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{\bf ON TWO--PARAMETER LINEARIZING TRANSFORMATIONS \\
     FOR UNIFORM TREATMENT OF TWO--BODY MOTION.} 
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{\it Luis Flor\'{\i}a.}
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Grupo de Mec\'anica Celeste, \\
Departamento de Matem\'atica Aplicada a la Ingenier\'{\i}a,\\
E. T. S. de Ingenieros Industriales,\\
Universidad de Valladolid, E--47 011 Valladolid, SPAIN.
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{\bf Abstract}
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The ideas of regularization and linearization of the differential equations 
governing perturbed dynamical systems have inspired the introduction of 
adequate transformations of dependent and/or independent variables. As well 
known, these processes of regularization and linearization play an important 
role in Celestial Mechanics. In this respect, the study of perturbed Keplerian 
systems constitutes a field in which these techniques have been put into 
action. A classical, general reference for this subject is Stiefel \& 
Scheifele (1971).

By generalizing previous research by Belen'kii (1981) and Cid, Ferrer \& Elipe 
(1983), and within the framework of the elliptic--type two--body motion, 
Ferr\'andiz \& Ferrer (1986) and Ferr\'andiz, Ferrer \& Sein--Echaluce (1987) 
replaced the physical time by a fictitious time (in fact, a family of them) as 
the new independent variable for the integration of perturbed Kepler problems. 
They used the method of linearization by means of time transformations on 
introducing a family of generalized anomalies defined via a differential 
relation in which the linearizing function depends on two homogeneous 
parameters that can be taken as functions of some orbital elements. In the 
case of pure Kepler problems, they analytically integrated the respective time 
transformations, in closed form, by means of elliptic integrals and functions. 
Their developments were originally intended to facilitate (on the basis of 
some radial intermediaries) the analytical treatment of artificial Earth 
satellite orbits under a zonal model of geopotential, and as a preconditioning 
of the problem prior to numerical integration. 

The present research deals with the analytical investigation of an appropriate 
modification and a systematic study of these generalized Sundman--type 
transformations allowing the extension of the results established by the said 
authors to a uniform treatment of two--body orbits under a {\em universal 
formulation} and representation of the two--body problem, in the sense of 
Stiefel \& Scheifele (1971), \S 11, or Battin (1987), \S 4.5 and \S 4.6. To 
this end, an essential role is played by certain families of special 
functions, the so--called {\em universal functions} (Battin 1987, \S 4.5 and 
\S 4.6) and the {\em Stumpff c--functions} (Stiefel \& Scheifele 1971, \S 11). 
As expected, throughout the required reckoning work when integrating the 
changes of time parameter under consideration, elliptic integrals and 
functions will also play an important role. Symbolic computation will allow us 
to develop and systematize our approach when perturbed motion is analyzed. \\

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\noindent {\bf KEY WORDS:} reparametrizing transformations, generalized 
anomalies, uniform treatment of two--body motion, universal functions, elliptic 
integrals and functions. \\

\noindent{\bf * Symposium Topic.} Mathematical Techniques: elliptic functions 
and algebraic manipulation.  

\begin{thebibliography}{99}
\bibitem{} Battin, R. H.: 1987, {\em An Introduction to the Mathematics and 
 Methods of Astrodynamics}. AIAA Education Series, American Institute of 
 Aeronautics and Astronautics, New York. 
\bibitem{} Belen'kii, I. M.: 1981, A Method of Regularizing the Equations of 
Motion in the Central Force--Field. {\em Celestial Mechanics} {\bf 23,} 9--32. 
\bibitem{} Cid, R., Ferrer, S. and Elipe, A: 1983, Regularization and 
 Linearization of the Equations of Motion in Central Force--Fields.  
{\em Celestial Mechanics} {\bf 31,} 73--80. 
\bibitem{} Ferr\^`ndiz, J. M. and Ferrer, S.: 1986, A New Integrated, 
 General  Time Transformation in the Kepler Problem. {\em Bulletin of the 
  Astronomical Institutes of Czechoslovakia} {\bf 37,} 226--229. 
\bibitem{} Ferr\^`ndiz, J. M., Ferrer, S. and Sein--Echaluce, M. L.: 1987, 
 Generalized Elliptic Anomalies. {\em Celestial Mechanics} {\bf 40,} 315--328.
\bibitem{stsch} Stiefel, E., L. and Scheifele, G.: 1971, {\em Linear and 
 Regular Celestial Mechanics}. Springer, Berlin, Heidelberg, New York. 
\end{thebibliography}

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\noindent {\bf The Author:} \\ 
\noindent Luis Flor\'{\i}a. \\
\noindent Grupo de Mec\'anica Celeste, \\
\noindent Departamento de Matem\'atica Aplicada a la Ingenier\'{\i}a, \\
\noindent Escuela T\'ecnica Superior de Ingenieros Industriales.\\
\noindent Universidad de Valladolid.\\ 
\noindent Paseo del Cauce, s/n.\\
\noindent E -- 47 011, Valladolid. Spain.\\
\noindent Phone: 34 -- 83 -- 42 33 94, and 34 -- 83 -- 42 33 97. \\ 
\noindent Fax: 34 -- 83 -- 42 33 10, and 34 -- 83 -- 42 34 06.  

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