共查询到20条相似文献，搜索用时 15 毫秒
1.
New Runge-Kutta algorithms are applied for determining the solution of a system of ordinary differential equations at any point within a given integrating step. In this paper we propose the application of these new algorithms in order to determine, with the smallest possible cost, the exact point of intersection of a symmetric orbit, with the axis or plane of symmetry, which appear in various problems of celestial mechanics. 相似文献
2.
The integration by recurrent power series of certain differential equations occurring in celestial mechanics is shown to be very much more efficient and accurate than that produced by classical one step methods. It is shown that for any such system of differential equations the machine time taken to carry out an integration is a minimum for a certain choice of the number of terms taken in the recurrent power series. In the two-body orbits considered this number is about 15. For the same accuracy criterion the power series is faster than the Runge-Kutta method of the fourth order by a factor which varies between 6 and 15 depending on the eccentricity of the orbit.
相似文献3.
D. G. Bettis 《Celestial Mechanics and Dynamical Astronomy》1973,8(2):229-233
A Runge-Kutta algorithm of order five is presented for the solution of the initial value problem where the system of ordinary differential equations is of second order and does not contain the first derivative. The algorithm includes the Fehlberg step control procedure.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972. 相似文献
4.
Numerical integration methods for orbital motion 总被引：1，自引：0，他引：1
O. Montenbruck 《Celestial Mechanics and Dynamical Astronomy》1992,53(1):59-69
The present report compares Runge-Kutta, multistep and extrapolation methods for the numerical integration of ordinary differential equations and assesses their usefulness for orbit computations of solar system bodies or artificial satellites. The scope of earlier studies is extended by including various methods that have been developed only recently. Several performance tests reveal that modern single- and multistep methods can be similarly efficient over a wide range of eccentricities. Multistep methods are still preferable, however, for ephemeris predictions with a large number of dense output points. 相似文献
5.
Victor Szebehely 《Celestial Mechanics and Dynamical Astronomy》1987,43(1-4):139-145
The concept of finite predictability of gravitational many-body systems is related to the non-deterministic nature of celestial mechanics and of dynamics, in general. The basic, fundamental reasons for the uncertainty of predictions are as follows: (1) the initial conditions are known only approximately since they are obtained either from observations or from approximate computations; (2) the equations of motion given by a selected model describe the actual system only approximately; (3) the physical constants of the dynamical system have error limits; (4) the differential equations of motion are non-integrable and numerical integration methods must be used for solution, generating errors in the final result at every integration step.In addition to these reasons, mostly depending on our techniques, there are some more fundamental reasons depending on the nature of the dynamical system investigated. These are the appearance of regions of instability, non-integrability and chaotic motion.Details, effects and controls of these regions for finite predictability are discussed for various dynamical systems of importance in celestial mechanics with special emphasis on planetary systems. 相似文献
6.
Paul E. Nacozy 《Astrophysics and Space Science》1971,14(1):40-51
The numerical integration of systems of differential equations that possess integrals is often approached by using the integrals to reduce the number of degrees of freedom or by using the integrals as a partial check on the resulting solution, retaining the original number of degrees of freedom.Another use of the integrals is presented here. If the integrals have not been used to reduce the system, the solution of a numerical integration may be constrained to remain on the integral surfaces by a method that applies corrections to the solution at each integration step. The corrections are determined by using linearized forms of the integrals in a least-squares procedure.The results of an application of the method to numerical integrations of a gravitational system of 25-bodies are given. It is shown that by using the method to satisfy exactly the integrals of energy, angular momentum, and center of mass, a solution is obtained that is more accurate while using less time of calculation than if the integrals are not satisfied exactly. The relative accuracy is ascertained by forward and backward integrations of both the corrected and uncorrected solutions and by comparison with more accurate integrations using reduced step-sizes. 相似文献
7.
