UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
DOCTORADO EN CIENCIAS (ASTROFÍSICA)
THE ROLE OF THE BAR PATTERN SPEED IN
SHAPING THE GALACTIC STRUCTURES
T   E   S   I   S
QUE PARA OPTAR POR EL GRADO DE:
DOCTOR EN CIENCIAS (ASTROFÍSICA)
PRESENTA:
LUIS ALBERTO GARMA OEHMICHEN
DIRECTORES DE TESIS:
HÉCTOR MANUEL HERNÁNDEZ TOLEDO
INSTITUTO DE ASTRONOMÍA - UNAM
LUIS ALBERTO MARTÍNEZ MEDINA
INSTITUTO DE ASTRONOMÍA - UNAM
CIUDAD UNIVERSITARIA, CDMX, FEBRERO, 2023
 
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Universidad Nacional Autónoma de México
Instituo de Astronoḿıa
Ciudad Universitaria
The role of the bar pattern speed in shaping
the galactic structures
T E S I S
que para obtener el t́ıtulo de:
Doctor en Astrof́ısica
presenta:
Luis Alberto Garma Oehmichen
Tutores:
Dr. Héctor Manuel Hernández Toledo
Dr. Luis Alberto Martinez Medina
Coyoacán, Ciudad de México 2023
instituto de astronomía 
Agradecimientos
A mi papá y mi mamá, por su apoyo incondicional y por asegurarse de que nunca me
nunca me faltara nada.
A mi hermano Alex, por acompañarme todos estos años con risas, discusiones y ocurrencias.
A mi esposa Amairani, por todo el cariño y amor. Que esta es una de las muchas cosas
que construiremos juntos.
A mis amigos, que hicieron que el viaje fuera emocionante y divertido. Gracias Alexia,
Ale, Edgar, Fransico, Arturo, Eli, Gibran, Tania, Abel, Carlos, Milton y Sonia.
Y por supuesto, a mis asesores Héctor y Luis. Gracias por las enseñanzas, la paciencia,
las horas de discusiones y de trabajo. Sin ustedes, esto no hubiera sido posible.
i
Notation
Mathematical expressions
ΩBar km kpc−1 s−1 Bar pattern speed
RCR kpc Corotation radius
RBar kpc Bar radius
RDep
Bar kpc Deprojected bar radius
R RCR/R
Dep
Bar Rotation rate
V flat
c km s−1 Disc flat circular velocity
QBar Bar gravitational torque
logM/M⊙ Logarithmic galactic mass
ρ(x, y, z) M⊙ kpc−3 Density profile
σ km s−1 Velocity dispersion
S Shear rate
rs Spearman correlation coefficient
p p-value
Acronyms
MaNGA Mapping Nearby Galaxies at Apache Point Observatory
DBSCAN Density-based spatial clustering of applications with
noise
ILR Inner Lindblad Resonance
OLR Outer Lindblad Resonance
MW Milky Way
TW Tremaine-Weinberg method
PA Position angle
ΛCDM Lambda Cold Dark Matter
ii
Contents
Agradecimientos i
Notation ii
Resumen vii
1 Introduction 1
§1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§1.2 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§1.3 Statement of originality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Galactic structures 3
§2.1 Cosmological context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
§2.2 Disc galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
§2.3 Bulges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
§2.4 Spiral arms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
§2.5 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Review on Stellar Bars 11
§3.1 Bar properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
§3.2 Stellar orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
§3.3 Bar formation and evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 12
§3.4 Secular evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
§3.5 Bar pattern speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
§3.6 Rotation rate and the fast bars problem . . . . . . . . . . . . . . . . . . . 19
iii
§3.7 The Milky Way bulge/bar . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
§3.8 The bar-spiral connection . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 The bar and the spiral structure 23
5 Bar Pattern Speed in Milky Way like Galaxies 39
6 Conclusions 61
§6.1 Main conclusions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
§6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
A Photometric Signature of Ultraharmonic Resonances in Barred Galaxies 64
B Galaxy Zoo: Kinematics of strongly and weakly barred galaxies 79
List of Figures
2.1 Total mass density by galactic components. Elliptical galaxies (red) dom-
inate the high stellar mass regime, while discs (blues and greens) have a
broad distribution from the intermediate to the lower mass regimes. Credit:
Image taken from Moffett et al. (2016). . . . . . . . . . . . . . . . . . . . . 4
2.2 The so-called “phase spiral” in the (z, Vz) plane around the Solar neighbour.
The pattern is best seen when colour coded by the median radial velocity
VR (left panel), the median azimuthal velocity Vφ (middle panel) and the
median angular momentum Lz (right panel). Credit: Left and Middle panels
taken from Antoja et al. (2018), Right panel adapted from Khanna et al.
(2019). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
iv
LIST OF FIGURES v
2.3 Examples of grand design spiral galaxies. Left panel : The spiral structure of
M51 (the whirlpool galaxy) was presumably formed by the tidal interaction
with its companion galaxy NGC 5195. Right panel :In NGC 1300 the grand
design structure is probably induced by the large bar. Credit: Hubble
material from NASA and ESA. . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Examples of ringed galaxies. Left panel: NGC 1512. The nuclear ring
is being fuelled by gas falling through the bar in structures called “dust
lanes”. An inner ring envelops the stellar bar. Right panel: The Cartwheel
galaxy. A collision with its neighbour near the centre, produced an outward
propagating wave where the ring is. The galaxy is in the intermediate
process of re-constructing spiral arms. Credit: Hubble and JWST material
from NASA and ESA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1 Two examples of the modelling done by Rautiainen et al. (2005) The first
two columns shows the deprojected H-band and B-band of the galaxy. The
last two columns show the gas morphology of the best fit model. . . . . . 16
3.2 Font-Beckman method. Left panel: Velocity field of UGC2709 from Hα
emission line. Middle panel: Residual velocity field after subtracting a 2D
rotational velocity field model. Right panel: Histogram of the number of
phase reversals in the residual velocity map, against galactocentric radius.
Adapted from Font et al. (2014). . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 The dark gaps method. Left panel: A test particle from Schwarz (1984)
of a strongly barred disc showing dark gaps forming around the L4,5 La-
grangian points. Buta (2017b) proposed these dark gaps are located at the
bar corrotation resonance. Using a high resolution N-body simulation Kr-
ishnarao et al. (2022), proposed the dark gaps are closer to the 4:1 ultra
harmonic resonance. Middle panel: Stellar surface brightness as a function
of the radius of the major and minor axis of a snapshot from the simulation.
Right panel: Difference between the major and minor axis, highlighting the
maximal difference that coincides with the 4:1 ultra-harmonic resonance. . 17
LIST OF FIGURES vi
3.4 A comparison of the bar pattern speed, corotation radii and R found in
various works. Credit: Tobias et al. in prep. . . . . . . . . . . . . . . . . . 19
Resumen
Las barras estelares son uno de los principales agentes conductores de la evolución secular
en galaxias de disco. Intercambian masa y momento angular tanto en sus resonancias
internas como en las externas. La localización de estas resonancias, depende directamente
de la velocidad patrón de barra y en la curva de rotación circular del disco. Por tanto, la
velocidad patrón es el parámetro más importante para evaluar los efectos de la barra en
el disco. Desafortunadamente, también es el parámetro más dif́ıcil de estimar.
En este proyecto doctoral, he decidido estudiar el problema de velocidad patrón de
barra usando dos aproximaciones diferentes pero complementarias: (i) Usando simulaciones
numéricas en potenciales estacionarios, (ii) Midiendo la velocidad patrón en una muestra
de galaxias Análogas a la Vı́a Láctea.
Las últimas décadas han visto grandes avances en los escenarios de formación y evolu-
ción de galaxias. Simulaciones de N-cuerpos (tanto de galaxias individuales como cos-
mológicas) han sido capaces de reproducir discos realistas, gracias a las constantes mejoras
en la resolución y el modelado de f́ısica de bariones. Son la herramienta ideal para estudiar
los procesos de evolución galáctica. Sin embargo, debido al alto costo computacional, es
muy dif́ıcil estudiar como pequeñas variaciones en la distribución de densidad puede llegar
a afectar procesos dinámicos internos. Es aqúı, donde simulaciones numéricas con poten-
ciales estacionarios presentan algunas ventajas. En el primer art́ıculo que presento en esta
tesis, exploramos como variar parámetros de la barra afecta a las propiedades de brazos es-
pirales inducidos. Usamos un conjunto de 108 simulaciones numéricas, donde exploramos
un espacio de parámetros que incluye razón de rotación de la barra R, longitud, masa
y elepticidad. Nuestros resultados muestran que los parámetros que mejor describen la
fuerza de la barra (torque gravitacional QBar, masa, longitud) están fuertemente correla-
vii
RESUMEN viii
cionados con la amplitud espiral. Por otro lado, los parámetros que describen la rotación
de la barra (velocidad patrón, razón de rotación) están correlacionados con el ángulo de
apertura de los brazos espirales.
Grandes avances observacionales se han logrado con la introducción de la técnica de
Espectroscopia Integral de Campo. La espectroscopia espacialmente resuelta permite el
estudio de estructuras galácticas complejas. En particular, revela el estado cinemático
de las estrellas y el gas, que pueden ser a su vez utilizados para estimar la velocidad
patrón de barra. El número de galaxias con mediciones de velocidad patrón de barra ha
incrementado constantemente desde el trabajo pionero por Aguerri et al. (2015) con 15
galaxias del catastro CALIFA.
En el segundo art́ıculo de esta tesis, hemos medido la velocidad patrón de 97 galaxias
del catastro MaNGA usando el método Tremaine-Weinberg. La muestra fue seleccionada
para ser similar a la Vı́a Láctea en masa estelar y morfoloǵıa. Encontramos 3 relaciones
importantes: (i) Discos con mayor masa estelar tienden a poseer barras más largas y con
menor velocidad patrón. (ii) Discos con altos valores de velocidad circular debeŕıan poseer
barras con velocidades patrón más altas, pero menor razón de rotación R. (iii) Galaxias
con mayor concentración de masa (sistemas más soportados por presión) están débilmente
correlacionados con velocidades patrón bajas pero mayores razones de rotación.
Abstract
Stellar bars are the main drivers of the secular evolution in disc galaxies. They exchange
mass and angular momentum at both, their internal and external resonances. The location
of these resonances, depend directly on the bar pattern speed and the disc circular rotation
curve. Thus, the bar pattern speed is the most important parameter to evaluate the effects
of the bar on the disc. Unfortunately, it is also the most difficult parameter to estimate.
For my doctoral project I decided to study the problem of the bar pattern speed
using two different but complementary approaches: (i) Using numerical simulations in
stationary potentials, (ii) Measuring the bar pattern speed in a sample of Milky Way
Analogue galaxies.
RESUMEN ix
Last decades have seen great advancements in the scenarios of formation and evolution
of galaxies. N-body simulations (both for individual galaxies and cosmological) are able to
reproduce realistic discs, thanks to the constant improvements in resolution and the mod-
elling of baryonic physics. They are the ideal tool for studing the processes of the galactic
evolution. However, because of the high computational cost, it is very difficult to study
how small variation in the density distribution can affect internal dynamical processes. It
is here where numerical simulations in analytical potentials present some advantages. In
the first paper I present in this thesis, we explore how different bar parameters affect the
properties of the induced spiral arms. We use a set of 108 numerical simulations, where
we explore a parameter space that includes the bar rotation rate R, length, mass and
ellipticity. Our results show that the parameters that best describe the bar strength (grav-
itational torque Q, mass, bar length) are strongly correlated with the spiral amplitude. On
the other hand, parameters that describe the bar rotation (pattern speed, rotation rate)
are correlated with the spirals pitch angles.
Great advancements have been made in observations with the introduction of the Inte-
gral Field Spectroscopy technique. The spatially resolved spectroscopy allows the study of
the complex and rich internal structures of galaxies. In particular, it reveals the kinematic
state of the stars and gas, which can be used to determine the bar pattern speed. The
number of galaxies with bar pattern speed measurements has been constantly increasing
starting with the pioneering work by Aguerri et al. (2015) with 15 galaxies in the CALIFA
survey.
In the second paper of this thesis, we have measured the bar pattern speed of 97
galaxies of the MaNGA survey using the Tremaine-Weinberg method. This sample was
selected to be similar to the Milky Way in stellar mass and morphology. We found three
important relations: (i) Higher stellar mass discs tend to host larger bars with lower bar
pattern speeds. (ii) High circular velocity discs should host bars with higher pattern speed,
but slower rotation rate R. (iii) Galaxies with high mass concentration (more pressure
supported systems) are weakly correlated with lower bar pattern speeds but faster rotation
rates.
Chapter 1
Introduction
1.1 Objective
In this doctoral project I study the evolution and interactions of stellar bars with other
galactic structures. More specifically, I try to understand the importance and effects of
the bar pattern speed, which is one of the most important parameters for the galactic
secular evolution. To achieve this objective, I studied the problem with different but
complementary approaches, that combine theory and observations.
1.2 Thesis Structure
The thesis starts with two introductory chapters. In Chapter 2 I present a general overview
on the galactic structures that can be found in disc galaxies. I cover how they are formed,
how they are classified and what is the current picture of those structures in the Milky
Way. In Chapter 3 I do a general review on the stellar bars. What are their most important
properties, how do they form and evolve, what stellar orbits support them, the problem
with fast bars and what is their connection with the spiral arms.
This doctoral project has resulted in two original papers where I led the research and
draft writing. In Chapter 4 I present the first one, on the bar-spiral coupling problem
using numerical experiments:
• L. Garma-Oehmichen; L. Martinez-Medina; H. Hernández-Toledo and I. Puerari.
1
CHAPTER 1. INTRODUCTION 2
How the properties of the bar affect those of induced spiral arms in numerical simu-
lations 2021MNRAS.502.4708G
In Chapter 5 I present the second paper on measuring the bar pattern speed in a sample
of Milky Way Analogue galaxies with the Tremaine-Weinberg method:
• L. Garma-Oehmichen; H. Hernández-Toledo; E. Aquino-Ort́ız; L. Martinez-Medina;
I. Puerari; M. Cano-Dı́az; O. Valenzuela; J. A. Vázquez-Mata; T. Géron; L. A.
Mart́ınez-Vázquez and R. R. Lane Bar pattern speed of Milky-Way Analogue galaxies.
2022MNRAS.517.5660G
Finally in Chapter 6 I present the main conclusions of the project, and include some
ideas for future work. As an additional product of my doctoral research I also contributed
as co-author in two papers. They are included in appendices A and B, where I briefly
discuss the contributions I made.
• D. Krishnarao; Z.J. Pace; E. D’Onghia; J. A. L. Aguerri; R. L. McClure; T. Peterken;
J. G. Fernández-Trincado; M. Merrifield; K. Masters; L. Garma-Oehmichen; N.
F. Boardman; M. Bershady; N. Drory and R. R. Lane Photometric Signature of
Ultraharmonic Resonances in Barred Galaxies 2022ApJ...929..112K
• T. Géron; R. J. Smethurst; C. Lintott; S. Kruk; K. L. Masters; B. Simmons; K. B.
Mantha; M. Walmsley; L. Garma-Oehmichen; N. Dory and R. R. Lane Galaxy Zoo:
Kinematics of strongly and weakly barred galaxies (submitted)
1.3 Statement of originality
I certify that the intellectual content of this thesis is the product of my own work and that
all the assistance received in preparing this thesis and sources have been acknowledged.
Chapter 2
Galactic structures
In this section I present a brief overview of the galactic structures found in disc galaxies
of the local universe, including some of the most recent trends in the field.
I start by introducing some cosmological context to understand in a broad way how
spheroids and discs are formed. I continue by describing the most common structures
found in spiral galaxies. More specifically: the disc itself, bulges, spiral arms, and rings.
The stellar bar will be presented in a separate chapter, as it is the main subject of the
thesis.
