Title: Helium-like atom as a classical three-body problem.

Author: T.J. Stuchi(1), A.C.B. Antunes(1) and M.A. Andreu(2).

Adress:
(1) Instituto de Fisica, Universidade Federal do Rio de Janeiro,
    Caixa Postal 68.528, 21945-970 Rio de Janeiro, RJ, Brazil.
    e-mail: tstuchi@if.ufrj.br
(2) Dept. Matematica Aplicada i Analisi, Univ. Barcelona.
    Gran Via 585, 08007--Barcelona (Spain).
    e-mail: mangel@maia.ub.es
 
Abstract:
We study the classical problem of the muonic helium atom, a helium atom
with one of its electrons  replaced by  a muon. Firstly, we establish the
connection of the model with the one-dimensional frozen planetary 
approximation of the helium atom. After that, we find that there is only 
an equilibrium configuration if the particles are allowed a non-zero angular
momentum with the alpha/muon distance restricted. We then present the
restricted muonic helium problem, an approximation for the movement of the
electron in the potential of the alpha/muon pair moving in a circular orbit.  
In this model we find that there is only an equilibrium point of the type
center-saddle-center. The planar and vertical Lyapunov families of periodic 
orbits and their stability parameters have been studied showing that the
planar family bifurcate into Halo orbits like its gravitational counterpart.
The most interesting feature is the stability of these Halo orbits for a 
range of energy values. The vicinity of the alpha particle is also studied
using a Levi--Civita regularization of the electron-alpha particle collision. 
The movement gets more and more chaotic as the energy increases. However,
we find that retrograde zones of bounded stability are possible for energies
higher than that of the equilibrium point, when the zero velocity curves open. 
