AA YYYY

# Spirals in galaxies

J. A. Sellwood1 and Karen L. Masters2 1Steward Observatory, University of
Arizona, Tucson, Arizona, USA; email<EMAIL_ADDRESS>2Departments of
Physics and Astronomy, Haverford College, Haverford, Pennsylvania, USA; email:
<EMAIL_ADDRESS>
###### Abstract

Spirals in galaxies have long been thought to be caused by gravitational
instability in the stellar component of the disk, but discerning the precise
mechanism had proved elusive. Tidal interactions, and perhaps bars, may
provoke some spiral responses, but spirals in many galaxies must be self-
excited. We survey the relevant observational data and aspects of disk
dynamical theory. The origin of the recurring spiral patterns in simulations
of isolated disk galaxies has recently become clear and it is likely that the
mechanism is the same in real galaxies, although evidence to confirm this
supposition is hard to obtain. As transient spiral activity increases random
motion, the patterns must fade over time unless the disk also contains a
dissipative gas component. Continuing spiral activity alters the structure of
the disks in other ways: reducing metallicity gradients and flattening
rotation curves are two of the most significant. The overwhelming majority of
spirals in galaxies have two- or three-fold rotational symmetry, indicating
that the cool, thin disk component is massive. Spirals in simulations of halo-
dominated disks instead manifest many arms, and consequently do not capture
the expected full spiral-driven evolution. We conclude by identifying areas
where further work is needed.

To appear in vol 60 of ARAA

###### doi:

10.1146/((please add article doi))

###### keywords:

disk galaxies, kinematics and dynamics, structure and evolution

††journal: Xxxx. Xxx. Xxx. Xxx.

###### Contents

  1. 1 INTRODUCTION
  2. 2 OBSERVED PROPERTIES OF SPIRALS
    1. 2.1 Modern Demographics of Spiral Arms in Galaxies
    2. 2.2 Rotational Symmetry
    3. 2.3 Pitch Angle Measurements
    4. 2.4 Amplitude Estimates of Spiral Arms
    5. 2.5 Do Spirals Trigger or Concentrate Star Formation?
    6. 2.6 Spirals in High Redshift Galaxies
    7. 2.7 The Spirals of the Milky Way
    8. 2.8 Summary of Observational Evidence
  3. 3 SPIRALS AS DRIVEN RESPONSES
    1. 3.1 Bar-driven Spirals
    2. 3.2 Tidally-driven Spirals
    3. 3.3 Self-excited Spirals
  4. 4 SPIRAL DYNAMICS
    1. 4.1 Preliminaries
      1. 4.1.1 A 2D Stellar Disk in a Rigid Halo
      2. 4.1.2 Disturbance Potential
    2. 4.2 Local Stability Analysis
      1. 4.2.1 Axisymmetric Stability
      2. 4.2.2 Dispersion Relations and Tightly-wrapped Spirals
      3. 4.2.3 Non-axisymmetric Responses
      4. 4.2.4 More Disk Dynamics
    3. 4.3 Lumps and Scratches
      1. 4.3.1 Polarization
      2. 4.3.2 Scratches to the DF
    4. 4.4 Modes in Galactic Disks
      1. 4.4.1 Cavity Modes
      2. 4.4.2 Edge and Groove Modes
  5. 5 THE ORIGIN OF SPIRALS IN GALAXIES
    1. 5.1 A Recurrent Cycle of Groove Modes
      1. 5.1.1 Disk Heating by Spirals
      2. 5.1.2 Sub-maximum disks
    2. 5.2 Other Theories
      1. 5.2.1 Quasi-steady Density Waves
      2. 5.2.2 Responses to Noise
      3. 5.2.3 Shearing Spirals
    3. 5.3 Observational Tests of Theoretical Ideas
  6. 6 GAS IN SPIRAL GALAXIES
    1. 6.1 Maintaining Spiral Activity
    2. 6.2 Gas Flows in Spiral Potentials
    3. 6.3 Flocculent Spirals
  7. 7 ROLE OF SPIRALS IN GALAXY EVOLUTION
    1. 7.1 Radial Migration
      1. 7.1.1 Mechanism of radial migration
      2. 7.1.2 Random Walk in Radius
      3. 7.1.3 Radial Migration in Sub-maximum Disks
      4. 7.1.4 Observational Evidence for Radial Migration
    2. 7.2 Flattening Rotation Curves
      1. 7.2.1 Smoothing Mechanism
      2. 7.2.2 Maximum Entropy State
    3. 7.3 Driving Turbulence in the ISM
    4. 7.4 Age-velocity Dispersion Relations
    5. 7.5 Galaxy Formation
  8. 8 SUMMARY AND CONCLUSIONS

##  1 INTRODUCTION

The question “why do large disk galaxies manifest spirals?” has a simple,
though rather unsatisfying, answer. Evolution in any rotationally supported
disk, be it an accretion disk, a disk in a proto-planetary system, a planetary
ring system, or a disk galaxy, is driven by outward angular momentum
transport. Some viscous-like property is required in shearing disks to extract
angular momentum from the inner parts and deposit it farther out. Because a
collisionless disk of stars is inviscid, outward transport of angular momentum
is accomplished by the gravitational stress from massive, trailing spiral
features. However, this explanation does not begin to address the much harder
question that is the topic of this review: how exactly are spiral patterns
created in disk galaxies?

Lord Rosse was the first to observe spiral arms in a galaxy; he published
(Rosse, 1846) a sketch of the pattern in Messier 51, i.e. the Whirlpool
Galaxy. However, it took many decades before spiral nebulae were recognized as
external galaxies having sizes comparable to that of the Milky Way, and still
longer to discover that the larger galaxies had flat rotation curves, and even
now the extent to which the central attraction is dominated by the stars and
gas in the disk is still debated (e.g. Posti et al., 2019, Macció et al.,
2020).

The prominence of young stars and HII regions at first suggested that spirals
were a gas-dynamical phenomenon, perhaps controlled by magnetic fields.
However, Bertil Lindblad (pronounced “Lindblard”) in the 1940s and 50s,
suggested that self-gravity of the stars could be important, and the idea was
taken up in earnest by the applied mathematicians of MIT and Harvard
University in the early 1960s. One of the first conferences JAS attended as a
graduate student was IAU Symp. 77, where he witnessed an intense discussion
about the cause of spiral patterns and resolution of the issues debated seemed
far off. Pasha (2002, 2004) gives a well-researched historical account of the
developing clash of ideas up to about this date.

By the early 1970s, computers had become powerful enough to follow the
collective dynamics of many thousands of self-gravitating particles. The first
simulations (Miller, Prendergast & Quirk, 1970, Hohl, 1971, Hockney &
Brownrigg, 1974) revealed that models bearing some resemblance to disk
galaxies could manifest spiral features for a short time, seeming to confirm
that they had captured the essential physics, but it has taken almost another
half-century to understand the mechanism that causes spiral instabilities in
simulations of isolated stellar disks (Sellwood & Carlberg, 2019). Although we
review this progress, it should be noted at the outset that we still lack
truly compelling evidence that any spirals in real galaxies are caused by the
mechanism that has been identified from the simulations.

Furthermore, it has been established that spirals in galaxies need not be
exclusively self-excited instabilities of an equilibrium disk, but they can be
the driven responses to passing companions, substructure in the halo, and
perhaps also the stellar bar that resides in the inner parts of a large
fraction of disk galaxies. The spiral responses of the disk to these driving
mechanisms may seem to beg the question: do we need a self-excitation
mechanism at all? We review the evidence that one is needed in §3.

Simulations have also revealed that spirals are important drivers of secular
evolution in galaxies, and that the present-day structure of disk galaxies is
not simply the result of initial conditions at the time of formation (e.g.
Kormendy & Kennicutt, 2004). Spiral-driven evolution can alter the
distribution of angular momentum within the disk, contribute to the increased
random motion of older disk stars, smooth rotation curves, assist galactic
dynamos, and cause a widespread diffusion of the orbit radii of stars, with
important consequences for the spatial and age distributions of metals among
the stars.

Toomre (1977) provided an insightful, though now rather dated, review of
spiral structure. Among other reviews, Athanassoula (1984) had a similar
interpretation, but Dobbs & Baba (2014) and Shu (2016) each embraced radically
differing perspectives. Binney & Tremaine (2008) provide a clear introduction
to some of the basic concepts and mathematical derivations of important
formulae, which we will rely upon in this review.

##  2 OBSERVED PROPERTIES OF SPIRALS

Setting aside dwarfs, the majority of galaxies in the local Universe are
spiral galaxies. This fact noticed in early galaxy surveys continues to hold
in the Galaxy Zoo crowd-sourced morphologies of close to a million galaxies
(e.g. Lintott et al., 2011, Willett et al., 2013).

We here give a brief review of the general nature of spiral arms in galaxies,
but our goal is to focus on properties of the patterns, such as arm
multiplicity, pitch angle, arm amplitude, etc., that could provide some
constraints on models. Several theories for the origin of spiral patterns,
reviewed in §5, invoke strong disk responses and/or swing-amplification (see
§4.2.3) and therefore make virtually identical predictions for most
observables, even though each such theory proposes a different origin for what
is being amplified. However, “quasi-stationary density waves” (§5.2.1, Bertin
& Lin, 1996, p9) are mildly amplified in general, and therefore the observed
properties of the spiral patterns are not expected to conform with the
predictions of swing-amplification in this case.

###  2.1 Modern Demographics of Spiral Arms in Galaxies

The Main Galaxy Survey (Strauss et al., 2002) of the Sloan Digital Sky Survey,
completed over a decade ago, provided well-resolved images in five color bands
of almost $250\,000$ galaxies. Yet progress on quantifying spiral structure
has been slow because measurements of the spiral patterns present a particular
challenge. Automated, or semi-automated, photometric decompositions generally
ignore, or azimuthally average, the spiral features, and most detailed
observational studies of spiral arms have been made in relatively modest
samples, necessitated by visual inspection, or other intensive image analysis.

Spiral patterns are complex structures, whose morpologies can differ in a
variety of ways, including amplitude (i.e. arm-inter-arm contrast), width,
pitch angle, number of arms, and patchiness. Pitch AngleThe angle, $\alpha$,
in the disk plane that a spiral arm makes with a circle at the same radius.
Hubble originally ordered spiral galaxies in a sequence as Sa, Sb or
Sc111Intermediate types Sab, Sbc and S0, Sd and Sm galaxies were added later
based on a combination of the prominence of the central bulge and the “the
spiral pitch angle and degree of resoution” of the spiral arms Hubble (1926,
p325). Disk galaxies lacking any spiral patterns, gas, or dust were classified
as S0, or lenticular. With no implication of an evolutionary sequence, Hubble
labeled disk galaxies to range from S0/Sa “early-type” to Sc/Sd “late-type”
spirals. The Hubble classification correlates with galaxy color, with
lenticulars and ellipticals being redder, while spirals are increasingly blue
towards the later types. Galaxies are bi-modal both in morphology and color,
but care should be taken with equating red with early-type and blue with
spiral as interesting sub-populations of red spirals and blue early types do
exist (e.g. Smethurst et al. 2022).

The Hubble sequence does not capture all the ways in which spiral arms vary in
appearance. Elmegreen & Elmegreen (1982) coined the apt word “flocculent” to
describe NGC 2841-type galaxies that had previously been identified as a
separate “division” by Sandage (1961) and by Kormendy & Norman (1979).
Elmegreen & Elmegreen (1982) went on to describe a set of arm-classes
from“grand design” to “flocculent”. Grand design patterns generally, though
not exclusively, have just two arms and are slightly more common among the
earlier Hubble types, FlocculentLiterally having a fleece like appearance,
meaning spiral arms that are short, closely spaced and fragmented Grand
designSpiral arms that are highly symmetric and continuous whereas the
occurrence of flocculents increases toward the later spirals. But the trend is
not strong; for example, Elmegreen & Elmegreen (1982) find a complete range
from flocculent to grand design among Sb-Sc galaxies.

The wavelength of observation strongly affects the appearance of spiral arms.
As older stars contribute relatively more light to redder bands, red and IR
images better reflect the underlying stellar population, while bright young
stars stand out in images in bluer bands. As a consequence, spiral arms appear
smoother in near-IR (NIR) images (e.g. Jarrett et al., 2003). Elmegreen et al.
(2011) found from the S4G sample (Sheth et al., 2010) that most optically
flocculent galaxies are still at least partially flocculent in the mid-IR
(MIR), although some galaxies which appear flocculent in blue light, are
revealed to have underlying grand design spirals in NIR images (Thornley,
1996). Such examples are rare, however, and Buta et al. (2010) used the S4G
survey to confirm the earlier conclusion by Eskridge et al. (2002) that
classifications of most spiral galaxies in the MIR are similar to those from
B-band images, being roughly one Hubble type earlier mainly because bulges
become more prominent.

One successful method of analyzing galaxy morphology in larger samples, has
been the use of crowd-sourcing (citizen science) to obtain quantitative visual
classifications (i.e. Galaxy Zoo, Lintott et al., 2011). Using Galaxy Zoo
classifications, in a series of papers, Hart et al. (2016, 2017a, 2017b, 2018)
provide a detailed look at the demographics and properties of over 6,000
galaxies having visible spiral arms in the redshift range $0.03<z<0.05$ and
with $r$-band magnitude $M_{r}<-21$ (or stellar masses $\mathrel{\hbox
to0.0pt{\lower 2.15277pt\hbox{$\mathchar 536\relax$}\hss}\raise
1.72218pt\hbox{$\mathchar 318\relax$}}10^{10}M_{\odot}$).

###  2.2 Rotational Symmetry

Few spiral galaxies manifest highly regular and completely symmetric spiral
patterns, but a rough symmetry can usually be picked out by eye. Hart et al.
(2016) found that the majority, 62% in their luminosity limited sample, of
spiral galaxies have two spiral arms, 20% of galaxies have three arms, 6.5%
have four arms, and a similar percentage has five or more. These proportions
were found to depend somewhat on galaxy properties, such that two-armed
spirals are more common in higher density environments and in redder disk
galaxies. These percentages from crowd-sourced visual inspection agree well
with findings from Fourier analysis of images of smaller samples (e.g. Davis
et al., 2012, Yu & Ho, 2018).

