共查询到20条相似文献，搜索用时 31 毫秒
1.
D. G. Bettis 《Celestial Mechanics and Dynamical Astronomy》1970,2(3):282-295
To examine the stabilizing effects of a modification of the classical finite difference methods of numerical integration the differential equations of perturbed Keplerian motion are integrated for two examples: an artificial satellite of the Earth, and Hill's variation orbit. The modified methods remove much of the instability that is inherent to the classical methods.Presented at the Conference on Celestial Mechanics. 相似文献
2.
Numerical integration of unstable differential equations should be avoided since a numerical error during thenth step produces erroneous initial values for the next step and thus deteriorates the subsequent integration in an unstable manner. A method is offered to stabilize the equations of motion corresponding to a given HamiltonianH by transformingH into a new HamiltonianH
* which is equivalent to the Hamiltonian of a harmonic oscillator. In contrast to other methods of stabilization the realm of canonical mechanics is thus not abandoned. Perturbations are discussed and as examples the Keplerian motion and the motion of a gyroscope are presented. 相似文献
3.
V. Szebehely 《Celestial Mechanics and Dynamical Astronomy》1976,14(4):499-508
A method is presented which transforms certain non-linear differential equations of dynamics into linear equations by introducing a new independent variable and by utilizing the integrals of motion. As examples of special interest the linearizations of unperturbed and perturbed Keplerian motions are discussed. 相似文献
4.
Bryan Brown 《Celestial Mechanics and Dynamical Astronomy》1977,16(2):229-259
Some of the results of an investigation into the long period behavior of the orbits of the Galilean satellites of Jupiter are presented. Special purpose computer programs were used to perform all the algebraic manipulations and series expansions that are necessary to describe the mutual interactions among the satellites.The disturbing function was expanded as a Poisson series in the modified Keplerian elements referred to a Jovicentric coordinate system. The differential equations for the modified Keplerian elements were then formed, and all short period perturbations were removed using Kamel's perturbation method. Approximate analytical solutions for these differential equations are derived, and the general form of the solutions are given. 相似文献
5.
We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potential that can be approximated to sufficient accuracy by expanding it to second order in the coordinates. After averaging over time to obtain the secular Hamiltonian, we use angular momentum and eccentricity vectors as elements. The method of variation of constants then leads to a set of equations of motion that are simple and regular, thus allowing efficient numerical integration. Some possible applications are briefly described. 相似文献
6.
Victor R. Bond 《Celestial Mechanics and Dynamical Astronomy》1974,10(3):303-318
The Newtonian differential equations of motion for the two-body problem can be transformed into four, linear, harmonic oscillator equations by simultaneously applying the regularizing time transformation dt/ds=r and the Kustaanheimo-Stiefel (KS) coordinate transformation. The time transformation changes the independent variable from time to a new variables, and the KS transformation transforms the position and velocity vectors from Cartesian space into a four-dimensional space. This paper presents the derivation of uniform, regular equations for the perturbed twobody problem in the four-dimensional space. The variation of parameters technique is used to develop expressions for the derivatives of ten elements (which are constants in the unperturbed motion) for the general case that includes both perturbations which can arise from a potential and perturbations which cannot be derived from a potential. These element differential equations are slightly modified by introducing two additional elements for the time to further improve long term stability of numerical integration.Originally presented at the AAS/AIAA Astrodynamics Specialists Conference, Vail, Colorado, July 1973 相似文献
7.
Martin Lara 《Celestial Mechanics and Dynamical Astronomy》2017,129(1-2):137-151
An implementation of the ideal frame formulation of perturbed Keplerian motion is presented which only requires the integration of a differential system of dimension 7, contrary to the 8 variables traditionally integrated with this approach. The new formulation is based on the integration of a scaled version of the Eulerian set of redundant parameters and slightly improves runtime performance with respect to the 8-dimensional case while retaining comparable accuracy. 相似文献
8.
J. M. A. Danby 《Celestial Mechanics and Dynamical Astronomy》1970,2(3):311-318
Matrix methods for computing perturbations of non-linear perturbed systems, as formulated by Alexeev, involve an expression for the full solution of the first variational equations of the system evaluated about a reference orbit. These cannot be immediately applied to a regularized system of equations where perturbations about Keplerian motion are considered since the solution of the variational equations of regularized Keplerian motion does not in general correspond to the solution of the variational equations of the unregularized equations. But, as Kustaanheimo and Stiefel have pointed out, the regularized equations of Keplerian motion should be excellent for the initiation of a perturbation theory since they are linear in form. This paper describes a method for applying Alexeev's theorem to a regularized system where full advantage is taken of the basic linear form of the unperturbed equations.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 17–23, 1969. 相似文献
9.
