太阳质量损失对流星群轨道改变的长期影响

将作者在变质量天体力学所得理论结果应用于太阳质量损失对流星群轨道根数变化的长期效应上。太阳质量损失包括光子辐射和太阳风造成的质量损失。利用G—M型变质量天体轨道根数变化方程的一阶和二阶解对15个流星群轨道半长轴、近日点距离、轨道周期和近日点经度因太阳质量损失造成的每世纪的长期改变效应做了数值计算,并得出计算结果。其计算结果表明,太阳质量损失使流星群轨道半长轴每世纪的改变效应较明显,它们同太阳距离的扩大影响值得关注,但对轨道周期的拉长每世纪的影响甚小,对近日点经度只有量级变化小到可以略而不计。     

Analytical expansions of torque-free motions for short and long axis modes

Torque-free motion of a rigid body is integrable and its solution is expressed in terms of elliptic functions and elliptic integrals. The conventional analytical expression of the solution, however, is complicated and not suitable for hand-calculation. Recently the rotational motions of small celestial bodies in the solar system are frequently investigated by numerically integrating the equations of motion instead of using the analytical solution, since the numerical evaluation of the analytical and exact solution is a little bit difficult. As the observational accuracy of the rotational motions of the small bodies in the solar system is quite low, what we need for the reduction of these observations are rough estimates of the period of Eulerian motion ( or the free precession period) and the amplitudes of the main periodic terms. Here we give simple analytical expansions of torque-free motions for short- and long-axis modes, which are correct up to the second-order of a small parameter. These expressions include only trigonometric functions and are easily evaluated by hand calculation for estimates of the essential quantities from which we can determine a global rotational motion of the torque-free motion. They can also be used as the zero-th order solution in a perturbation method, when the motion is perturbed by external torques.     

Secular Effects on Orbits of Binary Stars Induced by Temporal Variation of Gravitational Constant (Case for Elliptical Orbit)

In this paper we investigate the influence of a varying gravitation constant on the orbits of celestial bodies. Regarding the eccentric anomaly as an independent variable, we find the solutions to the perturbed equations of motion. In the first order solutions, we find the secular and periodic variations in semi-major axis. For the other orbital elements only periodic variations exhibit. However in the second order solutions, the longitude of periastron and the mean longitude have secular terms. Applying the calculations to six selected binaries, we give the numerical estimations of the variations of orbits. These results are then carefully compared and discussed.     

Poynting-Robertson effect and small eccentric orbits

Perturbation equations of celestial mechanics in application to solar electromagnetic radiation are investigated. Special attention is payed to nearly circular orbis. Results of Klaka and Kaufmannová (1992) are generalized and various initial semi-major axes are taken into acount. Time-averaged (during one period) eccentricity decreases only during the first 25% of the total time of inspiralling toward the Sun. The value 25% hods for all initial values of semi-major axes (initialy circular orbits are supposed). The orbits become more and more eccentric in the subsequent orbital evolution.     

Orbitally stable multistep methods

Lambert and Watson (1976) examine the family of symmetric linear multistep methods for the special second-order initial value problem, and connect the property of symmetry with a property of periodicity. The problems of celestial mechanics may be formulated as second-order initial value problems, but these frequently incorporate the first derivative explicity. It is common for such equations to be reduced to a system of first-order equations. Thus motivated, we utilize ideas from the aforementioned paper to determine the family of linear multistep methods for first-order initial value problems that possess an analogous property of periodicity. This family of orbitally stable methods is illustrated by examining the regularized equations of motion of an artificial earth satellite in an oblate atmosphere.     

The integrable cases of the planetary three-body problem at first-order resonance

This paper investigates the stability of the motion in the averaged planar general three-body problem in the case of first-order resonance. The equations of the averaged motion of bodies near the resonance surface is obtained and is analytically integrated by quadratures. The stability of the averaged motion is analytically investigated in relation to the semi-major axes, the eccentricities and the resonance phases. An autonomous second-order equation is obtained for the deviation of semiaxes from the resonance surface. This equation has an energy integral and is analytically integrated by quadratures. The quasi-periodic dependence on time with two-frequency basis of the averaged motion of bodies is found. The basic frequencies are analytically calculated. With the help of the mean functionals calculated along integral curves of the averaged problem the new analytic first integrals are constructed with coefficients periodic in time. The analytic conditions of librations of resonance phases are obtained.     

