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

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5. Vertical motion around [FORMULA] and [FORMULA]

For the sake of completeness, we now consider the vertical motion around the Lagrangian points [FORMULA] located on the x -axis at some distance from the bar ends. (The vertical motion close to the equilibrium point [FORMULA], so along the rotational z -axis, was described in MP87 and P87.)

As we see from Fig. 2, the points [FORMULA] remain simply unstable when varying [FORMULA] from 0 to 0.8, there is no change of instability type once the bar strength is non-zero. Contrary to [FORMULA], there is no transition S- [FORMULA]. Thus, no qualitative change in the phase space structure is expected around the orbits starting at [FORMULA] over the whole range of reasonable bar strengths.

The search of 3-dimensional periodic orbits around the equilibrium points proceeds as for [FORMULA], though owing to the strong simple instability of [FORMULA] more care must be adopted in choosing the initial trial conditions. Otherwise, the Newton algorithm would rarely converge toward the seeked periodic solutions. By means of a standard continuation method, we obtain two families of periodic orbits, symmetric with respect to the [FORMULA] plane, with initial conditions [FORMULA] and [FORMULA]. Some members of this family are shown in Fig. 3. The morphology of the whole family is little dependent on the bar strength. As for the orbits starting at [FORMULA], at low amplitudes these orbits also start as a twisted and bended 8-shape crossing the galactic plane. They extend around the whole bar as h increases, until joining the retrograde orbit family in the plane, often called [FORMULA] (R in P84) 4, and outside the corotation radius at a period-doubling bifurcation of the latter.

For the bar and disk parameters fixed as usual, we choose [FORMULA], the marginal stability value for [FORMULA]. Then [FORMULA]. The low-h evolution of their stability parameters [FORMULA] when increasing h is shown in Fig. 8. As for the families starting at [FORMULA], at high h the [FORMULA] -curves turn back to 0 in [FORMULA] when the families converge toward the retrograde main family in the galaxy plane.

[FIGURE] Fig. 8. Low-h part of the stability parameters [FORMULA] and [FORMULA] of the orbit family starting from [FORMULA] (thick curves) as a function of [FORMULA], and a bifurcating family (thin curves) in the range [FORMULA]. Stable orbits are those which have [FORMULA] and [FORMULA] above the dotted line, which corresponds to [FORMULA] (or, [FORMULA]).

A natural question that arises is whether there exist stable periodic orbits bifurcating from the unstable family associated to [FORMULA]. The answer is positive. At first, unstable bifurcating orbits begin at a critical orbit with [FORMULA], increase up to a maximum value and decrease down to zero, i.e. they end with being planar retrograde orbits, for which [FORMULA] (see Fig. 8). In between, critical orbits with [FORMULA] exist, as just visible in Fig. 8. Thus an interval with stable orbits does exist. Of course, from the critical points new periodic families must start. However, all these orbits are too far from the Lagrangian points to be relevant here, so we do not describe them any longer.

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998

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