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 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 2 d y dx (3) --- + 2 Omega_0 -- = 0 , 2 dt dtwhere 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_0for the axial ratio of the ellipse. In the solar neighborhood, alpha = ~0.7; thus the Sun and nearby disk stars are moving on elliptical epicycles which are squashed by ~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 with the definition of A_0 yields the same formula as 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).
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 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 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 k is a constant used to adjust the strength of the perturbation. No perturbation was applied to the y and z velocities. Frames made after 0.5, 1.0, and 1.5 rotation periods show the development of an open, two-armed spiral pattern which becomes more tightly wound with time (although not as tightly wound as a material spiral would become).
Last modified: March 7, 1995