The order of rotational symmetry of spiral patterns is predicted by swing
amplification theory (§4.2.3.2) to correlate, albeit with significant scatter,
inversely with the disk contribution to the central attraction (Sellwood &
Carlberg, 1984, Athanassoula et al., 1987). A two or three arm pattern is
indicative of a heavy, almost maximum disk,222The contribution to the central
attraction from an absolute maximum disk is as large as it can be without the
rotation curve requiring a hollow halo. while significantly sub-maximum disks
should manifest higher rotational symmetry. Furthermore, spiral patterns in
galaxies generally have higher multiplicity in the outer parts where the halo
contribution becomes more dominant. If the large number of spiral fragments in
flocculent galaxies are gravitationally-driven, then that would indicate a
very low mass, cool, sub-component of the disk, which may co-exist with an old
unresponsive hot disk (see §6.3).

The review by Jog & Combes (2009) reports that fully a third of spiral
galaxies exhibit significant asymmetry or lopsidedness. Lopsided instabilities
are predicted in galaxy disks that lack any significant halo (Zang, 1976,
Evans & Read, 1998b), but this seems an unlikely explanation. Therefore
asymmetries are typically attributed (e.g. Zaritsky & Rix, 1997) to some kind
of external forcing, such as tidal interactions or gas inflow.

###  2.3 Pitch Angle Measurements

Logarithmic spirals, which have a constant pitch angle with radius, frequently
fit spiral arms reasonably well (Kennicutt, 1981), although individual arms
can rarely be traced over a significant radial range. Measuring the pitch
angle of an arm, let alone an average over all arms in a galaxy, is
complicated by the patchy nature of spirals, kinks or branches that occur in
some arms, differences between pitch angles in multiple arms in a single
galaxy, and the uncertainty created by determining the inclination and center
of the galaxy. All these factors mean that visual estimates of spiral winding
are easy, but quantitative measurement of pitch angles is deceptively tricky
(see e.g. Díaz-García et al., 2019, Hewitt & Treuthardt, 2020, Yu & Ho, 2020,
for recent comparisons of measurement techniques). These complications,
together with sample selection, have prevented the emergence of a consensus on
correlations between spiral arm pitch angle and global or local galaxy
properties.

The classic Hubble sequence implies a correlation between bulge size and pitch
angle, and such a correlation has been found, albeit with large scatter
(Kennicutt, 1981, Davis et al., 2015, Yu & Ho, 2020), while others have
reported an absence of a strong correlation (Hart et al., 2017b, Masters et
al., 2019, Díaz-García et al., 2019, Lingard et al., 2021). Two recent papers
have attempted to correlate pitch angles with other galaxy properties in large
samples ($N>2000$; Hart et al. 2017b, Yu & Ho 2020), with both finding a
greater frequency of tightly wound arms in redder disks that have greater
stellar mass as well as galaxies with higher concentration and greater stellar
velocity dispersion.

Swing amplification predicts a loose correlation between pitch angle and
rotation curve slope: galaxies having rising rotation curves should have more
open arms while more tightly wrapped arms are expected where rotation curves
decline (§4.2.3.2 and Grand et al., 2013). A similar conclusion about the mass
distribution was reached by Roberts et al. (1975). Kennicutt (1981) and Seigar
et al. (2006) both noted a correlation between pitch angle and maximum
circular speed in the disk, but Seigar et al. (2006) linked the trend to the
impact of shear on pitch angle. However, Yu & Ho (2019) find a much weaker
correlation with shear, and they argue that the central slope of the rotation
curve is more important.

Spiral arms are said to trail when the ridge line of the spiral lags with
respect to the rotation direction as the radius increases, for which
$\alpha>0$ is conventional. Assessing whether spiral arms lead or trail
requires both kinematic data on which side of the minor axis is approaching
and which receding and a determination of which side of the galaxy’s major
axis in projection is physically tilted towards the observer, which is
generally determined through visual inspection of dust lanes. Historically,
Slipher (1922) used his first Doppler shift measurements of galaxy rotation to
conclude that all the spirals trailed in his small sample of galaxies. de
Vaucouleurs (1958) found trailing arms in all 17 galaxies for which he had
complete data. Pasha (1985) found just two cases of leading spirals among the
109 galaxies he examined, and both were tidally disturbed. While a handful of
other galaxies with leading arms have been found (e.g. Buta et al., 1992),
there have been no significant recent updates, and it is widely assumed that
trailing spirals are the norm.

###  2.4 Amplitude Estimates of Spiral Arms

Images of spiral galaxies in both NIR and MIR wavebands have been used to
estimate the mass contrast between the arm and inter-arm regions. For example,
Rix & Rieke (1993) measured the contrast for M51, the prototypical grand
design spiral, to be a factor of 2-3. Elmegreen et al. (2011) used S4G images
to survey arm contrasts across a wide range of spirals, finding a similar
range (0.3-1.3 magnitudes, or factors of 2-3), noting that grand design
spirals have larger contrasts than do flocculent spirals, and that the mean
contrast increases slightly toward later Hubble types. Querejeta et al. (2015)
corrected these 3.6-$\mu$m images for dust emission, which reduced arm-
interarm contrasts by some 10% (Bittner et al., 2017). Although dust
corrections are important, a concern with all these measurements is that
emission from young stars also remains bright in the MIR, causing surface
brightness differences to overestimate surface density variations. This
concern was addressed by Zibetti et al. (2009), who modeled the stellar
population pixel by pixel to estimate the stellar mass surface density in a
sample of nine galaxies from The Spitzer Infrared Nearby Galaxies Survey
(SINGS Kennicutt et al., 2003). The lower $M/L$ in the spiral arms found by
Zibetti et al. (2009) reduced the arm-interarm mass contrast from that in any
single photometric band; specifically they found the arm-interarm mass
contrast in NGC 4321 was half that in either $i$\- or H-band images.

If spiral arms represent significant mass over-densities, they should should
also cause detectable deviations from smooth circular motion of stars and gas.
High spatial and velocity resolution data are required to measure such non-
circular flows and only a few cases have been reported so far (e.g. Visser,
1978, Kranz et al., 2003, Shetty et al., 2007, Erroz-Ferrer et al., 2015). The
data reveal “wiggles” in the projected isovelocity contours across spiral
arms; this is evidence that the arms are massive enough to perturb the
circular gas flow by $\sim 20\;$km s-1. Spiral-driven streaming motions are
best interpreted by fitting models (see §5.3). With the current explosion of
kinematic measurements of galaxies from integral field spectrograph surveys,
as well as gas imaging data, this may be an area ripe for more systematic
study.

###  2.5 Do Spirals Trigger or Concentrate Star Formation?

The presence of spiral arms is observed to correlate with an enhancement in
star-formation rate (SFR), and spirals may trigger and/or concentrate star
formation (SF; Roberts, 1969, Kim et al., 2020). It has long been noted that
the average SF properties of galaxies vary systematically along the classic
Hubble spiral sequence (Kennicutt, 1998a), with later spirals having more gas
and enhanced SF relative to earlier spirals. Hart et al. (2017a) in agreement
with early work (e.g. Elmegreen & Elmegreen, 1982) also note a link, with
flocculent spirals tending to be bluer than grand design spirals, but conclude
the overall SFRs are similar. The SF efficiency (SFE) can be used to
distinguish enhanced SF due to increased gas density from an enhancement of
the SFR relative to density. Studies of the SFE of discs suggest it does not
vary much from arm to inter-arm (Foyle et al., 2010), although Yu et al.
(2021) demonstrate a correlation of SFE (or gas depletion timescales) with
spiral arm strength in a large sample.

Given the observed relationship between spirals and star formation, it remains
surprising that significant numbers of red, or quiescent (sometimes anemic)
spirals exist. The first mention of this effect was by van den Bergh (1976)
who found examples of gas stripped spirals in the Virgo and Coma clusters.
Using Galaxy Zoo, Masters et al. (2010) showed that optically red galaxies
include significant fractions of massive and/or early-type spirals, and even
6% of late-type spirals are red. In general these galaxies have some residual
star-formation (e.g. they are detectable in UV; e.g. Fraser-McKelvie et al.
2016), but it is significantly less than that expected for typical spirals of
the same size, demonstrating that spiral arms can be visible, at least for a
while, in the absence of significant star-formation (cf. §6.1).

###  2.6 Spirals in High Redshift Galaxies

The search for spiral patterns in high redshift galaxies is of interest to
learn when disk galaxies first became settled enough to develop them. The
exquisite resolution of the Hubble Space Telecope (HST) enabled observations
of galaxy morphology beyond the very local Universe, and a number of large
area surveys have provided data for understanding the galaxy population as a
whole. One complication with high redshift observations is band shifting; high
redshift images are often in bluer rest-frame bands in which local galaxies
also tend to look clumpier (§2.1). Also genuine spirals need to be
distinguished from “bridges and tails” (Toomre & Toomre, 1972) created by
tidal interactions between galaxies that are more common at higher redshift.

Early results painted a picture of high-redshift star-forming galaxies being
significantly more irregular and clumpy (e.g. Abraham et al. 1996) and having
larger velocity dispersions (e.g. Genzel et al., 2008) than local discs.
Elmegreen et al. (2009) searched for typical spirals in a sample of 200
galaxies out to $z\sim 1.4$ in two HST surveys, finding examples of all types
of local spirals (grand design, mixed and flocculent) alongside the more
typical high redshift clumpy galaxies. Going to even higher redshifts,
Elmegreen & Elmegreen (2014) perform a visual classification of galaxies in
the ultra-deep field, finding examples of grand design spirals out to at least
$z=1.8$, and flocculent types to $z=1.4$. They measured the arm-inter-arm
contrast in a high redshift grand design spiral, finding it comparable to that
in local spirals. Crowd-sourcing has also been used visually to classify the
largest high-$z$ samples from HST (Willett et al., 2017, Simmons et al., 2017)
providing hundreds of examples of spiral galaxies at high redshift. With the
imminent launch James Webb Space Telescope, we can expect higher resolution
and more sensitive images taken at longer wavelengths that may reveal settled
disks manifesting spirals at even higher redshift.

###  2.7 The Spirals of the Milky Way

Observing the spiral structure of our own Milky Way presents a particular
challenge because of our location within the disk. Progress has been made via
radio and IR observations, as well as through the kinematic and astrometric
observations of stars, especially the exquisite data from the Gaia mission. As
this topic has recently been reviewed by Vallée (2018) and by Shen & Zheng
(2020), we will give just a short summary here. Note that while observations
of the disk structure of the Milky Way are challenging, they uniquely provide
the full 6D phase space information of individual stars, offering the best
hope for constraining the formation mechanism of spirals, at least in the one
case of the Milky Way. GaiaAn astrometric satellite that is returning high
precision positions, proper motions, and photometry of billions of stars and
radial velocities of the brighter ones

Surveys of gas emission lines from the Galactic plane, both of the 21cm line
of neutral hydrogen and of various molecular lines, provide intensity and
kinematic information along the line of sight. Extracting information from
such data about the locations and streaming flows in spiral arms challenging
(see Peek et al., 2021, for a clear example). It is is best done by
constructing models; e.g. Yuan (1969) fitted a single global 2-armed spiral
while Li et al. (2016) fitted a bisymmetric spiral and separate bar flow to
the inner Galaxy only.

Reid et al. (2019) used very long baseline interferometry observations to map
the positions and motions of young, high-mass stars that appear to be maser
sources. Their data favor a four arm model for the Milky Way, with average
pitch angles for the major parts of the arms of $\alpha=10^{\circ}$, which is
surprising as four-armed spiral galaxies are rare (e.g. Hart et al. 2016 find
just 335 of their 6683 spirals have four arms). By contrast, only two major
arms were revealed in the Spitzer Space Telescope’s GLIMPSE (Galactic Legacy
Infrared Mid-Plane Survey Extraordinaire) survey (Churchwell et al., 2009),
which was based on NIR star counts of the old stellar population within the
disk and may be more representative of the mass distribution.

Khoperskov et al. (2020) find surface density variations in the local star
distribution from the Gaia second data release (DR2 Gaia collaboration, 2018)
stellar positions, but incompleteness among the faintest stars and a limited
volume make it hard to say much about either the mass amplitude or the global
pattern. However, the Gaia DR2 data also manifested ridges (or ripples) in the
$R-v_{\phi}$ distribution of stars that Eilers et al. (2020) interpreted as
the kinematic signature of spiral arms; they fitted a steady spiral model to
these data to estimate the arm relative amplitude as $\sim 10\%$ and pitch
angle to be $\sim 12^{\circ}$. Castro-Ginard et al. (2021) use star clusters
identified in Gaia EDR3 to map Galactic spiral arms, their pattern speeds and
to look for age gradients. They find a declining pattern speed with radius,
and a lack of age gradients downstream from the arms. Both Hunt et al. (2018)
and Sellwood et al. (2019) used the phase space distribution of stars from
Gaia DR2 to test theories for the origin spirals, which we discuss more fully
in §5.3.333The “snail shell in phase space” (Antoja et al., 2018, abstract),
one of the most interesting discoveries in the Gaia DR2 data, is probably
caused by excitation of a bending disturbance, rather than spiral activity.

This somewhat confusing picture of the spiral structure of our Galaxy
unfortunately complicates interpretation of spiral arm signatures in this one
case where we have a truly close-up view.

###  2.8 Summary of Observational Evidence

This short review of the observational evidence reveals a picture where sample
selection and details of measurements complicate any general conclusions. And
while observations of the Milky Way are revealing exquisite detail, they are
from just a single spiral galaxy. There is clearly plenty of observational
work still to be done to improve our understanding of spiral arms in galaxies.
For example questions which at present have limited, or conflicting results
include the following:

  * •

What is the distribution of pitch angles across the galaxy population?

  * •

What galaxy properties does the pitch angle physically correlate with?

  * •

What is the range of arm-interarm stellar mass contrast?

  * •

How large are the typical velocity perturbations caused by spiral arms?

  * •

Is SFE constant between arms and inter-arm regions in all types of spirals -
i.e. do spiral arms trigger, or just enhance SF via increased gas density?

  * •

Does the disk of the Galaxy outside the bar have 2- or 4-fold rotational
symmetry?

The ultimate, though still elusive, goal is to find observational evidence
that can discriminate among theories for the origin of spiral arms.

