Many disk galaxies contain a *bar*: a linear structure crossing
the disk. Here we analyze bars in terms of the orbits which occur in
the potential of a rotating bar, and discuss how such structures may
form and evolve.

Bars may be described as strong or weak, depending on the amplitude of the nonaxisymmetric part of the potential. Orbits in strong bars are most easily analyzed in a frame of reference which rotates with the bar, while those in weak bars can be studied by the epicyclic approximation, which illustrates the role of resonances.

The equation of motion for a particle in a frame of reference `r
= (x,y,z)` rotating at angular velocity `Omega_b` about the
`z` axis is

2 d r d | 0 1 0| (1) --- = - -- Phi_eff + 2 Omega_b |-1 0 0| v , 2 dr | 0 0 0| dtwhere

1 2 2 (2) Phi_eff = Phi(r) - - Omega_b R . 2The term proportional to

The rotating-frame analog of the total energy is the *Jacobi
integral*,

1 2 (3) E_J = - |v| + Phi_eff . 2This quantity is conserved by motion in a rotating frame of reference.

As an example of a strong bar, BT87 use the logarithmic potential

1 2 2 2 2 2 (4) Phi(x,y) = - v_0 ln(R_c + x + y / q ) , 2where

d d (5) -- Phi_eff = -- Phi_eff = 0 . dx dyThese equilibrium positions (in the rotating frame of reference) are called the

Motion in the vicinity of a Lagrange point `(x_L,y_L)` may be
studied by expanding the effective potential in powers of `x-x_L`
and `y-y_L` (BT87, Ch. 3.2.2). The key results are outlined here
(see also BT87, Ch. 4.6.3). In the special case of a non-rotating
potential with a finite core radius, a star near the L3 point executes
independent and generally incommensurate harmonic motions in the
`x` and `y` directions. For the case of a rotating
potential the motion may likewise be decomposed into the sum of two
periodic motions: one a retrograde motion about an epicycle, and the
other a prograde motion of the guiding center. Because two motions are
involved, it follows that orbits near the L3 point must have another
integral of motion in addition to `E_J`. Similar results are
obtained at the L4 and L5 points when these are stable.

Numerical integration of Eq. (1) provides a way to study orbits
which do not stay close to a Lagrange point (*e.g.* Contopoulos
& Papayannopoulos 1980). Just as in the earlier discussion of
orbits in triaxial systems, here too each closed, *stable* orbit
parents an orbit family. Close to the core of a barred potential the
only important orbit families are the prograde *x_1* family,
which is aligned with the bar, and the retrograde *x_4* family,
which is nearly circular. For slightly smaller values of
`-E_J` two new types of closed orbits may arise (BT87, Fig.
3-17): the stable *x_2* orbits and the unstable *x_3*
orbits. Both are elongated perpendicular to the bar, but only the
*x_2* orbits, which are rounder than *x_3* orbits of the
same `E_J`, can parent an orbit family. At yet-smaller values
of `-E_J` these perpendicular orbits disappear, and finally the
*x_1* orbits likewise vanish when `-E_J` is small enough
for the star to reach the L1 and L2 points. At comparable values of
`-E_J` one may also find closed orbits circling the L4 and L5
points.

To understand the effects of resonances, consider the *driven
harmonic oscillator*,

2 d x i Omega_1 t (6) --- + k x = alpha e , 2 dtwhere

i Omega_0 t i Omega_1 t (7) x(t) = e + A e ,where by direct substitution it follows that

2 (8) Omega_0 = k ,and

2 2 (9) A = alpha / (Omega_0 - Omega_1) .Thus if the driving force is zero the system oscillates at its natural frequency

Now consider the effect of a bar with pattern speed
`Omega_b` on a star moving in a circular orbit with angular
speed `Omega`. Relative to the star, the angular speed of the
bar is `Omega - Omega_b`, and because the bar is bisymmetric
the star feels a perturbation at *twice* this angular speed.
This perturbation is in resonance with the star's epicyclic frequency
`kappa` if

(10) 2 |Omega - Omega_b| = kappa .Because

(11) Omega_b = Omega + kappa/2 .Depending on the rotation curve and on the value of

(12) Omega_b = Omega - kappa/2 .Finally, there is the

(13) Omega_b = Omega .

To link these results with the above discussion of orbits in strong
bars, note that these resonances mark transitions between orbital
families. If the pattern speed of the bar is higher than the peak
value of `Omega - kappa/2` then no LIRs exist and the
*x_1* family extends all the way from the origin to the CR. The
*x_2* family, on the other hand, occurs only at those radii where
`Omega_b < Omega - kappa/2`. Because only the *x_1*
family is elongated with the bar, we may guess that bars in disk
galaxies have pattern speeds greater than the maximum value of
`Omega - kappa/2` (BT87, Ch. 6.5.1(a)).

- Contopoulos, G. & Papayannopoulos, Th. 1980,
*Astron. Ap.***92**, 33. - Sparke, L.S. & Sellwood, J.A. 1987,
*M.N.R.A.S.***225**, 633.

Joshua E. Barnes (barnes@zeno.ifa.hawaii.edu)

Last modified: March 20, 1995