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     

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.     

Eclipses of natural planetary satellites

A general method is given for predicting eclipse events for natural planetary satellites suitable for use on a large scale digital computer. The precision is sufficient to process photometric eclipse observations to improve natural satellite ephemerides. Expected accuracy improvement for Saturn's satellites should be an order of magnitude or better. Predicted eclipse times of satellites by Saturn and the Rings are given for the next decade.Presented at IAU Colloquium No. 28, Ithaca, New York, August, 1974.     

Quasi-critical orbits for artificial lunar satellites

We study the problem of critical inclination orbits for artificial lunar satellites, when in the lunar potential we include, besides the Keplerian term, the J ₂ and C ₂₂ terms and lunar rotation. We show that, at the fixed points of the 1-D averaged Hamiltonian, the inclination and the argument of pericenter do not remain both constant at the same time, as is the case when only the J ₂ term is taken into account. Instead, there exist quasi-critical solutions, for which the argument of pericenter librates around a constant value. These solutions are represented by smooth curves in phase space, which determine the dependence of the quasi-critical inclination on the initial nodal phase. The amplitude of libration of both argument of pericenter and inclination would be quite large for a non-rotating Moon, but is reduced to <0°.1 for both quantities, when a uniform rotation of the Moon is taken into account. The values of J ₂, C ₂₂ and the rotation rate strongly affect the quasi-critical inclination and the libration amplitude of the argument of pericenter. Examples for other celestial bodies are given, showing the dependence of the results on J ₂, C ₂₂ and rotation rate.     

Some orbital characteristics of lunar artificial satellites

In this paper we present an analytical theory with numerical simulations to study the orbital motion of lunar artificial satellites. We consider the problem of an artificial satellite perturbed by the non-uniform distribution of mass of the Moon and by a third-body in elliptical orbit (Earth is considered). Legendre polynomials are expanded in powers of the eccentricity up to the degree four and are used for the disturbing potential due to the third-body. We show a new approximated equation to compute the critical semi-major axis for the orbit of the satellite. Lie-Hori perturbation method up to the second-order is applied to eliminate the terms of short-period of the disturbing potential. Coupling terms are analyzed. Emphasis is given to the case of frozen orbits and critical inclination. Numerical simulations for hypothetical lunar artificial satellites are performed, considering that the perturbations are acting together or one at a time.     

The population of natural Earth satellites

We have for the first time calculated the population characteristics of the Earth’s irregular natural satellites (NESs) that are temporarily captured from the near-Earth-object (NEO) population. The steady-state NES size–frequency and residence-time distributions were determined under the dynamical influence of all the massive bodies in the Solar System (but mainly the Sun, Earth, and Moon) for NEOs of negligible mass. To this end, we compute the NES capture probability from the NEO population as a function of the latter’s heliocentric orbital elements and combine those results with the current best estimates for the NEO size–frequency and orbital distribution. At any given time there should be at least one NES of 1-m diameter orbiting the Earth. The average temporarily-captured orbiter (TCO; an object that makes at least one revolution around the Earth in a co-rotating coordinate system) completes (2.88 ± 0.82) rev around the Earth during a capture event that lasts (286 ± 18) d. We find a small preference for capture events starting in either January or July. Our results are consistent with the single known natural TCO, 2006 RH₁₂₀, a few-meter diameter object that was captured for about a year starting in June 2006. We estimate that about 0.1% of all meteors impacting the Earth were TCOs.     

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.     

A search for natural satellites of Mercury

We present the results of an imaging survey of Mercury's Hill sphere in search for objects dynamically bound to the planet, motivated by the existence of hermeocentric orbits that have been shown to be stable over 5 Myr or more. A six-day survey of Mercury's apparent vicinity from 6 to 140 Mercury radii, with full coverage between 19 and 73 Mercury radii, was performed with the Nordic Optical Telescope using ALFOSC in the R-band. The deepest limiting magnitude of 18.6 at a signal-to-noise-level of 3 corresponds to a hermeocentric object size of 0.5 km, while the brightest limiting magnitude corresponds to a size of 1.6 km. While two suspected sources were found, no hermeocentric objects could be confidently identified.     

Analytic short period lunar and solar perturbations of artificial satellites

The short period luni-solar theory of Kozai is generalized for arbitrary obliquity of the ecliptic and inclination of the moon's orbit to the ecliptic. Analytic first order lunar perturbations to the elements are derived. The theory is illustrated by an application to the communication satellite Intelsat 3F3.Presently at the Department of Environmental Sciences, University of Tel Aviv, Ramat Aviv, Israel.     

