Many disk galaxies contain a *bar*: a linear structure
crossing the disk. Bars can be analyzed in terms of the orbits which
occur in a rotating triaxial potential. They seem to form naturally
when density waves tunnel through the center of a disk, and may drive
slow evolutionary changes in the galaxies they inhabit.

Bars in disk galaxies are interesting for several reasons. First, they are very common; at least half of all disk galaxies have bars, often in association with rings, lenses, oval features, and splendid symmetric spiral structure (de Vaucouleurs & de Vaucouleurs 1964). Second, bars occur naturally in N-body simulations of rotating stellar disks (e.g. Sellwood 1981); so naturally, in fact, that the existence of disk galaxies without bars has provided evidence for unseen mass in galaxies like the Milky Way (Ostriker & Peebles 1973). Third, while bars themselves seem intrinsically stable, they may act as ``engines'' of evolutionary change in their host galaxies (Kormendy 1982).

Bars are the flattest and most triaxial stellar systems known.
Viewed face-on, most bars have axial ratios of between 2.5:1 and 5:1,
whereas few elliptical galaxies are flatter than 3:1. Compared to
ellipticals, bars have rather square outlines, and their surface
density profiles tend to be flat rather than steeply declining. Most
bars can be no thicker than the disks they inhabit, since we don't see
them as distinct from disks when viewed edge-on; their major-to-minor
axis ratios may be as extreme as 10:1. (*Edge-on bars could be
recognized with 21-cm observations; has anyone done this?*).

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 ^ (1) --- = - -- Phi_eff + 2 Omega_b z cross v , 2 dr dtwhere

1 2 2 2 (2) Phi_eff = Phi(r) - - Omega_b (x + y ) . 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 *x1* family,
which is aligned with the bar, and the retrograde *x4* 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 *x2* orbits and the unstable *x3*
orbits. Both are elongated perpendicular to the bar, but only the
*x2* orbits, which are rounder than *x3* 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
*x1* 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.

The bar is weak if it generates only a small fraction of the total
gravitational field. In such systems most stellar orbits can be
described by the epicyclic theory plus a weak driving force due to the
periodic motion of the bar. But even weak perturbations can add up if
they happen to be in *resonance* with the system they drive;
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
*x1* family extends all the way from the center to the CR. The
*x2* family, on the other hand, occurs only at those radii where
`Omega_b < Omega - kappa/2`. Because only the *x1*
family is elongated with the bar, we may guess that weak bars in disk
galaxies have pattern speeds greater than the maximum value of
`Omega - kappa/2` (BT87, Ch. 6.5.1(a)).

Bars are the endpoints of a bona-fide *instability* in disk
galaxies. They develop spontaneously in N-body models of rotating
stellar disks (Hohl 1971); indeed, it's not easy to devise stable
models for disk simulations. Normal-mode analysis (Zang 1976, Kalnajs
1978) and direct integration of the CBE (Inagaki, Nishida, &
Sellwood 1984) both confirm the reality of the bar instability.

The fact that rotating disk galaxies swing-amplify perturbations in
the form of leading spiral waves helps explain the bar instability
(Toomre 1981). Transient spirals result when the swing amplifier
selects leading waves from the ambient noise present in the disk and
amplifies them while shearing them into trailing spiral patterns.
Strictly speaking, this is not an instability; the spirals produced
are analogous to the fluctuating output of a high-gain amplifier
connected to a resistor at finite temperature. The swing amplifier is
stable because it can't amplify trailing spiral waves. But an
amplifier with a positive feedback loop can be and often is unstable.
Feedback occurs if trailing spirals can turn into leading spirals by
reflecting from a well-defined surface or by tunneling through the
center. Surface reflections play a role in instabilities of spinning
fluid bodies, but not in disk galaxies since disks lack definite
edges. On the other hand, inward-propagating trailing waves are
naturally transformed into outward-propagating leading waves if they
can get through the center of the disk (Toomre 1981; see BT87,
Fig. 6-20). Recall from the last lecture that in-going density waves
are absorbed if they encounter an ILR; thus to get through, the wave's
pattern speed must exceed the maximum of `Omega - kappa/2` so
that no ILR exists. This is the same condition on the pattern bar's
speed just reached by considering available orbits in weak bar
potentials. In numerical simulations the emerging bar in fact rotates
faster than the peak value of `Omega - kappa/2` (Sellwood 1981,
Sparke & Sellwood 1987); this is evidence that tunneling through
the center completes the feedback loop in bar-unstable disks.

