Generation of a secular system in the analytical theory for the motion of the moon

In order to generate an analytical theory of the motion of the Moon by considering planetary perturbations, a procedure of general planetary theory (GPT) is used. In this case, the Moon is considered as an addition planet to the eight principal planets. Therefore, according to the GPT procedure, the theory of the Moon’s orbital motion can be presented in the form of series with respect to the evolution of eccentric and oblique variables with quasi-periodic coefficients, which are the functions of mean longitudes for principal planets and the Moon. The relationship between evolution variables and the time is determined by a trigonometric solution for the independent secular system that describes the secular motion of a perigee and the Moon node by considering secular planetary inequalities. Principal planetary coordinates required for generating the theory of the motion of the Moon includes only Keplerian terms, the intermediate orbit, and the linear theory with respect to eccentricities and inclinations in the first order relative to the masses. All analytical calculations are performed by means of the specialized echeloned Poisson Series Processor EPSP.     

Quantitative predictions with detuned normal forms

The phase-space structure of two families of galactic potentials is approximated with a resonant detuned normal form. The normal form series is obtained by a Lie transform of the series expansion around the minimum of the original Hamiltonian. Attention is focused on the quantitative predictive ability of the normal form. We find analytical expressions for bifurcations of periodic orbits and compare them with other analytical approaches and with numerical results. The predictions are quite reliable even outside the convergence radius of the perturbation and we analyze this result using resummation techniques of asymptotic series.     

Analytical evaluation of the Voigt function using binomial coefficients and incomplete gamma functions

Using the binomial expansion theorem, the simple general analytical expressions are obtained for the Voigt function arising in various fields of physical research. As we will seen, the present formulation yields compact closed-form expressions which enable the ready analytical calculation of the Voigt function. The validity of this approximation is tested by other calculation methods. The series expansion relations established in this work are accurate enough in the whole range of parameters. The convergence rate of the series is estimated and discussed. Some examples of this methodology are presented.     

Analytical solutions for Euler parameters

Two new analytical solutions for Poinsot motion in terms of Euler parameters are derived. The first solution is a straightforward ‘universal’ (no branches) time series practical for short time motion calculations or as a basis for analytical continuation. The second, more involved solution is also universal but is not restricted to short times; it is in terms of circular, hyperbolic, and elliptic functions and elliptic integrals.     

Symbolic analytical developments of the zero pressure cosmological model of the universe

In the present paper, literal analytical solutions in power series forms are developed for the radius of curvature and the expansion velocity of the zero pressure cosmological models of the universe at any time t. Also, we develop literal analytical solutions in power series forms for the inverse problem of the zero pressure cosmological model, that is to find the time $t=\tilde{t}$ (say) at which the radius of curvature of the model $R=\tilde{R}$ (say) is known. The importance of these analytical power series representations is that, they are invariant under many operations because, addition, multiplication, exponent ion, integration, different ion, etc of a power series is also a power series. A fact which provides excellent flexibility in dealing with analytical as well as computational developments of the problems related to zero pressure cosmological models.For computational developments of these solutions, an efficient method using continued fraction theory is provided. By means of the present methods we able to analyze some known zero-pressure cosmological models, of these are Einstein and De Sitter models. In addition we also analyzed some other models by which one can know if the universe keep expanding forever, or will it reach a maximal size and then turn into contraction stage.     

Analytical lunar libration tables

We have developed a theory of the rotation of the Moon, for the purpose of obtaining libration series explicitly dependent upon lunar gravitational field model parameters. A nonlinear model is used in which the rigid Moon, whose motion in space is that of the main problem of lunar theory, and whose gravity potential is represented through its third degree harmonics, is torqued by the Earth and Sun. The analytical series are then obtained as Poisson series. Numerical comparisons with Eckhardt's solution are briefly exposed.     

A series expansion of the solid Earth tide effect on geopotential

We develop analytical series representing the main part of corrections to the geopotential coefficients caused by the solid Earth tides, where Love numbers are assumed to be frequency-independent. The series are compact, precise and valid over 1800 A.D.–2200 A.D. The maximum difference between the corrections given by the analytical series and their numerical values, obtained with use of the DE/LE-423 planetary/lunar ephemerides, does not exceed $0.7\times 10^{-12}$ . A new algorithm is proposed for calculating amplitudes of the additional variations of the geopotential coefficients for frequency dependence of Love numbers. It uses the representation of the Earth tide-generating potential in the standard HW95 format and takes into account the phase of tidal waves. Corrections of up to $2\times 10^{-12}$ to the published by the IERS Conventions (2010) amplitudes of the additional variations of the geopotential coefficients are suggested. Examples of use of the obtained series in analytical theories of motion of low-altitude STARLETTE and high-altitude ETALON-1 satellites are given.     

Specialized celestial mechanics systems for symbolic manipulation

Construction and application of the current high accuracy analytical theories of motion of celestial bodies necessitates the development of specialized software for the implementation of analytical algorithms of celestial mechanics. This paper describes a typical software package of this kind. This package includes a universal Poisson processor for the rational functions of many variables, a tensorial processor for purposes of relativistic celestial mechanics, a Keplerian processor valid for the solutions of the two body problem in the form of a Poisson series, Taylor expansions in powers of time and closed expressions, and an analytical generator of celestial mechanics functions, facilitating the immediate implementation of the present analytical methods of celestial mechanics. The package is completed with a numerical-analytical interface designed, in particular, for the fast evaluation of the long Poisson series.     

