The main problem of artificial satellite theory for small and moderate eccentricities

Perturbation techniques based on Lie transforms as suggested by Deprit were used as the theoretical foundation for programming the analytical solution of the Main Problem in Satellite Theory (all gravitational harmonics being zero exceptJ ₂). The collection of formulas necessary and sufficient to construct an ephemeris is given in the exposition. Short and long period displacements, as well as the secular terms, have been obtained up to the third order inJ ₂ as power series of the eccentricity. They result from two successive completely canonical transformations which it has been found convenient not to compose into a unique transformation. Division by the eccentricity appears nowhere in the developments-neither explicitly nor implicitly. The determination of the constants of motion from the initial conditions has been given an elementary solution that is both complete and explicit without being iterative. The program was developed by Rom from MAO's package of subroutines forMechanizedAlgebraicOperations. Reliability tests have been run in two instances: in-track errors for ANNA 1B are only 20 cm after 210 days in orbit, while for RELAY II, they are 2.4 m, even after 350 days in orbit.     

First-order secular planetary theory up to fourth power in eccentricities and inclinations

We construct a first-order secular general planetary theory, using the Jacobi-Radau set of origins, referring to common fixed plane and in terms of Poincaré canonical variables. We neglect powers higher than the fourth with respect to the eccentricities and sines of inclinations.     

Implementation of an explicit numerical method to the study of stability of asteroids with large eccentricities and inclinations

In this paper, we consider the secular variations in the restricted three bodies problem by implementing an explicit numerical technique for studying stability of equilibrium solutions. In the present case, the three bodies are the Sun, the Jupiter and an asteroid. Our results arise from studying some maximum stable orbital elements corresponding to the stable equilibrium solutions of the particular problem.     

Regularization and the artificial Earth satellite problem

Some of the properties of singularities of a system of differential equations, which includes several formulations of the artificial Earth satellite problem, are derived. Using them, it is shown that this problem cannot be regularized by using the current commonly used ideas and definitions of regularization.     

Equilibria of a charged artificial satellite subject to gravitational and Lorentz torques

The attitude dynamics of a rigid artificial satellite subject to a gravity gradient and Lorentz torques in a circular orbit are considered. Lorentz torque is developed on the basis of the electrodynamic effects of the Lorentz force acting on the charged satellite's surface. We assume that the satellite is moving in a Low Earth Orbit in the geomagnetic field, which is considered to be a dipole. Our model of torque due to the Lorentz force is developed for an artificial satellite with a general shape, and the nonlinear differential equations of Euler are used to describe its attitude orientation. All equilibrium positions are determined and conditions for their existence are obtained.The numerical results show that the charge q and radius ρ0of the center of charge for the satellite provide a certain type of semi-passive control for the attitude of the satellite. The technique for this kind of control would be to increase or decrease the electrostatic screening on the satellite. The results obtained confirm that the change in charge can affect the magnitude of the Lorentz torque, which can also affect control of the satellite. Moreover, the relationship between magnitude of the Lorentz torque and inclination of the orbit is investigated.     

Numerical calculations in the orbital determination of an artificial satellite for a long arc

Numerical methods are usually used for the computation of ephemerides with perturbations for the precise orbital determination of an artificial satellite. But their numerical stability will be encountered in a long arc. In this case the use the improved Encke special perturbation methods has been suggested. The results of this paper show that Encke's method does indeed have a certain effectiveness, but cannot yet completely resolve the numerical stability, and the more efficient method is to use the energy integral or its variational relation to control the growth of the along-track error in general numerical calculations so that the aim of stabilization can be achieved.     

Drag perturbations of artificial satellite orbits in a spherically symmetrical atmosphere

The equations characterizing the motion of an artificial satellite in a non-rotating spherically symmetrical atmosphere are integrated in the assumption of a linear variation of the density scale height with height, and using a new variable instead of the true anomaly. The secular perturbations in the semi-major axis and eccentricity are deduced.     

