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.     

Prediction of trajectories in the Earth's gravitational field with axial symmetry using KS uniform regular elements

In this paper, a motion prediction algorithm based on the KS regular elements is developed for the motion in the Earth's gravitational field with axial symmetry. The algorithm is of recursive nature and general in the sense that it could be applied for any conic motion whatever the number of zonal harmonic coefficientsN 2 may be. Applications of the algorithm for the problem of the final state prediction are illustrated by numerical examples of eight typical ballistic missiles for geopotential model with zonal harmonic terms up toJ ₃₆. A final state of any desired accuracy is obtained for each case study, a result which shows the efficiency and the flexibility of the algorithm.     

Construction d'orbites periodiques de satellites dans un systeme d'axes tournants,I

Résumé On développe une méthode de construction d'orbites périoldiques dans un système d'axes tournants, pour un satellite gravitant autour d'un sphéroide. Les orbites sont quasi circulaires,i est l'inclinaison sur le plan équatorial de la planète. Pour les petites inclinaisons, la solution est donnée jusqu'aux termes enJ ₂ ² etJ ₄.Ce modèle peut être appliqué aux satellites de Saturne. Des valeurs observées des longitudes des noeuds ascendants de Mimas et Téthys, on donne une estimation des valeurs deJ ₂ etJ ₄ du potentiel de Saturne. La valeur deJ ₂ est très sensible aux valeurs adoptées pour le rayon équatorial de la planète.

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Construction of periodic orbits of satellites in a moving system of axes, I

We give an algorithm for the construction of periodic orbits in a rotating frame for the cases of satellites moving around an oblate planet.The orbits are near to the circular case; the asymptotic developments of the periodic solutions are completely calculated for the termsJ ₂ andJ ₄ of the potential. The solutions for small inclinations are given up toJ ₂ ² .The families of solutions depend on three parameters: the semi-major axis, the inclination of the generating orbit and the initial position on this orbit.These solutions can be applied to the motion of the Saturnian satellites. From the observed longitudes of the ascending nodes of Mimas and Tethys, we estimate the valuesJ ₂ andJ ₄ of the Saturnian potential, the value ofJ ₂ very strongly depends on the adopted value of the planet's equatorial diameter.

     

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.     

An analytic satellite theory using gravity and a dynamic atmosphere

An analytical solution is given for the motion of an artifical Earth satellite under the combined influences of gravity and atmospheric drag. The gravitational effects of the zonal harmonicsJ ₂,J ₃, andJ ₄ are included, and the drag effects of any arbitrary dynamic atmosphere are included. By a dynamic atmosphere, we mean any of the modern empirical models which use various observed solar and geophysical parameters as inputs to produce a dynamically varying atmosphere model. The subtleties of using such an atmosphere model with an analytic theory are explored, and real world data is used to determine the optimum implementation. Performance is measured by predictions against real world satellites. As a point of reference, predictions against a special perturbations model are also given.     

Effect of 3rd-degree gravity harmonics and Earth perturbations on lunar artificial satellite orbits

In a previous work we studied the effects of (I) the J ₂ and C ₂₂ terms of the lunar potential and (II) the rotation of the primary on the critical inclination orbits of artificial satellites. Here, we show that, when 3rd-degree gravity harmonics are taken into account, the long-term orbital behavior and stability are strongly affected, especially for a non-rotating central body, where chaotic or collision orbits dominate the phase space. In the rotating case these phenomena are strongly weakened and the motion is mostly regular. When the averaged effect of the Earth’s perturbation is added, chaotic regions appear again for some inclination ranges. These are more important for higher values of semi-major axes. We compute the main families of periodic orbits, which are shown to emanate from the ‘frozen eccentricity’ and ‘critical inclination’ solutions of the axisymmetric problem (‘J ₂ + J ₃’). Although the geometrical properties of the orbits are not preserved, we find that the variations in e, I and g can be quite small, so that they can be of practical importance to mission planning.     

Long-term motion of resonant satellites with arbitrary eccentricity and inclination

A first-order, semi-analytical method for the long-term motion of resonant satellites is introduced. The method provides long-term solutions, valid for nearly all eccentricities and inclinations, and for all commensurability ratios. The method allows the inclusion of all zonal and tesseral harmonics of a nonspherical planet.We present here an application of the method to a synchronous satellite includingonly theJ ₂ andJ ₂₂ harmonics. Global, long-term solutions for this problem are given for arbitrary values of eccentricity, argument of perigee and inclination.     

