The Phase Space Structure of the General Sitnikov problem
J. Kallrath1 and R. Dvorak 2 1 BASF - Ludwigshafen, Germany 2 Inst. f. Astronomie - University of Vienna, Austria
The Sitnikov Problem is defined as follows:
Two primaries of equal mass (m1=m2) move on Keplerian
orbits, while an infinitesimally small mass is confined to
move on an axis (z-axis) perpendicular to the primaries' orbital plane.
In the ``General Sitnikov Problem'' (=SGP) all the symmetries are kept, but the third
body has also a mass and thus disturbs the primaries' orbit according to the
mass ratio mu=m3/(m1+m2). We derive the equations
of motion which we study in 2 different ways:
A purely numerically integrations provides the phase space structure
with the aid of properly chosen Poincar\'e surfaces of section (r versus
r dot, where r is the distance of the primaries to the
barycenter in cylindric
coordonates) for different mass ratio and different energies.
We concentrate on 5 different cases (mu = 0.01, 0.1, 1, 10, 100).
A determination of the Periodic Orbits and their Stability
for the 5 mentioned cases for special low order resonances leads to
additional knowledge of the complicated structure of the GSP.
For motions close to the escape energy most of phase space is filled
by stochastic motions for all mass ratios. The two extreme mass ratios
mu=0.01 and mu=100 cover the case of a planet in a double star system
and is a slightly disturbed SP, the seconbd case is a single star surounded
by two planets moving on opposit sides of their commoin barycenter. Although
the configuration is very unstable itself, their study give an interesting
insight into the dynamics of a low dimensional dynamical system.