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Mon. Not. R. Astron. Soc. 000, 1–16 (2002) Printed 23 December 2009 (MN LATEX style file v2.2)
Applying the Analytic Theory of Colliding Ring Galaxies
Curtis Struck1?
1Dept. of Physics and Astronomy, Iowa State Univ., Ames, IA 50011 USA
23 December 2009
ABSTRACT
An analytic theory of the waves in colliding ring galaxies was presented some years
ago, but the observations where not of sufficient quality then to make quantitative
comparisons. Well-resolved observations of a few systems are now available to make
such comparisons, and structure imaged in several dozen systems, derived from the
recent compilation of Madore, Nelson and Petrillo and the Galaxy Zoo project, can
further constrain the theory. Systems with two rings are especially useful for deriving
such constraints. After examining the implications of recent observations of ring sizes
and structure, I extend the analytic theory, investigate limiting cases, and present
several levels of approximation. The theory is especially simple in the case of nearly
flat rotation curves. I present observational comparisons for a few systems, including:
Arp 10, the Cartwheel and AM2136-492. The fit is quite good over a large range of
cases. For the Cartwheel there are discrepancies, but the areas of disagreement are
suggestive of additional factors, such as multiple collisions. A specific prediction of
the theory in the case of nearly flat rotation curves is that the ratio of the outward
velocity of successive rings approximately equals the ratio of ring sizes. Ring velocities
are also shown to scale simply with local circular velocities in this limit.
Key words: galaxies: dynamics – galaxies: interactions – galaxies: individual: Arp
10, The Cartwheel, AM2136-492, M31.
1 INTRODUCTION: BRINGING RING
GALAXY OBSERVATIONS, MODELING
AND THEORY TOGETHER
The conceptual theory of colliding ring galaxies as prop-
agating waves, generated by an impulsive disturbance, fol-
lowing a head-on collision was proposed by Lynds & Toomre
(1976), and supported by numerical models. One of the nice
properties of the Lynds and Toomre ring galaxy theory is
that it has a perturbation limit. When the companion-to-
primary galaxy mass ratio is small, or the relative velocity
is large, the impulsive disturbance will be small. In this case,
the structure of the target galaxy, including its global grav-
itational potential and its flattened disc, will not be greatly
perturbed. This fact provides a foundation for approximate
treatments of the ring waves.
An analytic model for stellar rings based on this
approximation, and making use of the theory of caus-
tic wavefronts, was developed by the present author and
collaborators in several papers beginning in the late
1980s (e.g., Struck-Marcell & Lotan (1990), Struck-Marcell
(1990)). This and other work published by the mid-1990s
on colliding ring galaxies was summarized in the review ar-
ticle of Appleton & Struck-Marcell (1996, henceforth ASM).
? E-mail: curt@iastate.edu
Since that time a number of new numerical models have been
published (see discussion and references in Sec. 4). These
have generally confirmed the early conceptual theory, and
provided new extensions. Most of the results of these numer-
ical models have been in accord with the qualitative predic-
tions of the analytic model. However, few detailed compar-
isons have been made to date.
The nonlinear dynamics of most galaxy collisions make
quantitative analytic modeling impossible. In the cases
where it is possible there could be substantial benefits to
comparing these predictions to observations and numerical
models. Where the analytic models and observations agree
numerical models can be checked. Where numerical and an-
alytic models agree they can provide powerful tools for in-
terpreting high resolution observations, and extending the
theory. I will provide specific examples of these general state-
ments in Sec 4.
In the case of colliding ring galaxies there have been
several barriers to the quantitative application of the ana-
lytic model since its development. The first barrier is that it
has not been clear whether the perturbation approximation
is applicable to any of the (few) well-studied ring galaxies.
This barrier may be best overcome by ignoring it, i.e. testing
the application of the analytic model. However, the second
barrier is more substantial and practical, except for a very
small number of cases, like the well-known Cartwheel ring,

2 Curtis Struck
sufficient observational data to allow detailed comparisons
have not been available. Moreover, the best-studied cases
(like the Cartwheel, see Sec. 4.2) are not the most symmet-
ric rings, and often their companions are not small enough
to obviously fit the perturbation approximation. However,
much more data has been acquired in the last decade. Ex-
amples are given in Sections 2 and 4.
The final barrier to the application of analytic rings the-
ory was also practical. As originally formulated the analytic
models had a number of parameters, and some of the qual-
itative behaviors depended sensitively on those parameters.
Moreover, the equations were moderately complex and the
formalism of caustic waveforms is not a commonplace in as-
tronomy. In retrospect, the development of the formalism,
e.g., as summarized in ASM, may have been overly broad.
In the time since a great deal has been learned about the
universal scaling properties of all types of galaxies. Observa-
tional surveys of large samples of galaxies like the SDSS have
allowed statistically meaningful average properties and dis-
persions around them to be determined. Large scale cosmo-
logical simulations of galaxy formation, like the Millennium
simulation (Springel et al. 2005), have greatly improved our
understanding of halo formation and buildup. This, in turn,
has allowed the testing of universal analytic halo profiles
against the numerical models.
For the purposes of this paper, the fundamental result
of most interest from among these advances is the univer-
sality of flat rotation curves in galaxy discs. This allows us
to considerably restrict the range of halo profiles for ring
galaxies and their companions. In essence we can make a
second perturbation approximation - that the flat rotation
curve potential is the standard, and others are generally a
small perturbation from it. We will see in Sec. 3 that this ap-
proximation allows us to considerably simplify the analytic
ring wave equations, essentially eliminate some parameters,
and derive some very specific quantitative predictions that
can be compared to the new generation of observations. The
generally favorable result of this comparison implies that we
have a very complete understanding of the simplest kind of
galaxy collision, and holds out the hope that analytic models
can be used to extend this understanding to less symmetric
cases.
2 OBSERVATIONAL RING
PHENOMENOLOGY
When the ASM review was written only one symmetric
ring galaxy had been studied in detail with high resolution,
multi-waveband observations − the Cartwheel. Even then,
there were indications that the Cartwheel was not the most
representative of the class. This will be discussed further in
Sec. 4. Meaningful comparisons between theory and observa-
tion require extensive data from at least a few representative
systems, and moderately resolved optical observations of a
much larger sample to determine average properties. Acquir-
ing such data is especially difficult for ring galaxies because
they are a rare subset of a rare class of galaxy (galaxies in-
volved in major collisions). Only a modest number have been
discovered in surveys, and many of these are quite distant
by the standards of low redshift studies.
As we will see in Sec. 4, detailed studies have been car-
ried out on several prominent systems, including HI den-
sity and kinematic mapping. New imaging collections have
also been published. The first such collections were de-
rived from the Arp (1966) and Arp & Madore (1987) at-
lases of peculiar galaxies. Few & Madore (1986) assembled
an extensive list of ring galaxies from the latter atlas. How-
ever, many of these objects were not colliding ring galaxies,
and images were not readily available for most of the list
objects. Recently, Madore, Nelson, & Petrillo (2009, hence-
forth MNP) have provided a refined list and images of about
100 probable colliding rings, with the selection based on
collision morphologies and the presence of a possible col-
lision partner. This very helpful resource motivated the
present study. A collection of multi-colour observations of
more than a dozen rings was also recently published by
Romano, Mayya, & Vorobyov (2008, also see the study of
Appleton & Marston 1997). More multicolour images have
been produced by the Sloan Digital Sky Survey (SDSS), and
classified by the Galaxy Zoo project on the internet.
As will be discussed in the next section, among the key
predictions of the analytic rings theory are the relative sizes
of successive rings and stellar ring widths. The ring widths
are still difficult to analyze since images of most systems
are only able to detect the young stars formed from the
gas component of the ring. Historically, successive ring sizes
have been hard to study because very few systems with mul-
tiple rings have been discovered. The MNP collection and
Galaxy Zoo objects provide some new, or at least relatively
unknown, examples, though the number remains very small.
However, the analytic theory shows that second and
third rings will generally be broader than the first. This
suggests that it will take longer for them to clearly separate
from an inner bulge or bar component. If the disc gas is
dispersed by strong star formation in the first ring wave,
then star formation may be weaker in subsequent waves,
which would also make them harder to detect. On the other
hand, the surface brightness in the inner regions of many
ring galaxies is apparently rather sharply truncated at an
outer edge, suggesting that we may be seeing the outer edge
of a second (or later) ring rather than a typical declining
bulge profile. In some cases, a partial (inner) ring arc is seen.
The Cartwheel, without a large obscuring bulge, appears to
a good example of these generalizations. In any case, the
outer edge of the inner light distribution provides an upper
limit to the size of an inner ring, and even limits on inner ring
size relative to the outer ring size can be usefully compared
to the theory.
In the next subsection I will present a subset of the
MNP rings, with a few additions, for which such ring size
comparisons can be made. The criteria for inclusion in this
subset are the following.
1. The ring is reasonably symmetric, elliptical, and the
majority of it is visible. These symmetry conditions are de-
signed to eliminate very warped rings or ring-like spirals,
which are the result of additional interaction variables be-
yond those considered by the simple analytic theory. The
visibility criterion is designed to eliminate systems where
most of the ring image is so faint it is hard to be certain
that it is in fact a symmetric ring produced by a direct,
symmetric collision. A large fraction of the MNP rings suffer
from such symmetry/visibility limitations. This is especially

Theory of Ring Galaxies 3
true when only a single sky survey image is available for the
system.
2. A related, but more specific condition is that it must
be possible to fit an ellipse to the outer ring. In principle, this
criterion eliminates rings, whose oval shape is not a circle or
projected circle. In practice, such cases are rare, though in
a few cases I have allowed a very liberal interpretation of
the ellipse fit. The most egregious example is Arp 143, the
triangular ring.