Mark T. Lane 《Celestial Mechanics and Dynamical Astronomy》1987,42(1-4):3-38
A first order analytical approximation of the tesseral harmonic resonance perturbations of the Keplerian elements is presented, and the mean elements (the Keplerian elements with the long period portions averaged out) will also be given in closed form. Finally the results of a numerical test, which compares the analytical solution against a numerical integration of the Lagrange equations of motion, will be summarized.This work was sponsored with the support of the Department of the Air Force under contract F19628-85-C-0002.The views expressed are those of the author and do not reflect the official policy or position of the U.S. Government. 相似文献
10.
J. Baumgarte 《Celestial Mechanics and Dynamical Astronomy》1976,13(1):105-109
In the publication Baumgarte and Stiefel (1974a) a stabilization of the Keplerian motion was offered by making use of a manipulation of the Hamiltonian. By this stabilization technique the given HamiltonianH(p
i,q
i) is replaced by a new HamiltonianH
*, which leads to Lyapunov-stable differential equations of motion.Whereas, in the quoted publication, the physical timet was used as the independent variable we now develop a generalization which allows to combine the stabilization with the introduction of a new independent variables. Such a fictitious times is popular for achieving an analytic step-size adaptation (Baumgarte and Stiefel, 1974c). Perturbations of Kepler motion are discussed. 相似文献
11.
Several integration schemes exist to solve the equations of motion of the N -body problem. The Lie-integration method is based on the idea to solve ordinary differential equations with Lie-series. In the 1980s, this method was applied to solve the equations of motion of the N -body problem by giving the recurrence formulae for the calculation of the Lie-terms. The aim of this work is to present the recurrence formulae for the linearized equations of motion of N -body systems. We prove a lemma which greatly simplifies the derivation of the recurrence formulae for the linearized equations if the recurrence formulae for the equations of motions are known. The Lie-integrator is compared with other well-known methods. The optimal step-size and order of the Lie-integrator are calculated. It is shown that a fine-tuned Lie-integrator can be 30–40 per cent faster than other integration methods. 相似文献
12.
N. Borderies 《Celestial Mechanics and Dynamical Astronomy》1977,16(3):291-308
This paper deals with the Adams-Moulton-Cowell multistep integrator, as described by Oestwinter and Cohen (1972). In order to evaluate the accuracy of the method, we started to test it in the case of the unperturbed two-body motion; numerical instability may arise by integrating first order systems. The accuracy is improved by applying a Sundmann transformation of the independent variable. The algorithm is then modified such that the equations of pure keplerian motion are integrated with respect to the new independent variable without truncation error; numerical experiments show the considerable improvement of accuracy and the reduction of computing time for Keplerian motion.If terms of the disturbing function of the Earth are added to the central potential, the time-transformation is less effective. With a modification of this time-transformation as given by Moynot in 1971, it is possible to reduce the propagation of the truncation error in the J2 problem. 相似文献
13.
The absolute X-ray flux from the whole disc of the sun in the wave length range 2 to 12 Å has been observed for a prolonged period by University of Iowa equipment on the earth-orbiting satellite Explorer 33 and the moon-orbiting satellite Explorer 35, both of the Goddard Space Flight Center of the National Aeronautics and Space Administration. The observations are continuing at the date of writing (July 1969). A comprehensive catalog of the flux F (2–12 Å) is being produced. The observational technique and the scheme of reducing data are described herein. Sample tabulations and plots are given. A catalog of tabular and graphical data with a time resolution of either 81.8 or 163.6 sec has been completed for the following periods: From Explorer 33: 2 July 1966 to 27 July 1967 From Explorer 35: 26 July 1967 to 18 September 1968 These blocks of data have been delivered to the National Space Science Data Center National Aeronautics and Space Administration Goddard Space Flight Center Greenbelt, Maryland 20771, U.S.A. and made available through that agency to interested workers in solar and ionospheric physics. Further blocks of data will be made available as they are completed. An abridged summary of principal flares is published in the monthly Solar-Geophysical Data of the U.S. Department of Commerce, Environmental Science Services Administration. 相似文献
14.
By Hamiltonian manipulation we demonstrate the existence of separable time‐transformed Hamiltonians in the extended phase‐space.
Due to separability explicit symplectic methods are available for the solution of the equations of motion. If the simple leapfrog
integrator is used, in case of two‐body motion, the method produces an exact Keplerian ellipse in which only the time‐coordinate
has an error. Numerical tests show that even the rectilinear N‐body problem is feasible using only the leapfrog integrator.
In practical terms the method cannot compete with regularized codes, but may provide new directions for studies of symplectic
N‐body integration.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
15.
