Numerical integration of the variational equations of satellite orbits

A recurrent power series (RPS) method is constructed for the numerical integration of the equations of motion together with the variational equations of N point masses orbiting around an oblate spheroid. By the term “variational equations” we mean the equations of the partial derivatives of the bodies’ position and velocity components with respect to the initial conditions, the relative masses and the spheroid's oblateness coefficients J₂ and J₄. The construction of recursive relations for the partial derivatives involved in the variational equations is based on partial differentiation of the corresponding recursive relations for the integration of the equations of motion. Since the number of the auxiliary variables needed for this complex system becomes tremendously large when N>1, special care must be taken during computer implementation, so as to minimize the amount of computer memory needed as well as the cost in CPU time. The RPS method constructed in this way is tested for N=1,…,4 using real initial conditions of the Saturnian satellite system. For various sets of satellites, we monitor the behaviour of all the corresponding partial derivatives. The results show a prominent difference in the behaviour of the partial derivatives between resonant and non-resonant orbital systems.     

Numerical integration methods for orbital motion

The present report compares Runge-Kutta, multistep and extrapolation methods for the numerical integration of ordinary differential equations and assesses their usefulness for orbit computations of solar system bodies or artificial satellites. The scope of earlier studies is extended by including various methods that have been developed only recently. Several performance tests reveal that modern single- and multistep methods can be similarly efficient over a wide range of eccentricities. Multistep methods are still preferable, however, for ephemeris predictions with a large number of dense output points.     

On the computation of the spherical harmonic terms needed during the numerical integration of the orbital motion of an artificial satellite

A method is presented for the accurate and efficient computation of the forces and their first derivatives arising from any number of zonal and tesseral terms in the Earth's gravitational potential. The basic formulae are recurrence relations between some solid spherical harmonics,V _n,m, associated with the standard polynomial ones.     

Numerical integration of periodic orbits in the main problem of artificial satellite theory

We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ ₂ problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ ₂ andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (–1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ ₂, as large as |J ₂|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.     

Second-order theory of the rotation of an artificial satellite

This work presents the expansion of the second-order of an analytical theory of the attitude evolution of an artificial satellite perturbed by given torques. The first-order of the theory has already been presented by the author in Celestial Mechanics39 (1986) 309–327. It is a theory that is valid under very general conditions including slow rotation and inequal axes of inertia. The present theory is suitable for any internal or external disturbing forces producing the torques. A formal solution is expanded in the second-order according to powers of a small parameter characteristic of the order of magnitude of the disturbing torques. These torques are expanded in Fourier series and the theory applies whatever is the length of these series. The coefficients of the solution are given by an iterative formation law. The comparison of the results with a numerical integration based upon a HIPPARCOS model shows that the second order has brought an improvement to the theory by at least one order of magnitude over the results of the first order.     

Theory of the motion of an artificial Earth satellite

An improved analytical solution is obtained for the motion of an artificial Earth satellite under the combined influences of gravity and atmospheric drag. The gravitational model includes zonal harmonics throughJ ₄, and the atmospheric model assumes a nonrotating spherical power density function. The differential equations are developed through second order under the assumption that the second zonal harmonic and the drag coefficient are both first-order terms, while the remaining zonal harmonics are of second order.Canonical transformations and the method of averaging are used to obtain transformations of variables which significantly simplify the transformed differential equations. A solution for these transformed equations is found; and this solution, in conjunction with the transformations cited above, gives equations for computing the six osculating orbital elements which describe the orbital motion of the satellite. The solution is valid for all eccentricities greater than 0 and less than 0.1 and all inclinations not near 0^o or the critical inclination. Approximately ninety percent of the satellites currently in orbit satisfy all these restrictions.     