##  3 SPIRALS AS DRIVEN RESPONSES

###  3.1 Bar-driven Spirals

Many spiral galaxies possess bars (e.g. Buta et al., 2015), and both theorists
(e.g. Toomre, 1969, Feldman & Lin, 1973) and observers (e.g. Kormendy &
Norman, 1979) have suggested bars as a driving mechanism for spirals. A bar
introduces a quadrupole component to the gravitational field of a galaxy that
can drive an open spiral response in a smooth, massless gas layer (e.g.
Sanders & Huntley, 1976, and dozens of subsequent papers), and perhaps also a
weak response in the stars (Athanassoula, 2012). A good case can be made for
bar driving in the fraction of barred galaxies that also possess
(pseudo-)rings (Buta & Combes, 1996). Open spirals are more common in the
majority of barred galaxies that lack outer rings, where we should expect any
bar-driven spirals to be both bisymmetric and to have the same pattern speed
as the bar. However, spirals in the outer disks of both simulations (e.g.
Sellwood & Sparke, 1988, Lieb et al., 2021) and observed barred galaxies
frequently have a different pattern speed from that of the bar, and therefore
cannot be simple driven responses. Since the quadrupole field decays rapidly
with distance from the bar, it is likely that these spirals are excited by
other means than by the bar.

Some nearby barred galaxies, such as NGC 1300 and NGC 1365, do have
beautifully regular bi-symmetric spirals joined to the ends of the bar. While
such cases may superficially support the idea that the spirals are driven by
the bar, Speights & Rooke (2016) found that the spirals in NGC 1365 have a
lower pattern speed than that of the bar, thereby ruling out the idea of
simple bar driving in that case. The appearance of the arms starting from the
bar end is not just a coincidence, however, since Sellwood & Sparke (1988)
reported that an apparent connection between the spiral and bar lasts for a
very large fraction of the beat period. Li et al. (2016) also fitted a pattern
speed for the spiral that was lower than that of the bar in their detailed fit
to the inner Milky Way. Font et al. (2014) present estimates of corotation
radii in a large sample of galaxies based on sign changes of the radial gas
flow (see §6.2), identifying multiple pattern speeds in 28 of the 32 barred
galaxies in their sample. Furthermore, some barred galaxies have a three-armed
pattern in the outer disk, which is inconsistent with bar driving; examples
from NIR images are M83 (Jarrett et al., 2003) and NGC 2336 (available in
NED).444NASA/IPAC Extragalactic Database, funded by the National Aeronautics
and Space Administration and operated by the California Institute of
Technology.

A number of papers report statistical evidence for or against the idea that
spirals can be driven by bars, but the conclusions are mixed. In studies of
MIR images, Salo et al. (2010), in a reversal of the previous conclusion by
several of the same authors (Buta et al., 2009), argued that the data favor
bar driven spirals, while Kendall et al. (2011) found that spirals in the
outer disks of barred galaxies are little different from those in apparently
unbarred galaxies. In Galaxy Zoo, Masters et al. (2019) reported that arms in
barred galaxies appear to be less tightly wound on average, but a more
detailed study (Lingard et al., 2021) ruled out any statistically significant
correlation, confirming the finding of Kendall et al. (2011).

In summary, spirals in some barred galaxies may be driven responses. There
are, however, many spirals in barred galaxies for which simple bar forcing is
clearly ruled out, and some other mechanism is required to excite them.

###  3.2 Tidally-driven Spirals

A spiral pattern may also be triggered by the tidal field of a passing
companion galaxy: M51 and M81 are particularly clear examples. The vigor of
the spiral response is generally enhanced by swing-amplification (Toomre,
1981, and §4.2.3.2). Salo & Laurikainen (2000) and Dobbs et al. (2010) report
detailed models of the M51 system that provide a reasonable match to most of
the observational data. Simulations have also shown that tidal encounters can
trigger bar formation (e.g. Peschken & Łokas, 2019, and references therein),
but it is unclear, as of this writing, what ranges of masses or orbits of
perturbers would excite spirals but not provoke bars.

Kendall et al. (2011) selected a sample of 13 galaxies from the SINGS survey
for which they were able to characterize the spiral pattern as either grand
design or having no well defined spiral. They found that “the presence of a
close companion” defined objectively “is (almost) a sufficient condition” for
grand design spirals (Kendall et al., 2011, p562), confirming the earlier
conclusion of Kormendy & Norman (1979) from optical images. They also note
that some galaxies that lack companions (according to their criteria) also
have grand design spirals, which must therefore be excited by other means.

Companions that may have excited spirals need not all be visible; dark halos
hosting few if any stars could also be responsible. Hierarchical galaxy
formation (see Somerville & Davé, 2015, for a review) indeed predicts that
galaxies are assembled from fragments that fall together, and that every halo
contains sub-halos having a range of masses and orbits about the main host.
Sawala et al. (2017) studied the survival of subhalos as disk galaxies form,
reporting a relative underdensity near the center and most remaining subhalos
that approach the disk have predominantly radial orbits. Very low-mass sub-
halos will have little effect on the disk, whereas massive sub-halos moving on
plunging orbits will disrupt the disk. While some spiral patterns probably are
responses to a sub-halo passing the disk, as suggested for example by Purcell
et al. (2011), to argue that the majority are tidally excited transient
responses would require repeated passages by subhalos in the appropriate mass
range, while the same galaxies have so far avoided encounters with slightly
more massive subhalos that would be disruptive or trigger bars. These
requirements seem hard to arrange, since the mass function of surviving
subhalos is a smooth power law (Sawala et al., 2017). It is more natural to
suppose another mechanism excites the majority of spirals.

###  3.3 Self-excited Spirals

Spirals are ubiquitous in disk galaxies having even a small fraction of gas,
many of which appear to lack bars or companions. While some patterns clearly
are tidal responses, and a few may be bar-driven, we conclude that spirals in
many disk galaxies must be self-excited. Furthermore, there can be no doubt
that spirals in simulations are also self-excited, since it is easy for the
experimenter to simulate completely isolated galaxies that lack bars, and yet
such models still spontaneously develop spiral patterns. Simulations therefore
offer a fruitful avenue for identifying the mechanism(s) for self-excitation.

##  4 SPIRAL DYNAMICS

Once the idea that spirals were gravitationally-driven density waves in the
stellar disks of galaxies took hold (circa 1964), the first step towards an
understanding of the mechanism was to examine the gravitational stability of
an axisymmetric stellar disk supported largely by rotation. The working
hypothesis was that smooth disks would possess spiral-shaped linear
instabilities, which would give rise to the patterns we observe. We review
aspects of spiral dynamics in this section and discuss current theories for
the origin of spirals in §5.

Table 1: Symbols used in this review Symbol | Meaning
---|---
$\alpha$ | pitch angle of a spiral
$\Phi$ | gravitational potential
$\Omega_{c}$ | local angular frequency of circular motion in the disk plane
$\kappa$ | local epicycle frequency (eq. 3.30 of Binney & Tremaine, 2008)
$E$, $L_{z}$ | specific energy and angular momentum of a star
$J_{R}$, $J_{\phi}$ | radial and azimuthal actions
$w_{R}$, $w_{\phi}$ | instantaneous phase angles of a star conjugate to the actions
$\Omega_{R}$, $\Omega_{\phi}$ | generalized frequencies of orbits of arbitrary eccentricity
$m$ | sectoral harmonic used in azimuthal Fourier analysis
$\lambda$, $k$ | wavelength and wavenumber of density waves
$\lambda_{\rm crit}$ | characteristic scale of gravitationally-driven disturbances in disks
$\Sigma$ | undisturbed surface mass density in the disk
$f_{d}$ | fraction of a full-mass disk mass that is active
$Q$ | a numerical indicator of local axisymmetric stability
$\sigma_{R}$, $\sigma_{\phi}$, $\sigma_{z}$ | components of the stellar velocity dispersion tensor in the disk
$\Omega_{p}$ | angular frequency of a rotating disturbance, aka pattern speed
$\omega$ | angular frequency of a wave $=m\Omega_{p}(+i\beta$)
$\beta$ | growth rate of an instability
$\Gamma$ | dimensionless shear rate
$X$ | dimensionless azimuthal wavelength
$N$ | number of particles in a simulation

###  4.1 Preliminaries

Here we list the principal simplifying assumptions that underlie most
theoretical analyses. Table 1 gives a glossary of the mathematical symbols
used in this review.

####  4.1.1 A 2D Stellar Disk in a Rigid Halo

To this day, all analyses have assumed a razor-thin disk as a necessary
simplifying approximation. Allowing 3D motion would not only add an extra
dimension, but would require perturbation analysis of a 3-integral equilibrium
distribution function (DF), when the third integral itself requires numerical
evaluation (e.g. Binney, 2016). Neglect of vertical motion in a thin, heavy
disk may be justified by the high vertical oscillation frequency of stars;
that part of their motion should be adiabatically invariant, and therefore
decoupled from low-frequency disturbances in the plane. Also the in-plane part
of the behavior in simulations that allow 3D motion generally resembles that
in others in which motion is confined to a plane (cf. Sellwood & Carlberg,
1984, 2014).

Stars moving in a flat axisymmetric disk have two classical integrals of
motion: the specific energy $E$ and specific angular momentum, $L_{z}$. We
will also occasionally make use of action-angle variables
$(J_{R},J_{\phi},w_{R},w_{\phi})$ because they enable an exact description of
orbits of arbitrary eccentricity. Actions in a 2D axisymmetric potential have
a very simple physical interpretation: the azimuthal action $J_{\phi}$ is
identical to $L_{z}$, while the radial action, $J_{R}$, also has the
dimensions of angular momentum and quantifies the degree of non-circular
motion of a star; $J_{R}=0$ for circular orbits and increases with the orbit
eccentricity. See Lynden-Bell & Kalnajs (1972, their §4) for a clear and
concise introduction to actions, angles, and frequencies,
$(\Omega_{R},\Omega_{\phi})\equiv(\dot{w}_{R},\dot{w}_{\phi})$, for motion in
a plane. Action-angle variablesActions are an alternative set of integrals,
angles specify the azimuthal and radial phases of a star

The deceptively simple equations that govern the dynamics of a smooth stellar
fluid are the collisionless Boltzmann equation (CBE) and Poisson’s equation
only; note that a collisionless fluid has no equation of state that relates
pressure to density. Since we will be interested principally in the disk
components of galaxies, we will consider the DF of disk stars only, while the
total potential is $\Phi=\Phi_{\rm disk}+\Phi_{\rm halo}+\Phi_{\rm gas}$,
where $\Phi_{\rm halo}$ arises from the bulge and halo components, and
$\Phi_{\rm gas}$ arises from the gaseous component, which does not obey the
CBE. To make progress, theorists have generally ignored $\Phi_{\rm gas}$,
effectively lumping it together with $\Phi_{\rm disk}$, which may be valid in
galaxies having a low gas mass fraction, and treated $\Phi_{\rm halo}$ as an
axisymmetric, fixed external field, which assumes that the bulge and halo are
decoupled from spiral dynamics in the disk. This last assumption was shown to
be adequate for spiral instabilities only recently (Sellwood, 2021), but does
not hold for bar instabilities (see §4.4.1.2).

####  4.1.2 Disturbance Potential

The principal challenge is presented by Poisson’s equation, for which there
are few known solutions outside of spherical symmetry, whereas we require the
potential of general non-axisymmetric density variations in a thin disk. The
rotational invariance of Poisson’s equation allows the field of each sectoral
harmonic, $m$, of the mass distribution to be computed independently, but the
radial part has no similar useful property. Sectoral harmonics of density and
potential remain separate at any amplitude, but the motions of the stars in
response to large potential variations generally create density variations of
several harmonics; therefore, analyses that are confined to a single harmonic
implicitly assume a disturbance of small amplitude.

A global solution for the potential of non-axisymmetric density variations in
a razor-thin disk can be obtained by expanding the surface density
distribution in some basis set of orthogonal functions, each of which has an
exact solution for the potential, as pioneered by Kalnajs (1971) and Clutton-
Brock (1972).

The WKB approximation treats a general wave-like disturbance as an infinite
plane wave in a razor-thin sheet. If the wave-vector lies in the
$x$-direction, the disturbed density amplitude, $\Sigma_{a}$, gives rise to
the disturbed potential $\Phi_{1}$

$\Phi_{1}(x,z)=-{2\pi G\Sigma_{a}\over|k|}e^{ikx-|kz|},$ (1)

(Binney & Tremaine, 2008, their equation 5.161). WKB approximationInvoked in
quantum mechanics by Wentzel, Kramers, and Brillouin This may be applicable to
spiral density waves in thin disks if curvature of the spiral can be
neglected. Formally, this would require the crest-to-crest wavelength,
$\lambda$, to be short compared with the distance to the center, $R$, so that
$|kR|\gg 1$, with the wavenumber $k\equiv 2\pi/\lambda$, but it “works
remarkably well” (Tremaine, private communication) as long as
$|kR|\mathrel{\hbox to0.0pt{\lower 2.15277pt\hbox{$\mathchar
536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar 318\relax$}}1$. Note this
density-potential relation holds for any angle of the wave to the radius
vector, and yields a surprisingly good approximation to the local
gravitational field near the center of a limited wave packet because
contributions to the field from the missing distant parts of the assumed
infinite wave are oscillatory and would have largely cancelled.

###  4.2 Local Stability Analysis

Dynamical studies of spiral waves in a part of the disk are described as
“local”. They include axisymmetric and tightly-wound spirals, as well as open
patterns.

####  4.2.1 Axisymmetric Stability

The WKB density-potential relation (1) was invoked by Toomre (1964) in his
classic study of gravitationally-driven disturbances in razor-thin stellar
disks. He showed that rotation stabilizes axisymmetric disturbances in a disk
lacking any random motion unless the local radial wavelength

$\lambda<\lambda_{\rm crit}\equiv{4\pi^{2}G\Sigma\over\kappa^{2}}.$ (2)

Short-wavelength Jeans instabilities are stabilized by random motion, and
Toomre found that all axisymmetric disturbances would be stable provided the
rms radial velocity,

$\sigma_{R}\geq\sigma_{R,\rm
crit}\simeq{3.358G\Sigma\over\kappa}\qquad\hbox{or}\qquad
Q\equiv{\sigma_{R}\over\sigma_{R,\rm crit}}\geq 1.$ (3)

Equation (3) applies to razor thin stellar disks. The constant 3.358, which
results from assuming an exact Gaussian velocity distribution among the stars,
is replaced by $\pi$ and $\sigma_{R}$ by the sound speed in the equivalent
stability criterion for gravitationally-driven, axisymmetric disturbances in a
thin, rotating gas sheet. A number of authors (e.g. Bertin & Romeo, 1989,
Romeo, 1992, Rafikov, 2001) have proposed modifications that take account of
finite disk thickness, in which the gravitational disturbance forces are
weaker, and/or a sheet containing both stars and gas. We present an example of
instability in a two-component disk in §6.3.