3. Another related condition is that the ring thickness
must not vary greatly with azimuth. Generally, this circum-
stance is the result of an off-centre collision, which will in-
troduce torques not considered in the simple theory (see
ASM). I have also relaxed the application of this restriction
in some cases, but a good example of a beautiful colliding
ring that was eliminated with this condition is the “Sacred
Mushroom” AM 1724-622 (Wallin & Struck-Marcell 1994).
4. A stringent requirement is that in addition to the
outer ring, an inner luminosity component must be present.
This eliminates many empty rings, like the western com-
ponent of Arp 147, a beautiful image of which was recently
produced by the Hubble Heritage Project. Most empty rings
probably come from late-type progenitors, without a sub-
stantial bulge or bar, and have a companion that is sub-
stantial enough to destroy a nuclear star cluster and central
gas disc (see the Hubble Heritage images of Arp 147 and
Arp 148). This condition also eliminates many systems that
are too young and small to have fully detached from their
central regions. The main reason for imposing this restric-
tion is that it is a necessary pre-condition for having an
inner ring in addition to an outer ring. Of course, in most
cases the inner component will be a bulge, a bar, or a ring
plus bulge or bar, rather than simply a nascent inner ring.
For the reasons given above the different possibilities gener-
ally cannot be distinguished without much higher resolution
observations than presently available.
5. A final symmetry requirement is that the central light
distribution should not be displaced too far from the ring
centre. Specifically, I generally require that the central light
core be closer to the ring centre than to the ring itself. This
requirement eliminates some very interesting galaxies, like
NGC 985. However, the rule has been stretched to include
the Lindsey-Shapley ring (AM0644-741) and VII Zw 466.
2.1 The MNP subsample
Application of these conditions reduces the MNP to a couple
of dozen objects, as listed, with the addition of a few non-
MNP objects, in Table 1. This is our primary sample of
possible multiple ring galaxies.
Two or three ellipses were fitted (manually, by eye) to
each object in this sample. In most cases, survey images from
MNP were used. These were originally derived from IIIa-
J photographic plates from the UK Schmidt southern sky
survey.When superior optical images were available, e.g., via
the NASA Extragalactic Database, they were used. Multi-
colour SDSS or Hubble Heritage images of several systems
were found there. In each case, the first ellipse was fitted
to the middle of the outer ring. I will be referring to the
“middle” or central ridge line of finite width rings many
times in this paper. To simply, I will henceforth call this the
RCC for ring central circle. A second, smaller ellipse was
Figure 1. An example of the ellipse fitting procedure. The inset
shows the image from Fig. 5 of MNP; the expanded image shows
the by eye ellipse fitting to the central ridge of the outer ring, and
the outer edge of inner luminous region. The semi-major axis of
the latter is taken as an upper limit to the inner ring radius.
fit either to the visible inner ring, or to the apparent outer
edge of the inner light distribution. In the former case, a
third ellipse was fit to the apparent outer edge of the inner
light distribution. An example of the fitted ellipses is shown
in Figure 1. The ring sizes given in Table 1 are generally
very close to those of MNP, I measured them all for internal
consistency with my measurements of sample objects that
were not included in MNP.
Generally, the innermost ellipse will bound the outer
edge of the innermost ring. When it actually coincides with
the outer edge of the innermost ring, it provides an overesti-
mate of the size of that ring. In most cases the innermost ring
is either buried within a bulge or a bar, has not formed, or
will not form. The latter case is most likely if the innermost
ring would be a third or higher order ring, where phase mix-
ing prevents the development of any distinguishable ring. In
all of these three cases the ellipse is substantially bigger than
the innermost ring, and the ratio of next larger to the inner
ellipse diameter will be a lower limit to the corresponding
ring size ratio.
Nonetheless, it is possible that the innermost ring is
larger than the central light distribution, but is so faint that
it is not detected on the available image. In this case we will
underestimate the inner ring size, and the ring size ratio will
be too large. Since the ring waves will generally incorporate
most of the stars in their radial domain, this circumstance
seems very unlikely unless the stellar surface density falls
unusually rapidly within the disc. However, it could occur if,
as in low surface brightness galaxies, the stars are confined
to a small central region, and the disc consists mostly of
gas. The outer ring could induce star formation, making it
visible, while the inner ring could have propagated past the
stellar core, and been too weak to induce star formation.
Again, this would seem to be a rare happenstance.
Another important assumption was adopted in estimat-
ing the ring size ratios of Table 1. It was assumed that the
longest axis of the ellipse represented the size of the ring,
as projected onto the plane of the sky. The same assump-

4 Curtis Struck
Table 1. Ring diameters.
Ring System a Outer ring Outer ring Ring size r2/r3 b Comment
diameter diameter ratio
(arcsec) c (kpc) d r1/r2
1. Cartwheel 58 35 4.2 1.8
AM0035-335
2. AM0425-421 55 16 > 2.1 -
3. AM0643-462 60 47 > 3.2 - Like AM0644-741
4. AM0644-741 97 41 > 3.2 - Lindsay-Shapley ring
5. AM1133-245 67 51 > 2.6 -
6. AM1135-284 24 8.6 > 2.3 -
7. AM 1323-222 88 26 2.0 > 1.8 Rings unclear e
8. AM1354-250 53 21 > 2.9 -
9. AM1358-221 53 38 > 3.8 - The western ring
10. AM1413-243 35 32 > 2.8 -
11. AM1434-783 40 12 3.5 - Spokes?
12. AM2100-725 27 - > 2.2 -
13. AM2132-535 32 - > 3.3 -
14. AM2136-492 72 76 1.8 > 2.1 Bulge or third ring?
15. AM2200-715 32 - > 2.5 - Bar or second ring?
16. AM2230-481 52 36 > 3.2 -
17. AM2238-541 32 - > 1.8 - Colliding or just barred?
18. Arp 318 40 11 > 2.3 -
19. Arp 10 44 27 > 3.3 -
20. Arp 147 17 11 3.2 - East galaxy. Second
ring uncertain.
21. ESO 200 - 29 33 > 2.1 -
IG009
22. Arp 143 87 23 > 3.3 - Messy ring.
23. NGC 2793 46 5.1 > 4.0 - Bar, not second ring.
24. IC 0614 34 23 > 2.1 -
25. VII Zw 466 21 20 3.0 - Very off-centre second ring.
26. II Hz 4 28 24 3.0 - Emerging second ring?
a Systems listed in the order of MNP, i.e., R.A. order for Arp-Madore systems followed by northern systems.
b These are based on the authors measurements, from various published images.
c All values from MNP, except for II Hz 4, which is from Marston & Appleton (1995). All values estimated
from longest axis.
d Values derived from previous column and (Galactocentric, GSR) distance given in NED.
e On the available image the rings are quite indistinct. They could be rings 2, 3, 4 with small spacings, as
assumed here. Alternately, the outer ring could be ring 1, and the inner rings could be the inner and outer
edges of a wide ring 2.
tion was made for inner ring(s). This assumption is based
on the notion that we have selected symmetric rings whose
true form is circular. Even in this restricted sample there is
evidence that this is not always the case, and that the shape
of some rings is due to more than just projection of a circle.
This evidence includes nuclei offset from the ring centre, and
a lack of alignment between the two ellipses. Extremes of the
former were not included in the sample. In most systems the
ellipses are fairly well aligned, but a significant fraction have
large misalignments.
In most of these last cases we do not actually resolve
an inner ring, so the inner ellipse may only be outlining a
bar component. An exception is the Cartwheel, which has
resolved and misaligned rings.
Figure 2 shows the computed ring ratios versus object
number in Table 1. The most obvious result apparent in
Figure 1 is that there are no ratios less than about 1.8 and
none above the 4.2 of the Cartwheel. Granted, most of the
points are lower limits, so that there could well be rings
with higher ratios. The points representing lower limits are
scattered throughout the range between these two limits.
Interestingly, all of the points seem to concentrate around
values slightly greater than 3.0 and about 2.0, and this is es-
pecially true of the points representing apparently resolved
multiple rings. However, beyond the fact that the points
cluster between values of 1.5 and 4.5, these results are not
very statistically significant. This is especially true given all
the caveats above about the systems with unresolved inner
rings. Yet when compared to the analytic theory below, they
are very suggestive, and certainly provide motivation for ob-
taining well-resolved observations of more ring galaxies.
As an aside we note that in
the spiral sample recently studied by
Mart´ınez-Garc´ıa, Gonza´lez-Lo´pezlira, & Bruzual-A (2009)
the diameter ratios of the outer Lindblad resonance to the
inner Lindblad resonance similarly range over factors of

Theory of Ring Galaxies 5
0 5 10 15 20 25 301
1.5
2
2.5
3
3.5
4
4.5
5
Ring Galaxy (System # from Table 1)
R
in
g
Si
ze
R
at
io
Figure 2. Ring size ratio versus object number from Table 1.
Plus signs mark measured values, upward pointing triangles give
lower limits.
about 2-3. Since both types of wave are associated with
epicyclic motions this is not entirely surprising. We also
recall that Athanassoula et al. (1982) studied the statistical
properties of double ringed galaxies from the large sample of
de Vaucouleurs & Buta (1980). They found that in barred
galaxies the average value of the ring diameter ratio was
about 2.2, while this value ranged more widely (from about
1.7 to 11.7) in non-barred galaxies. In most cases these rings
are probably located at Lindblad resonances in isolated
galaxies. These examples emphasize the nature of rings
cannot be determined from ring ratios alone. Kinematic
evidence or evidence of interactions are also needed.
