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					                                                                   o
                                                         Presentaci´n de un cartel en condiciones
                                                                                                    E. lautor
                                                                                                     a
                                                                                       Grupo de Mec´nica Espacial
                                                                                      Universidad de Zaragoza, Spain.
                                                                                             *****@unizar.es

                                                                         In this surface of section we have both direct and retrograde
Introduction                                                           orbits. It is possible to find out orbits which appear from some
The most common method to compute the gravitational field of
                                                                       islands, but they are too small to be shown on this figure. Below
the celestial bodies is by using expansions in spherical harmonics,
                                                                       we plot these orbits.
but these series do not converge. This fact is important if one
plans to study orbits around some irregular asteroids, Kuiper Belt
Objects, or natural satellites.


The potential of a plate
First works on the potential of a finite plate of homogeneous den-
sity σ were done by Kellogg[?] and MacMillan[?]. More recent
works are by Broucke[?] and Werner [?]. The last two authors
give an intrinsic formulation of the potential of a general trian-
gular plate.
  The potential function of a plate of n vertices in its same plane,
i.e. in the xy-plane, is:
                  n
       U = Gσ          (Ci,mod(i+1,n)Li,mod(i+1,n)/ri,mod(i+1,n))
                 i=1                                                    The stability index k of the left orbit from figure above is
 We consider the following plates:
• An equilateral triangle with the origin at the center of mass
  and with one of its vertices in the minus x axis.                    Periodic Orbits f2
• A square with its sides parallel to the axes.                        The limit curves of the surface of section of                      Future work
 In the figure above we have drawn the countour plots of these                                                                             We plan to study the trace of these orbits.
potentials.

                                                                                                                                          Acknowledgements
Periodic Orbits 1                                                                                                                         This work has been partially financed by


                                                                                                                                          References
                                                                                                                                          [1] O.D. Kellogg Foundations of potential theory, Dover,
                                                                                                                                              N.Y, 1929.
                                                                                                                                          [2] W.D. MacMillan. The theory of the potential. Dover,
                                                                                                                                              N.Y., 1930.
                                                                                                                                          [3] R. Broucke. Closed Form Expressions for Some Gravita-
                                                                                                                                              tional Potentials: Triangle, Rectangle, Pyramid and Polyhe-
                                                                                                                                              dron. In The Dynamics of Small Bodies in the Solar Sys-
                                                                                                                                              tem, A Major Key to Solar System Studies, B. A. Steves
                                                                                                                                              and A. E. Roy (eds.), pp. 321. Kluwer Academic Publishers,
                                                                                                                                              1999.



                                                                        We have found several symmetrical

				
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