Book Review

The aim of this book is to present techniques for the study of motion of solar system objects in highly eccentric orbits. Instead of using the usual anomalies (mean, true, eccentric), the authors define and use a new kind of anomaly, the elliptic anomaly.In this way, it is possible, in a theory using perturbation series expansions, to make the ratio: (accuracy)/(number of needed terms), higher than in the classical techniques. The book consists of six chapters. The first chapter deals with the elliptic anomaly in the two-body problem and the second chapter presents the general technique to construct first-order perturbation theory in elliptic function expansions. The next three chapters deal with applications of the new technique to artificial satellites and asteroids, in highly eccentric orbits. The last chapter describes the basic algorithms of the theory.The tools developed in the book demand the use of computer algebra, which is implemented by means of Mathematica 3.0.The book is well written and the new technique is clearly presented and related to the existing techniques, making it useful to all those who use analytical or semi-analytical methods for the study of highly eccentric motion. Celestial Mechanics at High Eccentricities, Gordon and Breach Publishers, US$95, GBP 59, EUR 79, ISBN 90-5699-212-0     

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.     

Infinitesimal Orbits Around the Libration Points in the Elliptic Restricted Three Body Problem

The objective of the present work is to develope explicit analytical expressions for the small amplitude orbits of the infinitesimal mass about the equilibrium points in the elliptic restricted three body problem. To handle this dynamical problem, the Hamiltonian for the elliptic problem is formed with the true anomaly and then with the eccentric anomaly as independent variables. The origin is then transformed to a fixed point and the Hamiltonian is developed up to O(4) in the eccentricity, e, (which plays the role of the small parameter of the problem) of the primaries. The integration of the model problem under consideration is undertaken by means of a perturbation technique based on Lie series developments, which leads to the solution of the canonical equations of motion.     

Motion near the triangular points in the elliptic restricted problem of three bodies

In this paper the first variational equations of motion about the triangular points in the elliptic restricted problem are investigated by the perturbation theories of Hori and Deprit, which are based on Lie transforms, and by taking the mean equations used by Grebenikov as our upperturbed Hamiltonian system instead of the first variational equations in the circular restricted problem. We are able to remove the explicit dependence of transformed Hamiltonian on the true anomaly by a canonical transformation. The general solution of the equations of motion which are derived from the transformed Hamiltonian including all the constant terms of any order in eccentricity and up to the periodic terms of second order in eccentricity of the primaries is given.     

The two-body problem in the (truncated) PPN — Theory

The solution of the two-body problem in the (truncated) PPN theory is presented. It is given in two different analytical forms (the Wagoner-Will and Brumberg representation) and by the method of osculting elements.Work supported in part by the Deutsche Forschungsgemeinschaft (DFG), SFB78.     

The mean anomaly in elliptic motion as random variable

In this paper we deal with the different means for some characteristic quantities and functions in elliptic motion. We then give some probability interpretations of the mean anomaly and discuss the free terms of certain expansions in the Fourier series related to the mean anomaly. Finally we give a therem illustrating the mathematical meaning of results obtained by these interpretations.     

Automated,closed form integration of formulas in elliptic motion

In some perturbation theories it is possible to avoid expansion of the perturbations in powers of the eccentricity, obtaining results in closed form by using the true or eccentric anomaly instead of the mean anomaly. This paper describes an algorithm (which has been programmed for the 6600 computer using the formula manipulation system TRIGMAN) for automatically performing the integrals which arise in these theories.     

Semi-analytical integration algorithms based on the use of several kinds of anomalies as temporal variable

One of the main problems in celestial mechanics is the management of long developments in Fourier or Poisson series used to describe the perturbed motion in the planetary system.In this work we shall develop a software package that is suitable for managing these objects. This package includes algorithms to obtain the inverse of the distance based on an iterative method, a set of integration algorithms according to several sets of temporal variables.This paper contains a comparative study on the use of the true, eccentric, and elliptic anomalies in semi-analytical methods on celestial mechanics.     

