Special perturbations employing osculating reference states

The concept of employing osculating reference position and velocity vectors in the numerical integration of the equations of motion of a satellite is examined. The choice of the reference point is shown to have a significant effect upon numerical efficiency and the class of trajectories described by the differential equations of motion. For example, when the position and velocity vectors on the osculating orbit at a fixed reference time are chosen, a universal formulation is yielded. For elliptical orbits, however, this formulation is unattractive for numerical integration purposes due to Poisson terms (mixed secular) appearing in the equations of motion. Other choices for the reference point eliminate this problem but usually at the expense of universality. A number of these formulations, including a universal one, are considered here. Comparisons of the numerical characteristics of these techniques with those of the Encke method are presented.     

Periodic solution of the nonlinear Sitnikov restricted three-body problem

The objective of this paper is to find periodic solutions of the circular Sitnikov problem by the multiple scales method which is used to remove the secular terms and find the periodic approximated solutions in closed forms. Comparisons among a numerical solution (NS), the first approximated solution (FA) and the second approximated solution (SA) via multiple scales method are investigated graphically under different initial conditions. We observe that the initial conditions play a vital role in the numerical and approximated solutions behaviour. The obtained motion is periodic, but the difference of its amplitude is directly proportional with the initial conditions. We prove that the obtained motion by the numerical or the second approximated solutions is a regular and periodic, when the infinitesimal body starts its motion from a nearer position to the common center of primaries. Otherwise when the start point distance of motion is far from this center, the numerical solution may not be represent a periodic motion for along time, while the second approximated solution may present a chaotic motion, however it is always periodic all time. But the obtained motion by the first approximated solution is periodic and has regularity in its periodicity all time. Finally we remark that the provided solutions by multiple scales methods reflect the true motion of the Sitnikov restricted three–body problem, and the second approximation has more accuracy than the first approximation. Moreover the solutions of multiple scales technique are more realistic than the numerical solution because there is always a warranty that the motion is periodic all time.     

Numerical Integration of Regular Equations of the Planar Motion of Terrestrial Bodies

The parameters of L matrices are applied to the numerical integration of regular equations describing the motion of minor bodies in the Solar System. The problem of the optimal choice of the regularizing change of variables is formulated in the context of the numerical integration of the equations of motion using the Runge–Kutta–Fehlberg method. Arbitrary perturbations are taken into account. This problem is completely solved in the case of planar motion. The solution of the optimization problem reduces the amount of computations needed to determine the vector of perturbing accelerations. Results of numerical integrations are given.     

Orbital Evolution of the Sun–Jupiter–Saturn–Uranus–Neptune Four-Planet System on Long-Time Scales

The four-planet problem is solved by constructing an averaged semi-analytical theory of secondorder motion by planetary masses. A discussion is given of the results obtained by numerical integration of the averaged equations of motion for the Sun–Jupiter–Saturn–Uranus–Neptune system over a time interval of 10 Gyr. The integration is based on high-order Runge–Kutta and Everhart methods. The motion of the planets is almost periodic in nature. The eccentricities and inclinations of the planetary orbits remain small. Short-period perturbations remain small over the entire interval of integration. Conclusions are drawn about the resonant properties of the motion. Estimates are given for the accuracy of the numerical integration.     

Supernova remnants and expanding supershells in inhomogeneous moving medium

We present the results of calculations of blast waves propagation in plane-stratified medium, taking into account gravitational forces, pre-explosion gas motion and different laws of energy input into the cavity. The numerical hydrodynamic scheme based on infinitely thin-layer approximation is described. The results are compared with two-dimensional axisymmetric numerical calculations. The evolution of a superbubble in plane-stratified galactic disk is discussed. It is shown that late SNR structure is highly dependent on pre-supernova-interstellar gas relative motion.     

Numerical determination of stability regions for orbital motion in uniformly rotating second degree and order gravity fields

The stability of orbital motion about a uniformly rotating arbitrary second degree and order gravity field is investigated. A normalized form of the equations of motion are derived and analyzed. A numerical stability criteria is proposed and used to evaluate the stability of initially near-circular orbits in the equatorial plane of the body. Regions of stable and unstable motion are clearly delineated, and are seen to be strongly related to resonances between the mean motion and the body rotation rate.     