2.2 The Galaxy Zoo sample
The Arp and Arp-Madore catalogs illustrate the value of
careful searches of survey images to discover examples of
rare galaxy types, like the colliding rings. The Sloan Dig-
ital Sky Survey, with images of many millions of galaxies
presents a great opportunity for such work, but one beyond
the capabilities of any small group of investigators. Fortu-
nately, the Galaxy Zoo project has mustered a much larger
group of amateur investigators to do the sorting. There are
two levels of sorting in the project. The first is the general
classification carried out by any interested participant. The
second is a more informal sort by participants of specialized
internet forums within the project. One of these focusses on
colliding ring galaxies.
From the many objects accumulated in that forum by
October 2009 I’ve selected a dozen for a second sample, see
Table 2. These objects (including one found by the author
in the course of general classification) generally satisfy the
5 criteria applied to the MNP sample above. Of course, the
MNP sample is already a select subset of the Few & Madore
(1986) sample designed to increase the likelihood of them be-
ing colliding rings. Thus, I added a couple of selection crite-
ria like those used in the MNP sample. Specifically,“theta-
rings,” with a bar-like structure extending across the ring
0 5 10 151
1.5
2
2.5
3
3.5
4
4.5
5
Ring Galaxy (System # from Table 2)
R
in
g
Si
ze
R
at
io
Figure 3. Ring size ratio versus object number from Table 2.
As in Fig. 2 plus signs mark measured values, upward pointing
triangles give lower limits.
were excluded, and each object had to have a plausible com-
panion within a few diameters. Here plausible means of ap-
propriate size so that is was not likely a foreground or back-
ground object, and was substantial enough to be a possible
cause of the ring wave.
Table 2 lists the SDSS designations of these objects and
alternate names. The latter show that most of these objects
could already be found in earlier catalogs. However, in many
cases images of the quality of the SDSS are needed to recog-
nize them as ring galaxies. Table 2 also gives ring diameters
and ratios determined as described in the previous subsec-
tion. The ring size ratios and limits for this sample are shown
in Figure 3. The range of values there is very similar to that
of Figure 2, given the limited numbers in both samples. Yet
while this second sample is smaller than the first, it has
a larger fraction of apparently resolved inner rings. Thus,
there are more crosses in Fig. 3 than Fig. 2, and these con-
firm that the typical size ratio of the first to second ring is
about 3.
3 SIMPLE ANALYTIC THEORY AND
APPLICATIONS
In order to understand the suggestive results of the preced-
ing section, and provide more specific predictions to guide
future observations, we reexamine the analytic theory in this
section. This discussion closely parallels that of Sec. 4.2.1 of
ASM, with modifications and extensions. The analytic the-
ory is based on several fundamental approximations: 1. the
collision is perfectly symmetric (i.e., the orbit of the com-
panion is along the rotation axis of the target disc), 2. the
Impulse Approximation describes the disturbance, and 3.
the epicyclic approximation gives the post-collision motion
of stars in the ring galaxy disc. An ancillary approximation
is that the target disc is not significantly disturbed in the
direction perpendicular to its initial plane. Additionally, we
will not consider gas dynamical effects. Alternatives to the

6 Curtis Struck
Table 2. Galaxy Zoo sample ring diameters.
Ring System Outer ring Outer ring Ring size r2/r3 a Comment
diameter diameter ratio
(arcsec) b (kpc) c r1/r2
1. J090225.39 33 53 > 2.4 -
+553633.2
2. J092603.26 31 18 2.9 > 1.7
+124403.7
(UGC 05025)
3. J105007.28 52 25 > 3.3 -
+362030.5
(UGC 05936)
4. J110003.87 43 25 3.1 > 2.7
+172527.6
(PGC 033141)
5. J125255.13 67 51 > 2.6 - Two double rings
+320451.2 in system.d
(KUG 1250+323) See next row.
6. J125302.93 34 16 3.2 > 2.8 Diameter in kpc
+320625.2 assumes same
(KUG 1250+323) distance as 5.
7. J133547.36 18 22 2.4 > 2.4 Broad, faint outer
+455037.5 ring.
8. J145556.45 23 10 4.6 > 3.3 Wide outer ring.
+115229.4
(KPG 445B)
9. J153227.65 28 23 3.0 > 2.2
+414842.3
(CGCG 222-022)
10. J160153.01 39 33 2.5 > 2.5
+452107.0
(PGC 056751)
11. J172430.82 42 26 > 2.9 -
+565434.9
(CGCG 277-042)
12. J230658.93 78 33 2.8 > 2.3 NGC 7489 is
+225611.3 nearby, with
(IC 5285) similar redshift.
a These are based on the authors measurements, from SDSS images.
b All values estimated from longest axis.
c Values derived from previous column and (Galactocentric, GSR) distance given in NED.
d In the primary, the two rings, and the outer edge of the presumed third are all aligned. The outer ring of
the companion is very faint.
impulse approximation were considered in ASM, and the an-
alytic precessing elliptical orbits considered in Struck (2006)
could be used in place of epicyclic orbits. However, in neither
case would gains in accuracy offset the added complications
to the formalism.
In this section I will partially re-interpret the ASM dis-
cussion by focusing on the limiting case of flat rotation curve
(FRC) galaxies, both the ring galaxy and its companion. The
analytic formalism is particularly simple in this limit, which
in addition to its physical significance to galaxies, means
that it provides a good baseline state for comparing other
nearby states (modestly rising or falling rotation curves).
We begin by considering the amplitude of the impulsive dis-
turbance that drives the waves.
3.1 Disturbance amplitudes in flat rotation curve
galaxies
Before the collision we assume that all orbits in the tar-
get disc are circular. According to the Impulse Approxima-
tion, immediately after the collision the stellar orbits are
unchanged except for the addition of a radial velocity com-
ponent resulting from the inward acceleration due to the
companion’s gravity. After the collision, this component is
given by,
vr(t) = ∆vr cos (κ(q)t), (1)
where ∆vr is the impulsive disturbance, q is the initial un-
perturbed radius of the given star, and κ(q) is the radial
epicyclic frequency at radius q. The velocity impulse is ap-
proximately given by,

Theory of Ring Galaxies 7
∆vr = a∆t =
−GMc(q)
q2 ∆t, (2)
where a is the average acceleration on a target disc star,
Mc(q) is the companion mass interior to the radius q, and
∆t is the time interval that the companion spends within
a distance q/2 of the target centre, exerting this pull on
the disc star. Then, ∆t ≈ q/vrel where vrel is the relative
velocity of the galaxies at impact, and is assumed constant
over the relatively brief time interval ∆t. If the companion
galaxy is an FRC galaxy (like the target), with a circular
velocity of vcir2, then we have,
∆vr = −v
2
cir2
q ∆t =
−v2cir2
vrel
= constant. (3)
Because for a flat rotation curve κ ∼ 1/q, the relative ampli-
tude A = (−∆vr)/(qκ) is also constant. Both conceptually
and for formal manipulations this is a great simplification.
3.2 Finding ring radii
For the moment we will set aside the result of the last sub-
section that in the FRC case (both galaxies) the disturbance
amplitude is independent of radius in order to present the
formalism in a more general context. Following ASM (Eq.
4.1) we can approximate the rotation curve of the target
galaxy as a power-law,
vcir = vγ
(
r
γ
)1/n
, (4)
where γ is a reference radius, and vγ is the azimuthal velocity
at that radius. I will assume that the same form applies to
the rotation curve of the companion, but with a different
power-law index (n′) and a different scale velocity (vγc).
In the theory of ASM (and Struck-Marcell & Lotan
1990) nonlinear ring waves are bounded by sharp inner and
outer edges where orbiting stars pile up before reversing
their radial motion. The ring itself is a region of enhanced
stellar density where stellar orbits cross, which they do not
do in the rarefied zones outside the rings. Formally, the ring
edges are caustics where the density goes to infinity. These
caustics are defined as regions where the first-order deriva-
tive of radius with respect to initial radius (∂r/∂q) goes to
zero. (I.e., the radial compression goes to infinity.) Gener-
ally, the caustics are born at a finite radius, where the two
edges meet in a cusp, and ∂r/∂q = ∂2r/∂2q = 0. At times
after the cusp appears, these two equalities are satisfied at
different radii. Specifically, the second derivative condition
defines the centre of the ring, while the first derivative con-
dition defines the inner and outer limits. For nearly FRC
cases the ring width is not large, however.
For estimating the relative sizes of different rings we are
most interested in the location of the RCCs (ring central cir-
cles) of finite width rings , and thus, in the second derivative
condition. It can be written (see ASM),
( 2
n′
)
κt cos (κt)−
( 1
n − 1
)
(κt)2 sin (κt)
=
( 1
n − 2n′
)
( 1
n − 1
)
(
qκ
|∆vr|
)
=
( 1
n − 2n′
)
( 1
n − 1
)
1
Aγ
(
q
γ
) 1
n
− 2
n′
,
with Aγ =
(
|∆vr|
qκ
)
q=γ
, (5)
where again n′ is the rotation curve index of the companion,
corresponding to n in Eq. (4) for the primary. When the
rotation curve is not flat the impulsive perturbation ∆vr in
the target disc varies with radius, and is a power-law for a
power-law rotation curve. Specifically, Eq. (2) implies that
∆vr ∼ q2/n
′ , which accounts for the final factor in Eq. (5).
To simplify the last exponent in Eq. (5) it is convenient to
define,
1
n′′ =
1
n −
2
n′ . (6)
Equation (5) appears complex. However, we will see that in
physically relevant cases it is quite manageable.