A class of semi-analytical solutions for a rotating toroidal plasma

In this paper, the problem of stationary MHD flow for a rotating toroidal plasma is investigated by assuming that the entropy is a surface quantity. Then, the system of ideal MHD equations is reduced to a single second-order elliptic partial differential equation known as the modified Grad-Shafranov (or Maschke-Perrin) equation. Under the assumption that both the function,P _s andf ² are quadratic polynomials of the flux function, a class of semi-analytical solutions is obtained for a plasma contained in a perfectly conducting toroidal boundary with a rectangular cross section. The flux function, poloidal current and the generalized pressure are obtained and discussed for relevant values of the parameters.     

A Method for the Construction of a Restricted Motion in an Arbitrary Central Field

The well-known problem of motion in a central field integrable in quadratures is considered. The force function of the problem depends only on the particle distance to the chosen coordinate origin. In the general case of an arbitrary central force, a rigorous analytical solution of the problem cannot be obtained due to the complexity of the integrals. In this paper we propose a semi-analytical method of constructing an approximate solution for the case where the distance varies in a limited range that allows the time dependences of the polar coordinates to be obtained using elliptic functions and integrals. As an example, we consider the model problems of the perturbed motion of hypothetical Jovian and lunar equatorial satellites as well as the problem of the motion of a single star in the principal plane of a galaxy. The methodical accuracy has been estimated by a comparison with the numerical solution.     

A discussion of the solution for the motion of Pluto

In a previous paper, a semi-analytical solution for the long-term motion of Pluto was presented. The present paper contains: (1) a comparison of the present solution with the solution by Williams and Benson; (2) a discussion of the effect of the near resonance between Pluto and Uranus; and, (3) a calculation of the librational period of the eccentricity, inclination and perihelion.The semi-analytical solution is shown to agree very closely with the long-term solution for Pluto obtained by Williams and Benson using numerical integration of the averaged equations of motion. A small difference between the two solutions is attributed to neglecting the eccentricity and inclination of Neptune in the semi-analytical solution.     

Development Of The Hamiltonian Function Using The Jacobi System Of Reference

In this paper we describe two different methods to expand the second term of the planetary Hamiltonian function. The Jacobi system of coordinates is adopted leading to a unique evaluation of the Hamiltonian. Previous analytical or semi-analytical planetary theories suffer from the drawback of computing the perturbation function for each planet, which is quite cumbersome. The inclinations of planets are referred to a common fixed plane and the longitudes to a common origin. This is necessary when we deal with n > 2 planets. The treatment is straightforward, and no complexities appear throughout the analysis. This revised version was published online in July 2006 with corrections to the Cover Date.     

Generalized elliptic anomalies

A two-parameter time transformationdt=r ^3/2(α₀+α₁ r)^?1/2 dτ is proposed, where τ is the radial distance while α₀ and α₁ are, if not constants, at least conservative functions of positions and velocities. In Keplerian systems, the quadrature implied by the transformation may by carried out by elliptic functions. When α₀=0, τ is the eccentric anomaly; if α₁=0, then τ is the intermediate or elliptic anomaly. Considering several values of α₀ and α₁, numerical examples of the relation of thegeneralized elliptic anomaly τ with the classical and elliptic anomalies are given. Application of this transformation to some perturbed Kepler problems is briefly outlined.     

Account of additional perturbations in canonical system solutions obtained by Lie transforms

The analytical or semi-analytical solution of some canonical system (basic theory) constructed by the perturbation technique based on the Lie transforms is supposed to be known. A method allowing one to obtain the solution of perturbed canonical or non-canonical system using the known basic theory is discussed.     

