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.     

The Impact of the Static Part of the Earth's Gravity Field on Some Tests of General Relativity with Satellite Laser Ranging

In this paper we calculate explicitly the classical secular precessions of the node and the perigee of an Earth artificial satellite induced by the even zonal harmonics of the static part of the geopotential up to degree l = 20. Subsequently, their systematic errors induced by the mismodelling in the even zonal spherical harmonics coefficients J _l are compared to the general relativistic secular gravitomagnetic and gravitoelectric precessions of the node and the perigee of the existing laser-ranged geodetic satellites and of the proposed LARES. The impact of the future terrestrial gravity models from CHAMP and GRACE missions is discussed as well. Preliminary estimates with the recently released EIGEN-1S gravity model including the first CHAMP data are presented.     

Analytical approach using KS elements to short-term orbit predictions including J 2

Analytical solutions using KS elements are derived. The perturbation considered is the Earth's zonal harmonic J ₂. The series expansions include terms of fourth power in the eccentricity. Only two of the nine KS element equations are integrated analytically due to the reasons of symmetry. The analytical solution is suitable for short-term orbit computations. Numerical studies show that reasonably good estimates of the orbital elements can be obtained in one step of 10 to 30 degrees of eccentric anomaly for near-Earth orbits of moderate eccentricity. For application purposes, the analytical solution can be effectively used for onboard computation in the navigation and guidance packages, where the modelling of J ₂ effect becomes necessary.     

Electron-cyclotron maser emission from the sun and stars: Variations with plasma temperature and density

Very bright and highly circularly polarized radio bursts from the Sun, the planets, flare stars, and close binary stars are attributed to the electron-cyclotron maser instability. The mode and frequency of the dominant radiation from the maser instability is shown to be dependent on the plasma temperature and the ratio _p / _e of the plasma frequency to the electron-cyclotron frequency. For the emission from the Sun _p / _e is probably greater than 0.3 and for 0.3 < _p / _e < 2 the emission can be either in the x-mode at the second harmonic or in the o- and/or z-modes at the fundamental. For higher _p / _e , the emission moves to higher harmonics of _e with the emission being predominately in the z-mode when _p / _e > 3.Proceedings of the Workshop on RadioContinua during Solar Flares, held at Duino (Trieste), Italy, 27–31 May, 1985.     

Analytical short-term orbit predictions withJ 3 andJ 4 in terms of KS elements

Analytical theory for short-term orbit motion of satellite orbits with Earth's zonal harmonicsJ ₃ andJ ₄ is developed in terms of KS elements. Due to symmetry in KS element equations, only two of the nine equations are integrated analytically. The series expansions include terms of third power in the eccentricity. Numerical studies with two test cases reveal that orbital elements obtained from the analytical expressions match quite well with numerically integrated values during a revolution. Typically for an orbit with perigee height, eccentricity and inclination of 421.9 km, 0.17524 and 30 degrees, respectively, maximum differences of 27 and 25 cm in semimajor axis computation are noted withJ ₃ andJ ₄ term during a revolution. For application purposes, the analytical solutions can be used for accurate onboard computation of state vector in navigation and guidance packages.     

Extension of the solution of Kepler's equation to high eccentricities

The classic Lagrange's expansion of the solutionE(e, M) of Kepler's equation in powers of eccentricity is extended to highly eccentric orbits, 0.6627 ... <e<1. The solutionE(e, M) is developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of the expansion are given in terms of the derivatives of the Bessel functionsJ _n (ne). The expansion is convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is a positive real number, which is calculated numerically.     

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.     

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.     

On the perturbation of the anomalistic period of a highly eccentric orbit satellite due to the zonal harmonics

In this note a simple formula is given for the perturbation of the anomalistic period of a highly eccentric orbit due to the zonal harmonics. This perturbation depends essentially only on the semi-major axisa, the eccentricitye (or pericentre radius r =a(1-e)) and the latitude of the pericentre.     

Analysis of 27 satellite orbits to determine odd zonal harmonics in the geopotential

The odd zonal harmonics in the geopotential are the terms independent of longitude and antisymmetric about the Equator: they define the ‘pear-shape’ effect. The coeffecients J₃, J₅, J₇,…of these harmonics have been evaluated by analysing the variations in eccentricity of 27 orbits covering wide range of inclinations. We use again most of the orbits from our previous (1969) evaluations, but we now have the advantage of 3 accurate orbits at inclinations between 60° and 66°, where the variations in eccentricity become very large, and 3 near-equatorial orbits, at inclinations between 3° and 15°, whereas previously there were none at inclinations lower than 28°. The new data lead to much more accurate and reliable values for the coeffecients. Our recommended set, which terminates at J₁₇, is

$\text{10}{}^9\text{J}{}_3\text{= ?2531 ± 7}\text{10}{}^9\text{J}{}_{11}\text{= 159 ± 16}\text{J}{}_5\text{= ?246 ± 9}\text{J}{}_{13}\text{= ?131 ± 22}\text{J}{}_7\text{= ?326 ± 11}\text{J}{}_{15}\text{= ?26 ±24}\text{J}{}_9\text{= ?94 ± 12}\text{J}{}_{17}\text{= ?258 ± 19}$

. With this new set of values the pear-shape tendency of the Earth amounts to 44.7 m at the poles, instead of the previous 40 m, though the new geoid is within 1 m of the old at latitudes away from the poles.     

