# 18. Theories of Spiral Structure

## Astronomy 626: Spring 1997

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.

## 18.1 Material Arms & Density Waves

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.

## 18.2 Epicyclic Theory

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 x increases outward from the center, and y increases in the direction of rotation.

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
```
and
```             2
d y             dx
(3)         --- + 2 Omega_0 -- = 0 ,
2             dt
dt
```
where A_0 is Oort's `constant' evaluated at R_0. These linearized equations have solutions of the form
```(4)         x(t) = alpha cos(kappa t) ,

(5)         y(t) = - sin(kappa t) ,
```
which describe an ellipse about the guiding center. The sign of y(t) is such that the motion about the ellipse is retrograde with respect to the galactic rotation. This follows from conservation of angular momentum: when the star is at radii R > R_0 it must drift backward with respect to the guiding center since both have the same specific angular momentum.

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

```                      kappa
(6)         alpha = ---------
2 Omega_0
```
for the axial ratio of the ellipse. In the solar neighborhood, alpha is about 0.7; thus the Sun and nearby disk stars are moving on elliptical epicycles which are squashed by about 30% in the radial direction (BT87, Ch. 3.2.3).

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 1.3 radial oscillations per orbit about the Galactic Center (BT87, Ch. 3.2.3).

## 18.3 Kinematic Spiral Waves

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.

## 18.4 Swing Amplification

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.

## 18.5 Alternatives

### Quasi-Stationary Spiral Structure

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| ----- ,
2
```
to be independent of R, where |nu| = omega/kappa is the dimensionless frequency given by the WKB dispersion relation for nearly-axisymmetric density waves (T77, Fig. 4). This is possible, in principle, because |nu| depends on the local radial wavelength lambda.

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).

### Chaotic Spirals

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).

### Tidal Spirals

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 k is a constant used to adjust the strength of the perturbation. No perturbation was applied to the y and z velocities. Shown below after only 1.5 rotation periods is the open, two-armed spiral density wave created by this tidal perturbation. ## References

• 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.

## Homework

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)