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.     

A numerical investigation of secular terms of the planetary disturbing function

The secular terms of the planetary disturbing function are given, after elimination of short period terms by von Zeipel's transformation. The adequacy of this expansion up to terms of eighth order in the inclination and eccentricity is investigated by numerical processes, as a function of the Keplerian elementsa, e andi. The eccentricityé of the outer planet, is taken equal to zero. It is concluded that for values ofi which are not small the inclusion of additional terms in the expression for the disturbing function, results to drastic changes in its values, while larger values ofe do not have an equaly large effect on the disturbing function.     

Elimination of the perigee in the satellite problem

Developed in this paper is a new approach to an analytic satellite theory which is based on Deprit's elimination of the parallax. The first step in the theory is the elimination of the parallax canonical transformation which eliminates the short period terms in the perturbations to within a factor of (1/r)². A new approach is then taken. The perigee terms are eliminated while retaining the short period terms in (1/r)². A Delaunay normalization of the short period terms in the (1/x)² factor is then constructed to complete the theory.     

The elimination of short-periodic terms in a Uranus-Neptune general planetary theory by von Zeipel's method

We eliminate by the method of von Zeipel the short-period terms in a first order-with respect to planetary masses—general planetary Uranus-Neptune theory. We exclude in the expansion terms of eccentricities and sines of inclinations higher than the third power.Our variables are the Poincaré canonical variables. We use the Jacobi-Radau set of origins, and we refer the planes of the osculating ellipses to a common fixed plane, the longitudes to a common origin. The short-periodic terms arising from the indirect and principal parts of the disturbing functions, are eliminated separately. The Fourier series of the principal part of the disturbing function, is reduced to the sum of only the first three terms.     

A second order Jupiter-Saturn planetary theory

We present a second order secular Jupiter-Saturn planetary theory through Poincaré canonical variables, von Zeipel's method and Jacobi-Radau referential. We neglect in our expansions terms of power higher than the fourth with respect to eccentricities and sines of inclinations. We assume that the disturbing function is composed of secular and critical terms only. We shall deriveF _2si and writeF _2s in terms of Poincaré canonical variables in Part II of this problem.     

Long-Term Evolution of Orbits About A Precessing Oblate Planet: 1. The Case of Uniform Precession

It was believed until very recently that a near-equatorial satellite would always keep up with the planet’s equator (with oscillations in inclination, but without a secular drift). As explained in Efroimsky and Goldreich [Astronomy & Astrophysics (2004) Vol. 415, pp. 1187–1199], this misconception originated from a wrong interpretation of a (mathematically correct) result obtained in terms of non-osculating orbital elements. A similar analysis carried out in the language of osculating elements will endow the planetary equations with some extra terms caused by the planet’s obliquity change. Some of these terms will be non-trivial, in that they will not be amendments to the disturbing function. Due to the extra terms, the variations of a planet’s obliquity may cause a secular drift of its satellite orbit inclination. In this article we set out the analytical formalism for our study of this drift. We demonstrate that, in the case of uniform precession, the drift will be extremely slow, because the first-order terms responsible for the drift will be short-period and, thus, will have vanishing orbital averages (as anticipated 40 years ago by Peter Goldreich), while the secular terms will be of the second order only. However, it turns out that variations of the planetary precession make the first-order terms secular. For example, the planetary nutations will resonate with the satellite’s orbital frequency and, thereby, may instigate a secular drift. A detailed study of this process will be offered in a subsequent publication, while here we work out the required mathematical formalism and point out the key aspects of the dynamics. In this article, as well as in (Efroimsky 2004), we use the word ‘‘precession’’ in its most general sense which embraces the entire spectrum of changes of the spin-axis orientation -- from the long-term variations down to the Chandler Wobble down to nutations and to the polar wonder.     

