The transformational behaviour of perturbation theories

The transformational behaviour of Hori's noncanonical perturbation theory (Hori 1971) as well as that of the theory of Krylov-Bogoliubof-Mitropolsky is studied. An integration procedure of the perturbation equations is based on the transformation properties that have been established.     

A keplerian method to estimate perturbations in the restricted three-body problem

We develop a new and fast method to estimate perturbations by a planet on cometary orbits. This method allows us to identify accurately the cases of large perturbations in a set of fictitious orbits. Hence, it can be used in constructing perturbation samples for Monte Carlo simulations in order to maximize the amount of information. Furthermore, the estimated perturbations are found to yield a good approximation to the real perturbation sample. This is shown by a comparison of the perturbations obtained by the new estimator with the results of numerical integration of regularized equations of motion for the same orbits in the same dynamical model: the three-dimensional elliptic restricted three-body problem (Sun-Jupiter-comet).     

Kepler discretization in regular celestial mechanics

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.     

Perturbations in rectangular coordinates by iteration

An investigation has been made on computing orbits with Picard's method of successive approximations. The perturbations are integrated in the form of a general displacement from a fixed Keplerian reference orbit. Several variation-of-parameters methods are obtained for the integration of the displacement equation. These variation-of-parameters methods could be used as special perturbation or general perturbation methods. The present paper investigates the applications as iterative numerical perturbation techniques. Four different formulations are proposed. They have been implemented on a computer with Chebychev series and their respective advantages and disadvantages are analyzed. Connections with other known perturbation methods are also described.This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by the National Aeronautics and Space Administration.     

An intermediate matching technique for solving two point boundary value problems using the perturbation method

The perturbation method, a numerical method for solving two point boundary value problems (TPBVP), is modified to attempt to improve inherent instability and sensitivity problems associated with the method. The desired solution to the TPBVP is divided into two time intervals. The differential equations required to define a solution to the two point boundary value problem are integrated independently over these shorter segments rather than consecutively over the entire trajectory. The independent integration of the differential equations over approximately half of the trajectory instead of the entire trajectory substantially decreases sensitivity and stability properties associated with the numerical integration. The equations for both time segments can be integrated simultaneously. By this procedure, a system of twice the dimension of the original problem is integrated for a period of time equal to half of the time interval for the original problem. To show the effectiveness of the method, two impulse trajectories which minimize the total velocity increment required to transfer a spacecraft from an Earth orbit into a lunar orbit are calculated.     

倾角函数及其导数的定积分计算方法

给出了一种倾角函数及其导数的定积分计算方法,表达式十分简单,其计算精度:倾角函数可达10^-15,导数可达10^-13,可与Gooding方法相媲美.该方法的稳定性和适用倾角范围均较好,可供倾角函数的最高阶数L_max≤50时使用.     

Regularization of Equations of Motion of Celestial Bodies. 2. Correction of the KS-Coordinate System in Numerical Integration

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.     

On the Numerical Stability of Some Symplectic Integrators

In this paper, we analyze the linear stabilities of several symplectic integrators, such as the first-order implicit Euler scheme, the second-order implicit mid-point Euler difference scheme, the first-order explicit Euler scheme, the second-order explicit leapfrog scheme and some of their combinations. For a linear Hamiltonian system, we find the stable regions of each scheme by theoretical analysis and check them by numerical tests. When the Hamiltonian is real symmetric quadratic, a diagonalizing by a similar transformation is suggested so that the theoretical analysis of the linear stability of the numerical method would be simplified. A Hamiltonian may be separated into a main part and a perturbation, or it may be spontaneously separated into kinetic and potential energy parts, but the former separation generally is much more charming because it has a much larger maximum step size for the symplectic being stable, no matter this Hamiltonian is linear or nonlinear.     

Propagation of two dimensional cylindrical fast magnetoacoustic solitary waves in a warm dust plasma

A set of multi-fluid equations and Maxwell’s equations are carried out to investigate the properties of nonlinear fast magnetoacoustic solitary waves with the combined effects of dusty plasma pressure and transverse perturbation in the bounded cylindrical geometry. The reductive perturbation method has been applied to the dynamical system causeway and the derived two dimensional cylindrical Kadomtsev–Petviashvili equation (CKP) predicts different natures of solitons in complex plasma. Under a suitable coordinate transformation the CKP equation can be solved analytically. The change in the soliton structure due to mass of dust, ion temperature, ion density, and dust temperature is studied by numerical calculation of the CKP equation. It is noted that the dust cylindrical fast magnetoacoustic solitary waves in warm plasmas may disappear slowly because of an increase in dust mass. The present analysis could be helpful for understanding the nonlinear ion-acoustic solitary waves propagating in interstellar medium and pulsar wind,which contain an excess of superthermal particles.     