2.1 Cosmological context
Naively speaking, galaxies can be defined by two components: spheroids and discs. Spheroids
are systems dominated by old stellar population, dust free, low star formation, pressure
supported, and possess little sub-structure. In contrast disc systems are populated by
intermediate-to-young stellar populations, rotationally supported, usually dusty and star
forming. Discs possess a rich variety of sub-structures, including bars, rings and spiral
arms. All of these can be classified into additional sub-categories.
In the current standard cosmological model, ΛCDM, galaxies reside in cold dark matter
halos and structure grows hierarchically by mergers between galaxies (White & Rees,
1978). The resulting structure depends on the gas content of the merging galaxies. Gas
loses angular momentum by internal gravitational torques produced by the merger. This
3
CHAPTER 2. GALACTIC STRUCTURES 4
Figure 2.1: Total mass density by galactic components. Elliptical galaxies (red)
dominate the high stellar mass regime, while discs (blues and greens) have a broad
distribution from the intermediate to the lower mass regimes. Credit: Image taken from
Moffett et al. (2016).
gas falls to the centre and triggers the formation of stars and stellar wind feedback. This
feedback allows a large fraction of the gas to retain angular momentum and re-form the
disc after the merger (Hopkins et al., 2009; Athanassoula et al., 2016; Peschken et al.,
2020).
In the local universe, the stellar mass density is almost equally divided between spheroids
and discs (Moffett et al., 2016; Driver et al., 2022). Figure 2.1 shows the total mass density
of spheroids and discs derived by Moffett et al. (2016). They used a photometric decompo-
sition analysis in the GAMA survey to derive the stellar mass function by morphological
components. Of the total mass density budget, elliptical galaxies contribute ∼ 35%, bulges
∼ 15%, S0-Sa discs ∼ 20%, Sab-Scd galaxies ∼ 22% and Sd-Irr ∼ 6%.
CHAPTER 2. GALACTIC STRUCTURES 5
2.2 Disc galaxies
Many astronomical systems have discs shapes for the same reason: conservation of angu-
lar momentum. Gas radiates energy, but not angular momentum, so the state of lowest
energy is a flat disc (Binney & Tremaine, 2008). Moreover, for rotating discs it is energet-
ically favourable to transfer angular momentum outward and mass inward (Lynden-Bell
& Kalnajs, 1972; Lynden-Bell & Pringle, 1974). In galactic discs (where the stellar com-
ponent is a collisionless system) this happens through gravitational torques produced by
non-axisymmetric features like bars and spiral arms (Tremaine & Weinberg, 1984b).
Galactic discs are rotationally supported systems where the centripetal acceleration
equals the radial acceleration due to gravity:
V 2(R)
R
=
∣
∣
∣
∣
∂Φ
∂R
∣
∣
∣
∣
(2.1)
where Φ is the total gravitational potential and V (R) is the orbital azimuthal velocity
as a function of the galactocentric radius. It can be traced from the neutral hydrogen
well beyond the edge of the stellar distribution (Bosma, 1981). The mass distribution
of the galaxy dictates the shape of the rotation curve. Purely exponential discs should
host rotation curves that decrease with the square root of the radius. However, most
disc galaxies posses a flat rotation curve (Rubin et al., 1980; Bosma, 1981). This can
be explained by the presence of a large dark matter halo build by hierarchical clustering
(Navarro et al., 1997), or by modifying the Newtonian laws of motion (Milgrom, 1983).
Disc galaxies can be classified by their Hubble morphological type, which distinguishes
galaxies by the presence of a bar (S, SAB and SB for unbarred, weakly barred and barred
galaxies respectively) and the interplay between the degree of winding of the spiral arms
and prominence of the bulge (Sa to Sd, where ‘a’ are more tightly wound and bulge promi-
nent, and ‘d’ are more open and practically bulge-less) (Hubble, 1926; de Vaucouleurs,
1959). Sa galaxies are called early-type spiral galaxies, whilst Sc-Sd are called late-type
spiral galaxies. However, contrary to what their name suggests, the trends in star forma-
tion rate, bulge-to-disc ratio and stellar populations age indicate an evolutionary sequence
from Sd to Sa (Kennicutt, 1998). The Hubble sequence also holds a strong relation with the
CHAPTER 2. GALACTIC STRUCTURES 6
Figure 2.2: The so-called “phase spiral” in the (z, Vz) plane around the Solar neighbour.
The pattern is best seen when colour coded by the median radial velocity VR (left panel),
the median azimuthal velocity Vφ (middle panel) and the median angular momentum Lz
(right panel). Credit: Left and Middle panels taken from Antoja et al. (2018), Right
panel adapted from Khanna et al. (2019).
environment, shifting from early types in high density regions to late types in low density
regions (Blanton & Moustakas, 2009). This could be related to the gas accretion, which is
suppressed in dense regions (van de Voort et al., 2017), but is fed by cold filamentary gas
in low density regions (Fakhouri & Ma, 2010).
The surface brightness of discs usually obeys a single exponential law on the radial
direction (de Jong, 1996). However, most discs are better described with a broken expo-
nential law, where the radial surface brightness changes abruptly in the outer parts (Pohlen
& Trujillo, 2006). A down-bending profile is called a Type II disc, and an up-bending pro-
file is called a Type III (Erwin et al., 2008). The physical origin of these breaks is not
clear, however it could to be related to the star formation efficiency (Kennicutt, 1989),
transitions between the cool an warm phases of the inter-stellar medium (Schaye, 2004),
the angular momentum redistribution induced by non-axisymmetric structures like bars
and spirals (Debattista et al., 2006) or the halo spin (Herpich et al., 2017).
The second data release of the Gaia mission (Gaia Collaboration et al., 2018) revealed
that the Milky Way disc possess a very rich and complex velocity field that is far from
equilibrium. This is clearly evidenced by the “phase spiral” presented by Antoja et al.
(2018) in the vertical velocity Vz vs. vertical position z plane. The spiral pattern is best
seen when encoded by either the median radial velocity VR, median azimuthal velocity
Vφ or the median angular momentum Lz (Khanna et al., 2019) (see Figure 2.2). The
CHAPTER 2. GALACTIC STRUCTURES 7
phase spiral suggests the Milky Way disc was perturbed relatively recently, presumably
by the passage of the Sagittarius dwarf galaxy ∼ 0.5 Gyr ago (Binney & Schönrich, 2018;
Bland-Hawthorn et al., 2019). Nonetheless, internal processes like the bar buckling are
also capable of producing a phase-space spiral feature (Khoperskov et al., 2019).
2.3 Bulges
The formation of bulges is currently a subject of major debate in galaxy formation theory.
Some resemble elliptical galaxies (classical bulges): they are compact spheroids, with high
surface brightness and high velocity dispersion. However they do not follow the same mass-
size relation (Gadotti, 2009; Laurikainen et al., 2010). They are thought to be formed in
the early universe via major mergers (Hopkins et al., 2009), accretion of small satellites
(Aguerri et al., 2001), coalescene of clumps (Bournaud et al., 2007; Ceverino et al., 2010)
or by the monolithic collapse of a primordial gas (Eggen et al., 1962; Tacchella et al., 2016).
Other bulges reassemble exponential discs (the so-called pseudo-bulges): they are
rotationally-supported, have flatter and more elongated surface brightness profiles, and
are build mostly of disc stellar populations. These inner stellar components are build by
gas funnelled to the centre through the bar (Sormani et al., 2015). They can display a
boxy/peanut/X-shape structure formed by the buckling of the bar. They are consistent
with a bar-driven formation scenario (Gadotti et al., 2020).
The Milky Way bulge exhibits some characteristics associated with both classical and
pseudo bulges. For example, it is dominated by old, low metalicity stellar populations with
high velocity dispersion (Sit & Ness, 2020). However, it has been shown with observations
and simulations that these old populations are consistent the same kinematically warm
population from the thick disc (Di Matteo, 2016; Fragkoudi et al., 2018, 2020), so there
is no need for a classical bulge. Moreover, the MW bar/bulge possess clear signatures of
cylindrical rotation in metal rich stars (Howard et al., 2009), and the an extended X-shape
structure (McWilliam & Zoccali, 2010; Ness & Lang, 2016) suggesting it has been mostly
build by in-situ processes, such as the bar buckling instability (Martinez-Valpuesta et al.,
2006).
CHAPTER 2. GALACTIC STRUCTURES 8
2.4 Spiral arms
Spiral arms are one of the primary sites of star formation in disc galaxies. They can be
characterised by the number of arms, the pitch angle, strenght, shape, pattern speed and
lifetime (Dobbs & Baba, 2014). Some of these properties are inherited from the disc.
For example, the susceptibility of the disc to perturbations (measured by Toomre stability
parameter Q) could be related to the number of arms as they can amplify local instabilities
(Goldreich & Lynden-Bell, 1965; Toomre, 1981; Hart et al., 2018). The differential rotation
speed of the disc and local shear could be related to the pitch angle (Seigar et al., 2005,
2006). However, the measurements have great scatter, and some properties could be better
explained by the presence of other structures like a stellar bar (Salo et al., 2010).
A great variety of spirals can be observed in the local universe. They can be classified in
3 main categories: flocculent, multi-arm and grand design. Flocculent spirals are granular
and patchy with many short arm sections (Elmegreen & Elmegreen, 1984). Their origin is
probably local, rather than global, as they do not have a spiral pattern in the old stellar
disc (Buta et al., 2015). Their origin could be related to local patches of star formation
that are sheared out by the differential rotation of the disc (Binney & Tremaine, 2008).
In contrast, grand design spirals are characterised by having two symmetrical long
spiral arms supported by an underlying old stellar component. They are presumed to be
formed by tidal interactions, by bar instabilities or be self-excited as shown by numerous
simulations (e.g. see review from Sellwood & Masters, 2021). In Figure 2.3 I show two
examples: M51 where the spirals are presumably formed by the tidal interactions with
NGC 5195 and NGC1300 where the bar is probably driving the spirals. They are expected
to be long-lasting structures, maintained by density waves (Lin & Shu, 1964). These kind
of spirals are not material in nature, but rather are sites of greater density, where stars
pass most time of their orbits. The manifold theory has been proposed as an alternative
mechanism, where the supporting stars come from chaotic orbits at the unstable Lagrange
points near the end of the bar (Voglis et al., 2006; Romero-Gómez et al., 2006, 2007).
In contrast to the density wave theory, stars move along the spiral arms in the manifold
theory.
CHAPTER 2. GALACTIC STRUCTURES 9
Figure 2.3: Examples of grand design spiral galaxies. Left panel : The spiral structure of
M51 (the whirlpool galaxy) was presumably formed by the tidal interaction with its
companion galaxy NGC 5195. Right panel :In NGC 1300 the grand design structure is
probably induced by the large bar. Credit: Hubble material from NASA and ESA.
Multi-arm spirals are a middle category that is closer to grand design. A statistical
analysis of the host galaxies shows that multi-arm and grand design galaxies share most
of their fundamental properties (Bittner et al., 2017).
2.5 Rings
Galactic rings can be classified in three types: nuclear, inner and outer rings. Nuclear rings
are sites of intense star formation, found at the centres of barred galaxies (Benedict et al.,
2002). The Milky Way is likely to host one, at the Central Molecular Zone (Sormani et al.,
2018). Inner rings are larger features that usually envelop the bar if present. They are
relatively common structures as around 35% of nearby galaxies host inner rings (Comerón
et al., 2014). Outer rings are larger and diffuse, and as the name suggests are located
in the outskirts of the galaxy. They are usually made by tightly warped spiral arms, or
“pseudo-rings” (Buta, 2017a).
Rings are usually associated with important bar resonances, like the inner and outer
Lindblad resonances as well as the inner and outer 4:1 resonances (Buta & Combes, 1996).
The manifold theory also offers an explanation for their origin in the form of chaotic orbits
(Romero-Gómez et al., 2015). Some rings however, are presumably produced by close head-
on galactic collisions in disc galaxies (Appleton & Struck-Marcell, 1996). Such is the case of
the famous “Cartwheel galaxy” or the Arp 147 interacting galaxies. The collision produces
CHAPTER 2. GALACTIC STRUCTURES 10
Figure 2.4: Examples of ringed galaxies. Left panel: NGC 1512. The nuclear ring is
being fuelled by gas falling through the bar in structures called “dust lanes”. An inner
ring envelops the stellar bar. Right panel: The Cartwheel galaxy. A collision with its
neighbour near the centre, produced an outward propagating wave where the ring is. The
galaxy is in the intermediate process of re-constructing spiral arms. Credit: Hubble and
JWST material from NASA and ESA.
an outward propagating density wave, that triggers star formation (Higdon, 1996).
In Figure 2.4 I show two examples of ringed galaxies. NGC 1512 is a beautiful example
of how secular processes can re-shape the morphology of the galaxy. This galaxy possess
a nuclear and an inner ring, both interacting with the stellar bar. The former is feeding
from gas funnelled by the bar through structures called “dust lanes”. The bar causes the
gas to shocks and streamline to the central region, feeding the nuclear ring (Athanassoula,
1992).
Chapter 3
Review on Stellar Bars
In this chapter I present a more extensive review on stellar bars. This section intends to
give a general theoretical context for the papers presented in Chapters 4 and 5.
3.1 Bar properties
Stellar bars are found at the centres of approximately 30 per cent of disc galaxies in
the local Universe (Sellwood & Wilkinson, 1993; Menéndez-Delmestre et al., 2007). This
fraction increases up to 70 per cent when accounting for small and weak bars that can
only be seen in the near infrared, hidden behind dust (Eskridge et al., 2000; Marinova &
Jogee, 2007).
Bars exist in a great variety of shapes, sizes and environments. Most of their proper-
ties are strongly tied to the stellar mass and morphology of the host galaxy. For instance,
the bar fraction (the likelihood of hosting a large-scale bar) and bar length are strongly
dependent on the galaxy mass (Nair & Abraham, 2010; Masters et al., 2012; Erwin, 2018).
Early-type spiral galaxies tend to host longer and stronger bars than their late-types coun-
terparts (Méndez-Abreu et al., 2012; Dı́az-Garćıa et al., 2016; Erwin, 2018). These early-
type bars also tend to be more prolate shaped (Dı́az-Garćıa et al., 2016; Méndez-Abreu
et al., 2018) and are better described by a flat density profile (Elmegreen & Elmegreen,
1985; Kim et al., 2015). The difference could be related to the gas content, as gas-rich
galaxies produce weaker bars and much later in time (Masters et al., 2012; Athanassoula
11
CHAPTER 3. REVIEW ON STELLAR BARS 12
et al., 2013).
3.2 Stellar orbits
The bar is mostly supported by regular resonant stellar orbits located inside corotation,
the most important being the 2:1 “x1” orbit family (Contopoulos & Papayannopoulos,
1980). Near the centre, between the Inner Lindblad Resonances (ILR) reside the “x2”
family orbits, which are more round and perpendicular to the bar. Nonetheless, it is now
known that chaotic orbits are also important building blocks for the bar. For example,
by modelling N-body simulation snapshots, various authors have shown that regular and
sticky orbits 1 can eventually transform to chaotic orbits that support the X-shaped/boxy
structure (Voglis et al., 2007; Harsoula & Kalapotharakos, 2009; Chaves-Velasquez et al.,
2017). Moreover, chaotic orbits near the Lagrangian points could be responsible for the
support of the spiral structure according to the “Manifold” theory (Voglis et al., 2006;
Romero-Gómez et al., 2006, 2015).
The multiple resonances produced by bars and spiral arms also have a profound effect on
the disc stellar orbits. Simulations show that near corotation, stars can scatter inwards or
outwards without changing their ellipticity in a process called “radial migration” (Sellwood
& Binney, 2002; Kubryk et al., 2015a). Since stars preserve the circularity of their orbits,
this process does not contribute to the radial heating of the disc, but should affect the
metallicity distribution (Martinez-Medina et al., 2017).
