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 v is the velocity in the rotating frame and the effective potential is
1 2 2 2 (2) Phi_eff = Phi(r) - - Omega_b (x + y ) . 2The term proportional to Omega_b in Eq. (1) represents the Coriolis force, which comes about through conservation of angular momentum; thus a particle moving outward with velocity v_x along the x axis accelerates at -2 Omega_b v_x in the y direction. The term proportional to Omega_b^2 in Eq. (2) represents the outward-directed centrifugal pseudoforce present in a rotating frame of reference. When Omega_b = 0 we recover the equations of motion for a non-rotating frame.
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 v_0 is the asymptotic circular speed, R_c is the core radius, and q is the axial ratio of the potential; q < 1 for a bar elongated along the x axis. A contour plot of Phi_eff for this potential (BT87, Fig. 3-13) displays five places where
d d (5) -- Phi_eff = -- Phi_eff = 0 . dx dyThese equilibrium positions (in the rotating frame of reference) are called the Lagrange points. One, conventionally known as the L3 point, occupies the origin and occurs even in a non-rotating potential. Points L1 and L2 fall along the x axis, while L4 and L5 lie along the y axis; these are the places where the centrifugal pseudoforce balances gravity. L3 is a minimum of Phi_eff and hence is always stable, while L1 and L2 are saddle points and thus are always unstable. The points L4 and L5, though they mark maxima of Phi_eff, are stable for a logarithmic barred potential.
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 k is the spring constant, alpha is the amplitude of the driving force, and Omega_1 is the driving frequency. The solution has the form
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 Omega_0, while if |alpha| > 0 there is also an oscillation at the driving frequency Omega_1. As Omega_1 -> Omega_0 the amplitude of the driven response diverges. Notice also that the sign of the response changes on moving through a resonance -- there is a phase shift of pi radians.
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 Omega and kappa depend on radius, there are specific radii in the disk where Eq. (10) is satisfied. At the outer Lindblad resonance (OLR),
(11) Omega_b = Omega + kappa/2 .Depending on the rotation curve and on the value of Omega_b, there may be zero, one, or two inner Lindblad resonances (ILRs) where
(12) Omega_b = Omega - kappa/2 .Finally, there is the corotation resonance (CR) where
(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
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).
Last modified: April 8, 1997