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1.
The ordinary spinor differential Equation (20) of the unperturbed Kepler motion is obtained from the classical equation of motion (19) if one uses the spinor regularization (9) and postulates an essential subsidiary condition (10). A natural generalization for the Kepler motion follows by dropping this subsidiary conditions; it is the 8-parameter set of solutions of the spinor equation of motion (20). The sixteen natural extensive integrals (30)–(35) for this generalized Kepler motion are here deduced by means of the relativistic motors (2), (7) of the Spinor Ring Algebra. These integrals form, with respect to the Poisson bracket operation, a 15-dimensional Lie algebra (40)–(44), closely related to the Lie algebras in quantum mechanics.Dedicated to Professor G. Järnefelt on his 70th anniversary.  相似文献   

2.
A stabilization of the classical equations of two-body motion is offered. It is characterized by the use of the regularizing independent variable (eccentric anomaly) and by the addition of a control-term to the differential equations. This method is related to the KS-theory (Stiefel, 1970) which performed for the first time a stabilization of the Kepler motion. But in contrast to the KS-theory our method does not transform the coordinates of the particle. As far as the theory of stability and the numerical experiments are concerned we restrict ourselves to thepure Kepler motion. But, of course, the stabilizing devices will also improve the accuracy of the computation of perturbed orbits. We list, therefore, also the equations of the perturbed motion.  相似文献   

3.
The singularity of the Kepler motion can be eliminated by means of the spinor regularization. The extensive integrals of the Kepler motion form a Lie algebra with respect to the Poisson bracket operation. Mayer-Gürr has shown that in the caseH>0 the corresponding Lie group is the multiplicative group of all real 4×4 unimodular matrices SL(4,R). Kustaanheimo has posed the problem of the identification of the corresponding Lie groups in the elliptic and parabolic cases. We solve this problem here and use the opportunity to introduce the concept of the Clifford algebra which is needed in our solution.  相似文献   

4.
This paper is an attempt to bring unity in the study of the classical Kepler problem by combining, through simple vectorial and quaternionic techniques, its two peculiar aspects: the determination of the constants of the motion and the regularization at the origin.Research supported by the Consiglio Nazionale delle Ricerche of Italy (C.N.R.-G.N.F.M.).  相似文献   

5.
As an outcome of our previous notes [13, 14] on the quaternion regularization of the classical Kepler problem and pre-quantization of the Kepler manifold we show, first, that both the cross product of two quaternions and the cross product of their anti-involutes are susceptible of a simple geometrical representation in the ordinary 3-dimensional euclidean spaceR 3 and, secondly, that they satisfy anSO(4)-invariant relation that implies projection of curves from the quaternion space onto the spaceR 3. ThisSO(4)-invariance allows—in the particular case of orthogonal quaternions of equal norm—a straight derivation: (i) of the correspondence between the free motion on the surface of a sphereS 3 and the physical elliptical Kepler motion (collisions included) on a plane denoted by w ; (ii) of the celebrated Kepler equation and (iii) of the Levi-Civita regularizing time transformation. With (i) and (ii) we recover some of Györgyi's [3] results. The aforesaid orbital plane w and the orbital plane *, arrived at independently by exploiting the Kustaanheimo-Stiefel regularizing transformation, are shown to be inclined exactly at an angle characterizing the ratio of the semi-axes of the elliptical orbits and intimately related to the cross product representation. Thus the eventual superimposition of the two planes confirms the intimate connection between the various regularization procedures—transforming the classical Kepler problem into the geodesic flow onS 3—and the Fock's procedure for the quantum theoretical Kepler problem of the hydrogen atom (accidental degeneracy).This research was supported by the Consiglio Nazionale delle Ricerche of Italy (C.N.R.-G.N.F.M.).  相似文献   

6.
A massive neutrino halo will exert important effects on the motion of the galactic stars inside. In this paper, we give a fully relativistic treatment of the galactic velocity curve in the presence of a massive halo. The results are different from the classical Kepler motion, and provides a possible means of verifying the existence of neutrino halos.  相似文献   

