Direct Calculation Methods of Hansen Coefficients and Their Derivatives

Seven direct calculation methods of Hansen coefficients and their derivatives are reviewed. The computational efficiencies of these methods are compared, and their computational stabilities are analyzed. We show that the recursion relations of Hansen coefficients can be used to determine the stabilities of calculation results. Finally, it is pointed out that Wnuk's method (double precision computation) and McClain's methods (quadruple precision computation) are stable, which can be used to calculate orbit perturbations. Because of small orbital eccentricities of most satellites, the perturbation calculations without singularities are required, and McClain's first method (quadruple precision computation) is recommended.     

Hansen系数及其导数的直接计算方法

回顾总结了7种Hansen系数及其导数的直接计算方法,比较分析了这些方法的计算效率和计算稳定性.研究表明:Hansen系数的递推关系可以用来判别计算结果的稳定性.最后指出, Wnuk方法(双精度计算)和McClain方法(4精度计算)是稳定的,可以用来计算人造卫星轨道摄动.由于大多数人造卫星采用小偏心率轨道,需要计算无奇点摄动,推荐使用McClain方法1 (4精度计算).     

Recursive calculation of hansen coefficients

Hansen coefficients are used in expansions of the elliptic motion. Three methods for calculating the coefficients are studied: Tisserand's method, the Von Zeipel-Andoyer (VZA) method with explicit representation of the polynomials required to compute the Hansen coefficients, and the VZA method with the values of the polynomials calculated recursively. The VZA method with explicit polynomials is by far the most rapid, but the tabulation of the polynomials only extends to 12th order in powers of the eccentricity, and unless one has access to the polynomials in machine-readable form their entry is laborious and error-prone. The recursive calculation of the VZA polynomials, needed to compute the Hansen coefficients, while slower, is faster than the calculation of the Hansen coefficients by Tisserand's method, up to 10th order in the eccentricity and is still relatively efficient for higher orders. The main advantages of the recursive calculation are the simplicity of the program and one's being able to extend the expansions to any order of the eccentricity with ease. Because FORTRAN does not implement recursive procedures, this paper used C for all of the calculations. The most important conclusion is recursion's genuine usefulness in scientific computing.     

A New Method to Determine the Derivatives of the Laplace Coefficients

The determination of the secular variations of the orbital elements of objects in N-body systems is based on the literal development of the perturbing function. The development makes use of the Laplace coefficients and their derivatives. In this paper a new method is described for the analytical computation of the derivatives of the Laplace coefficients. It is an explicit formula in the sense that it only contains the Laplace coefficients and the parameter on which the Laplace coefficients depend. The advantage of this method is that it is unnecessary to calculate all the derivatives up to the desired order. It is enough to calculate the Laplace coefficients. Easy coding is a further benefit of the method and it provides more accurate numerical results. The paper describes in detail the application of the method through an example and gives comparison with former methods.This revised version was published online in October 2005 with corrections to the Cover Date.     

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.     

Numerical Computation of Hansen-like Expansions

Fourier expansions of elliptic motion functions in multiples of the true, eccentric, elliptic and mean anomalies are computed numerically by means of the fast Fourier transform. Both Hansen-like coefficients and their derivatives with respect to eccentricity of the orbit are considered. General behavior of the coefficients and the efficiency (compactness) of the expansions are investigated for various values of eccentricity of the orbit. This revised version was published online in July 2006 with corrections to the Cover Date.     

Note on the Generalized Hansen and Laplace Coefficients

Recently, Breiter et al. [Celest. Mech. Dyn. Astron., 2004, 88, 153–161] reported the computation of Hansen coefficients X _k ^{γ ,m} for non-integer values of γ. In fact, the Hansen coefficients are closely related to the Laplace b _s ^(m), and generalized Laplace coefficients b _s,r ^(m) [Laskar and Robutel, 1995, Celest. Mech. Dyn. Astron., 62, 193–217] that do not require s,r to be integers. In particular, the coefficients X ₀ ^{γ ,m} have very simple expressions in terms of the usual Laplace coefficients b _{γ +2} ^(m), and all their properties derive easily from the known properties of the Laplace coefficients.     

