We present the results of a solution of the Earth’s rotation built with analytical solutions of the planets and of the Moon’s motion. We take into account the influence of the Moon, the Sun and all the planets on the potential of the Earth for the zonal harmonics Cj,0 for j from 2 to 5, and also for the tesseral harmonics C2,2, S2,2C3,k, S3,k for k from 1 to 3 and C4,1, S4,1. We determine three Euler angles ψ, ω, and φ by calculating the components of the torque of the external forces with respect to the geocenter in the case of the rigid Earth. The analytical solution of the precession-nutation has been compared to a numerical integration over the time span 1900–2050. The differences do not exceed 16 μas for ψ and 8 μas for ω whereas the contribution of the tesseral harmonics reaches 150 μas in the time domain.

The equations of the translatory motion of the major planets and the Moon and the Poisson equations of the Earth’s rotation in Euler parameters are reduced to the secular system describing the evolution of the planetary and lunar orbits (independent of the Earth’s rotation) and the evolution of the Earth’s rotation (depending on the planetary and lunar evolution).

In this paper, we consider the problem of the rotation of the Earth, using a stationary triaxial gyrostat as a model. The problem is formulated by means of dimensionless canonical variables of Serret-Andoyer, referred to the mean ecliptic of date, in a similar way to Kinoshita (1977). We choose the constant components of the gyrostatic momentum in such a way that the period of the polar motion corresponds to Chandler’s period. Finally, the problem is integrated by means of Deprit’s perturbation method.

Secular changes in the rotational motion of a planet due to dissipation of energy in its core are investigated. It is assumed that the effect of a liquid core is equivalent to the action on a “frozen” planet of a special type of nonconservative torque. The Bogoliuboff-Mitropolsky method is used to explain the secular effects. The following qualitative features of the evolution of the motion are established: the fast rotation of a planet gradually becomes slower and the inclination of its equator to the orbital plane gradually increases.

The necessity to elaborate a theory of nutation and precession matching the accuracy of very modern techniques as Very Long Baseline Interferometry and Lunar Laser Ranging led recently to various works. We discuss here the good agreement between those related to the nutation when considering the Earth as a solid body. In comparison we show the uncertainty concerning the modelisation of the transfer function leading to theoretical determination of the nutation coefficients when including dominant geophysical characteristics.

This paper presents the theoretical study of precession and nutation of Mars in a rigourous way. For this work we choose a natural reference system, based on the concept of non-rotating origin, and the appropriate canonical variables. Then, we solve the equations of the problem by taking into account the effects of the Sun, the Earth, Jupiter, and satellites of Mars, Phobos and Deimos.

Averaging methods are convenient tools for studying long-periodic variations of the motion of artificial satellites. The main lines of a semi-analytical theory of the mean motion are given. We show how, when coupled with a careful reduction of the tracking data, this theory allows to determine parameters related to the temporal variations of the Earth gravity field (e.g. the amplitude of 18.6 years tide and the secular variation of even zonal harmonics). The theory is also very useful for other applications such as mission analysis.

LAGEOS I is a high-density geodetic satellite launched by NASA on 4 May 1976 (Johnson et al., 1976). Using a network of laser ranging stations, GSFC/NASA has maintained extremely accurate information on the orbital motions of LAGEOS I, and the later LAGEOS II and on the time-dependent evolution of their orbital parameters. The development of short-pulse laser ranging systems, and better models for atmospheric refractive effects, have dramatically improved the ability to locate the spacecraft or measure geodetic position and terrestrial crustal motion, but these systems do not measure the satellite rotational motion or gyroscopic effects, thought initially unimportant. Primarily as a result of technical strides in orbit determination, it is now recognized that the spin-motion is critical to understanding weak but important interactions with the environment.

In the frame of European cooperation among some stations (Brussels, Praha, Cagliari, Torino, Penc, Borowiec, Teddington, Metsahovi, Besançon and San Fernando) the method of time comparisons by a TV link has been used for orbit determination of the Eutelsat-F2 satellite. Observational stations are equipped with commercial TV receivers and high stability time services. The technique of observations is based on the registration of arrival times of synchronous pulses broadcast by the satellite. For this type of observations a special orbital software package (TOP-COGEOS) has been developed, which allows the determination of the dynamical state of the satellite and some additional parameters connected to applied observational technique.

The paper presents the results of the orbit determination of geosynchronous satellites derived from 1921 positional observations of a few dozens of satellites obtained at the double wide-angle astrograph DWA (D=0.4m, F=2m) of the Main Astronomical Observatory of the National Academy of Sciences of Ukraine in 1985–1991. The observations cover the longitude range from 19° W to 80° E. Orbital inclination of the observed objects is less than 3°. The mean values of the rms residuals are 066 in R.A. and 101 in Dec. over all processed orbital arcs.