Solutions and periodicity of satellite relative motion under even zonal harmonics perturbations

Finding relative satellite orbits that guarantee long-term bounded relative motion is important for cluster flight, wherein a group of satellites remain within bounded distances while applying very few formationkeeping maneuvers. However, most existing astrodynamical approaches utilize mean orbital elements for detecting bounded relative orbits, and therefore cannot guarantee long-term boundedness under realistic gravitational models. The main purpose of the present paper is to develop analytical methods for designing long-term bounded relative orbits under high-order gravitational perturbations. The key underlying observation is that in the presence of arbitrarily high-order even zonal harmonics perturbations, the dynamics are superintegrable for equatorial orbits. When only J ₂ is considered, the current paper offers a closed-form solution for the relative motion in the equatorial plane using elliptic integrals. Moreover, necessary and sufficient periodicity conditions for the relative motion are determined. The proposed methodology for the J ₂-perturbed relative motion is then extended to non-equatorial orbits and to the case of any high-order even zonal harmonics (J _2n , n ≥ 1). Numerical simulations show how the suggested methodology can be implemented for designing bounded relative quasiperiodic orbits in the presence of the complete zonal part of the gravitational potential.     

Hamiltonian Dynamics of a Gyrostat in the N-Body Problem: Relative Equilibria

We consider the non-canonical Hamiltonian dynamics of a gyrostat in Newtonian interaction with n spherical rigid bodies. Using the symmetries of the system we carry out two reductions. Then, working in the reduced problem, we obtain the equations of motion, a Casimir function of the system and the equations that determine the relative equilibria. Global conditions for existence of relative equilibria are given. Besides, we give the variational characterization of these equilibria and three invariant manifolds of the problem; being calculated the equations of motion in these manifolds, which are described by means of a canonical Hamiltonian system. We give some Eulerian and Lagrangian equilibria for the four body problem with a gyrostat. Finally, certain classical problems of Celestial Mechanics are generalized.     

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.     

Integrable approximation of J 2-perturbed relative orbits

Most existing satellite relative motion theories utilize mean elements, and therefore cannot be used for calculating long-term bounded perturbed relative orbits. The goal of the current paper is to find an integrable approximation for the relative motion problem under the J ₂ perturbation, which is adequate for long-term prediction of bounded relative orbits with arbitrary inclinations. To that end, a radial intermediary Hamiltonian is utilized. The intermediary Hamiltonian retains the original structure of the full J ₂ Hamiltonian, excluding the latitude dependence. This formalism provides integrability via separation, a fact that is utilized for finding periodic relative orbits in a local-vertical local-horizontal frame and determine an initialization scheme that yields long-term boundedness of the relative distance. Numerical experiments show that the intermediary-based computation of orbits provides long-term bounded orbits in the full J ₂ problem for various inclinations. In addition, a test case is shown in which the radial intermediary-based initial conditions of the chief and deputy satellites yield bounded relative distance in a high-precision orbit propagator.     

Estimation of low degree geopotential coefficients using SLR data

Geodetic satellites have been providing the low frequency part of the geopotential models used for precise orbit determination purposes (e.g. JGM3, EGM96, …). Nevertheless they can be used to estimate the temporal variation of selected coefficients, helping to clarify the complex interrelations in the earth-ocean-atmosphere system. In this paper we present the two years long analysis of SLR data from the seven available geodetic satellites (Lageos I–II, Stella, Starlette, Ajisai, Etalon I–II) to recover monthly estimates of low degree geopotential coefficients; the results are obtained analysing the satellites separately and in proper combination. An accurate modelling of the satellite orbits is required in order to separate the geopotential coefficients: we assume as a priori geopotential the JGM3 model together with its associated tides and we take care of non-gravitational effects on the satellites by means of proper empirical estimated accelerations. The time series of the estimated coefficients (J₂, J₃, J₄, J₅) are inspected to detect the sub-annual perturbations related to seasonal variation of mass distribution. Huge residual seasonal signals in the orbit of Stella indicate a strong model deficiency related to the Sun's influence on the environment. The remaining six satellites are homogeneously modelled and build up a three cycles per year oscillation on J₂ and a seasonal oscillation (1 year and six month periods) revealed on the J₄. The origin and possible causes of these signals are further discussed in the text. We also present a preliminary estimate, using twelve years of Lageos-I and Lageos-II observations, that is compared with previous obtained values.     