Theorie generale planetaire etendue au cas de la resonance et application au systeme des satellites galileens de Jupiter

We consider a system of planets defined by a given distribution of mean mean motions and masses: we represent the osculating elliptic elements of their heliocentric orbits by quasi-periodic functions of time, through a method adapted to the commensurability case; these functions are the sum of the general solution of a critical system, expressed in long-period terms, and of a particular solution. As in the B. Brown's method (applied to the galilean satellites), the critical system contains the secular terms, the longperiod terms (great inequalities), and the resonant terms; the particular solution consists of short-period terms only, whose amplitude is an explicit function of the solution of the critical system.If all the long-period terms in the critical system are harmonic of one fundamental term, we can perform a simple change of variables which transforms the critical system in an autonomous one, and thus we reduce the resolution to an eigenvalue problem. Applying that to the galilean satellites of Jupiter and neglecting the solar perturbations, we obtain a differential system with constant coefficients, whose linear part concerns all the variables (including the major-axes and the mean longitudes) and gives, as a first approximation, the great inequalities, the free oscillations and the libration; nevertheless this solution agrees already with known results, but should be improved by taking into account the non-linear parts and the solar terms in a new approximation.

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Proceedings of the Conference on Analytical Methods and Ephemerides: Theory and Observations of the Moon and Planets. Facultés universitaires Notre Dame de la Paix, Namur, Belgium, 28–31 July, 1980     

Prediction of satellite orbits contraction due to diurnally varying oblate atmosphere and altitude-dependent scale height using KS canonical elements

A new non-singular analytical theory for the motion of near-Earth satellite orbits with the air drag effect is developed in terms of uniformly regular KS canonical elements. Diurnally varying oblate atmosphere is considered with variation in density scale height dependent on altitude. The series expansion method is utilized to generate the analytical solutions and terms up to fourth-order terms in eccentricity and c (a small parameter dependent on the flattening of the atmosphere) are retained. Only two of the nine equations are solved analytically to compute the state vector and change in energy at the end of each revolution, due to symmetry in the equations of motion. The important drag perturbed orbital parameters: semi-major axis and eccentricity are obtained up to 500 revolutions, with the present analytical theory and by numerical integration over a wide range of perigee height, eccentricity and inclination. The differences between the two are found to be very less. A comparison between the theories generated with terms up to third- and fourth-order terms in c and e shows an improvement in the computation of the orbital parameters semi-major axis and eccentricity, up to 9%. The theory can be effectively used for the re-entry of the near-Earth objects, which mainly decay due to atmospheric drag.     

Collisional processes in stellar systems

The theory of collisional relaxation in stellar systems is discussed in terms of an expansion in powers of 1/N, the inverse of the total number of stars. The results are expressed in terms of the concept of gravitational polarization.     

The Fourier-Chebychev approximation for time series with a great many terms

A new type of approximation is proposed to evaluate Fourier series with a great many terms. It combines Fourier series and Chebychev polynomials to represent time series over a finite time interval. On one side, the high frequencies are approximate multiples of a basic frequency to represent short periodic terms; on the other side, the slowly variable functions of the time, on a given interval, which contain terms with long periods, are approximated by Chebychev polynomials. Application is made in case of the Lunar Theory ELP which contains about 30 000 trigonometric terms. The computing time necessary to evaluate coordinates is very much reduced when we give to series the intermediary representation inF-T form, as compared to a direct substitution of the time in the arguments.One shows also on a numerical example, that the Fourier-Chebychev approximation smoothly degrades outside its range of representation.Bureau des Longitudes. Equipe de Recherche Associée au CNRS     

Satellite orbits perturbed by direct solar radiation pressure: General expansion of the disturbing function

An expression is derived for the solar radiation pressure disturbing function on an Earth satellite orbit which takes into account the variation of the solar radiation flux with distance from the Sun's centre and the absorption of radiation by the satellite. This expression is then expanded in terms of the Keplerian elements of the satellite and solar orbits using Kaula's method. The Kaula inclination functions are replaced by an equivalent set of modified Allan inclination functions.The resulting expression reduces to the form commonly used in solar radiation pressure perturbation studies (e.g. Aksnes, 1976), when certain terms are neglected. If, as happens quite often in practice, a satellite's orbit is in near-resonsnce with certain of these neglected terms, these near-resonant terms can cause changes in the satellite's orbital elements comparable to those produced by the largest term in Aksnes's expression. A new expression for the solar radiation pressure disturbing function expansion is suggested for use in future studies of satellite orbits perturbed by solar radiation pressure.     