Adiabatic consideration of the motion of charged particles in superposed dipole and uniform magnetic fields

The motion of a charged particle is studied within a magnetic field. This field consists of two separate fields; a dipole and a uniform magnetic field, parallel to dipole's magnetic moment. The present study is maintained by means of the adiabatic theory. We use a numerical integration of the equations of motion and give comparative results between the adiabatic theory and the numerical integration. The previous results are applied to the case of the Earth's open magnetosphere. Diagrams and tables support this application.     

Elements of spin motion

For use in numerical studies of rotational motion, a set of elements is introduced for the torque-free rotational motion of a rigid body around its barycenter. The elements are defined as the initial values of a modification of the Andoyer canonical variables. A computational procedure is obtained for determining these elements from the combination of the spin angular momentum vector and a triad defining the orientation of the rigid body. A numerical experiment shows that the errors of transformation between the elements and variables are sufficiently small. The errors increase linearly with time for some elements and quadratically for some others.     

Time regularization of an Adams-Moulton-Cowell algorithm

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.     

Accurate Analytical Calculation of Effects of Rorations of the Central Planet on a Staellite's Orbit

Satellite orbital perturbations due to many rotations of the planet-fixed reference frame are calculated by a general analytical method. For the International Terrestrial Reference Frame (ITRF) the effects of the Earth irregular rotation, precession, nutation, and polar motion are considered. Gravity coefficients of the Earth potential expansion are expressed in an inertial Celestial Reference Frame (CRF) as functions of the set of standard constant coefficients derived in the ITRF and of the rotation angles between the CRF and ITRF. The analytical motion theory uses time dependent gravity coefficients, and the Lagrange motion equations are integrated in the CRF, as it is done by numerical methods. Comparison of the proposed analytical method with a numerical one is presented. Motion of the ETALON-1 geodetic satellite perturbed by the geopotential (36*36) and by the full effects of the Earth irregular rotation, precession, nutation and polar motion is predicted. The r.m.s. difference between the satellite's coordinates calculated by both methods over a year interval is 2 cm. This revised version was published online in July 2006 with corrections to the Cover Date.     

任意倾角的共轨运动研究

共轨运动天体与摄动天体的半长径相同,处于1:1平运动共振中.太阳系内多个行星的特洛伊天体即为处于蝌蚪形轨道的共轨运动天体,其中一些高轨道倾角特洛伊天体的轨道运动与来源仍未被完全理解.利用一个新发展的适用于处理1:1平运动共振的摄动函数展开方式,对三维空间中的共轨运动进行考察,计算不同初始轨道根数情况下共轨轨道的共振中心、共振宽度,分析轨道类型与初始轨道根数的关系.并将分析方法所得结果与数值方法的结果相互比较验证,得到了广阔初始轨道根数空间内共轨运动的全局图景.     

Analytical description of charged particle motion in a reconnecting current sheet

We have obtained an analytical solution to the equation of motion in the guiding center approximation for nonrelativistic charged particles in a reconnecting current sheet with a three-component magnetic field. Given the electric field attributable to magnetic reconnection, the solution describes stable and unstable three-dimensional particle orbits. We have found the domain of input parameters at which the motion is stable. A physical interpretation of the processes affecting the stability of the motion is given. Charge separation is shown to take place in the sheet during the motion: oppositely charged particles are localized mostly in different regions of the current sheet. A formula is derived for the particle energy in stable and unstable orbits. The results obtained by numerical and analytical methods are compared.     

