Computation of elliptic Hansen coefficients and their derivatives

The problem of computation of elliptic Hansen coefficients and their derivatives is considered for constructing a motion theory of an artificial Earth satellite with large eccentricity. An algorithm for analytical and numerical computation of these coefficients and their derivatives is described. The recurrence relations for derivatives of the first and second order and initial values for recurrences are obtained. As an example, numerical values of some elliptic Hansen coefficients are given for the orbit with eccentricityk=0.74.     

Convergence of the expansions of the disturbing functions in the planar three-body planetary problem

In the framework of the planar three-body planetary problem, conditions are found for the absolute convergence of the expansions of the disturbing functions in powers of the eccentricities, with coefficients represented by trigonometric polynomials with respect to the mean, eccentric, or true anomaly of the inner planet. It is found that using the eccentric or true anomaly as the independent variable instead of the mean anomaly (or time) extends the holomorphy domain of the principal part of the perturbation functions. The expansions of the second parts converge in open bicircles, which admit values of the eccentricity of the inner planet in excess of the Laplace limit.     

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.     

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.     

Analytical orbit predictions with air drag using KS uniformly regular canonical elements

A new nonsingular analytical theory for the motion of near Earth satellite orbits with the air drag effect is developed for long term motion in terms of the KS uniformly regular canonical elements by a series expansion method, by assuming the atmosphere to be symmetrically spherical with constant density scale height. The series expansions include up to third order terms in eccentricity. 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. Numerical comparisons of the important orbital parameters semi major axis and eccentricity up to 1000 revolutions, obtained with the present solution, with KS elements analytical solution and Cook, King-Hele and Walker's theory with respect to the numerically integrated values, show the superiority of the present solution over the other two theories over a wide range of eccentricity, perigee height and inclination.     

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).     

Hansen系数递推的效率

探讨了偏心率函数的递推方法的效率, 给出了一种成批递推方法, 其计算效率明显优于直接计算方法, 而且递推是正向的. 采用该成批递推方法递推时, 偏心率函数的量级从小到大变化, 可以保证递推的精度.     

Computational Efficiency of the Recursion of Hansen Coefficents

Computational efficiency of the recursion of eccentricity functions is investigated, and a kind of batch recursion method is given. Its computational efficiency is significantly superior to the direct calculation method. Moreover, this kind of batch recursion is forward so that the magnitudes of eccentricity functions experience from small to large change in the recursive process. Hence in this way the high accuracy of the recursion of eccentricity functions can be guaranteed.     

High-order solutions of invariant manifolds associated with libration point orbits in the elliptic restricted three-body system

High-order analytical solutions of invariant manifolds, associated with Lissajous and halo orbits in the elliptic restricted three-body problem (ERTBP), are constructed in this paper. The equations of motion of ERTBP in the pulsating synodic coordinate system have five equilibrium points, and the three collinear libration points as well as the associated center manifolds are unstable. In our calculation, the general solutions of the invariant manifolds associated with Lissajous and halo orbits around collinear libration points are expressed as power series of five parameters: the orbital eccentricity, two amplitudes corresponding to the hyperbolic manifolds, and two amplitudes corresponding to the center manifolds. The analytical solutions up to arbitrary order are constructed by means of Lindstedt–Poincaré method, and then the center and invariant manifolds, transit and non-transit trajectories in ERTBP are all parameterized. Since the circular restricted three-body problem (CRTBP) is a particular case of ERTBP when the eccentricity is zero, the general solutions constructed in this paper can be reduced to describe the dynamics around the collinear libration points in CRTBP naturally. In order to check the validity of the series expansions constructed, the practical convergence of the series expansions up to different orders is studied.     

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.     

Contraction of satellite orbits using KS uniformly regular canonical elements in an oblate diurnally varying atmosphere

A new non-singular analytical theory for the motion of near Earth satellite orbits with the air drag effect is developed in terms of the Kustaanheimo and Stiefel (KS) uniformly regular canonical elements, by assuming the atmosphere to be oblate diurnally varying with constant density scale height. The series expansions include up to third-order terms in eccentricity and c (a small parameter dependent on the flattening of the atmosphere). 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. Numerical comparisons of the important orbital parameters semimajor axis and eccentricity up to 1000 revolutions, obtained with the present solution, with the third-order analytical theories of Swinerd and Boulton and in terms of the KS elements, with respect to the numerically integrated values, show the superiority of the present solution over the other two theories over a wide range of eccentricity, perigee height and inclination.     

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.     

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.     

