Grupo de Mecanica Celeste, Departamento de Matematica Aplicada a la
Ingenieria,
E. T. S. de Ingenieros Industriales, Universidad de Valladolid,
E-47 011 Valladolid, Spain.
Within the elliptic-type two-body problem, to achieve an analytical step-size regulation in numerical integration of highly eccentric orbits, Brumberg (1992) proposed the use of orbital length of arc as independent variable. In performing a reparametrization of time, he replaced t by a pseudo-time introduced via a generalized Sundman transformation, and made to fit his construction into a more general pattern resembling that of two-parameter time transformations of Ferrandiz & Ferrer (1986) and Ferrandiz et al. (1987). See also Floria (1995). In this context, the parameters on which the transformation depends are usually taken as functions of orbital elements. Comparison with the results of Ferrandiz and his collaborators shows that the fictitious time used by Brumberg does not belong to the class of the generalized elliptic anomalies considered by them, and such an independent variable cannot be obtained from the proposals due to these authors, although it is related to t by an equation similar to theirs.
Brumberg himself (1992, p.325 and Formula [14]) stated that his transformation is applicable to any kind of Keplerian orbit. Nevertheless, neither a proof nor the least hint was adduced in this sense. In fact, his derivation was drastically limited to, and strictly based on, the explicit consideration of geometrical and dynamical properties holding for ellipses. We consider that a justification and a rigorous extension of his approach is pertinent for future practical applications, specially for the derivation of analytical expressions when constructing a perturbation theory for highly-eccentric orbits.
Since the approach taken by Brumberg is not limited to elliptic motion, this research is devoted to a general and systematic derivation of this adapted time parameter within a universal formulation of the two-body problem: we intend to modify and adapt his elliptic motion treatment so as to take into account other cases of Keplerian orbits, and recover his formulae under this universal approach.
Accordingly, the motivating point at the origin of the present study was the question whether (and how) Brumberg's (1992) study could be rendered applicable to the orbital length of arc in the case of non--elliptic Keplerian motion. These questions are answered on the stage of a universal formulation of two-body motion, on the basis of which we can analytically integrate the corresponding time transformation, in closed form, by means of elliptic integrals and functions.
In this line, this research continues previous work (Floria 1995) within the universal approach to the change of time variable for the treatment of Keplerian-like systems.
Universal-like functions (Battin 1987, § 4.5 and § 4.6; Stiefel & Scheifele 1971, § 11; Stumpff 1959, Chapter V, § 41) provide an adequate and powerful tool for the study of problems of orbital motion, and particularly for a compact representation and treatment of analytical solutions of the two-body problem. These functions are generalizations of the standard trigonometric and hyperbolic functions. It must be mentioned that the use of universal functions, along with appropriate changes of integration variable, will allow us to reduce the integration of the reparametrizing transformation to that of some algebraic functions, which leads to elliptic integrals and functions.
Key words: arc length, reparametrization of motion, uniform treatment of two--body problems, universal functions, elliptic integrals and functions.