequency (diagonal and non-diagonal, stable and unstable); the second-order resonance, which is non-diagonal and stable, and the third-order resonance which is generically unstable, except for three points in the parameters' space, corresponding to stable equilibria.We discuss a perturbed version of Kokoriev and Kirpichnikov model, and we find that if the perturbation is small and depends on the coordinates only, the triangular equilibria persist, except if for the unperturbed equilibria the first-order resonance occurs. We show that the resonances of the order higher than two are also preserved if the perturbation acts. 相似文献
2.
Krzysztof Goździewski 《Celestial Mechanics and Dynamical Astronomy》1998,70(1):41-58
In this paper we consider the problem of motion of an infinitesimal point mass in the gravity field of an uniformly rotating
dumb-bell. The aim of our study is to investigate Liapunov stability of Lagrangian libration points of this problem. We analyze
the stability of libration points in the whole range of parameters ω, μ of the problem. In particular, we consider all resonance
cases when the order of resonance is not greater than five.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
3.
We consider the problem of the motion of a zero-mass body in the vicinity of a system of three gravitating bodies forming a central configuration.We study the case where two gravitating bodies of equal mass lie on the same straight line and rotate around the central body with the same angular velocity. Equations for calculating the equilibrium positions in this system have been derived. The stability of the equilibrium points for a system of three gravitating bodies is investigated. We show that, as in the case of libration points for two bodies, the collinear points are unstable; for the triangular points, there exists a ratio of the mass of the central body to the masses of the extreme bodies, 11.720349, at which stability is observed. 相似文献
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5.
Andrzej J. Maciejewski Maria Przybylska 《Celestial Mechanics and Dynamical Astronomy》2016,126(4):297-311
This paper discusses the dynamics of systems of point masses joined by massless rigid rods in the field of a potential force. The general form of equations of motion for such systems is obtained. The dynamics of a linear chain of mass points moving around a central body in an orbit is analysed. The non-integrability of the chain of three masses moving in a circular Kepler orbit around a central body is proven. This was achieved thanks to an analysis of variational equations along two particular solutions and an investigation of their differential Galois groups. 相似文献
6.
A. Migus 《Earth, Moon, and Planets》1980,23(4):391-427
We have developed a theory of the rotation of the Moon, for the purpose of obtaining libration series explicitly dependent upon lunar gravitational field model parameters. A nonlinear model is used in which the rigid Moon, whose motion in space is that of the main problem of lunar theory, and whose gravity potential is represented through its third degree harmonics, is torqued by the Earth and Sun. The analytical series are then obtained as Poisson series. Numerical comparisons with Eckhardt's solution are briefly exposed. 相似文献
7.
The motion of two mutually attracting triaxial rigid bodies has been considered. Thirty six particular solutions corresponding to the libration points and analogous to the points Spoke, Arrow and Float (Duboshin, 1959) have been found. The stability of these libration points has been discussed in two categories of cases. In the first category, different shapes of the bodies have been taken and in the second category, the mass and the linear dimensions of one of the bodies have been taken small in comparison to the other. 相似文献
8.
In this paper we have proved the existence of libration points for the generalised photogravitational restricted problem of three bodies. We have assumed the infinitesimal mass of the shape of an oblate spheroid and both of the finite masses to be radiating bodies and the effect of their radiation pressure on the motion of the infinitesimal mass has also been taken into account. It is seen that there is a possibility of nine libration points for small values of oblateness, three collinear, four coplanar and two triangular. 相似文献
9.
《New Astronomy》2021
We study the non-collinear libration points in the frame work of photo-gravitational circular restricted three-body problem with Stokes drag acting as a dissipative force and considering the more massive primary as a radiating body and the less massive primary as a triaxial rigid body. The combined effects of radiation pressure and Stokes drag on the existence and stability of non-collinear libration points is analyzed. It is found that there exist two non-collinear libration points and are asymptotically stable in the interval 0.6149 ≤ q ≤ 1 for μ = 0.01, where q and μ are the radiation factor and mass ratio, respectively. 相似文献
10.
V. T. Kondurar 《Celestial Mechanics and Dynamical Astronomy》1974,10(3):327-344
The present paper is a direct continuation of the paper (Duboshin, 1973) in which was proved the existence of one kind of Lagrange (triangle) and Euler (rectilinear) solutions of the general problem of the motion of three finite rigid bodies assuming different laws of interaction between the elementary particles of the rigid bodies. In particular, Duboshin found that the general problem of three rigid bodies permits such solutions in which the centres of mass of the bodies always form an equilateral triangle or always remain on one straight line, and each body possesses an axial symmetry and a symmetry with respect to the plane of the centres of mass and rotates uniformly around its axis orthogonal to this plane. The conditions for the existence of such solutions have also been found. The results in Duboshin's paper have greatly interested the author of the present paper. In another paper (Kondurar and Shinkarik, 1972) considering a more special problem, when two of the three bodies are spheres, either homogeneous or possessing a spherically symmetric distribution of the densities or of the material points, and the third is an axially symmetrical body possessing equatorial symmetry, the present author obtained analogous solutions of the ‘float’ type describing the motion of the indicated dynamico-symmetrical body in assuming its passive gravitation. In the present paper new Lagrange solutions of the considered general problems of three rigid bodies of ‘level’ type are found when the axes of geometrical and mechanical symmetry of all three bodies always lie in the triangle plane, and the bodies themselves rotate inertially around the symmetry axis, independently of the parameters of the orbital motion of the centres of mass as in the ‘float’ case. The study of particular solutions of the general problem of the translatory-rotary motion of three rigid bodies, which are a generalization of Lagrange solutions, is in the author's opinion, a novelty of some interest for both theoretical and practical divisions of celestial mechanics. For example, in recent times the problem of the libration points of the Earth-Moon system has acquired new interest and value. A possible application which should be mentioned is that to the orbits of artificial satellites near the triangular libration points to serve as observation stations with the aim of specifying the physical parameters in the Earth-Moon system (e.g., the relation of the Earth's mass to the Moon's mass for investigating the orientation of the satellite, solar radiation, etc.). 相似文献
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12.
