4.2. Varieties of Symmetric Rings
To begin to appreciate the range of symmetric ring morphologies we
show a number of different radius-time plots in
Figure 16. The first
panel has a rising rotation curve and the amplitude
A() is large, so
both the first and second rings are orbit crossing waves. On the other
hand, the perturbation amplitude falls with increasing radius, so both
these caustic waves pinch off at finite radii. The second panel shows
a case with a declining rotation curve and constant perturbation
amplitude across the disk and is similar to
Figure 3. In this case
both first and second waves are caustic rings that do not pinch
off. The third panel shows a case whose rotation curve rises somewhat
more rapidly than a solid body curve. This is not a physically very
realistic case, but the inward propagating waves are an interesting
result, perhaps relevant to galaxy formation.
![]() |
Figure 16. Examples of kinematic models of ring waves for different values of primary and companion structural parameters m and n defined in the text. Radius vs. time is plotted for a number of collisionless particles as in figure 7. In all the cases shown the amplitude A (see equation (4.6) and discussion following) has a value of 0.3. For the effects of varying amplitudes see Figure 3 of Struck-Marcell and Lotan 1990. In plot a) (the upper left), n = 10, m = 1, i.e., a quite flat primary rotation curve, with a perturbation amplitude that declines with radius. In plot b), (upper right), n = - 2, m = 0, a constant perturbation amplitude and a declining rotation curve. In c) (lower left) n = 0.5, m = 1, a declining perturbation amplitude and a primary rotation curve that rises more steeply than a solid body curve. In d) (lower right) n = 10, m = - 0.2, i.e. perturbation amplitude that rises with radius, corresponding to a very extended companion. |
The success of the kinematic caustics theory is that the
morphological variety of Figure 16 can be
accounted for by the caustics Equation (4.13). This is demonstrated by
Figure 17, which
consists of a montage of the solutions of Equation (4.13) as a
function of the parameters m, and n. In the low amplitude
cases (e.g., A()
0.1) there are few
caustic rings, although there will be
significant orbit crowding, so these are not shown.
Figure 17 shows
the A(
) = 0.3 cases,
i.e., relatively large amplitude cases. We will
briefly consider some examples. First of all, in the m = 0 case the
right-hand-side of Equation (4.13) is constant, so the phases
t must
also be independent of radius. Thus, one only has to solve Equation
(4.13) once for the phases of the inner and outer ring edges. At any
given time the Lagrangian radius of the edges can be determined from
the equations
t =
constant, and then the Eulerian radii r(t) are
given by Equation (4.5). This case was considered in detail in
Struck-Marcell and Lotan
(1990).
When n = 1 the primary has a solid-body rotation curve, so that following the (impulsive) collision all stars in the disk should execute synchronous radial oscillations. Thus, there should be no rings, for rings are generally the result of orbital dispersion, which leads to orbit crowding. However, the caustic rings can result from an amplitude gradient in this case. Stars within two different annuli reach their minimum at the same time, but only if the amplitude of the stars in the outer annulus is greater than those in the inner annulus will there be orbit crossings.
Next consider rings at relatively large radii
(q >> ),
when n > 1
(i.e., the rotation curve rises less steeply with radius than for
solid body rotation). In this case the second term on the
left-hand-side of Equation (4.13) will usually dominate, since the
first term is less than or of the order of unity. Then we also require
that
cos(
t) < 1, or
(2p - 1)(
/2) <
t < (2p
+ 1)(
/2), for
p = 1, 2, 3 .... In general, the first few rings will pinch off at
radii that are not too large. In fact,
Figure 17 shows that first ring
caustics are rare, the first rings are usually weak.
If n < 0, the primary disk has a declining rotation curve. In this case the absolute value of the (1/n - 1) term is greater than in the rising rotation curve case, and rings tend to be more robust and broader. Physically this is simply because this case is still farther from the solid body case, and the orbital dispersion is correspondingly greater. When 0 < n < 1, the rotation curve rises more steeply than the solid body case and the rings propagate inward. To achieve this, the mass distribution would have to rise extremely rapidly with radius, for example if the galaxy contained a central deficiency of matter.
If m < 0, then the perturbation amplitude increases outward. Normally this is not physically realistic unless the "companion" is larger than the ring galaxy. In this case outward propagating rings become very broad with time.
Figure 17 also provides information on ring propagation speeds through the disk. Rings in disks with flat rotation curves tend to have a nearly constant propagation velocity. This is often assumed in estimates of ring ages from measured expansion velocities.
To summarize, several generalizations can be derived from Figure 17 and Equation (4.13).