####  4.2.2 Dispersion Relations and Tightly-wrapped Spirals

Kalnajs (1965) derived a dispersion relation for axisymmetric waves in a 2D
stellar disk that may be rewritten as

$\omega^{2}=\kappa^{2}-2\pi G\Sigma|k|{\cal F}.$ (4)

This relation states that the frequency of the disturbance, $\omega$, is
decreased from the unforced epicycle frequency, $\kappa$, by the self-gravity
of the wave. Dispersion relationA relation between wave number $k$ and
frequency $\omega$ for self-consistent waves The “reduction factor” ${\cal
F}\leq 1$ (given by Binney & Tremaine, 2008, their Appendix K) depends upon
$Q$, $k$, and $\omega$, and quantifies the extent to which the self-gravity
term is weakened by random motion. Note, eq. (4) contains the same essential
dynamics as the study by Toomre (1964): in particular, ${\cal F}=1$ for a cold
disk ($Q=0$), giving the stability condition on $k$ that is equivalent to eq.
(2). Also ${\cal F}$ remains small enough that $\omega^{2}\geq 0$ for all $k$
when $Q\geq 1$.

A similar relation was derived independently by Lin & Shu (1966), but in order
to relate it to spiral waves they also equated the wave frequency, $\omega$,
to the Doppler-shifted frequency at which stars encounter an $m$-fold
symmetric spiral $\omega=m(\Omega_{p}-\Omega_{c})$. Here, $\Omega_{p}$ is the
pattern speed, and $\Omega_{c}$ is the circular angular frequency, which
varies with radius, and the factor $m$ appears because a star encounters $m$
wavecrests in a full turn relative to the pattern. By equating this forcing
frequency to the frequency of axisymmetric waves in the disk, Lin & Shu (1966)
made the additional assumption that the wave-vector of the spiral is closely
radial, which is known as the tight-winding approximation. Henceforth, we will
denote the WKB dispersion relation for tightly wrapped waves,

$[m(\Omega_{p}-\Omega_{c})]^{2}=\kappa^{2}-2\pi G\Sigma|k|{\cal F},$ (5)

as the Lin-Shu dispersion relation, or LSDR for short.

Figure 1: A collection of orbits each drawn in a frame that rotates at its own
rate $\Omega_{c}-\kappa/2$, so that all close to make a bi-symmetric ellipse.
The left panel shows the major axes all aligned, whereas the ellipses are
rotated by successive amounts in the other panels. After Kalnajs (1973), with
permission.

Kalnajs (1973) argued that one can think of a bisymmetric spiral density wave
as being composed of closed orbits, as shown in Figure 1, each of which
precesses at its angular rate $\Omega_{c}-\kappa/2$. However, the unforced
angular precession rate varies with radius and the initially aligned orbits in
the left-hand panel would wind over time, albeit at a rate that is much slower
than the shear rate in the disk. The achievement of Lin & Shu (1966) was to
show, within the limitations of their approximations, that self-gravity could
be used to adjust the precession rates to create a pattern of a particular
pitch angle, or wavenumber $k$, given by eq. (5), that would not shear.
Unfortunately, neither those authors, nor anyone subsequently, has been able
explain how such a steady pattern could be established and maintained.

Additionally, the tight-winding approximation excludes swing amplification,
which is a vital piece of spiral dynamics. This phenomenon was first revealed
by Goldreich & Lynden-Bell (1965) and Julian & Toomre (1966), who pioneered a
proper treatment of open spirals in a local approximation (see §4.2.3.2). The
LSDR implies a “forbidden region” (Binney & Tremaine, 2008, their §6.2.5)
around the CR that cannot support steady density waves for any $Q>1$, but
swing-amplified waves in fact have peak amplitude precisely where the LSDR
predicts steady waves should be evanescent. CRCorotation resonance where
$\Omega_{p}=\Omega_{c}$

Furthermore, eq. (5) holds equally for both leading and trailing spirals, and
therefore provides no explanation for the preference for trailing spirals,
which is both observed (§2.3) and required for outward angular momentum
transport. Swing amplification also provides the reason that trailing spirals
are preferred.

Binney & Tremaine (2008, their §6.2.2) present a detailed discussion of the
LSDR despite its limited applicability to spirals in galaxies. Its predictions
for short waves usefully yield some qualitative indications of spiral
behavior, but the fundamentally different character of LSDR waves when
$\lambda\mathrel{\hbox to0.0pt{\lower 2.15277pt\hbox{$\mathchar
536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar 318\relax$}}0.5\lambda_{\rm
crit}$, known as the “long wave branch” of the relation, is of little value
for galaxy disks.

####  4.2.3 Non-axisymmetric Responses

There is no known general stability criterion for non-axisymmetric
disturbances in rotationally supported stellar disks, and very few models have
been shown to be globally stable. However, before describing global modes we
first introduce two closely-related aspects of non-axisymmetric responses to
perturbations in otherwise stable disk models: wakes and swing amplification.

#####  4.2.3.1 Wakes

The disk surrounding a co-orbiting density excess develops a trailing spiral
response (Julian & Toomre, 1966, Binney, 2020). Since both these papers are
highly mathematical, it is easy to lose sight of the physics of why this
happens, which we therefore illustrate in Figure 2.

Figure 2: (a) Orbits in the sheared sheet that flow past a co-orbiting,
softened point mass ($\epsilon=0.2$) at the coordinate origin. (b) The
smoothed net disturbance density of the massless orbits in the left panel. (c)
The net response when disk self-gravity is included in a disk having
sufficient random motion such that $Q=1.4$. The unit of length in panels (a)
and (b) is $GM/V_{0}^{2}$, where $M$ is the perturbing mass and $V_{0}$ the
circular speed, whereas in panel (c) it is $\lambda_{\rm crit}$ (eq. 2).
Colors are relative to the maximum overdensity.

Both papers consider disturbances in the “sheared sheet” (Hill, 1878) and
adopt a flat rotation curve model. The approximation focuses on a rectangular
patch of the disk whose center orbits at the local circular speed and is
sufficiently small, relative to the distance to the disk center, that the
curvature of the rectangle can be neglected. Sheared sheetAn approximation
first invoked by G W Hill for a Kepler potential The $x$-direction is radial
and the $y$\- azimuthal, while disk material moves in a steady shear flow to
the right as $x$ increases and to the left for negative $x$. Figure 2a shows
how the flow is disturbed by the gravitational attraction of a co-orbiting
mass, which remains fixed at the origin of these coordinates. The dotted lines
mark the positions of massless particles at equal time intervals that enter
the frame on circular (i.e. straight) orbits but are deflected as they pass
the mass. The particles that pass at a distance experience mild impulses that
create epicyclic motion, whose effect is both diminished and shifted farther
downstream for faster moving orbits (well-spaced dots) having larger impact
parameters. However, particles whose impact parameters lie within the “Hill
radius” (described in Binney & Tremaine, 2008, their chapter 8) follow
horseshoe orbits that cause them to cross the corotation radius and reverse
their apparent motion in this moving frame. Hill radiusRegion in which the
gravitational field is dominated by the perturbing mass

Figure 2b presents the smoothed combined density of six times as many orbits
each sampled 10 times more often than those illustrated in the left panel. The
twisted ridge of the net response density of these massless particles results
purely from their superposed orbits in the disturbed flow. The deflections
scale with the perturber mass, which therefore sets the spatial scale of the
upper panels.

Figure 2c includes the self-gravity of the disk response as calculated by the
method of Julian & Toomre (1966), which adds substantially to the mass of the
wake, and in this case the spatial scale is in units of $\lambda_{\rm crit}$
(eq. 2). The similarity in appearance between the response of the cold
massless disk in the top right panel, and that in the heavy, warm ($Q=1.4$)
disk reveals that the wake is induced by the gravitational deflections of the
stars as they pass the mass, augmented by the disk response. In this case, the
spatial extent of the wake is determined by the self-gravity of the response,
while the density scale varies in proportion to the perturbing mass.

Figure 2 is drawn for a flat rotation curve. The spiral response is more open
where the rotation curve rises and less open where it falls.

#####  4.2.3.2 Swing amplification

The closely related phenomenon of swing amplification was discovered
independently by Goldreich & Lynden-Bell (1965) for a gaseous disk with self-
gravity, the year before the stellar dynamical treatment of Julian & Toomre
(1966). Both papers present a local treatment in the sheared sheet, but Figure
3, which is reproduced from Toomre (1981), gives a vivid illustration of the
process in a global calculation.

This Figure results from a linearized, global perturbation analysis of a
$Q=1.5$ Mestel disk in which the surface density is reduced by $f_{d}=0.5$ so
that only half the central attraction comes from the disk, while a rigid halo
makes up the other half. Mestel diskA disk having the surface density
$\Sigma(R)=V_{0}^{2}/(2\pi GR)$, giving rise to a circular orbit speed,
$V_{0}$, that is constant from $0\leq R\leq\infty$ The first panel illustrates
an imposed leading, 2-arm spiral wave packet, and the subsequent panels show
its evolution at intervals of half a rotation period at the corotation radius,
$R_{\rm CR}$, marked by the dotted circle.555Since the pitch angle of the
disturbance changes with time, it does not have the same pattern speed at
every radius and $R_{\rm CR}$ is therefore deduced from an average pattern
speed. Toomre (1981) offers a long, insightful explanation for the vigorous
amplification that requires some tenacity to follow. We describe other aspects
of the behavior seen in this Figure in §4.2.4.

Figure 3: Swing amplification in the Mestel disk. Numbers indicate the time
sequence in units of half a rotation period. The initially imposed leading
spiral unwinds at first and amplifies dramatically as it swings from leading
to trailing. The contours are of fractional overdensity only, underdense
regions are not contoured. Abbreviations: CR, corotation resonance; ILR, inner
Lindblad resonance; OLR, outer Lindblad resonance. Figure adapted from Toomre
(1981), with permission.

The factor by which an initially far leading disturbance is amplified depends
on three dimensionless parameters: $Q$, $X$, and $\Gamma$. The familiar $Q$
was defined above (eq. 3), while the other two are:

  * •

$\Gamma\equiv-(R/\Omega_{c})d\Omega_{c}/dR$. The reasonable range for galaxies
$0\leq\Gamma<1.5$, where $\Gamma=0$ for uniform rotation and $\Gamma=1.5$ for
a Kepler potential. Note also that $\Gamma=1$ for a flat rotation curve.

  * •

$X\equiv\lambda_{\phi}/\lambda_{\rm crit}$, with $\lambda_{\phi}=2\pi R_{\rm
CR}/m$. As before, $m$ is the rotational symmetry of the pattern and the
gravitational yardstick, $\lambda_{\rm crit}$, was defined in eq. (2).

In a disk with $\Gamma=1$ and $Q=1.2$, the amplification factor may vary from
less than 2 for $X>3$ to greater than 100 for $1<X<2$. Because $\lambda_{\rm
crit}$ varies with the disk surface density, $m=2$ disturbances are vigorously
amplified in heavy disks and feebly so in strongly sub-maximum disks where
$X\mathrel{\hbox to0.0pt{\lower 2.15277pt\hbox{$\mathchar
536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar 318\relax$}}3$. However,
vigorous swing-amplification can still occur in a sub-maximum disk for higher
values of $m$. Taking the self-similar Mestel disk as a simple example, we
find $X=2/(mf_{d})$, which implies the rotational symmetry of a strongly swing
amplified spiral is $1/f_{d}\mathrel{\hbox to0.0pt{\lower
2.15277pt\hbox{$\mathchar 536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar
316\relax$}}m\mathrel{\hbox to0.0pt{\lower 2.15277pt\hbox{$\mathchar
536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar 316\relax$}}2/f_{d}$. Quite
generally, swing-amplified spiral patterns should be more multi-armed when the
disk mass fraction is low (Sellwood & Carlberg, 1984, Athanassoula et al.,
1987, Hart et al., 2018).

The amplification factor varies even more strongly with $Q$ since, at fixed
$X=1.5$ and $\Gamma=1$, it exceeds 100 for $Q=1.2$, while it is less than 10
for $Q=2$. Thus perturbations of any wavelength in disks having
$Q\mathrel{\hbox to0.0pt{\lower 2.15277pt\hbox{$\mathchar
536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar 318\relax$}}2$ produce only
mild responses.

While Toomre (1981) chose to evaluate the expected amplification in the
important case of a flat rotation curve ($\Gamma=1$), we note that the range
of $X$ for vigorous amplification is increased in declining rotation curves,
and reduced when the rotation curve rises. Vigorous amplification occurs for
$1.5\mathrel{\hbox to0.0pt{\lower 2.15277pt\hbox{$\mathchar
536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar 316\relax$}}X\mathrel{\hbox
to0.0pt{\lower 2.15277pt\hbox{$\mathchar 536\relax$}\hss}\raise
1.72218pt\hbox{$\mathchar 316\relax$}}4$ when $\Gamma=1.5$ (i.e. Keplerian)
and the amplitude peaks at more strongly trailing angles, whereas for
$\Gamma=0.5$, the preferred range is $0.5\mathrel{\hbox to0.0pt{\lower
2.15277pt\hbox{$\mathchar 536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar
316\relax$}}X\mathrel{\hbox to0.0pt{\lower 2.15277pt\hbox{$\mathchar
536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar 316\relax$}}1.5$ and the
spirals are more open. Naturally, the range of $X$ for which amplification can
occur shrinks to zero in a uniformly rotating disk ($\Gamma=0$), since
disturbances are not sheared.

#####  4.2.3.3 Connection Between Swing Amplification and Wakes

As already noted, the physics of wake formation is intimately connected with
swing-amplification, and indeed the formulations of both Julian & Toomre
(1966) and of Binney (2020) calculate the disk response to a co-orbiting
perturber as the superposition of a continuous stream of shearing waves. The
source of the waves is the perturbing mass; a point mass in 2D can be
represented by a uniform spectrum of plane waves of all possible pitch angles.
The leading components of this spectrum introduce forcing terms into the shear
flow, creating leading disturbances that amplify as they swing to trailing.
Since the spectrum is continuous, the superposed responses create a steady
trailing wake, as was illustrated in Figure 2.

Note that swing amplification, illustrated in Figure 3, is computed for the
$m=2$ sectoral harmonic only, whereas the wake response to a co-orbiting
perturber is summed over all possible azimuthal wavelengths $0<X<\infty$, each
of which produces a steady response. Clearly, the response is dominated by
wavelengths that are most strongly swing-amplified, i.e. $1<X<2$. While the
wake response is indeed caused by swing-amplification, we will try to reserve
those words to describe features whose pitch angle evolves, as in Figure 3,
and to use the phrase “supporting response” to describe steady or growing
features in the surrounding disk, as in Figure 2). The vigor of the supporting
response varies with the parameters $X$, $Q$, and $\Gamma$ in exactly the same
manner as for swing-amplification.

