Periodic orbits in the copenhagen problem in three dimensions
Roger Broucke
University of Texas, USA
The present work is motivated by the fact that several planets
of other stars have recently been discovered, in particular some
cases for binary stars. It is thus justified to attempt to
determine the regions of stability where habitable planets may
eventually exist.
We compute families of periodic orbits in the restricted
three-body problem, with two primaries of equal mass, and in three
dimensions. Similar work was performed in Copenhagen by
E. Stromgren in the first half of the previous century and later
by M. Henon and several others.
It is well known that the Restricted Problem has some
symmetries with respect to the syzygy line. In the case of equal
masses, there even are additional symmetries. More precisely,
there are four types of symmetry, according to the orthogonality
of the orbits with:
1.the x-axis, 2.the xoz-plane. 3.the yoz-plane, 4.the y-axis.
As these conditions may be realized at the end of an orbit as
well, there result 10 possible types of symmetric periodic orbits.
We found examples of all 10 types. For all these periodic orbits,
it is sufficient to integrate only half revolutions, er even
quarter revolutions in some cases.
We especially pay attention to the stability of the periodic
orbits. We determine stability by computing the eigenvalues of the
6-by-6 monodromy matrix of the variational equations or by a
modified method which results in a 4-by-4 matrix. The case of all
4 eigenvalues on the unit-circle corresponds to stable periodic
orbits, while there also are six types of unstable configurations,
resulting in several bifurcations that were discovered..
Actually, we found that many of the families of Periodic
Orbits originate as a bifurcation out of the two-dimensional orbits
at the stability transition, as was shown by M. Henon in his work
on the Copenhagen problem. We used several such transition orbits
as a starting point for the generation of the three-dimensional
families.