3.3 Bar formation and evolution
How bars are formed and why some galaxies do not develop a bar remains unclear. Early
numerical simulations showed that cold discs are naturally unstable to bar formation
(Miller et al., 1970; Hohl, 1971; Efstathiou et al., 1982; Sellwood & Wilkinson, 1993).
In contrast, dynamically hotter discs with large stellar velocity dispersion can suppress
the bar instability (Athanassoula & Sellwood, 1986; Combes et al., 1990; Athanassoula,
1Sticky orbits are chaotic orbits that wander long times near the borders of the stability region in the
system phase space. Eventually they become completely chaotic.
CHAPTER 3. REVIEW ON STELLAR BARS 13
2003). Bars can also be tidally induced by close flybys and mergers (Berentzen et al.,
2004; Martinez-Valpuesta et al., 2017). It is expected that bars are formed, destroyed and
reformed multiple times in the early universe due the frequency of mergers, flybys and
cold gas accretion (Kraljic et al., 2012; Bi et al., 2022). Once the disc settles and becomes
dynamically cold, bars are expected to remain as long-lived robust structures that evolve
through slow secular processes (Kraljic et al., 2012; Gadotti et al., 2015; Fragkoudi et al.,
2020; de Sá-Freitas et al., 2022). Exploring galaxies from the COSMOS survey Kim et al.
(2021) showed that the relation in size between bar and the disc remains almost constant
over the cosmic time (z / 0.84), suggesting a strong coupling between both structures.
The properties of the dark matter halo are also important for bar formation and evo-
lution. Several N-body simulations have shown that more concentrated halos are able to
form stronger and larger bars (Debattista & Sellwood, 1998, 2000; Athanassoula & Misiri-
otis, 2002). Triaxial halos can induce the formation of bars (Valenzuela et al., 2014), but
such bars are weaker compared to bars in spherical halos (Athanassoula et al., 2013). A
spinning halo can suppress bar formation, by being unable to absorb angular momentum
with the same efficiency (Long et al., 2014; Collier et al., 2018; Rosas-Guevara et al., 2022).
Spinning halos can also change the radial extent of the disc (Grand et al., 2017), affecting
the interpretation of some bar instability criteria (Izquierdo-Villalba et al., 2022).
Shortly after their formation, most bars pass through a violent episode called the“buck-
ling instability” where the bar distorts vertically out of the disc (Combes & Sanders, 1981;
Raha et al., 1991). After the buckling phase, the bar weakens significantly and reassembles
a boxy/peanut/x-shaped bulge when viewed from the side (Martinez-Valpuesta & Shlos-
man, 2004; Erwin & Debattista, 2017). As the galaxy evolves, the bar is constantly braking
against the dark matter halo via dynamical friction, resulting in a monotonic decline in
the bar pattern speed while also increasing in size and strength (Tremaine & Weinberg,
1984b; Debattista & Sellwood, 2000; Martinez-Valpuesta et al., 2006).
CHAPTER 3. REVIEW ON STELLAR BARS 14
3.4 Secular evolution
Different physical processes affect the mass distribution and kinematics in galactic discs.
To mention a few: gas accretion, interaction with neighbouring galaxies, feedback pro-
cesses due to star formation or nuclear activity, and internal dynamical processes. Some
of these processes occur quickly and episodically, while others slowly and steadily. The
evolution due to the latter is called secular evolution (Kormendy & Kennicutt, 2004), and
is responsible for the structural richness observed in the disc galaxies of the local universe.
The collective processes between stars, interstellar medium, dark matter and companion
galaxies give rise to a great diversity of bulges, bars, rings and spiral arms (see for example
NGC 1512 in Figure 2.3).
Bars are the main drivers of the galaxy secular secular evolution (Weinberg, 1985;
Sellwood, 2014; Dı́az-Garćıa et al., 2016). They transfer angular momentum from inner
resonances to those outside of corotation (Lynden-Bell & Kalnajs, 1972; Tremaine & Wein-
berg, 1984b; Athanassoula, 2003). Most of the angular momentum is absorbed by the dark
matter halo (Weinberg, 1985; Debattista & Sellwood, 2000) and in a smaller fraction by
the bulge (Kataria & Das, 2019).
As a result, bars induce substantial gaseous flows to the galaxies centres. Gas piles up
on the leading edges of bars where it shocks (Athanassoula, 1992) and plunges to the x2
disc in the central part (Sormani et al., 2015; Fragkoudi et al., 2016; Sormani et al., 2018).
Observations of barred galaxies show a clear increase in the concentration of molecular gas
(Sakamoto et al., 1999; Jogee et al., 2005) and star formation rate in the central regions
(Ellison et al., 2011; Chown et al., 2019). If the processes is not balanced with the inflow
of cosmological gas, the bar can deplete the gas supply in the disc, causing the so called
“bar quenching” (Masters et al., 2012). This scenario is supported by observations of the
specific star formation rate (Cheung et al., 2013) colour (Gavazzi et al., 2015; Kruk et al.,
2018), gas fraction (Newnham et al., 2020) star formation histories (Fraser-McKelvie et al.,
2020), and statistical properties of the galaxies (Géron et al., 2021).
A theoretical consequence of the large scale gas flows is the flattening of the metallicity
profile. The inward gas flow dilutes the higher metal content of the central region, while the
CHAPTER 3. REVIEW ON STELLAR BARS 15
outwards flows enrich the outer metal poor disc regions (Cavichia et al., 2014; Kubryk et al.,
2015b). However, this behaviour does not appear in most observations (Sánchez-Blázquez
et al., 2014; Pérez-Montero et al., 2016; Sánchez-Menguiano et al., 2016), and has only been
observed in low luminosity (low mass) galaxies (Zurita et al., 2021). Observations from ?
show that the flattening of the metallicity occurs along the bar major axis, suggesting a
localized influence of the bar.
3.5 Bar pattern speed
Most of the secular evolution induced by the bar depends to an extent on its pattern
speed (Ωbar), or its rotational frequency. Most methods developed to measure Ωbar require
some modelling. Some use the gas flow induced by the bar, by matching observations with
hydrodynamical simulations (Sanders & Tubbs, 1980; Hunter et al., 1988; England et al.,
1990; Weiner et al., 2001; Pérez et al., 2004; Zánmar Sánchez et al., 2008; Rautiainen et al.,
2008). An example is shown in Figure 3.1 from Rautiainen et al. (2005). In this paper,
the authors build a mass (potential) model of the galaxy based on a H-band photometry
decomposition. They simulate a disc of collisionless star particles and inelastically colliding
gas particles. Using the bar pattern speed as a free parameter, they look for the model
that best reproduces the morphological features of the galaxy.
Some methods try to estimate the pattern speed by first estimating the location of the
corotation resonance. For example, the “Font-Beckman” method identifies corotation as
the region where the radial velocity of the gas changes from going inwards to outwards
(Font et al., 2011, 2014, 2017). The method requires a 2D rotation model of the gas,
that is subtracted from the original velocity map to obtain a residual non-circular velocity
map. Then, corotation (and other resonances) should appear at the radii with most phase
reversals. Figure 3.2 shows an example of the method applied to the galaxy UGC2709.
The authors suggest that the multiple peaks in the phase reversal histogram correspond
to the interlocking of resonances between different pattern speeds.
Other methods are based on the location and shape of different morphological features
like dark gaps in ringed galaxies (Buta, 2017b; Krishnarao et al., 2022), the offset of
CHAPTER 3. REVIEW ON STELLAR BARS 16
Figure 3.1: Two examples of the modelling done by Rautiainen et al. (2005) The first
two columns shows the deprojected H-band and B-band of the galaxy. The last two
columns show the gas morphology of the best fit model.
Figure 3.2: Font-Beckman method. Left panel: Velocity field of UGC2709 from Hα
emission line. Middle panel: Residual velocity field after subtracting a 2D rotational
velocity field model. Right panel: Histogram of the number of phase reversals in the
residual velocity map, against galactocentric radius. Adapted from Font et al. (2014).
CHAPTER 3. REVIEW ON STELLAR BARS 17
Figure 3.3: The dark gaps method. Left panel: A test particle from Schwarz (1984) of a
strongly barred disc showing dark gaps forming around the L4,5 Lagrangian points. Buta
(2017b) proposed these dark gaps are located at the bar corrotation resonance. Using a
high resolution N-body simulation Krishnarao et al. (2022), proposed the dark gaps are
closer to the 4:1 ultra harmonic resonance. Middle panel: Stellar surface brightness as a
function of the radius of the major and minor axis of a snapshot from the simulation.
Right panel: Difference between the major and minor axis, highlighting the maximal
difference that coincides with the 4:1 ultra-harmonic resonance.
dust lanes (Athanassoula, 1992; Sánchez-Menguiano et al., 2015), changes in the phase of
spirals (Puerari & Dottori, 1997; Aguerri et al., 1998), the position of rings (Buta, 1986;
Rautiainen & Salo, 2000; Patsis et al., 2003).
For example, Figure 3.3 shows two different interpretations of the dark gaps method.
Using simulations in a collisionless disc with test particles, Schwarz (1984) showed that in
the presence of a strong bar, the L4,5 Lagrangian points become unstable. They generate
chaotic motion and depopulation of stars in the area around them. Buta (2017b) proposes
the dark gaps that are often seen in ringed galaxies, can be associated with the Lagrange
points L4,5, and can be used as tracers for the corotation resonance. However, this picture
was challenged by Krishnarao et al. (2022), who used a high-resolution N-body simulation
to follow the location of the dark gaps in a non-ringed galaxy. They found the location of
the dark gaps is actually closer to the 4:1 ultra-harmonic resonance.
Nonetheless, all these methods have strong assumptions behind and are limited by their
modelling techniques. The only direct method for estimating Ωbar is the so called Tremaine
& Weinberg (1984a) (hereafter TW) method. The method uses the surface brightness and
line-of-sight velocity of a tracer that satisfies the continuity equation. Until recently, most
CHAPTER 3. REVIEW ON STELLAR BARS 18
measurement were made with long-slit spectroscopy in early-type spiral galaxies (Kent,
1987; Merrifield & Kuijken, 1995; Debattista et al., 2002; Aguerri et al., 2003; Corsini
et al., 2003; Debattista & Williams, 2004; Corsini et al., 2007). With the advent of the
integral field spectroscopy technique, the TW method has been applied to an increasing
number of galaxies. Starting with the CALIFA survey, Aguerri et al. (2015) measured
Ωbar in 15 strong barred galaxies, finding no trend with the morphological type. Guo et al.
(2019) used a sample of 53 galaxies from the MaNGA survey and studied the effects of the
galaxy position angle and inclination on the determination of Ωbar (see also Debattista,
2003). Cuomo et al. (2019) continued the measurements from Aguerri et al. (2015) within
the CALIFA survey, finding that both weakly and strongly barred galaxies have similar
values of Ωbar. In Garma-Oehmichen et al. (2020), we used a sample of 15 MaNGA galaxies
and 3 CALIFA galaxies to study different sources of uncertainty and identify systematic
errors in the method. Williams et al. (2021) applied the TW-method to 19 galaxies from
the PHANGS-MUSE survey. They found ISM tracers produce erroneous signals in the
TW integrals because of their clumpy distribution.
In Chapter 5 I present the estimates of Ωbar in a new sample of 97 Milky-Way analogue
galaxies from the MaNGA survey. Submitted for publication, Tobias et al. (in prep) have
used the TW method in a sample of 225 MaNGA weakly and strongly barred galaxies.
Figure 3.4 shows a comparison of the distributions of the bar pattern speed, corotation
radius, and R ratio ordered by sample size. It is interesting that as the sample size increases
the median value of R seems to be slowly shifting to slower values of the bar. However,
it is not easy to tell if this shifting is statistically significant. Larger samples are more
representative of the large variety of galaxy and bar types. Nonetheless, some decisions in
the method are strongly author dependent and are subject to biases. For example, there
is no consensus in the astronomical community on how to measure the bar length. In
Garma-Oehmichen et al. (2020) we modelled the bar length using a uniform distribution
between the maximum ellipticity radius RBar,ǫ and the change of position angle radius
RBar,PA. Two years later in Garma-Oehmichen et al. (2022), we changed our modelling to
a log-normal distribution with mean value at RBar,ǫ and 2-sigma upper limit at RBar,PA.
This distribution allows for shorter bars and thus, explains the shift in the R distribution
CHAPTER 3. REVIEW ON STELLAR BARS 19
Figure 3.4: A comparison of the bar pattern speed, corotation radii and R found in
various works. Credit: Tobias et al. in prep.
in our works.
3.6 Rotation rate and the fast bars problem
The rotation rate parameter R = RCR/RBar (the ratio between corotation and the radius
where the bar ends) gives a dynamical interpretation to the bar. Various studies have
shown the x1 orbits that support the bar cannot extend beyond the corotation resonance
(Contopoulos & Papayannopoulos, 1980; Athanassoula, 1980). The faster the bar rotates
in relation to the disc, the closer R goes to 1. This ratio is used to classify bars as slow
(R > 1.4), fast (1 < R < 1.4) and the theoretically un-physical ultra-fast (R < 1).
Most recent measurements using the TW-method have found that most bars are con-
sistent with a fast classification (with a concerning number of ultra-fast bars) (Guo et al.,
2019). This has led to a growing tension with the ΛCDM cosmological framework, where
bars tend to slow down excessively because of dynamical friction with the dark matter
halo (Algorry et al., 2017; Peschken &  Lokas, 2019; Roshan et al., 2021a,b).
Fragkoudi et al. (2021) showed that fast bars are not incompatible with the ΛCDM
framework. However, in order for bars to remain fast, they need to live in more baryon-
dominated galaxies with higher stellar-to-dark matter ratios. Frankel et al. (2022) suggest
cosmological simulations are actually producing shorter bars, but they are not slower than
observations. Also, they note that comparing observations with simulations is not a trivial
CHAPTER 3. REVIEW ON STELLAR BARS 20
task.
Other studies suggest the problems come from incorrect estimates of the bar radius. For
example, Cuomo et al. (2020) noticed that most of the ultrafast bars are found in spiral
galaxies. The overlap of the spiral arms with the bar could produce an overestimation
of the bar radius, hence giving an apparent rotation rate R<1. Using hydrodynamical
simulations, Hilmi et al. (2020) estimated that observations could be overestimating the
bar radius by more than 15%, depending ono the relative orientation between the bar and
the spiral arms.
3.7 The Milky Way bulge/bar
In the last decade, several attempts have been made to estimate the bar pattern speed of
our galaxy. Arguably, the best estimates have been derived using the made-to-measure
method, which matches the density of red clump giants with N-body models (Portail
et al., 2017; Pérez-Villegas et al., 2017; Clarke et al., 2019; Clarke & Gerhard, 2022).
Other attempts have used an adaptation of the TW method (Bovy et al., 2019), kinematic
maps of the disc and bulge stars (Sanders et al., 2019) and N-body simulations of MW-like
galaxies (Tepper-Garcia et al., 2021; Kawata et al., 2021). All these recent measurements
seem to converge to a pattern speed of Ωbar ∼ 30−40 km s−1 kpc−1. This value corresponds
to a corotation radius of RCR ∼ 5 − 6 kpc and R ∼ 1.2, making the MW consistent with
containing a fast bar.
Although these theoretical models have relatively small errors (around ±3 − 5 km s−1
kpc−1 in the bar pattern speed), it is important to notice that the kinematic features
observed in the Solar neighbourhood can be explained in multiple ways. For instance,
different bar resonances can induce similar kinematic features (Kawata et al., 2021), which
overlap with the signatures caused by transient spiral arms (Hunt et al., 2018) or orbiting
satellite dwarf galaxies like Sagittarius (Khanna et al., 2019).
