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1.
Bruno Cordani 《Celestial Mechanics and Dynamical Astronomy》2000,77(3):185-200
The usual action-angle variables for the Kepler Problem (the Delaunay variables) are not globally defined, leaving out some
orbits (circular orbits or those lying on the xy-plane). Moreover they are trascendental functions of the physical variables, making it quite difficult to write the perturbed
Hamiltonian. The way-out proposed here is to pass to a 8-dimensional rank-6 Poisson manifold, that is, to parametrize the
state of the Kepler Problem with two 4-dimensional vectors mutually orthogonal and of equal norm.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
2.
Sergey Bolotin 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):343-371
We consider the plane restricted elliptic 3 body problem with small mass ratio and small eccentricity and prove the existence
of many periodic orbits shadowing chains of collision orbits of the Kepler problem. Such periodic orbits were first studied
by Poincaré for the non-restricted 3 body problem. Poincaré called them second species solutions. 相似文献
3.
We investigate a method to compute a finite set of preliminary orbits for solar system bodies using the first integrals of
the Kepler problem. This method is thought for the applications to the modern sets of astrometric observations, where often
the information contained in the observations allows only to compute, by interpolation, two angular positions of the observed
body and their time derivatives at a given epoch; we call this set of data attributable. Given two attributables of the same body at two different epochs we can use the energy and angular momentum integrals of
the two-body problem to write a system of polynomial equations for the topocentric distance and the radial velocity at the
two epochs. We define two different algorithms for the computation of the solutions, based on different ways to perform elimination
of variables and obtain a univariate polynomial. Moreover we use the redundancy of the data to test the hypothesis that two
attributables belong to the same body (linkage problem). It is also possible to compute a covariance matrix, describing the uncertainty of the preliminary orbits which results
from the observation error statistics. The performance of this method has been investigated by using a large set of simulated
observations of the Pan-STARRS project. 相似文献
4.
Kazunori Iwasaki Keiji Ohtsuki 《Monthly notices of the Royal Astronomical Society》2007,377(4):1763-1771
Planetesimals encountering with a planet cannot be captured permanently unless energy dissipation is taken into account, but some of them can be temporarily captured in the vicinity of the planet for an extended period of time. Such a process would be important for the origin and dynamical evolution of irregular satellites, short-period comets, and Kuiper-belt binaries. In this paper, we describe the basic formulation for the study of temporary capture of planetesimals from heliocentric orbits using three-body orbital integration, such as the definition of the duration and rate of temporary capture, and present results in the case of low random velocity of planetesimals. In the case of planetesimals initially on circular orbits, we find that planetesimals undergo a close encounter with the planet before they become temporarily captured. When planetesimals are scattered by the planet into the vicinity of one of periodic orbits around the planet, the duration of temporary capture tends to be extended. Typically, these capture orbits are in the retrograde direction around the planet. We evaluate the rate of temporary capture of planetesimals, and find that the ratio of this rate to their collision rate on to the planet increases with increasing semimajor axis of the planet. Similar results are obtained for planetesimals with non-zero but small random velocities, as long as Kepler shear dominates the relative velocity between the planet and planetesimals. For larger initial random velocities of planetesimals, temporary capture in both prograde and retrograde directions with much longer duration becomes possible. 相似文献
5.
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
6.
Maria Dina Vivarelli 《Celestial Mechanics and Dynamical Astronomy》1985,36(4):349-364
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
7.
8.
G. F. Gronchi D. Farnocchia L. Dimare 《Celestial Mechanics and Dynamical Astronomy》2011,110(3):257-270
The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of
a celestial body, which may be obtained either by optical or by radar observations. We write polynomial equations for this
problem, which can be solved using the powerful tools of computational Algebra. An algorithm to decide if the linkage of two short arcs is successful, i.e. if they belong to the same observed body, is proposed and tested numerically. This
paper continues the research started in Gronchi et al. (Celest. Mech. Dyn. Astron. 107(3):299–318, 2010), where the angular momentum and the energy integrals were used. The use of a suitable component of the Laplace–Lenz vector
in place of the energy turns out to be convenient, in fact the degree of the resulting system is reduced to less than half. 相似文献
9.
The accelerated Kepler problem (AKP) is obtained by adding a constant acceleration to the classical two-body Kepler problem.
This setting models the dynamics of a jet-sustaining accretion disk and its content of forming planets as the disk loses linear
momentum through the asymmetric jet-counterjet system it powers. The dynamics of the accelerated Kepler problem is analyzed
using physical as well as parabolic coordinates. The latter naturally separate the problem’s Hamiltonian into two unidimensional
Hamiltonians. In particular, we identify the origin of the secular resonance in the AKP and determine analytically the radius
of stability boundary of initially circular orbits that are of particular interest to the problem of radial migration in binary
systems as well as to the truncation of accretion disks through stellar jet acceleration. 相似文献
10.