This paper formulates a circular restricted four body problem (CRFBP), where the three primaries are set in the stable Lagrangian
equilateral triangle configuration and the fourth body is massless. The analysis of this autonomous coplanar CRFBP is undertaken,
which identifies eight natural equilibria; four of which are close to the smaller body, two stable and two unstable, when
considering the primaries to be the Sun and two smaller bodies of the Solar System. Following this, the model incorporates
‘near term’ low-thrust propulsion capabilities to generate surfaces of artificial equilibrium points close to the smaller
primary, both in and out of the plane containing the celestial bodies. A stability analysis of these points is carried out
and a stable subset of them is identified. Throughout the analysis the Sun-Jupiter-asteroid-spacecraft system is used, for
conceivable masses of a hypothetical asteroid set at the libration point L
4. It is shown that eight bounded orbits exist, which can be maintained with a constant thrust less than 1.5 × 10−4 N for a 1000 kg spacecraft. This illustrates that, by exploiting low-thrust technologies, it would be possible to maintain
an observation point more than 66% closer to the asteroid than that of a stable natural equilibrium point. The analysis then
focusses on a major Jupiter Trojan: the (624) Hektor asteroid. The thrust required to enable close asteroid observation is
determined in the simplified CRFBP model. Finally, a numerical simulation of the real Sun-Jupiter-(624) Hektor-spacecraft
is undertaken, which tests the validity of the stability analysis of the simplified model. 相似文献
13.
GP. Horedt 《Astrophysics and Space Science》1973,22(2):321-327
We show within the framework of the restricted three body problem that
- Only in the immediate neighbourhood of the Lagrangian pointsL 4 andL 5 the distribution of a cloud of particles tends to become uniform under the influence of random stochastic perturbations. No consequences can be derived from this fact for a tendency of dispersion of clouds librating at arbitrary distances around the Lagrangian points.
- From an elementary inspection of the Jacobi integral we cannot conclude that mutual completely inelastic collisions tend to drive the particles away from the vicinity of the libration points.
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15.
George Voyatzis John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》2006,95(1-4):259-271
We study the dynamics of 3:1 resonant motion for planetary systems with two planets, based on the model of the general planar three body problem. The exact mean motion resonance corresponds to periodic motion (in a rotating frame) and the basic families of symmetric and asymmetric periodic orbits are computed. Four symmetric families bifurcate from the family of circular orbits of the two planets. Asymmetric families bifurcate from the symmetric families, at the critical points, where the stability character changes. There exist also asymmetric families that are independent of the above mentioned families. Bounded librations exist close to the stable periodic orbits. Therefore, such periodic orbits (symmetric or asymmetric) determine the possible stable configurations of a 3:1 resonant planetary system, even if the orbits of the two planets intersect. For the masses of the system 55Cnc most of the periodic orbits are unstable and they are associated with chaotic motion. There exist however stable symmetric and asymmetric orbits, corresponding to regular trajectories along which the critical angles librate. The 55Cnc extra-solar system is located in a stable domain of the phase space, centered at an asymmetric periodic orbit. 相似文献
16.
S. M. El-Shaboury 《Earth, Moon, and Planets》1990,48(3):233-242
In this paper the photogravitational forces restricted of three bodies be considered. We have assumed the infinitesimal mass of the shape of an axisymmetric body and one the finite masses be spherical luminous body while the other be an axisymmetric body non-luminous body. It is seen that there is a possibility of nine libration points for small values of oblatenesses. 相似文献
17.
Andrzej J. Maciejewski Maria Przybylska Leon Simpson Wojciech Szumiński 《Celestial Mechanics and Dynamical Astronomy》2013,117(3):315-330
This paper discusses a constrained gravitational three-body problem with two of the point masses separated by a massless inflexible rod to form a dumbbell. This problem is a simplification of a problem of a symmetric rigid body and a point mass, and has numerous applications in Celestial Mechanics and Astrodynamics. The non-integrability of this system is proven. This was achieved thanks to an analysis of variational equations along a certain particular solution and an investigation of their differential Galois group. Nowadays this approach is the most effective tool for study integrability of Hamiltonian and non-Hamiltonian systems. 相似文献
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19.
The stability of collinear and triangular libration points is investigated in the photogravitational elliptic restricted three-body problem, in which two primary bodies emit light energy simultaneously. The conditions of stability of the collinear and triangular libration points are obtained based on a linearized set of equations of perturbed motion for various values of the eccentricity of the Keplerian orbits and the mass ratio of the primary bodies. The maximal numerical value is defined for the eccentricity at which a stable libration point can still exist. It is demonstrated how the parametric resonance causes an instability of collinear and triangular libration points; the evolution of the origination of the instability zones is traced. The minimal eccentricity value is found at which zones of instability of triangular libration points arise. 相似文献
20.
The motion around the collinear libration points in the restricted three body problem is unstable. But there exist conditionally stable periodic orbits around these points. Special-purpose space probes located in the vicinity of these points (e.g., ISEE-3, SOHO) can benefit from this dynamical property, in regard to maintaining the orbit in position and the energy required of placing the probe in position. As an example, we study in this paper the launch and orbital control of a space probe around the L1 libration point in the system consisting of the Sun and the Earth-Moon. We present some theoretical and numerical simulations’ results, which may serve as a basis for the realization of such a space probe in future. 相似文献