The future of artificial satellite theories Hybrid ephemeris compression model

Since the time of Newton, astrodynamics has focused on the analytical solution of orbital problems. This was driven by the desire to obtain a theoretical understanding of the motion and the practical desire to be able to produce a computational result, Only with the advent of the computer did numerical integration become a practical consideration for solving dynamical problems. Although computer technology is not yet to the point of being able to provide numerical integration support for all satellite orbits, we are in a transition period which is being driven by the unprecedented increase in computational power, This transition will affect the future of analytical, semi-analytical and numerical artificial satellite theories in a dramatic way, In fact, the role for semi-analytical theories may disappear. During the time of transition, a central site may have the capacity to maintain the orbits using numerical integration, but the user may not have such a capacity or may need results in a more timely manner, One way to provide for this transition need is through the use of some type of satellite ephemeris compression. Through the combined use of a power series and a Fourier series, good quality ephemeris compression has been achieved for 7 day periods, The ephemeris compression requires less than 40 terms and is valid for all eccentricities and inclinations.     

The transition from elliptic to hyperbolic orbits in the two-body problem by slow loss of mass

Transition from elliptic to hyperbolic orbits in the two-body problem with slowly decreasing mass is investigated by means of asymptotic approximations.Analytical results by Verhulst and Eckhaus are extended to construct approximate solutions for the true anomaly and the eccentricity of the osculating orbit if the initial conditions are nearly-parabolic. It becomes clear that the eccentricity will monotonously increase with time for all mass functions satisfying a Jeans-Eddington relation and even for a larger set of functions. To illustrate these results quantitatively we calculate the eccentricity as a function of time for Jeans-Eddington functionsn=0(1) 5 and 18 nearly-parabolic initial conditions to find that 93 out of 108 elliptic orbits become hyperbolic.     

A note on independent variables for restricted three-body problems

In studies of the elliptic restricted three-body problem, the true anomaly of the motion of the primaries is often used as the independent variable. The equations of motion then show invariancy in form from the circular case. It is of interest whether other independent variables exist, such that the invariant form of the equations is maintained. It is found that true anomaly is the only such variable.     

The Lissajous transformation II. Normalization

Normalization of a perturbed elliptic oscillator, when executed in Lissajous variables, amounts to averaging over the elliptic anomaly. The reduced Lissajous variables constitute a system of cylindrical coordinates over the orbital spheres of constant energy, but the pole-like singularities are removed by reverting to the subjacent Hopf coordinates. The two-parameter coupling that is a polynomial of degree four admitting the symmetries of the square is studied in detail. It is shown that the normalized elliptic oscillator in that case behaves everywhere in the parameter plane like a rigid body in free rotation about a fixed point, and that it passes through butterfly bifurcations wherever its phase flow admits non isolated equilibria.     

Study on Two Methods for Nonlinear Force-Free Extrapolation Based on Semi-Analytical Field

In this paper, two semi-analytical solutions of force-free fields (Low and Lou, Astrophys. J. 352, 343, 1990) have been used to test two nonlinear force-free extrapolation methods. One is the boundary integral equation (BIE) method developed by Yan and Sakurai (Solar Phys. 195, 89, 2000), and the other is the approximate vertical integration (AVI) method developed by Song et al. (Astrophys. J. 649, 1084, 2006). Some improvements have been made to the AVI method to avoid the singular points in the process of calculation. It is found that the correlation coefficients between the first semi-analytical field and extrapolated field using the BIE method, and also that obtained by the improved AVI method, are greater than 90% below a height 10 of the 64×64 lower boundary. For the second semi-analytical field, these correlation coefficients are greater than 80% below the same relative height. Although differences between the semi-analytical solutions and the extrapolated fields exist for both the BIE and AVI methods, these two methods can give reliable results for heights of about 15% of the extent of the lower boundary.     

The role of true anomaly in ballistic capture

A massless particle may perform a ballistic capture about a primary when two or more gravitational attractions are considered. The dynamics governing the ballistic capture depend on the mutual position of the primaries, if these are let to revolve in eccentric orbits. This paper studies the effect of the primaries true anomaly on the ballistic capture about the smaller primary in the planar elliptic restricted three-body problem. The dynamics of the Hill curves are studied, and the conditions for a favorable capture are derived. It is shown that these lead to regular, quasi-stable post-capture orbits. This is confirmed by numerical simulations implementing the concept of capture set: a set of initial conditions that generates ballistic capture orbits with a prescribed stability number.