Forum Springer Astron. Astrophys.
Forum Whats New Search Orders

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

Previous Section Next Section Title Page Table of Contents

3. Equilibrium and stability diagram

The equations of motion used are those defined by the Hamiltonian H, in a frame of reference rotating about the z -axis with the frequency [FORMULA], and the x -axis aligned with the long axis of the bar, that is,


where [FORMULA], [FORMULA] are the canonical variables and the expression of H is


with [FORMULA] the potential described in Sect. 2. Since [FORMULA] is symmetric in z, it follows that [FORMULA] and the equilibrium points lie in the plane [FORMULA]. More specifically, the Lagrangian points [FORMULA] on the x axis are obtained from the equation [FORMULA], and the Lagrangian points [FORMULA] on the y -axis from [FORMULA].

Linearizing the equations of motion at [FORMULA] and around a stationary point, we obtain the variational equations,


where the [FORMULA] 's denote the linear variations and the potential second derivatives [FORMULA], [FORMULA], [FORMULA], and [FORMULA] are constants evaluated at the stationary point. The terms [FORMULA], [FORMULA] do not appear in Eq. (6) since [FORMULA] vanishes at [FORMULA].

It is clear from Eq. (6) that the pair of equations for [FORMULA], [FORMULA] separates from the rest and corresponds to an independent bounded harmonic motion if [FORMULA]. In sufficiently thin disks [FORMULA] is equivalent to requiring the positivity of mass at [FORMULA] by Poisson's equation. Therefore we conclude that at each Lagrangian point a family of periodic orbits must start and continue transversally to the galaxy plane. Contrary to the axial orbits passing through the centre [FORMULA], these transverse orbits will not remain straight at non-infinitesimal amplitudes and their shape will be less obvious to determine.

Before computing these orbits it is useful to calculate their stability at low amplitude, which is just the stability of the relevant Lagrangian point. As determined in P90, one has to calculate the second derivatives of the potential [FORMULA], [FORMULA] for a given pattern speed [FORMULA] to find whether the motion around them is characterized by stability or any other type of instability.

In Fig. 2 we plot the position of the Lagrangian points in the relevant stability diagram discussed in P90 (see Fig. [1], p. 57). When [FORMULA] increases from 0 to 0.8, the points [FORMULA] remain simply unstable, moving toward the upper left of the figure; but the points [FORMULA] see a transition from stability to complex instability (at [FORMULA]), moving downward across the transition curve. This is a crucial property of strong bars found in N -body simulations (P90).

[FIGURE] Fig. 2. Stability diagram which delineates the stable and unstable regions of the plane [FORMULA]. S stands for stable, U simple unstable, and [FORMULA] complex unstable (notation as in Contopoulos & Magnenat 1985). The position of the Lagrangian points is indicated when the bar mass [FORMULA] varies from 0 (on the S-U marginal line) to 0.8 (at the tip of the mark).

Previous Section Next Section Title Page Table of Contents

© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998