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1.
In this paper, we study the linear stability of the relative equilibria for homogeneous and quasihomogeneous potentials. First, in the case the potential is a homogeneous function of degree −a, we find that any relative equilibrium of the n-body problem with a>2 is spectrally unstable. We also find a similar condition in the quasihomogeneous case. Then we consider the case of three bodies and we study the stability of the equilateral triangle relative equilibria. In the case of homogeneous potentials we recover the classical result obtained by Routh in a simpler way. In the case of quasihomogeneous potentials we find a generalization of Routh inequality and we show that, for certain values of the masses, the stability of the relative equilibria depends on the size of the configuration.  相似文献   

2.
The motion of a point mass in the J 2 problem is generalized to that of a rigid body in a J 2 gravity field. The linear and nonlinear stability of the classical type of relative equilibria of the rigid body, which have been obtained in our previous paper, are studied in the framework of geometric mechanics with the second-order gravitational potential. Non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are obtained through a Poisson reduction process by means of the symmetry of the problem. The linear system matrix at the relative equilibria is given through the multiplication of the Poisson tensor and Hessian matrix of the variational Lagrangian. Based on the characteristic equation of the linear system matrix, the conditions of linear stability of the relative equilibria are obtained. The conditions of nonlinear stability of the relative equilibria are derived with the energy-Casimir method through the projected Hessian matrix of the variational Lagrangian. With the stability conditions obtained, both the linear and nonlinear stability of the relative equilibria are investigated in details in a wide range of the parameters of the gravity field and the rigid body. We find that both the zonal harmonic J 2 and the characteristic dimension of the rigid body have significant effects on the linear and nonlinear stability. Similar to the classical attitude stability in a central gravity field, the linear stability region is also consisted of two regions that are analogues of the Lagrange region and the DeBra-Delp region respectively. The nonlinear stability region is the subset of the linear stability region in the first quadrant that is the analogue of the Lagrange region. Our results are very useful for the studies on the motion of natural satellites in our solar system.  相似文献   

3.
We outline some aspects of the dynamics of an infinitesimal mass under the Newtonian attraction of three point masses in a symmetric collinear relative equilibria configuration when a repulsive Manev potential (\(-1/r +e/r^{2}\)), \(e>0\), is applied to the central mass. We investigate the relative equilibria of the infinitesimal mass and their linear stability as a function of the mass parameter \(\beta \), the ratio of mass of the central body to the mass of one of two remaining bodies, and e. We also prove the nonexistence of binary collisions between the central body and the infinitesimal mass.  相似文献   

4.
In this paper, we discuss dust motion and investigate possible mass transfer of charged particles in a binary asteroid system, in which the asteroids are electrically charged due to solar radiation. The surface potential of the asteroids is assumed to be a piecewise function, with positive potential on the sunlit half and negative potential on the shadow half. We derive the nonautonomous equations of motion for charged particles and an analytic representation for their lofting conditions. Particle trajectories and temporary relative equilibria are examined in relation to their moving forbidden regions, a concept we define and discuss. Finally, we use a Monte Carlo simulation for a case study on mass transfer and loss rates between the asteroids.  相似文献   

5.
It has been suggested that pressure shifts are the sole cause for the systematic blue shift of solar fraunhofer lines (Hart, 1974). To check this we evaluate the significance of pressure shifts for Fe i lines. The observed wavelength shifts of a large number of lines are compared with the shifts calculated for the Lennard-Jones potential following Hindmarsh et al. (1967). The Lennard-Jones potential for the interatomic forces yield pressure shifts caused by neutral hydrogen, which explain only a small fraction of the observed blue shift. It is also shown that the quadratic Stark effect contributes insignificantly to the position of Fe i lines. Table I summarizes the average line shifts for iron lines at the center of the solar disk after correction for pressure shifts.Operated by the Association of Universities for Research in Astronomy, Inc. under contract AST 74-04129 with the National Science Foundation.  相似文献   

6.
This paper concerns the dynamics of a rigid body of finite extent moving under the influence of a central gravitational field. A principal motivation behind this paper is to reveal the hamiltonian structure of the n-body problem for masses of finite extent and to understand the approximation inherent to modeling the system as the motion of point masses. To this end, explicit account is taken of effects arising because of the finite extent of the moving body. In the spirit of Arnold and Smale, exact models of spin-orbit coupling are formulated, with particular attention given to the underlying Lie group framework. Hamiltonian structures associated with such models are carefully constructed and shown to benon-canonical. Special motions, namely relative equilibria, are investigated in detail and the notion of anon-great circle relative equilibrium is introduced. Non-great circle motions cannot arise in the point mass model. In our analysis, a variational characterization of relative equilibria is found to be very useful. Thereduced hamiltonian formulation introduced in this paper suggests a systematic approach to approximation of the underlying dynamics based on series expansion of the reduced hamiltonian. The latter part of the paper is concerned with rigorous derivations of nonlinear stability results for certain families of relative equilibria. Here Arnold's energy-Casimir method and Lagrange multiplier methods prove useful. This work was supported in part by the AFOSR University Research Initiative Program under grant AFOSR-87-0073, by AFOSR grant 89-0376, and by the National Science Foundation's Engineering Research Centers Program: NSFD CDR 8803012. The work of P.S. Krishnaprasad was also supported by the Army Research Office through the Mathematical Sciences Institute of Cornell University.  相似文献   