Analytical theory of a lunar artificial satellite with third body perturbations

We present here the first numerical results of our analytical theory of an artificial satellite of the Moon. The perturbation method used is the Lie Transform for averaging the Hamiltonian of the problem, in canonical variables: short-period terms (linked to l, the mean anomaly) are eliminated first. We achieved a quite complete averaged model with the main four perturbations, which are: the synchronous rotation of the Moon (rate ), the oblateness J ₂ of the Moon, the triaxiality C ₂₂ of the Moon ( ) and the major third body effect of the Earth (ELP2000). The solution is developed in powers of small factors linked to these perturbations up to second-order; the initial perturbations being sorted ( is first-order while the others are second-order). The results are obtained in a closed form, without any series developments in eccentricity nor inclination, so the solution apply for a wide range of values. Numerical integrations are performed in order to validate our analytical theory. The effect of each perturbation is presented progressively and separately as far as possible, in order to achieve a better understanding of the underlying mechanisms. We also highlight the important fact that it is necessary to adapt the initial conditions from averaged to osculating values in order to validate our averaged model dedicated to mission analysis purposes.     

Poisson equations of rotational motion for a rigid triaxial body with application to a tumbling artificial satellite

Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w ₀)², wheren is the satellite's orbital mean motion andw ₀ its initial rotational angular speed.     

Computer applications to lunar and artificial satellite laser ranging

The University of Texas McDonald Observatory has long been a pioneer in acquiring laser ranging data, a data type which has substantially improved our knowledge of the dynamics of the earth-moon system as well as various aspects of geophysics and general relativity. (See Mulholland, 1980; Shelus, 1985; Shelus, 1987.) The McDonald Laser Ranging System (MLRS) is one of only 2 laser ranging stations world-wide having the capability of routine data acquisition on both lunar and artificial satellite targets (Shelus,IEEE, 1985). In this paper we discuss the current applications of modern computer technology to the problems of acquiring and reducing that ranging data. As technology continues to improve, the logical upgrade is the replacement of obsolescent station minicomputers with the resource-rich environment of micro-computers. The goal is to allow the automation of many station ranging functions as well as the enhancement of onsite data quality control, filtering, and analysis. Plans for such upgrades and their implications for dynamical astronomy are discussed.     

Study of alpha component dynamics in the solar wind using the Prognoz satellite

The analysis of solar wind He⁺⁺ and H⁺ ion distribution functions, collected over five months by the satellite Prognoz 1, shows that these are in general maxwellian but that often tails appear at higher speeds. The existing relation V-T, the observation of ratios of T/Tp 3.83 and V/Vp 1.035 give evidence of preferential He⁺⁺ ion heating and acceleration. The criteria for heating by dissipation of hydromagnetic waves proposed by Barnes and Hung (1973) are tested experimentally. Finally, multifluid models are likely to predict certain observations such as dependence of the velocity ratio V/Vp on the solar wind flux.     

Second-order theory of the rotation of an artificial satellite

This work presents the expansion of the second-order of an analytical theory of the attitude evolution of an artificial satellite perturbed by given torques. The first-order of the theory has already been presented by the author in Celestial Mechanics39 (1986) 309–327. It is a theory that is valid under very general conditions including slow rotation and inequal axes of inertia. The present theory is suitable for any internal or external disturbing forces producing the torques. A formal solution is expanded in the second-order according to powers of a small parameter characteristic of the order of magnitude of the disturbing torques. These torques are expanded in Fourier series and the theory applies whatever is the length of these series. The coefficients of the solution are given by an iterative formation law. The comparison of the results with a numerical integration based upon a HIPPARCOS model shows that the second order has brought an improvement to the theory by at least one order of magnitude over the results of the first order.     

Theory of the motion of an artificial Earth satellite

An improved analytical solution is obtained for the motion of an artificial Earth satellite under the combined influences of gravity and atmospheric drag. The gravitational model includes zonal harmonics throughJ ₄, and the atmospheric model assumes a nonrotating spherical power density function. The differential equations are developed through second order under the assumption that the second zonal harmonic and the drag coefficient are both first-order terms, while the remaining zonal harmonics are of second order.Canonical transformations and the method of averaging are used to obtain transformations of variables which significantly simplify the transformed differential equations. A solution for these transformed equations is found; and this solution, in conjunction with the transformations cited above, gives equations for computing the six osculating orbital elements which describe the orbital motion of the satellite. The solution is valid for all eccentricities greater than 0 and less than 0.1 and all inclinations not near 0^o or the critical inclination. Approximately ninety percent of the satellites currently in orbit satisfy all these restrictions.     

