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.     

The equations of motion of an artificial satellite in nonsingular variables

The equations of motion of an artificial satellite are given in nonsingular variables. Any term in the geopotential is considered as well as luni-solar perturbations up to an arbitrary power ofr/r, r being the geocentric distance of the disturbing body. Resonances with tesseral harmonics and with the Moon or Sun are also considered. By neglecting the shadow effect, the disturbing function for solar radiation is also developed in nonsingular variables for the long periodic perturbations. Formulas are developed for implementation of the theory in actual computations.     

An alternate transition from the Lagrangian of a satellite to equations of motion

A procedure is outlined for a conservative system which makes it possible to go from a Lagrangian of a librating system to the corresponding equations of motion in the Eulerian form. The transition does not require a choice of rotational coordinates and makes use of angular velocities and direction cosines directly. The procedure thus synthesizes attractive features of two classical approaches and has far reaching consequences: it is particularly useful in formulating equations of motion for complex flexible systems of contemporary interest. For the case of a satellite with two flexible plate-type appendages, for example, the approach reduced the formulation time to one-third. The basic mathematical concepts are briefly touched upon in the beginning which help explain the subsequent development.     

On the artificial satellite theory

Some of the basic ideas of an analytical orbiter theory which is being developed by Hubert Claes in Namur are presented.The theory is based on the Lie transform technique and will be expressed in a closed form up to second order. The inclusion of additional terms of the third order (expanded in power series of the eccentricity) will be considered.Special attention is being given to the choice of the elements and to the final form of the theory. Three main criteria are used. The removal of the virtual singularities of small inclination and eccentricity. The simplicity of the final form of the theory once the elements have been given their numerical values. The numerical stability of the evaluation of the theory.     

A semi-analytic theory for the motion of a lunar satellite

A semi-analytical solution to the problem of the motion of a satellite of the moon is presented. Perturbative effects which are considered include those due to the attraction of the moon, earth, and sun, the non-sphericity of the moon's gravitational field, coupling of lower-order terms, solar radiation pressure, and physical libration. Short-period terms and intermediate-period terms, terms with the period of the moon's longitude, are produced by means of von Zeipel's method; it is proposed to obtain the secular perturbations, and those depending only on the argument of perilune, by numerical integration of the equations of motions. The short-period terms and intermediate-period terms are developed up to second order, where first order is 10^–2. The secular perturbations and perturbations dependent on the argument of perilune are obtained to third order.     

Analytical theory of motion of phoebe,the ninth satellite of saturn

The literal solution of the restricted three body problem obtained by the authors up to the eleventh order with respect to the minor parameter is applied to the investigation of the motion of Phoebe, the ninth satellite of Saturn. As distinct from the existing analytical theories of the motion of the satellite, in the present paper the planetary perturbations are taken into account. A comparison with the modern numerical theory of the motion of Phoebe has shown that the new analytical theory of the satellite motion represents observations with the same degree of accuracy.     

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.     

On the dynamics of the satellite galaxies in NGC 5044

The NGC 5044 galaxy group is dominated by a luminous elliptical galaxy that is surrounded by ∼160 dwarf satellites. The projected number density profile of this dwarf population deviates within ∼1/3 of the virial radius from a projected Navarro, Frenk and White (NFW) profile, which is assumed to approximate the underlying total matter distribution. By means of a semi-analytic model, we demonstrate that the interplay between gravitation, dynamical friction and tidal mass loss and destruction can explain the observed number density profile. We use only two parameters in our models: the total to stellar mass fraction of the satellite haloes and the disruption efficiency. The disruption efficiency is expressed by a minimum radius. If the tidal radius of a galaxy (halo) falls below this radius, it is assumed to become unobservable. The preferred parameters are an initial total to stellar mass fraction of ∼20 and a disruption radius of  4 kpc  . In that model, about 20 per cent of all the satellites are totally disrupted on their orbits within the group environment. Dynamical friction is less important in shaping the inner slope of the number density profile because the reduction in mass by tidal forces lowers the impact of the friction term. The main destruction mechanism is tide. In the preferred model, the total B -band luminosity of all disrupted galaxies is about twice the observed luminosity of the central elliptical galaxy, indicating that a significant fraction of stars are scattered into the intragroup medium. Dwarf galaxy satellites closer to the centre of the NGC 5044 group may exhibit optical evidence of partial tidal disruption. If dynamical friction forces the satellite to merge with the central elliptical, the angular momentum of the satellite tends to be removed at the apocentre passage. Afterwards, the satellite drops radially towards the centre.     