General planetary theory in elliptic functions

It is shown that the first-order general planetary theory, i.e. the theory without secular terms, developed in (Brumberg and Chapront, 1973) may be re-constructed and presented by the series in powers of the eccentricity and inclination variables with the closed form coefficients expressed in terms of elliptic functions. The intermediate solution of the zero degree in eccentricities and inclinations has been given explicitly with the aid of elliptic functions and the Hansen type quadratures with trigonometric function kernels. In determining the first and higher degree terms in eccentricities and inclinations one meets the Hansen type quadratures with elliptic function kernels. The secular evolution is described by the autonomous polynomial differential system.     

Analytic properties of Hansen coefficients

Hansen’s coefficients in the theory of elliptic motion with eccentricity e are studied as functions of the parameter η = (1 − e ²)^1/2. Their analytic behavior in the complex η plane is described and some symmetry relations are derived. In particular, for every Hansen coefficient, multiplication by suitable powers of e and η results in an entire analytic function of η. Consequently, Hansen’s coefficients can be in principle computed by means of rapidly convergent series in powers of η. A representation of Hansen’s coefficients in terms of two entire functions of e ² follows.      

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.     

Numerical computation of incomplete elliptic integrals of a general form

We present an algorithm to compute the incomplete elliptic integral of a general form. The algorithm efficiently evaluates some linear combinations of incomplete elliptic integrals of all kinds to a high precision. Some numerical examples are given as illustrations. This enables us to numerically calculate the values and the partial derivatives of incomplete elliptic integrals of all kinds, which are essential when dealing with many problems in celestial mechanics, including the analytic solution of the torque-free rotational motion of a rigid body around its barycenter.     

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 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.     

Analytical solutions for the rotating polytrope

Analytical solutions are constructed for the polytropen=1. An algorithm is devised to determine the numerical values of coefficients. These are compared with existing values determined from purely numerical schemes. The usefulness of the approach is discussed together with numerical strategies for this type of problem.     

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.     

Some Recurrence Relations Between Hansen Coefficients

A new system of recurrence relations for Hansen coefficients is obtained. This system gives a connection between only those coefficients which are included in the disturbing function of planetary or satellite motion and allows to compute efficiently the Hansen coefficients for perturbations both from internal and external bodies. The recurrence process can be realized both from high to low and from low to high harmonical terms of the disturbing function. The corresponding algorithms of evaluation of Hansen coefficients are presented. The efficiency of the obtained system of recurrence relations is discussed.     

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.     

Trigonometric series representations of the orbital inclination function

In this paper, new trigonometric series representations of the orbital inclination functionF _imp (i) in multiples of cosines or sines will be established for all possible values ofl, m, andp. For such representations, the literal analytical expressions and the recurrence formulae satisfied by their coefficients will be established. Moreover, an economic algorithm for the table formulation of these coefficients for the possible values ofl, m, andp is given. Finally, numerical examples of the representations forl=2(1)4;m=0(1)l;p=0(1)l are also included.     

On ‘out of plane’ equilibrium points in the elliptic restricted three-body problem with radiating and oblate primaries

We locate and examine the stability of the ‘out of plane’ equilibrium points, L _6,7 of an infinitesimal body in the field of stellar-oblate binary systems moving in elliptic orbits around their common center of mass. Their positions and stability depend on the oblateness as well as radiation coefficients of the primaries and the eccentricity of their orbits. A numerical application of this problem for the systems: Gamma Leporis and Altair are given.     

On the expansions of intermediate orbit for general planetary theory

Intermediate orbit for general planetary theory is constructed in the form of multivariate Fourier series with numerical coefficients. The structure and efficiency of the derived series are illustrated by giving various statistical properties of the coefficients.The ability of the recently proposed elliptic function approach to compress the Fourier series representing the intermediate orbit is investigated. Our results confirm that when mutual perturbations of a pair of planets are considered the elliptic function approach is quite efficient and allows one to compress the series substantially. However, when perturbations of three or more planets are under study the elliptic function approach does not give any advantages.