Motion of artificial satellites in the set of Eulerian redundant parameters,II

In this paper the full recurrent power series solution is established for J ₂-gravity perturbed motion in terms of the Eulerian redundant parameters. Applications of the method for the problem of the final state prediction are illustrated by numerical examples of some typical ballistic missiles, a final state of very high accuracy is obtained for each case study.     

Motion of artificial satellites in the set of Eulerian redundant parameters (III)

In this paper, the classical and generalized Sundman time transformations are used to establish new generating set of differential equations of motion in terms of the Eulerian redundant parameters. The implementation of this set on digital computers for the commonly used independent variables is developed once and for all. Motion prediction algorithms based on these equations are developed in a recursive manner for the motions in the Earth's gravitational field with axial symmetry whatever the number of the zonal harmonic terms may be. Applications for the two types of short and long term predictions are considered for the perturbed motion in the Earth's gravitational field with axial symmetry with zonal harmonic terms up to J ₃₆ . Numerical results proved the very high efficiency and flexibility of the developed equations.     

Oblateneess and drag effects on the motion of satellites in the set of Eulerian redundant parameters

In this paper, the perturbed motion of the artificial satellites under the effects of the earth's oblateness and atmospheric drag will be established. These equations will be expressed in terms of the Eulerian redundant parameters. Applications of the method for the perturbed motion are illustrated by numerical examples for some test cases of the orbits.     

On the tidal effects in the motion of artificial satellites

The general perturbations in the elliptic and vectorial elements of a satellite as caused by the tidal deformations of the non-spherical Earth are developed into trigonometric series in the standard ecliptical arguments of Hill-Brown lunar theory and in the equatorinal elements ω and Ω of the satellite. The integration of the differential equations for variation of elements of the satellite in this theory is easy because all arguments are linear or nearly linear in time. The trigonometrical expansion permits a judgment about the relative significance of the amplitudes and periods of different tidal ‘waves’ over a long period of time. Graphs are presented of the tidal perturbations in the elliptic elements of the BE-C satellite which illustrate long term periodic behavior. The tidal effects are clearly noticeable in the observations and their comparison with the theory permits improvement of the ‘global’ Love numbers for the Earth.     

Problems of critical inclination and commensurability in the motion of artificial satellites

We discuss the resonance problems resulting from critical inclination and commensurability in the motion of an artificial satellite. We consider the perturbation due to the Earth's asphericity on the average model and we derive the equilibrium solution and the corresponding region of libration and compare with the actual motion of a satellite. We find that the resonance plays a stabilizing role in the motion and that the luni-solar perturbations have a significant effect on the orbital resonance of 24-h synchronous satellites.     

The effect of the dynamical parameters in the motion of the Galilean satellites of Jupiter

The construction of an analytical theory of the motion of the Galilean satellites of Jupiter requires that we keep track of the dynamical parameters, that is, the masses of the satellites, and the harmonic coefficients of the potential of the planet J₂ and J₄. This is realized here. But as in other theories the solution becomes partly numerical from the resolution of an autonomous system. The aim of this paper is to present a method to obtain developped solutions of this autonomous system. In these solutions the proper motions of the pericenters and nodes are obtained as short series developped in the neighbourhood of a numerical solution. We have used these results to obtain complementary terms in the general solution which give a complete representation of the motions with respect to the dynamical parameters.     

The motion of the Martian satellites

An extensive analysis of the motion of Phobos and Deimos from 1877 to 1973 has been fulfilled. The new values of the parameters of the orbital model first developed by Struve have been determined for both satellites. The new sets of the orbital parameters compete with the solutions of similar accuracy found by Wilkins and Sinclair. A secular acceleration in longitude of Phobos is found to be equal to +(0.107±0.011)×10^?7 deg day^?2. The value of the acceleration is little affected when one or another group of oppositions is omitted. The acceleration of Deimos is determined with great uncertainty: +(0.06±0.34)×10^?9 deg day^?2. Values found for the orbital parameters seem to be in good agreement since the mass, oblateness and coordinates of the pole of Mars inferred from the motion of each satellite have similar values in both cases.     

Relativistic effects in the motion of artificial satellites: The oblateness of the central body II

The motion of artificial satellites in the gravitational field of an oblate body is discussed in the post — Newtonian framework using the technique of canonical Lie transformations. Two Lie transformations are used to derive explicit results for the longperiodic and secular perturbations for satellite orbits in the Einstein case.     