CHAPTER 3. REVIEW ON STELLAR BARS 21
3.8 The bar-spiral connection
Spiral arms also play an important role in the secular evolution of galactic discs. Similar
to stellar bars, spirals also exchange angular momentum near their resonances. There is a
large body of evidence, theoretical and observational, that radial migration is a ubiquitous
process in galaxies with spiral arms, central bar, or both (Minchev & Famaey, 2010; Vera-
Ciro et al., 2014; Hayden et al., 2015; Loebman et al., 2016; Martinez-Medina et al.,
2017; Daniel & Wyse, 2018). Spiral arms also induce dynamical heating in the disc,
increasing the velocity dispersion with time (Holmberg et al., 2009; Roškar et al., 2013;
Martig et al., 2014) and modifying the velocity ellipsoid, an effect that depends on the
nature and morphological properties of the spiral pattern (Jenkins & Binney, 1990; Gerssen
& Shapiro Griffin, 2012; Martinez-Medina et al., 2015).
In most barred galaxies, bars co-evolve with spiral arms, and in some cases, they
could drive the spiral structure (Salo et al., 2010). This is especially apparent in two-
armed grand design galaxies, where most spirals appear to be connected to the ends of
the bar. How these structures interact remains a matter of debate. Kormendy & Norman
(1979) first suggested that strong bars or galaxy companions can lead to the formation of
spiral density waves. There are different theoretical scenarios on how the pattern speeds
of both structures are related (see e.g. Dobbs & Baba, 2014). If bars and spirals are
strongly coupled, the pattern speeds and amplitude of both structures should be strongly
correlated. This strong coupling is expected from the spiral density theory (Lin & Shu,
1964), and the manifold theory where spirals are formed by stars in escaping orbits around
the unstable Lagrangian points at the end of the bar (Romero-Gómez et al., 2006, 2007,
2015; Athanassoula et al., 2009). Observational evidence of a strong coupling comes from
the correlation between the torques and density amplitudes of both structures (Buta et al.,
2003; Block et al., 2004; Buta et al., 2005; Salo et al., 2010; Bittner et al., 2017; Dı́az-Garćıa
et al., 2019).
Another scenario proposes that the pattern speeds are related by a non-linear mech-
anism (Tagger et al., 1987). This is suggested through indirect measurements of the bar
pattern speed with the Font-Beckman method (Font et al., 2014, 2017).
CHAPTER 3. REVIEW ON STELLAR BARS 22
Finally, a third scenario where bar and spirals are decoupled structures, is supported by
observations of the galaxy NGC 1365, where Speights & Rooke (2016) found results con-
sistent with bar and spiral patterns being dynamically distinct features. The observational
evidence to distinguish these scenarios is limited though, and hard to obtain due to the
great uncertainties in estimating the pattern speed of both structures (Garma-Oehmichen
et al., 2020).
The last scenario is supported by the vast majority of N-body simulations, where spirals
are transient features, which can be formed recurrently in time (Sellwood, 2011; Grand
et al., 2012; D’Onghia et al., 2013; Mata-Chávez et al., 2019). Numerical simulations
with steady bar potentials have found that the gas particles settle in steady-state trailing
spirals (Sanders & Huntley, 1976; Athanassoula, 1992; Wada, 1994; Rodriguez-Fernandez
& Combes, 2008). Collisionless star particles can also produce prominent spirals and rings
near the Outer Lindblad Resonance (Schwarz, 1981; Bagley et al., 2009).
Chapter 4
The bar and the spiral structure
In this chapter I present the first original research paper of the thesis.
There are many ways for studying stellar bars: observations, N-body simulations or
dynamical models. While observations provide the ground truth that models aim to re-
produce, they only provide a single snapshot of the complex process of galactic evolution.
N-body simulations are an ideal playground for a detailed dynamical analysis. However
they are computationally expensive, and depend critically on the modelled physics. Usu-
ally this kind of studies focus on one or a few runs, tailored to study some aspect of galactic
evolution.
Numerical simulations in steady potential are an ideal tool for the statistical analy-
sis. They are computationally cheap, as they ignore the particle-to-particle interactions.
However, this simplicity is also their main disadvantage. Ignoring self-gravity means they
should not be used to study complex galactic evolution processes where changing poten-
tials might be important. Instead, they are a great tool for studying dynamical processes
near resonances, responses to small density perturbations and orbital studies.
In this work, we use a set of numerical simulations to study how the bar properties
can affect in a first order the spiral arms. To achieve this, we create a set of simulations,
where we change the length, ellipticity, mass and rotation rate of the simulated bar.
23
MNRAS 502, 4708–4722 (2021) doi:10.1093/mnras/stab333
Advance Access publication 2021 February 9
How the bar properties affect the induced spiral structure
L. Garma-Oehmichen,1‹ L. Martinez-Medina ,1 H. Hernández-Toledo1 and I. Puerari2
1Instituto de Astronomı́a, Universidad Nacional Autónoma de México, Apartado Postal 70-264, 04510 CDMX, México
2Instituto Nacional de Astrofı́sica, Optica y Electrónica, Apdo. Postal 51 y 216, 72000 Puebla, México
Accepted 2021 January 31. Received 2021 January 29; in original form 2021 January 16
ABSTRACT
Stellar bars and spiral arms coexist and co-evolve in most disc galaxies in the local Universe. However, the physical nature of this
interaction remains a matter of debate. In this work, we present a set of numerical simulations based on isolated galactic models
aimed to explore how the bar properties affect the induced spiral structure. We cover a large combination of bar properties,
including the bar length, axial ratio, mass, and rotation rate. We use three galactic models describing galaxies with rising, flat,
and declining rotation curves. We found that the pitch angle best correlates with the bar pattern speed and the spiral amplitude
with the bar quadrupole moment. Our results suggest that galaxies with declining rotation curves are the most efficient forming
grand design spiral structure, evidenced by spirals with larger amplitude and pitch angle. We also test the effects of the velocity
ellipsoid in a subset of simulations. We found that as we increase the radial anisotropy, spirals increase their pitch angle but
become less coherent with smaller amplitude.
Key words: galaxies: disc – galaxies: evolution – galaxies: kinematics and dynamics – galaxies: structure.
1 I N T RO D U C T I O N
Stellar bars inhabit a large fraction of disc galaxies in the local
Universe, in a great variety of shapes, sizes, and environments. They
promote the galaxy secular evolution by exchanging mass, energy,
and angular momentum with stars and gas across the disc (Weinberg
1985; Kormendy & Kennicutt 2004; Sellwood 2014; Dı́az-Garcı́a,
Salo & Laurikainen 2016), transporting angular momentum from
the inner bar resonances to those outside co-rotation (Lynden-Bell &
Kalnajs 1972; Tremaine & Weinberg 1984; Athanassoula 2003) and
redistributing stars in the disc by radial heating and radial migration
(Monari et al. 2016; Martinez-Medina et al. 2017). Bars can also
induce gas inflow to the central regions via shock waves, forming
gaseous structures like dust-lanes and rings (Athanassoula 1992;
Hernquist & Mihos 1995; Martinet & Friedli 1997; Kim, Seo &
Kim 2012; Sormani et al. 2018; Seo et al. 2019). Some moving
groups in the solar neighbourhood have their dynamical origin under
the influence of the Galactic bar (Pérez-Villegas et al. 2017) and
through its induced resonances shape the stellar velocities in the
solar neighbourhood (Fux 2001).
The importance of these phenomena depends on the morphological
and dynamical characteristics of the bar. In late-type galaxies, bars
tend to be smaller in relation to their discs (Méndez-Abreu et al. 2012;
Dı́az-Garcı́a et al. 2016; Erwin 2018), more oblate shaped (Méndez-
Abreu et al. 2012; Dı́az-Garcı́a et al. 2016), and prone to having
an exponential density profile (Elmegreen & Elmegreen 1985; Kim
et al. 2015). In contrast, bars in early-type galaxies tend to be larger,
prolate shaped, and with a flat density profile. Such co-relations
with the galaxy type have not been observed in measurements of the
bar pattern speed (hereafter Bar), neither with the dimensionless
⋆ E-mail: lgarma@astro.unam.mx
rotation rate (hereafter parameter R) defined as the ratio between
the bar length (hereafter a) and the co-rotation resonance (hereafter
RCR; Aguerri et al. 2015; Cuomo et al. 2019; Guo et al. 2019; Garma-
Oehmichen et al. 2020). Nonetheless, it has been observed that the
Bar seems to correlate with the galaxy luminosity and total stellar
mass (Cuomo et al. 2020; Garma-Oehmichen et al. 2020).
Spiral arms also play an important role in the secular evolution
of galactic discs. By exchanging angular momentum, spirals churn
stars, and gas in the disc (Sellwood & Binney 2002). There is a large
body of evidence, theoretical and observational, that radial migration
is a ubiquitous process in galaxies with spiral arms, central bar, or
both (Minchev & Famaey 2010; Vera-Ciro et al. 2014; Hayden et al.
2015; Loebman et al. 2016; Martinez-Medina et al. 2017; Daniel &
Wyse 2018). Spiral arms also induce dynamical heating in the disc,
increasing the velocity dispersion with time (Holmberg, Nordström
& Andersen 2009; Roškar, Debattista & Loebman 2013; Martig,
Minchev & Flynn 2014), and modifying the velocity ellipsoid, an
effect that depends on the nature and morphological properties of the
spiral pattern (Jenkins & Binney 1990; Gerssen & Shapiro Griffin
2012; Martinez-Medina et al. 2015).
In most barred galaxies, bars co-evolve with spiral arms, and in
some cases, they could drive the spiral structure (Salo et al. 2010).
This is especially apparent in two-armed grand design galaxies,
where most spirals appear to be connected to the ends of the
bar. How these structures interact remains a matter of debate.
Kormendy & Norman (1979) first suggested that strong bars or
galaxy companions can lead to the formation of spiral density waves.
There are different theoretical scenarios on how the pattern speed of
both structures is related (see e.g. Dobbs & Baba 2014). If bars
and spirals are strongly coupled, the pattern speed and amplitude of
both structures should be strongly correlated. This strong coupling
is expected from the spiral density theory (Lin & Shu 1964), and
the manifold theory where spirals are formed by stars in escaping
C© 2021 The Author(s)
Published by Oxford University Press on behalf of Royal Astronomical Society
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How the bar affects the spiral structure 4709
orbits around the unstable Lagrangian points at the end of the bar
(Romero-Gómez et al. 2006, 2007, 2015; Athanassoula, Romero-
Gómez & Masdemont 2009). Observational evidence of a strong
coupling comes from the correlation between the torques and density
amplitudes of both structures (Buta, Block & Knapen 2003; Block
et al. 2004; Buta et al. 2005; Salo et al. 2010; Bittner et al. 2017;
Dı́az-Garcı́a et al. 2019). Another scenario proposes that the pattern
speeds are related by a non-linear mechanism (Tagger et al. 1987),
as suggested through indirect measurements of the bar pattern
speed (Font et al. 2017). Finally, a third scenario where bar and
spirals are decoupled structures, is supported by observations of the
galaxy NGC 1365, where Speights & Rooke (2016) found results
consistent with bar and spiral patterns being dynamically distinct
features. The observational evidence to distinguish these scenarios
is limited though, and hard to obtain due to the great uncertainties in
estimating the pattern speed of both structures (Garma-Oehmichen
et al. 2020).
The vast majority of N-body simulations show that spirals are
transient or multi-arm features, that can be formed recurrently in
time (Sellwood 2011; Grand, Kawata & Cropper 2012; D’Onghia,
Vogelsberger & Hernquist 2013; Mata-Chávez et al. 2019). Numeri-
cal simulations with steady bar potentials have found that the gas par-
ticles settle in steady-state trailing spirals (Sanders & Huntley 1976;
Athanassoula 1992; Wada 1994; Rodriguez-Fernandez & Combes
2008). Collisionless star particles can also produce prominent spirals
and rings near the Outer Lindblad Resonance (Schwarz 1981; Bagley,
Minchev & Quillen 2009).
In this paper, we use a set of numerical simulations based on galac-
tic potential models with one million test particles to explore how
the bar properties affect the response spiral arms. Our simulations
cover a large bar parameter space, tailored to explore a large number
of combinations of bar properties (size, shape, mass, and pattern
speed). We also test the effect of discs with different shapes of
rotation curves and velocity ellipsoids. To characterize the response
spiral arms, we identify the spiral particles using the density-based
clustering algorithm DBSCAN. This let us obtain clean measurements
of the spiral amplitude and pitch angle.
The paper is organized as follows. In Section 2, we describe the
galactic potential models, and the space of parameters to explore. In
Section 3, we describe how we use the algorithm DBSCAN to detect
the spiral overdensities, and how we estimate the properties of the
induced spiral arms. In Section 4, we show how the bar parameters
affect the spiral arms properties. In Section 5, we explore the effects
of different galactic models and the rotation curve shape. In Section 6,
we present the effects of the velocity ellipsoid. In Section 7, we
discuss the relative importance of the bar and disc properties in
predicting the spiral properties. Finally, in Section 8, we discuss our
results and present our conclusions.
2 SI M U L AT I O N S
2.1 Galactic models
The mass distribution of a galaxy could play a significant role in the
properties of the spiral arms. Using a sample of 94 galaxies, Biviano
et al. (1991) observed that galaxies with steeper, rising rotation curves
tend to host flocculent spirals, while galaxies with flat rotation curves
have grand design spirals. Seigar et al. (2005, 2006) found that the
pitch angle strongly correlates with the rate of shear in the disc
defined as
S =
1
2
(
1 −
R
V
dV
dR
)
. (1)
They suggested that spirals in rising rotation curves galaxies have
greater pitch angles compared to galaxies with flat or declining
rotation curves. However, this relationship has been questioned by
Kendall, Clarke & Kennicutt (2015) and Yu et al. (2018), who failed
to observe such correlation (see also Dı́az-Garcı́a et al. 2019).
To test the effects of different rotation curves, we used three
galactic potential models with similar enclosed mass at ∼20 kpc,
but different mass distributions. We will refer to these models as
flat, rising, and declining, to figure out the shape of their rotation
curve. The flat model is the well-known Allen & Santillan (1991)
potential comprising a Plummer bulge, a Miyamoto Nagai disc, and
a spherical halo that reproduces the nearly flat Milky Way rotation
curve. The density profiles of the bulge, disc, and halo (referred with
the sub-indexes B, D, and H, respectively) are
ρB(r) =
3b2
BMB
4π
(
r2 + b2
B
)5/2
(2)
ρD(R, z) =
b2
DMD
4π
(
R2aD + 3
(
z2 + b2
D
)1/2
)
(
R2 +
(
aD +
(
z2 + b2
D
)1/2
)2
)5/2
×
(
aD +
(
z2 + b2
D
)1/2
)2
(
z2 + b2
D
)3/2
(3)
ρH(r) =
MH
4πaHr2
(
r
aH
)1.02 (
2.02 + (r/aH)1.02
(1 + (r/aH)1.02)2
)
, (4)
where r is the spherical radius coordinate, (R, z) are cylindrical
coordinates, MB, MD, and MH are the masses, and bB, aD, bD, aH are
the characteristic scales of each component.
For the rising model, we reduced the central mass concentration
by extending the bulge scale radius to match the spherical halo and
doubled the disc scale radius. In the declining model, we reduce the
disc scale radius by a factor of 3/4. All galactic models share the
same mass for each component. We show the parameters used for
the galactic models in Table 1. Fig. 1 shows the rotation curves of
the three axisymmetric models.
2.2 Bar model
To simulate the bar potential, we use a Ferrers ellipsoid of index n =
2 (Binney & Tremaine 1987). The density distribution in Cartesian
coordinates is
ρ(x, y, z) =
{
105
32πabc
MBar(1 − m2)2, m < 1
0, m > 1,
(5)
where MBar is the bar mass and m2 = (x/a)2 + (y/b)2 + (z/c)2.