7.
Saari's Conjecture for the Planar Three-Body Problem with Equal Masses   总被引：1，自引：1，他引：0  
In the N-body problem, it is a simple observation that relative equilibria (planar solutions for which the mutual distances between the particles remain constant) have constant moment of inertia. In 1970, Don Saari conjectured that the converse was true: if a solution to the N-body problem has constant moment of inertia, then it must be a relative equilibrium. In this note, we confirm the conjecture for the planar three-body problem with equal masses. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

8.
This paper discusses relative equilibria (or steady motions) and their stability for the dynamics of the system of two spring-connected masses in a central gravitational field. The system can be regarded as a simplified model for the Tethered Satellite System (TSS), where the tether is modeled by a (linear or nonlinear) spring. In the previous studies of the TSS problem, it was typically assumed that the center of mass is located at the massive one of the two end-masses, and moves on a great-circle orbit. However, for the simple system treated in this paper, it is proved that nongreat-circle relative equilibria do exist. Some fundamental concepts of the dynamics of an arbitrary assembly moving in a central gravitational field are discussed. The notion of steady motions used in engineering literature is linked with the notion of relative equilibria in geometric mechanics. Numerical computations show some interesting nongreat-circle relative equilibria for the spring-connected system. Radial relative equilibria, which correspond to the station-keeping mode for TSS, are then introduced. Within the framework of symmetry and reduction, their stability properties are investigated by adopting the reduced energy-momentum method, which takes the advantage of the intrinsic symmetry structure. It is shown that for practical configurations, the system at radial relative equilibria is stable if some conditions are satisfied.This work was partially supported by the National Science Council, Republic of China, under grant NSC-83-0208-M-002-082. The authors would like to thank W.-T. Chou for some computational assistance.  相似文献   

9.
The motion of a point mass in the J 2 problem has been generalized to that of a rigid body in a J 2 gravity field for new high-precision applications in the celestial mechanics and astrodynamics. Unlike the original J 2 problem, the gravitational orbit-rotation coupling of the rigid body is considered in the generalized problem. The existence and properties of both the classical and non-classical relative equilibria of the rigid body are investigated in more details in the present paper based on our previous results. We nondimensionalize the system by the characteristic time and length to make the study more general. Through the study, it is found that the classical relative equilibria can always exist in the real physical situation. Numerical results suggest that the non-classical relative equilibria only can exist in the case of a negative J 2, i.e., the central body is elongated; they cannot exist in the case of a positive J 2 when the central body is oblate. In the case of a negative J 2, the effect of the orbit-rotation coupling of the rigid body on the existence of the non-classical relative equilibria can be positive or negative, which depends on the values of J 2 and the angular velocity Ω e . The bifurcation from the classical relative equilibria, at which the non-classical relative equilibria appear, has been shown with different parameters of the system. Our results here have given more details of the relative equilibria than our previous paper, in which the existence conditions of the relative equilibria are derived and primarily studied. Our results have also extended the previous results on the relative equilibria of a rigid body in a central gravity field by taking into account the oblateness of the central body.  相似文献   

10.
We study the non-linear stability of the equilibria corresponding to the motion of a particle orbiting around a finite straight segment. The potential is a logarithmic function and may be considered as an approximation to the one generated by elongated celestial bodies. By means of the Arnold's theorem for non-definite quadratic forms we determine the orbital stability of the equilibria, for all values of the parameter k of the problem, resonant cases included.  相似文献   

11.
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12.
We hereby study the stability of a massless probe orbiting around an oblate central body (planet or planetary satellite) perturbed by a third body, assumed to lay in the equatorial plane (Sun or Jupiter for example) using a Hamiltonian formalism. We are able to determine, in the parameters space, the location of the frozen orbits, namely orbits whose orbital elements remain constant on average, to characterize their stability/unstability and to compute the periods of the equilibria. The proposed theory is general enough, to be applied to a wide range of probes around planet or natural planetary satellites. The BepiColombo mission is used to motivate our analysis and to provide specific numerical data to check our analytical results. Finally, we also bring to the light that the coefficient J 2 is able to protect against the increasing of the eccentricity due to the Kozai-Lidov effect and the coefficient J 3 determines a shift of the equilibria.  相似文献   

13.
We consider the non-canonical Hamiltonian dynamics of a gyrostat in Newtonian interaction with n spherical rigid bodies. Using the symmetries of the system we carry out two reductions. Then, working in the reduced problem, we obtain the equations of motion, a Casimir function of the system and the equations that determine the relative equilibria. Global conditions for existence of relative equilibria are given. Besides, we give the variational characterization of these equilibria and three invariant manifolds of the problem; being calculated the equations of motion in these manifolds, which are described by means of a canonical Hamiltonian system. We give some Eulerian and Lagrangian equilibria for the four body problem with a gyrostat. Finally, certain classical problems of Celestial Mechanics are generalized.  相似文献   