7.
There exist many comets with near-parabolic orbits in the Solar System. Among various theories proposed to explain their origin, the Oort cloud hypothesis seems to be the most reasonable (Oort, 1950). The theory assumes that there is a cometary cloud at a distance 103 – 105 AU from the Sun and that perturbing forces from planets or stars make orbits of some of these comets become of near-parabolic type. Concerning the evolution of these orbits under planetary perturbations, we can raise the question: Will they stay in the Solar System forever or will they escape from it? This is an attractive dynamical problem. If we go ahead by directly solving the dynamical differential equations, we may encounter the difficulty of long-time computation. For the orbits of these comets are near-parabolic and their periods are too long to study on their long-term evolution. With mapping approaches the difficulty will be overcome. In another aspect, the study of this model has special meaning for chaotic dynamics. We know that in the neighbourhood of any separatrix i.e. the trajectory with zero frequency of the unperturbed motion of an Hamiltonian system, some chaotic motions have to be expected. Actually, the simplest example of separatrix is the parabolic trajectory of the two body problem which separates the bounded and unbounded motion. From this point of view, the dynamical study on near-parabolic motion is very important. Petrosky's elegant but more abstract deduction gives a Kepler mapping which describes the dynamics of the cometary motion (Petrosky, 1988). In this paper we derive a similar mapping directly and discuss its dynamical characters.  相似文献   

8.
We compute the normal forms for the Hamiltonian leading to the epicyclic approximations of the (perturbed) Kepler problem in the plane. The Hamiltonian setting corresponds to the dynamics in the Hill synodic system where, by means of the tidal expansion of the potential, the equations of motion take the form of perturbed harmonic oscillators in a rotating frame. In the unperturbed, purely Keplerian case, the post-epicyclic solutions produced with the normal form coincide with those obtained from the expansion of the solution of the Kepler equation. In all cases where the perturbed problem can be cast in autonomous form, the solution is easily obtained as a perturbation series. The generalization to the spatial problem and/or the non-autonomous case is straightforward.  相似文献   

9.
In this paper, a special extrapolation method for the numerical integration of perturbed Kepler problems (given in KS-formulation) is worked out and analyzed in detail. The underlying so-called Kepler discretization isexact for the pure (elliptic) Kepler motion. A numerically stable realization is presented together with a backward error analysis: this analysis shows that the effect of the arising rounding errors can be regarded as a small perturbation inferior to the physical perturbation. For test purposes, a well-known example describing the motion of an artificial Earth satellite in an equator plane subject to the oblateness perturbation is used to demonstrate the efficiency of the new extrapolation method.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.  相似文献   

10.
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11.
The long-term monitoring and high photometric precision of the Kepler satellite will provide a unique opportunity to sound the stellar cycles of many solar-type stars using asteroseismology. This can be achieved by studying periodic changes in the amplitudes and frequencies of the oscillation modes observed in these stars. By comparing these measurements with conventional ground-based chromospheric activity indices, we can improve our understanding of the relationship between chromospheric changes and those taking place deep in the interior throughout the stellar activity cycle. In addition, asteroseismic measurements of the convection zone depth and differential rotation may help us determine whether stellar cycles are driven at the top or at the base of the convection zone. In this paper, we analyse the precision that will be possible using Kepler to measure stellar cycles, convection zone depths and differential rotation. Based on this analysis, we describe a strategy for selecting specific targets to be observed by the Kepler Asteroseismic Investigation for the full length of the mission, to optimize their suitability for probing stellar cycles in a wide variety of solar-type stars.  相似文献   

12.
Geometric algebra is introduced as a general tool for Celestial Mechanics. A general method for handling finite rotations and rotational kinematics is presented. The constants of Kepler motion are derived and manipulated in a new way. A new spinor formulation of perturbation theory is developed.This work was partially supported by JPL under contract with the National Aeronautics and Space Administration.  相似文献   

13.
The Fokker-Planck equation for small stochastic changes to particles in Kepler orbits has to be formulated in terms of the integrals of motion. We generalize the modelling of proton and electron collisional perturbations to gas particles on trajectories through the solar system in order to include both spatial and velocity diffusion. The general solution is obtained in terms of a 4-dimensional normal distribution. Treatment of the singularity in the Fokker-Planck operator reduces the dimensionality by one. In addition to extending earlier results for anisotropic collisional heating in the thermal approximation, the present formulation gives the changes in density due to the mean repulsive force and to perturbations of trajectories (spatial diffusion). The net diffusion is almost everywhere towards the sun and the density increase is significant in the downstream hydrogen wake, particularly where destructive depletion is strong and gravitational focussing weak.  相似文献   