Method of ignoring the systematic variations of stellar parallaxes over the celestial sphere in a kinematic analysis of stellar proper motions

A method for determining the velocity field parameters free from the distortions due to the systematic variations of stellar parallaxes over the celestial sphere is proposed. The method is based on the approximation of parallaxes as a function of coordinates on the sphere using spherical harmonics and can be applied in those cases where the solar motion cannot be eliminated from the stellar proper motions. Numerical experiments have shown that our method is able to obtain accurate coordinates of the solar apex and to calculate the kinematic parameters of the Ogorodnikov-Milne model to within three coefficients of the decomposition of parallaxes into first-order spherical harmonics. Examples of applying the method to the stellar proper motions of the Hipparcos catalogue, which admits checking the results using trigonometric parallaxes, are provided. Such a check has been found to yield a positive result only for nearby stars at heliocentric distances that do not exceed 400 pc and for which the parallaxes were determined with a relative error of at least 30%. An interesting feature of this method is the possibility to construct the shape of the figure which is formed by the deviations of the parallaxes from the sphere corresponding to the average parallaxes of the stars under consideration. It should be specially emphasized that all of this is done in the complete absence of information about the stellar parallaxes. The “solar terms” of the stellar proper motions that are formed by the products of the parallaxes by the solar motion components relative to the centroid of stars are the main source of information about the parallaxes here.     

On derivation of EIH (Einstein–Infeld–Hoffman) equations of motion from the linearized metric of general relativity theory

The half-century old idea of Infeld to use the variational principle of the general relativity field equations is reminded to show that the commonly employed EIH (Einstein–Infeld–Hoffman) equations of motion may be derived from the linearized (weak-field) metric alone. Based on it, the linearized metric might be sufficient for post-Newtonian celestial mechanics and astrometry enabling one to derive the post-Newtonian equations of motion and rotation of celestial bodies as well as the post-Newtonian equations of light propagation within the general relativity framework.     

Exchange orbits: an interesting case of co-orbital motion

In this investigation we treat a special configuration of two celestial bodies in 1:1 mean motion resonance namely the so-called exchange orbits. There exist—at least—theoretically—two different types: the exchange-a orbits and the exchange-e orbits. The first one is the following: two celestial bodies are in orbit around a central body with almost the same semi-major axes on circular orbits. Because of the relatively small differences in semi-major axes they meet from time to time and exchange their semi-major axes. The inner one then moves outside the other planet and vice versa. The second configuration one is the following: two planets are moving on nearly the same orbit with respect to the semi-major axes, one on a circular orbit and the other one on an eccentric one. During their dynamical evolution they change the characteristics of the orbit, the circular one becomes an elliptic one whereas the elliptic one changes its shape to a circle. This ‘game’ repeats periodically. In this new study we extend the numerical computations for both of these exchange orbits to the three dimensional case and in another extension treat also the problem when these orbits are perturbed from a fourth body. Our results in form of graphs show quite well that for a large variety of initial conditions both configurations are stable and stay in these exchange orbits.     

Secular influence of change in the heliocentric gravitation constant <Emphasis Type="Italic">GM</Emphasis><Subscript>⊙</Subscript> on evolution of orbits of Meteor Streams

The Secular influence of the change in the heliocentric gravitational constant on the evolution of orbits of Meteor Streams is examined by using the method of celestial mechanics with variable mass and variable gravitational constant. The change in the heliocentric gravitational constant includes the combined changes in the sun’s mass and gravitational constant obtained from the modern observation of planets and spacecraft. The perturbation equations are solved by expanding series with mean anomaly. The solutions of the secular and periodic variation of orbital elements are derived. The theoretical results for the secular variables of the semi-major axes, solar distances at perihelion and orbital periods are given for three Meteor Streams: Dracorids, Quadrantids, and Ursids. The numerical results are shown in Table 2. The discussion and conclusion are drawn.     

