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
soon wind them up into very tightly-coiled spirals. The expected
pitch angle of material arms in a spiral galaxy like the Milky Way is
only about `0.25` degrees (BT87, Ch. 6.1.2). In fact, pitch
angles measured from photographs range from about `5` degrees
for Sa galaxies to `20` degrees for Sc galaxies (Kennicutt
1981). The most likely implication is that *spiral arms 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' 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 dtand

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,

Substituting Eqs. (4), (5), & (6) into Eq. (2), we obtain

2 2 (7) kappa = 4 (Omega_0 - A_0 Omega_0) ,which is equivalent to the formula given in the previous lecture. The Sun and nearby disk stars make about

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 illustrated by Fig. 2 of Toomre (1977; hereafter T77).

By superimposing ovals of different sizes, one can produce a wide
variety of spiral patterns. If `Omega-kappa/2` were
independent of `R`, such patterns would persist indefinitely
because all the superimposed ovals would precess at the same rate. In
fact, plausible disk galaxy models have circular velocity profiles
which yield `Omega-kappa/2` fairly constant over a range of
radii (*e.g.*, Fig. 6-10 of BT87). Compared to material arms,
density waves in the Milky Way should wind up about six times less
rapidly, yielding predicted pitch angles of about `1.4`
degrees. This is an improvement, but still inconsistent with most
observed pitch angles. Moreover, this kinematic model has neglected
the self-gravity of spiral structures, so it can't be telling the
whole story.

*The subject of swing amplification is covered 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 below is an
N-body simulation of a disk galaxy with a central bulge (yellow), an
exponential disk (blue), and a dark halo (red), evolved for about
`1.5` rotation periods. Apart from Poissonian fluctuations,
this disk was initially featureless; the spiral pattern which develops
is due to swing-amplified particle noise.

With time, however, the spiral patterns in numerical simulations tend to fade away as perturbations due to spiral features boost the random velocities of disk stars (e.g. Sellwood & Carlberg 1984). 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, N-body experiments fall short of explaining the spiral patterns of real galaxies, which persist for many tens of rotations.

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 tricky; 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).

To maintain spiral structure in N-body experiments it's necessary to counteract the increasing random motions of disk stars. One way to do this is to mimic the effects of star formation by constantly adding new stars on circular orbits. Sellwood & Carlberg (1984) present simulations in which the disk is assumed to grow by ongoing gas accretion; the accreted mass is added to the model in the form of particles on initially circular orbits. If the mass accreted per rotation is about 1.5% of the disk's initial mass, the disk can maintain an open spiral pattern similar to the spiral patterns of typical Sc galaxies. Further implications of this accretion hypothesis are reviewed by Toomre (1990).

Tides between galaxies provoke a two-sided response, much like the
ocean's response to the tidal pull of the Moon and the Sun. Since
classic two-armed `grand-design' spiral galaxies like M51 and M81 are
evidently interacting with close companions, it's very likely that
these galaxies owe their symmetric spirals to tidal interactions
(Toomre & Toomre 1972). Tidal perturbations, swing-amplified in
differentially-rotating disks, can indeed 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

- Kennicutt, R.C. 1981,
*A. J.***86**, 1847. - Lin, C.C. & Shu, F.H. 1964,
*Ap. J.***140**, 646. - Lin, C.C. & Shu, F.H. 1966,
*Proc. Nat. Acad. Sci.***55**, 229. - Sellwood, J.A. & Carlberg, R.G. 1984,
*Ap. J.***282**, 61. - Toomre, A. 1977,
*Ann. Rev. Astr. Ap.***15**, 437 (T77). - 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. - Toomre, A. & Toomre, J. 1972,
*Ap. J.***179**, 623.

Due date: 4/8/97

18. In class I gave a heuristic justification of the second term in Eq. (3) in terms of conservation of angular momentum, arguing that a star which is displaced outward should lag behind the guiding center. Provide similar heuristic justifications for the second and third terms in Eq. (2).

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

Last modified: April 2, 1997