Analytical treatment of air drag and earth oblateness effects upon an artificial satellite

A development of an analytical solution for the motion of an artificial Earth satellite subject to the combined effects of Earth gravity and air drag is presented. The atmospheric model takes into account a linear variation of the density scale height with altitude, the rotation and the oblateness of the atmosphere. The perturbation theory is based upon Lie transforms. The secular and long-periodic terms as well as the short-periodic effects are included in the theory which is valid for small to moderate eccentricities and for all values of the inclination.Belgian National Fund for Scientific Research     

Numerical continuation of families of frozen orbits in the zonal problem of artificial satellite theory

In the zonal problem of a satellite around the Earth, we continue numerically natural families of periodic orbits with the polar component of the angular momentum as the parameter. We found three families; two of them are made of orbits with linear stability while the third one is made of unstable orbits. Except in a neighborhood of the critical inclination, the stable periodic (or frozen) orbits have very small eccentricities even for large inclinations.     

Dependence on the observational time intervals and domain of convergence of orbital determination methods

In the framework of the orbital determination methods, we study some properties related to the algorithms developed by Gauss, Laplace and Mossotti. In particular, we investigate the dependence of such methods upon the size of the intervals between successive observations, encompassing also the case of two nearby observations performed within the same night. Moreover we study the convergence of Gauss algorithm by computing the maximal eigenvalue of the jacobian matrix associated to the Gauss map. Applications to asteroids and Kuiper belt objects are considered.     

Air drag effect on the motion of an artificial Earth satellite

Two methods have been used to compute and compare the perturbations in perigee distance for an artificial Earth satellite. The two methods have used different air density models. The first (Helali, 1987) used the TD model, formulated by Sehnel (1986a), which contains terms that describe all the principal changes of the thermospheric density due to solar activity, geomagnetic activity, and the height. The second method (Davis, 1963) used a model of the density which takes into account the rotation of the atmosphere, the bulging atmosphere and the height. For different values of eccentricities from 0.001 to 0.05 we computed the perturbations P _r in the perigee distance at different heights from 200 to 350 km for both methods. The results show a good agreement for the computed values of P _r for different values of e (0 < e 0.02) in both methods at perigee heights from 250 to 350 km. Meanwhile, for perigee heights smaller than about 250 km we found a maximum difference in P _r amounting to 20 metres/revolution for e = 0.005 and 0.01.     

On the determination of the long period tidal perturbations in the elements of artificial earth satellites

In the present article we develop the theory of the long period tidal effects in the motion of artificial satellites assuming the variability of elastic parameters of the Earth (Love numbers) across the parallels. The dependence of Love numbers on the longitude produces perturbations of the period of one day or less and hence is neglected in the present theory. In this respect we follow in the footsteps of Kaula (1969). If the deviations ofk ₂ andk ₃ from pure constants are not taken into consideration, then the perturbations caused by the variability ofk ₂ andk ₃ across the parallels will be misinterpreted as the perturbations caused byk ₄...-terms, and the spurious values ofk ₄... will be deduced. It is extremely doubtful, however, that the real effects caused byk ₄,k ₅,..., are significant enough to be detected. The short period effects with the period of the revolution of the satellite, or less, were removed from the differential equations for the variation of elements of the satellite by the averaging over the orbit of the satellite. These differential equations are in the form convenient for numerical integration over a long interval of time and also suitable for developing the tidal effects into trigonometric series with the arguments ω, Ω of the satellite andl, l′, F, D, Γ of the Moon. The numerical integration can be performed using some simple quadrature formula, without resorting to a predictor-corrector system.     

Orbit determination for a radar mapping satellite of Venus

'Geology and Tectonics of Venus', special issue edited by Alexander T. Basilevsky (USSR Acad. Sci. Moscow), James W. Head (Brown Univ., Providence), Gordon H. Pettengill (M.I.T., Cambridge, Mass.) and R. S. Saunders (J.P.L., Pasadena).     

Effects of motion of the equatorial plane on the orbital elements of an earth satellite

Exact differential equations relating the perturbations to satellite orbital elements by the motion of the Earth's equatorial plane are derived, and they are solved to second order in precession. The system proposed in a previous paper (Kozai, 1960), in which the inclination and the argument of perigee are referred to the equator of date and the longitude of the ascending node is measured from a fixed point along a fixed plane and then along the equator of date, can still be recommended for precise studies of satellite motion even when the second-order perturbations are taken into account.     

