First-order theory of weakly eccentric orbital motion

Starting from the analytical theory of perturbed circular motions presented in Celestial Mechanics (Bois, 1994), this paper presents an extended resolution valid also for small eccentricity orbits. The solution is of the first order of a small parameter characterizing the magnitude of disturbing forces. The solution has the form of Fourier series with the coefficients given by iterative formation laws. The solution is free from singularities due to small eccentricity or inclination. As an example of numerical application the equatorial artificial satellite orbits are analyzed. For some high satellite orbits with small eccentricity the difference between the numerical integration and the analytical model does not exceed few centimeters per one revolution.On leave from Astronomical Observatory of A. Mickiewicz University, Soneczna 36, PL60-286 Pozna, Poland.     

First-order theory of satellite attitude motion application to HIPPARCOS

This work aims at finding an analytic solution corresponding to the attitude evolution in space of a satellite submitted to disturbing torques. This paper presents a basic frame applicable to any perturbed rotation satellite, and a method of resolution leading to a formal solution which is given here to the first order. Thus, the main problem is the slow rotation of a body with three unequal axes of inertia, essentially submitted to a dominant solar radiation pressure torque, with the axis pointing far away from a position of equilibrium. The comparison of the results with a numerical integration based upon a HIPPARCOS model is convincing.     

Numerical continuation of families of frozen orbits in the zonal problem of artificial satellite theory

In the zonal problem of a satellite around the Earth, we continue numerically natural families of periodic orbits with the polar component of the angular momentum as the parameter. We found three families; two of them are made of orbits with linear stability while the third one is made of unstable orbits. Except in a neighborhood of the critical inclination, the stable periodic (or frozen) orbits have very small eccentricities even for large inclinations.     

Spectral decomposition of geopotential,earth and ocean tidal perturbations in linear satellite theory

The spectra of geopotential, Earth and ocean tidal perturbations on a satellite can be obtained using Kaula's linear theory, or an extension thereof, as summations of terms depending on four indices l, m, p, q. In this work algorithms are presented that generate the equivalence classes induced by the composition rule of frequency on the set of all (l, m, p, q) combinations up to a maximum degree L and maximum value Q of the last index. These algorithms eliminate the need to search the set of frequencies when the linear theory is programmed on a computer.     

Analytical treatment of air drag and earth oblateness effects upon an artificial satellite

A development of an analytical solution for the motion of an artificial Earth satellite subject to the combined effects of Earth gravity and air drag is presented. The atmospheric model takes into account a linear variation of the density scale height with altitude, the rotation and the oblateness of the atmosphere. The perturbation theory is based upon Lie transforms. The secular and long-periodic terms as well as the short-periodic effects are included in the theory which is valid for small to moderate eccentricities and for all values of the inclination.Belgian National Fund for Scientific Research     

Canonical Floquet theory

Classical Floquet theory is reviewed with careful attention to the case of repeated eigenvalues common in Hamiltonian systems. Floquet theory generates a canonical transformation to modal variables if the periodic matrix can be made symplectic at the initial time. It is shown that this symplectic normalization can always be carried out, again with careful attention to the degenerate case. The periodic modal vectors and canonical modal variables can always be chosen to be purely real. It is possible to introduce real valued action-angle variables for all modes. Physical interpretation of the canonical degenerate normal modal variables are offered. Finally, it is shown that this transformation enables canonical perturbation theory to be carried out using Floquet modal variables.     

Short Term Evolution of Artificial Satellites

When the elimination of the parallax and the elimination of the perigee is applied to the zonal problem of the artificial satellite, a one-degree of freedom Hamiltonian is obtained. The classical way to integrate this Hamiltonian is by applying the Delaunay normalization, however, changing the time to the perturbed true anomaly and the variable to the inverse of the distance, the Hamilton equations become a perturbed harmonic oscillator. In this paper we apply the Krylov—Bogoliubov—Mitropolsky (KBM) method to integrate the perturbed harmonic oscillator as an alternative method to the Delaunay normalization. This method has no problem with small eccentricities and inclinations, and shows good numerical results in the evaluation of ephemeris of satellites.This revised version was published online in October 2005 with corrections to the Cover Date.     

An averaging method to study the motion of lunar artificial satellites I: Disturbing Function

We describe a semi-analytical averaging method aimed at the computation of the motion of an artificial satellite of the Moon. In this paper, the first of the two part study, we expand the disturbing function with respect to the small parameters. In particular, a semi-analytic theory of the motion of the Moon around the Earth and the libration of the lunar equatorial plane using different reference frames are introduced. The second part of this article shows that the choice of the canonical Poincaré variables lead to equations in closed form without singularities in e = 0 or I = 0. We introduce new expressions that are sufficiently compact to be used for the study of any artificial satellite. This revised version was published online in July 2006 with corrections to the Cover Date.     