During an observation campaign in winter 94/95 astrometric positions from Meteosat 4 and 5 were acquired at the Zimmerwald observatory using a CCD camera mounted in the prime focus of the 0.5 m Satellite Laser Ranging telescope. The measurements cover a time interval of four months, their precision is of the order of .

The modeling of radiation pressure for the small, cylindrically shaped satellites is relatively easy and they are therefore excellent objects to probe the geopotential. The orbital parameters and the radiation pressure coefficients for the two satellites as well as the resonant coefficients C22, S22 of the geopotential were determined by a single least square adjustment procedure including all the Zimmerwald observations. The relative errors estimated for the terms C22 and S22 are of the order of 1 ÷ 3 · 10−4.

We consider the motion of a spherically-symmetric balloon satellite perturbed by the Earth’s oblateness and solar radiation pressure. For equatorial satellite orbits and neglecting the Earth obliquity, the orbit-averaged equations for eccentricity and longitude of pericenter are integrable in quadratures (Krivov and Getino, 1996). The instability zone associated with the saddle separatrix in the phase space has been found and explored in depth. For semimajor axes about two Earth’s radii, and for area-to-mass ratios in the order of several tens cm2g−1, the amplitude and period of eccentricity oscillations may change nearly twofold under a small change of initial conditions or force parameters. We then restore the actual Earth obliquity of 235 and consider a spatial (non-integrable) problem. Near the saddle separatrix, a stochasticity zone appears that leads to large unpredictable eccentricity variations. The quasirandom motions of space balloons are investigated in terms of two-symbol (0-1) sequences by methods of stochastic celestial mechanics.

The space environment is presently dominated by man-made debris, for particles larger than 1 mg. A comprehensive survey of the debris population from 1 mg to the larger sizes in view of the recent data from radar and optical observations, and from the analysis of materials retrived from space is given.

A brief description of the major source and sink mechanisms acting on the debris population is given, along with a very short introduction to the two models for the long term evolution developed by the group in Pisa in the last years.

The results of the long term evolution analysis are presented in some detail. A likely scenario of the future space activities leads to a large growth of mmsize particles due to several catastrophic collisions. The simulation highlights the necessity of more realistic explosion models, since the current ones overestimate the 10 cm-sized fragments.

An enlarged version of this paper can be found at the CNUCE Spaceflight Dynamics Group Web site: http://apollo.cnuce.cnr.it/~rossi/homerossi.html.

Using the computational advantages offered by the formulation of orbital problems with respect to the ideal frame, with or without regularization, we consider the multi–revolution methods to obtain numerically the short and long–term propagation of the resulting debris clouds after a breakup event of an operational satellite. The problem posed and solved in this form gives high efficiency. We think that software with these features will serve two purposes: it will support studies of the hazards of space debris for operational orbital space systems and also will help to create debris analysis programs more transferable and reliable.

The problems of modeling of the rotational motion of the Earth are considered in the framework of general relativity. Both, rigid and deformable bodies are discussed. Rigorous definitions of the tensor of inertia, Tisserand-like axes and the angular velocity of rotation of an extended deformable body moving and rotating in external gravitational fields are proposed in the first post-Newtonian approximation. The implications of these post-Newtonian definitions on modeling of Earth rotation are analyzed.

We consider the problem of the attitude dynamics of a gyrostat under no external torques and constant internal spins. We introduce coordinates to represent the orbits of constant angular momentum as a flow on a sphere. This new representation shows that the problem is a particular case in the class of dynamical systems defined by a Hamiltonian that is a polynomial of at most degree two in a base of the Lie algebra so(3).

In this paper a new method of construction of the perturbation motion theory of celestial bodies, based on the averaging principle in view of frequency resonances, is stated. The first approximation of the asymptotic theory is the exact solution of the dynamics averaging equations, in which are included “secular” and “long-periodic” terms. The high-degree approximations are the exact solution of a known Krylov-Bogoliubov generalized equation. It is shown that these iterations are expressed in the analytical form by multiple Fourier series.

A uniform treatment of the two-body problem leads to a differential time transformation to introduce the arc length along the orbit as the independent variable. The transformation is integrated in terms of the classical anomalies.

Many quite different theories and methods of celestial mechanics are referred to as semi-analytical or semi-numerical. Their common feature is the trade-off of a generality for a computational efficiency. Four main ideas behind semi-analytical theories are identified and reviewed: series contraction, analytical support of numerical integration, piecewise solutions, and numerical evaluation of action-angle variables. The role of numerical analysis tools in analytical theories is briefly discussed.

In this communication, we use Chebyshev series for integrating orbital motions. The nice properties that Chebyshev polynomials have, such as giving good approximations of functions in the Chebyshev norm, easy handling of their algebra with algebraic manipulators, allowing very big step sizes for integration and giving the solution in the form of polynomials, make these polynomials very attractive in orbit computations.