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1.
Vladislav V. Sidorenko Alessandra Celletti 《Celestial Mechanics and Dynamical Astronomy》2010,107(1-2):209-231
This paper is devoted to the dynamics in a central gravity field of two point masses connected by a massless tether (the so called “spring–mass” model of tethered satellite systems). Only the motions with straight strained tether are studied, while the case of “slack” tether is not considered. It is assumed that the distance between the point masses is substantially smaller than the distance between the system’s center of mass and the field center. This assumption allows us to treat the motion of the center of mass as an unperturbed Keplerian one, so to focus our study on attitude dynamics. A particular attention is given to the family of planar periodic motions in which the center of mass moves on an elliptic orbit, and the point masses never leave the orbital plane. If the eccentricity tends to zero, the corresponding family admits as a limit case the relative equilibrium in which the tether is elongated along the line joining the center of mass with the field center. We study the bifurcations and the stability of these planar periodic motions with respect to in-plane and out-of-plane perturbations. Our results show that the stable motions take place if the eccentricity of the orbit is sufficiently small. 相似文献
2.
This paper contains an analysis of the attitude stability of a spinning axisymmetric satellite whose mass center moves in any known planar periodic orbit of the restricted three-body problem while the spin axis remains normal to the orbit plane. A procedure based on Floquet theory is developed for constructing attitude instability charts, and examples of these are presented for two stable periodic orbits of the Earth-Moon system—one direct and one retrograde. The physical significance of these instability predictions is then explored by means of numerical integration of the full nonlinear equations of motion. Finally, an analysis based on averaging is performed, leading to approximate instability charts and indicating a possible connection between certain orbital-attitude resonance conditions and unstable attitude motions. 相似文献
3.
We deal with the stability problem of planar periodic motions of a satellite about its center of mass. The satellite is regarded a dynamically symmetric rigid body whose center of mass moves in a circular orbit.By using the method of normal forms and KAM theory we study the orbital stability of planar oscillations and rotations of the satellite in detail. In two special cases we investigate the orbital stability analytically by introducing a small parameter. In the general case, numerical calculations of Hamiltonian normal form are necessary. 相似文献
4.
Hanlun Lei Christian Circi Emiliano Ortore Bo Xu 《Celestial Mechanics and Dynamical Astronomy》2018,130(8):53
In this work, periodic attitudes and bifurcations of periodic families are investigated for a rigid spacecraft moving on a stationary orbit around a uniformly rotating asteroid. Under the second degree and order gravity field of an asteroid, the dynamical model of attitude motion is formulated by truncating the integrals of inertia of the spacecraft at the second order. In this dynamical system, the equilibrium attitude has zero Euler angles. The linearised equations of attitude motion are utilised to study the stability of equilibrium attitude. It is found that there are three fundamental types of periodic attitude motions around a stable equilibrium attitude point. We explicitly present the linear solutions around a stable equilibrium attitude, which can be used to provide the initial guesses for computing the true periodic attitudes in the complete model. By means of a numerical approach, three fundamental families of periodic attitudes are studied, and their characteristic curves, distribution of eigenvalues, stability curves and stability distributions are determined. Interestingly, along the characteristic curves of the fundamental families, some critical points are found to exist, and these points correspond to tangent and period-doubling bifurcations. By means of a numerical approach, the bifurcated families of periodic attitudes are identified. The natural and bifurcated families constitute networks of periodic attitude families. 相似文献
5.
The collinear equilibrium position of the circular restricted problem with the two primaries at unit distance and the massless body at the pointL 3 is extended to the planar three-body problem with respect to the massm 3 of the third body; the mass ratio μ of the two primaries is considered constant and the constant angular velocity of the straight line on which the three masses stay at rest is taken equal to 1. As regards periodic motions ‘around’ the equilibrium pointL 3, four possible extensions from the restricted to the general problem are presented each of them starting with a simple or a doubly periodic orbit of the family α of the Copenhagen category (μ=0.50). Form 3=0.10, μ=0.50 (i.e. for fixed masses of all three bodies) the characteristic curve of the extended family α is found. The qualitative differences of the families corresponding tom 3=0 andm 3=0.10 are discussed. 相似文献
6.