Prediction of satellite orbits contraction due to diurnally varying oblate atmosphere and altitude-dependent scale height using KS canonical elements

A new non-singular analytical theory for the motion of near-Earth satellite orbits with the air drag effect is developed in terms of uniformly regular KS canonical elements. Diurnally varying oblate atmosphere is considered with variation in density scale height dependent on altitude. The series expansion method is utilized to generate the analytical solutions and terms up to fourth-order terms in eccentricity and c (a small parameter dependent on the flattening of the atmosphere) are retained. Only two of the nine equations are solved analytically to compute the state vector and change in energy at the end of each revolution, due to symmetry in the equations of motion. The important drag perturbed orbital parameters: semi-major axis and eccentricity are obtained up to 500 revolutions, with the present analytical theory and by numerical integration over a wide range of perigee height, eccentricity and inclination. The differences between the two are found to be very less. A comparison between the theories generated with terms up to third- and fourth-order terms in c and e shows an improvement in the computation of the orbital parameters semi-major axis and eccentricity, up to 9%. The theory can be effectively used for the re-entry of the near-Earth objects, which mainly decay due to atmospheric drag.     

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.     

About Tidal Evolution of Quasi-Periodic Orbits of Satellites

Tidal interactions between Planet and its satellites are known to be the main phenomena, which are determining the orbital evolution of the satellites. The modern ansatz in the theory of tidal dissipation in Saturn was developed previously by the international team of scientists from various countries in the field of celestial mechanics. Our applying to the theory of tidal dissipation concerns the investigating of the system of ODE-equations (ordinary differential equations) that govern the orbital evolution of the satellites; such an extremely non-linear system of 2 ordinary differential equations describes the mutual internal dynamics for the eccentricity of the orbit along with involving the semi-major axis of the proper satellite into such a monstrous equations. In our derivation, we have presented the elegant analytical solutions to the system above; so, the motivation of our ansatz is to transform the previously presented system of equations to the convenient form, in which the minimum of numerical calculations are required to obtain the final solutions. Preferably, it should be the analytical solutions; we have presented the solution as a set of quasi-periodic cycles via re-inversing of the proper ultra-elliptical integral. It means a quasi-periodic character of the evolution of the eccentricity, of the semi-major axis for the satellite orbit as well as of the quasi-periodic character of the tidal dissipation in the Planet.     

The relative motion of two spheroidal rigid bodies

We consider two spheroidal rigid bodies of comparable size constituting the components of an isolated binary system. We assume that (1) the bodies are homogeneous oblate ellipsoids of revolution, and (2) the meridional eccentricities of both components are small parameters.We obtain seven nonlinear differential equations governing simultaneously the relative motion of the two centroids and the rotational motion of each set of body axes. We seek solutions to these equations in the form of infinite series in the two meridional eccentricities.In the zero-order approximation (i. e., when the meridional eccentricities are neglected), the equations of motion separate into two independent subsystems. In this instance, the relative motion of the centroids is taken as a Kepler elliptic orbit of small eccentricity, whereas for each set of body axes we choose a composite motion consisting of a regular precession about an inertial axis and a uniform rotation about a body axis.The first part of the paper deals with the representation of the total potential energy of the binary system as an infinite series of the meridional eccentricities. For this purpose, we had to (1) derive a formula for representing the directional derivative of a solid harmonic as a combination of lower order harmonics, and (2) obtain the general term of a biaxial harmonic as a polynomial in the angular variables.In the second part, we expound a recurrent procedure whereby the approximations of various orders can be determined in terms of lower-order approximations. The rotational motion gives rise to linear differential equations with constant coefficients. In dealing with the translational motion, differential equations of the Hill type are encountered and are solved by means of power series in the orbital eccentricity. We give explicit solutions for the first-order approximation alone and identify critical values of the parameters which cause the motion to become unstable.The generality of the approach is tantamount to studying the evolution and asymptotic stability of the motion.Research performed under NASA Contract NAS5-11123.     