The parameters a, b, c are the semi-axes length in the x, y, and z
directions, respectively, with a > b > =c. The forces produced by
this distribution are described in Pfenniger (1984).
We introduce the bar adiabatically as a smooth function of time
by transferring mass from the bulge to the bar. We used the fifth-
degree polynomial described in equation (4) of Dehnen (2000), which
guarantees a smooth transition to the barred state. We set the time
growth of 500 Myr, similar to Romero-Gómez et al. (2015).
Orbital and dynamical studies have shown that self-consistent bars
cannot extend beyond their co-rotation radius RCR, where the bar
and the disc rotate at the same angular speed. Beyond RCR stellar
orbits change their orientation, becoming perpendicular to the bar
(Athanassoula 1980). Also, the density of resonances increases near
RCR leading to chaotic behaviour in the phase space (Contopoulos
MNRAS 502, 4708–4722 (2021)
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4710 L. Garma-Oehmichen et al.
Table 1. Parameters of the galactic models.
Galactic model MB bB MD aD bD MH aH
(1010 M⊙) (kpc) (1010 M⊙) (kpc) (kpc) (1010 M⊙) (kpc)
(1) (2) (3) (4) (5) (6) (7) (8)
Rising 1.406 12.0 8.561 10.636 0.25 10.709 12.0
Flat (Allen–Santillan) 1.406 0.387 8.561 5.318 0.25 10.709 12.0
Declining 1.406 0.387 8.561 3.988 0.25 10.709 24.0
Note. Col. (1): Galactic model. Col. (2): Bulge mass. Col. (3): Bulge scale length. Col. (4): Disc mass. Col. (5): Disc
radial scale length. Col. (6): Disc vertical scale length. Col. (7): Halo mass. Col. (8): Halo scale length.
Figure 1. Rotation curves of the three axisymmetric galactic models used in
this work.
Table 2. Bar parameter space.
Parameter Values
(1) (2)
a (kpc) 2.659, 5.318, 7.977
b/a 0.3, 0.6
R 0.9, 1.2, 1.5
MBar (1010 M⊙) 0.703, 1.406
c/a 0.3
Note. Col. (1): Bar parameters. From top to bottom:
Length, axial ratio, rotation rate, mass, and vertical axial
ratio. Col. (2): Values explored.
& Papayannopoulos 1980). Because of its physical importance, the
dimensionless parameter R = RCR/a is used to parametrize the bar
rotation rate. Bars are classified kinematically as ‘slow’ if R >1.4,
and ‘fast’ if R<1.4. The theoretically impossible case of R<1 is
referred as ‘ultrafast’.
In this work, we explore the following bar parameter space: (i)
Three bar radius corresponding to a = 0.5, 1.0, 1.5 times the disc
radial scale radius of the Allen–Santillan model. (ii) Two axial ratios
between the minor and major axis b/a = 0.3, 0.6, corresponding
to prolate and oblate bars, respectively. (iii) Two bar masses that
correspond to a complete and a half-mass transfer between the bulge
and the bar. (iv) Three values for the dimensionless parameter R,
exploring the scenario of slow, fast, and ultrafast bars. We do not
explore the effects of the vertical axial ratio, which was set to c/a
= 0.3 for all our simulations. Table 2 shows a summary of the bar
parameter space. Our simulations explore 36 combinations for the bar
model, and three galactic potential models, yielding a total number
of 108 simulations. All our simulations use 1 million test particles.
Given the bar length a and the parameter R, the pattern speed Bar
is estimated by evaluating axisymmetric angular velocity curve at
the corotation resonance RCR = R × a. However, as the bar is being
introduced, the mass distribution and the location of the resonances
changes. This is especially important in the rising model, where
the bar formation increases the central mass concentration. In those
cases, we use the unstable Lagrange point L1 as a proxy for the
location of RCR (Binney & Tremaine 2008). If the relative difference
between L1 and the axisymmetric RCR is greater than 10 per cent, we
correct the value of Bar until both distances match. From hereafter,
when we refer to R we are using the L1/a ratio, accounting for the
bar mass redistribution.
2.3 Initial conditions and orbit integration
A natural space to study spirals is the (θ , ln R) plane, where θ is
the azimuthal angle and R the cylindrical radius. In this space, a
logarithmic spiral can be described with the straight line equation
(Lin & Shu 1964):
R(θ ) = R0 × eθ tan α, (6)
where R = R0 at θ = 0 and α is the pitch angle.
However, a disc populated with test particles uniformly distributed
in the (θ , ln R) space would have few particles in the inner disc.
Instead, we choose to set the initial spatial distribution to be uniform
in the (θ , log10(R + 1)) space, which keeps the logarithmic spacing
in the radius, populates the inner disc and highlights the spiral
overdensities over a uniform background. The resulting density
distribution resembles that of a single exponential disc. For the
vertical dimension, we choose a usual sech2(z/z0) law distribution
(van der Kruit & Searle 1981) with scale height z0 = 0.25 kpc.
We generated the initial velocities using the Hernquist (1993)
moments method. This procedure constrains the shape of the velocity
ellipsoid, by assuming the radial and vertical dispersion are propor-
tional and constant throughout the disc (σ 2
R ∝ σ 2
z ). For simplicity,
we set the velocity dispersion ratio to σ z/σ R = 1. In Section 6, we
explore the effect of other normalization constants with greater radial
velocity dispersion for a subset of simulations.
Before introducing the bar, we first integrate the test particles
orbits in the axisymmetric potential for 3 Gyr so they relax and
reach the statistical equilibrium (Romero-Gómez et al. 2015). We
performed the integration with a time adaptive fifth-order Runge–
Kutta integrator using the FORTRAN subroutines ODEINT and RKQS
(Press et al. 2007). We followed the Jacobi energy and vertical
velocity distribution for a subset of test particles to confirm the
stability.
Figs 2 and 3 show snapshots of a simulation using the flat
model and bar parameters a = 5.318 kpc, b/a = 0.3, R = 1.2,
MBar = 1.406 × 1010 M⊙ in the (x, y) and (θ , log10(R + 1)) planes,
MNRAS 502, 4708–4722 (2021)
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250 
200 
~ 150 
~ 
5100 
~ 
............ 
Flat (Allen-Santillan) 
Rising 
Declining 
01---__ -_-_-_-_-_-_~ 
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 
R [kpc] 
How the bar affects the spiral structure 4711
Figure 2. Snapshots of a simulation in the flat model and bar properties a = 5.318 kpc, b/a = 0.3, R = 1.2, MBar = 1.406 × 1010 M⊙. We masked particles
inside corotation RCR to make spirals more visible. The OLR is shown with a white segmented line. The bar is shown with a black ellipse. The bar is introduced
adiabatically by transferring mass from the bulge component up to 500 Myr. The spiral amplitude reaches a maximum at 1000 Myr, and decreases afterwards.
Figure 3. Same as Fig. 2, but in the (θ , log10(R + 1)) plane. The white segmented line highlights the location of the OLR. Notice that in the first snapshot, the
particles are uniformly distributed.
respectively. Both figures are in the bar rotating reference frame. The
snapshots are separated in time steps of 250 Myr. The first snapshot
corresponds to the relaxed disc distribution. The bar stops growing in
mass at 500 Myr. We show the corotation resonance with a white solid
line. The Inner and Outer Lindblad Resonances (ILR and OLR from
hereafter) are shown with white segmented lines. These resonances
were estimated using the axisymmetric angular velocity curve and
the epicyclic frequency curve.
3 ESTIMATING THE PROPERTIES OF TH E
SPIRAL ARMS
3.1 Detecting overdensities using DBSCAN
To detect the spiral overdensities, we use the density-based clustering
algorithm DBSCAN (Ester et al. 1996). The algorithm works by
classifying each point in a given space as ‘core’, ‘member’, or ‘noise’,
MNRAS 502, 4708–4722 (2021)
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15 
10 
U 
a. 
O ~ 
>-
-5 
-10 
-15 
15 
10 
U 
a. 
O ~ 
>-
-5 
-10 
-15 
1.20 
1.15 
;::t 
1.10 
+ 
~ 1.05 
i 1.00 
0.95 
0.90 
1.20 
1.15 
;::t 
1.10 
+ 
~ 1.05 
i 1.00 
0.95 
0.90 
O 
-10 O 10 
X [kpc] 
2 4 
(} [rad] 
-10 
6 O 
O 
X [kpc] 
4 
(} [rad] 
10 -10 
6 O 
O 
X [kpc] 
4 
(} [rad] 
10 -10 O 
X [kpc] 
6 O 2 4 
(} [rad] 
10 
6 
14 
12 
10 
N 
'C 
Q.. 
O 
O 
8 
........ 
<: 
8 <: 
6 
4 
4712 L. Garma-Oehmichen et al.
Figure 4. Detecting spiral overdensities with DBSCAN. We use a snapshot from the simulation with bar parameters a = 5.318 kpc, b/a = 0.3, Mbar =
1.406 × 1010 M⊙, R = 1.2, in the flat galactic model. First panel: We select particles between RCR and the OLR + 4 kpc where the spiral arms are formed. We
rescale the data so distances in radius and angle are the same. We mask the particles in the range [OLR −2ǫ, OLR] shown with two red horizontal lines. At the
peak spiral density, we draw the two squares of side 2ǫ. Second panel: Distribution of the masked particles versus the scaled angle. We fit a cosine function to
get the amplitude and peaks of the spiral overdensity. We count the number of test particles around the peaks ±ǫ, shown with red vertical lines. We determine
mincnt by multiplying this count by π/8. Third panel: Using the estimated parameters (ǫ, mincnt), DBSCAN identifies three clusters in the data. The points
classified as ‘core’ are shown with slightly bigger dots, so they appear as a solid coloured area. ‘Member’ points surround the core area in an ǫ size envelope.
Notice this plot is in the (θ , ln R) space. We use a piecewise linear function to fit the identified clusters shown with two white lines. We measure the pitch angle
using the greatest slope in the piecewise fit. Fourth panel: Same as the third panel, but in the (x,y) plane.
Figure 5. Spiral arm amplitude as a function of time, for the family of
simulations with bar parameters a = 5.318 kpc and R = 1.2 in the flat model.
The solid black line shows the bar formation (500 Myr), the segmented and
dotted lines show 1 and 2 dynamical times at the OLR after the bar formation,
respectively.
based on two parameters: (i) ǫ which specifies the distance between
two points to be considered ‘neighbours’, and (ii) mincnt which
specifies the minimum number of neighbours a point must have to be
classified as a ‘core’ of a cluster. All points in the ǫ-neighbourhood
of a core, that do not satisfy the mincnt condition are classified as
‘members’ of the cluster. Finally, all points that do not inhabit a
cluster are classified as ‘noise’.
DBSCAN is especially useful to find arbitrarily shaped structures
and does not require knowing a priori the number of clusters in the
data. The use of the algorithm has been increasing in astronomy
in recent years. To mention a few applications: identifying lensed
features in residual images (Paraficz et al. 2016), classifying eclipsing
binaries light curves (Kochoska et al. 2017), detecting low-surface
brightness galaxies in the Virgo cluster (Prole et al. 2018), and finding
open clusters in the Gaia data (Castro-Ginard et al. 2018, 2020).
Choosing meaningful parameters (ǫ, mincnt) is challenging when
the size and density of the clusters are unknown. The parameter ǫ is
related to the spatial resolution of the clusters and can be determined
from their expected size. The spiral arms produced in our simulations
typically have widths of ∼0.6 kpc, so we choose to fix the value of ǫ to
0.3 kpc. The parameter mincnt is related to the expected density, or in
this case, the spiral amplitude. However, the amplitude changes with
time and between simulations. We performed the following steps to
determine an appropriate value of mincnt at any given snapshot. We
show an example of the procedure in Fig. 4:
(i) We mask all particles between RCR and OLR + 4 kpc, where
we expect the spiral arms to be located. We rescale the data, so the
distances in log (R + 1) and θ are the same in an Euclidean metric.
We also rescale the value of ǫ. The first panel of Fig. 4 shows a
snapshot of a simulation where the spiral arms have already formed,
and the data have been rescaled.
(ii) A second mask is used to select particles that lie in the radial
range R = [OLR − 2ǫ, OLR]. In the first panel of Fig. 4, the two red
horizontal lines delimit this range.
(iii) All our simulations produced two symmetrical spirals arms.
We bin the masked particles in 64 bins and fit a simple cosine
wavefunction f(θ ) = Acos (2θ + φ) + C. This is equivalent to the m
= 2 Fourier component of the azimuthal light profile near the OLR.
We use the normalized amplitude of the fit, A/C, as a proxy of the
spiral amplitude. In the second panel of Fig. 4, we show the masked
particles distribution and the cosine fit.
(iv) We count the number of particles around the peaks ±ǫ,
illustrated with two red segmented lines in the second panel of Fig. 4.
The resulting count is the expected number of particles in two squares
of size 2ǫ at the peak of the spirals. We show these squares in the
first panel of Fig. 4.
(v) We multiply the resulting count by π /8 which is the ratio
between the area of a circle with radius ǫ and two squares of side 2ǫ.
This results in our estimation for mincnt.
Once we have estimated (ǫ, mincnt), DBSCAN can detect the spiral
overdensities as shown in the third and fourth panels of Fig. 4. In this
example, the algorithm detected three clusters, shown in different
colours. We plot the core points with slightly bigger dots, so they
appear as a solid coloured region. Member points surround the core
with an envelope of radius ǫ. Notice that the pink cluster is just a
continuation of the blue spiral, but displaced by radians. In such
cases, we join the separated clusters by manually adding the 2π
rotation.
MNRAS 502, 4708–4722 (2021)
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1
1.5 
1.0 
.... 
+ 0.5 
~ 
i 0.0 
al 
~ -0.5 
Vl 
-1.0 
-1.5 
0.6 
0.5 
Q) 
-g 0.4 
:!: 
c.. 
E 0.3 
oC( 
~ 0.2 
'c.. 
Ul 
0.1 
0.0 
-1 O 
Scaled angle 8 
o 500 
350 
300 
250 
200 
150 
100 
50 
-1 .5 - 1.0 - 0.5 0 .0 0 .5 1.0 1.5 
Scaled Angle 
:-- Bar Formation 
:- - 1 Dynamical Time 
2 Dynamical Time 
: . bla=0.3 M.,,=0.703 
1000 1500 2000 
Time [Myr] 
• bla=0.3 M • .,=1.406 
• bla=0.6 M.,,=0.703 
• bl.=0.6 M.,,=1.406 
2500 3000 
2.6 
2.5 
_2.4 
íJ 
~2 . 3 
ex: 
~2.2 
2 .1 
2.0 
1.9 
4 
8 [Radl 
g: 
~ 
).. 
10 
-5 
-10 
-10 -5 O 10 
x [kpcl 
How the bar affects the spiral structure 4713
Figure 6. Spiral properties versus bar parameters in the rising galactic model. Top row: Relations with the average pitch angle. Bottom row: Relations with
the spiral amplitude. The grey segmented lines join simulations that share the same bar parameters, except for the one that is being plotted. The Spearman
correlation coefficient and the corresponding p-value are shown at the top of each panel.
Figure 7. Same as Fig. 6, but in the flat model.
Not every simulation produces spirals as clearly as in the example
shown in Fig. 4. Some models produce very weak spiral arms that
are almost indistinguishable from the background. In those cases, we
reduce the value of mincnt manually until the algorithm can detect
the underlying spiral structure.
3.2 Estimating the pitch angle
Although we identify the spiral overdensities in the (θ , log10R + 1)
plane, the measurement of the pitch angle is done in the (θ , ln R)
space, as described by equation (6) (i.e. arctan of the slope of the
overdensities). In some cases, the spirals wind up at the outer radius,
MNRAS 502, 4708–4722 (2021)
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30 
E: 25 .. 
D 
:1 20 
~ 
~ 15 
• f'lO 
~ 
.. 5 
0.7 
o .• 
• B 0.5 
:a ,¡; O., 
f 0.3 
'c. 