####  4.2.4 More Disk Dynamics

The take home message from Figure 3 is the phenomenon of swing amplification,
but it also illustrates several other important aspects of spiral dynamics
that will factor into our discussions of spiral modes (§4.4) and theories
(§5).

#####  4.2.4.1 Group velocity

Not only does the initial spiral in Figure 3 change its pitch angle and
amplitude over time, but the wave packet inside CR travels outwards when
leading and, later, inwards when trailing. Recall that the group velocity of a
wave packet is $v_{g}\equiv\partial\omega/\partial k$, which may be calculated
from a dispersion relation. Using the LSDR, Toomre (1969) showed that a short
wavelength packet propagates radially towards corotation when the wave is
leading and away from corotation when it is trailing. For completeness, the
sign of the group velocity on the long wave branch of the LSDR is predicted to
be the opposite for all cases of leading/trailing and inside/outside
corotation from those on the short wave branch. However, as noted at the end
of Section 4.2.2, the LSDR provides a poor representation of long waves in
heavy disks.

Employing the local apparatus of Julian & Toomre (1966), Toomre (1969) also
demonstrated the radial propagation of an impulsively excited wave packet. His
numerical solutions confirmed the prediction from the LSDR when the wave was
tightly wrapped but, when open, part of the disturbance propagated across
corotation to the outer disk, as also occurred in his later global calculation
(Figure 3).

#####  4.2.4.2 Lindblad resonance damping

As the wave packet in Figure 3 travels inward at late times, it becomes ever
more tightly wrapped and is eventually absorbed. Stars at any radius in the
disk experience forcing by a spiral disturbance but, except near resonances,
their orbits vary adiabatically as a small-amplitude wave packet passes over
them, leaving no lasting change. For near circular orbits, a Lindblad
resonance arises when the forcing frequency $\omega\equiv
m(\Omega_{p}-\Omega_{c})=\pm\kappa$. ILRInner Lindblad resonance OLROuter
Lindblad resonance The negative sign is for the ILR, where stars overtake the
wave, and the positive is for the OLR where the wave overtakes the stars, at
the local epicycle frequency in both cases. Action-angle variables (§4.1.1)
describe orbits of arbitrary eccentricity, for which the resonance condition
becomes $m(\Omega_{p}-\Omega_{\phi})=l\Omega_{R}$, with $l=\pm 1,0$ for OLR,
ILR and CR respectively.

A star in Lindblad resonance may either gain or lose random energy, depending
on both its previous epicycle size and the phase difference between the star
and the wave. Lynden-Bell & Kalnajs (1972) showed that, to second order, the
distribution of resonant stars gains random energy on average at Lindblad
resonances causing the wave to be damped (Mark, 1974). However, there are the
following two caveats:

  * •

The second order increase in random motion, though tiny for weak disturbances,
does cause a lasting change to the phase-space density of disk stars, creating
a scratch in the DF that turns out to be important (see §4.3).

  * •

Perturbation theory predicts resonance damping of small amplitude waves, but
larger amplitude waves cause stars to become trapped in the resonance (see
§4.4.1).

Stars near the CR move slowly relative to the pattern, and may therefore gain
or lose angular momentum, depending on their phase relative to the potential
maximum. But they neither change their random energy (see §4.2.4.4 below), nor
damp the wave.

#####  4.2.4.3 Angular momentum transport

Formally, wave action density is carried at the group velocity, but Toomre
(1969, privately assisted by Kalnajs) showed it to be equivalent to angular
momentum. The wave packet inside corotation in Figure 3 has a positive
(outward) group velocity when it is leading and a negative group velocity when
trailing. The part of the disturbance outside corotation is less clear from
the figure, but the group velocity there is outward in the later trailing
evolution. As a trailing wave carries angular momentum outward, it may seem
paradoxical that the group velocity inside corotation is inward.

However, spiral disturbances, such as that in Figure 3, cannot have any net
angular momentum in a disk when no external torque is applied. Thus as the
disturbance develops, it reduces the angular momentum of the inner disk while
it increases that of the outer. Therefore the positive group velocity outside
corotation carries positive angular momentum outwards, while the group
velocity carries a disturbance of negative angular momentum inwards in the
inner disk, and thus a trailing spiral carries angular momentum outwards
everywhere. This explanation was given by Lynden-Bell & Kalnajs (1972), but
their identification of corotation as being the radius where the sign of the
angular momentum stored in the wave changes is not always correct: edge modes
for example (see §4.4.2) are mostly confined within the CR, and require the
sign change to lie well interior to that radius in order that the disturbances
have no net angular momentum.

Unfortunately, this is not the whole story; a Reynolds-type stress, which
Lynden-Bell & Kalnajs (1972, p20) called “lorry transport,” is a second radial
transport mechanism. It is of most relevance to angular momentum transport in
the opposite sense from the gravity torque on the long wave branch of the
LSDR, and is probably of less importance for spirals in galaxies.

Figure 4: The Lindblad diagram showing possible values of $E$ and $L_{z}$ for
a razor-thin Mestel disk model. Circular orbits lie along the full-drawn curve
and eccentric orbits fill the region above it. Angular momentum and energy
exchanges between particles and a steadily rotating disturbance move them
along lines of slope $\Omega_{p}$, as shown by the arrows. The dotted and
dashed lines are the loci of resonances, where
$m(\Omega_{p}-\Omega_{\phi})=l\Omega_{R}$, for an $m=2$ perturbation of
arbitrary $\Omega_{p}$.

#####  4.2.4.4 Angular momentum changes at resonances

Stars not only gain random motion on average at the Lindblad resonances, but
they also absorb the incoming angular momentum (Lynden-Bell & Kalnajs, 1972).
In fact, Jacobi’s integral, $E_{J}\equiv E-\Omega_{p}L_{z}$, which is
conserved in a rotating, steady non-axisymmetric potential (§3.3.2 of Binney &
Tremaine, 2008), requires that $\Delta E=\Omega_{p}\Delta L_{z}$, where
$\Delta E$ and $\Delta L_{z}$ are respectively the changes to the specific
energy and angular momentum of a star. Possible changes due to one pattern all
have the same slope in the Lindblad diagram, as illustrated by the arrows in
Figure 4, which are directed away from the circular orbit curve at both
Lindblad resonances. The sense of the vectors at the Lindblad resonances in
Figure 4 illustrate the net changes, averaged over the stellar distribution
(§4.2.4.2). The locus of circular orbits is curved in the indicated sense for
any shearing model, but the curve becomes a straight line for a uniformly
rotating disk, for which no energy can be extracted from the potential by
angular momentum redistribution.

Sellwood & Binney (2002) derived a useful first order relation between the
angular momentum exchanged at a resonance $\Delta L_{z}$ and the change of
radial action:

$\Delta J_{R}={l\over m}\Delta L_{z}\qquad\hbox{at resonances}.$ (6)

$\Delta J_{R}$ is positive at both Lindblad resonances, since the loss of
$L_{z}$ at the ILR, where $l=-1$, allows the star to settle deeper into the
potential well, freeing up energy for random motion to increase. Thus a spiral
disturbance may extract angular momentum from stars at the ILR and deposit it
at the OLR, increasing $J_{R}$, or heating the stars, at both resonances.

Notice that the sectoral harmonic $m$ appears in the denominator of eq. (6),
implying that a unit change $\Delta L_{z}$ causes less heating for
disturbances of higher $m$. This is because the Lindblad resonances lie closer
to the CR, and the outward transport of angular momentum extracts less energy
from the potential when it is carried over a shorter radial distance.

Stars may also exchange angular momentum with the wave at CR. Because the
vectors at this resonance are parallel to the tangent to the circular orbit
curve in Figure 4, stars neither gain nor lose random energy at this resonance
to first order, as setting $l=0$ in eq. (6) confirms, implying that $\Delta E$
is exactly balanced by the change of energy associated with the radial change
to the star’s guiding center caused by $\Delta L_{z}$. Guiding centerThe
steadily orbiting point about which the star librates in its motion around an
axisymmetric galaxy In a disk that is approximately uniform across CR, the
gainers and losers at that resonance roughly balance, leading to little net
angular momentum change. However, where the density of stars near the CR
decreases steeply with $L_{z}$, at an outer edge say (see §4.4.2), the CR
becomes the principal angular momentum sink.

#####  4.2.4.5 Wave action

As noted above, the quantity that is transported at the group velocity is wave
action density. The textbook example of wave action conservation is of a wave
packet travelling along a whip that has a decreasing mass per unit length: the
displacement amplitude of the packet increases as it travels to the thin end
of the whip.

As spiral waves travel radially in a disk, their amplitude is indeed affected
by the changing disk surface density, but changes to the group velocity, which
slows as the wave approaches a Lindblad resonance, and the geometric change to
the area that the wave occupies are also important. The focusing of an inward
travelling wave causes its relative overdensity to increase, and conversely
the relative overdensity decreases as an outward travelling wave spreads over
a larger area. Thus the later fate of the wave packet in Figure 3 appears to
concentrate in the inner disk, while the outward travelling wave beyond
corotation disappears under the lowest contour level. This effect is
particularly pronounced for 2-arm spirals, since the Lindblad resonances,
which limit the radial extent of the pattern, lie closer to corotation for
patterns of higher rotational symmetry. Note also that disturbance amplitudes
in the sheared sheet, such as that in Figure 2, are symmetric across
corotation, since there the disk is assumed to be uniform and curvature is
neglected.

#####  4.2.4.6 Super-reflection

An alternative description of swing amplification is that the outgoing leading
wave super-reflects off the corotation resonance in a three wave interaction
(Mark, 1976, Goldreich & Tremaine, 1978, Drury, 1980). This means that the
incident leading wave from the inner galaxy reflects as an amplified trailing
wave that propagates radially inward while conservation of wave action
requires that the reflection also excites a transmitted trailing wave that
propagates outward. This concept is useful for our discussions of cavity modes
in §§4.4.1 & 5.2.1.

###  4.3 Lumps and Scratches

The collisionless Boltzmann equation embodies an idealization that phase space
is smooth; in other words, the discrete nature of stars can be neglected and
the stellar fluid is continuous. The number of stars in galaxy disks is large
enough that this assumption holds quite well (see Sellwood, 2014, for
caveats). However, galaxies contain mass clumps such as star clusters and
giant molecular clouds (GMCs), and the number of particles employed in a
simulation is generally several orders of magnitude fewer than the number of
stars in a galaxy disk. Thus both clumps in real galaxies and shot noise in
simulations give rise to inhomogeneities within the disk.

The noise spectrum in a flattened, shearing distribution of randomly
distributed gravitating masses inevitably contains leading wave components
that are strongly amplified as the shear carries them from leading to
trailing. This behavior has two important consequences.

####  4.3.1 Polarization

The first consequence of swing-amplified shot noise is that each heavy
particle develops a trailing wake (Figure 2) both towards and away from the
disk center. The wake exists in both the background disk, and among the heavy
particles themselves. Thus the distribution of heavy particles becomes
polarized, with their two-point correlation function being greater along the
direction of the wake and lower in other directions. Since they are no longer
randomly distributed, the amplitude of all components of the noise spectrum is
enhanced, causing subsequent noise-induced fluctuations to be stronger,
although linear theory predicts this enhancement should asymptote in a few
epicycle periods to a mean steady excess over the level expected from
uncorrelated noise (Julian & Toomre, 1966, Toomre & Kalnajs, 1991).

####  4.3.2 Scratches to the DF

The collectively amplified response to any one component of the noise also
launches a coherent wave in the disk that propagates away from corotation
(Toomre, 1969, Figure 3) until it reaches a Lindblad resonance where it is
absorbed (§4.2.4.2). On average, particles lose $L_{z}$ at the ILR and gain at
the OLR (§4.2.4.4), and this outward transfer of $L_{z}$ allows the wave to
extract energy from the potential enabling the scattered particles to acquire
additional random energy at both resonances (Figure 4). The larger amplitude
waves, in particular, therefore depopulate stars originally having near
circular orbits over the narrow region of each Lindblad resonance, thereby
creating a “scratch” in the DF (Sellwood & Carlberg, 2019, p117) and (Sridhar,
2019) that affects subsequent activity.

It is important to realize that linear theory neglects this second order
effect by assumption, i.e. it does not allow for changes to the equilibrium
state. In fact, Sellwood (2012) found the amplitudes of successive episodes of
uncorrelated swing amplified noise in a stable disk model rose steadily as a
result of scratches to the DF. The Lindblad resonance absorption of each
traveling wave caused an abrupt change to the impedance of the disk at which
subsequent traveling waves were partially reflected. Swing amplification of
the weak reflected leading wave gave a further boost to the amplitude, which
led to ever deeper scratches as the evolution proceeded (Sellwood & Carlberg,
2014). Fouvry & Pichon (2015) successfully applied second order perturbation
theory to calculate this series of events, which continued until the partial
reflections became strong enough that the disk was able to support an unstable
mode (Sellwood, 2012, De Rijcke et al., 2019), and coherent growth to large
amplitude began.

Here we have used the word scratch to describe quite mild changes to the DF
from weak disturbances that can cause partial reflections of subsequent waves
propagating radially within the disk. But scattering at a Lindblad resonance
by a larger amplitude spiral could also carve a deeper feature that seeds a
groove mode (§4.4.2) instead, and this appears to be the more usual behavior
(Sellwood & Carlberg, 2019). Even larger amplitude waves that encounter an ILR
cause particles to become trapped (§4.4.1).

###  4.4 Modes in Galactic Disks

A normal mode of any system is a self-sustaining, sinusoidal disturbance of
fixed frequency and constant shape, save for a possible uniform rotation; the
frequency would be complex if the mode were to grow or decay. The perturbed
surface density of a mode in a galaxy disk is the real part of

$\delta\Sigma(R,\phi,t)=A_{m}(R)e^{i(m\phi-\omega t)},$ (7)

where $\omega=m\Omega_{p}+i\beta$ is now allowed to be complex with $\beta$
being the growth rate. The complex function $A_{m}(R)$, which is independent
of time, describes the radial variation of amplitude and phase of the mode.

Stability analysis of a system supposes small amplitude perturbations about
the equilibrium state, which is linearized by discarding any terms that
involve products of small quantities – see Kalnajs (1971) for a careful
formulation. The self-consistency requirement that the surface density
variations give rise to the disturbance potential that produced them leads to
a matrix, the eigenvalues of which are the normal modes of the disk (Kalnajs,
1977, Polyachenko, 2005, Jalali, 2007, De Rijcke & Voulis, 2016). The
equilibrium is linearly unstable if any of the resulting modes have a positive
growth rate, since the disturbance exponentiates out of the noise until the
neglected second and higher order terms become no longer negligible.

