Luis Floría

Canonical Elements and Keplerian-Like Solutions for Intermediary Orbits of Satellites of an Oblate Planet

Within the framework of the Canonical Formalism in the extended phase space,a general Hamiltonian is investigated that covers a wide class of radial intermediaries accounting for themajor secular effects due to a planets oblateness perturbations.An analytical, closed-form solution for this generic Hamiltonian is developed in terms of elementary functions via the corresponding Hamilton-Jacobi equation. The analytical solution so obtained can be contemplated according to a simple geometrical and dynamical interpretation in Keplerian language by means of the usual relations characterizing elliptic elements along ahypothetic Keplerian motion.Appropriate choices for the terms appearing in the proposed Hamiltonian lead to recovering the analogues of some well-known, classical radial intermediaries (those introduced by Deprit and the one built by Alfriend and Coffey), but also certain new ones derived by Ferrándiz for the Main Problem in the Theory of Artificial Satellites of the Earth. In any case, the results are also applicable to problems dealing with orbital motion of other planetary satellites.The generality of this pattern leads to asystematic obtaining of solutions to the considered intermediaries: special choices of the Hamiltonian yield the correspondinganalytical solution to the respective intermediary problem.

Perturbed Gylden Systems and Time-Dependent Delaunay-Like Transformations

We study the possibilities and limitations of the application of generalized Delaunay-like transformations (in the 6-dimensional phase space) and TR-like mappings (in the 8-dimensional, extended phase space) to perturbed two-body problems with a time-varying Keplerian parameter µ(t), that is, to Gylden-type systems. For the sake of theoretical completeness, both negative- and positive-energy motion (with nonstationary coupling parameter) are, in principle, considered. Our developments are intended to introduce canonical variables parallelling the classical ones of Delaunay and the Delaunay-Similar variables of Scheifele.

Uniform Development of a TR-Transformation to Generalized Universal Variables

We investigate a general approach to the derivation of the Delaunay-Similar (DS) canonical TR-variables (with the true anomaly as the independent variable), originally introduced by Scheifele, in order to render the construction applicable to the study of any kind of two-body orbit. Our analysis also allows for certain classes of perturbations compatible with a basic Keplerian-like dynamical structure. Accordingly, universal TR-elements are obtained for a wide class of perturbed Keplerian systems.

New Intermediaries for the Main Problem in Satellite Theory

After reviewing the original approach leading to the introduction of intermediaries in Satellite Theory, a general procedure to define intermediaries for the Main Problem in this Theory is proposed. This procedure is susceptible to an intuitive interpretation analogous to solving a simple puzzle. The application of this method to the Main Problem allows us not only to recover the well known classical intermediaries but also to obtain several completely new ones, all admitting simple solutions.

Solving a Gylden-Meščerskij System in Delaunay-Scheifele-Like Variables

For an analytical treatment of Gylden-Mescerskij systems, in order to reduce the Gylden model to a perturbed Keplerian system, a canonical transformation from extended polar nodal variables to universal Delaunay-Similar variables is developed. This TR-like transformation is then followed by a change of independent variable. Our considerations are based on Deprit’s approach to Scheifele’s reducing TR-mapping and Deprit’s treatment of the Gylden problem.

Gylden-Like Systems and DEF-Mappings

We apply the linearizing Deprit-Elipe-Ferrer (DEF) mapping (including time t into the transformation equations) to a two-body problem with time-varying Keplerian parameter (a Gylden-type system), whose Hamiltonian is presented in homogeneous canonical formulation. After changing the independent variable by a true-like anomaly, we derive regularized second-order equations for the coordinates, resembling harmonic equations. Reduction to harmonic oscillators may not be achieved.

Zonal Harmonics of the Gravity Field in DEF-Variables

To take advantage of the linear and regular formulation and treatment of Celestial Mechanics problems (Kustaanheimo & Stiefel 1965; Stiefel & Scheifele 1971; Deprit, Elipe & Ferrer 1994), Sharaf & Saad (1997) have given an analytical expansion of the Earth’s gravitational zonal potential in terms of Kustaanheimo-Stiefel (KS) regular elements (Stiefel & Scheifele 1971, § 19), with special emphasis on its application to elliptic-type two-body orbits and, consequently, using a generalized (elliptic) eccentric anomaly as the independent variable.

Generalized Gylden-Type Systems in Universal DS-Like TR-Variables

Scheifele(1970) applied Delaunay-Sirmlar (DS) elliptic Keplerian elements (with the true anomaly as the independent variable) to the J 2 Problem in Artificial Satellite Theory, making an element of the true anomaly. (1981) views Scheifele’s TR-mapping as an extension of Hill’s transformation from a 6-dimensional phase space to an enlarged, 8-dimensional one. To adapt this approach to elliptic-type two-body problems with a time-varying Keplerian parameter μ(t), Fioria (1997, §3, §4) treated a Gylden system (Deprit 1983) and derived “Delaunay-Similar” variables via a TR-like transformation. Now we extend our treatment to perturbed Gylden systems, and modify the TR-map to deal with any kind of two-body orbit. We work out our generalization and the resulting variables within a unified pattern whatever the type of motion, in terms of universal functions (Stiefel & Scheifele 1971, §11; Battin 1987, §4.5, §4.6) and auxiliary angle-like parameters.

On Reduction of the Nonstationary Two-Body Problem to Oscillator Form

We deal with a perturbed Gylden system in homogeneous canonical formulation. After a simple extension of the Burdet-Ferrandiz (BF) transformation to an enlarged phase space, we convert the canonical equations of motion (derived from the transformed Hamiltonian of the problem) into a set of four second-order differential equations for the position-like variables. Attempts to establish analogies with conservative cases suggest that, although linearization cannot be assured, regularization of the equations of motion is attainable in a slightly modified BF formulation.

A Regularization of the Nonstationary Two-Body Problem Under the Maneff Perturbing Potential

Within the framework of classical mechanics, the Maneff model of gravitational potential constitutes a nonrelativistic modification of Newton’s gravitational law which can be successfully used to accurately account for the secular motion of the pericentre of some celestial bodies, at least in the Solar System (e.g. the advance of the perihelion of the inner planets, or the motion of the perigee of the Moon.) We are concerned with the two-body problem as contemplated in classical celestial mechanics, and concentrate on the so-called Gylden systems, say two-body problems with variable Keplerian parameter,μ(t). On such systems, we superimpose perturbing effects emanating from a Maneff-type nonrelativistic gravitational potential, and we intend to arrive at regularized equations of motion for the resulting dynamical system.