3.3 Ring sizes in the FRC case
When the rotation curves of the two galaxies are flat, n,n′
and n′′ go to infinity. In this limit, Eq. (5) reduces to the
very simple form
(κt)2 sin (κt) = 0. (7)
Then, the RCCs of successive rings are found at epicyclic
phases of κt = pi, 3pi, 5pi... At these phases the radii of ring
RCCs are the same as their unperturbed radii, that is, r =
q−Aq sin (κt) = q (where A is given by the last of Eqs. (5),
but here for any q value). In this case κ =
√
2vcir/q, so the
radii of successive rings are given by
qri =
√
2vcirt
(2i− 1)pi , for i = 1, 2, 3... (8)
and the relative sizes of successive rings are given by,
qr1
qr2
= 3, qr2qr3
= 53 , ...
qri
qr(i+1)
= 2i+ 12i− 1 . (9)
The remarkable simplicity of this case, and this result in par-
ticular, was not appreciated previously. In non-FRC cases
the term on the right-hand-side of Eq. (5) does not disap-
pear. The extra dependence on q in that term means that
the ratios of ring sizes will vary with time or with the ring
size. Moreover, in such cases the RCC phase, κt, of a ring
will not be a multiple of pi, and the r − q relation will also
depend on time or additional factors in q. There are also
dependences on the parameters, n, n′, and A in these cases.
On the other hand, most galaxy rotation curves are only
moderately rising or falling if they are not flat, so all these
additional dependences may not have a great effect, and the
FRC case may be a good guide. Nonetheless, before attempt-
ing to compare to observation, it is worth considering what
effect they do have on the ring size ratios.
3.4 Approximate ring sizes in non-FRC cases
The discussion of the previous section suggests that in ana-
lyzing Eq. (5) we should consider κt as the primary variable,
and the other variables as parameters, which when varied
over reasonable ranges have a moderate effect on solutions
κt. In essence, the question to consider is how far can the
κt values for RCCs vary from (odd) multiples of pi, since

8 Curtis Struck
this determines the ring positions and size ratios (i.e., via
generalized versions of Eqs. (8) and (9))?
We can begin to answer this question by considering
the ranges of the rotation curve exponents n and n′. First,
note that n = 1 is the solid body case. Rings do not form
in the solid body case. Values of n between 0 and 1 imply a
rotation curve that rises more steeply than solid body, and
yield unphysical inward propagating waves (see ASM). Since
rotation curves do not generally rise very steeply outside
of core regions of galaxies, it seems reasonable to restrict
consideration to positive values of n (or n′) greater than a
few.
Negative values of n and n′ correspond to falling ro-
tation curves. The Keplerian case corresponds to n = −2.
In general, we expect falling rotation curves in galaxies to
be much less steep than that, so we can expand our restric-
tions to absolute values of n (or n′) greater than a few. By
the definition of n′′, its value will be similarly constrained,
and the q-dependence on the right-hand-side of Eq. (5) will
always be weak.
There are limits on the constant terms on the right-
hand-side of Eq. (5) as well. The amplitude A must be
greater than about 0.1 in order for there to be a significant
ring wave, and the constant factors will be of order unity or
less, depending on the value of n′.
When |n|, |n′| > 10, the second term on the left-hand-
side of Eq. (5) must be small because the all other terms
in the equality are. The solution for κt must be close to a
multiple of pi as in the FRC case. Only when these exponents
take values between 3 and 10 (or their negatives) do we
expect any significant deviation from the FRC case, subject
to the adopted restrictions.
This kind of argument can be pursued from another
direction, and the conclusion can be generalized. We begin
with a generalization of Eq. (9) for the ratios of ring radii
to the non-FRC case,
qr1
qr2
=
(
φr2
φr1
)1−1/n
, (10)
where I adopt the terms φr1 and φr2 for the phases κt of
the RCC of the first and second ring waves. These phases
are odd multiples of pi in the FRC case, but need not be in
general. Also in the FRC case, φr2 = 2pi + φr1. Since the
RCC phase of any ring will depend on q in the general case,
this relation will also not be adhered to. However, we expect
that it will be approximately true since the additional q-
dependences of Eq. (5) in the non-FRC cases are moderate.
Nonetheless, even small differences in its value can have a
significant effect on the analytic solution.
Generally, the first-to-second ring ratio will be large
when φr1 is small. It is physically implausible that φr1 <
pi/2, i.e., where the ring centre would be an at epicyclic
phase corresponding to the initial radial infall of stars. In
this extreme case the ring ratio (with φr2 = 2pi+φr1) would
be less than 5.0, depending on the value of n. At a more
moderate phase of 3pi/4 (i.e., in terms of deviations from
the canonical phase of pi), the ring ratio would be less than
3.7, not much greater than 3.0.
Taking another point of view we can ask, what does it
take to get a ring phase of about pi/2 < φr1 < 3pi/4, which is
needed to get a large ring ratio? This question leads us back
to Eq. (5). If we also assume that both n and n′ are in the
(interesting) range of 3-10 (or the corresponding negative
range), then the first term on the left-hand-side of Eq. (5)
is usually smaller than the second. (Exceptions occur in a
small parameter range.) Moreover, the constant term on the
right-hand-side of the equation is of order unity. With these
simplifications, the equation can be written as,
−
( 1
n − 1
)
(φr1)2 sin (φr1) ≈
(
q
γ
)1/n′′
. (11)
With the adopted constraints on n and φr1, the left-hand-
side is always positive and greater than about 2.6 (e.g., when
n = 3 and φr1 = 3pi/4). If n′′ is negative, then this equation
can only be satisfied when q  γ. If γ is viewed as a core ra-
dius (see below), then there can only be a large ring ratio at
very small radii in this limit. If n′′ is positive, then the equa-
tion can be satisfied at q > γ. However, with the adopted
constraints we expect that 1/n′′ < 1, and usually much less
than 1.0. Thus, we usually expect that this equation is sat-
isfied, and we can get relatively large ring ratios, only when
q  2.6γAγ (with the constant from Eq. (5) restored for
a better estimate). Since galaxy discs usually don’t extend
much beyond a few core radii, this limit is also unphysical
unless Aγ is very small. Thus, regardless of the sign of n′′,
the ring ratio is unlikely to exceed 3.0 by very much over
the interesting range of radii in the primary disc.
An optimal exception to this generalization would have
n = 3 (steeply rising primary rotation curve), n′ = −3/2
(centrally concentrated companion), and n′′ = 1. However,
these values push our limits and we emphasize again that
such exceptions occupy a small area of the relevant param-
eter space. In most realistic cases the ring ratios will be
similar to the FRC case.
This discussion covers the question of maximal ring size
ratios; minimal ring size ratios require that the ring middle
(RCC) phase be greater than the relevant odd multiple of
pi for the given ring number. By an argument analogous
to the one above for the lowest phase, I conclude that the
largest physically meaningful phase for the centre of the first
ring wave is 3pi/2. Then the lowest value of the first-to-
second ring size ratio (assuming φr2 = 2pi + φr1) is about
(7/3)2/3 ≈ 1.8 (with the same restrictions on the values of
n and n′ as above). Again, it would be difficult to push to
this limit, except in special cases.
3.5 Approximating ring sizes in specific systems
The discussion of the previous subsections suggests a se-
quence of successive approximations for using Eq. (5) to
estimate ring spacings. The first approximation is simply
to use Eqs (8) and (9), which are based on the assumption
that all terms of Eq. (5) are very small. This will be a good
approximation in the case that both collision partners have
rotation curves that are close to flat. Unfortunately, the ro-
tation curves of very few ring galaxies, and as far as I know
none of their companions, have been measured. If the rota-
tion curves are not very flat, then this can be a rather bad
approximation.
A second approximation is like that used in Eq. (11)
where we assume that the first term on the left-hand-side of
Eq. (5) is negligible compared to the second. This is valid

Theory of Ring Galaxies 9
if, for example, n′ is large (positive or negative). We do not,
however, want to assume as in Eq. (11) that the coefficient
of the right-hand-side of Eq. (5) is of order unity. (E.g., as
it is not for large n and n′.) Rather we can look at the ratio
of the versions of Eq. (5) for two successive rings. In that
case the right-hand-side reduces to a power of the ratio of
unperturbed ring radii, e.g., qr1/qr2. Using a measured ring
size ratio, Eq. (10) for φr2 this equation can be solved for
φr1 (assuming we know the values of n and n′, or that they
are very large).
This approximation also assumes that the measured
ring size ratio rr1/rr2 ≈ qr1/qr2. If this is not the case,
it could be refined by successive approximations using the
derived values of φ. This approximation will be applied to
the Arp 10 ring galaxy in the next section.
A final approximation is not to neglect either of the
terms on the left-hand-side of Eq. (5), but to use the ratio
of the equation for two rings, as in the second approxima-
tion, to simplify the right-hand-side. Again Eq. (10) and an
estimate of the ring size ratio qr1/qr2 is used, but the result-
ing equation for φr1 is much more complex.
In the case of both the second and third approximations,
after φr1 is determined we can then apply Eq. (5) to the first
ring and solve for the amplitude A. Interestingly, in many
cases, there are a large range of solutions that fail at this
point because they yield negative values of A. In specific
systems values of A can also be constrained by kinematic
observations.
In sum, we see that while the caustic ring wave equa-
tions of ASM are very well defined when we know the val-
ues of the several parameters, the situation is more complex
when the observables provide less information. However, we
can get some interesting limits in many cases. We will con-
sider the application to specific systems in the next section.
3.6 Ring expansion speeds in the FRC case
We conclude this section with some simple kinematic results.
In the text above Eq. (8) we gave an expression for the
epicyclic frequency in an FRC disc. We can multiply that
expression by time t and apply it to a ring RCC to get the
ring radius as a function of time and the central ring phase.
For the ith ring we have,
r = vrit, where, vri =
√
2vγ
φri
. (12)
vri is the ring velocity, defined as the slope of the r − t
relation; vγ is the scale velocity given by Eq. (4), which we
can identify with the circular velocity in the FRC case. The
second equation of Eqs. (12), gives a very beautiful result,
that the ratio of the ring velocity to the scale (or circular)
velocity is a constant divided by the ring epicyclic phase.
Specifically, for a first ring with phase φr1 = pi, the ring
expansion speed is about 0.45 times the circular velocity.