Contraction of near-Earth satellite orbits using uniformly regular KS canonical elements in an oblate atmosphere with density scale height variation with altitude

A new non-singular analytical theory for the contraction of near-Earth satellite orbits under the influence of air drag is developed in terms of uniformly regular Kustaanheimo and Stiefel (KS) canonical elements using an oblate atmosphere with variation of density scale height with altitude. The series expansions include up to fourth power in terms of eccentricity and c (a small parameter dependent on the flattening of the atmosphere). 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. It is observed that the analytically computed values of the semi-major axis and eccentricity are consistent with the numerically integrated values up to 500 revolutions over a wide range of the drag-perturbed orbital parameters. The theory can be effectively used for re-entry of near-Earth objects.     

Some expansions of the elliptic motion to high eccentricities

Some classic expansions of the elliptic motion — cosmE and sinmE — in powers of the eccentricity 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 are given in terms of the derivatives of Bessel functions with respect to the eccentricity. The expansions have the same radius of convergence (e*) of the extended solution of Kepler's equation, previously derived by the author. Some other simple expansions — (a/r), (r/a), (r/a) sinv, ..., — derived straightforward from the expansions ofE, cosE and sinE are also presented.     

Sensitivity study of high eccentricity orbits for Mars gravity recovery

By linear perturbation theory, a sensitivity study is presented to calculate the contribution of the Mars gravity field to the orbital perturbations in velocity for spacecrafts in both low eccentricity Mars orbits and high eccentricity orbits(HEOs). In order to improve the solution of some low degree/order gravity coefficients, a method of choosing an appropriate semimajor axis is often used to calculate an expected orbital resonance, which will significantly amplify the magnitude of the position and velocity perturbations produced by certain gravity coefficients. We can then assess to what degree/order gravity coefficients can be recovered from the tracking data of the spacecraft. However, this existing method can only be applied to a low eccentricity orbit, and is not valid for an HEO. A new approach to choosing an appropriate semimajor axis is proposed here to analyze an orbital resonance. This approach can be applied to both low eccentricity orbits and HEOs. This small adjustment in the semimajor axis can improve the precision of gravity field coefficients and does not affect other scientific objectives.     

General-altitude transformations between geocentric and geodetic coordinates

Formulas for the general-altitude (height above the ellipsoid) transformation from geocentric to geodetic coordinates and vice versa are derived. The set of four formulas is expressed in each of two useful forms: series expansions in powers of the Earth's flattening and series expansions in powers of the Earth's eccentricity. The error incurred in these expansions is of the order of one part in 3×10⁷.     

Line formation in moving media: Asymptotic expansions of some special functions

Line formation in the spectrum of a moving medium with a spherical geometry is considered. In the Sobolev approximation there are some special functions that determine the source function and the force of radiation pressure in the line. The most important case is that of a small dimensionless velocity gradient (i.e., a large dimensionless Sobolev length τ) and a small ratio β of the opacity in the continuum to the opacity in the line. Until now there has been no detailed analytical information about the asymptotic behavior of these functions. For the case of a Doppler profile of the absorption coefficient, we clarify the nontrivial structure of their total asymptotic expansions for τ » 1, β « 1, and arbitrary Βτ. We give an algorithm for obtaining all the coefficients of these expansions and give explicit expressions for the first few coefficients. We also compare the asymptotic expansions with the numerical calculations of these functions available in the literature. We also briefly consider the case of a power-law decrease in the absorption coefficient in the line wing (and, in more detail, the case of Lorentz wings of the Voigt profile).     

Some orbital characteristics of lunar artificial satellites

In this paper we present an analytical theory with numerical simulations to study the orbital motion of lunar artificial satellites. We consider the problem of an artificial satellite perturbed by the non-uniform distribution of mass of the Moon and by a third-body in elliptical orbit (Earth is considered). Legendre polynomials are expanded in powers of the eccentricity up to the degree four and are used for the disturbing potential due to the third-body. We show a new approximated equation to compute the critical semi-major axis for the orbit of the satellite. Lie-Hori perturbation method up to the second-order is applied to eliminate the terms of short-period of the disturbing potential. Coupling terms are analyzed. Emphasis is given to the case of frozen orbits and critical inclination. Numerical simulations for hypothetical lunar artificial satellites are performed, considering that the perturbations are acting together or one at a time.     

Tesseral harmonic perturbations in radial transverse and binormal components

The formulae for the perturbations in radial, transverse and binormal components of the Earth artificial satellite motion have been derived. Perturbations due to the tesseral part of the geopotential are considered. The geopotential expressed in terms of the orbital elements has the form proposed by Wnuk (1988). The formulae for the perturbations have been obtained using the Hori (1966) method. They can be effectively applied in calculation of the perturbations in the components including the coefficients of the high order and degree tesseral harmonics. The derived formulae reveal no singularities at zero eccentricity.