Joseph J. F. Liu Philip M. Fitzpatrick 《Celestial Mechanics and Dynamical Astronomy》1975,12(4):463-487
Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w
0)2, wheren is the satellite's orbital mean motion andw
0 its initial rotational angular speed. 相似文献
7.
R. C. Calleja E. J. Doedel A. R. Humphries A. Lemus-Rodríguez E. B. Oldeman 《Celestial Mechanics and Dynamical Astronomy》2012,114(1-2):77-106
We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordinary differential equations. Specifically, we consider the circular restricted three-body problem (CR3BP), which models the motion of a satellite in an Earth–Moon-like system. The CR3BP has many well-known families of periodic orbits, such as the planar Lyapunov orbits and the non-planar vertical and halo orbits. We compute the unstable manifolds of selected vertical and halo orbits, which in several cases leads to the detection of heteroclinic connections from such a periodic orbit to invariant tori. Subsequent continuation of these connecting orbits with a suitable end point condition and allowing the energy level to vary leads to the further detection of apparent homoclinic connections from the base periodic orbit to itself, or the detection of heteroclinic connections from the base periodic orbit to other periodic orbits. Some of these connecting orbits are of potential interest in space mission design. 相似文献
8.
Solar sails are a proposed form of spacecraft propulsion using large membrane mirrors to propel a satellite taking advantage of the solar radiation pressure. To model the dynamics of a solar sail we have considered the Earth–Sun Restricted Three Body Problem including the Solar radiation pressure (RTBPS). This model has a 2D surface of equilibrium points parametrised by the two angles that define the sail orientation. In this paper we study the non-linear dynamics close to an equilibrium point, with special interest in the bounded motion. We focus on the region of equilibria close to SL 1, a collinear equilibrium point that lies between the Earth and the Sun when the sail is perpendicular to the Sun–sail direction. For different fixed sail orientations we find families of planar, vertical and Halo-type orbits. We have also computed the centre manifold around different equilibria and used it to describe the quasi-periodic motion around them. We also show how the geometry of the phase space varies with the sail orientation. These kind of studies can be very useful for future mission applications. 相似文献
9.
John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1975,12(3):255-276
A method is developed to study the stability of periodic motions of the three-body problem in a rotating frame of reference, based on the notion of surface of section. The method is linear and involves the computation of a 4×4 variational matrix by integrating numerically the differential equations for time intervals of the order of a period. Several properties of this matrix are proved and also it is shown that for a symmetric periodic motion it can be computed by integrating for half the period only.This linear stability analysis is used to study the stability of a family of periodic motions of three bodies with equal masses, in a rotating frame of reference. This family represents motion such that two bodies revolve around each other and the third body revolves around this binary system in the same direction to a distance which varies along the members of the family. It was found that a large part of the family, corresponding to the case where the distance of the third body from the binary system is larger than the dimensions of the binary system, represents stable motion. The nonlinear effects to the linear stability analysis are studied by computing the intersections of several perturbed orbits with the surface of sectiony
3=0. In some cases more than 1000 intersections are computed. These numerical results indicate that linear stability implies stability to all orders, and this is true for quite large perturbations. 相似文献
10.
Proceeding with our investigation into the motion of a particle influenced by the electromagnetic field of three celestial bodies of a magnetic-dipole nature we give here for the first time the analytical expressions of periodic solutions around a planar equilibrium point. These relations are expansions of the planar equations of motion in series of second order power of a parameter in the vicinity of equilibria. The above analytical expressions of periodic solutions give the first members of the family of periodic orbits which emanate from a stable equilibrium point. The whole family can then be calculated using a predictor-corrector algorithm. 相似文献
11.
