![]() | Annu. Rev. Astron. Astrophys. 2004. 42:
603-683 Copyright © 2004 by Annual Reviews. All rights reserved |
2.2. Dynamics of Barred Galaxies: The Importance of Resonances
To understand bar-driven evolution, we need to dip briefly into the dynamics of bars. An in-depth review is provided by SW93. Here we need a primer on the nature and importance of orbital resonances.
Seen from an inertial frame, an orbit in a galactic disk is an unclosed rosette. That is, there are a nonintegral number of radial oscillations for every revolution around the center. However, in a frame of reference that rotates at the average angular velocity of the star, the star's mean position is fixed and its radial oscillation makes it move in a small ellipse around that mean position 2 It is an ellipse and not just radial motion because the star revolves faster than average near pericenter and slower than average near apocenter. Any global density pattern such as a bar that rotates at the above angular velocity will pull gravitationally on the star in essentially the same way at all times and will therefore make large perturbations in its orbit. "Corotation" is the strongest of a series of resonances in which the pattern repeatedly sees the star in the same way.
For example, there is another rotating frame in which the star executes
two radial oscillations for each circuit around the center. If a bar rotates
at this angular velocity, it sees the stellar orbit as closed, roughly
elliptical, and centered on the galactic center
(Figure 4). This is called
inner Lindblad resonance (ILR). It occurs when the pattern speed of the
bar is
p =
-
/ 2, where
is the average
angular velocity of revolution of the star and
is its frequency of
radial oscillation. The limit of small radial oscillations is called the
epicyclic approximation; then
2 =
(2V / r)(V / r + dV / dr),
where V(r) is the circular-orbit rotation curve
(Mihalas & Routly
1968
provide a particularly transparent discussion).
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Figure 4. (Top) Frequencies
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Outer Lindblad resonance (OLR) is like inner Lindblad resonance, except
that the star drifts backward with respect to the rotating frame while it
executes two radial oscillations for each revolution:
p =
+
/2.
Resonances are important for several reasons.
Figure 4 shows generic
frequency curves and the most important periodic orbit families in a barred
galaxy. We can begin to understand how a self-consistent bar is
constructed by exploiting the fact that
-
/ 2 varies only slowly
with radius
(except near the center, if there is an ILR). Calculations of orbits in a
barred potential show that the main "x1" family of
orbits is elongated parallel to the bar between ILR and corotation. Bars
are largely made of x1
orbits and similar, non-periodic orbits that are trapped around them by
the bar's self-gravity. Typical x1 orbits are shown in
the bottom panel of Figure 4. They are not
nearly circular, but the essence of their behavior
is captured if we retain the language of the epicyclic approximation and say
that orbits of different radii look closed in frames that rotate at
different angular velocities
-
/ 2. But if
-
/ 2
varies only a little with radius,
then it is possible to pick a single pattern speed
p in
which the orbits precess almost together. If they precessed exactly
together, then one could make a bar by aligning elongated orbits as in
the bottom panel of the figure. Since
-
/ 2 is not quite
constant, it is the job of self-gravity to make the orbits precess not
approximately but exactly together.
This idea was used to understand self-consistent bars by
Lynden-Bell &
Kalnajs (1972)
and by
Lynden-Bell (1979)
and to demystify spiral structure by
Kalnajs (1973) and by
Toomre (1977b).
They were following in the pioneering footsteps of Bertil Lindblad
(1958,
1959: see Section 20).
Calculations of orbits in barred potentials reveal other orbit families
(e.g.,
Contopolous &
Mertzanides 1977;
Athanassoula 1992a,
b;
SW93),
only a few of which are relevant here. Next in importance is the
x2 family,
which lives interior to ILR and which is oriented perpendicular to the bar
(Figure 4, bottom). Between corotation and OLR,
the principal orbits are
elongated perpendicular to the bar, and outside OLR, they are again oriented
parallel to the bar. Near corotation is the 4:1 ultraharmonic resonance in
which a star executes 4 radial oscillations for every revolution:
p =
-
/ 4. We will need these
results in the following sections.
The important consequence is emphasized by Sellwood & Wilkinson (1993): "Not only do the eccentricities of the orbits increase as exact resonance is approached, but the major axes switch orientation across all three principal resonances, making the crossing of orbits from opposite sides of a resonance inevitable" (bottom panel of Figure 4). This is important mainly when the orbits are very noncircular, as in strongly barred galaxies. Now, orbits that cross are no problem for stars. But gas clouds that move on such orbits must collide near resonances. Dissipation is inevitable, and the consequence is an increase in the gas density and hence star formation. This heuristic discussion helps to explain the numerical results reviewed in the following sections, in which gas tends to build up in rings and to form stars there.
2 It is an ellipse and not just radial motion because the star revolves faster than average near pericenter and slower than average near apocenter. Back.