The Ω dependence in equations of motion

The equations of motion governing the evolution of a collisionless gravitating system of particles in an expanding universe can be cast in a form which is almost independent of the cosmological density parameter, Ω, and the cosmological constant, Λ. The new equations are expressed in terms of a time variable τ=ln D , where D is the linear rate of growth of density fluctuations. The dependence on the density parameter is proportional to ε=Ω^−0.2−1 times the difference between the peculiar velocity (with respect to τ) of particles and the gravity field (minus the gradient of the potential); or, before shell-crossing, times the sum of the density contrast and the velocity divergence. In a one-dimensional collapse or expansion, the equations are fully independent of Ω and Λ before shell crossing. In the general case, the effect of this weak Ω dependence is to enhance the rate of evolution of density perturbations in dense regions. In a flat universe with Λ7ne;0, this enhancement is less pronounced than in an open universe with Λ=0 and the same Ω. Using the spherical collapse model, we find that the increase of the rms density fluctuations in a low-Ω universe relative to that in a flat universe with the same linear normalization is ∼0.01ε(Ω)〈δ³〉, where δ is the density field in the flat universe. The equations predict that the smooth average velocity field scales like Ω^0.6, while the local velocity dispersion (rms value) scales, approximately, like Ω^0.5. High-resolution N -body simulations confirm these results and show that density fields, when smoothed on scales slightly larger than clusters, are insensitive to the cosmological model. Haloes in an open model simulation are more concentrated than haloes of the same M /Ω in a flat model simulation.     

The relativistic equations of motion for a satellite in orbit about a finite-size,rotating Earth

The relativistic equations of motion are derived for N self-gravitating, rotating finite bodies. These equations are then applied to the near-Earth satellite orbit determination problem. The apparent change of the shape of the Earth from the Earth centered frame to the Solar System barycentric frame changes the value of the Newtonian potential term in the metric. This in turn leads to a simplification of the equations of motion in the barycentric frame.     

On the motion of a satellite in resonance with its rotating planet

The influence of resonance perturbations due to the gravitational field of an oblate planet on its satellite whose motion is commensurable with rotation of the planet has been investigated. It has been shown that in special case of the critical inclination or circular orbit the Lagrange equations can be integrated for all resonance terms simultaneously. The method is applied to the investigation of the motion of the 12-hour communication and navigation satellites of the Molniya and Navstar type. The computations has been performed by the use of four models of the geopotential.     

On quasi-satellite periodic motion in asteroid and planetary dynamics

Applying the method of analytical continuation of periodic orbits, we study quasi-satellite motion in the framework of the three-body problem. In the simplest, yet not trivial model, namely the planar circular restricted problem, it is known that quasi-satellite motion is associated with a family of periodic solutions, called family f, which consists of 1:1 resonant retrograde orbits. In our study, we determine the critical orbits of family f that are continued both in the elliptic and in the spatial models and compute the corresponding families that are generated and consist the backbone of the quasi-satellite regime in the restricted model. Then, we show the continuation of these families in the general three-body problem, we verify and explain previous computations and show the existence of a new family of spatial orbits. The linear stability of periodic orbits is also studied. Stable periodic orbits unravel regimes of regular motion in phase space where 1:1 resonant angles librate. Such regimes, which exist even for high eccentricities and inclinations, may consist dynamical regions where long-lived asteroids or co-orbital exoplanets can be found.     