Accurate analytical representation of Pluto modern ephemeris

An accurate development of the latest JPL’s numerical ephemeris of Pluto, DE421, to compact analytical series is done. Rectangular barycentric ICRF coordinates of Pluto from DE421 are approximated by compact Fourier series with a maximum error of 1.3 km over 1900–2050 (the entire time interval covered by the ephemeris). To calculate Pluto positions relative to the Sun, a development of rectangular heliocentric ICRF coordinates of the Solar System barycenter to Poisson series is additionally made. As a result, DE421 Pluto heliocentric positions by the new analytical series are represented to an accuracy of better than 5 km over 1900–2050.     

An analytic method to determine future close approaches between satellites

The calculation of the times of future close approaches between pairs of satellites has been formulated using analytical techniques. The resulting analytical equations are solved using numerical iterative techniques similar to solving Kepler's equation. A solution is obtained in a very efficient manner by use of a series of prefilters which eliminate many cases from further consideration. The method is valid for all values of eccentricities less than one and all relative geometries between the two orbits. This approach produces results in a very efficient and reliable manner.     

Radius of the Roche equipotential surfaces

In literature, there is no exact analytical solution available for determining the radius of Roche equipotential surfaces of distorted close binary systems in synchronous rotation. However, Kopal (Roche Model and Its Application to Close Binary Systems, Advances in Astronomy and Astrophysics, Academic Press, New York 1972) and Morris (Publ. Astron. Soc. Pac. 106:154, 1994) have provided the approximate analytical solutions in the form of infinite mathematical series. These series expressions have been commonly used by various authors to determine the radius of the Roche equipotential surfaces, and hence the equilibrium structures of rotating stars and stars in the binary systems. However, numerical results obtained from these approximating series expressions are not very accurate. In the present paper, we have expanded these series expressions to higher orders so as to improve their accuracy. The objective of this paper is to check, whether, there is any effect on the accuracy of these series expressions when the terms of higher orders are considered. Our results show that in most of the cases these expanded series give better results than the earlier series. We have further used these expanded series to find numerically the volume radius of the Roche equipotential surfaces. The obtained results are in good agreement with the results available in literature. We have also presented simple and accurate approximating formulas to calculate the radius of the primary component in a close binary system. These formulas give very accurate results in a specified range of mass ratio.     

Integration of the elliptic restricted three-body problem with Lie series

The method of Lie series is used to construct a solution for the elliptic restricted three body problem. In a synodic pulsating coordinate system, the Lie operator for the motion of the third infinitesimal body is derived as function of coordinates, velocities and true anomaly of the primaries. The terms of the Lie series for the solution are then calculated with recurrence formulae which enable a rapid successive calculation of any desired number of terms. This procedure gives a very useful analytical form for the series and allows a quick calculation of the orbit.The project is supported by the Austrian Fonds zur Förderung der wissénschaftlichen Forschung under Project No. 4471.     

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.     

A resonance problem of two degrees of freedom

An analytical theory is presented for determining the motion described by a Hamiltonian of two degrees of freedom. Hamiltonians of this type are representative of the problem of an artificial Earth satellite in a near-circular orbit or a near-equatorial orbit and in resonance with a longitudinal dependent part of the geopotential. Using the classical Bohlin-von Zeipel procedure the variation of the elements is developed through a generating function expressed as a trigonometrical series. The coefficients of this series, determined in ascending powers of an auxiliary parameter, are the solutions of paired sets of ordinary differential equations and involve elliptic functions and quadrature. The first order solution accounts for the full variation of the resonance terms with the second coordinate.     

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.     

月球探测器过渡轨道的短弧定轨方法

论述的短弧定轨,是指在无先验信息情况下又避开多变元迭代的初轨计算方法,它需要相应的动力学问题有一能反映短弧内达到一定精度的近似分析解．探测器进入月球引力作用范围后接近月球时可以处理成相对月球的受摄二体问题,而在地球附近,则可处理成相对地球的受摄二体问题,但在整个过渡段的力模型只能处理成一个受摄的限制性三体问题．而限制性三体问题无分析解,即使在月球引力作用范围外,对于大推力脉冲式的过渡方式,相对地球的变化椭圆轨道的偏心率很大(超过Laplace极限),在考虑月球引力摄动时亦无法构造摄动分析解．就此问题,考虑在地球非球形引力(只包含J2项)和月球引力共同作用下,构造了探测器飞抵月球过渡轨道段的时间幂级数解,在此基础上给出一种受摄二体问题意义下的初轨计算方法,经数值验证,定轨方法有效,可供地面测控系统参考．     

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 ofG will be established for anyx _i; withn, N positive integers and |_i|<1 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 by-products of the analyses are two important periodic integrals developed analytically and computationally.     

Accurate analytical nutation series

In this letter we present the first accurate analytical nutation series, deduced from the Hamiltonian theory by the authors. They provide the highest accuracy ever obtained by any analytical nutation series, since the deviation in CEP (celestial ephemeris pole) offsets with respect to that of the IERS Conventions 1996 is kept below 1 mas in the time domain, in spite of still lacking ocanic corrections.     

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.     

The long period behavior of the orbits of the Galilean satellites of Jupiter

Some of the results of an investigation into the long period behavior of the orbits of the Galilean satellites of Jupiter are presented. Special purpose computer programs were used to perform all the algebraic manipulations and series expansions that are necessary to describe the mutual interactions among the satellites.The disturbing function was expanded as a Poisson series in the modified Keplerian elements referred to a Jovicentric coordinate system. The differential equations for the modified Keplerian elements were then formed, and all short period perturbations were removed using Kamel's perturbation method. Approximate analytical solutions for these differential equations are derived, and the general form of the solutions are given.