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.     

The effect of the Earth’s oblateness on the Moon’s physical libration in latitude

The Moon’s physical libration in latitude generated by gravitational forces caused by the Earth’s oblateness has been examined by a vector analytical method. Libration oscillations are described by a close set of five linear inhomogeneous differential equations, the dispersion equation has five roots, one of which is zero. A complete solution is obtained. It is revealed that the Earth’s oblateness: a) has little effect on the instantaneous axis of Moon’s rotation, but causes an oscillatory rotation of the body of the Moon with an amplitude of 0.072″ and pulsation period of 16.88 Julian years; b) causes small nutations of poles of the orbit and of the ecliptic along tight spirals, which occupy a disk with a cut in a center and with radius of 0.072″. Perturbations caused by the spherical Earth generate: a) physical librations in latitude with an amplitude of 34.275″; b) nutational motion for centers of small spiral nutations of orbit (ecliptic) pole over ellipses with semi-major axes of 113.850″ (85.158″) and the first pole rotates round the second one along a circle with radius of 28.691″; c) nutation of the Moon’s celestial pole over an ellipse with a semi-major axis of 45.04″ and with an axes ratio of about 0.004 with a period of T = 27.212 days. The principal ellipse’s axis is directed tangentially with respect to the precession circumference, along which the celestial pole moves nonuniformly nearly in one dimension. In contrast to the accepted concept, the latitude does not change while the Moon’s poles of rotation move. The dynamical reason for the inclination of the Moon’s mean equator with respect to the ecliptic is oblateness of the body of the Moon.     

Formulae for the statistical interrelationships between several orbital parameters of particles in the Oort Cloud and other orbiting aggregates

Assuming the motion of particles in an orbiting aggregate (e.g., the Oort Cloud is unperturbed Keplerian, the mean joint density of distance and speed depends only upon the densities of eccentricity and semi-major axis length. We derive a formula for the mean joint density of distance and speed in terms of these densities. Also provided are formulae which, given an observed mean joint density of distance and speed, permit the computation of the corresponding semi-major axis length and eccentricity densities. The results of this paper permit one to derive the structure of an orbiting aggregate given a minimum of information.Box 58421 Houston, Texas 77058     

The formal integrals of the restricted planar problem of the three bodies

The first integrals of motion of the restricted planar circular problem of three bodies are constructed as the formal power series in r^1/2, r being the distance of a moving particle from the primary. It is shown that the coefficients of these series are trigonometric polynomials of an angular variable. Some particular solutions have been found in a closed form. The proposed method for constructing the formal integrals can be generalized to a spatial problem of three bodies.     

Semi-analytical integration algorithms based on the use of several kinds of anomalies as temporal variable

One of the main problems in celestial mechanics is the management of long developments in Fourier or Poisson series used to describe the perturbed motion in the planetary system.In this work we shall develop a software package that is suitable for managing these objects. This package includes algorithms to obtain the inverse of the distance based on an iterative method, a set of integration algorithms according to several sets of temporal variables.This paper contains a comparative study on the use of the true, eccentric, and elliptic anomalies in semi-analytical methods on celestial mechanics.     

Astronomical measurements and coordinate conditions in relativistic celestial mechanics

Determination of dynamical effects from the equations of motion and calculation of ephemerides in terms of measurable quantities on the basis of the equations of light should be performed in one and the same coordinate system. The choice of coordinate system is arbitrary. For illustration we consider coplanar circular motions of the Earth and one of the inner planets in the solar gravitational field described by the generalized three-parametric Schwarzschild metric. Specific values of the metric parameters characterize the adopted gravitational theory, as well as a definite coordinate system (for example, isotropic or standard coordinates). Coordinates of the planets and radii of the orbits are coordinate-dependent quantities and cannot be directly reconciled with measurable quantities such as the round-trip transit times of radar signals or the angular distance between the planet and the distant fixed source (quasar). These ephemeris data may be calculated in terms of the initial measured values independently of the employed coordinate system. Relativistic ephemeris corrections should be taken into account both in radar reflection measurements and astrometric observations.