The geomagnetic effects on the motion of an electrically charged artificial satellite

The orbital effects of the Lorentz force on the motion of an electrically charged artificial satellite moving in the Earth's magnetic field are determined. The geomagnetic field is considered as a multipole potential field and the satellite electrical charge is supposed to be constant. The relativistic perturbations of the main geomagnetic field are discussed briefly. The results are concentrated on the determination of the secular changes, and numerical values are computed for the case of the LAGEOS satellite. The results are discussed in the context of a possible detection of the Lense-Thirring effect analyzing the orbital perturbations of the LAGEOS and LAGEOS X satellites.     

Numerical determination of stability regions for orbital motion in uniformly rotating second degree and order gravity fields

The stability of orbital motion about a uniformly rotating arbitrary second degree and order gravity field is investigated. A normalized form of the equations of motion are derived and analyzed. A numerical stability criteria is proposed and used to evaluate the stability of initially near-circular orbits in the equatorial plane of the body. Regions of stable and unstable motion are clearly delineated, and are seen to be strongly related to resonances between the mean motion and the body rotation rate.     

Length of arc as independent argument for highly eccentric orbits

For analytic step regulation in numerical integration of highly eccentric orbits it is proposed to use the orbital arc length of a moving particle as independent argument.     

Effects of physical librations of the Moon on the orbital elements of a lunar satellite

Physical librations of the Moon are small cyclic perturbations with periods of one month and longer, and amplitudes of 100 arc seconds or less. These cause the selenographic axes fixed in the true Moon to have a different orientation than similar axes fixed in the mean Moon.Physical librations have two types of effects of present interest. If the orbital elements of a lunar satellite are referred to selenographic axes in the true Moon as it rotates and librates, then the librations cause changes in the orientation angles (node, inclination, and periapsis argument of the satellite) large enough that long-period perturbation theory cannot be used without compensation for such geometrical effects. As a second effect, the gravitational potential of the Moon is actually wobbled in inertial space, a condition not included in the potential expression used in perturbation theory.This paper gives data on the magnitude of the physical librations, the geometrical effects on the orbital elements and the equivalent changes in the coefficients in the potential. It is shown that geometrical effects can be accommodated either by using an inertial axes system or by compensating for the lunar librations and precession when the selenographic axes are used. Further, it is shown that physical effects are small and negligible for all but the most exacting endeavors.     

Sensitivity of the orbital content of a model stellar system to the potential approximation used to describe it

We have classified orbits in a stationary triaxial stellar system created from a cold dissipationless collapse of 100,000 particles. In order to integrate the orbits, two potential approximations with different fitting functions were used in turn. We found that the relative amount of chaotic versus regular orbits does depend on the chosen approximation of potential, even though both models resulted in very good fits of the underlying exact potential. On the other hand, the content of regular orbits, i.e., its distribution among main families, does not strongly depend of the potential approximation, being therefore a more robust signature of the gravitational system under study.     

Analytical theory of a lunar artificial satellite with third body perturbations

We present here the first numerical results of our analytical theory of an artificial satellite of the Moon. The perturbation method used is the Lie Transform for averaging the Hamiltonian of the problem, in canonical variables: short-period terms (linked to l, the mean anomaly) are eliminated first. We achieved a quite complete averaged model with the main four perturbations, which are: the synchronous rotation of the Moon (rate ), the oblateness J ₂ of the Moon, the triaxiality C ₂₂ of the Moon ( ) and the major third body effect of the Earth (ELP2000). The solution is developed in powers of small factors linked to these perturbations up to second-order; the initial perturbations being sorted ( is first-order while the others are second-order). The results are obtained in a closed form, without any series developments in eccentricity nor inclination, so the solution apply for a wide range of values. Numerical integrations are performed in order to validate our analytical theory. The effect of each perturbation is presented progressively and separately as far as possible, in order to achieve a better understanding of the underlying mechanisms. We also highlight the important fact that it is necessary to adapt the initial conditions from averaged to osculating values in order to validate our averaged model dedicated to mission analysis purposes.