Satellite orbital precessions caused by the first odd zonal J 3 multipole of a non-spherical body arbitrarily oriented in space

An astronomical body of mass M and radius R which is non-spherically symmetric generates a free space potential U which can be expanded in multipoles. As such, the trajectory of a test particle orbiting it is not a Keplerian ellipse fixed in the inertial space. The zonal harmonic coefficients J ₂,J ₃,… of the multipolar expansion of the potential cause cumulative orbital perturbations which can be either harmonic or secular over time scales larger than the unperturbed Keplerian orbital period T. Here, I calculate the averaged rates of change of the osculating Keplerian orbital elements due to the odd zonal harmonic J ₃ by assuming an arbitrary orientation of the body’s spin axis \(\hat{\boldsymbol{k}}\) . I use the Lagrange planetary equations, and I make a first-order calculation in J ₃. I do not make a-priori assumptions concerning the eccentricity e and the inclination i of the satellite’s orbit.     

The critical inclination problem with small eccentricity

A qualitative solution is presented of the critical inclination problem in artificial satellite theory for motions in which the orbits are nearly circular. The effects of all the zonal harmonics are taken into account, and bothshallow anddeep resonance regimes are considered. An investigation of the (e sing,e cosg)-plane reveals that six fundamentally different types of phase-plane portraits exist. These portraits illustrate the long-term behaviour of the eccentricity and line of apsides.     

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.     

Orbits near critical inclination,including lunisolar perturbations

An improved theory is presented of long period perigee motion for orbits near the critical inclinations 63.4° and 116.6°. Inclusion of lunisolar perturbations andall measured zonal harmonic coefficients from a recent Earth model are significant improvements over existing theories. Phase portraits are used to depict the interaction between eccentricity magnitude and argument of perigee. The Hamiltonian constant can be chosen as the parameter to display a family of phase plane trajectories consisting of libration, circulation, and asymptotic motion along separatrices near equilibrium points. A two parameter family of phase portraits is defined by the other two integrals, the average semimajor axis and component of angular momentum resolved along the Earth's polar axis. There are regions of the parameter space where the stability and total number of equilibria can change, or two separatrices can coalesce. These phenomena signal large qualitative changes in phase portrait topology. Numerical studies show that lunisolar perturbations control stability of equilibria for orbits with semimajor axes exceeding 1.4 Earth radii. Moreover, a theory which includes lunisolar perturbations predicts larger maximum fluctuations in eccentricity and faster oscillations near stable equilibria compared to a theory which models only the zonal harmonics.     

Analytical expansion of the earth's zonal potential in terms of ks regular elements

In this paper, the expansion of the Earth's zonal potential is established analytically in terms of KS regular elements whatever the power of the eccentricity e(e < l) and the number of the zonal harmonics may be.     

Expansions of (r/a)m cos jv and (r/a)m sin jv to high eccentricities

Expansions of the functions (r/a)^cos jv and (r/a)m ^sin jv of the elliptic motion are extended to highly eccentric orbits, 0.6627 ... <e<1. The new expansions are developed in powers of (e–e*), wheree* is a fixed value of the eccentricity. The coefficients of these expansions are expressed in terms of the derivatives of Hansen's coefficients with respect to the eccentricity. The new expansions are convergent for values of the eccentricity such that |e–e*|<(e*), where the radius of convergence (e*) is the same of the extended solution of Kepler's equation. The new expansions are intrinsically related to Lagrange's series.     

Multiple expansion of the tidal potential

The Earth tidal deformation causes an additional gravitational potential. Its effect on the Moon orbital motion has been studied by several authors.In this contribution, we develop this additional potential without specifying the inertial frame chosen.For this purpose, we use the properties of the representation of rotation groups in 3 dimensions space. We finally obtain the interaction potential between the distorted Earth and the Moon which is a necessary preliminary to the study of the evolution of the Earth-Moon system.Nomenclature T.R.O Tide raising object - (, , ) Spherical coordinates of the T.R.O. - (J, _E ) Earth spin axis orientation. _E is the longitude of the ascending node of Earth's equator on thexy-plane - (a ,I ,e , , ,M ) Elliptics elements of the T.R.O     

Variations of perturbations in perigee height with eccentricity for artificial Earth's satellites due to air drag

The variations of perturbations in perigee distance for different values of the orbital eccentricity for artificial Earth's satellites due to air drag have been studied. The analytical solution of deriving these perturbations, using the TD model (Total Density) have been applied, Helali (1987). The Theory is valid for altitudes ranging from 200 to 500 km above the Earth's surface and for solar 10.7 cm flux. Numerical examples are given to illustrate the variations of the perturbations in perigee distance with changing eccentricity (e < 0.2). A stronge perturbations in the perigee distance have been shown when the eccentricity in the range 0.001 <e < 0.05, especially for perigee distance 200 km.