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Astron. Astrophys. 334, 829-839 (1998)

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6. Lagrangian orbits in the plane

An other question that could be raised is whether the vertical and radial instability found around [FORMULA] 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 [FORMULA] 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 [FORMULA]. 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 [FORMULA] is small ([FORMULA]), 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 [FORMULA] approaches [FORMULA], 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 [FORMULA]. This could be expected when knowing that [FORMULA] undergoes an inverse Hopf-like bifurcation, because then the volume of stable invariant tori around the stable point shrinks (down to zero) before reaching the transition to complex instability.

[FIGURE] Fig. 9. Radial (top) and vertical (bottom) extensions of orbits sampled at regular time intervals and starting on the corotation ellipse (shown by light-gray areas in the top four frames), with various angles [FORMULA] from the x -axis. The initial velocities in the rotating frame are similar to the velocity dispersion in a barred galaxy at this radius. Clearly, orbits starting near the complex unstable [FORMULA] can diffuse to more than 2 kpc above the bar region, while most of the orbits diffuse to large R and moderate [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998