Note that the swing amplified response to a perturbation, such as in Figure 3,
is not a mode both because the shape changes with time and its amplitude does
not grow exponentially. Also the wake response to an imposed co-orbiting mass
clump, Figure 2, is not a mode because, to first order, it would disperse if
the clump were removed (e.g. Sellwood & Carlberg, 2021), and it therefore is
not self-sustaining. Both are simply linear responses of the disk to
hypothesized imposed disturbances. However, they are both very helpful
concepts when trying to understand the mechanisms of self-sustaining modes.

####  4.4.1 Cavity Modes

Normal modes can be standing wave oscillations that exist between two
reflecting barriers, as in organ pipes and guitar strings, which are generally
described as cavity modes in galaxy disks. The prime example in galaxies is
the bar-forming mode, for which a reflection takes place at the center while a
super-reflection takes place at corotation that causes exponential growth
(Toomre, 1981, Binney & Tremaine, 2008, and §4.2.4.6). Overtones also exist,
but generally have lower growth rates than the fundamental mode (Toomre, 1981)
and are therefore less important. Instabilities of this type in a smooth disk
are possible only if the inward traveling wave can avoid an ILR, since linear
theory (Mark, 1974, and §4.2.4.2) predicts that any small-amplitude
disturbance that encounters an ILR will be absorbed, and therefore damped. For
$m\geq 2$, an ILR must be present for any reasonable pattern speed when the
center is dense, and therefore the only small-amplitude cavity modes that are
possible in a featureless disk of this kind can only be for $m=1$ (Zang, 1976,
Evans & Read, 1998a, b).

But $\Omega_{c}-\kappa/2$ has a maximum value in mass models that have gently-
rising inner rotation curves, and linear bar-forming instabilities avoid
resonance damping as long as $\Omega_{p}>(\Omega_{c}-\kappa/2)_{\rm max}$. The
dominant mode of several bar-unstable models has been identified in
simulations, with excellent quantitative agreement of both the frequency and
mode shape (Sellwood & Athanassoula, 1986, Earn & Sellwood, 1995, Khoperskov
et al., 2007). The non-linear evolution of the dominant mode is a bar in the
inner disk and a hot, mildly responsive outer disk.

#####  4.4.1.1 Large-amplitude Trapping

Note that despite the linear theory prediction that an ILR should inhibit the
bar instability, simulations having dense centers often form bars anyway.
Efstathiou et al. (1982) emphasized this point, but a similar result has been
reported in numerous other simulations. Many barred galaxies are also observed
to have dense bulges (e.g. Masters et al., 2011). The damping of a disturbance
by an ILR is a prediction of small-amplitude perturbation theory, but a finite
amplitude disturbance at an ILR can cause particles to become trapped in the
resonance, as noted in §4.2.4.2. Swing-amplified shot noise (see §4.3) can
create sufficiently large amplitude trailing spirals to overwhelm the ability
of the ILR to damp them. In this case, the outcome of trapping can be a large
amplitude bar, as demonstrated by Sellwood (1989a). Simulations are able to
reproduce the predicted linear stability (e.g. Sellwood & Evans, 2001), but
only when they are set up carefully, employ sufficient particles that swing-
amplified shot noise can be damped, and are terminated before the noise
amplitude builds up (Sellwood, 2012).

#####  4.4.1.2 Stabilization by Halos

Despite years of effort, we do not understand how bars are prevented from
forming in galaxies that lack a dense center. Historically, Ostriker & Peebles
(1973) argued that massive halos stabilize disks against bars, which works
because the swing amplification parameter $X>3$ for $m=2$ in sub-maximum
disks, causing patterns having $m>2$ to be favored instead (see §4.2.3.2).
However, this strategy also inhibits bi-symmetric spirals for the same reason,
and there are a number of galaxies, M33 being a prominent example, that have
dominant 2-arm spirals and no bar. Indeed, Sellwood, Shen & Li (2019) could
find no satisfactory explanation for the absence of a bar in M33, despite a
systematic exploration of many possible avenues.

The challenge presented by the apparent stability of unbarred galaxies is
further compounded because the bar-forming instabilities of a disk in a
responsive halo are more vigorous than when the disk is embedded in an
equivalent rigid halo (e.g. Athanassoula, 2002, Saha & Naab, 2013, Berrier &
Sellwood, 2016). Since the disturbance in the disk couples to a responsive
halo at small-amplitude, the bar instability should be thought of as a mode of
the combined disk+halo system.666This first-order halo response differs from
dynamical friction, which is second order (Binney & Tremaine, 2008). As such,
it violates one of the assumptions of spiral theory set out in §4.1.1.
Fortunately, Sellwood (2021) found that a rigid halo is an adequate
approximation for groove modes, and therefore this assumption may be violated
only for bar-forming modes. Responsive haloA halo component composed of
collisionless massive particles in equilibrium with the disk

####  4.4.2 Edge and Groove Modes

Galactic disks can also support another class of mode: edge modes (Toomre,
1981, Papaloizou & Lin, 1989) and groove modes (Sellwood & Lin, 1989, Sellwood
& Kahn, 1991) are the best known examples, but ridges and other features are
also destabilizing (see §7.2.1). Although bona fide modes, they are not
standing waves, and instead the pattern speed is tied to the circular orbital
frequency near the radius of the feature in the angular momentum density in
the disk.

#####  4.4.2.1 Edge mode

In the case of the edge mode, a small non-axisymmetric distortion of a disk
where the surface density decreases steeply, moves high density material out
to places where the equilibrium density is lower, and conversely at other
azimuthal phases. On their own, such infinitesimal co-orbiting distortions
would be neutrally stable and therefore of no interest. But as described above
(§4.2.3.1) and illustrated in Figure 2, a cool surrounding disk responds
vigorously to a co-orbiting mass excess, creating a trailing wake that extends
radially far into the shear flow on either side of the perturbing mass. An
outward bulge on the edge therefore excites a strong supporting response from
the interior disk that is not balanced by the exterior response because the
equilibrium density drops rapidly with radius at the edge. The forward
attraction of the interior wake on the bulging edge increases its angular
momentum, causing it to rise farther outward, and therefore to grow
exponentially as it rotates. Toomre (1989, pp 153-4) indicated instability
requires not only that $Q\mathrel{\hbox to0.0pt{\lower
2.15277pt\hbox{$\mathchar 536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar
316\relax$}}2$ and $X\mathrel{\hbox to0.0pt{\lower 2.15277pt\hbox{$\mathchar
536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar 316\relax$}}3$, but also that
“the radial distance over which the disc density undergoes most of its rapid
change should be no larger than about one-quarter of … $\lambda_{\rm crit}$”
(eq. 2).

In a disk with random motion, the crucial gradient is in the angular momentum
density, while random motion may spread out the surface density gradient. In
this case, the above argument still applies to the guiding centers, with the
epicyclic librations of the stars blurring the density variations and thereby
reducing the growth rate.

#####  4.4.2.2 Groove mode

A groove in a disk is effectively two closely spaced edges, which however give
rise to a single mode because the distortions on each edge are gravitationally
coupled. Again, it is the supporting response of the surrounding disk, or
equivalently swing-amplification, that causes the groove mode to have a
substantial radial extent and to grow rapidly. Sellwood & Kahn (1991) were
able to obtain reasonable quantitative agreement between their local analytic
predictions and global simulations.

Figure 5: The shape of the unstable mode fitted to data from simulation G of
Sellwood & Carlberg (2019), in which a groove had been created by hand by
adding random motion to particles having near circular orbits near
$R=2.93R_{0}$. The distance scales are in units of $R_{0}$, the central radius
of the inner disk cutout, the solid circle marks the radius of CR and the
dotted circles are the radii of the Lindblad resonances.

A disk with a groove can still be an axisymmetric, well-mixed (i.e.
stationary) equilibrium, but the DF is no longer a smooth function of the
integrals. More so than for the edge mode, epicyclic blurring can mask a
groove in surface density almost entirely, since the radial width of the
groove in angular momentum is generally smaller than the epicyclic radius of a
typical disk star. Furthermore, a deficiency need not extend to stars of large
radial action; Sellwood & Carlberg (2019) provoked an instability in an
otherwise stable $Q=1.5$, $f_{d}=0.5$ Mestel disk, simply by adding random
energy to particles on near circular orbits over a narrow range centered on
$L_{z}=2.93V_{0}R_{0}$. Thus their “groove” was merely a deficiency of nearly
circular orbits, caused by selected particles being given additional random
energy. The shape of the resulting unstable mode, which was determined by
fitting for $A_{2}(R)$ in eq. (7) to data from the simulation, is shown in
Figure 5. Corotation for the mode, which has a fitted frequency of
$\omega=0.656\pm 0.005+(0.017\pm 0.003)i$, is at $R=3.06R_{0}$, which is just
outside the radius of the groove they introduced to the DF.

The mode shape (Figure 5) has a slight kink at the radius of the groove.
Sellwood & Kahn (1991) presented mode shapes for groove instabilities in cold
disks, i.e. lacking random motion, that had much more pronounced kinks, which
reflect the mode mechanism. Distortions to the two edges of the groove attract
each other, and the exchange of angular momentum causes both distortions to
grow when the disturbance on the outer edge of the groove leads that on the
inner edge. Thus, unlike a single edge, a groove is unstable even in the
absence of a disk supporting response. Of course, the disk supporting response
increases the mode growth rate and creates a large-scale disturbance. The
“groove” in a disk with random motion is a feature in the angular momentum
distribution; the mechanism is unchanged, although random motions blur the
sharp features reported by Sellwood & Kahn (1991) into the mild kink visible
in Figure 5, and reduce the growth rate.

Sellwood & Kahn (1991) also reported that CR for their groove modes lay just
outside the groove, whereas their local analysis predicted it should lie at
the groove center. This minor difference is caused by curvature in global
modes; it decreases for modes of higher sectoral harmonics, and disappears in
the $m\rightarrow\infty$ limit of the sheared sheet.

##  5 THE ORIGIN OF SPIRALS IN GALAXIES

Since spirals are ubiquitous in large disk galaxies containing a modest gas
fraction, and also develop spontaneously in simulations of isolated disks, we
argued in §3 that some mechanism is needed to excite them. Unfortunately,
early normal mode analyses of apparently reasonable models of featureless
disks did not identify any promising spiral modes. On the one hand, models
where the rotation curve rose gently from the center are dominated by vigorous
bar-forming instabilities (Hohl, 1971, Kalnajs, 1978). On the other hand,
smooth disk models having a dense (bulge-like) center and a moderate halo have
no instabilities whatsoever (Toomre, 1981).777Fully self-gravitating disks in
cusped potentials suffer from lop-sided instabilities (Zang, 1976, Evans &
Read, 1998a, b) that cannot be blocked by an ILR, since $\Omega_{c}-\kappa<0$
everywhere. However, they are inhibited by a moderate halo fraction. Spiral
modes are still favored today, but it took a long time to realize that the
relevant instabilities are provoked by local deficiencies in the stellar DF,
and that the earlier failure stemmed from the apparently innocuous assumption
that the DF should be a smooth function of the integrals.

As noted in §2, few spiral galaxies manifest highly regular spiral patterns.
Individual arms can rarely be traced over a significant radial range and
bifurcations and branches are common. Spiral patterns that develop
spontaneously in simulations of isolated, unbarred stellar disks generally
give this impression also; thus an understanding of the mechanism for spiral
generation in the simulations may provide a useful guide to the behavior in
galaxies.

Miller et al. (1970) and Hohl (1971) first reported spiral patterns appearing
spontaneously in simulations of collisionless particle disks, apparently
confirming that they are a collective phenomenon of many body Newtonian
dynamics. Subsequent simulations have progressed from their $N\sim 10^{5}$
particles confined to a plane to $N\mathrel{\hbox to0.0pt{\lower
2.15277pt\hbox{$\mathchar 536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar
318\relax$}}10^{8}$ moving in 3D, but the qualitative spiral behavior has not
changed. As $N$ is increased, the relative amplitude of shot noise, which
varies as $N^{-1/2}$, is reduced, enabling spiral patterns to be traced to
collective modes that stand clear of the noise (§4.3).

###  5.1 A Recurrent Cycle of Groove Modes

It is now clear that the spontaneous development of spiral patterns in
simulations of isolated and unbarred disks results from a recurrent cycle of
groove modes (Sellwood & Carlberg, 2014, 2019). The conceptual breakthrough of
this discovery is that it discards the assumption of a DF that is a smooth
function of the integrals, which was entrenched in all early work. Instead,
the DF possesses a groove, or a deficiency over a narrow range of $L_{z}$,
that seeds a linear instability (§4.4.2.2). Furthermore, the nonlinear
evolution of a groove instability creates new grooves at the Lindblad
resonances of the original mode, thereby setting up a recurrent cycle. This
behavior can occur in collisionless particle disks only and does not have an
analog in gas disks, for example.

Power spectra (Sellwood & Athanassoula, 1986) taken from the simulations
(Sellwood, 1989b, Roškar et al., 2012, Minchev et al., 2012, Sellwood &
Carlberg, 2014, 2019) reveal that the changing appearance of the spirals
results from the superposition of several separate waves, each having a
constant pattern speed over a broad radial range. The amplitudes of the
individual disturbances grow and decay, but each is detectable over a period
of several rotations at the corresponding corotation radius. These waves are
modes that differ fundamentally from those in other theories (e.g. §5.2.1)
because they are supported by a vigorous disk response, they do not last for
nearly as long, and fresh instabilities develop to maintain spiral activity.

Although the individual modes have constant shapes and pattern speeds, the
spiral appearance of a simulation changes continuously. This is because the
disk supports several modes at any one time, each having a different pattern
speed and perhaps also angular periodicity, as well as a time varying
amplitude. The superposition of several modes causes the pitch angle of
individual arm features to decrease with time (Sellwood & Carlberg, 2021) (see
Supplemental Video 1),888 temporary url:
http://www.physics.rutgers.edu/$\sim$sellwood/supp material.html while fresh
patterns come to the fore, and the detailed appearance of the overall pattern
changes radically in less than one disk rotation.

