On Relativistic Equations of Motion of an Earth Satellite

For numerical integration of the geocentric equations of motion of Earth satellites in the general relativity framework one may choose now between rather simple equations involving in their relativistic dynamical part only the Earth-induced terms and very complicated equations taking into account the relativistic third-body action. However, it is possible quite easily to take into account the relativistic indirect third-body perturbations and to neglect much lesser direct third-body perturbations. Such approach is based on the use of the Newtonian third-body perturbations in geocentric variables with expressing them in the relativistic manner in terms of the barycentric arguments. Together with it, to extend the known results for the spheroid model of the Earth, the Earth-induced terms are treated in great detail by including the non-spin part of the Earth vector-potential and the Earth triaxial non-sphericity.This revised version was published online in October 2005 with corrections to the Cover Date.     

Jerk in Planetary Systems and Rotational Dynamics,Nonlocal Motion Relative to Earth and Nonlocal Fluid Dynamics in Rotating Earth Frame

Some results following from the implications of nonlocal-in-time kinetic energy approach introduced recently by Suykens in the framework of rotational dynamics and motion in a non-inertial frame are discussed. Their roles in treating aspects concerning the nonlocal motion relative to Earth, the free-fall problem, the Foucault pendulum and the motion of a massive body in a rotating tube are analyzed. Governing nonlocal equations of fluid dynamics in particular the nonlocal-in-time Navier–Stokes equations are constructed under the influence of Earth rotation. Their properties are analyzed and a number of features were revealed and discussed accordingly.     

弹性地球各种几何轴和物理轴的定义

基于经典的弹性地球自转动力学理论,建立了极移和章动的联合动力学方程。由此给出了弹性地球各种几何轴和物理轴(Tisserand轴、自转轴、瞬时形状轴、角动量轴、CEP和CIP轴)的极移、岁差章动的动力学方程,明确了各种轴的定义及其之间的理论关系。理论研究表明,联合动力学方程要比经典动力学方程综合性强易于理解,可同时求解极移和章动,特别是在文[1]理论中出现的倾斜模(TOM),在此只是作为了一个特解而存在。     

Relativistic effects for near-earth satellite orbit determination

The relativistic formulations for the equations which describe the motion of a near-Earth satellite are compared for two commonly used coordinate reference systems (RS). The discussion describes the transformation between the solar system barycentric RS and both the non-inertial and inertial geocentric RSs. A relativistic correction for the Earth's geopotential expressed in the solar system barycentric RS and the effect of geodesic precession on the satellite orbit in the geocentric RS are derived in detail. The effect of the definition of coordinate time on scale is also examined. A long-arc solution using 3 years of laser range measurements of the motion of the Lageos satellite is used to demonstrate that the effects of relativity formulated in the geocentric RS and in the solar system barycentric RS are equivalent to a high degree of accuracy.     

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.     

Approximate Analytical Theory of Satellite Orbit Prediction Presented in Spherical Coordinates Frame

A comparative review of analytic theories for the motion of Earth satellites in quasi-circular orbits written in the spherical coordinate frame is presented. The theory of motion is developed for satellites in quasi-circular and quasi-equatorial orbits subjected to geopotential, luni-solar and solar radiation pressure force perturbations. The intermediate orbit is Keplerian and the equations of motion are solved by the Lyapunov–Poincaré small parameter method. Both resonant and non-resonant cases are considered. The results can be useful for the development of a complete theory of weakly eccentric orbits.     

The equations of relative motion in the orbital reference frame

The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill–Clohessy–Wiltshire equations. Circular motion is not, however, a solution when the Earth’s flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the \(J_2\) effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the \(J_2\) perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a \(J_2\)-perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill–Clohessy–Wiltshire equations for circular reference motion, or the de Vries/Tschauner–Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the \(J_2\) perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession of the node and the argument of perigee, i.e., of the entire orbital plane. Here we provide a derivation of the exact equations of relative motion by expressing the angular velocity of the RTN frame in terms of the state vector of the reference spacecraft. As such, these equations are completely general, in the sense that the orbit of the reference spacecraft need only be known through its ephemeris, and therefore subject to any force field whatever. It is also shown that these equations reduce to either the Hill–Clohessy–Wiltshire, or the Tschauner–Hempel equations, depending on the level of approximation. The explicit form of the equations of relative motion with respect to a \(J_2\)-perturbed reference orbit is also introduced.     