Determining the nature of orbits in a three‐dimensional galaxy model hosting a BL Lacertae object

A three‐dimensional dynamical model for a galaxy hosting a BL Lacertae object is constructed. The model consists of a logarithmic potential representing an elliptical host galaxy with a bulge of radius cb and a dense massive nucleus. Using numerical experiments, we try to distinguish between regular and chaotic motion in both 2D and 3D system. In particular, we investigate how the basic parameters of our model, such as the mass of the nucleus, the internal perturbation and the flattening parameters influence the amount and the degree of chaos. Interesting correlations are presented for both 2D and 3D dynamical models. Our numerical results are explained and supported using elementary theoretical arguments and analytical calculations. Of particular interest is the local integral of motion which have been found to exist in the vicinity of stable periodic points. The obtained numerical outcomes of the present research are linked and also compared with several data derived from observations. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Keplerian motion and gyration

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.     

A semi-analytical method to study perturbed rotational motion

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.     

Construction of the semi-analytical and numerical solutions to the problem of rotational motion of the moon

New high-precision, semianalytical and numerical solutions to the problem of the rotational motion of the Moon are obtained, for use in the long 418.9-year time frame. The dynamics of the rotational motion of the Moon is studied numerically using the Rodrigues-Hamilton parameters, relative to the fixed ecliptic for the epoch J2000. The results of the numerical solution to the problem under study are compared with a compiled semianalytical theory of Moon rotation (SMR). The initial conditions for the numerical integration have been taken from the SMR. The comparative discrepancies derived from the comparison between the numerical solutions and the SMR do not exceed 1.5″ on the time-scale of 418.9 yr. The investigation of the comparative discrepancies between the numerical and semianalytical solutions is performed using the least squares and spectral analysis methods in the Newtonian case. All the periodic terms describing the behavior of the comparative discrepancies are interpreted as the corrections to the semianalytical SMR theory. As a result, the series are constructed to describe the rotation of the Moon (MRS2010) in the time interval under study. The numerical solution for the Moon’s rotation has been obtained anew, with new initial conditions calculated using MRS2010. The discrepancies between the new numerical solution and MRS2010 do not exceed 20 arc milliseconds on the time-scale of 418.9 years. The results of the comparison suggest that that the MRS2010 series describe the rotation of the Moon more correctly than the SMR series.     

Effects of random solar-wind variations in the motion of small particles in the solar system

The evolution of the orbit of a small particle affected by regular and irregular components of the solar wind is examined. If the irregularity is taken into account, the pattern of motion may qualitatively change on large time scales, because the general integrals of motion are not conserved. The diffusion along the eccentricity of the orbit is most important. In certain cases, it can lead to the escape of a particle from the solar region. Corresponding numerical estimates are given.     

Efficient numerical routines to find the averaged motion of small dust particles under the Poynting-Robertson force

The orbit-averaged differential equations of motion of dust particles under gravity, radiation pressure and Poynting-Robertson drag were given by Wyatt and Whipple (1950). An integral of motion enables the system of two equations in semi-major axis a and eccentricity e to be reduced to one equation, the solution of which is presented here in terms of analytical formulae. An efficient numerical algorithm to compute the solution is given. Listings of two FORTRAN routines are included.     

Analytical theory of motion of phoebe,the ninth satellite of saturn

The literal solution of the restricted three body problem obtained by the authors up to the eleventh order with respect to the minor parameter is applied to the investigation of the motion of Phoebe, the ninth satellite of Saturn. As distinct from the existing analytical theories of the motion of the satellite, in the present paper the planetary perturbations are taken into account. A comparison with the modern numerical theory of the motion of Phoebe has shown that the new analytical theory of the satellite motion represents observations with the same degree of accuracy.     

Low-eccentricity motion of asteroids near the 2/1 Jovian resonance

(903) Nealley moves on an orbit of low eccentricity with a mean motion that is slightly larger than the 2/1 value of resonance. This orbit and some related fictious orbits are studied by numerical integrations of the four-body problem Sun-Jupiter-Saturn-asteroid over an interval of 110000 yr. The author's experience on related cases of resonance allows a study of the variation of suitably defined orbital parameters. The long-term evolution of the orbits is compared with earlier predictions. Some of the librating orbits are temporarily captured in a secondary resonance that refers to three-dimensional motion and is demonstrated by a special example.