This paper discusses the formulation and the numerical integration of large systems of differential equations occurring in the gravitational problem ofn-bodies.Different forms of the pertinent differential equations of motion are presented, and various regularizing and smoothing transformations are compared. Details regarding the effectiveness and the efficiency of the Kustaanheimo-Stiefel and of other methods are discussed. In particular, a method is described in which some of the phase variables are treated in the regularized system and others in the ordinary system. This mixed method of numerical regularization offers some advantages.Several numerical integration techniques are compared. A high order Runge-Kutta method, Steffensen's method, and a finite difference method are investigated, especially with regard to their adaptability to regularization.The role of integrals and integral invariants is displayed in controlling the accuracy of the numerical integration.Numerical results are described with 5, 25 and 500 bodies participating. These examples compare the various integration techniques, several regularization methods and different logics in treating binaries. 相似文献
8.
We construct an idealized spherically symmetric relativistic model of an exploding object within the framework of the theory of surface layers in GR. A Vaidya solution for a radially radiating star is matched through a spherical shell of dust to a Schwarzschild solution. The (incomplete) equations for the motion of the spherical shell of dust and the radiation density of the Vaidya solution, as given by the matching conditions, are reduced to a first-order system and a general analysis of the characteristics of the motion is given. This system of differential equations is completed, adding a relation between the unknowns which represents the simplest way to avoid an unphysical singularity in the motion. The results of a numerical integration of the equations are presented in two cases which we think may have some relationship to stellar explosions. A comparative set of results for other solutions is also given, and some possible generalizations of the model are pointed out. 相似文献
9.
Sławomir Breiter 《Celestial Mechanics and Dynamical Astronomy》1996,65(4):345-354
Once the generating function of a Lie-type transformation is known, canonical variables can be transformed numerically by application of a Runge-Kutta type integration method or any other appropriate numerical integration algorithm. The proposed approach avails itself of the fact, that the transformation is defined by a system of differential equations with a small parameter as the independent variable. The integration of such systems arising in the perturbation theories of Hori and Deprit is discussed. The method allows to compute numerical values of periodic perturbations without deriving explicitly the perturbation series. This saving of an algebraic work is achieved at the expense of multiple evaluations of the generator's derivatives. 相似文献
10.
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
11.
The power-limited solar electric propulsion system is considered more practical in mission design. An accurate mathematical model of the propulsion system, based on experimental data of the power generation system, is used in this paper. An indirect method is used to deal with the time-optimal and fuel-optimal control problems, in which the solar electric propulsion system is described using a finite number of operation points, which are characterized by different pairs of thruster input power. In order to guarantee the integral accuracy for the discrete power-limited problem, a power operation detection technique is embedded in the fourth-order Runge-Kutta algorithm with fixed step. Moreover, the logarithmic homotopy method and normalization technique are employed to overcome the difficulties caused by using indirect methods. Three numerical simulations with actual propulsion systems are given to substantiate the feasibility and efficiency of the proposed method. 相似文献
12.
Some new Runge-Kutta and Runge-Kutta-Nystrom algorithms are presented for the solution of ordinary differential equations of the initial value type. The methods are compared with others in integrating the equations of motion of the two body problem and are shown to offer advantages in efficiency. It is also demonstrated that the new methods can be tuned to achieve some measure of global error control. 相似文献
13.
Maria Gousidou-Koutita 《Earth, Moon, and Planets》1985,32(1):21-45
The present study deals with numerical modeling of the elliptic restricted three-body problem as well as of the perturbed elliptic restricted three-body (Earth-Moon-Satellite) problem by a fourth body (Sun). Two numerical algorithms are established and investigated. The first is based on the method of the series solution of the differential equations and the second is based on a 5th-order Runge-Kutta method. The applications concern the solution of the equations and integrals of motion of the circular and elliptical restricted three-body problem as well as the search for periodic orbits of the natural satellites of the Moon in the Earth-Moon system in both cases in which the Moon describes circular or elliptical orbit around the Earth before the perturbations induced by the Sun. After the introduction of the perturbations in the Earth-Moon-Satellite system the motions of the Moon and the Satellite are studied with the same initial conditions which give periodic orbits for the unperturbed elliptic problem. 相似文献
14.
C. N. Frangakis 《Astrophysics and Space Science》1973,23(1):17-42
The long period problem provides the initial conditions for numerical computation of close periodic solutions separated in three categories. For each type of commensurability a number of periodic solutions are computed and their stability is studied by computing the characteristic exponents of the matrizant. The Runge-Kutta method for the solution of differential equations of motion was used in all cases. The results obtained are presented for a four cases of commensurability. 相似文献
15.