Ul 0.2 
O., 
rs= -0.56, P = 4e-Q4 
~ \ .. _ .. _._ .• 
~ ----------- . 
~ .. ; : ~ -
2.66 5.32 7.98 
Bar length [kpc] 
rs= 0.16, p = 4e-Ol 
rs= -0.16, P = 4e-Ql rs= 0.18, P = 3e-OI rs= -0.64, p = 4e-Q5 
/\ 
~ ~ .~. ~ 
0.703 1.406 0.9 1.2 1.5 
Axis ratio bla Bar Mass [lCIOM e ] parameter'R. 
rs= -0.23, p = 2e-OI rs= 0.45. p = 7e-03 rs= -0.39. P = 2e-02 
.-_.-------..--"'" 
2.66 5.32 7.98 0.3 0.6 0.703 1.406 0.9 1.2 1.5 
Bar length [kpcl Axis ratio bla 
,. rs= -0.65. P = 2e-05 rs= -0.13, P = 4e-Ol 
E 14 :: =~=~~=~=~== : 
~12 
~ --------
~ ': ";: -==--~~ : 
1 : E.'-· :~--- -. -1 
.- --_:--:-------=---=--- ~ 
2~-- __ --_'------_-----_ 
• ~ 
0.8 
il 
~O.6 
,¡; 
i!! 0.4 -.. 
'" 0.2 
2.66 5.32 7.98 
Bar length [kpe] 
rs= 0.64, p = 3e-OS 
O., O .• 
Axis ratio bla 
rs= -0.24, p = 2e-Ql 
::: ~_._------- . 
......--.'=--­.. ,-----._- ~ .. -.:::::::::.~ 
_ .~ -:::: ::-._ ~ 
~ ~~~ ~ 
Bar Mass [lOIOM a ] ParameterR 
's"" 0.26, p"" le-Ol 's"" -0.65, p"" le-05 
0.703 1.406 0.9 1.2 1.5 
Bar Mass [lOloM a ] Parameter:R. 
's"" 0.55, P = 6e-04 's= -0.40, P = le-02 
------ "",---
~~~ ~ ~ --- ~ . 
~ ~ .. ~ ! .. ~ ~~ 
2.66 5.32 7.98 0.3 0.6 0.703 1.406 0.9 1.2 1.5 
Bar length [lepe] Axis ratio bla Bar Mass [lOIOM a ] Parameter:R. 
90 
80 
40 
30 
20 
.0 
80 
70 
.01 
50J 
40 
30 
20 
4714 L. Garma-Oehmichen et al.
Figure 8. Same as Figs 7 and 6, but in the declining model.
Table 3. Spearman correlation coefficient and statistical significance with the spiral properties.
Pitch angle Amplitude
Parameter Rising Flat Declining Rising Flat Declining
(1) (2) (3) (4) (5) (6) (7)
a − 0.56 (4 × 10−4) − 0.65 (2 × 10−5) − 0.60 (1 × 10−4) 0.16 (4 × 10−1) 0.64 (3 × 10−5) 0.85 (3 × 10−11)
b/a − 0.16 (4 × 10−1) − 0.13 (4 × 10−1) − 0.17 (3 × 10−1) − 0.23 (2 × 10−1) − 0.24 (2 × 10−1) − 0.20 (2 × 10−1)
MBar 0.18 (3 × 10−1) 0.26 (1 × 10−1) 0.22 (2 × 10−1) 0.45 (7 × 10−3) 0.55 (6 × 10−4) 0.36 (3 × 10−2)
R − 0.64 (4 × 10−5) − 0.65 (1 × 10−5) − 0.67 (6 × 10−6) − 0.39 (2 × 10−2) − 0.40 (1 × 10−2) − 0.14 (4 × 10−1)
Bar 0.83 (1 × 10−9) 0.88 (1 × 10−12) 0.84 (1 × 10−10) 0.02 (9 × 10−1) 0.39 (2 × 10−2) 0.71 (1 × 10−6)
QBar − 0.42 (1 × 10−2) − 0.48 (3 × 10−3) − 0.45 (6 × 10−3) 0.36 (4 × 10−2) 0.82 (1 × 10−9) 0.95 (2 × 10−19)
Shear − 0.44 (1 × 10−2) − 0.82 (8 × 10−10) − 0.75 (1 × 10−7) 0.00 (1 × 100) 0.44 (7 × 10−3) 0.73 (4 × 10−7)
Note. Col. (1): Parameter. Col. (2): Pitch angle in the rising model. Col. (3): Pitch angle in the flat model. Col. (4): Pitch angle in the declining model.
Col. (5): Spiral amplitude in the rising model. Col. (6): Spiral amplitude in the flat model Col. (7): Spiral amplitude in the declining model.
forming a ring-like structure just outside of the OLR. These kind
of rings are common in numerous simulations with test particles
(Schwarz 1981, 1984; Bagley et al. 2009) and barred galaxies (Buta
& Crocker 1991; Buta & Combes 1996; Buta 2017). The snapshot
shown in Fig. 4 is an example of such behaviour. To separate the
rings from the spiral, we fit a piecewise linear function and use the
slope of the spiral segment to estimate the pitch angle. The fitted
piecewise function is shown with white lines in the third and fourth
panels of Fig. 4. Notice how the change in pitch angle cannot be
distinguished ‘by eye’ in the (x, y) plane. We measure the pitch angle
in both spirals using the whole cluster and only core points. From
hereafter, when we refer to the pitch angle we are using the average
from these measurements.
3.3 Estimating the spiral amplitude
We use the normalized amplitude of the fitted cosine wavefunction
(second panel in Fig. 4) as a proxy of the spiral amplitude. In
general, all our simulations showed the same trend: as the bar is
being introduced, the spiral amplitude increases as a function of time
until it reaches a maximum and slowly decreases. In Fig. 5, we show
the spiral amplitude as a function of time of four simulations in the
flat galaxy model that share the same bar length a and parameter R
(and thus, the same Bar), but vary in mass and axial ratio. In most
simulations, the amplitude peaks between 1 or 2 dynamical times at
the OLR after the bar has formed. After 3 to 4 dynamical times the
spiral structure becomes more diffuse. Simulations that share a and
R, usually have spirals that peak in the same snapshot, except for
some simulations in the rising model, that are more sensitive to the
bar potential.
In the next section, we discuss which bar parameters produce
stronger spirals, but from this figure is clear that the mass and shape
of the bar are strongly co-related to the spiral amplitude.
4 R E L AT I O N S B E T W E E N T H E BA R
PARAMETERS AND THE SPIRAL PROPERTIES
The spiral arms produced in our simulations are the collective result
of the bar perturbing the stellar epicyclic orbits. Thus, our spiral arms
are produced by the density-wave mechanism and do not account for
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E: 20.0 
.!! 17.5 
m 
.:¡ 15.0 
~ 12.5 
;;: 
Qj 10.0 
m 
~ 7.5 
~ 5.0 
rs= -0.60, p = le-04 rs= -0.17, P = 3e-Ol 
:: ~=.~==-:=:.::...---=-=- -: 
rs= 0.22, P = 2e-Ol 
: ==~=:::::::=-=:::::::: : 
2.5 ~ _______ ~ ~ _______ ~ 
• ~ 
0.8 
'ª 0.6 
o. 
~ 
~ 0.4 
.c. 
'" 0.2 
2.66 5.32 7.98 o., 0.6 
Bar length [kpc] 
rs= 0.85, p "" 3e-ll 
0.703 1.406 
Bar Mass [lOlOM 0 ] 
rs"" 0.36, p = 3e-02 
----. ~ 
~ : ~ ~~':...- ~ 
~~ 
~ ":;:::;" .. ~~ 4-
.;;:--
2.66 5.32 7.98 0.3 0.6 0.703 1.406 
Bar length [kpc] Axis ratio bla Bar Mass [lOlOM 0] 
rs= -0.67, P = 6e-06 
1.2 1.5 
ParameterR 
rs= -0.l4, P = 4e-Ol 
1.2 1.5 
ParameterR 
90 
80 
70 
60~ 
l 
soJ 
40 
,o 
20 
How the bar affects the spiral structure 4715
Figure 9. Pitch angle and spiral amplitude versus bar pattern speed in the three galactic models (rising in the left column, flat in the middle, and declining in
the right). Simulations are colour-coded according to their R parameter (blue for slow, red for fast, and green for ultrafast bars). The more massive bars are
shown with diamonds, while their less massive counterpart are shown with crosses.
the effects of self-gravity. Our results should be interpreted as the
initial spiral perturbation that arises from the presence of different
kinds of bars. Spiral arms produced this way, are expected to form
near the OLR (Athanassoula 1980), and be tightly wound with small
pitch angles.
Because of the great variety of parameters we are exploring,
the spirals arms form at different times, amplitudes, and locations
throughout the disc. To make the comparison between different
simulations as fair as possible, we use the measurements of the
snapshot where spirals are at their greatest amplitude.
4.1 Observed correlations
In Figs 6–8, we show the average pitch angle and the spiral amplitude
versus the bar parameters of our simulations in the rising, flat, and
declining galactic models, respectively. We join with grey dotted
lines the simulations that share the same bar parameters except for
the one that is being plotted. The Spearman correlation coefficient
rS and the corresponding statistical significance p are shown at the
top of each panel. We colour the results with the bar pattern speed.
To facilitate the interpretation of our results, Table 3 summarizes all
Spearman correlation coefficients that we mention through the text.
In all galactic models, we found a strong anticorrelation of the
pitch angle with the bar length (rS ∼ −0.6) and the R parameter (rS
∼ −0.65). At a first approximation, it would appear that the mass and
shape of the bar do not correlate with the pitch angle, with a small
rS coefficient and p-values >0.05. However, the connected models
display a clear downward trend with b/a and an upper trend with
MBar. We performed a Student’s t-test to see if these slight differences
between connected simulations could happen by random chance. We
could reject the null hypothesis in both cases in all galactic models
(p ∼ 1 × 10−4 for b/a and p ∼ 1 × 10−5 for MBar).
The relations with the spiral amplitude does seem to depend more
on the galactic models. The correlation with the bar length is non-
existent in the rising model (rS = 0.16), but becomes strong in the
flat and declining models (rS = 0.64 and rS = 0.85, respectively).
In contrast, the relation with the R parameter does not seem
significant in the declining model (rS = −0.14) but becomes a weak
anticorrelation in the rising and flat models (rS ∼ −0.40 in both
cases). The connected simulations in all three models show a clear
upward trend with MBar and a downward trend with the ratio b/a,
which we were able to confirm with the Student t-test, rejecting the
null-hypothesis. The strong trends with the bar mass and axial ratio
can also be observed in amplitude versus time plot in Fig. 5.
Some of the observed correlations are related in the form of a
third relationship. For example, the bar length a and parameter R
are intimately related with Bar and the disc rotation curve V(R):
Bar = V (RCR)/RCR = V (RCR)/(R × a). (7)
Thus, the strong trends we observe among the pitch angle, the
bar length, and R are probably a consequence of the much stronger
correlation between the pitch angle and the bar pattern speed, (rS
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Rising Flat Declining 
_30 
rs= 0.83, p = le-09 rs= 0.88, p = le-12 rs= 0.84, p = le-lO 
:!... • 
.!!! 25 
'" c: 
« 20 • .<: 
u • ~ 
t c:: 15 • (IJ .. ¡ '" ~ .... • f. ttJI 
t t • e 10 • ~ 5 ., I • ..... 
20 40 60 80 100 20 40 60 80 20 40 60 80 
(l.a, [km/s/kpcl (l.a, [km/s/kpcl (l.a, [km/s/kpcl 
rs= 0.02, p = ge-01 rs= -0.39, p = 2e-02 rs= -0.71, p = le-06 
• • 
_ Slow 
0.8 _ Fast 
(IJ •• • • • • • _ Ultra-fast 
'C • g 0.6 •• •• le MBar =O.703 
a. .. ~ • . t ... • • MBar = 1.406 
E • • « • ..... t • ••• • ;;; 0.4 ):.' . • • • ~ • ••• • • • • • 'C, ;,c . : • • • • • • '" • .. • • • • • • • 0.2 • • • • • • • , • • JI • • • 
20 40 60 80 100 20 40 60 80 20 40 60 80 
(l.a, [km/s/kpcl (l •• , [km/s/kpcl (l •• , [km/s/kpcl 
4716 L. Garma-Oehmichen et al.
Figure 10. Pitch angle and spiral amplitude versus bar quadrupole moment in the three galactic models.
∼ 0.85) in all three galaxy models. The relation between the spiral
properties and Bar is shown in Fig. 9 for the three galactic models.
The colours and shape of the dots are used to distinguish models in
R and MBar, respectively. The relation between the spiral amplitude
and Bar depends on the galactic model, as this relation is non-
existent in the rising model, but becomes a strong anticorrelation in
the declining model.
The bar length a, mass MBar, and axial ratio a/b are related to the
bar quadrupole moment (hereafter QBar). For a Ferrers ellipsoid of
index n QBar is
QBar = MBara
2
(
1 −
b2
a2
)
/ (5 + 2n) . (8)
Same as with other bar properties, in Fig. 10 we show the cor-
relations between the spiral properties and the quadrupole moment
in the three galactic models. Our results show a remarkably strong
correlation between the spiral amplitude and QBar in the flat and
declining models. Thus, the observed correlations between spiral
amplitude and a, b/a, and MBar could be a consequence of the more
stronger correlation with QBar. Nonetheless, this correlation does not
seem to be as important in the rising model (as with a, b/a, and
MBar).
The relationship between QBar and the pitch angle is more complex.
The general trend is negative (rS ∼ −0.45 in the three models).
However, if we look only at simulations with the same pattern speed
as shown by the colour code (i.e. simulations with same bar length and
R parameter), the positive relation with the mass (and anticorrelation
with b/a) becomes visible.
5 I N F L U E N C E O F T H E ROTAT I O N C U RV E
We study the effects of the rotation curve in the spiral properties, by
comparing simulations with the same bar parameters, but different
galaxy model. In Fig. 11, we show the pitch angles of the flat models
versus the rising and declining models. As in Fig. 9, the colours and
shape are used to distinguish the R and MBar, respectively. We do not
observe a significant difference between the rising and flat models,
except for one outlier that corresponds to an ultrafast, small, massive
bar. In comparison, the declining simulations produce spirals with
consistently larger pitch angles. The effect also appears to be more
significant with the ultrafast bars, as those differ more significantly
from the 1:1 relation. We include the best linear fit for reference.
Similarly, in Fig. 12 we compare the spiral amplitude between the
three galactic models. The rising model tends to produce weaker
spirals. The difference becomes more significant with the more
massive bars. On the other hand, the declining and flat models have
similar spiral amplitudes for all the bar models.
We also estimate the shear rate S (equation 1) at the OLR in all
our simulations. Values of S < 0.5 correspond to a rising region in
the rotation curve, S = 0.5 to a flat region, and S > 0.5 to a declining
region. The presence of the bar changes the shape of the rotation curve
from the axisymmetric model. We estimate the perturbed rotation
curve by averaging V (R) =
√
Rdφ/dR over 10 angles uniformly
distributed between 0 and π/2 radians.
In Fig. 13, we show the correlations between the spiral properties
and the local shear rate. We found a strong anticorrelation with the
pitch angle, but only in the flat and declining models. The relations
with the spiral amplitude are weaker, except on the declining models,
where we observe a strong correlation.
MNRAS 502, 4708–4722 (2021)
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• 
,g¡ 25 
'" <: 
« 20 
.<: 
H 
;;: 15 
" '" ~ 10 
~ « 5 
0.8 
" 'C 
::J 
~ 0.6 
c. 
E « 
;¡; 0.4 
~ 
'c. 