Armando Majorana 《Celestial Mechanics and Dynamical Astronomy》1981,25(3):267-270
In this paper we study a particular four-body problem: three bodies revolve around their center of mass in circular orbits under the influence of their mutual gravitational attraction, while a fourth body moves in the plane defined by the three bodies but non influencing their motion. The linear stability of the eight equilibrium points is studied, and it is found that it depends on the values of the masses. 相似文献
12.
13.
关于星座小卫星的编队飞行问题 总被引：3，自引：0，他引：3
从轨道力学角度来看星座小卫星编队飞行和星星跟踪中的伴飞，遵循着如下动力学机制：(1)在各小卫星绕地球运动过程中轨道摄动变化的主要特征决定了星-星之间的空间构形，(2)当星星之间相互距离较近时，在退化的限制性三体问题(实为限制性二体问题)中，共线秤动点附近的条件周期运动亦可在一定时间内制约星-星之间的空间构形．将具体阐明这两种动力学机制的原理和相应的星星之间的相对构形，并用仿真计算来证实这两种动力学机制的适用范围，为星座小卫星编队飞行和在伴飞运动过程中进行轨控提供理论依据和具体的轨控条件． 相似文献
14.
In this paper we prove the existence of ring-type bounded motion in an isolated system consisting of a massive point particle and a homogeneous cube. We study the case of planar motion where the particle moves in a symmetry plane of the cube and we use a rotating frame of reference with its center at the mass center of the cube and its axes coinciding with the symmetry axes of the cube. We prove that, for negative values of the total energy and properly chosen values of the total angular momentum, the relative distance of the bodies has an upper and a lower bound-i.e., the regions of possible motion lie inside an annulus around the cube (motion inside a ring or an island). 相似文献
15.
John D. Hadjidemetriou Dionyssia Psychoyos George Voyatzis 《Celestial Mechanics and Dynamical Astronomy》2009,104(1-2):23-38
We study orbits of planetary systems with two planets, for planar motion, at the 1/1 resonance. This means that the semimajor axes of the two planets are almost equal, but the eccentricities and the position of each planet on its orbit, at a certain epoch, take different values. We consider the general case of different planetary masses and, as a special case, we consider equal planetary masses. We start with the exact resonance, which we define as the 1/1 resonant periodic motion, in a rotating frame, and study the topology of the phase space and the long term evolution of the system in the vicinity of the exact resonance, by rotating the orbit of the outer planet, which implies that the resonance and the eccentricities are not affected, but the symmetry is destroyed. There exist, for each mass ratio of the planets, two families of symmetric periodic orbits, which differ in phase only. One is stable and the other is unstable. In the stable family the planetary orbits are in antialignment and in the unstable family the planetary orbits are in alignment. Along the stable resonant family there is a smooth transition from planetary orbits of the two planets, revolving around the Sun in eccentric orbits, to a close binary of the two planets, whose center of mass revolves around the Sun. Along the unstable family we start with a collinear Euler–Moulton central configuration solution and end to a planetary system where one planet has a circular orbit and the other a Keplerian rectilinear orbit, with unit eccentricity. It is conjectured that due to a migration process it could be possible to start with a 1/1 resonant periodic orbit of the planetary type and end up to a satellite-type orbit, or vice versa, moving along the stable family of periodic orbits. 相似文献
16.
We study the motion of an infinitesimal mass point under the gravitational action of three mass points of masses μ, 1–2μ and
μ moving under Newton's gravitational law in circular periodic orbits around their center of masses. The three point masses
form at any time a collinear central configuration. The body of mass 1–2μ is located at the center of mass. The paper has
two main goals. First, to prove the existence of four transversal ejection–collision orbits, and second to show the existence
of an uncountable number of invariant punctured tori. Both results are for a given large value of the Jacobi constant and
for an arbitrary value of the mass parameter 0<μ≤1/2.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
17.