L. Benet T. H. Seligman D. Trautmann 《Celestial Mechanics and Dynamical Astronomy》1998,71(3):167-189
We study the scattering motion of the planar restricted three‐body problem for small mass parameters μ. We consider the symmetric
periodic orbits of this system with μ = 0 that collide with the singularity together with the circular and parabolic solutions
of the Kepler problem. These divide the parameter space in a natural way and characterize the main features of the scattering
problem for small non‐vanishing μ. Indeed, continuation of these orbits yields the primitive periodic orbits of the system
for small μ. For different regions of the parameter space, we present scattering functions and discuss the structure of the
chaotic saddle. We show that for μ < μc and any Jacobi integral there exist departures from hyperbolicity due to regions of
stable motion in phase space. Numerical bounds for μc are given.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
11.
Alessandra F. S. Ferreira Antônio F. B. A. Prado Othon C. Winter Denilson P. S. Santos 《Astrophysics and Space Science》2018,363(2):24
The objective of the present paper is to derive a set of analytical equations that describe a swing-by maneuver realized in a system of primaries that are in elliptical orbits. The goal is to calculate the variations of energy, velocity and angular momentum as a function of the usual basic parameters that describe the swing-by maneuver, as done before for the case of circular orbits. In elliptical orbits the velocity of the secondary body is no longer constant, as in the circular case, but it varies with the position of the secondary body in its orbit. As a consequence, the variations of energy, velocity and angular momentum become functions of the magnitude and the angle between the velocity vector of the secondary body and the line connecting the primaries. The “patched-conics” approach is used to obtain these equations. The configurations that result in maximum gains and losses of energy for the spacecraft are shown next, and a comparison between the results obtained using the analytical equations and numerical simulations are made to validate the method developed here. 相似文献
12.
A. J. Ferrari 《Celestial Mechanics and Dynamical Astronomy》1973,7(1):46-76
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
13.
Kenneth R. Meyer 《Celestial Mechanics and Dynamical Astronomy》1981,23(1):69-81
This paper shows that there exist two families of periodic solutions of the restrictedN-body problem which are close to large circular orbits of the Kepler problem. These solutions are shown to be of general elliptic type and hence are stable. If the restricted problem admits a symmetry, then there are symmetric periodic solutions which are close to large elliptic orbits of the Kepler problem. 相似文献
14.
Using the continuation method we prove that the circular and the elliptic symmetric periodic orbits of the planar rotating
Kepler problem can be continued into periodic orbits of the planar collision restricted 3-body problem. Additionally, we also
continue to this restricted problem the so called “comet orbits”.
An erratum to this article can be found at 相似文献
15.
In the present paper we give some numerical results about natural families of periodic orbits, which emanate from limiting orbits around the equilateral equilibrium points of the Restricted Three-Body Problem, when the mass ratio is greater than Routh's critical one. 相似文献
16.
A two-point boundary value problem of the Kepler orbit similar to Lambert’s problem is proposed. The problem is to find a
Kepler orbit that will travel through the initial and final points in a specified flight time given the radial distances of
the two points and the flight-direction angle at the initial point. The Kepler orbits that meet the geometric constraints
are parameterized via the universal variable z introduced by Bate. The formula for flight time of the orbits is derived. The admissible interval of the universal variable
and the variation pattern of the flight time are explored intensively. A numerical iteration algorithm based on the analytical
results is presented to solve the problem. A large number of randomly generated examples are used to test the reliability
and efficiency of the algorithm. 相似文献
17.
We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus T3 on the phase space of the Kepler problem, computing its associated momentum map and using the geometry associated with this structure. A central feature in this derivation is the identification of the mean anomaly as the angle variable for a symplectic S
1 action on the union of the non-degenerate elliptic Kepler orbits. This approach is geometrically more natural than traditional ones such as directly solving Hamilton–Jacobi equations, or employing the Lagrange bracket. As an application of the new derivation, we give a singularity free treatment of the averaged J
2-dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J
2-Hamiltonian is a collective Hamiltonian of the T3 momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J
2 effect as geometric phases. 相似文献
18.
This paper deals with the Restricted Three Body Problem (RTBP) in which we assume that the primaries are radiation sources
and the influence of the radiation pressure on the gravitational forces is considered; in particular, we are interested in
finding families of periodic orbits under theses forces.
By means of some modifications to the method of numerical continuation of natural families of periodic orbits, we find several
families of periodic orbits, both in two and three dimensions. As starters for our method we use some known periodic orbits
in the classical RTBP.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献
19.
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
20.
K. Uldall Kristiansen P. L. Palmer M. Roberts 《Celestial Mechanics and Dynamical Astronomy》2010,106(4):371-390
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