VI 
0.2 
Rising 
rs= -0.42, p = le-02 
• 
• ..,. • •••• • • 
JI II :¡ .: I 
• 
I 
Flat 
rs= -0.48, p = 3e-03 
t' • • • • • --, ... .. • I . , • • 
Declining 
rs= -0.45, p = 6e-03 
>. 
• 
• • • ~ ... • •• 
I I :¡ .: • • • • • • 
o 1000 2000 3000 4000 O 1000 2000 3000 4000 O 
Quadrupole moment Q 
1000 2000 3000 4000 
Quadrupole moment Q Quadrupole moment Q 
rs= 0.36, p = 4e-02 rs= 0.82, p = le-09 rs= 0.95, p = 2e-19 
• • 
• • I I • • • • • , l' I • • I • • • .. • • a - • . I! •• I •• • • • I • :.. . -: • •• • • ••• • .. .. •• • -. .. ' 
' . • r , . , 
o 1000 2000 3000 4000 O 1000 2000 3000 4000 O 1000 2000 3000 4000 
Quadrupole moment Q Quadrupole moment Q Quadrupole moment Q 
90 
How the bar affects the spiral structure 4717
Figure 11. Comparison of the pitch angle between the three galactic models.
For reference, the slopes of the best fits are m = 0.92 in the top panel
(excluding the outlier) and m = 1.26 in the bottom one.
6 I N F L U E N C E OF THE VELOC ITY ELLIPSOID
Disc heating mechanisms play an important role in the secular
evolution of galactic discs, altering the stellar kinematics and
increasing the random motion of stars. It has been shown that the
shape of the velocity ellipsoid, i.e. the ratio of the vertical and radial
velocity dispersion σ z/σ R is strongly correlated with the Hubble type,
with late-type galaxies being more anisotropic and early-types being
more isotropic (van der Kruit & de Grijs 1999; Gerssen & Shapiro
Griffin 2012). However, these results have been questioned by Pinna
et al. (2018), who found a wide range of dispersion ratios from the
literature around σ z/σ R = 0.7 ± 0.2. In the solar neighbourhood,
σ z/σ R ranges from ∼0.4 to ∼0.9 depending on the stellar age, mean
orbital radii, or even the dynamical modelling (Mackereth et al. 2019;
Nitschai, Cappellari & Neumayer 2020).
Figure 12. Comparison of the spiral amplitude between the three galactic
models. For reference, the slopes of the best fits are m = 0.39 in the top panel
and m = 0.90 in the bottom one.
To study the effects of the velocity ellipsoid on the formation
of spiral arms, we re-simulate a subset of eight galaxies in the flat
galactic model. We build new initial conditions where the velocity
dispersion relation is set to σ z/σ R = 0.5 and a more extreme case
σ z/σ R = 0.33. We chose the subset of simulations to cover a wide
range of pitch angles and spiral amplitudes produced by the original
simulations with isotropic velocity ellipsoid.
The spiral arms produced by the new simulations are still located
near the OLR, and peak in amplitude at the same time as the σ z/σ R
= 1 simulation. In Fig. 14, we show a side-by-side comparison of
the snapshot we presented in Fig. 4, but simulated with different
velocity ellipsoids. As we decrease the σ z/σ R ratio, the response
spirals increase their pitch angle, but result in a less coherent, wider
structure with smaller amplitude.
MNRAS 502, 4708–4722 (2021)
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25 
20 
~ 
° ~ 
gIS 
. ¡¡; 
a: 
C1 
c: 
10 
S 
O 
20 
15 
:S 10 
u 
Q¡ 
CI 
S 
O 
, .. , .. ,.. 
<o. 
Average Pitch Angle [0] 
• ... ~ .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , . ..;' 
.~ 
.¡, 
.¡, 
~ 
• 
_ Slow 
_ Fast 
_ Ultra-fast 
S 10 15 20 25 30 
Flat (Allen-Santillan) [0] 
......... 
.. 
••• + •• + 
.... 
... +. 
• M.a, = 0.703 
• M.a, = 1.406 
1:1 Relation 
Best fit 
o l' 
~----~--~--~----~ 
o S 10 15 20 
Flat (Allen-Santillan) [0] 
C1 
c: .¡¡; 
0.8 
0.6 
a: 0.4 
0.2 
Average Spiral Amplitude 
.......... _ Slow 
_ Fast 
_ Ultra-fast ...... 
••• + •• + 
•••• • 
•• • •••• • .. . ... ... 
.+ "","'" .. % ...... •••• . ... . 
......... !! ~ ; . . • • 
.,. . 
..... • 
. 
0.0 ... _. --~--~--~--~--
0.8 
C1 0.6 
c: 
·2 
u 
~ 0.4 
0.2 
0.0 0.2 0.4 0.6 
Flat (Allen-Santillan) 
• M.a, = 0.703 
0.8 
.. 
..+ J .... ~ .. , .. , , 
• M.a, = 1.406 
1: 1 Relation 
Best fit .. :" .. , .. , 
.t­«e . 
;~ . • 
• 
~ .. 
0.0 -I-'----~-~--~--~--
0.0 0.2 0.4 0.6 0.8 
Flat (Allen-Santillan) 
4718 L. Garma-Oehmichen et al.
Figure 13. Pitch angle and spiral amplitude versus shear rate measured at the OLR in the three galactic models.
In Fig. 15, we show the pitch angle and spiral amplitude as
a function of the velocity dispersion ratio. We connect with grey
segmented line simulations that share the same bar model. Is clear,
from this small subset that the effects the velocity ellipsoid has on the
spiral arms are substantial. The more radially heated discs produced
very low amplitude spirals. In one simulation, we were not able to
detect any spiral structure from the background.
7 BAR VERSUS DISC PROPERTIES
So far, we have explored how different bar and disc parameters relate
with the induced pitch angle and spiral amplitude independently.
Using solely their Spearman correlation coefficient and the statistical
significance, the parameters that best predict the pitch angle and
amplitude are the Bar and QBar, respectively. Nonetheless, the
differences observed between galactic models suggest the shear or
galactic model itself, could also be important features for predicting
the spiral properties. Also, some of these parameters are related to
each other and cannot be treated as independent variables.
To rank the relative importance of each parameter compared to the
rest, we used a random forest regressor (Breiman 2001), trained
to predict the spiral properties of our simulations using all bar
parameters, the shear rate, and the galactic model. The model works
by averaging the prediction of multiple decision trees, which try to
predict the target variable using a flowchart-like decision, ranking
the input variables.
We set the number of decision trees to 1000 and the maximum
number of features used by each tree to 4. This prevents overfitting,
and increases the relative importance of features that are highly
correlated. For example, if Bar is considered an important variable
to predict the pitch angle, a decision tree that does not train with
this feature should weight R more heavily, as these two are highly
correlated.
As a proof of concept, we trained several models using an 80/20
train-test split that consistently achieved a coefficient of determina-
tion r2 > 0.80 for both the pitch angle and the spiral amplitude. Since
the purpose of these models is to get the relative importance of each
feature, we re-train the models using all our simulation data. We used
the permutation feature importance technique; where the relevance
of each feature is estimated by the difference between the r2 score of
the original data (r2
baseline) and the score after randomly shuffling such
feature (r2
permuted) (Breiman 2001). In Fig. 16, we show the feature
importance estimated after shuffling each feature 50 times.
The most important features for predicting the pitch angle are
related to the bar rotation rate. This was expected, as Bar and R
showed the strongest correlations with the pitch angle. In comparison,
all other parameters are not consider as important, as these only
change the r2 score of the prediction by 0.1 or less. This shows
that the relative strong correlation with the bar length was actually
a consequence of the much more stronger relations with the bar
frequency. Notice in particular, that the ‘Declining’ feature is
considered more important than the Flat and Rising features. This is
probably due to the declining simulations having consistently higher
pitch angles as we showed in Fig. 11.
For the spiral amplitude, the most important features are related to
the bar strength. Again, this was expected from the strong correlations
of QBar, MBar, and a. In contrast with the pitch angle, the amplitude
model considers the bar frequency (R and Bar) and the disc rotation
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rs= -0.82, p = 8e-10 
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0.50 0.55 0.60 
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0.65 
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How the bar affects the spiral structure 4719
Figure 14. Bar induced spiral arms in discs with different velocity ellipsoids. As in Fig. 4, this is the view of the test particles between RCR and the OLR +
4 kpc at the snapshot where the spiral amplitude is maximum. The top row shows the view in the xy-plane, bottom row shows the (θ , log10(R + 1))plane. Test
particles classified by DBSCAN as part of clusters are coloured.
curve (S and Rising) to be as important as the bar mass and length.
This could be because of the rising simulations forming consistently
spirals with lower amplitudes as we showed in Fig. 12.
Our random forest models do not take into account the velocity
dispersion ratio of the stellar disc. However, it is clear from the
simulations we studied in Section 6 that the dynamical temperature of
the disc is an important feature to consider when predicting the spiral
properties. Perhaps equally or more important than the perturber
properties or the shape of the velocity curve. How these effects
complement or affect the relations discussed, will be explored in a
future work.
8 D I S C U S S I O N A N D C O N C L U S I O N S
Since our simulations lack the effects of self-gravity, our results
cannot predict how the system would evolve after the initial pertur-
bation. Our spirals have typical lifetimes between 2 and 3 dynamical
times at the OLR after the bar formation. However, we point out
that self-gravity and star formation could increase and highlight the
spiral amplitude. It is expected that these spirals would eventually
disappear and fragment into smaller substructures as several N-body
simulations have shown.
The evolution of the pitch angle is unclear. In the Lin–Shu density
wave picture, the pitch angle remains constant in time and depends
on the global galaxy properties. Several N-body simulations have
shown that the pitch angle decreases with time (Grand, Kawata &
Cropper 2013; Pettitt & Wadsley 2018), while others have shown
it remains roughly constant, independently of their origin (Mata-
Chávez et al. 2019). Nonetheless, Mata-Chávez et al. (2019) also
observed an increase in the pitch angle after the buckling of the
bar, probably because of the mass redistribution. Recently, Pringle
& Dobbs (2019) proposed the pitch angle evolves as a decreasing
function of time, from an initial measurable maximum αmax to a
minimum αmin, as evidenced from the uniform distribution of cot α
of galaxies observed by Yu et al. (2018). Our results suggest that,
when the spirals are induced by a bar perturber, the initial pitch angle
is not random nor has a fixed maximum for all galaxies. It mainly
depends on the bar pattern speed.
In the majority of our simulations, the spirals are tightly wound. In
order to produce wide open spirals in barred galaxies like NGC 1365,
NGC 1672, and NGC 2903 other interaction mechanisms should be
considered. For example, the different nature of spiral arms has also
been explained by invariant manifolds (Athanassoula 2012; Romero-
Gómez et al. 2015), galaxy encounters (Pettitt, Tasker & Wadsley
2016), evolution of the mass distribution (Mata-Chávez et al. 2019),
or a shearing spiral pattern (Speights & Westpfahl 2011).
Although the shear and the rotation curve shape did not appear
to be powerful predictors for the spiral properties (in comparison
with the bar), they seem to be important on how the disc responds
to the perturbations. Our simulations show that bars in galaxies with
MNRAS 502, 4708–4722 (2021)
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(J"z /(J"R = 1 (J"z / (J"R = 0.5 (J"z / (J"R = 0.33 
10 10 10 
5 5 5 
U U U 
a. O a. O a. O 
~ ~ ~ 
>- >- >-
-5 -5 -5 
-10 -10 -10 
-10 -5 O 5 10 -10 -5 O 5 10 -10 -5 O 5 10 
X [kpcl X [kpcl X [kpcl 
2.6 2.6 2.6 
2 .5 2 .5 2.5 
~2 . 4 ~2 . 4 ~2.4 
U U U 
~2.3 ~2.3 ~2.3 
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~2.2 
E 
~2.2 
E 
~2.2 
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2 .1 2 .1 2.1 
2.0 2.0 2.0 
1.9 1.9 1.9 
2 4 6 2 4 6 2 4 6 
fJ [Radl fJ [Radl fJ [Radl 
4720 L. Garma-Oehmichen et al.
Figure 15. Pitch angle and spiral amplitude as a function of the velocity
dispersion ratio σ z/σR. Simulations that share the same bar model are
connected with the grey segmented lines.
rising rotation curves are less efficient forming a grand design spiral
structure. We found evidence in the weaker relationship between
the spiral amplitude and QBar, and the 1:1 comparison of the spiral
amplitude between galaxy models with same bar parameters. In
contrast, bars in galaxies with declining rotation curves appear to
be the most efficient, having the strongest response with QBar, and
consistently producing spirals with higher pitch angles compared
with the other two galactic models.
These results are consistent with observations. Using galaxies
from the Spitzer Survey of Stellar Structure in Galaxies (S4G),
Bittner et al. (2017) showed that the distribution of flocculent
galaxies is statistically different from the multi-arm and grand
design galaxies. Flocculent spirals are more common in late-type
galaxies, with weaker bars and less concentrated bulges. The low
mass concentration suggests these galaxies should have a slowly
rising rotation curve. N-body simulations have shown that galaxies
with rapidly rising rotation curves form strong flat bars, whereas
exponential bars are more typical of slowly rising rotation curves
(Combes & Elmegreen 1993; Athanassoula & Misiriotis 2002).
Dı́az-Garcı́a et al. (2019) measured the pitch angle, spiral strength,
and bar strength of galaxies in the S4G survey. They observed the
same strong relationship between the spiral and bar strength, even
in flocculent galaxies, independently of the method used for the
measurement (see also Buta et al. 2003; Block et al. 2004; Salo
et al. 2010). The relationship with flocculent galaxies is consistently
weaker (i.e. bars in flocculent galaxies are associated with weaker
spirals). They also observed a positive weak correlation between
the spiral strength and pitch angle (see their figs 17 and E.1).
However, the dispersion in the pitch angle measurements is quite
large. We do not find any correlation between the spiral properties
Figure 16. Permutation feature importance of two random forest regressors
trained to predict the pitch angle (top panel) and spiral amplitude (bottom
panel). Each feature was permuted 50 times, to produce a distribution of r2
scores. The orange line shows the median value of the distribution. The box
limits are the 25 and 75 percentiles. The lines extend to the maximum and
minimum values. Outliers of the distribution are shown with open circles.
independently of the galaxy model used (rS = 0.01, p = 0.88 for all
our simulations). Nonetheless, this could be an effect of the spiral
arm acting on itself via self-gravity that we cannot capture in our
simulations.
It is possible that our major results could be generalized to
other kinds of spiral-producing perturbations (galaxy encounters,
tri-axial dark matter haloes, giant molecular clouds, etc.). That is,
the frequency of the perturber relates to the spiral pitch angle, and
the strength of the perturbation relates to the spiral amplitude (see
e.g. Pettitt et al. 2016; Pettitt & Wadsley 2018).
It is clear that the shape of the velocity ellipsoid and the dynamical
state of the disc have an important role in predicting the spiral proper-
ties and how the disc responds to different gravitational perturbations.
Simulations by Athanassoula & Sellwood (1986) showed that bars
delay their growth in dynamically hotter discs. In our tests, spirals
generated in discs with isotropic ellipsoids had a greater amplitude,
but a smaller pitch angle. As we increased the anisotropy with greater
radial dispersion, the spirals unwind, but reduced their amplitude. Is
possible that an optimal range of anisotropy is required for discs
to be able to form grand design spirals. How the velocity ellipsoid
affects the relations we discussed with the bar and the rotation curve
remains unclear, and will be explored in a future work.
AC K N OW L E D G E M E N T S
We thank the referee C. Struck for his useful comments that
improved this paper. We acknowledge DGTIC-UNAM for providing
MNRAS 502, 4708–4722 (2021)
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........ ... ..... ---- ------.. ......... . -----::---------W- ______ __ - ____ • 
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How the bar affects the spiral structure 4721
HPC resources on the Cluster Supercomputer Miztli. LGO and
LMM acknowledge support from PAPIIT IA101520 grant. LGO
acknowledges support from CONACYT scholarship.
