7. Diffusion from the corotation radius
After having described the instabilities affecting the main periodic orbits around the Lagrangian points, a natural question is to characterize the neighbouring phase space: how fast and how far in R and are the neighbouring chaotic orbits diffusing from the corotation radius? Are there some additional constraints to the Jacobi integral (cf. Fig. 1)?
To get a first insight into these questions, we have integrated several sets of orbits starting on 91 regularly spaced points on an elliptic arc aligned with the bar potential and joining to :
The bar mass is chosen slightly larger () than the marginal stability value (). The total mass remains fixed to 1. The situation would correspond to a bar that has grown up to slightly overshoot the stability condition.
Here we show the initial velocities of orbits of the set starting with a modulus of in the rotating frame, and various directions:
Other starting values have been used with similar results: faster initial conditions lead to larger diffusion, and vice-versa. A velocity of 0.07 matches conservatively a velocity dispersion of slightly less than at the corotation radius. For example, if the Galactic bar corotation lies at and the velocity dispersion squared follows an exponential decrease with the same scale-length as the surface density one, then the expected velocity dispersion at corotation would reach 2-3 times the solar value at kpc for a disk scale-length of 4 kpc.
The orbits are integrated either up to 10 Gyr or when they first reach a radial distance of from the origin, which is the most frequent case. They are sampled at regular time intervals of 10 Myr.
The main result is that most of the trajectories diffuse fast, in one or two rotation periods beyond twice the corotation radius. The horizontal diffusion is clearly constrained inside corotation to a few "channels" for orbits starting near (Fig. 9, top).
As visible in Fig. 9, bottom, the envelope of diffusion in z beyond corotation is approximately linearly increasing with R: to . The additional constraint well beyond corotation is the "third integral" (Contopoulos 1963) 5. As visible when comparing the frame in Fig. 9, bottom, to the frames, the confinement due to an effective integral is stronger for orbits starting around than around .
A few orbits diffuse differently, rather vertically at more than while above the bar. Fig. 9, bottom, shows clearly that the complex instability associated to and its associated chaotic phase-space structure allows stars to diffuse higher in z and also above the bar in the inner stellar halo, than the simple instability associated to .
The average surface density of these diffusing, mostly chaotic looking orbits is asymptotically tending toward an exponentially decreasing radial distribution, as already described in P85b.
© European Southern Observatory (ESO) 1998
Online publication: June 2, 1998