Analytical expressions of the Earth's position and velocity for the calculation of apparent positions

In order to calculate the apparent places of stars, the barycentric and heliocentric positions of the Earth and the barycentric velocity of the Earth are required. It is shown that several tens of trigonometric terms for their expressions are sufficient for analyses of optical observations which have an observational accuracy of the order of 0.001. We also give the analytical expressions of the position and velocity of the Earth with higher precision so that we can use them in the analyses of precise astrometry such as VLBI observations.     

A note on relative motion in the general three-body problem

It is shown that the equations of the general three-body problem take on a very symmetric form when one considers only their relative positions, rather than position vectors relative to some given coordinate system. From these equations one quickly surmises some well known classical properties of the three-body problem such as the first integrals and the equilateral triangle solutions. Some new Lagrangians with relative coordinates are also obtained. Numerical integration of the new equations of motion is about 10 percent faster than with barycentric or heliocentric coordinates.     

1988至1992各年度IERS综合河外射电源表的重新归算

鉴于ＩＥＲＳ综合河外射电源表在建立ＩＡＵ协议天球参考架方面的重要意义和它们的不足之处，本文重新组建了１９８８至１９９２各年度的综合河外射电源表，称之为Ｃａｔ系列表。并依据综合表的组建原理，对重新计算得到的Ｃａｔ系列表进行了检验，结果表明Ｃａｔ系列表克服了ＩＥＲＳ系列表中坐标系不统一的缺点，指向的定义及维持亦较之ＩＥＲＳ系列表有较大程度地改观。比较结果显示，ＩＥＲＳ系列表中部分源的坐标存在毫角秒量级的偏差，各年度综合表的指向与ＩＥＲＳ预期指向亦有不同程度的偏离，个别轴的最大指向偏差至０．６ｍａｓ。同时指出，１９９３年ＩＥＲＳ综合表的指向亦有待讨论。     

Angular momentum and gyroscope in flat space-time theory of gravitation

The study of a previously proposed theory of gravitation in flat space-time (Petry, 1981a) is continued. A conservation law for the angular momentum is derived. Additional to the usual form, there must be added a term coming from the spin of the gravitational field. The equations of motion and of spin angular momentum for a spinning test particle in a gravitational field are given. An approximation of the equations of the spin angular momentum in the rest frame of the test particle is studied. For a gyroscope in an orbit of a rotating massive body (e.g., the Earth) the precession of the spin axis agrees with the result of Einstein's general theory of relativity.     

The rotational motion of an earth orbiting gyroscope according to the Einstein theory of general relativity

The classical Poisson equations of rotational motion are used to study the attitude motions of an Earth orbiting, rapidly spinning gyroscope perturbed by the effects of general relativity (Einstein theory). The center of mass of the gyroscope is assumed to move about a rotating oblate Earth in an evolving elliptic orbit which includes all first-order oblateness effects produced by the Earth.A method of averaging is used to obtain a transformation of variables, for the nonresonance case, which significantly simplifies the Poisson differential equations of motion of the gyroscope. Longterm solutions are obtained by an exact analytical integration of the simplified transformed equations. These solutions may be used to predict both the orientation of the gyroscope and the motion of its rotational angular momentum vector as viewed from its center of mass. The results are valid for all eccentricities and all inclinations not near the critical inclination.This paper represents a part of the author's Ph. D. dissertation for the Mathematics Department, Auburn University.     

Motion of a space station. I

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.     

Kinematical modeling of the Earth rotation,focusing on the Oppolzer terms in a rigid Earth and the Oppolzer-like terms in an elastic Earth