Improvement on the second-order solution of artificial satellite orbits

Three methods are proposed in this paper to lessen the complexity in the derivation and the resulting expressions of second-order analytical solutions of artificial satellite orbits while retaining the advantages of analytical solutions. Of the three, the one combining the theories of elliptical perturbation and of intermediate orbits is noteworthy for its simplicity. It can also be used with advantage in first-order and third- or higher order solutions.     

Numerical simulation of the Moon's rotation in a rigorous relativistic framework

This paper describes a numerical simulation of the rigid rotation of the Moon in a relativistic framework.Following a resolution passed by the International Astronomical Union(IAU) in 2000,we construct a kinematically non-rotating reference system named the Selenocentric Celestial Reference System(SCRS) and give the time transformation between the Selenocentric Coordinate Time(TCS) and Barycentric Coordinate Time(TCB).The post-Newtonian equations of the Moon's rotation are written in the SCRS,and they are integrated numerically.We calculate the correction to the rotation of the Moon due to total relativistic torque which includes post-Newtonian and gravitomagnetic torques as well as geodetic precession.We find two dominant periods associated with this correction:18.6 yr and 80.1 yr.In addition,the precession of the rotating axes caused by fourth-degree and fifth-degree harmonics of the Moon is also analyzed,and we have found that the main periods of this precession are 27.3 d,2.9 yr,18.6 yr and 80.1 yr.     

Some properties of the dumbbell satellite attitude dynamics

The dumbbell satellite is a simple structure consisting of two point masses connected by a massless rod. We assume that it moves around the planet whose gravity field is approximated by the field of the attracting center. The distance between the point masses is assumed to be much smaller than the distance between the satellite’s center of mass and the attracting center, so that we can neglect the influence of the attitude dynamics on the motion of the center of mass and treat it as an unperturbed Keplerian one. Our aim is to study the satellite’s attitude dynamics. When the center of mass moves on a circular orbit, one can find a stable relative equilibrium in which the satellite is permanently elongated along the line joining the center of mass with the attracting center (the so called local vertical). In case of elliptic orbits, there are no stable equilibrium positions even for small values of the eccentricity. However, planar periodic motions are determined, where the satellite oscillates around the local vertical in such a way that the point masses do not leave the orbital plane. We prove analytically that these planar periodic motions are unstable with respect to out-of-plane perturbations (a result known from numerical investigations cf. Beletsky and Levin Adv Astronaut Sci 83, 1993). We provide also both analytical and numerical evidences of the existence of stable spatial periodic motions.     

Libration dynamics and stability of electrodynamic tethers in satellite deorbit

This paper studies libration dynamics and stability of deorbiting nano-satellites by short and bare electrodynamic tethers. A critical aspect of satellite deorbit by an electrodynamic tether is to maintain the tether aligned with the local vertical and stable while subjected to external perturbations. The dynamics of electrodynamic tether system in deorbit application is divided into the orbital motion of the center of system’s mass and the tether libration motion relative to that center. Major space environmental perturbations including the current-induced electrodynamic force, atmospheric drag, oblateness effect of the Earth, irregularity of geomagnetic field, variable plasma density, solar radiation pressure, and lunisolar gravitational attractions are considered in the dynamic analysis. Quantitative analyses are provided in order to characterize the order of the perturbative torques during the deorbit process. A single index is derived from the libration energy to stabilize the libration motion by regulating the current in the tether through simple on-off switching. Numerical results show that the libration dynamics of an electrodynamic tether has significant impacts on the deorbit process and the electrodynamic tether cannot effectively deorbit satellites without libration stability control. The proposed current regulation strategy is simple and very effective in stabilizing libration motion of an electrodynamic tether.