The Third Integral in a Self-consistent Galactic Model

We apply the theory of the third integral to a self-consistent galactic model, generated by the collapse of a N-body system. The final configuration after the collapse is a stationary triaxial system, that represents an almost prolate non-rotating elliptical galaxy with its longest axis in the z-direction. This system is represented by an axisymmetric potential V plus a small triaxial perturbation V ₁. The orbits in the potential V are of three types: box orbits, tube orbits (corresponding to various resonances), and chaotic orbits.The intersections of the box and tube orbits by a Poincaré surface of section z=0 are closed invariant curves. The main tube orbits are like ellipses and form an island of stability on the (R,R) plane.We calculated the third integral I in the potential V for the general non-resonant case and for various resonant cases. The agreement between the invariant curves of the orbits and the level curves of the third integral is good for the box and tube orbits, if we truncate the third integral at an appropriate level. As expected the third integral fails in the case of chaotic orbits. The most important result is the form of the number density F on the Poincaré surface of section. This function decreases exponentially outwards for the box orbits, like Fexp(–bI), while it is constant, as expected, for the chaotic orbits. In the case of the island of the main tube orbits it has a minimum at the center of the island. This can be explained by the form of the near elliptical orbits that are elongated along R, thus they fail to support a self-consistent galaxy, which is elongated along the z-axis.