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Astrophys Space Sci (2014) 349:117123 DOI 10.1007/s10509-013-1628-6
O R I G I NA L A RT I C L E
Generalization of Levi-Civita regularization in the restricted three-body problem
R. Roman I. Szcs-Csillik
Received: 7 August 2013 / Accepted: 2 September 2013 / Published online: 22 September 2013 Springer Science+Business Media Dordrecht 2013
Abstract A family of polynomial coupled function of n degree is proposed, in order to generalize the Levi-Civita regularization method, in the restricted three-body problem. Analytical relationship between polar radii in the physical plane and in the regularized plane are established; similar for polar angles. As a numerical application, trajectories of the test particle using polynomial functions of 2, 3, . . . , 8 degree are obtained. For the polynomial of second degree, the Levi-Civita regularization method is found.
Keywords Celestial mechanics Regularization
Restricted three-body problem
1 Introduction
The regularization (in Celestial Mechanics) is a transformation of space and time variables, in order to eliminate the singularities occurring in equations of motion. As Szebehely show (see Szebehely 1967), the purpose of regularization is to obtain regular differential equations of motion and not regular solutions.
The regularization was introduced by Levi-Civita in (1906) in plane, and generalized by Kustaanheimo and Stiefel in (1965) in space. At the beginning, the regularization was developed for studying the singularities of Kepler motion, for analyzing the collisions of two point masses, and
R. Roman (B) I. Szcs-Csillik
Astronomical Institute of Romanian Academy, Astronomical Observatory Cluj-Napoca, Str. Ciresilor No. 19,400487 Cluj-Napoca, Romaniae-mail: mailto:[email protected]
Web End [email protected]
I. Szcs-Csillike-mail: mailto:[email protected]
Web End [email protected]
for improving the numerical integration of near-collision orbits. Many studies of the regularization problem are in the restricted three-body problem, where there are two singularities. We can regularize local (one of them), or global. Birkhoff (1915), Thiele (1896), Burrau (1906), Lematre (1955), Arenstorf (1963), rdi (2004), Szcs-Csillik and Roman (2012), and many other researchers studied the regularization of the restricted three-body problem.
In order to obtain the regularized equations of motion, one introduces a generating function S, which depends on two harmonic and conjugated functions f and g. But there are many harmonic and conjugated functions. Using different couples of polynomial functions, one can obtain different methods of regularization. For the polynomial of second degree...