An elimination of the critical terms of a first-order Uranus-Neptune theory by Hori's method

We eliminate the 1:2 critical terms — after a previous elimination of the short period terms — in the Hamiltonian of a first order U-N theory. We take into account terms of degree 0, 1, 2, 3, 4 in the eccentricity-inclination. We apply for this elimination the Hori-Lie technique through the Poincaré canonical variables and the Jacobi coordinates. The purely principal first order secular U-N Hamiltonian admits a complete solution. We obtained the U-N equations of motion generated by the principal first order long period U-N Hamiltonian which will be solved later. This part III is closely related to the two previous papers (Kamel, 1982, 1983).     

The construction of a third-order secular analytical J-S-U-N theory by the Hori-Lie technique

We expand both parts, the principal and indirect, of the Hamiltonian function up to the third order in the masses for the four major planets Jupiter-Saturn-Uranus-Neptune. Accordingly we write down the secular terms ofF ₁,F ₂,F ₃ and the critical terms ofF ₁,F ₂ in terms of the canonical variables of H. Poincaré neglecting powers higher than the second inH, K, P, Q.     

Third-order Uranus-Neptune planetary theory

In this part we calculate the secular and critical terms arising from the indirect part of the classical planetary Hamiltonian for Uranus and Neptune. We neglect in our expansions powers higher than the second in the eccentricity-inclination. Our required results, are expressed in terms of Poincaré variables.     

Analytic formulas for the mass-transfer rate and the evolution of a close binary system of neutron (degenerate) stars

We derive approximate analytic relations between the mass-transfer rate in a close binary system described in terms of the Roche potential and its basic parameters, such as the total mass of the binary, the radius of its circular orbit, the mass of the mass-losing component, and the degree of its Roche lobe overfilling. Using simplifying assumptions (conservative mass transfer, a short relaxation time of matter on the mass-gaining component compared to the mass-transfer time scale, adiabaticity and quasi-stationarity of the mass flow through the Lagrangian point L ₁) allows the evolution of a binary system of neutron (degenerate) stars to be described in terms of two ordinary differential equations. This makes it possible to qualitatively analyze the evolution process, which is useful in those cases where the evolution of a close binary system must be investigated in general terms, for example, in terms of the scenario for the transformation of the collapse of a rotating presupernova core into a supernova explosion proposed by Imshennik and Nadyozhin (1992) and Imshennik (1992).     

An analytic solution for theJ 2 perturbed equatorial orbit

An analytic solution for theJ ₂ perturbed equatorial orbit is obtained in terms of elliptic functions and integrals. The necessary equations for computing the position and velocity vectors, and the time are given in terms of known functions. The perturbed periapsis and apoapsis distances are determined from the roots of a characteristic cubic.     

Third-order Jupiter — Saturn planetary theory

The construction of a third order J-S theory is presented. The Hori theory of planetary perturbations is employed. No Critical J-S terms due to the 2:5 commensurabilities and its multiples exist, when we take into account the periodic terms of order 0, 1, 2 with respect to the eccentricity- inclination. In this case the Lie series transformation degenerates and is meaningless. The J-S equations of motion for secular perturbations are solved when we neglect in our treatment, the Poisson terms of degree > 2 in the Poincaré canonical variables H _u , K _u , P _u Q _u (u = 1, 2). The Jacobi-Radau referential is adopted, and the theory is expressed in terms of the canonical variables of H. Poincaré.Now at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, U.S.A.     

A second-order Uranus-Neptune theory by Hori's method

We shall establish a second order - with respect to a small parameter which is of the order of planetary masses - Uranus-Neptune canonical planetary theory. The construction will be through the Hori-Lie perturbation theory. We perform the elliptic expansions by hand, taking into account powers 0, 1, 2 of the eccentricity-inclination. Only the principal part of the planetary Hamiltonian will be taken into consideration. Our theory will be expressed in terms of the canonical variables of Henri Poincaré, referring the planetary coordinates to the Jacobi-Radau system of origin. Only U- N critical terms will be assumed as the periodic terms.     

Determination of secular terms in general planetary theory

A method to calculate secular terms of the two parts of the planetary disturbing function— when it is expressed in terms of the true anomalies or the eccentric anomalies instead of the mean anomalies - is described. Also an alternative method is outlined.     

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.