14.
We consider the problem of 4 bodies of equal masses in R 3 for the Newtonian r−1 potential. We address the question of the absolute minima of the action integral among (anti)symmetric loops of class H 1 whose period is fixed. It is the simplest case for which the results of [4] (corrected in [5]) do not apply: the minima cannot be the relative equilibria whose configuration is an absolute minimum of the potential among the configurations having a given moment of inertia with respect to their center of mass. This is because the regular tetrahedron cannot have a relative equilibrium motion in R 3 (see [2]). We show that the absolute minima of the action are not homographic motions. We also show that if we force the configuration to admit a certain type of symmetry of order 4, the absolute minimum is a collisionless orbit whose configuration ‘hesitates’ between the central configuration of the square and the one of the tetrahedron. We call these orbits ‘hip-hop’. A similar result holds in case of a symmetry of order 3 where the central configuration of the equilateral triangle with a body at the center of mass replaces the square. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

15.
We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ 2 throughJ 9. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J 2 only). In particular, we examine the manner in which the odd zonalJ 3 breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J 4+J 2 2 =0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions.  相似文献   

16.
In this paper we have studied in detail (numerically) the trajectories of charged particles in a magnetic field (dipolar at infinity) associated with a static star in the framework of Rosen's bimetric theory of gravity. It was found that there do exist potential wells that allow possible trapping of particles in stable orbits both on and off the equatorial plane. A particularly interesting feature that has shown up is the fact that the characteristics of the effective potential wellV eff depend on the ratio of the magnetic field strength parameter λ, and the angular momentumL of the charged particle. For values lower than a critical (λ/L) c the potential well lies within the regionr≤2m.  相似文献   

17.
In this paper we consider the two-body problem of a spherical pseudo-rigid body and a rigid sphere. Due to the rotational and “re-labelling” symmetries, the system is shown to possess conservation of angular momentum and circulation. We follow a reduction procedure similar to that undertaken in the study of the two-body problem of a rigid body and a sphere so that the computed reduced non-canonical Hamiltonian takes a similar form. We then consider relative equilibria and show that the notions of locally central and planar equilibria coincide. Finally, we show that Riemann’s theorem on pseudo-rigid bodies has an extension to this system for planar relative equilibria.  相似文献   

18.
Energy and stability in the Full Two Body Problem   总被引：1，自引：0，他引：1  
The conditions for relative equilibria and their stability in the Full Two Body Problem are derived for an ellipsoid–sphere system. Under constant angular momentum it is found that at most two solutions exist for the long-axis solutions with the closer solution being unstable while the other one is stable. As the non-equilibrium problem is more common in nature, we look at periodic orbits in the F2BP close to the relative equilibrium conditions. Families of periodic orbits can be computed where the minimum energy state of one family is the relative equilibrium state. We give results on the relative equilibria, periodic orbits and dynamics that may allow transition from the unstable configuration to a stable one via energy dissipation.   相似文献   

19.
Stability of the planar full 2-body problem   总被引：1，自引：0，他引：1  
The stability of the Full Two-Body Problem is studied in the case where both bodies are non-spherical, but are restricted to planar motion. The mutual potential is expanded up to second order in the mass moments, yielding a highly symmetric yet non-trivial dynamical system. For this system we identify all relative equilibria and determine their stability properties, with an emphasis on finding the energetically stable relative equilibria and conditions for Hill stability of the system. The energetically stable relative equilibria always correspond to the classical “gravity gradient” configuration with the long ends of the two bodies pointed at each other, however there always exists a second equilibrium in this configuration at a closer separation that is unstable. For our model system we precisely map out the relations between these different configurations at a given value of angular momentum. This analysis identifies the fundamental physical constraints and limitations that exist on such systems, and has immediate applications to the stability of asteroid systems that are fissioned due to a rapid spin rate. Specifically, we find that all contact binary asteroids which are spun to fission will initially lie in an unstable dynamical state and can always re-impact. If the total system energy is positive, the fissioned system can disrupt directly from this relative equilibrium, while if it is negative the system is bound together.  相似文献   

20.
In a previous work we studied the effects of (I) the J 2 and C 22 terms of the lunar potential and (II) the rotation of the primary on the critical inclination orbits of artificial satellites. Here, we show that, when 3rd-degree gravity harmonics are taken into account, the long-term orbital behavior and stability are strongly affected, especially for a non-rotating central body, where chaotic or collision orbits dominate the phase space. In the rotating case these phenomena are strongly weakened and the motion is mostly regular. When the averaged effect of the Earth’s perturbation is added, chaotic regions appear again for some inclination ranges. These are more important for higher values of semi-major axes. We compute the main families of periodic orbits, which are shown to emanate from the ‘frozen eccentricity’ and ‘critical inclination’ solutions of the axisymmetric problem (‘J 2 + J 3’). Although the geometrical properties of the orbits are not preserved, we find that the variations in e, I and g can be quite small, so that they can be of practical importance to mission planning.  相似文献   

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