Relativistic effects in the motion of artificial satellites: The oblateness of the central body I

The motion of artificial satellites in the gravitational field of an oblate body is discussed in the parametrized-post-Newtonian (PPN) framework with parameters and . Analytical expressions for first order post-Newtonian short periodic, long periodic and secular perturbations of orbital elements are given.     

The visual appearance of artificial satellites

The paper lists the stellar magnitudes of satellites observed from the U.K. during 1965. Statistically derived values are given which describe the fluctuating or constant magnitudes and the rates of flashing.     

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.     

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.     

Relative motion of near orbiting satellites

The relative motion of two particles on adjacent orbits about the same primary has been investigated under the condition that both motions have the same period. The geometrical properties of the relative displacement and velocity traces, on representative planes, are studied. A complete state of the motion is given; and, the range and range-rate variations, over one or more orbits, are described.It has been found that cusps appear on some of the traces provided that a proper relationship exists between the eccentricity and inclination. (Here, one particle moves on a circular path while the second moves on an ellipse.) The conditions for which cusps appear are given, and typical traces are shown.     

Ground traces of artificial earth satellites

This paper considers the ground trace of an artificial earth satellite. It determines the effects of the trace caused by perturbations due to atmospheric drag, the oblateness of the earth, and the moon and the sun as a third body.The necessary mathematical relations giving these perturbations which are available in literature are utilized (Betz, 1967; Brouwer and Clemence, 1961; Brouwer and Hori, 1961; Danby, 1962; Escobal, 1965; Kentet al., 1963; Kozai, 1962). Those relations unavailable elsewhere are derived.The computation was done by programming in FORTRAN language and utilizing an IBM 360/65.Captain, USAF     

Lunar perturbations of artificial satellites of the earth

Lunisolar perturbations of an artificial satellite for general terms of the disturbing function were derived by Kaula (1962). However, his formulas use equatorial elements for the Moon and do not give a definite algorithm for computational procedures. As Kozai (1966, 1973) noted, both inclination and node of the Moon's orbit with respect to the equator of the Earth are not simple functions of time, while the same elements with respect to the ecliptic are well approximated by a constant and a linear function of time, respectively. In the present work, we obtain the disturbing function for the Lunar perturbations using ecliptic elements for the Moon and equatorial elements for the satellite. Secular, long-period, and short-period perturbations are then computed, with the expressions kept in closed form in both inclination and eccentricity of the satellite. Alternative expressions for short-period perturbations of high satellites are also given, assuming small values of the eccentricity. The Moon's position is specified by the inclination, node, argument of perigee, true (or mean) longitude, and its radius vector from the center of the Earth. We can then apply the results to numerical integration by using coordinates of the Moon from ephemeris tapes or to analytical representation by using results from lunar theory, with the Moon's motion represented by a precessing and rotating elliptical orbit.     

Stability of artificial and natural satellites

A quantitative measure of stability based on Hill's definition is evaluated for direct and retrograde satellite orbits. These orbits are known as Poincaré's first kind in the restricted problem of three bodies. Onsets of possible instabilities and captures are established. A critical (maximum) value of the satellite's orbital radius is found for stability as a remarkably simple function of the massparameter. The results are applied to the natural satellites of the solar system.     

Two problems about the motion of low-moon-orbit satellites

A detailed theoretical analysis on the orbital lifetime and frozen orbit of low-moon-orbit satellites (LMOS) is carried out, and their relationships with the orbital inclination, as well as some mutual relationships are presented. Taking account of the main perturbing sources of low-orbit satellites, we carried out numerical simulations under a comprehensive force model, and the results not only confirm the correctness of the theoretical analysis, but also provide some valuable insights on the orbital design of LMOS.     

On a semi-inverse problem in the motion of gyrostat satellites

In this paper we study the equilibrium orientation of a gyrostat satellite in the gravity field of a point mass. Direct problem is to find all possible equilibrium orientation when the relative angular momentum vector is given. Inverse problem is to find this relative angular momentum in order to obtain equilibrium in a given orientation. Semi-inverse problem is solved here when some parameters (but not all) giving orientation of the satellite are chosen arbitrarily, giving for what choices real solutions occur.