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1.
This paper presents a Hamiltonian approach to modelling spacecraft motion relative to a circular reference orbit based on
a derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling
of high-order terms and orbital perturbations within the context of the Clohessy–Wiltshire solution. First, the Hamiltonian
is partitioned into a linear term and a high-order term. The Hamilton–Jacobi equations are solved for the linear part by separation,
and new constants for the relative motions are obtained, called epicyclic elements. The influence of higher order terms and
perturbations, such as Earth’s oblateness, are incorporated into the analysis by a variation of parameters procedure. As an
example, closed-form solutions for J2-invariant orbits are obtained. 相似文献
2.
Aaron J. Rosengren Daniel J. Scheeres 《Celestial Mechanics and Dynamical Astronomy》2014,118(3):197-220
We consider sets of natural vectorial orbital elements of the Milankovitch type for perturbed Keplerian motion. These elements are closely related to the two vectorial first integrals of the unperturbed two-body problem; namely, the angular momentum vector and the Laplace–Runge–Lenz vector. After a detailed historical discussion of the origin and development of such elements, nonsingular equations for the time variations of these sets of elements under perturbations are established, both in Lagrangian and Gaussian form. After averaging, a compact, elegant, and symmetrical form of secular Milankovitch-like equations is obtained, which reminds of the structure of canonical systems of equations in Hamiltonian mechanics. As an application of this vectorial formulation, we analyze the motion of an object orbiting about a planet (idealized as a point mass moving in a heliocentric elliptical orbit) and subject to solar radiation pressure acceleration (obeying an inverse-square law). We show that the corresponding secular problem is integrable and we give an explicit closed-form solution. 相似文献
3.
The instability of axisymmetric flows of inviscid compressible fluid with respect to two-dimensional infinitesimal perturbations with the nonconservation of angular momentum is investigated by numerically integrating the differential equations of hydrodynamics. The compressibility is taken into account for a homentropic flow with an adiabatic index varying over a wide range. The problem has been solved for two angular velocity profiles of an initial axisymmetric flow. In the first case, a power-law rotation profile with a finite enthalpy gradient at the flow edges has been specified. For this angular velocity profile, we show that the instability of sonic and surface gravity modes in a nearly Keplerian flow, when a radially variable vorticity exists in the main flow, can be explained by the combined action of the Landau mechanism and mode coupling. We also show that including a radially variable vorticity makes the limiting exponent in the rotation law at which the unstable surface gravity modes vanish dependent on the fluid compressibility. In the second case, a Keplerian rotation law with a quasi-sinusoidal deviation has been specified in such a way that the enthalpy gradient vanished at the flow edges. We have found than the sonic modes are then stabilized and the flow is unstable only with respect to the perturbations that also exist in an incompressible fluid. 相似文献
4.
Numerical simulations of the magnetic reconnection process in a current sheet show that, in some cases, MHD shocks appear to be attached to edges of the sheet. The appearance of the shocks may be considered to be a result of splitting of the sheet. In the present paper we suppose that this splitting takes place in consequence of non-evolutionarity of the reconnecting current sheet as a discontinuity. The problem of time evolution of small perturbations does not have a unique solution for a non-evolutionary discontinuity, and it splits into other (evolutionary) discontinuities. Such an approach allows us to determine conditions under which the splitting of the-sheet occurs. The main difficulty of this approach is that a current sheet is not reduced to a classified 1D discontinuity, because inhomogeneity of flow velocity inside the sheet is two-dimensional. To formulate the non-evolutionarity problem, we solve the linear MHD equations inside and outside the sheet and deduce linearized 1D boundary conditions at its surface. We show that for large enough conductivity, small perturbations exist which interact with the sheet as with a discontinuity. Then we obtain a non-evolutionarity criterion, with respect to these perturbations, in the form of a restriction on the flow velocity across the surface of the sheet. 相似文献
5.
M. Moons 《Celestial Mechanics and Dynamical Astronomy》1984,34(1-4):263-273
A theory of the libration of the Moon, completely analytical with respect to the harmonic coefficients of the lunar gravity field, was recently built (Moons, 1982). The Lie transforms method was used to reduce the Hamiltonian of the main problem of the libration of the Moon and to produce the usual libration series p1, p2 and . This main problem takes into account the perturbations due to the Sun and the Earth on the rotation of a rigid Moon about its center of mass. In complement to this theory, we have now computed the planetary effects on the libration, the planetary terms being added to the mean Hamiltonian of the main problem before a last elimination of the angles. For the main problem, as well as for the planetary perturbations, the motion of the center of mass of the Moon is described by the ELP 2000 solution (Chapront and Chapront-Touze, 1983). 相似文献
6.