Eqs. (12) also imply that the ratio of successive ring
velocities equals the inverse of the ratio of the ring RCC
phases. Thus, according to Eq. (10), in the FRC case, the
ratio of successive ring velocities equals the ring size ratio,
Ri.,
vri
vr(i+1)
= qriqr(i+1)
= Ri. (13)
Note that in the FRC case the actual ring radius ri equals
the initial value qi.
These basic predictions of the analytic theory can be
checked against observation in any system. Alternately,
given the current paucity of kinematic observations they can
be used to predict relative ring speeds from high-resolution
images. Examples and more details are given in the following
section. The generalization to non-FRC cases is discussed in
the Appendix.
4 CONFRONTING THEORY AND
OBSERVATION
In the previous section I showed that in the symmetric, per-
turbation limit, most cases of physical interest have ring size
ratios very close to the fixed values of the FRC case. In Sec.
2, we looked at the available data on ring sizes or upper lim-
its. At the present time, this data is limited, and in many
cases of minimal quality. Nonetheless, the comparison with
the theory is favorable. First of all, there are no ring size ra-
tios greater than 4.5, well below the theoretical maximum of
about 5.0. Similarly, there are none below about 1.7. In the
case of second and third rings we would expect some lower
values, but there are probably few of these in the sample
(though they could be mistaken for first and second rings,
see Sec. 4.3).
Secondly, for the measured values, as opposed to the
limiting values, there appears to be concentrations at val-
ues slightly greater than 3.0 and slightly less than 2.0. The
latter few values include the inner second and third rings of
the Cartwheel, so we know that they are not first rings. At
the present time, the statistical significance of these appar-
ent results is hardly worth calculating. Nonetheless, the data
offer no contradictions to the theory, and the theory’s pre-
dictions are sufficiently definite to compare to better data in
the future. There are a few individual cases that merit more
detailed discussion now.
4.1 The case of Arp 10
Bizyaev, Moiseev, & Vorobyov (2007, henceforth BMV)
have done a detailed imaging and Fabry-Perot kinematic
study of the Arp 10 ring galaxy. The latter analysis is of
higher resolution than the earlier 21 cm kinematic study of
Charmandaris & Appleton (1996). The outer ring is not as
symmetric as we would like for comparison to the theory.
The inner second ring is not clearly separated from the core
or bulge light in the broad band imagery. The Hα image of
BMV does not show it as a very distinct ring. From the Mt.
Palomar image of the Arp Atlas I estimate that the first and
second rings have a size ratio of > 3.3. Evidently based on
the Hα image, BMV favor a value > 4.
Since BMV find that Arp 10 has a flat rotation curve,
we would expect values of the ring size ratio to be close to
3.0. The higher values are somewhat surprising. According
to the theory they are certainly too high for the outer ring
to be a second ring. This is in agreement with the numerical
model of BMV.
Are the ring ratio estimates so high that they contradict
the theory? To answer this question we apply the theory in
the second approximation discussed in the previous section.

10 Curtis Struck
To keep this calculation relatively simple, we begin with the
assumptions that the ring size ratio is 3.3 (the lower value),
and that the rotation curve of the primary is flat (1/n ≈ 0).
Then, the specific form taken by Eq. (10) in this case is,
qr1
qr2
= 3.3 = φr2φr1
. (14)
The ratio of the two versions of Eq. (5) for the first and
second rings reduces to,
2 cosφr1 + n′φr1 sinφr1
6.6 cos (3.3φr1) + 10.89n′φr1 sin (3.3φr1)
= 3.3−2/n
′
. (15)
In the second approximation we neglect the cosine terms,
and this reduces further to,
sinφr1 = 10.89 sin (3.3φr1). (16)
Given the relatively large coefficient on the right-hand-side,
the sine term on that side must be small, so 3.3φr1 ≈ 3pi.
Specifically, φr1 = 2.848, so the second approximation yields
a result about 10% smaller than that of the first approxima-
tion φr1 = pi.
If we use these results in Eq. (5) for the first ring we find
that n′A ≈ 0.85. Thus, if n′ is large (e.g., n′ = 10), then the
amplitude A is small (e.g., A = 0.085). If we redo the calcu-
lation with an assumed ring ratio of 4.0, we find φr1 = 2.345,
and n′A ≈ 0.51, somewhat smaller. Given the small appear-
ance of the companion in this system, and the high relative
velocity inferred by BMV, these estimates seem plausible,
at least plausible enough to justify a further quantitative
estimate.
Equations (3), (4), and (5) can be combined to yield,
Aγ = 1√2
v2cir2
v2cir
vcir
vrel
= 1√
2
(M2
M1
)
q=γ
vcir
vrel
(17)
BMV estimate the mass ratio of the two galaxies at 1/4,
the mean circular velocity at a bit less than 300 km s−1, and
the relative velocity at impact of about 800 km s−1 (includ-
ing a slowing since impact to the observed 480 km s−1).
Combining these numbers we get an observationally based
estimate of about Aγ = 0.066, which is very consistent with
the analytic estimates for a value of n′ ≈ 10. This estimate
is also consistent with the observation that the companion
is much less luminous than the primary.
Using the results of Sec. 3.6 the analytic model can also
be compared to the observed kinematic values. Using Eqs.
(12) and (13) we find that when the ring size ratio is 3.3
(φr1 = 2.848, vγ = 300km s−1), the outer and inner ring
expansion velocities are (150, 45) km s−1. When the ring
ratio is 4.0 (φr1 = 2.345), the two expansion velocities are
(180, 45) km s−1. BMV find that the radial velocities of
emission line regions in the outer ring range from 30-110
km s−1, and those in the inner ring are about 25 km s−1,
both significantly lower.
However, there are several reasons to suspect that the
comparison is not so direct. Firstly, there is some uncer-
tainty about the inclination of the Arp 10 disc, see BMV and
Charmandaris & Appleton (1996). Moreover, little is known
about the warping of that disc, which could help account for
the range of velocities seen in the outer ring. Another impor-
tant factor is that the measured emission regions are very
likely to have a different epicyclic phase, and thus a differ-
ent radial velocity, than the RCC. They could either just
be in a modestly different phase of their epicyclic motion,
or dissipative interactions could have changed that phase
and their radial velocity. Shocks are likely for gas clouds
in the ring. Vorobyov & Bizyaev (2003) make similar ar-
guments for the idea that emission line observations yield
underestimates of ring speeds, based on their observational
and numerical studies of the Cartwheel.
Interestingly, BMV provide another way to estimate the
ring speeds. This is based on the strength of a number of spe-
cific spectral lines, which depends on the admixture of stellar
populations at each radius. The age of the younger popula-
tions, whose formation was induced by the rings, depends
on the time since ring passage. Given this age and positional
information, BMV were able to derive a best-fit kinematic
model to the spectral index data. The ring velocities in this
model were (180, 46) km s−1. These velocities have substan-
tial uncertainties (see BMV), however, their agreement with
the analytic results is excellent.
Granted the considerable uncertainties, the simple an-
alytic model provides a good match for the Arp 10 observa-
tions. Nonetheless, more accurate data would be very helpful
in firming up our understanding of the dynamics of the Arp
10 interaction.
4.2 The unique, mysterious, and atypical
Cartwheel
The Cartwheel is a more complex case for several reasons.
First of all, the size ratio between the first and second ring
is the largest known among classical ring galaxies. This is
one of the features that makes it so visually spectacular,
but also severely constrains the analytic models. Secondly,
21cm kinematic observations (Higdon 1996) indicate a ris-
ing, not flat, rotation curve. This means the analytic mod-
els are more complicated, as well as further constrained by
this fact. Thirdly, based on the radial velocity profile given
by Higdon, the perturbation amplitude is evidently quite
strong. This means that the azimuthal velocities are also
significantly affected by the collisional perturbation, so the
rotation curve may not give an accurate impression of the
gravitational potential. E.g., the true rotation curve may be
somewhat flatter. This case pushes the limits of the pertur-
bation approximations that the analytic models are based
on. That said, it is still worth trying to fit a model, both to
see how the analytic models perform at their limits, and to
see the differences between this case and FRC cases.
The second and third items of the previous paragraph
suggest opposite approaches. In accord with the second item,
I initially explored rapidly rising rotation curve models for
the Cartwheel disc. However, it proved essentially impossi-
ble to produce a satisfactory model. I then explored models
with more moderate rotation curves. Though possible these
models are still quite strongly constrained.
Before continuing we should note that Amram et al.
(1998) presented a Fabry-Perot study of the Hα emission
in the Cartwheel and companion G2. Their results differ in
several ways from Higdon’s. Most significantly, while their
rotation curve for the approaching side of the Cartwheel
disc is quite similar to Higdon’s, that for the receding side is
much flatter. They observed relatively little emission in the
region between the rings, so their rotation curves are much
more uncertain in that critical region. They also adopted a

Theory of Ring Galaxies 11
mean inclination that is 10◦ different than Higdon’s. Both
papers consider different possible potential centre points,
another source of uncertainty. The primary lessons we take
from these results are that there are likely to be local velocity
variations within the disc beyond those implied by circular
rotation, radial ringing, and a modest amount of warping.
Given the currently available data, substantial kinematic un-
certainties remain.
As in the case of Arp 10, we start with an estimate of
ring size ratio. Here I take the value R1 = 4.3. (Higdon 1996
obtains a larger value of 4.4, which is within our measure-
ment uncertainties.) A value somewhat larger than that of
Table 1 is chosen for reasons that will become apparent at
the end of this example. I also set the values n = n′ = 6.
This choice is close to the smallest value of n which yields
a value of φr2 between (5/2)pi and 3pi, when φr1 is between
pi/2 and pi, and the ring ratio is as large as assumed. These
conditions are also required for a solution to the equations
below. The value of n′ is less constrained, so the present
choice is somewhat arbitrary.