Under perturbations from outer bodies, the Earth experiences changes of its angular momentum axis, figure axis and rotational axis. In the theory of the rigid Earth, in addition to the precession and nutation of the angular momentum axis given by the Poisson terms, both the figure axis and the rotational axis suffer forced deviation from the angular momentum axis. This deviation is expressed by the so-called Oppolzer terms describing separation of the averaged figure axis, called CIP (Celestial Intermediate Pole) or CEP (Celestial Ephemeris Pole), and the mathematically defined rotational axis, from the angular momentum axis. The CIP is the rotational axis in a frame subject to both precession and nutation, while the mathematical rotational axis is that in the inertial (non-rotating) frame. We investigate, kinematically, the origin of the separation between these two axes—both for the rigid Earth and an elastic Earth. In the case of an elastic Earth perturbed by the same outer bodies, there appear further deviations of the figure and rotational axes from the angular momentum axis. These deviations, though similar to the Oppolzer terms in the rigid Earth, are produced by quite a different physical mechanism. Analysing this mechanism, we derive an expression for the Oppolzer-like terms in an elastic Earth. From this expression we demonstrate that, under a certain approximation (in neglect of the motion of the perturbing outer bodies), the sum of the direct and convective perturbations of the spin axis coincides with the direct perturbation of the figure axis. This equality, which is approximate, gets violated when the motion of the outer bodies is taken into account.     

Relative motion of satellites exploiting the super-integrability of Kepler’s problem

This paper builds upon the work of Palmer and Imre exploring the relative motion of satellites on neighbouring Keplerian orbits. We make use of a general geometrical setting from Hamiltonian systems theory to obtain analytical solutions of the variational Kepler equations in an Earth centred inertial coordinate frame in terms of the relevant conserved quantities: relative energy, relative angular momentum and the relative eccentricity vector. The paper extends the work on relative satellite motion by providing solutions about any elliptic, parabolic or hyperbolic reference trajectory, including the zero angular momentum case. The geometrical framework assists the design of complex formation flying trajectories. This is demonstrated by the construction of a tetrahedral formation, described through the relevant conserved quantities, for which the satellites are on highly eccentric orbits around the Sun to visit the Kuiper belt.     

Non-linear equations for the rotation of a viscoelastic planet taking into account the influence of a liquid core

Non-linear equations governing the temporal evolution of the vector of instantaneous rotation are developed for an Earth with a homogeneous mantle having a viscoelastic Maxwell rheology and with a homogeneous inviscid fluid core.This general theory is investigated using the angular momentum theorem applied to the coupled core-mantle system. It allows to study the influence upon the planetary rotation of a quasi-rigid rotational motion in the liquid core. It also enables to investigate the consequences of excitation sources (e.g. pressure), located at the core-mantle interface. Especially, the influence of viscoelastic variations in the inertia tensors resulting from the rotation itself or from various excitation sources are detailed with the help of a Love number formalism. The equations of the linear theory for an elastic Earth with a liquid core, and the non-linear theory for a viscous planet with a quasi-fluid behavior are shown to be particular cases of our generalized system of equations. Some planetological applications may be derived from the quasi-fluid approximation.     

Application of the stroboscopic method to the stellar three-body problem

The stellar three-body problem has been approached by directly integrating the equations of motion in the orbital elements. The problem is set up in a barycentric chain with the orbits as perturbed ellipses.The integration was performed using the semi-analytical stroboscopic method, which is particularly useful for solving differential systems that depend on several slow variables and one fast, angular variable. Perturbation theory is applied and the solution for a particular order is obtained by way of successive approximations on the fundamental period of the fast variable.     

国际地球参考框架:现状与未来

地球参考框架是研究地球整体和局部运动以及探讨地球各圈层运动机制的一个重要基准。简要回顾地球参考架的发展和现状，主要分析当前取得的成绩及其存在问题和局限性，着重提出未来地球参考架的目标和若干建议。     

On the Evolution of Balloon Satellite Motions in a Plane Restricted Three-Body Problem with Light Pressure

We investigate the evolution of high Earth satellite orbits under gravitational perturbations from the Sun and light pressure forces, without the Earth shadow effect. We present the disturbing function of the problem provided that the satellite is a sphere. The mean value of the disturbing function in the absence of resonances between the mean unperturbed motion of the satellite and the mean motion of the Sun has also been obtained. The semimajor axis of the satellite orbit and the mean value of the disturbing function are shown to be integrals of the averaged osculating equations. TheHill version of the problem, whereby the distance to the satellite is much smaller than the Earth–Sun distance, has been studied in detail: we have constructed the phase portraits of the oscillations at various parameters and described three types of quasiperiodic satellite trajectories—librational and rotational trajectories as well as Earth collision trajectories. Numerical simulations of trajectories have allowed the additional effects caused by light pressure to be described: the displacement of the bounded trajectory of the satellite as a whole relative to the trajectory of the classical three-body problem into a region more distant from the Sun.