《New Astronomy》2007,12(6):490-496
To explore the dynamics of a test particle in the near-Mercury’s environment, the orbital motion of an orbiter around Mercury is considered. Different perturbing forces, namely the Mercurian gravity field, the solar radiation pressure, the solar wind and the coronal mass ejections, are taken into account. The order of magnitude of each perturbing term is assessed. The equations of motion in canonical representation are obtained. The Hamiltonian in terms of Hansen coefficients is expressed. A procedure for solution is presented. The short and long periodic terms are removed from the Hamiltonian and the solution is obtained. Long periodic perturbations on the orbital dynamics of an orbiter around Mercury due to the solar events are found as revealed by Eq. (26) in the text. Resonance cases are discussed and the different resonant inclinations are obtained. A procedure for the computation of the position and velocity is presented. 相似文献
7.
In this paper the two-degree of freedom problem of a geosynchronous artificial satellite orbiting near the critical inclination is studied. First a local approach of this problem is considered. A semi-numerical method, well suited to describe the perturbations of a non-trivial separable system, is then applied such that surfaces of section illustrating the global secular dynamics are obtained. The results are confirmed by numerical integrations of the full Hamiltonian.Research Assistant for the Belgian National Fund for Scientific Research 相似文献
8.
Efi Meletlidou Simos Ichtiaroglou Fabian Josef Winterberg 《Celestial Mechanics and Dynamical Astronomy》2001,80(2):145-156
We prove that Hill's lunar problem does not possess a second analytic integral of motion, independent of the Hamiltonian. In order to obtain this result, we avoid the usual normalization in which the angular velocity of the rotating reference frame is put equal to unit. We construct an artificial Hamiltonian that includes an arbitrary parameter b and show that this Hamiltonian does not possess an analytic integral of motion for in an open interval around zero. Then, by selecting suitable values of , b and using the invariance of the Hamiltonian under scaling in the units of length and time, we show that the Hamiltonian of Hill's problem does not possess an integral of motion, analytically continued from the integrable two–body problem in a rotating frame. 相似文献
9.
In this work we reveal for the first time that in the three dipole problem only asymmetric periodic orbits exist.For these periodic orbits — planar and three dimensional — of a charged particle moving under the influence of the electromagnetic field of the three dipoles we give their symplectic relations using the Hamiltonian formulation which is related to the symplectic matrix. Also we study the properties of the symplectic matrix and we give the relations there are among the variations of a periodic solution. These relations have been used to check the accuracy of numerical integration of equations of first order variations. 相似文献
10.
Giuseppe Pucacco 《Celestial Mechanics and Dynamical Astronomy》2004,90(1-2):109-123
We study strongly and weakly integrable 2-dimensional Hamiltonian systems with velocity dependent potentials. We determine
the set of conditions which must be satisfied in order to allow the existence of an independent second invariant polynomial
in the momenta. We then investigate the linear case for which a complete solution of the problem can be obtained. We recover
the classical set of linear strongly integrable systems and provide several new examples of weakly integrable systems whose
equations of motion can be explicitly solved at a fixed value of the energy. 相似文献
11.
Continuing a work initiated in an earlier publication (Yamada et al. in Phys Rev D 91:124016, 2015), we reexamine the linear stability of the triangular solution in the relativistic three-body problem for general masses by the standard linear algebraic analysis. In this paper, we start with the Einstein–Infeld–Hoffmann form of equations of motion for N-body systems in the uniformly rotating frame. As an extension of the previous work, we consider general perturbations to the equilibrium, i.e., we take account of perturbations orthogonal to the orbital plane, as well as perturbations lying on it. It is found that the orthogonal perturbations depend on each other by the first post-Newtonian (1PN) three-body interactions, though these are independent of the lying ones likewise the Newtonian case. We also show that the orthogonal perturbations do not affect the condition of stability. This is because these do not grow with time, but always precess with two frequency modes, namely, the same with the orbital frequency and the slightly different one due to the 1PN effect. The condition of stability, which is identical to that obtained by the previous work (Yamada et al. 2015) and is valid for the general perturbations, is obtained from the lying perturbations. 相似文献
12.
We consider numerical integration of nearly integrable Hamiltonian systems. The emphasis is on perturbed Keplerian motion, such as certain cases of the problem of two fixed centres and the restricted three-body problem. We show that the presently known methods have useful generalizations which are explicit and have a variable physical timestep which adjusts to both the central and perturbing potentials. These methods make it possible to compute accurately fairly close encounters. In some cases we suggest the use of composite (instead of symplectic) alternatives which typically seem to have equally good energy conservation properties.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
13.
