Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, elliptic expansions in terms of the sectorial variables _j ⁽ⁱ⁾ introduced by the author in Paper IV (Sharaf, 1982) to regularize the highly oscillating perturbation force of some orbital systems will be established analytically and computationally for the ninth, tenth, eleventh, and twelfth categories according to our adopted scheme of presentation drawn up in Paper V (Sharaf, 1983). For each of the elliptic expansions belonging to a category, literal analytical expressions for the coefficients of its trigonometric series representation are established. Moreover, some recurrence formulae satisfied by these coefficients are also established to facilitate their computation, and numerical results are included to provide test examples for constructing computational algorithms. Finally, the first collection of completed elliptic expansions in terms of _j ⁽ⁱ⁾ so explored will be given in Appendix A for the guidance of the reader.     

Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, elliptic expansions in terms of the sectorial variables _j ⁽ⁱ⁾ introduced recently in Paper IV (Sharaf, 1982) to regularize highly oscillating perturbations force of some orbital systems will be established analytically and computationally for the fifth and sixth categories. For each of the elliptic expansions belonging to a category, literal analytical expressions for the coefficients of its trigonometric series representation are established. Moreover, some recurrence formulae satisfied by these coefficients are also established to facilitate their computations; numerical results are included to provide test examples for constructing computational algorithms.     

Expansion theory for elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, we arrive at the end of the second step of our regularization approach, and in which, elliptic expansions in terms of the sectorial variables _j ⁽ⁱ⁾ introduced by the author in Paper IV (Sharaf, 1982b) to regularize the highly oscillating perturbation force of some orbital systems will be established analytically and computationally for the thirteenth, fourteenth, fifteenth, and sixteenth categories according to our adopted scheme of presentation drawn up in Paper V (Sharaf, 1983). For each of the elliptic expansions belonging to a category, literal analytic expressions for the coefficients of its trigonometric series representation are established. Moreover, some recurrence formulae satisfied by these coefficients are also established to facilitate their computations, and numerical results are included to provide test examples for constructing computational algorithms. Finally, the second and the last collection of completed elliptic expansion will be given in Appendix B, such that, the materials of Appendix A of Paper VIII (Sharaf, 1985b) and those of Appendix B of the present paper provide the reader with the elliptic expansions in terms of _j ⁽ⁱ⁾ so explored for the second step of our regularization approach.     

Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, elliptic expansions in terms of the sectorial variables θ _j ⁽ⁱ⁾ introduced in Paper IV (Sharaf, 1982) to regularise highly oscillating perturbation force of some orbital systems will be explored for the first four categories. For each of the elliptic expansions belonging to a category, literal analytical expressions for the coefficients of its trigonometric series representation are established. Moreover, some recurrence formulae satisfied by these coefficients are also established to facilitate their computations, numerical results are included to provide test examples for constructing computational algorithms.     

Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, the third step of the author's regularization approach will be started by establishing the expansions of the functionX _n ^(r) (, ,u) in terms of the sectorial variables _j ⁽ⁱ⁾ introduced in Paper IV (Sharaf, 1982) to regularize the highly-oscillating perturbation force of some orbital systems. The literal analytical expressions for the Fourier expansion of the function will be explored in terms of _j ⁽ⁱ⁾ for anyn positive integer,r any real number whatever the types and the number of sectors forming the divisions situation of the elliptic orbits may be. The basic computational materials of the theory will also be given and for which the method of solution, the recurrence formulae, and the general computational sequence for the coefficients are considered.     

Expansions of elliptic motion based on elliptic function theory

New expansions of elliptic motion based on considering the eccentricitye as the modulusk of elliptic functions and introducing the new anomalyw (a sort of elliptic anomaly) defined byw=u/2K–/2,g=amu–/2 (g being the eccentric anomaly) are compared with the classic (e, M), (e, v) and (e, g) expansions in multiples of mean, true and eccentric anomalies, respectively. These (q,w) expansions turn out to be in general more compact than the classical ones. The coefficients of the (e,v) and (e,g) expansions are expressed as the hypergeometric series, which may be reduced to the hypergeometric polynomials. The coefficients of the (q,w) expansions may be presented in closed (rational function) form with respect toq, k, k=(1–k ²)^1/2,K andE, q being the Jacobi nome relatedk whileK andE are the complete elliptic integrals of the first and second kind respectively. Recurrence relations to compute these coefficients have been derived.on leave from Institute of Applied Astronomy, St.-Petersburg 197042, Russia     

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.     

Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, the time transform and the explicit exact forms of the time will be established in terms of the sectorial variables _j ⁽ⁱ⁾ introduced in Paper IV (Sharaf, 1982) to regularize the highly oscillating perturbation force. Simple recurrence formulae are given to facilitate the computations. The formulations are general in the sense that they are valid whatever the types and the number of sectors forming the divisions situation of the elliptic orbit may be. Moreover, the constants of integration for the explicit forms of the time are determined in a way that it gives for these forms its generality during any revolution of the body in its Keplerian orbit.     

Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, literal analytical expressions for the coefficients of the Fourier series representation ofF will be established for anyx _i ; withn, N positive integers 1 and | _i | fori=1, 2,...n. Moreover, the recurrence formulae satisfied by these coefficients will also be established. Illustrative analytical examples and a full recursive computational algorithm, with its numerical results, are included. The applications of the recurrence formulae are also illustrated by their stencils. As a by-product of the analyses is an integral which we may call a complete elliptic integral of thenth kind, in which the known complete elliptic integrals (1st, 2nd and 3rd kinds) are special cases of it.     

Expansion of the perturbing function of a satellite restricted four-body problem

A satellite four-body problem is the problem of motion of an artificial satellite of a planet in a region of the space where perturbations due to the gravitational field of the planet are of the same order as perturbations due to influences of two perturbing bodies. In this paper an expansion of the perturbing function into a Fourier series in terms of angular Keplerian elements ( _j , _j ,M _j :j=0,1,2) (designations are standard) is obtained taking into account a sharp commensurability of the typen/ ₀=(p+q)/p (n is the mean motion of the artificial satellite and ₀ is the angular velocity of rotation of the planet,p andq are integers).The coefficients of the Fourier series are the functions of the positional Keplerian elements (a _j ,e _j ,i _j ;j=0, 1, 2) (designations are standard) and, in particular, are series in terms ofe _j that, generally speaking, can be written out to an accuracy ofe _j ¹⁹ .The expansion obtained can be used for the construction of a semianalytical theory of motion of resonant satellites on the basis of conditionally periodic solutions of the restricted four-body problem.     

Parameter distribution of small periodic librations about the equilateral points of the elliptic restricted problem

As previously shown (Rabe, 1970), two classes of small periodic librations exist, in the plane, elliptic restricted problem, for an infinite sequence of easily specified oscillation frequenciesZ _j . The present paper considers the dependence ofZ on the eccentricitye of the primary motion, in addition to its dependence on the mass parameter , and determines the resulting relations between ande, for any given periodic frequencyZ _j . These relationships are obtained from the conditionD (Z _j ,, e)=0, where the basic determinantD has been expanded up to terms of orderZ ²⁰, ⁵, ande ⁴.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.     

Fourier techniques for an analysis of eclipsing binary light curves: I

The Fourier techniques developed so far for an analysis of eclipsing binary light curves have been re-discussed. The Fourier coefficients for the analysis have been derived in a simple form of series expansions, in terms of eclipse elements, valid for any type of eclipse (regardless of whetherr ₁r ₂).These coefficients may be utilized to solve the eclipse elements in terms of the observed characteristics of the light curves. A general relation between the observed quantitiesl and , and the eclipse elementsr _1,2,i andL ₁ has also been given in the form of series expansions which can be used for the synthesis of the light curves.     