We will use the third approximation in this example,
because the first and second are not sufficiently accurate. In
this case, the RCC equation (i.e., Eq. (5)), becomes,
φr1
3 cosφr1 +
5φ2r1
6 sinφr1 =
1
5Aγ
(
qri
γ
)−1/6
. (18)
Using Eq. (10) for the phase of the second ring, the ratio of
RCC (ring middle) equations for the first and second rings
(corresponding to Eq. (15) in the Arp 10 case) is,
cosφr1 + 2.5φr1 sinφr1
5.7565 cos (5.7565φr1) + 82.84φr1 sin (5.7565φr1)
= 4.3−1/6 = 0.784, (19)
where a very precise value of the ratio of phases (5.7565) is
used because of the sensitivity to the sine term in the denom-
inator. The solution of this equation is φr1 = 1.623, which
can then be substituted into Eq. (18). However, we cannot
immediately solve this for the amplitude, because there is
also a dependence on qr1. Specifically, Eq. (18) reduces to
qr1
γ =
(
0.0497
Aγ
)6
. (20)
The length γ is a free parameter from the mathematical
point of view. Physically, it was used by ASM for two pur-
poses. First, it was introduced as a scale length in the rota-
tion curve of the ring galaxy. In this context it is natural to
assume that it is the smallest radius at which the power-law
rotation curve applies. One expects deviations in the galaxy
core, so γ can be viewed as a core radius. Secondly, γ was
also used as a scale length in the radial dependence of the
perturbation amplitude. Use of the same scale length was a
simplification. As can be seen from Eq. (2) and the second
of Eqs. (5), the radius dependent perturbation depends on
the structure of both the companion and the primary. The
double use of the same scale length is justified only if both
galaxies have about the same scale length, e.g., core size. We
will continue to make that assumption for convenience.
As discussed above, if γ is a core radius, we generally
expect that the ring radii will be of the order of several core
radii. Another aspect of the identification of γ with the core
scale is that it implies that the radius of the extremum of
the perturbation is at about the same radius, which seems at
least physically plausible. The point here is that an analytic
model that requires, say qr1/γ  10 or qr1/γ  0.1 for
reasonable amplitudes Aγ , is somewhat unphysical.
To meet that requirement, we see that Eq. (20) suggests
a value for Aγ , of slightly less than 0.0497. Specifically, for
Aγ in the range 0.035 - 0.040, qr1/γ is in the range 3.7 - 8.2.
The latter range seems reasonable, so we adopt the former
range for the amplitude.
Because we clearly observe the second ring in the
Cartwheel it is worth computing a few more things. First
recall that the ring radii are not actually the qri, but rather
ri = qri(1 − A(q) sin (φri)). Using the values derived above
we find that r1/r2 = 4.2, little different from the assumed
value of R1, but in agreement with Table 1.
The HST imaging of the Cartwheel reveals ring/spiral
structure inside of the second ring. In Table 1 I have iden-
tified this as a third ring and estimated its size, despite its
uncertain nature. The calculation above can be repeated to
derive the parameters of a third ring in this sample model.
The specific form of the RCC equation and other details are
straightforward extensions of those above, except that in this
case we wish to solve for the ring ratio (R2) rather than the
phase of the outermost of the pair (which is now the given
quantity). We find the φr3 = 4.981pi and R2 ≈ 1.5. Table 1
shows that our estimate from the image is about 1.8. Given
that the third ring is not well formed, that we know noth-
ing about the rotation curve in the innermost regions, and
that the collision was probably not symmetric on these very
small scales (about 2 kpc) the discrepancy between these
two numbers is not great. The result lends some support to
the notion that the inner arc is indeed part of a third ring.
At this point our example model seems relatively suc-
cessful in comparison to the observed ring morphology, but
we should compare its predicted perturbation amplitude to
the observations. Higdon (1996) estimates that the mass
of each of the individual companions is less than 6% that
of the Cartwheel, which sounds quite close to our ampli-
tude estimate. However, the comparison is not that sim-
ple. Higdon’s indicative masses are derived from formula of
Balkowski (1973), which uses HI line widths and an optical
disc size. (A kind of early Tully-Fisher relationship seems to
be the basis of this formula.) If the distant companion G3 is
the ring-making collision partner, as suggested by Higdon,
then we note that its HI distribution does not extend much
beyond the fairly compact optical disc, and Higdon believes
that most of its gas (and stars?) has been removed to form
a long bridge to the Cartwheel. In that case, it seems quite
possible that as a result of the encounter both factors in the
mass formula may underestimate the mass of the compan-
ion halo, which perturbed the Cartwheel disc. Companion
G2 does have an extensive HI disc, so the indicative mass
may be more reliable if it was the collision partner. However,
its mass is less than half that of G3, and would seem to be
too low to be the sole cause of the Cartwheel’s rings. Com-
panions G1 and G2 may be interacting with each other, and
may have both collided with the Cartwheel. Their combined
indicative mass is close to that of G3. G1 has no detected
HI gas, and its mass estimate is based on its optical velocity

12 Curtis Struck
dispersion, which is likely to underestimate the mass of an
extended halo.
In sum, it seems possible that the mass ratio of the col-
lision partners was up to a few times larger than the 6%
estimate of Higdon (1996). According to Eq. (17), the per-
turbation amplitude also depends on a numerical factor and
the ratio of the Cartwheel’s circular velocity to the collision
velocity. Higdon estimates the current relative velocity of
G3 and the Cartwheel to be about 360 km s−1. As in the
case of Arp 10 it was probably considerably higher at the
time of impact. The Cartwheel’s circular velocity at the ring
is close to 300 km s−1 (Higdon 1996, Amram et al.s value
is about 10% lower), which implies a velocity ratio in Eq.
(17) of about 0.5. The factor is roughly the same if G1 and
G2 were the collision partners. In the case of our example
model, the numerical factor in Eq. (11) is about 0.5, instead
of 2−1/2 as in the FRC case (see below and Appendix). Com-
bining factors gives an observational amplitude estimate of
6% times a few times two factors of 0.5, which is very close
to the model amplitude Aγ of about 4%. The only problem
is that in using the total masses and the circular velocity at
the ring radius I have effectively computed the amplitude at
the ring. The model amplitude at that radius is about 0.04
x 3.7 = 0.15 (or alternately 0.045 × 1.8 = 0.08). Given the
large uncertainties, these numbers may be consistent, but at
this point the comparison is not reassuring.
As in the case of Arp 10 we can look to the kinematics of
the Cartwheel for more information. For the present model,
the ring speed equation, analogous to the second of Eqs. (12)
is,
vri
vγ
= 1.83φri
(
q
γ
)1/6
. (21)
According to Higdon (1996) the outer ring radius is at
about 16.5 kpc. With the factor qr1/γ ≈ 3.7 from above
we have γ ≈ 4.4 kpc (only slightly larger than the radius
of the second ring). At that radius, Higdon’s rotation curve
gives us a circular velocity of about 90 km s−1 (or up to
190 km s−1 in Amram et al. 1998). Using these values, and
the value for the phase derived above (1.623), we get an ex-
pansion velocity of 126 km s−1 for the outer ring (or up
to 170 km s−1 for Amram’s receding side). Higdon esti-
mates a value of about 50 km s−1. Noting the tendency
for ring speeds derived from emission line velocities to be
low, Vorobyov & Bizyaev (2003) suggest a value of close to
100 km s−1. In any case our model value is too high, though
not by much in the latter case. The real problem with the
kinematics is that it is in stark contrast to Higdons HI ro-
tation curve. Given the value of the circular velocity at the
r = γ, the model predicts a value of 110 km s−1 at the ring
radius. Higdon finds a value close to 270 km s−1. His rota-
tion curve is fairly steeply rising at all radii, while the model
is much flatter, and the formalism cannot account for such
a large ring ratio with such a rising rotation curve. Amram
et al.s receding side rotation curve is more consistent with
the model, except for a large predicted ring speed.
On balance the model is not too bad. It is consistent
with the large ring spacing, and the development of a third
ring, like the observed inner arc. However, it suggests a per-
turbation amplitude that is too large at the outer ring ra-
dius, and it fails to match the observed kinematics. Steeply
rising rotation curves in all or part of the disc are not com-
mon in galaxies, though low surface brightness galaxies often
have moderately rising ones. Such rotation curves are more
common among disturbed galaxies, e.g., see the study of
Virgo spirals by Rubin, Waterman, & Kenney (1999).
This kinematic failure may provide a clue to the
Cartwheel’s development. The models would be less con-
strained and the amplitude discrepancy could be relieved if
the rotation curve was flatter throughout and the first two
rings were not so widely spread. I conjecture that this was
the case when G3 hit the Cartwheel and started the forma-
tion of the rings, but that a second, prograde and off-centre
encounter with G1/G2 disturbed the kinematics in a manner
like that of some of the Virgo spirals. Parts of the outer disc
may have been spun up, and outward, helping to account
for the large diameter of the outer ring.
There are several collateral facts in support of this pic-
ture. First the HI bridge to G3 is long, suggesting that it has
been some time since impact (Higdon 1996). It traces back
to nearly the centre of the Cartwheel, and to produce three
nearly symmetric rings the impact must have been very near
the centre. Secondly, the X-ray bridge towards G1/G2 is
much shorter (Gao et al. 2003), suggesting a more recent
encounter. The extensive gas disc of G2 would likely have
been destroyed in a direct encounter with the Cartwheel
disc, so it probably did not collide directly with that disc.