Sandrine D’Hoedt Benoît Noyelles Julien Dufey Anne Lemaître 《Celestial Mechanics and Dynamical Astronomy》2010,107(1-2):93-100
The resonant rotation of Mercury can be modelised by a kernel model on which we can add perturbations. Our kernel model is a two-degree of freedom one written in Hamiltonian formalism. For this kernel, we consider that Mercury is solid and rotates on a Keplerian orbit. By introducing the perturbations due to the other planets of the Solar System, it appears that, in a particular case, our slow degree of freedom may enter into a 1:1 resonance with the Great Inequality of Jupiter and Saturn. Actually, as the moments of inertia of Mercury are still poorly known, this phenomenon cannot be excluded. 相似文献
14.
Francesco Fassò 《Celestial Mechanics and Dynamical Astronomy》1995,62(1):43-69
This paper deals with Hamiltonian perturbation theory for systems which, like Euler-Poinsot (the rigid body with a fixed point and no torques), are degenerate and do not possess a global system of action-angle coordinates. It turns out that the usual methods of perturbation theory, which are essentially local being based on the construction of normal forms within the domain of a local coordinate system, are not immediately usable to study perturbations of these systems, since degeneracy makes impossible to control that the system does not fall into a singularity of the coordinates. To overcome this difficulty, we develop a global formulation of Hamiltonian perturbation theory, in which the normal forms are globally defined on the phase space manifold. The key for this study lies in the geometry of the fibration by the invariant tori of an integrable degenerate Hamiltonian system, which is described by some generalizations of the Liouville-Arnol'd theorem and is reviewed in the paper. As an application, we provide a global formulation of Nekhoroshev's theorem on the stability for exponentially long times. 相似文献
15.
16.
Carol Williams 《Celestial Mechanics and Dynamical Astronomy》1987,43(1-4):35-35
An attempt is made to find a representation of planetary perturbations which does not require a Fourier expansion. For the planetary type three body problem a sequence of canonical transformations is given based on Landen transformations coupled with a rescaling of the time. The expansion is carried out to the order 2 in the Hamiltonian. 相似文献
17.
K. Uldall Kristiansen M. Vereshchagin K. Goździewski P. L. Palmer R. M. Roberts 《Celestial Mechanics and Dynamical Astronomy》2012,112(2):169-190
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.
Yoshio Kubo 《Celestial Mechanics and Dynamical Astronomy》1990,50(2):165-187
Hamiltonian mechanics is applied to the problem of the rotation of the elastic Earth. We first show the process for the formulation of the Hamiltonian for rotation of a deformable body and the derivation of the equations of motion from it. Then, based on a simple model of deformation, the solution is given for the period of Euler motion, UT1 and the nutation of the elastic Earth. In particular it is shown that the elasticity of the Earth acts on the nutation so as to decrease the Oppolzer terms of the nutation of the rigid Earth by about 30 per cent. The solution is in good agreement with results which have been obtained by other, different approaches. 相似文献
19.
Position and velocity perturbations in the orbital frame in terms of classical element perturbations
Stefano Casotto 《Celestial Mechanics and Dynamical Astronomy》1993,55(3):209-221
The transformation of classical orbit element perturbations to perturbations in position and velocity in the radial, transverse and normal directions of the orbital frame is developed. The formulation is given for the case of mean anomaly perturbations as well as for eccentric and true anomaly perturbations. Approximate formulas are also developed for the case of nearly circular orbits and compared with those found in the literature. 相似文献
20.
Martín Lara Jesús F. Palacián Ryan P. Russell 《Celestial Mechanics and Dynamical Astronomy》2010,108(1):1-22
Preliminary mission design for planetary satellite orbiters requires a deep knowledge of the long term dynamics that is typically
obtained through averaging techniques. The problem is usually formulated in the Hamiltonian setting as a sum of the principal
part, which is given through the Kepler problem, plus a small perturbation that depends on the specific features of the mission.
It is usually derived from a scaling procedure of the restricted three body problem, since the two main bodies are the Sun
and the planet whereas the satellite is considered as a massless particle. Sometimes, instead of the restricted three body
problem, the spatial Hill problem is used. In some cases the validity of the averaging is limited to prohibitively small regions,
thus, depriving the analysis of significance. We find this paradigm at Enceladus, where the validity of a first order averaging
based on the Hill problem lies inside the body. However, this fact does not invalidate the technique as perturbation methods
are used to reach higher orders in the averaging process. Proceeding this way, we average the Hill problem up to the sixth
order obtaining valuable information on the dynamics close to Enceladus. The averaging is performed through Lie transformations
and two different transformations are applied. Firstly, the mean motion is normalized whereas the goal of the second transformation
is to remove the appearance of the argument of the node. The resulting Hamiltonian defines a system of one degree of freedom
whose dynamics is analyzed. 相似文献