## 6. Lagrangian orbits in the planeAn other question that could be raised is whether the vertical and radial instability found around as the bar strength increases, has some correspondence in the long and short period orbits circulating around the Lagrangian points in the galaxy plane, the Lagrangian periodic orbits. These orbits and their stability in the plane have been investigated particularly by Contopoulos (1988), who finds an amazing wealth of structures, such as spiral characteristics. In P84, where were stable, we investigated briefly the vertical stability of these orbits without finding any significant vertical instability. Here we calculate the radial and vertical stability of the long and short period Lagrangian orbits for the parameters of the barred galaxy model, varying . We describe only briefly the general trends, without entering into details, because a full description of these orbits would become rapidly unwidely, without bringing better insight to the paper topic. When is small (), these orbits are far from being vertically unstable. As the bar strength increases, they become more and more radially unstable over an increasing portion of their characteristic. Vertical instability does occur in a significant way as approaches , but for orbits which are already much more unstable radially. The level of vertical instability is always significantly lower than the radial instability. Thus we conclude that the Lagrangian orbits reflect and slightly
anticipate the general instability at . This
could be expected when knowing that undergoes
an inverse Hopf-like bifurcation, because then the volume of stable
invariant tori around the stable point shrinks (down to zero)
© European Southern Observatory (ESO) 1998 Online publication: June 2, 1998 |