The recurrence mechanism, which was clearly demonstrated by Sellwood &
Carlberg (2019), is as follows: as a groove mode saturates, the angular
momentum stored in the wave (§4.2.4.3) drains at the group velocity (§4.2.4.1)
onto the Lindblad resonances where it is absorbed (§4.2.4.2), scattering
resonant stars to more eccentric orbits (§4.2.4.4), thereby depopulating
another part of the DF over a narrow range of $L_{z}$ and low $J_{R}$, and
seeding a fresh groove instability having a new pattern speed. The initial
groove to seed such a cycle in a real galaxy could be caused by resonance
scattering as, say, an orbiting mass clump settles into the disk or by the
near passage of a small companion or, in the unlikely circumstance that
neither of these events happen, spiral disturbances could bootstrap out of the
noise (Sellwood, 2012).

Sellwood & Carlberg (2019) showed that scattering at both the Lindblad
resonances of any one mode created grooves in the DF that seeded fresh groove-
type instabilities, with corotation for each subsequent mode being close to
the newly-carved grooves. Thus a new instability could be either closer to or
farther from the disk center and, moreover, need not have the same angular
symmetry as the original. Even starting from very contrived initial conditions
that supported a single instability only, the disk quickly developed many new
instabilities that caused the usual apparent transient spiral evolution.

A recurrent cycle of groove modes has been firmly established in simulations,
but it is not easy to find evidence that it operates in real galaxies. The
best evidence is that the distribution of particles in action space (Sellwood
& Carlberg, 2014) acquired multiple scattering features resembling those in
the Gaia data from the local Milky Way. See §5.3 for this and other possible
tests.

####  5.1.1 Disk Heating by Spirals

Note that scattering of stars at Lindblad resonances not only carves grooves,
but increases the general level of random motion in the disk, thereby
rendering the disk less responsive to future instabilities. Thus spiral
activity in a purely stellar disk is self-limiting, and simulations of massive
disks suggest it fades on a time-scale of some ten disk rotations (Sellwood &
Carlberg, 1984, 2014). Spiral activity can persist “indefinitely” (Sellwood &
Carlberg, 2014, p3) if the disk is cooled, as discussed in §6.1.

####  5.1.2 Sub-maximum disks

A slower heating rate was reported by Fujii et al. (2011) and others in their
simulations of sub-maximum disks, and those authors wrongly blamed the more
rapid heating reported by Sellwood & Carlberg (1984) on collisional
relaxation. Sellwood & Carlberg (2014) dismissed that idea and explained
instead that the amount of Lindblad resonance heating, i.e. $\Delta J_{R}$ for
outward transport of a given $\Delta L_{z}$, decreases with increasing $m$
(eq. 6). Therefore less rapid heating is expected in halo-dominated disks that
favor more multi-arm spirals (see §4.2.3.2).

###  5.2 Other Theories

A number of other theories have been proposed for the origin of spirals in
galaxies. Here we review three of them.

####  5.2.1 Quasi-steady Density Waves

The ubiquity of spirals in galaxies led many astronomers (e.g. Oort, 1962) to
favor long-lived spiral patterns, since they would not require constant
regeneration. This preference was met by the widely-cited theory of quasi-
steady waves promoted in the book by Bertin & Lin (1996) and the review by Shu
(2016). Following Mark (1977), these authors argued that “grand design”
spirals in galaxies are manifestations of a cavity-type (i.e. WASER) mode in a
sub-maximum disk that is dynamically cool over most of the disk, but which
also possesses a “$Q$ barrier” (Bertin & Lin, 1996, p212) both to provide an
inner turning point and to shield the ILR. WASERwave amplification by
stimulated emission of “radiation” The mildly-unstable mode persists for many
tens of galactic rotations and becomes quasi-steady due to dissipative shocks
in the gas; they also allowed that superposition of a second mode may be
needed in some cases. As noted in §4.2.3.2, strong swing-amplification occurs
for $1\mathrel{\hbox to0.0pt{\lower 2.15277pt\hbox{$\mathchar
536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar 316\relax$}}X\mathrel{\hbox
to0.0pt{\lower 2.15277pt\hbox{$\mathchar 536\relax$}\hss}\raise
1.72218pt\hbox{$\mathchar 316\relax$}}2$. By considering only bi-symmetric
disturbances in sub-maximum disks, Bertin et al. (1989) exploited the mild
disk response when $X>3$ in order to obtain slowly-growing spiral modes in
their stability analysis of many galaxy models. Note that their mode
calculations included a global solution for the gravitational field and
invoked a fluid model for the disk, which is valid away from Lindblad
resonances.

Simulations by Sellwood (2011) of one of the cases presented by Bertin et al.
(1989) confirmed that a single, slowly-growing mode was present when
disturbance forces were restricted to $m=2$. The basic state of the
collisionless particle disk did not evolve in this restricted simulation while
the mild instability grew slowly. Not surprisingly, however, Sellwood (2011)
also found much more vigorous instabilities appeared when higher sectoral
harmonics contributed to disturbance forces, and the contrived $Q$-profile of
the disk, which was designed to support the $m=2$ mode, was rapidly changed.
The onset of disk heating by these multi-armed disturbances was increasingly
delayed as larger numbers of particles were employed because those
instabilities took longer to grow from the decreased shot noise. The inclusion
of gas cooling, which his simulations omitted, would have slowed the disk
heating rate, and allowed the multi-arm activity to persist indefinitely (see
§6). Therefore, the slowly growing bi-symmetric spiral modes in halo-dominated
disks determined by Bertin et al. (1989) would indeed be overwhelmed by true
vigorous instabilities having $m>2$.

Furthermore, the Gaia DR2 data (Gaia collaboration, 2018) revealed a rich
level of substructure in the phase space distribution of stars near the Sun,
indicating that the local disk of the Milky Way is far from the settled, well-
mixed state invoked by Bertin & Lin (1996). Sellwood et al. (2019) argued it
seemed unlikely that a spiral instability could flourish in the observed
disequilibrium state of the Milky Way disk, and also showed that rival
theories would naturally create some of the observed features in phase space,
whereas quasi-steady modes would not. Thus if the Bertin-Lin mechanism for
spiral generation were somehow to operate in the Milky Way, some other recent
and/or on-going disturbances would be required to create the observed
unrelaxed phase space (Sellwood et al., 2019) without interfering with the
spirals. Consequently, the theory is now beset with multiple serious issues.

####  5.2.2 Responses to Noise

Toomre (1990) abandoned the idea of spirals as normal modes, and advocated
instead that a collection of massive clumps in the disk, each of which becomes
dressed with its own wake (§§4.2.3.1 & 4.3), would create a “kaleidoscope” of
shearing spiral patterns. Local simulations of this process by Toomre (1990)
and Toomre & Kalnajs (1991) employed a modest number of particles confined to
a shearing patch, in which the particles themselves were the “massive clumps”.
D’Onghia et al. (2013) conducted global simulations of a sub-maximum disk
composed of $10^{8}$ star particles, embedded in a rigid halo, to which they
added a sprinkling of heavy particles. Since responses in their sub-maximum
disk favored $6\mathrel{\hbox to0.0pt{\lower 2.15277pt\hbox{$\mathchar
536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar 316\relax$}}m\mathrel{\hbox
to0.0pt{\lower 2.15277pt\hbox{$\mathchar 536\relax$}\hss}\raise
1.72218pt\hbox{$\mathchar 316\relax$}}12$ (D’Onghia et al., 2013, §4.2.3.2),
the seed particles induced evolving multi-arm spiral patterns in the stars. In
separate experiments they also tried a single perturber, which they removed
after its wake had developed, and reported continued spiral activity without
additional forcing. As linear theory predicts that a wake in a stable disk
should decay once the driving term is removed, D’Onghia et al. (2013)
attributed the continuing activity to non-linear effects.

However, the perturber might have seeded unstable modes that would have
continued to grow after it was removed. Sellwood & Carlberg (2021) therefore
reproduced their experiment, but were unable to find any coherent modes in the
on-going activity. Taking one step further, these authors tried a single co-
orbiting mass in the stable (Toomre, 1981) half-mass Mestel disk model in
which responses are most vigorous for $2\leq m\leq 4$. In this case, they
found the disk had acquired several discrete instabilities after the
perturbing mass was removed; this was in contrast to the behavior in the low-
mass disk. Sellwood & Carlberg (2021) were able to show that the supporting
responses at $m=2$ & 3 carved isolated grooves in the heavier disk, but
scattering at higher $m$ resonances in the halo-dominated disk blurred
together to create a broad feature that was not destabilizing.

Toomre (1990) and D’Onghia et al. (2013) suggest that “ragged” (Toomre, 1977,
p473) spirals in galaxies result from responses to co-orbiting giant molecular
clouds, massive star clusters, etc., and to the lingering disk responses
should any disperse. Although their numerical results are sound, it is
unlikely that their proposed mechanism accounts for the observed spirals in
galaxies for several reasons. First, the heaviest perturbing mass,
$10^{7}\,M_{\odot}$, that D’Onghia et al. (2013) employed produced only a
modest wake within a narrow annulus in their halo-dominated disk. Second,
D’Onghia et al. (2013) found that the disk response to a collection of
randomly placed heavy particles was multiple spiral arms, not one that was
predominantly 2- or 3-armed. Third, clumps massive and numerous enough that
their associated wakes produce large-amplitude and radially-extensive spiral
patterns would scatter disk stars causing rapid heating of the disk so that
the responses would fade quickly unless the disk were cooled (see §6.1)
aggressively, and the necessary cooling (Toomre, 1990) seems rather extreme.

Spirals in real galaxies (§2) generally have greater amplitude, radial extent,
and lower rotational symmetry than those in the simulations of D’Onghia et al.
(2013), suggesting that more massive clumps in a more massive disk would be
needed. But it is likely that more massive disks readily support unstable
spiral modes, as discussed above (§5.1), obviating the need to stretch
responses to mass clumps into a full theory for spirals in galaxies.

####  5.2.3 Shearing Spirals

A number of authors (see the review by Dobbs & Baba, 2014, for early
references) and also Kawata et al. (2014), Baba (2015), Kumamoto & Noguchi
(2016), Michikoshi & Kokubo (2018, 2020), have argued that spiral patterns are
hardly density waves at all, but wind more tightly over time at a rate that is
almost as rapid as if they were material features. These papers report
evidence that swing-amplification plays a prominent role in the development of
the spirals, as was first noted by Sellwood & Carlberg (1984, their Fig. 3).
In fact all agree that the pitch angle of individual features in simulations
decreases over time, that spiral patterns change continuously and differ
beyond recognition after a single disk rotation.

However, this apparent behavior can be manifested by the superposition of two
or morelong-lived, uniformly-rotating patterns, as was convincingly
demonstrated by Sellwood & Carlberg (2021). They presented an animation,
available as Supplemental Video 1, showing the time evolution of the net
disturbance density when two steady wake responses having differing pattern
speeds were superposed. All that is required for the appearance to resemble
swing amplification is that the steady waves partially overlap in their radial
extent and the peak amplitude of the pattern having the higher pattern speed
lies interior to that of the slower. Since the dynamical clock runs faster
towards the center, this second requirement is almost inevitably
satisfied.999See Lieb et al. (2021) for a counter example, in which the bar
was slowed by dynamical friction, causing the faster rotating spiral in the
outer disk of their models to appear to alternate between leading and
trailing.

We stress that power spectra extracted from simulations over a few disk
rotations almost always reveal that apparently shearing, transient waves are
decomposed into a few steadily rotating waves each extending over a radial
range centered near the CR. We described the origin and nature of these
underlying modes above (§5.1).

Note that if the only process were swing amplification, the input noise would
merely be amplified by a factor (§4.2.3.2), causing the resulting spiral
amplitudes to scale as $N^{-1/2}$. Sellwood & Carlberg (2014) reported that
initial spiral amplitudes indeed scaled as $N^{-1/2}$, but the amplitudes
quickly rose to a level that was independent of the number of particles, which
they varied over several orders of magnitude. They also noted that it took
longer to reach the common amplitude as $N$ was increased. The most natural
explanation of their findings is that their models were linearly unstable to
spiral modes that exponentiate out of the noise, which is reduced as $N$ is
increased, and the modes saturate at a common amplitude due to nonlinear
effects.

None of the papers that claim spiral patterns in simulations to be simply
swing-amplified transients have reported the effect of varying the number of
particles by a few decades, yet the visible spirals in their simulations have
similar amplitudes even in experiments having several million particles. It
seems highly likely that the spirals they have reported were created by
instabilities and the shearing patterns and apparent swing amplification
resulted from superposition of some number of unstable modes.

###  5.3 Observational Tests of Theoretical Ideas

Observational evidence, reviewed in §2.4, indicates that most spirals are
density waves. Both NIR photometry and 2D velocity maps clearly suggest they
are quasi-sinusoidal density variations in the underlying stellar disk that
are massive enough to create non-axisymmetric streaming flows in the gas. Shu
(2016) reviews multiple papers that attempt to fit spiral models to nearby
galaxies. However, this exercise tells us nothing about the origin of the
density waves and, since gas rapidly adjusts to changes in the spiral
potential, the flow pattern is also insensitive to spiral lifetimes.

Spiral arms have long been predicted to trigger star-formation either via
shocks to the gas clouds (Roberts, 1969), or simply because the gas flow
converges (§6.2 and Kim et al., 2020). When the spiral has a fixed pattern
speed over a broad radial range, the streaming speed of stars and gas relative
to the wave increases with radial distance from corotation, and leads to
shallower stellar age gradients among the newly-formed stars downstream from
the spiral arm; this was first proposed by Dixon (1971) and restated by Dobbs
& Pringle (2010). While correct, we emphasize that even swing amplified
disturbances that shear at close to the material rate for a while, develop
wave-like properties in the later stages (Figure 3) when gas and stars stream
through the arms. The contrasting prediction of quasi-steady spiral structure
is that the pattern speed is constant over the entire radial range of the
spiral. Foyle et al. (2011) rule out age gradients downstream from the spiral
in their sample, but others (Chandar et al., 2017, Yu & Ho, 2018, Miller et
al., 2019, Peterken et al., 2019) claim to have detected them. However, none
of these careful studies was able to establish a fixed pattern speed over the
entire radial extent of the spiral.

A further suggestion from the same authors (Pringle & Dobbs, 2019) is that
$\cot\alpha$, with $\alpha$ being the pitch angle of the spiral, should have a
uniform distribution across some range of $\alpha$ and over spiral arms in
many galaxies if spirals wind up over time. Lingard et al. (2021) applied this
test to a sample of 200 galaxies, finding $\cot\alpha$ values that were
consistent with a uniform distribution over the range
$15^{\circ}\mathrel{\hbox to0.0pt{\lower 2.15277pt\hbox{$\mathchar
536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar
316\relax$}}\alpha\mathrel{\hbox to0.0pt{\lower 2.15277pt\hbox{$\mathchar
536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar 316\relax$}}50^{\circ}$.