The mass of G1 seems too low for it to have produced the
rings alone (though it may be underestimated). It seems
much more likely that this pair hit the outer disc of the
Cartwheel (each galaxy at somewhat different times), possi-
bly removing gas from G1. The Cartwheel’s rings are ellipti-
cal, but their major axes point in different directions, and so
the ellipse orientations are not simply the result of projec-
tion onto the sky. They may be the result of disc warping,
but that is less likely for the two innermost rings. Radius
dependent torques in a second encounter may be a better
explanation. Finally, we note that the unusual spokes would
be much easier to account for in a scenario with external
torques.
It will take detailed numerical modeling to test this two-
hit conjecture. While it is more complicated than a single-hit
picture, it does seem an economical hypothesis for account-
ing for a range of Cartwheel peculiarities. Most obviously
it would account for the rarity of such a large ring spacing.
Another comparable case in Table 1 is NGC 2793, one of
the most asymmetric rings of the collection, so perhaps also
a victim of strong torques. KUG 445B, object 8 in Table 2,
has a large ring spacing and is not asymmetric. However,
it has a significantly larger neighbor. If this is the collision
partner, then the disturbance may have been very nonlinear
(Aγ >> 1).
A number of numerical models of the Cartwheel
have been published in the last twenty years. None of
them include two collisions, nor account for the range of
phenomena discussed above. (In fairness, most of them
were designed only to fit a subset of Cartwheel phe-
nomenology, e.g., spokes or multiple rings.) Specifically,
the models of Hernquist & Weil (1993), Bosma (2001), and
Horellou & Combes (2001) do not reproduce the ring spac-
ing. Those of Struck-Marcell & Higdon (1993) do, but have
a steeply rising rotation curve in a rigid halo potential. Their
outer ring is very weak, which may be another reflection of

Theory of Ring Galaxies 13
the difficulties in making an analytic model with a steeply
rising rotation curve. Vorobyov & Bizyaev (2003) present a
two-dimensional model with similar characteristics. None of
these models attempts to match the shape of the Cartwheel
rings. The structural complexities of the Cartwheel, and the
dynamic complications suggested by the scenario above, are
likely to continue to challenge numerical modelers for some
time.
In this case we have learned a couple of things about
the analytic models. First, although they have a number of
adjustable parameters, to paraphrase Pauli, they are good
enough to be proven wrong, or at least inadequate. Secondly,
the way a model fails, or even the difficulties in getting a fit,
can provide clues to what is missing from the model. This
information can be very complementary to that provided by
numerical models.
4.3 AM2136-492 (ESO236-29) and other closely
spaced rings
As a third example consider the AM2136-492 system, which
has two large rings that are much closer together than the
previous two examples. This system has the smallest mea-
sured size ratio in Table 1 (1.8), and it is nearly impossi-
ble to see the rings as the first and second ones within the
context of the assumptions underlying the analytic theory.
Of course, they may not both be the result of a collision.
However, there is no bar component to hold responsible for
the inner ring. Nonetheless, this could be a resonance ring
which has persisted after the dissolution of a bar. On the
other hand, the enormous physical sizes of these rings argues
against this (see Table 1). MNP identify a possible compan-
ion at about two diameters projected separation. This is
about the separation we would expect if the two rings were
in fact the second and third rings. The analytic models and
comparisons to other systems in Table 1 both suggest that
this possibility is much more consistent with the small ring
spacing, assuming the rings are the result of a collision.
This hypothesis immediately raises a few questions.
Firstly, where is the first ring? If it were typically more than
three times the size of the outer ring, it would extend most
of the distance to the putative companion, and thus, may
well have propagated off the primary disc. If not, it may be
too faint to detect on the survey image. The present outer
(e.g., second) ring is already quite faint. This raises the cou-
ple questions why is the second ring so faint and why is the
third ring so thin? In typical analytic models of ASM the
opposite of both circumstances is usually true. In the MNP
sample faint outer rings are in the minority.
At the least it would seem to require a very weak pertur-
bation, and thus, a small or fast companion. In the present
case, the companion is not so distant that we have any rea-
son to believe that the relative velocity is especially large.
On the other hand, the companion appears significantly less
luminous than the primary. If it is experiencing an enhanced
rate of SF (unknown at present), then the mass ratio may
be even smaller than the luminosity ratio. In sum, the ob-
servations are limited, and quantitative tests would require
kinematic data.
The apparently strong SF in the third ring, together
with its thinness, raise another issue. It is likely that the
third ring is the first caustic ring, or nearly so, with ac-
tual stellar orbital crossings in radius, rather than just orbit
compressions. The gas compressions associated with caustic
formation, which is qualitatively like the breaking of a wave,
are much stronger than those associated with mere orbital
compressions. This could account for the differences in SF
in the two rings.
Strong caustics are associated with wide stellar rings,
though the gas may still be found in thin shocks (see ASM).
We see very few noticeably wide stellar rings in the objects
of Table 1. For outer rings this may mean that they have
propagated beyond the stellar disc in many cases. For inner
rings, width compromises our ability to see them as distinct
rings until they propagate out of the core, so there may be
a number of indiscernible ones. In any case, it appears that
many of the outer rings in the sample are either not caus-
tics, or just barely satisfy the orbit-crossing condition. Given
that, one may wonder about the validity of the theory, since
the relevant equations are “caustic” equations. Based on the
success of the theory in providing reasonable models for real
systems it appears that the caustic RCC equation does pro-
vide a valid representation of the compression centre, even
when the wave does not “break.”
Another possible member of an AM2136-492 class
is the z = 0.5, radio galaxy 16V31 studied by
Roche, Lowenthal, & Koo (2002). The faintness and low res-
olution of the HST WFPC2 image of this distant object
make it hard to be completely confident that it is a collid-
ing ring galaxy. The distance and lack of a visible candidate
companion would disqualify it from inclusion in the MNP
catalog. However, it may have two symmetric and similar
rings, and with the sizes given by the above authors, the
size ratio is 1.8, like AM2136-492. In contrast to the latter,
in the rings have similar surface brightnesses, with the outer
one being somewhat brighter. The possibility that these are
again second and third rings is consistent with the compan-
ion having had time to move far enough away to be off the
image. To learn anything more about this system would re-
quire the sensitivity and resolution of HST ACS or WFPC3
imaging.
Another similar system is ESO 381-47 recently studied
with radio, optical and UV observations by Donovan et al.
(2009). In their disc and bulge subtracted residual image
there appear to be three nearly circular rings. The outer ring
is star-forming in the outer gas disc of the galaxy. The mid-
dle ring is apparently the widest, though the inner ring may
not be fully formed, and may be comparably wide. The size
ratios of both the outer-to-middle and the middle-to-inner
are about 2. This is interesting, because it would imply that
the outer ring is the second ring and the inner ring is the
fourth. The galaxy is part of a group, with several possible
companions. They all appear to be considerably less mas-
sive, and so, could be responsible for a weak perturbation,
which gave rise to a system of weak, thin rings. Because the
rings appear to be nearly face-on there is little kinematic
information, including evidence for outward propagation of
the ring waves. Donovan et al. consider a number of possible
causes for the rings besides a collision, including accretion
of the outer disc gas. Given this considered uncertainty, I
have chosen not to include it in Table 1.
And finally, another possible system is NGC 1961, as
described by a recent preprint by Combes et al. (2009). The
visible ring ratio is about 2. It appears possible that an-

14 Curtis Struck
other inner ring is forming. The collisional model provided
by Combes et al. suggests that we are looking at a first and
second (and possibly third) rings. The galaxy is quite asym-
metric, so the analytic theory may not be applicable.
4.4 Mapelli et al.s rings
Recently, Mapelli et al. (2008) have shown that the proper-
ties of some well-known giant low surface brightness (GLSB)
galaxies can be accounted for as the result of what we might
call extreme ring-making collisions. What I mean by this
term is that in their simulations the companion is massive,
more than half the mass of the primary, and significantly
more compact, so the effective mass ratio contribution to
the perturbation amplitude in the disc is probably more than
unity. The relative velocity at impact is the escape velocity
and the rotation curves are roughly flat, so Eq. (17) applies,
and we can estimate the amplitude as A > 0.5.
I have not included Mapelli et al.s GLSB galaxies in
Table 1 because the objects are very unusual compared to
other ring galaxies. However, Mapelli et al.s comparison be-
tween observations and numerical models is convincing. The
physical size of these rings is very large compared to reso-
nance rings, for example. Thus, the interpretation that these
galaxies are rings seems likely, even though in the models
their structure looks unusual compared to other ring galaxy
simulations. On close inspection, it appears that the outer
ring in these models is very broad, in fact, extending across
more than half the disc. This, and a huge expansion of the
disc as the ring wave reaches its outer parts, is what the
analytic theory would predict for a large amplitude pertur-
bation. This situation is in stark contrast to that of the thin
rings of ESO 381-47 discussed in the last subsection.
The analytic theory is not really valid in the limit of
large amplitudes. E.g., the epicyclic approximation to the
orbits is probably not accurate. However, we can again test
the limits of the theory and its usefulness by applying it
to a Mapelli-type example. To begin, consider the caustic
equation, i.e., the equation that determines the inner and
outer caustic edges. It is (see ASM),
(
1− 1n′′
)
sinφ+
( 1
n − 1
)
φ cosφ = 1Aγ
(
q
γ
)1/n′′
. (22)
In the present case we assume the n, n′, and n′′ all are
large, and that Aγ = 0.5. Then the equation becomes simply
sin (φ) − φ cos (φ) = 2. The solutions for the first ring are
φ = 2.15 and 3.95. With these epicyclic phases we find that
the inner caustic edge is rmin = 0.58q, the RCC is about
r = q, and the outer edge is rmax = 1.36q. Thus, the ring
width ∆r/r = 0.78 in this example. Certainly a ring that
is most of the disc when it propagates to the outer parts
merits the description extreme. Nonetheless, the analytic
theory still provides a useful approximation.