To cure disk galaxies of the bar instability, one can

- lower the gain of the swing-amplifier by placing much of the mass in a dark halo (Ostriker & Peebles 1973),
- cut the feedback loop by increasing the central velocity dispersion, so that in-going waves are heavily damped before they emerge, or
- cut the feedback loop by increasing the central bulge mass, so
the maximum
`Omega - kappa/2`increases and ILRs exist for all pattern speeds (Toomre 1981) (*what limits the maximum possible pattern speed?*).

Once formed, stellar bars are quite robust; they typically persist
for the duration of N-body experiments with no more evolution than
might be expected from two-body relaxation. But a bar can interact
with the galaxy it lives in, and both the bar and its host may change
as a result. Over time, bars in disk simulations rotate more slowly
as gravitational torques transfer angular momentum from the bar to the
surrounding material (e.g. Sellwood 1981). They also grow somewhat
longer, and this is a natural consequence of their slowing down, since
bars tend to end at the CR, which moves out as `Omega_b`
decreases. Bars may also slow down by interacting with dark halos
(Sellwood 1980, Weinberg 1985). As the bar slows down some *x1*
orbits may be replaced by members of the perpendicular *x2* orbit
family.

A bar has a strong effect on any interstellar material present in a
galaxy (e.g. Prendergast 1983, Athanassoula 1992). Our own galaxy may
provide the closest example (e.g. Gerhard 1996). Streams of gas can't
flow through each other, so the gas tends to ``seek out'' closed,
non-intersecting orbits. In practice, gas orbits in barred potentials
never quite find such orbits, and weak shocks mark the places where
gas streams continue to intersect; such shocks often manifest as dust
lanes on the leading edges of stellar bars like the one in NGC 1300
(Roberts, Huntley, & van Albada 1979; see BT87, Fig. 6-28).
Passing through such a shock, the gas suffers an *irreversible*
change; consequently it can't stay in its old orbit, but must slowly
spiral inward. Should ILRs be present, the gas may accumulate on
*x2* orbits instead of flowing all the way to the center of the
galaxy. Gas accumulating at the CR may form the ``inner'' rings often
seen at the ends of bars. Likewise, gas in the outer part of a galaxy
may accumulate at the OLR (Schwarz 1981). If such rings are places
where gas collects, it's no coincidence that they have blue colors and
show signs of ongoing star formation (Kormendy 1982).

- Athanassoula, E. 1992,
*M.N.R.A.S.*,**259**, 345. - Contopoulos, G. & Papayannopoulos, Th. 1980,
*Astron. Ap.***92**, 33. - de Vaucouleurs, G. & de Vaucouleurs, A. 1964,
*Reference Catalogue of Bright Galaxies*. - Efstathiou, G., Lake, G., & Negroponte, J. 1982,
*M.N.R.A.S.*,**199**, 1069. - Gerhard, O.E. 1996, in
*Unsolved Problems of the Milky Way*, eds. L. Blitz & P. Teuben, p. 79. - Hohl, F. 1971,
*Ap. J.*,**168**, 343. - Inagaki, S., Nishida, M.T., & Sellwood, J.A. 1984,
*M.N.R.A.S.*,**210**, 589. - Kalnajs, A.J. 1978, in
*Structure and Properties of Nearby Galaxies*, ed. E.M. Berkhuijsen & R. Wielebinski, p. 113. - Kormendy, J. 1982. in
*Morphology and Dynamics of Galaxies*, ed. L. Martinet & M. Mayor, p. 113 (K82). - Ostriker, J.P. & Peebles, P.J.E. 1973,
*Ap. J.*186, 467. - Prendergast, K.H. 1983, in
*Internal Kinematics & Dynamics of Galaxies*, ed. E. Athanassoula, p. 215. - Roberts, W.W., Huntley, J.M., & van Albada, G.D. 1979,
*Ap. J.***233**, 67. - Schwarz, M.P. 1981,
*Ap. J.***247**, 77. - Sellwood, J.A. 1980,
*Astr. Ap.***89**, 296. - Sellwood, J.A. 1981,
*Astr. Ap.***99**, 362. - Sparke, L.S. & Sellwood, J.A. 1987,
*M.N.R.A.S.***225**, 633. - Toomre, A. 1981, in
*The Structure and Evolution of Normal Galaxies*, eds. M. Fall & D. Lynden-Bell, p. 111. - Weinberg, M.D. 1985,
*M.N.R.A.S.***213**, 451. - Zang, T.A. 1976, Ph.D. thesis, Massachusetts Institute of Technology.

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

Last modified: April 8, 1997