## 1. IntroductionIn contrast with elliptical galaxies, which rotate slowly, barred galaxies must possess several stationary points in their stellar part, the Lagrangian points, because their bar rotate rapidly, as generally admitted. In addition to the potential minimum at the galaxy centre, these galaxies have four additional stationary points in the rotating frame of the stellar bar, along the major and minor axes in the galaxy plane (see e.g., Binney & Tremaine 1987, p. 137). The corotation region therefore plays a fundamental role in the dynamics in barred galaxies (and other rotating patterns).
The most important topological constraint in a barred galaxy is
given by the zero velocity surface of the only global integral of the
system, the Jacobi integral To gain further understanding on the bar dynamics, one can adopt the famous Poincaré general strategy for conquering non-integrable systems: one should first characterize the stability of the fixed points, then explore the properties of the periodic orbits associated with the fixed points. In turn, the stability properties of the periodic orbits give insight on their neighbourhood: stable periodic orbits are mostly surrounded by quasi-periodic orbits, while unstable periodic orbits by chaotic orbits. In principle, the process can extend to higher order periodic orbits; in practice the complexity of the high-order orbits in non-integrable systems rapidly confuses the situation. Our experience has been that the knowledge of the main families of
periodic orbits, gained by the work of many people, is invaluable not
only to understand real barred galaxies, but also to develop insight.
For example one can immediately picture out what is happening in
complex situations developing in The stable periodic orbits trapping effectively stars are particularly important to understand the structure of these galaxies and the motion of its stars, but the unpopulated periodic orbits are important too for predicting the fate of small mass perturbations, such as infalling dwarf galaxies or gas clouds. Historically, the order of study of barred galaxies has not
followed exactly the Poincaré plan. Because the properties of
the central fixed point are trivial in the principal plane, people
have immediately concentrated the efforts around the orbits trapping
mass: the periodic orbits associated with the circular orbits in the
vanishing-bar limit, first in the galaxy plane, and then in 3D. Only
later did we look at the radial orbits along the rotation axis, so
associated with the central point (Martinet & Pfenniger 1987;
Pfenniger 1987, hereafter MP87 and P87). In systems with 3 degrees of
freedom we have the phenomenon of On the other hand the properties of the Lagrangian points remained
for many years insufficiently understood, although the orbits in the
plane developing around them were investigated (Contopoulos 1981,
1988; Contopoulos & Grosbol 1989), exhibiting an intimidating
wealth of structures. The precise conditions of stability of these
points being not clear, we investigated its properties some years
later (Pfenniger 1990, hereafter P90). It turned out that in the
typical situations of barred galaxies, these points are either always
unstable (around the Lagrangian points and
at the ends of the bar major axis
So today there remains still an unexplored major region of phase space of barred galaxies: the vertical orbit structure developing from the Lagrangian points. This is the purpose of this study to sketch their main properties. In Sect. 2, we consider a barred galaxy model made of a Ferrers bar embedded in a Miyamoto-Nagai disk, with a variable bar parameter. In Sect. 3, the derivation of the equilibrium points and the stability diagram are briefly recalled. In Sect. 4, we describe how the transition from stability to complex instability appears for the Lagrangian points when the bar strength is increased. The propagation of this transition on the family of periodic orbits around these points is also given, and we determine the orientation of the bifurcation at the transition. In Sect. 5, the Lagrangian points are considered. They are always simply unstable. The periodic orbits around them and their bifurcations are numerically computed and some stable orbits are found; nevertheless, these orbits are not useful for understanding the galactic structure. In Sect. 6, we discuss briefly the horizontal and vertical stability of the long and short periodic orbits circling around . Finally, in Sect. 7, we consider the non-linear effect of instability on sample orbits and their diffusion across the galaxi, and discuss in Sect. 8 some implications of the found results for galaxy secular evolution. © European Southern Observatory (ESO) 1998 Online publication: June 2, 1998 |