14.
In a previous note we have shown that the KS-transformation, introduced by Kustaanheimo and Stiefel into Celestial Mechanics for the regularization of the Kepler problem, may be formulated in terms of hypercomplex numbers as the product of a quaternion and its anti-involute, thus representing a particular morphism of the real algebra of quaternions-having for image the physical configuration space of the Kepler problem. In the present note we show, first, that this formulation allows a straight derivation of the Hopf fibering of the sphere S3 (characterized by unit quaternions) having the base space given by the sphere S2 (characterized by unit vectors), and secondly that the KS-transformation allows the quantization of the symplectic manifold S2 in the sense of Souriau, the associated quantum manifold S3 having a contact structure given by the bilinear relation characteristic of the KS-theory. Furthermore, after presenting a natural extension of the hypercomplex KS-transformation to the full phase space of the Kepler problem, we show that this extension allows the quantization of the manifold of Kepler orbits of fixed negative energy (manifold diffeomorphic to the symplectic product S2×S2). The energy levels satisfy a well known quantum integrality condition and the associated quantum manifold is diffeomorphic to the product manifold S3×S3 quotiented by a suitable equivalence relation.Research supported by the Consiglio Nazionale delle Ricerche of Italy, Gruppo per la Fisica-Matematica.  相似文献   

15.
The equations of motion of the planar three-body problem split into two parts, called an external part and an internal part. When the third mass approaches zero, the first part tends to the equations of the Kepler motion of the primaries and the second part to the equations of motion of the restricted problem.We discuss the Hill stability from these equations of motion and the energy integral. In particular, the Jacobi integral for the circular restricted problem is seen as an infinitesimal-mass-order term of the Sundman function in this context.  相似文献   

16.
This paper builds upon the work of Palmer and Imre exploring the relative motion of satellites on neighbouring Keplerian orbits. We make use of a general geometrical setting from Hamiltonian systems theory to obtain analytical solutions of the variational Kepler equations in an Earth centred inertial coordinate frame in terms of the relevant conserved quantities: relative energy, relative angular momentum and the relative eccentricity vector. The paper extends the work on relative satellite motion by providing solutions about any elliptic, parabolic or hyperbolic reference trajectory, including the zero angular momentum case. The geometrical framework assists the design of complex formation flying trajectories. This is demonstrated by the construction of a tetrahedral formation, described through the relevant conserved quantities, for which the satellites are on highly eccentric orbits around the Sun to visit the Kuiper belt.  相似文献   

17.
The classical Kepler Problem consists in the determination of the relative orbital motion of a secondary body (planet) with respect to the primary body (Sun), for a given time. However, any natural system tends to have minimum energy and is subjected to differential gravitational or tidal forces (called into play mainly due to the finite size and deformability of the secondary body). We formulate the Kepler Problem taking into account the finite size of the secondary body and consider an approximation which tends towards minimum energy orbits, by increasing the dimensionality of the problem. This formulation leads to a conceivable natural explanation of the fact that the planetary orbits are characterized by small eccentricities.  相似文献   

18.
The Kepler problem including radiation pressure and drag is treated. The equation of the orbit is derived and the scalar and vector integrals of motion are obtained by direct operation on the vector form of the equation of motion.  相似文献   

19.
A new method has been devised to determine the spherical harmonic coefficients of the lunar gravity field. This method consists of a two-step data reduction and estimation process. In the first step, a weighted least-squares empirical orbit determination scheme is applied to Doppler tracking data from lunar orbits to estimate longpperiod Kepler elements and rates. Each of the Kepler elements is represented by an independent function of time. The long-period perturbing effects of the Earth, Sun, and solar radiation are explicitly modeled in this scheme. Kepler element variations estimated by this empirical processor are then ascribed to the non-central lunar gravitation features. Doppler data are reduced in this manner for as many orbits as are available. In the second step, the Kepler element rates are used as input to a second least-squares processor that estimates lunar gravity coefficients using the long-period Lagrange perturbation equations.Pseudo Doppler data have been generated simulating two different lunar orbits. This analysis included the perturbing effects of the L1 lunar gravity field, the Earth, the Sun, and solar radiation pressure. Orbit determinations were performed on these data and long-period orbital elements obtained. The Kepler element rates from these solutions were used to recover L1 lunar gravity coefficients. Overall results of this controlled experiment show that lunar gravity coefficients can be accurately determined and that the method is dynamically consistent with long-period perturbation theory.  相似文献   

20.
For Vinti's dynamical problem, there is proposed a new form of solution wherein all three coordinates are expressed in terms of one independent variable. The formulae for the three co-ordinates are clear generalizations of the corresponding formulae for the Kepler problem while the independent variable corresponds to the true anomaly. The solution is completed by the relation connecting the independent variable with time: the latter is a generalization of the well known Kepler time-angle relationship. From the form and method of solution the main qualitative features of the motion are readily derived.  相似文献   

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