New observational techniques and precise orbit determination of artificial satellites

Modern observational techniques using ground-based and space-based instrumentation have enabled the measurement of the distance between the instrument and satellite to better than one centimeter. Such high precision instrumentation has fostered applications with centimeter-level requirements for satellite position knowledge. The determination of the satellite position to such accuracy requires a comparable modeling of the forces experienced by the satellite, especially when classical orbit determination methods are used. Geodetic satellites, such as Lageos, in conjunction with high precision ground-based laser ranging, have been used to improve for modeling of forces experienced by the satellite. Space-based techniques, such as Global Positioning System (GPS), offer alternatives, including kinematic techniques which require no modeling of the satellite forces, or only rudimentary models. This paper will describe the various techniques and illustrate the accuracies achieved with current satellites, such as TOPEX/POSEIDON, GPS/MET and the expectations for some future satellites.     

The escape of natural satellites from Mercury and Venus

It is suggested that the slow rotations of Mercury and Venus may be connected with the absence of natural satellites around them. If Mercury or Venus possessed a satellite at the time of formation, the tidal evolution would have caused the satellite to recede. At a sufficiently large distance from the planet, the Sun's gravitational influence makes the satellite orbit unstable. The natural satellites of Mercury and Venus might have escaped as a consequence of this instability.     

The escape of natural satellites from Mercury and Venus

Kumar's (1977) suggestion that the slow rotations of Mercury and Venus are in part due to natural satellites that subsequently escaped is discussed. A more useful criterion for the escape of such satellites than that previously proposed is derived, and it is shown that this distance is sufficiently small for Mercury and Venus to make the escape of satellites a likely possibility.     

Machine derivation of the tesseral perturbation of artificial satellites

A non-conservative Lie transformation is used to establish the theory of tesseral perturbation including the cross terms from the zonal harmonic J₂ with the tesseral harmonics. The formulae for the perturbations are derived with a computer. The storage of the Poisson series is effected through a one-to-one correspondence between the multi-dimensional index of a term in the series and the store address of the coefficient of that term. Rules for storing some typical series in celestial mechanics in computers are also given.     

Long-term evolution of the orbits of natural satellites

This review presents the recent works devoted to the construction or the improvement of the theories of motion of all natural planetary, satellites (except the Moon). The knowledge of the long-term evolution of these motions is strongly dependent on the accuracy of current theories. With the increasing precision of the ground-based observations, and with the past and future space missions, most of the theories have been or have to be revisited, taking into account more and more disturbing effects and specially tidal dissipation. These studies are often made difficult by the resonant behaviour of the system. We emphasize here tidal evolution in resonance. In the Jovian and Saturnian systems, tidal actions might explain the observed resonant state, as well as the heating of the satellites up to the softening and the resurfacing of some of them. However in the case of the Uranian satellites., no true resonance appears in spite of an observational evidence of tidal effects in resurfacing Ariel and Miranda, and new works try to expalin these differences.     

A photographic technique for the precise observation of artificial satellites and some results from data on high satellites

The construction and operation of a tilting-mirror satellite camera, adapted for use with a 24 in. reflecting telescope, are described. Some results are given from observations of the balloon satellite 1963 30D, and a value for the air density at a height of 3500 km derived.     

Attitude stability of artificial satellites subject to gravity gradient torque

The stability of the rotational motion of artificial satellites is analyzed considering perturbations due to the gravity gradient torque, using a canonical formulation, and Andoyer’s variables to describe the rotational motion. The stability criteria employed requires the reduction of the Hamiltonian to a normal form around the stable equilibrium points. These points are determined through a numerical study of the Hamilton’s equations of motion and linear study of their stability. Subsequently a canonical linear transformation is used to diagonalize the matrix associated to the linear part of the system resulting in a normalized quadratic Hamiltonian. A semi-analytic process of normalization based on Lie–Hori algorithm is applied to obtain the Hamiltonian normalized up to the fourth order. Lyapunov stability of the equilibrium point is performed using Kovalev and Savchenko’s theorem. This semi-analytical approach was applied considering some data sets of hypothetical satellites, and only a few cases of stable motion were observed. This work can directly be useful for the satellite maintenance under the attitude stability requirements scenario.     

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.