Periodic Solutions in the Ring Problem 总被引：3，自引：0，他引：3
T.J. Kalvouridis 《Astrophysics and Space Science》1999,266(4):467-494
In this article we investigate the symmetric periodic motions of a particle of negligible mass in a coplanar N-body system
consisting of ν=N-1 peripheral equidistant bodies of equal masses and a central body with a different mass. This system which
we refer to as the ring problem, is a simple model approximating the planar problem of N bodies.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
18.
John D. Hadjidemetriou George Voyatzis 《Celestial Mechanics and Dynamical Astronomy》2011,111(1-2):179-199
We present families of symmetric and asymmetric periodic orbits at the 1/1 resonance, for a planetary system consisting of a star and two small bodies, in comparison to the star, moving in the same plane under their mutual gravitational attraction. The stable 1/1 resonant periodic orbits belong to a family which has a planetary branch, with the two planets moving in nearly Keplerian orbits with non zero eccentricities and a satellite branch, where the gravitational interaction between the two planets dominates the attraction from the star and the two planets form a close binary which revolves around the star. The stability regions around periodic orbits along the family are studied. Next, we study the dynamical evolution in time of a planetary system with two planets which is initially trapped in a stable 1/1 resonant periodic motion, when a drag force is included in the system. We prove that if we start with a 1/1 resonant planetary system with large eccentricities, the system migrates, due to the drag force, along the family of periodic orbits and is finally trapped in a satellite orbit. This, in principle, provides a mechanism for the generation of a satellite system: we start with a planetary system and the final stage is a system where the two small bodies form a close binary whose center of mass revolves around the star. 相似文献
19.
K. E. Papadakis 《Astrophysics and Space Science》2009,323(3):261-272
In this paper we study the asymptotic solutions of the (N+1)-body ring planar problem, N of which are finite and ν=N−1 are moving in circular orbits around their center of masses, while the Nth+1 body is infinitesimal. ν of the primaries have equal masses m and the Nth most-massive primary, with m
0=β
m, is located at the origin of the system. We found the invariant unstable and stable manifolds around hyperbolic Lyapunov
periodic orbits, which emanate from the collinear equilibrium points L
1 and L
2. We construct numerically, from the intersection points of the appropriate Poincaré cuts, homoclinic symmetric asymptotic
orbits around these Lyapunov periodic orbits. There are families of symmetric simple-periodic orbits which contain as terminal
points asymptotic orbits which intersect the x-axis perpendicularly and tend asymptotically to equilibrium points of the problem spiraling into (and out of) these points.
All these families, for a fixed value of the mass parameter β=2, are found and presented. The eighteen (more geometrically simple) families and the corresponding eighteen terminating
homo- and heteroclinic symmetric asymptotic orbits are illustrated. The stability of these families is computed and also presented. 相似文献
20.
Rodolpho Vilhena de Moraes Regina Elaine Santos Cabette Maria Cecília Zanardi Teresinha J. Stuchi Jorge Kennety Formiga 《Celestial Mechanics and Dynamical Astronomy》2009,104(4):337-353
The stability of the rotational motion of artificial satellites is analyzed considering perturbations due to the gravity gradient
torque, using a canonical formulation, and Andoyer’s variables to describe the rotational motion. The stability criteria employed
requires the reduction of the Hamiltonian to a normal form around the stable equilibrium points. These points are determined
through a numerical study of the Hamilton’s equations of motion and linear study of their stability. Subsequently a canonical
linear transformation is used to diagonalize the matrix associated to the linear part of the system resulting in a normalized
quadratic Hamiltonian. A semi-analytic process of normalization based on Lie–Hori algorithm is applied to obtain the Hamiltonian
normalized up to the fourth order. Lyapunov stability of the equilibrium point is performed using Kovalev and Savchenko’s
theorem. This semi-analytical approach was applied considering some data sets of hypothetical satellites, and only a few cases
of stable motion were observed. This work can directly be useful for the satellite maintenance under the attitude stability
requirements scenario. 相似文献