Position and velocity perturbations for the determination of geopotential from space geodetic measurements

Although space geodetic observing systems have been advanced recently to such a revolutionary level that low Earth Orbiting (LEO) satellites can now be tracked almost continuously and at the unprecedented high accuracy, none of the three basic methods for mapping the Earth’s gravity field, namely, Kaula linear perturbation, the numerical integration method and the orbit energy-based method, could meet the demand of these challenging data. Some theoretical effort has been made in order to establish comparable mathematical modellings for these measurements, notably by Mayer-Gürr et al. (J Geod 78:462–480, 2005). Although the numerical integration method has been routinely used to produce models of the Earth’s gravity field, for example, from recent satellite gravity missions CHAMP and GRACE, the modelling error of the method increases with the increase of the length of an arc. In order to best exploit the almost continuity and unprecedented high accuracy provided by modern space observing technology for the determination of the Earth’s gravity field, we propose using measured orbits as approximate values and derive the corresponding coordinate and velocity perturbations. The perturbations derived are quasi-linear, linear and of second-order approximation. Unlike conventional perturbation techniques which are only valid in the vicinity of reference mean values, our coordinate and velocity perturbations are mathematically valid uniformly through a whole orbital arc of any length. In particular, the derived coordinate and velocity perturbations are free of singularity due to the critical inclination and resonance inherent in the solution of artificial satellite motion by using various types of orbital elements. We then transform the coordinate and velocity perturbations into those of the six Keplerian orbital elements. For completeness, we also briefly outline how to use the derived coordinate and velocity perturbations to establish observation equations of space geodetic measurements for the determination of geopotential.     

Using the transition to action variables of the newtonian problem in the numerical solution of the N-body problem

We propose an approach for overcoming the problem of close encounters in collisional systems, globular and open star clusters. As is well known, the numerical integration step in such systems, for example, during the formation of close binary stars, begins to fragment and the rate of calculations goes down to a complete stop. We show that using the perturbation theory in the proposed approach, one can isolate the singularity and to increase considerably the integration step without losing the physical effects that affect significantly the evolution of star clusters.     

A numerical method to determine the electromagnetic field of the pulsar magnetosphere with inclined magnetic moment

A numerical method to determine the electromagnetic field of a steadily rotating magnetosphere with an inclined magnetic moment under a given boundary condition on an arbitrary shaped boundary surface is presented. The region may include the light cylinder. The present method, together with a companion method giving particle motion and creation, makes an iterative scheme to obtain a global model of the pulsar magnetosphere. A key problem for explaining the particle acceleration in pulsars is to solve field-aligned electric field in an accelerating region bounded by an ideal-MHD region. The present method is fit to connect a solution for the non-ideal-MHD region with another solution for the ideal-MHD region on a boundary surface whose location should also be solved (i.e., a floating boundary). The integration scheme is based on the boundary element method and it has great advantage as compared with other methods like the finite difference method and the Fourier transformation method.     

Stabilization by modification of the Lagrangian

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.     

A Monte Carlo investigation of Jovian perturbations on short-period comet orbits

We present statistical distributions of Jovian perturbations on short-period comet orbits resulting from accurate numerical integrations. Our sample of 60, 000 cometary orbits with low inclinations and random orientations is characterized by perihelia between 0 and 7 AU and aphelia between 4 and 13 AU. The perturbations considered are those experienced because of Jupiter's gravitation per orbital revolution by the comets. Regularization and accurate step-length control in the numerical integration gives statistical results appreciably different from those computed by Rickman and Vaghi (1978). Their use of a crude method of integration led to erroneous results for close encounters. Strong asymmetries of the $\text{δ(}\text{1}\text{a}\text{)}$ distributions, in particular for the extreme tails, are observed for perihelion- or aphelion-tangent orbits. These orbits are also shown to experience the strongest energy perturbations on the average. Some results concerning the perturbations of Tisserand parameters are indicated. The perturbation distributions for the angular elements are described and discussed. The role of the minimum distance from Jupiter as an indicator of perturbations is investigated.     