With such a large perturbation we would not expect
these rings to be planar, and the numerical models con-
firm a large warping. This means that gas clouds can pursue
their epicyclic oscillations without colliding in the ring with
their counterparts at very different initial radii, just like the
stars. Local, low velocity collisions, between clouds are still
possible. Thus, we do not expect the gas to pile up at a
strong shock front, and star formation to occur through-
out the broad ring. This is what Mapelli et al.s simulations
show, albeit with some pileup at the caustics (see their Fig.
8). These galaxies are well worth further detailed study.
Finally, I would note that the analysis above could be
done in reverse for an observed system, like that of previ-
ous systems. That is, with observed values of rmin, rmax
and the ring centre (q), the edge phases and the amplitude
could be estimated from specific instances of Eq. (22) and
the epicyclic orbit equation (for the ring centre radius). The
derived amplitude could be compared to the observed com-
panion as a model check. With observationally derived kine-
matic information, the indices n and n′ would not have to be
assumed. With observations of a second, inner ring and the
ring spacing, precise ring centre phases could be derived, as
well as obtaining redundant information on the amplitude
to check the model consistency. All of which is to say that
information on the positions, widths and kinematics of mul-
tiple rings would completely specify the analytic model for
any system, and provide redundant checks.
4.5 Andromeda shows how to break the rules
Block et al. (2006) have recently proposed that M31, the
Andromeda Galaxy, might be a colliding ring galaxy, with
both an inner and outer ring easily visible in Spitzer in-
frared observations. Because of its proximity this should be
an ideal object for detailed study. Unfortunately, the nu-
merical model presented by Block et al. indicates that it
is not. The Andromeda rings have a size ratio of about 7,
well outside the range predicted by the models above for a
first and second ring. The animation of the numerical model
(provided to the author by F. Combes), shows that in the
inner disc the ring waves interact with pre-existing spirals
and a bar. The interaction is so strong it makes it difficult
to distinguish the second and higher ring waves.
It is possible that the observed inner ring is a third
wave, while the second wave is not visible due to the wave
interference. In that case, the large size ratio is compatible
with analytic models for the ratio of a first to third ring.
However, in the animation it appears that the second ring
is almost captured within the inner bar, so it may not be
accurate to describe the interaction as simply a disappearing
second ring.
The lesson of this simulation is that in the case of a
small companion, generating weak disturbances in a disc
dominated by strong pre-existing waves, the ring waves can
interact too strongly to propagate in the classical manner.
They may not even be visible. The problem is likely exacer-
bated in the case of off-centre collisions. In survey observa-
tions it would be hard to resolve an inner ring with such a
small relative size. This may be one reason why other cases
have not been documented. Additionally, the outer ring will
generally be weak in such cases, making them hard to iden-
tify as ring galaxies. Nonetheless, it is a very interesting class
of object.
5 SUMMARY AND CONCLUSIONS
This study began in Sec. 2 with the assembly of a sam-
ple of a few dozen symmetric ring galaxies, largely drawn

Theory of Ring Galaxies 15
from the sample of MNP. Although the amount and qual-
ity of data available for most of these objects is very lim-
ited, the optical imagery could be used to estimate the size
ratio of successive rings, or give limits thereof. The result
was a quite restricted range of variation of that quantity
and evidence for a bimodal distribution, albeit with few
data points and many limiting values (see Fig. 2). Although
these results are very tentative, they are in qualitative ac-
cord with the analytic models of ring galaxies developed by
Struck-Marcell & Lotan (1990) and ASM, and together with
detailed observations of a few specific systems, motivated re-
newed consideration of those models.
In Sec. 3 the analytic theory was reviewed and specific
expressions were presented for ring spacings and ring kine-
matics. The importance of the FRC case was emphasized,
both because it is common, and because the analytic model
is particularly simple in that case, and in similar cases with
modestly rising or falling rotation curves. The peaks of the
observed bimodal distribution correspond to the size ratios
of first to second, and second to third rings in FRC discs,
according to the analytic theory.
In Sec. 4 the theory was applied to the study of a few
systems which have been extensively observed, and which
have multiple rings. The general procedure begins by using
the observed ring size ratio and Eq. (10) to derive the ratio
of the epicyclic phases of the RCCs. The ratio of the two
RCC equations (two versions of Eq. (5)) provides a second
equation from which to derive those two phases. The mag-
nitude of the perturbation amplitude cancels in this latter
equation. After the phases have been obtained, either of the
individual ring RCC equations can be used to derive the per-
turbation amplitude. This can be compared to fundamental
constraints (it should lie between 0 and 1), and to the ob-
served luminosity ratio to judge accuracy of the model.
In this procedure the form of the rotation curves of the
two galaxies is input at the outset, and the accuracy of that
initial estimate is judged by the computational outcomes.
Most ring size ratios will only allow reasonable models for
limited ranges of the rotation curve indexes n and n′ and
limited ranges of the amplitude Aγ . This point is well il-
lustrated by the examples of Sec. 4. The models are much
more constrained when kinematic observations are available,
when a third ring is visible, or when caustic widths can be
measured. An example of the latter case is provided by the
extreme rings of Mapelli et al. (2008).
The Arp 10 system, the first example of Sec 4, could be
fit quite well by an analytic model, though there are some
complications in the observed kinematics. All of the various
observations of the Cartwheel system, the second example,
could not be fit very well by an analytic model. The primary
difficulties stem from the disc kinematics, and together with
other clues, they inspire a two-collision conjecture, described
above. The third example, AM2136-492 could also be well
fit by an analytic model, but only if the observed rings are
the second and third. As discussed in Sec. 4.3 that result
tells us some very interesting things about that relatively
unstudied system, and caustic rings generally.
The range of wave strengths from AM2136-492 to
Mapellis et al.s rings is probably very close to the maximum.
Perturbations much less than the former case would hardly
generate a visible ring. Perturbations much more than the
latter case, would be greater than unity and would likely
destroy the disc rather than ring it. It is encouraging that
the analytic models are evidently useful over the whole of
this range.
I conclude by emphasizing two points. First, no other
type of colliding galaxy, nor system of waves within a galaxy
disc can be modeled as simply, with a few algebraic equa-
tions, as the systems described in the preceding sections.
The symmetry of ring galaxies is special and allows them to
be used for fundamental tests of our understanding of the
dynamics of galaxy discs.
Secondly, as is evident from the MNP paper and the
literature in general, ring galaxies are under-observed. If the
data available for all of the galaxies in Table 1 was compara-
ble to that of Arp 10 and the Cartwheel, many fundamental
questions could be addressed using the special properties of
rings, including the following. What are the typical relative
velocities in direct collisions, how many lead to merger, and
on what timescale? How do the answers to these questions
compare to cosmological simulations? What range of com-
panion masses and collision parameters leads to significant
disturbances in the target disc? How does the SF in a ring
wave depend on the wave amplitude? How much does it de-
pend on other parameters, like gas surface densities? What
kind of star clusters are formed as a function of wave ampli-
tude, and how do they evolve behind the wave? Given the
answer to these questions, what is the net SF induced by
ring waves in their passage through a galaxy disc?
A number of these questions are of general relevance
to the problem of how galaxy discs evolve and are evolved
by waves. In other situations where waves are induced by
less symmetric interactions or bars, numerical models can
be compared to observations to shed light on these ques-
tions. However, for ring galaxies the analytic models offer
additional information and checks on the interpretation of
the observation. The potential for substantial scientific re-
turn seems high for observations of ring galaxy structure
and kinematics that are able to resolve multiple rings.
ACKNOWLEDGMENTS
I am grateful to B. J. Smith and F. Combes for com-
ments on an early version of this paper, and to Phil Ap-
pleton for many years of insights on ring galaxies. I am
also grateful to an anonymous referee whose comments
contribued considerably. Many thanks also to the Galaxy
Zoo(www.galaxyzoo.org) organizers and participants, espe-
cially the colliding rings forum group, for their work in
finding more objects. This research has made use of the
NASA/IPAC Extragalactic Database (NED), which is oper-
ated by the Jet Propulsion Laboratory, California Institute
of Technology, under contract with the National Aeronautics
and Space Administration.
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APPENDIX A: GENERAL RING EXPANSION
SPEEDS
In Sec. 3.6 we considered the ring expansion speeds in the
FRC case. The expression for ring speeds in discs with gen-
eral power-law rotation curves is somewhat more compli-
cated. The discussion of the Cartwheel above provided one
example of its use, so I present the general form here for
reference.
We can begin by substituting the equation for the
epicyclic frequency into the epicyclic phase expression,
φri = κt =
√
2
(
1 + 1n
)1/2 vγt
γ
(
qi
γ
) 1
n
−1
. (A1)
Then we differentiate with respect to time (after slight re-
arrangement to make sure the explicit time dependence is
eliminated), to get,
vqi
vγ
= 1vγ
dq
dt =
√
2
(
1 + 1n
)
(
1− 1n
)
φri
(
qi
γ
)1/n
. (A2)
This is the speed with which the wave moves through disc
annuli specified by their unperturbed radius q.
Analogous to Eq. (13), an expression for the ratio of the
speeds of two successive ring waves is
vqi
vq(i+1)
=
(
φi+1
φi
)
qri
qr(i+1)
1/n
=
(
qri
qr(i+1)
)
n2+n−1
n(n−1)
, (A3)
where Eq. (10) was used to obtain the last equality. Note
that for large n the last exponent is approximately 1, as in
Eq. (13), so that simple scaling is approximately retained for
nearly flat rotation curves for q-radii and q-velocities. To ob-
tain the value of the physical radii (ri) and velocities requires
solving for φi(q) for the given potential as described in the
text. Expressions for these quantities also include amplitude
dependences, which introduce further complications. Given
the state of the observations, deriving explicit expressions in
potentials that differ substantially from the FRC case does
not seem justified at present.
This paper has been typeset from a TEX/ LATEX file prepared
by the author.