Both these tests could possibly distinguish the classic density wave theory of
a single large-scale spiral mode (see §5.2.1) from other models, but winding
spirals are predicted in all three of the other mechanisms discussed in §§5.1
& 5.2. Thus, a different kind of test is required to discriminate among
theories of how the spiral disturbances are excited. Currently, the only
foreseeable such test can be made within the Milky Way, and requires the
exquisite data from Gaia.

As noted above, the second data release from Gaia (Gaia collaboration, 2018),
has revealed extensive substructure in the phase space distribution of stars
near the Sun. Hunt et al. (2018) showed that some of the features in the
velocity distribution could be reproduced by the winding spiral model.
However, Sellwood et al. (2019) converted the coordinates to action-angle
variables, finding a number of coherent features in action space that sloped
to smaller $L_{z}$ with increasing $J_{R}$, as expected from ILR scattering.
They also found a highly non-uniform distribution in angles, which is clear
evidence that the stellar distribution is not well-mixed, and has therefore
been subjected to recent disturbances. However, the features in action space
are unaffected by phase mixing and should therefore endure, although
scattering by molecular clouds may gradually blur them. These authors
experimented with idealized models of possible perturbations and concluded
that the observed features were somewhat more consistent with transient spiral
modes, than a simple model of a winding spiral. It is also likely that some,
though not all, of the features in action-space were created by resonances
with the bar of the Milky Way (Monari et al., 2019).

Thus Gaia DR2 has not provided sufficient evidence to discriminate
conclusively among the different theories for the origin of the spirals.
However, it is to be hoped that future releases with more precise measurements
extending to greater distances from the Sun may one day afford a decisive
test.

##  6 GAS IN SPIRAL GALAXIES

Our discussion so far has ignored the gas component (except as possible mass
clumps), even though our description of the observations (§2) noted that at
least a small gas fraction seemed almost essential for isolated galaxies to
possess spiral patterns. Here we discuss three roles for gas in spiral
galaxies.

###  6.1 Maintaining Spiral Activity

Both stars and gas clouds are scattered by the spirals (§5.1.1), but while
stellar random motions cannot be damped, those of the gas component are.
Individual clouds collide dissipatively, with the collision energy being
radiated, which drives them toward non-intersecting streamlines that are
circular in an axisymmetric potential.

Unfortunately, simulations are unable to model gas properly because the
dynamical processes of spiral formation occur on spatial scales that are many
orders of magnitude greater than would be required to capture the full
physical behavior of the clumpy, multi-phase interstellar medium (ISM). The
sub-grid physics of fragmentation, star-formation, feedback, heating, cooling,
shocks, turbulence, metallicity increases, magnetic fields, etc., can be
modeled only by adopting ad hoc rules. However, as far as spiral dynamics is
concerned, more or less any rule that mimics dissipation prolongs spiral
activity (e.g. Sellwood & Carlberg, 1984, Carlberg & Freedman, 1985, Toomre,
1990, Roškar et al., 2012, Aumer et al., 2016). Cosmological simulations of
galaxy formation also mimic gas physics and support mild spiral patterns (see
§7.5).

Not only do the gas clouds themselves dissipate random energy, but new stars
are formed with the kinematics of this lower velocity dispersion component.
Thus the crucial low velocity dispersion population of stars is augmented,
thereby maintaining the responsiveness of the star plus gas disk and enabling
spiral activity to persist. Without replenishment, star formation would
eventually consume the gas, with much of it being locked away in low mass
stars having essentially infinite lifetimes. However, a drizzle of infalling
gas onto the galaxy disk, over and above any possible fountain flow resulting
from star formation activity (see e.g. Roberts-Borsani & Saintonge, 2019), not
only replenishes the gas, but it also gradually raises the disk surface
density, which diminishes $Q$ (eq. 3) and makes the disk more responsive.
Sellwood & Carlberg (1984) found that a rate of gas infall and star formation
of a few solar masses per Earth year over the entire disk of a galaxy would
provide sufficient cooling to balance the heating by moderate spiral activity,
which is consistent with the requirement to maintain star formation rates,
first noted by Larson, Tinsley & Caldwell (1980). Thus the observation (e.g.
Oort, 1962, and §2) that almost all spiral patterns are seen in galaxies that
contain gas and are forming stars can be understood by this argument.

Hierarchical galaxy formation (reviewed by Somerville & Davé, 2015) indeed
predicts late infall both as cooling of shock-heated gas and in cold flows
onto the disks of galaxies in the field, which is responsible for the inside-
out growth of galaxy disks. Galaxies in large clusters, however, may have not
only their ISM stripped by their relative motion through the hot intra-cluster
gas, but also their disks are deprived of fresh infalling cool gas, which is
at least part of the reason that clusters host many S0 galaxies and few
spirals, as proposed by Gunn (1982). However, there are two reasons that
cluster S0s should not have the properties of field disk galaxies that have
merely been deprived of gas: first, galaxies in a cluster originated in a
denser environment than those in the field, causing them to have generally
larger classical bulges (from hierarchical merging) and second, accretion of
gas to grow the disk will have stopped at an earlier stage, causing them to
have less extensive disks, on average. Note that S0 galaxies also exist in the
field, and Fraser-McKelvie et al. (2018) propose two mechanisms for their
origin: faded spirals for low mass S0s, and mergers to create those of higher
mass.

###  6.2 Gas Flows in Spiral Potentials

Figure 1, in §4.2 above, was also used by Kalnajs (1973) to illustrate gas
streamlines in spirals, since cold gas will settle onto the illustrated
ballistic orbits if they can be nested without intersecting, although a shock
must intervene where orbits cross. From these diagrams, one can see that the
flow converges as the gas approaches the spiral and, if the gas overtakes the
wave (inside CR), it flows inwards in the arms, whereas outward flow along the
arm is expected outside corotation (Kalnajs, 1973). This sign change of the
radial flow velocity within spiral arms was exploited by Font et al. (2014) to
identify the radii of CRs in many galaxies. Note that these closed streamlines
create flows in the opposite sense between the arms, and there is no net
inflow or outflow, at least in the absence of shocks.

Although offering valuable insight, this picture is highly idealized, and the
detailed dynamics of the ISM matters a great deal. Since global simulations
cannot begin to model local star formation, feedback, etc., Kim, Kim &
Ostriker (2020) adopt an intermediate course, and try to build a more detailed
picture of ISM behavior in a small part of the disk of a spiral galaxy that is
subject to an imposed spiral perturbation. Their still idealized model
predicts that stars are preferentially formed in spiral arms, as a result of
the converging gas flow. They also show that supernova feedback blows holes in
the ISM, creating chimneys, and the shear flow carries the rims of the larger
holes to create features that match the observed spurs and feathers.

###  6.3 Flocculent Spirals

The theories reviewed above (§5) addressed the origin of patterns having
moderate numbers of spiral arms. However, flocculent galaxies have many short
spiral arm segments (§2.1 and Sandage, 1961); the prototype is NGC 2841, for
which the spiral fragments stand out in blue light, while NIR images are
almost featureless (Block et al., 1996). NGC 5055 is also flocculent in blue
light, but some IR images reveal an underlying 2-arm spiral that was confirmed
kinematically by Thornley & Mundy (1997).

Figure 6: A simulation having a two-component Mestel disk. (a) shows the
massive warm component at the same instant as the low-mass very cool component
(b), which manifests a flocculent spiral pattern. The radius of the disk is
$20R_{0}$, and the color scale reports the logarithm of the surface density in
units of $V_{0}^{2}/(GR_{0})$.

Elmegreen et al. (2003) relate flocculence to turbulence in the ISM, but a
more interesting dynamical explanation was proposed earlier by Elmegreen &
Thomasson (1993), who suggested that flocculent patterns arise through
gravitational instabilities in a low-mass cool disk component. They presented
simulations of a low-mass disk embedded in a massive halo that manifested
flocculent spirals. Here we show (Figure 6) that a two component disk behaves
in a similar manner. The model employed to create Figure 6 has a half-mass
($f_{d}=0.5$), $Q=1.5$, Mestel disk, while the cool disk, also composed of
collisionless particles, has one-tenth the mass ($f_{d}=0.05$) and an initial
$Q=0.05$, which is a deliberately low value in order to mimic the dynamical
responsiveness of a gas-rich component. The two components are dynamically
decoupled at first, and the supporting response favors sectoral harmonics
$20\mathrel{\hbox to0.0pt{\lower 2.15277pt\hbox{$\mathchar
536\relax$}\hss}\raise 1.72218pt\hbox{$\mathchar 316\relax$}}m\mathrel{\hbox
to0.0pt{\lower 2.15277pt\hbox{$\mathchar 536\relax$}\hss}\raise
1.72218pt\hbox{$\mathchar 316\relax$}}40$ in the cool component. The cool disk
quickly creates flocculent spirals, perhaps driven by mass clumps as suggested
by Toomre (1990) and DVH13 (§5.2.2), while the warm disk, which would have
been stable (Toomre, 1981) in the absence of the cool disk has a mild spiral
with a few arms; some coupling between the two components is apparent where
the cool component has heated significantly in the inner disk by the time
shown ($t=100R_{0}/V_{0}$).

A two-component disk like this could perhaps arise naturally if the old disk
were starved of fresh gas for a while, and then suddenly accreted a
substantial gas component. The flocculent instabilities would trigger star
formation in the arm segments perhaps giving rise to a galaxy resembling NGC
2841. A prediction of this model is that there should be a dearth of
intermediate age stars in the disk of a flocculent galaxy.

##  7 ROLE OF SPIRALS IN GALAXY EVOLUTION

As noted at the outset of this review, spiral activity is a driver of
evolution in a disk galaxy. Thus the present day structure of galaxy disks is
not merely the result of initial conditions at the time of their formation,
but it has been changed, at least in part, by internally-driven evolution, as
has long been argued by Kormendy (1979) and reviewed by Kormendy & Kennicutt
(2004), although they stressed the role of bars over spirals.

###  7.1 Radial Migration

For years, attention focused on Lindblad resonance scattering by spirals, and
changes at corotation went unreported. Sellwood & Binney (2002) were therefore
surprised to find that a transient spiral mode causes greater angular momentum
changes to stars at the CR than occur at the Lindblad resonances. The angular
momentum gains and losses by different stars at the CR generally roughly
balance.

Figure 7: (a) The distribution of changes to the angular momentum, $\Delta
j_{z}$, by 0.5 Gyr later for the given $j_{z}$ at 6.5 Gyr. The vertical lines
mark the estimated $j_{z}$ of corotation for the dominant three spiral modes
over the time interval 6 to 7 Gyr. (b) Two examples of orbits from the same
simulation. The colors of the lines in subpanels i and iii change with time as
indicated in subpanels ii and iv, which show the instantaneous radius of the
particle. Figure adapted from Roškar et al. (2012), with permission.

An example from a more realistic simulation is shown in Figure 7a (adapted
from Roškar et al., 2012, who used $j_{z}$ for $L_{z}$). The distribution of
changes manifests an inclined ridge in the middle of the range of $L_{z}$
indicating $\Delta L_{z}$ values for some particles that range up to $\sim
L_{z}/2$ at the start of this interval. The vertical lines mark the estimated
$L_{z}$ values for corotation of three separate spiral modes in the disk in
their simulation over a longer interval, 6 to 7 Gyr, but the middle wave
dominates over the time interval reported in Figure 7a (see their Fig. 6). It
is clear that many particles having smaller $L_{z}$ values than the middle
vertical (yellow) line increased their initial $L_{z}$ to rise outwards across
corotation, and others initially having larger $L_{z}$ values lost similar
amounts also to sink inside corotation. The numbers of gainers and losers were
similar, and the slope of the ridge indicates that there was a tendency for
particles that crossed CR to end up equally as far in $L_{z}$ from the
resonance as before.

Figure 7b presents two orbits of particles in the same simulation that have
experienced substantial radial migration. The orbit in the left pair of panels
at first migrates inwards at $t\sim 4\;$Gyr and then outwards by a larger
amount at $t\sim 6\;$Gyr, as does the orbit in the right pair of panels.
Notice that in all three instances, migration is rapid and occurs with no
significant increase in the size of the orbit’s epicycle, indicated by the
short period oscillations.

These $L_{z}$ changes, which completely dwarf those at the Lindblad
resonances, had not attracted attention because they do not heat the disk (eq.
6), and stars largely change places in a dynamically neutral manner. However,
radial mixing of stars with different chemical abundances has important
consequences for modeling the radial distribution of elements in a galaxy disk
(Roškar et al., 2008, Schönrich & Binney, 2009).

####  7.1.1 Mechanism of radial migration

Stars near corotation move slowly with respect to the spiral perturbation and
therefore experience almost steady forcing from the wave, which allows large
changes to build up – a process that is analogous both to surfing on ocean
waves and to Landau damping in plasmas, although the consequences differ.
Stars orbiting just behind the density excess are attracted forward by it and
gain angular momentum. However, the result of gaining angular momentum is that
the star moves onto an orbit having a larger guiding center radius, and its
angular frequency about the center decreases. If the star were just inside
corotation, and gaining on the density excess, the change can cause it to rise
to a radius just outside corotation where it begins to fall behind. This
behavior is described as a horseshoe orbit; Figure 2a includes a few example
orbits whose motion reverses in the rotating frame. Conversely, stars just
ahead of the perturbation are pulled back, lose angular momentum and sink
inwards, where they orbit at higher frequency. Those outside corotation, where
the perturbation gains on them, could lose enough angular momentum to cross
corotation and begin to run ahead of the wave. As long as the gradient
$\partial f/\partial L_{z}|_{J_{R}}$ is fairly shallow, roughly equal numbers
of stars gain as lose, and they largely change places.

Were the spiral potential to maintain a fixed amplitude, stars on horseshoe
orbits would be described as trapped. As they move slowly with respect to the
wave, stars would take a long time to reach the next density maximum, where
the changes just described would be exactly undone. However, if the amplitude
of a transient spiral mode has decayed by the time the star reaches the next
density peak, it will no longer be trapped and will continue to move with a
lasting change to its angular momentum.

The radial extent of the region where horseshoe changes occur varies as the
cube root of the perturbation amplitude (eq. 8.91 of Binney & Tremaine, 2008),
and therefore widens as a perturbation grows. Sellwood & Binney (2002) found
the spiral was strong for less than half the horseshoe period for most trapped
stars, which consequently undergo a single change.

The process affects stars with small peculiar velocities most strongly, since
greater epicyclic motion leads to less coherent forcing by the spiral