On the use of the autonomous Birkhoff equations in Lie series perturbation theory

In this article, we present the Lie transformation algorithm for autonomous Birkhoff systems. Here, we are referring to Hamiltonian systems that obey a symplectic structure of the general form. The Birkhoff equations are derived from the linear first-order Pfaff–Birkhoff variational principle, which is more general than the Hamilton principle. The use of 1-form in formulating the equations of motion in dynamics makes the Birkhoff method more universal and flexible. Birkhoff’s equations have a tensorial character, so their form is independent of the coordinate system used. Two examples of normalization in the restricted three-body problem are given to illustrate the application of the algorithm in perturbation theory. The efficiency of this algorithm for problems of asymptotic integration in dynamics is discussed for the case where there is a need to use non-canonical variables in phase space.     

Motion near the 3/1 resonance of the planar elliptic restricted three body problem

The global semi-numerical perturbation method proposed by Henrard and Lemaître (1986) for the 2/1 resonance of the planar elliptic restricted three body problem is applied to the 3/1 resonance and is compared with Wisdom's perturbative treatment (1985) of the same problem. It appears that the two methods are comparable in their ability to reproduce the results of numerical integration especially in what concerns the shape and area of chaotic domains. As the global semi-numerical perturbation method is easily adapted to more general types of perturbations, it is hoped that it can serve as the basis for the analysis of more refined models of asteroidal motion. We point out in our analysis that Wisdom's uncertainty zone mechanism for generating chaotic domains (also analysed by Escande 1985 under the name of slow Hamiltonian chaotic layer) is not the only one at work in this problem. The secondary resonance _p = 0 plays also its role which is qualitatively (if not quantitatively) important as it is closely associated with the random jumps between a high eccentricity mode and a low eccentricity mode.     

Explicit adaptive symplectic integrators for solving Hamiltonian systems

We consider Sundman and Poincaré transformations for the long-time numerical integration of Hamiltonian systems whose evolution occurs at different time scales. The transformed systems are numerically integrated using explicit symplectic methods. The schemes we consider are explicit symplectic methods with adaptive time steps and they generalise other methods from the literature, while exhibiting a high performance. The Sundman transformation can also be used on non-Hamiltonian systems while the Poincaré transformation can be used, in some cases, with more efficient symplectic integrators. The performance of both transformations with different symplectic methods is analysed on several numerical examples.     

An extended canonical perturbation method

In this investigation, a procedure is described for extending the application of canonical perturbation theories, which have been applied previously to the study of conservative systems only, to the study of non-conservative dynamical systems. The extension is obtained by imbedding then-dimensional non-conservative motion in a 2n-dimensional space can always be specified in canonical form, and, consequently, the motion can be studied by direct application of any canonical perturbation method. The disadvantage of determining a solution to the 2n-dimensional problem instead of the originaln-dimensional problem is minimized if the canonical transformation theory is used to develop the perturbation solution. As examples to illustrate the application of the method, Duffing's equation, the equation for a linear oscillator with cubic damping and the van der Pol equation are solved using the Lie-Hori perturbation algorithm.This research was supported by the Office of Naval Research under Contract N00014-67-a-0126-0013.     

Partial derivatives used in trajectory estimation

The Peano-Baker method is applied to the integration of the variational equations to produce the partial derivatives used in satellite navigation. In this method the analytic form of the state transition partial derivatives can be factored so that numerical integration is applied only to the departures from a simplified analytical model.The advantage of using the Peano-Baker approach rather than direct integration of the variational equations is that with the Peano-Baker method numerical integration can be performed adequately with low order formulae and relatively large step sizes. Numerical results are indicated.     

A velocity scaling method with least-squares correction of several constraints

A new scale transformation to the integrated velocity vector is designed to monitor the accumulation of numerical errors in several integrals of motion. The scale factor is derived from the least-squares correction that minimizes the sum of the squares of the errors of these integrals. In order to preserve an invariant, we employ the velocity scaling method for rigorously satisfying the constraint. When adjusting many constants, the new scheme like other existing methods is valid to typically reduce the integration errors below those of an uncorrected integrator. Via integral invariant relations, the new method is also able to treat slowly-varying quantities, such as the Keplerian energy and the Laplace vector, for a perturbed Keplerian problem or each of multiple bodies in the solar system dynamics. Consequently it does nearly agree with the rigorous dual scaling method in the sense of drastically improving the integration accuracy. As one of its advantages, the implementation of the new method is significantly easier than that of other methods. In particular, the method can be simply applied to a complicated dynamical system with some constraints.