Joachim W. Baumgarte 《Celestial Mechanics and Dynamical Astronomy》1976,13(2):247-251
In order to reduce the error growth during a numerical integration, a method of stabilization, of the differential equations of the Keplerian motion is offered. It is characterized by the use of the eccentric anomaly as independent variable in such a way that the time transformation is given by a generalized Lagrange formalism. The control terms in the equations of motion obtained by this modified Lagrangian give immediately a completely Lyapunov-stable set of differential equations. In contrast to other publications, here the equation of time integration is modified by a control term which leads to an integral which defined the time element for the perturbed Keplerian motion.This paper was supported by the National Research Council and the National Aeronautics and Space Administration and also by the Deutsche Forschungsgemeinschaft. It was presented at the Flight Mechanics/Estimation Theory Symposium, Goddard Space Flight Center, Greenbelt, Md., April 15–16, 1975. 相似文献
16.
The idea of using various L-matrices in numerical integration of the regular equations, which describe the motion of small bodies of the Solar System, is developed. The problem of the optimal position of the radius vector and velocity at numerical integration in the KS-coordinate system is posed. The solution of this problem, which reduces the number of calculations of the vector of perturbing accelerations, is given. The transformation providing this optimal solution is suggested, and the results of numerical integration are given. 相似文献
17.
An analytical approach to small amplitude solutions of the extended nearly circular Sitnikov problem
J. Hagel 《Celestial Mechanics and Dynamical Astronomy》2009,103(3):251-266
The model of extended Sitnikov Problem contains two equally heavy bodies of mass m moving on two symmetrical orbits w.r.t the centre of gravity. A third body of equal mass m moves along a line z perpendicular to the primaries plane, intersecting it at the centre of gravity. For sufficiently small distance from the
primaries plane the third body describes an oscillatory motion around it. The motion of the three bodies is described by a
coupled system of second order differential equations for the radial distance of the primaries r and the third mass oscillation z. This problem which is dealt with for zero initial eccentricity of the primaries motion, is generally non integrable and
therefore represents an interesting dynamical system for advanced perturbative methods. In the present paper we use an original
method of rewriting the coupled system of equations as a function iteration in such a way as to decouple the two equations
at any iteration step. The decoupled equations are then solved by classical perturbation methods. A prove of local convergence
of the function iteration method is given and the iterations are carried out to order 1 in r and to order 2 in z. For small values of the initial oscillation amplitude of the third mass we obtain results in very good agreement to numerically
obtained solutions. 相似文献
18.
R. Vilhena de Moraes 《Celestial Mechanics and Dynamical Astronomy》1988,45(1-3):281-284
A semi-analytical method is presented to study the system of differential equations governing the rotational motion of an artificial satellite. Gravity gradient and non gravitational torques are considered. Operations with trigonometric series were performed using an algebraic manipulator. Andoyer's variables are used to describe the rotational motion. The osculating elements are transformed analytically into a mean set of elements. As the differential equations in the mean elements are free of fast frequency terms, their numerical integration can be performed using a large step size. 相似文献
19.
The three-dimensional inverse problem is investigated. A quasi-linear system of partial differential equations is derived for the determination of the potential. The solution of this system is studied by a method of differential geometry. A necessary condition for the solution is derived and the determination of the potential is reduced to algebraic equations written in vectorial form. A few examples are also given. 相似文献
20.
The problem of optimizing the interplanetary trajectories of a spacecraft (SC) with a solar electric propulsion system (SEPS) is examined. The problem of investigating the permissible power minimum of the solar electric propulsion power plant required for a successful flight is studied. Permissible ranges of thrust and exhaust velocity are analyzed for the given range of flight time and final mass of the spacecraft. The optimization is performed according to Portnyagin’s maximum principle, and the continuation method is used for reducing the boundary problem of maximal principle to the Cauchy problem and to study the solution/ parameters dependence. Such a combination results in the robust algorithm that reduces the problem of trajectory optimization to the numerical integration of differential equations by the continuation method. 相似文献