Expansion theory for the elliptic motion of arbitrary eccentricity and semi-major axis

In this paper of the series, the literal analytical expressions for the Fourier expansion of the Earth's spherical harmonic potential will be explored in terms of the sectorial variables _j ⁽ⁱ⁾ introduced in Paper IV (Sharaf, 1982) to regularize the highly-oscillating perturbation force of some orbital systems.Now at the Department of Astronomy, King Abdulaziz University Jeddah, Saudi Arabia.     

Fourier analysis of the light curves of eclipsing variables. XIII

New expressions for the fractional loss of light _l ⁰ have been derived in the simple forms of rapidly converging expansions to the series of Chebyshev polynomials, Jacobi polynomials, and Kopal'sJ-integrals. In these expansions, which are a supplement to those given by Kopal (1977b), variablesk andh occur in different products that simplify the numerical computation. The treatment follows the new definition of _l ⁰ which has been recently developed by Kopal (1977a).     

Expansion of the planetary mutual distance raised to any negative power by the method of symbolic differential operators

We present a literal approach to evaluate ^–s necessary for the construction of high order planetary theories. This approach is valid to be applied on very large scale digital computers with standard Poisson series programs, for high order and high degree planetary theories. We apply the method of symbolic differential operators for single variable functions, and the binomial theorem expansions, for the evaluation of ^–s . We utilize Laplace coefficients and its derivatives to carry out the development, without dealing with Newcomb operators or Hansen's coefficients.     

Numerical investigation of all possible equilibria of dual spin satellites

This paper is the continuation of a previous work [6] in which we have obtained the set of all possible equilibria of a gyrostat satellite attracted by n points mass by solving two algebraical equations P₁=0 and P₂=0. It results that there is a maximum of 24 isolated equilibrium orientations for the satellite. Sufficient conditions of stability for these relative equilibria are given.Here we consider only the elementary case n=1. We show that the coefficients of the two algebrical equations depend on four parameters j₁, j₃, K and v². The two first parameters depend only on the direction of the internal angular momentum of the rotors, the third being only function of the principal moments of inertia of the satellite and the last parameter is a decreasing function of one of the components of . We show that the two polynomials P₁ and P₂ are unvariant within two transformations of the parameters j₁ and j₃. It is then possible to reduce the range of variation of these parameters.For some particular values of the parameters, it is possible to give the minimum number of real roots of equations P₁=0 and P₂=0. In general cases, a computing program is written to obtain the number of real roots of these equations according to the values of the parameters. We show that among the roots found, few of them corresponds to stable equilibrium orientations.     

Fourier analysis of the light curves of eclipsing variables,XII

The aim of the present paper will be to make use of the expressions, established in Paper XI, for the fractional loss of light _l ⁰ of arbitrarily limb-darkened stars in the form of Hankel transforms of zero order, in order to evaluate the explicit forms of the _l ⁰'s for different types of eclipses (Section 2), as well as of the momentsA _2mof the respective light curves (Section 3)-in a closed form; or in terms of expansions that converge under all circumstances envisaged. Particular attention will be directed to a connection between these expansions and other functions already available in tabular form; or to alternative forms amenable to automatic computation.     

Application of the Jacobi integral for the investigation of satellite librations around the centre of mass in a gravitational field

This paper deals with a three-dimensional rotationally and dynamically symmetrical satellite. The centre of mass of the satellite moves in a circular orbit. The existence of two first integrals of motion enables one to transform the system of differential equations to a special form facilitating the choice of the zero-approximation solution. The angles of precession and nutation as well as the amplitude functionk ²(t) are taken as variables of the motion. The first approximation solution is constructed for the case of spatial libration of the satellite axis of dynamical symmetry about the position of stable equilibrium. The series representing the functionk ²(t) is fast convergent due to the fast convergence of the expansions for elliptic functions.     

Radius of convergence of Lie series for some elliptic elements

For equatorial orbits about an oblate body, we show that the Lie series for the elliptic elementse,f,l and diverge when the oblateness exceeds a critical multiple of the transformed eccentricity constant. The use of similar truncated series expansions for such elliptic elements by Brouwer accounts for the first-order errors at low eccentricity in his derived coordinates for an artificial satellite.