《大气一号》气球卫星轨道倾角变化分析

引起《大气一号》两颗气球卫星（ＤＱ－１Ａ和ＤＱ－１Ｂ）轨道倾角变化的摄动因素主要是太阳光压摄动、大气旋转和日月引力摄动。太阳光压摄动引起气球卫星轨道倾角增大，平均每天变化约０．００１７，大气旋转引起轨道倾角减小，平均每天变化不到０．０００１，但随着高度下降，变化量亦增大，陨落前达０．００２。本文根据卫星轨道摄动理论，给出气球卫星轨道倾角变化的一种定量分析方法，得到的分析结果为：（１）由太阳光压摄动     

The problem of critical inclination combined with a resonance in mean motion in artificial satellite theory

In this paper the two-degree of freedom problem of a geosynchronous artificial satellite orbiting near the critical inclination is studied. First a local approach of this problem is considered. A semi-numerical method, well suited to describe the perturbations of a non-trivial separable system, is then applied such that surfaces of section illustrating the global secular dynamics are obtained. The results are confirmed by numerical integrations of the full Hamiltonian.Research Assistant for the Belgian National Fund for Scientific Research     

适用于人造卫星轨道数值积分的线性多步法的研究

总结了多年来构造适合卫星轨道计算的线性多步积分方法的研究进展;介绍了在构造积分公式的过程中选取伪根的三条原则。结合具体算例,对推荐的适用于不同类型卫星轨道的五线性多积分公式的性能与对称文坛及科威尔方法作了详细地比对和评述。     

An element formulation for perturbed motion about the center of mass

The perturbed motion of a rigid body about its center of mass, is formulated in terms of the six elements:l, the magnitude of the angular momentum vector;h, the total energy; and , two linear functions of the independent variable; and ₁ and ₁, two Euler angles that orientate the inertial frame with respect to the unperturbed solution. Solutions from the element formulation and the original Euler equations are numerically compared using shuttle-type data. For applied torques smaller than a given magnitude, the element formulation produced the following results: (1) larger step sizes in the numerical integration of the differential equations, resulting in an overall computational time-saving, and (2) more significant figures of accuracy in the computation of the variables describing the state of the rigid body.     

Quantitative perturbation theory by successive elimination of harmonics

We revisit some results of perturbation theories by a method of successive elimination of harmonics inspired by some ideas of Delaunay. On the one hand, we give a connection between the KAM theorem and the Nekhoroshev theorem. On the other hand, we support in a quantitative fashion a semi-numerical method of analysis of a perturbed system recently introduced by one of the authors.     

Second-order state transition for relative motion near perturbed,elliptic orbits

This paper develops a tensor and its inverse, for the analytical propagation of the position and velocity of a satellite, with respect to another, in an eccentric orbit. The tensor is useful for relative motion analysis where the separation distance between the two satellites is large. The use of nonsingular elements in the formulation ensures uniform validity even when the reference orbit is circular. Furthermore, when coupled with state transition matrices from existing works that account for perturbations due to Earth oblateness effects, its use can very accurately propagate relative states when oblateness effects and second-order nonlinearities from the differential gravitational field are of the same order of magnitude. The effectiveness of the tensor is illustrated with various examples.     

Superosculating intermediate orbits: Fourth-and fifth-order tangencies to trajectories of perturbed motion

The theory of superosculating intermediate orbits previously suggested by the author is developed. A new class of orbits with a fourth-order tangency to the actual trajectory of a celestial body at the initial time is constructed. Orbits with a fifth-order tangency have been constructed for the first time. The motion in the constructed orbits is represented as a combination of two motions: the motion of a fictitious attracting center with a variable mass and the motion relative to this center. The first motion is generally parabolic, while the second motion is described by the equations of the Gylden—Mestschersky problem. The variation in the mass of the fictitious center obeys Mestschersky’s first and combined laws. The new orbits represent more accurately the actual motion in the initial segment of the trajectory than an osculating Keplerian orbit and other existing analogues. Encke’s generalized methods of special perturbations in which the constructed intermediate orbits are used as reference orbits are presented. Numerical simulations using the approximations of the motions of Asteroid Toutatis and Comet P/Honda—Mrkos—Pajdu?áková as examples confirm that the constructed orbits are highly efficient. Their application is particularly beneficial in investigating strongly perturbed motion.     

Relativistic theory of the Moon's motion

Relativistic corrections for the elements and coordinates of the Moon have been obtained in the framework of the PPN-formalism. The influence of the coordinate conditions on the observational effects was studied.     

An averaging method to study the motion of lunar artificial satellites II: Averaging and Applications

On this, the second part of a two part study (Steichen, 1998) we further develop a semi-analytical theory for a lunar artificial satellite. This theory is obtained by averaging analytically the Hamiltonian function over period up to a month. The averaged equations are then numerically integrated. The solution is free from singularities at e = 0 and I = 0 and is not expanded in powers of these variables. In the last section, the analytic work is applied to characteristic examples to validate the method used. This revised version was published online in July 2006 with corrections to the Cover Date.     

A new set of integrals of motion to propagate the perturbed two-body problem

A formulation of the perturbed two-body problem that relies on a new set of orbital elements is presented. The proposed method represents a generalization of the special perturbation method published by Peláez et al. (Celest Mech Dyn Astron 97(2):131–150, 2007) for the case of a perturbing force that is partially or totally derivable from a potential. We accomplish this result by employing a generalized Sundman time transformation in the framework of the projective decomposition, which is a known approach for transforming the two-body problem into a set of linear and regular differential equations of motion. Numerical tests, carried out with examples extensively used in the literature, show the remarkable improvement of the performance of the new method for different kinds of perturbations and eccentricities. In particular, one notable result is that the quadratic dependence of the position error on the time-like argument exhibited by Peláez’s method for near-circular motion under the $J_{2}$ perturbation is transformed into linear. Moreover, the method reveals to be competitive with two very popular element methods derived from the Kustaanheimo-Stiefel and Sperling-Burdet regularizations.