That rotating disk galaxies should exhibit spiral structure is
scarcely surprising, but the nature of the spiral patterns is not
completely understood -- probably because there is no *unique*
cause of spiral structure.

Because disk galaxies rotate differentially, the orbital period is
an increasing function of radius `R`. Thus *if* spiral
arms were material features then differential rotation would would
wind them up into very tightly-coiled spirals within a few Gyr. In
fact, few spiral arms can be traced much more than one or two times
around a galaxy. The most likely implication is that *spirals are
not material features*.

The other possibility is that spiral arms are *density waves*;
in this case the stars which make up a given spiral arm are constantly
changing. Observational and numerical evidence lends strong support
to the idea of spiral density waves.

Just as water molecules in the ocean do not move very far in
response to a passing wave, the stars in a disk galaxy need not move
far from their unperturbed orbits to create a spiral density wave. To
describe the *local* motions of stars in a disk we study the
equations of motion for small perturbations from a circular orbit.
The result is a description of stellar motion in terms of
*epicycles*.

Let `x` and `y` be a `not-quite-Cartesian' (Toomre
1981) coordinate system which moves about the center of the galaxy
with the angular velocity `Omega_0 = Omega(R_0)` of a circular
orbit at radius `R_0`. In terms of `R` and
`theta`,

(1) x = R - R_0 , y = R_0 (theta - Omega_0 t) ;thus

In this coordinate system, the linearized equations of motion for a star near the guiding center are

2 d x dy (2) --- - 2 Omega_0 -- = 4 Omega_0 A_0 x , 2 dt dt 2 d y dx (3) --- + 2 Omega_0 -- = 0 , 2 dt dtwhere

(4) x(t) = alpha cos(kappa t) , (5) y(t) = - sin(kappa t) ,which describe an ellipse about the guiding center. The sign of

Substituting Eqs. (4) & (5) into Eq. (3), we obtain

kappa (6) alpha = --------- 2 Omega_0for the axial ratio of the ellipse. In the solar neighborhood,

2 2 (7) kappa = 4 (Omega_0 - A_0 Omega_0) ,which with the definition of

One application of epicycles is the construction of *kinematic
spiral waves*. For example, consider a ring of test particles on
similar epicyclic orbits with their guiding centers at the same radius
`R_0`. Let the initial phases of the epicycles be such that at
`t = 0` the particles define an oval. As time moves forward
the guiding centers travel around the galaxy with angular velocity
`Omega_0`, but the stars at the ends of the oval are being
carried backward with respect to their guiding centers, so the form of
the oval advances more slowly. The precession rate or `pattern speed'
of the oval is

(8) Omega_p = Omega - kappa/2 .This point is nicely illustrated by Fig. 2 of Toomre (1977).

By superimposing ovals of different sizes, one can produce a nice
variety of spiral patterns (*e.g.*, Fig. 6-11 of BT87). If
`Omega-kappa/2` were independent of `R`, such patterns
would retain their forms because all the superimposed ovals would
precess at the same rate. In fact, plausible models for the Milky Way
have circular velocity profiles which yield `Omega-kappa/2`
fairly constant over a range of radii (*e.g.*, Fig. 6-10 of
BT87). But spiral patterns are seen even where `Omega-kappa/2`
does depend on `R`, and this kinematic model has neglected the
self-gravity of spiral structures, so it cannot tell the whole
story.

*The subject of swing amplification is covered very nicely by
Toomre (1981), and you should see this review for details; a copy has
been placed in the A626 binder on the reserve shelf.*

In numerical experiments, swing amplification of particle noise can
bring forth trailing multi-armed spiral patterns. Shown here is an N-body model of a galaxy with a
central bulge (yellow), an exponential disk (blue), and a dark halo
(red). Apart from Poissonian fluctuations due to particle noise, this
disk is featureless. However, it does not remain featureless when
evolved forward in time. Frames made after
`0.5`,
`1.0`, and
`1.5`
rotation periods (measured at `R=3 r_0`, where `r_0`
is the exponential scale-length of the disk) show the development of a
trailing multi-armed spiral.

In time, however, the spiral patterns seen in numerical simulations die away as perturbations due to spiral features boost the random velocities of disk stars. Once the disks become too `hot', random stellar velocities reduce the gain of the swing-amplifier and prevent the amplification of small fluctuations. In this respect, the N-body experiments fail to explain the spiral patterns of real galaxies, which have persisted for many rotation periods.

The key assumption of the QSSS hypothesis is that spiral structures
simply rotate at constant pattern speed `Omega_p` without
significant evolution (Lin & Shu 1964, 1966). To arrange such a
spiral, we require the effective precession speed

kappa (9) Omega_eff = Omega - |nu| ----- , 2to be independent of

The mathematical details are pretty complex. Suffice it to say that this is a self-consistent problem, and that where a solution can be found it is unique. Thus the real advantage of the QSSS is that it provides a definite set of predictions for a given spiral system.

The WKB analysis of QSSS gets into trouble at *resonances*
where responses become very large and linear theory breaks down. The
three most important resonances are the *Outer Lindblad
Resonance* (OLR), where `Omega_p = Omega+kappa/2`, the
*Corotation Resonance* (CR), where `Omega_p = Omega`, and
the *Inner Lindblad Resonance(s)* (ILR), where `Omega_p =
Omega-kappa/2`. In particular, the ILR can *absorb* the
inward-propagating density waves, much like ocean waves break and
dissipate energy when they reach a beach (T77).

*See Toomre (1990) for an up-to-date discussion.*

Tides between galaxies provoke a two-sided response. Such
perturbations, if further swing-amplified in differentially-rotating
disks, may produce striking `grand-design' spiral patterns. In the
experiment shown here, an artificial tide was applied by taking the unperturbed disk above and instantaneously
replacing each `x` velocity with

(10) v_x -> v_x + k x ,where

- Lin, C.C. & Shu, F.H. 1964,
*Ap. J.***140**, 646. - Lin, C.C. & Shu, F.H. 1966,
*Proc. Nat. Acad. Sci.***55**, 229. - Toomre, A. 1977,
*Ann. Rev. Astr. Ap.***15**, 437. - Toomre, A. 1981, in
*The Structure and Evolution of Normal Galaxies*, eds. M. Fall & D. Lynden-Bell, p. 111. - Toomre, A. 1990, in
*Dynamics and Interactions of Galaxies*, ed. R. Wielen, p. 292.

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

Last modified: March 7, 1995