Prediction of satellites in Earth's gravitational field with axial symmetry using Burdet's regularized theory

In this paper, economical and stable recurrence formulae for the Earth's zonal potential and its gradient for Burdet's regularized theory will be established for any number N of the zonal harmonic coefficients. A general recursive computational algorithm based on these formulae is also established for the initial value problem of Burdet oscillator for the prediction of artificial satellites in the Earth's gravitational field with axial symmetry. Applications of the algorithm for the problem of the final state prediction are illustrated by numerical examples of three test orbits each for two geopotential models corresponding to N = 2 and N = 36.A final state of any desired accuracy is obtained for each case study, a result which shows the flexibility of the algorithm.Dept. of Astronomy, KAU     

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.     

Satellite theory

In this paper dynamical characteristics of satellites are outlined by classifying the satellites into three categories according to the values of the solar tidal factor (n/n)² which is the disturbing factor due to the sun and the oblateness factor of the primary planetJ ₂/a ². For inner satellites (n/n)² is much smaller thanJ ₂/a ² and there are several pairs among them, for which the mean motions are commensurable to each other, and for some of them secular accelerations in the mean longitudes have been detected. For outer satellites (n/n)² is much larger and the solar perturbations are dominant. For intermediary satellites the motion of the pole of the orbital plane is not so simple as those of the satellites of the other categories.     

Analytical solution to the perturbedJ 2 motion of Artificial Satellite in terms of Euler parameters

In this paper an analytical solution for the differential equations governing the motion of Artificial Satellite under the oblateness of the earth will be developed. We start with the differential equations in terms of Euler parameters. To compact algebra, we introduce the Cayley-Klein type complex variables. Comparison between numerical and analytical final states withJ ₂ will be given for a test case.     

Canonical Modelling of Relative Spacecraft Motion Via Epicyclic Orbital Elements

This paper presents a Hamiltonian approach to modelling spacecraft motion relative to a circular reference orbit based on a derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling of high-order terms and orbital perturbations within the context of the Clohessy–Wiltshire solution. First, the Hamiltonian is partitioned into a linear term and a high-order term. The Hamilton–Jacobi equations are solved for the linear part by separation, and new constants for the relative motions are obtained, called epicyclic elements. The influence of higher order terms and perturbations, such as Earth’s oblateness, are incorporated into the analysis by a variation of parameters procedure. As an example, closed-form solutions for J₂-invariant orbits are obtained.     

Frozen orbits for satellites close to an Earth-like planet

We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ ₂ throughJ ₉. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J ₂ only). In particular, we examine the manner in which the odd zonalJ ₃ breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J ₄+J ₂ ² =0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions.     

Analytical Solution of Coupled Perturbation of Tesseral Harmonic Terms of Mars's Non-Spherical Gravitational Potential

The non-spherical gravitational potential of the planet Mars is sig- nificantly different from that of the Earth. The magnitudes of Mars’ tesseral harmonic coefficients are basically ten times larger than the corresponding val- ues of the Earth. Especially, the magnitude of its second degree and order tesseral harmonic coefficient J_2,2 is nearly 40 times that of the Earth, and approaches to the one tenth of its second zonal harmonic coefficient J₂. For a low-orbit Mars probe, if the required accuracy of orbit prediction of 1-day arc length is within 500 m (equivalent to the order of magnitude of 10−4 standard unit), then the coupled terms of J2 with the tesseral harmonics, and even those of the tesseral harmonics themselves, which are negligible for the Earth satellites, should be considered when the analytical perturbation solution of its orbit is built. In this paper, the analytical solutions of the coupled terms are presented. The anal- ysis and numerical verification indicate that the effect of the above-mentioned coupled perturbation on the orbit may exceed 10⁻⁴ in the along-track direc- tion. The conclusion is that the solutions of Earth satellites cannot be simply used without any modification when dealing with the analytical perturbation solutions of Mars-orbiting satellites, and that the effect of the coupled terms of Mars's non-spherical gravitational potential discussed in this paper should be taken into consideration.