I. M. Belen'kii 《Celestial Mechanics and Dynamical Astronomy》1981,23(1):9-32
This paper deals with a method of regularization and linearization of the equations of motion in the central force-field, when the potential is given.This method of regularization of the equations of motion is known (Sundman, 1913), and is based on the transformation of time by means of introducing a new independent variable.In this article a condition has been obtained for the regularizing function when the potential is given.Some examples of the perturbed Keplerian motions are discussed. 相似文献
16.
In this first part of the work we develop the equations of motion of a triaxial space station in orbit around the oblate Earth. A first order solution of the problem is presented and the method of complete integration of the system is outlined up to second order of approximation. The zero order part of the Hamiltonian includes both the kinetic and potential energy (Earth's Newtonian attraction) of the station, while the motion in the vicinity of a specific configuration is assumed.The solution leads to deviations on the attitude introduced by the oblateness of the Earth. Such attitude is an exact solution of the equations of the station when its center of mass moves in an elliptic Keplerian orbit.The explicit expressions of the complete solution, discussion of other possible effects on the motion and numerical comparisons will be presented in the second part of the work. 相似文献
17.
Wagner Sessin 《Celestial Mechanics and Dynamical Astronomy》1986,39(2):173-180
In this paper a method for the integration of the equations of the extended Delaunay method is proposed. It is based on the equations of the characteristic curves associated with the partial differential equation of Delaunay-Poincaré. The use of the method of characteristics changes the partial differential equation for higher order approximations into a system of ordinary differential equations. The independent variable of the equations of the characteristics is used instead of the angular variables of the Jacobian methods and the averaging principle of Hori is applied to solve the equations for higher orders. It is well known that Jacobian methods applied to resonant problems generally lead to the singularity of Poincaré. In the ideal resonance problem, this singularity appears when higher order approximations of the librational motion are considered. The singularity of Poincaré is non-essential and is caused by the choice of the critical arguments as integration variables. The use of the independent variable of the equation of the characteristics in the place of the critical angles eliminates the singularity of Poincaré. 相似文献
18.
Stefano Casotto 《Celestial Mechanics and Dynamical Astronomy》2018,130(8):52
The Keplerian differential state transition matrix (KDSTM) is a fundamental tool in investigations of the sensitivity of orbital evolution to changes in initial conditions, in perturbation analysis, as well as in targeting and rendezvous operations. Several different forms of the KDSTM are available in the literature. They differ in the choice of state space variables, as well as in derivation methods. Here, a new method for constructing the KDSTM is presented, which is based on the well-known theorem on the differentiability of the solution of a system of ordinary differential equations with respect to initial conditions. A peculiarity of the method is that it allows the direct construction of analytical expressions for both the direct and the inverse fundamental matrices needed to form the KDSTM. The KDSTM is first built in the inertial reference frame and then transformed to the orbital, or Hill reference frame. The resulting expressions contain the full set of Keplerian elements and are hence readily extensible to perturbed Keplerian reference motion. The results are compared with some of the best known KDSTM’s available in the literature, with which they are proven to be fully equivalent, despite their sometimes dramatically different appearance. 相似文献
19.
M. Vitins 《Celestial Mechanics and Dynamical Astronomy》1978,17(2):173-192
An equivalence between Keplerian motion and harmonic oscillations has been established by Burdet by using essentially the true anomaly as a new independent variable. In this paper, a relation between these oscillator equations and the motion of a gyroscope is derived. Important numerical and analytical consequences are discussed. 相似文献
20.
Joseph W. Gangestad George E. Pollock James M. Longuski 《Celestial Mechanics and Dynamical Astronomy》2010,108(2):125-145
A spacecraft that generates an electrostatic charge on its surface in a planetary magnetic field will be subject to a perturbative
Lorentz force. Active modulation of the surface charge can take advantage of this electromagnetic perturbation to modify or
to do work on the spacecraft’s orbit. Lagrange’s planetary equations are derived using the Lorentz force as the perturbation
on a Keplerian orbit, incorporating orbital inclination and true anomaly for the first time for an electrostatically charged
vehicle. The planetary equations reveal that orbital inclination is a second-order effect on the perturbation, explaining
results found in earlier studies through numerical integration. All of the orbital elements are coupled, but the coupling
notably does not depend on the magnitude of the electrostatic charge or on the strength of the magnetic field. Analytical
expressions that characterize this coupling are tested with a propellantless escape example at Jupiter. A closed-form solution
exists that constrains the set of equatorial orbits for which planetary escape is possible, and a sufficient condition is
identified for escape from inclined orbits. The analytical solutions agree with results from the numerically integrated equations
of motion to within a fraction of a percent. 相似文献