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

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1. Introduction

In 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).

[FIGURE] Fig. 1a-c. Topology constraint to motion existing in a barred galaxy due to the Jacobi integral H, shown in the half-space [FORMULA]. The bar major axis is aligned with the x -axis, and the rotation axis is the z axis. The shaded surface shows the zero-velocity surface beyond which no motion does exist for three particular values h of the Hamiltonian H. Top: At low h, particles are confined either inside a lenticular shape in the bar region, or outside a surface similar to a one-sheet hyperboloid. Middle: At higher h the two surfaces join at the corotation locus allowing orbits to crossing it. Bottom: At still higher h forbidden regions above the galactic poles persist for bound particles.

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 H (the Hamiltonian expressed in coordinates co-rotating with the bar). If the value of this integral is lower than the value necessary to co-rotate with the Lagrangian point near the end of the bar, motion is confined either inside a lenticular shape, or outside a one-sheet hyperboloidal surface beyond the corotation radius (Fig. 1, top). For increasing values of H, the lenticular and hyperboloidal surfaces meet and merge, which opens a hole around [FORMULA] allowing to cross corotation and eventually to escape (Fig. 1, middle). This hole widens to the whole corotation circle at still higher values (Fig. 1, bottom). At any value of this integral there always exists a forbidden region above the bar that slowly recesses as H increases.

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 N -body simulations of barred galaxies (e.g., Pfenniger & Friedli 1991).

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 complex instability which is not possible in systems with 2 degrees of freedom. This kind of instability appears, for example, in the Elliptic Restricted Three-Body Problem (Broucke 1969), in the problem of planetary systems (Hadjidemetriou 1975), or in galactic dynamics (Magnenat 1982; Contopoulos & Magnenat 1985; Martinet & Pfenniger 1987; Pfenniger 1985b, 1990; see also Pfenniger 1995 for a short review).

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 [FORMULA] and [FORMULA] at the ends of the bar major axis 1), or not far from being marginally complex unstable around the Lagrangian points [FORMULA] and [FORMULA] along the minor axis in the self-consistent models issued from N -body simulations. A marginal state of fixed points means that profound changes of the phase space structure around these points can be expected for a small change of the bar parameters.

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 [FORMULA] 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 [FORMULA] 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 [FORMULA] 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 [FORMULA]. 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.

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998