On the roto-translatory motion of a satellite of an oblate primary

We deal with the problem of the motion of a triaxial satellite of an oblate primary of larger mass. We show that the treatment is simplified by using a canonical set of variables previously introduced by the authors, that allows a drastic reduction in the expansions of the potential. A general method to avoid the appearance of virtual singularities when the angles between certain planes are small is designed. Our approach is applicable either to natural or artificial satellites.     

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     

A linear solution of the equations of motion of an earth-orbiting satellite based on a Lie-series

Using Hill's variables, an analytical solution of a canonical system of six differential equations describing the motion of a satellite in the gravitational field of the earth is derived. The gravity field, expanded into spherical harmonics, has to be expressed as a function of the Hill variables. The intermediary is chosen to include the main secular terms. The first order solution retains the highly practical formal structure of Kaula's linear solution, but is valid for circular orbits and provides of course a spectral decomposition of radius vector and radial velocity. The resulting eccentricity functions are much simpler than the Hansen functions, since a series evaluation of the Kepler equation is avoided. The present solution may be extended to higher order solutions by Hori's perturbation method.     

The fifth-order analytical solution of the equations of motion of a satellite in orbit around a non-spherical planet

A high-precise analytical theory of a satellite in orbit around a non-spherical planet has been developed. The Poisson's small parameter method has been used. All secular and short-periodic perturbations proportional up to and including a product of five arbitrary harmonic coefficients of the planetary potential expansion are calculated. Long-periodic perturbations are derived with the accuracy of up to the fourth-order, inclusive. The influence of the high-order perturbations on the motion of ETALON-1 satellite has been investigated. The results of comparison of the numerical and analytical integration of the equations of its motion over a five year interval are as follows:

Third-order secular Solution of the variational equations of motion of a satellite in orbit around a non-spherical planet

We constructed an analytical theory of satellite motion up to the third order relative to the oblateness parameter of the Earth (J ₂). Equations of secular variations was developed for the first three orbital elements (a, e, i) of an artificial satellite. The secular variations are solved in a closed form.     

On the radial intermediaries and the time transformation in satellite theory

Taking advantage of the radial intermediaries and the regularization and linearization methods, the zonal Earth satellite theory is studied in the polar nodal canonical set of variables (, , ,R, ¡,N).The variable is eliminated in the first order of the Hamiltonian by applying Deprit's method. Then, the elimination of the perigee is carried out by another canonical transformation. As a consequence, a new radial intermediary, which contains all theJ _2n(n1) harmonics, is given. A comparison with the previous radial intermediaries of Cid and Lahulla, Deprit and Alfriend and Coffey is made.Finally, a regularizing transformation which allows us to linearize part of the radial intermediary is proposed, and an analytical study of this process is presented.     

The motion of a lunar satellite

Presented in this theory is a semianalytical solution for the problem of the motion of a satellite in orbit around the moon. The principal perturbations on such a body are due to the nonspherical gravity field of the moon, the attraction of the earth, and, to a lesser degree, the attraction of the sun. The major part of the problem is solved by means of the celebrated von Zeipel Method, first successfully applied to the motion of an artificial earth satellite by Brouwer in 1959. After eliminating from the Hamiltonian all terms with the period of the satellite and those with the period of the moon, it is suggested to solve the remaining problem with the aid of numerical integration of the modified equations of motion.This theory was written in 1964 and presented as a dissertation to Yale University in 1965. Since then a great deal has been learned about the gravity field of the moon. It seems that quite a number of recently determined gravity coefficients would qualify as small quantities of order two. Hence, according to the truncation criteria employed, they should be considered in the present theory. However, the author has not endeavored to update the work accordingly. The final results, therefore, are incomplete in the lunar gravitational perturbations. Nevertheless, the theory does give the largest such variations and it does present the methods by which perturbations may be derived for any gravity terms not actually developed.