DATA AVA IL A BILITY
The data that support the findings of this study are available from the
corresponding author, upon reasonable request.
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This paper has been typeset from a TEX/LATEX file prepared by the author.
MNRAS 502, 4708–4722 (2021)
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Chapter 5
Bar Pattern Speed in Milky Way like
Galaxies
In this chapter I present the second original research paper of this thesis, which has been
accepted to the Monthly Notices of the Royal Astronomical Society.
Here, we use observations to study the problem of the bar pattern speed. This work is
a natural continuation of Garma-Oehmichen et al. (2020) which was my masters project.
In that work we performed measurements of the bar pattern speed with the Tremaine-
Weinberg method in a sample of 18 galaxies from both MaNGA and CALIFA surveys
(Garma-Oehmichen et al., 2020). We studied the error sources that arise from the geo-
metric nature of the TW method, namely from the centre, slits length and position angle
estimation.
Various lessons were learned that we apply in this new paper. Firstly, we do not assume
independence between error sources. Instead our new estimation takes into account all
error sources in the same Monte Carlo procedure. Secondly, we propose a new way to
weight independent measurements of the disc PA. All methods for estimating the PA are
prone to biases that are difficult to correct. We choose to weight our PA measurements
based on the linearity of the TW integrals.
We choose a sample of 97 galaxies that resemble the Milky Way in stellar mass and
morphological type. Studying the properties of our Galaxy has always been difficult.
The extinction produced by the dust, and the stars distance uncertainties being the most
39
CHAPTER 5. BAR PATTERN SPEED IN MILKY WAY LIKE GALAXIES 40
important challenges. By studying galaxies with similar properties we circumvent these
limitations and determine where Milky Way may lie within the statistics.
In recent years, numerous studies have measured the bar pattern speed of our Galaxy,
converging to a value around ∼ 35 − 40 km s−1 kpc−1. Adding the distributions of our
measurements we get our results are in good agreement with the current estimates.
MNRAS 000, 1–20 (2022) Preprint 7 November 2022 Compiled using MNRAS LATEX style file v3.0
SDSS IV MaNGA: Bar pattern speed in Milky Way Analogue galaxies
L. Garma-Oehmichen,1★ H. Hernández-Toledo,1 E. Aquino-Ortíz,2 L. Martinez-Medina,1 I. Puerari,3
M. Cano-Díaz,1 O. Valenzuela,1 J. A. Vázquez-Mata,4,5 T. Géron,6 L. A. Martínez-Vázquez,1 and R. Lane7
1Instituto de Astronomía, Universidad Nacional Autónoma de México, Apartado Postal 70-264, CDMX, 04510, México
2Instituto de Astrofísica, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, 782-0436 Macul, Santiago, Chile.
3Instituto Nacional de Astrofísica, Optica y Electrónica, Apdo. Postal 51 y 216, 72000 Puebla, Puebla, México
4Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México, Ciudad Universitaria, CDMX, 04510, México
5Instituto de Astronomía sede Ensenada, Universidad Nacional Autónoma de México, Km 107, Carret. Tij.-Ens., Ensenada, 22060, BC, México
6Oxford Astrophysics, Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK.
7 Centro de Investigación en Astronomía, Universidad Bernardo O’Higgins, Avenida Viel 1497, Santiago, Chile
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Most secular effects produced by stellar bars strongly depend on the pattern speed. Unfortunately, it is also the most difficult
observational parameter to estimate. In this work, we measured the bar pattern speed of 97 Milky-Way Analogue galaxies
from the MaNGA survey using the Tremaine-Weinberg method. The sample was selected by constraining the stellar mass and
morphological type. We improve our measurements by weighting three independent estimates of the disc position angle. To
recover the disc rotation curve, we fit a kinematic model to the HU velocity maps correcting for the non-circular motions
produced by the bar. The complete sample has a smooth distribution of the bar pattern speed (Ω0A = 28.14+12.30
−9.55 km s−1
kpc −1), corotation radius ('' = 7.82+3.99
−2.96 kpc) and the rotation rate (R = 1.35+0.60
−0.40). We found two sets of correlations:
(i) between the bar pattern speed, the bar length and the logarithmic stellar mass (ii) between the bar pattern speed, the disc
circular velocity and the bar rotation rate. If we constrain our sample by inclination within 30° < 8 < 60° and relative orientation
20° < |%38B2 − %10A | < 70°, the correlations become stronger and the fraction of ultra-fast bars is reduced from 20% to
10% of the sample. This suggest that a significant fraction of ultra-fast bars in our sample could be associated to the geometric
limitations of the TW-method. By further constraining the bar size and disc circular velocity, we obtain a sub-sample of 25
Milky-Way analogues galaxies with distributions Ω0A = 30.48+10.94
−6.57 km s−1 kpc−1, '' = 6.77+2.32
−1.91 kpc and R = 1.45+0.57
−0.43, in
good agreement with the current estimations for our Galaxy.
Key words: galaxies: disc – galaxies: evolution – galaxies: kinematics and dynamics – galaxies: structure
1 INTRODUCTION
Stellar bars exist in a great variety of shapes, sizes, and galactic
environments. Most of their properties are strongly tied to the stellar
mass and morphology of their host galaxy. For instance, the bar
fraction (the likelihood of hosting a large-scale bar) and bar length
are strongly dependent on the galaxy mass (Nair & Abraham 2010;
Masters et al. 2012; Erwin 2018). Early-type galaxies host stronger
bars than their late-types counterparts. They are larger in relation to
their disc (Méndez-Abreu et al. 2012; Díaz-García et al. 2016a; Erwin
2018), prolate shaped (Díaz-García et al. 2016a; Méndez-Abreu et al.
2018) and have a flat density profile (Elmegreen & Elmegreen 1985;
Kim et al. 2015). Moreover, the size ratio between the bar and the
disc remains constant over the cosmic time, suggesting an efficient
coupling between both structures (Pérez et al. 2012; Kim et al. 2021).
Bars are one of the main drivers of the galaxy secular evolu-
tion (Weinberg 1985; Kormendy & Kennicutt 2004; Sellwood 2014;
Díaz-García et al. 2016b). Numerous analytical and numerical stud-
ies show they are efficient at transferring angular momentum from
★ E-mail: lgarma@astro.unam.mx
their inner resonances to those outside of corotation via dynamical
friction (Lynden-Bell & Kalnajs 1972; Tremaine & Weinberg 1984a;
Athanassoula 2003). This angular momentum is mostly absorbed by
the dark matter halo (Weinberg 1985; Debattista & Sellwood 2000),
and in a smaller fraction by the bulge (Kataria & Das 2019).
As a result, the bar is expected to induce substantial gaseous flows
to the galaxy centres (Sormani et al. 2015). Observations of barred
galaxies show a clear increase in the concentration of molecular gas
(Sakamoto et al. 1999; Jogee et al. 2005) and in the star formation
rate (Ellison et al. 2011; Chown et al. 2019) in the central regions.
If the process in not balanced with the inflow of cosmological gas,
the bar can deplete the gas supply in the disc, causing the so-called
“bar quenching” (Masters et al. 2012). This scenario is supported
by observations of the specific star formation rate (Cheung et al.
2013), colour (Gavazzi et al. 2015; Kruk et al. 2018), gas fraction,
(Newnham et al. 2020), star formation histories (Fraser-McKelvie
et al. 2020) and statistical properties of the galaxies (Géron et al.
2021).
A theoretical consequence of the large scale gas flows is the flat-
tening of the metallicity profile (Cavichia et al. 2014; Kubryk et al.
2015b). However, this behaviour does not appear in most obser-
© 2022 The Authors
2 L. Garma-Oehmichen et al.
vations (Sánchez-Blázquez et al. 2014; Pérez-Montero et al. 2016;
Sánchez-Menguiano et al. 2016), and has only been observed in low
luminosity (low mass) galaxies (Zurita et al. 2021).
The bar is mostly supported by regular resonant stellar orbits lo-
cated inside corotation. The most important being the x1 family (Con-
topoulos & Papayannopoulos 1980). Nonetheless, chaotic orbits are
also important building blocks for the bar. For example, by modelling
N-body simulation snapshots, various authors have shown that reg-
ular and sticky orbits can eventually transform to chaotic orbits that
support the X-shaped/boxy structure (Voglis et al. 2007; Harsoula
& Kalapotharakos 2009; Chaves-Velasquez et al. 2017). Moreover,
chaotic orbits near the Lagrangian points could be responsible for
the support of the spiral structure according to the Manifold theory
(Voglis et al. 2006; Romero-Gómez et al. 2006, 2015). The strong
correlation between the strengths of the bar and spiral arms sug-
gest both structures could be intimately coupled (Salo et al. 2010;
Díaz-García et al. 2019; Garma-Oehmichen et al. 2021).
The multiple resonances produced by bars and spiral arms also
have a profound effect on the disc stellar orbits. Simulations show
that near corotation, stars can scatter inwards or outwards without
changing their orbital ellipticity in a process called “radial migration”
(Sellwood & Binney 2002; Kubryk et al. 2015a). Since stars preserve
their circular orbits, this process does not contribute to the radial
heating of the disc, but should affect the metallicity distribution
(Martinez-Medina et al. 2017). The rearrangement of stars can also
trigger the creation of moving groups (Pérez-Villegas et al. 2017)
and resonant trapping stars in the disc and the stellar halo (Quillen
et al. 2014; Moreno et al. 2015).
All these secular effects depend in great extent on the bar pattern
speed (hereafter Ω10A ). Unfortunately, it is the most difficult obser-
vational parameter to estimate. Most methods developed to measure
Ω10A require some modelling. Some use the gas flow induced by
the bar, for example, by matching observations with hydrodynamical
simulations (Sanders & Tubbs 1980; Hunter et al. 1988; England
et al. 1990; Weiner et al. 2001; Pérez et al. 2004; Zánmar Sánchez
et al. 2008; Rautiainen et al. 2008), or studying the residual gas ve-
locity field after subtracting a rotation model (Font et al. 2011, 2014,
2017). Other methods are based on the location and shape of differ-
ent morphological features like dark gaps in ringed galaxies (Buta
2017; Krishnarao et al. 2022), the offset of dust lanes (Athanassoula
1992; Sánchez-Menguiano et al. 2015), changes in the phase of spi-
rals (Puerari & Dottori 1997; Aguerri et al. 1998; Sierra et al. 2015)
or the position of rings (Buta 1986; Rautiainen & Salo 2000; Patsis
et al. 2003).
The only direct method for estimating Ω10A is the so-called
Tremaine & Weinberg (1984b) (hereafter TW) method (see Section
3). It requires the surface brightness and line-of-sight (LOS) velocity
of a tracer that suffices the continuity equation. Until recently, most
measurement using the TW-method were made with long-slit spec-
troscopy in early-type galaxies (e.g. Kent 1987; Merrifield & Kuijken
1995; Gerssen et al. 1999; Debattista et al. 2002; Aguerri et al. 2003;
Corsini et al. 2003; Debattista & Williams 2004; Corsini et al. 2007).
With the advent of the integral field spectroscopy technique, the
TW method has been applied to an increasing number of galaxies.
Aguerri et al. (2015) measured Ω10A in 15 strong barred galaxies
from the survey CALIFA and found no trend with the morpholog-
ical type. Guo et al. (2019) used a sample of 53 galaxies from the
MaNGA survey and studied the effects of the galaxy position an-
gle and inclination on the determination of Ω10A (see also Debat-
tista (2003)). Cuomo et al. (2019) continued the measurements from
Aguerri et al. (2015) within the CALIFA survey, finding weakly
barred galaxies have similar values of Ω10A as the strongly barred
galaxies. In Garma-Oehmichen et al. (2020) (hereafter G20), we
used a sample of 15 MaNGA galaxies and 3 CALIFA galaxies to
study different uncertainty sources and identify systematic errors in
the method. Williams et al. (2021) applied the TW-method to 19
galaxies from the PHANGS-MUSE survey. They found ISM trac-
ers produce erroneous signals in the TW integrals because of their
clumpy distribution.
The rotation rate parameter R = ''/'0A (the ratio between
corotation and the bar radius) gives a dynamical interpretation to the
bar. The physical significance of this ratio comes from the fact that
the stellar orbits that support the bar cannot extend beyond corotation
(Contopoulos & Papayannopoulos 1980; Athanassoula 1980). Thus,
the faster the bar rotates in relation to the disc, the closer R goes to 1.
This ratio is commonly used to classify bars as slow (R > 1.4), fast
(1 < R < 1.4) and the theoretically un-physical ultra-fast (R < 1).
Recent measurements using the TW-method find that most bars
are consistent with a fast classification (with a concerning amount of
ultra-fast bars) (Guo et al. 2019). This has led to a growing tension
with the ΛCDM cosmological framework, where bars slow down
excessively because of dynamical friction with the dark matter halo
(Algorry et al. 2017; Peschken & Łokas 2019; Roshan et al. 2021a,b).
Nonetheless, Fragkoudi et al. (2021) showed that the bars can remain
fast in more baryon dominated galaxies with higher stellar-to-dark
matter ratios.
The properties of the dark matter halo are incredibly important
for the bar formation and evolution. Several N-body simulations
have shown that more concentrated halos are able to form stronger
and larger bars (Debattista & Sellwood 1998, 2000; Athanassoula
& Misiriotis 2002). Triaxial halos can induce the formation of bars
(Valenzuela et al. 2014) but produce weaker bars compared to spher-
ical halos (Athanassoula et al. 2013). Recent works have shown the
importance of the halo spin. For example, a spinning halo can ef-
fectively suppress bar formation, by being unable to absorb angular
momentum with the same efficiency (Long et al. 2014; Collier et al.
2018; Rosas-Guevara et al. 2022). Spinning halos can also change
the radial extent of the disc (Grand et al. 2017), affecting the bar
instability criteria (Izquierdo-Villalba et al. 2022).
Several attempts have been made in the last decade to estimate
Ω10A in the Milky Way (MW). Arguably, the best estimations have
been derived using the made-to-measure method, which matches the
density of red clump giants with N-body models of barred galaxies
(Portail et al. 2017; Pérez-Villegas et al. 2017; Clarke et al. 2019;
Clarke & Gerhard 2022). Other attempts have used an adaptation
of the TW method (Bovy et al. 2019), or kinematic maps of disc
and bulge stars, along with the continuity equation (Sanders et al.
2019). Other methods are based on N-body simulations of MW-like
galaxies (Tepper-Garcia et al. 2021; Kawata et al. 2021). All these
measurements seem to converge to a pattern speed of Ω10A ∼ 35
- 40 km s−1 kpc−1. This value corresponds to a corotation radius
'2A ∼ 5 − 6 kpc, and R ∼ 1.2, therefore classifying the MW bar in
the fast category.
In this paper, we tie together the bar pattern speed of our galaxy in
the context of extra-galactic measurements. We choose a sample of
galaxies that reassemble the Milky Way in morphology and stellar
mass. In G20 we argued some uncertainties in the TW method were
still not well understood (see also Discussion in Guo et al. (2019)),
and many improvements could be made. In line with this, we weight
various independent measurements of the disc position angle (PA),
we changed the distributions for sampling various parameters and
change the rotation curve model. Throughout our procedure, we
highlight possible systematic errors and decisions that could be bias-
MNRAS 000, 1–20 (2022)
Bar pattern speed in MWA galaxies 3
ing our measurements. All together, this has made our measurements
more robust.
We organised the paper as follows: In Section 2 we introduce
the MaNGA survey and the data processing packages used. Sec-
tion 3 gives an overview of the Tremaine-Weinberg method, and
discuss its limitations. In Section 4 we present our sample se-
lection. In Section 5 we show our measurement procedures and
discuss the error estimation. We present the complete sample re-
sults in section 6 and discuss future improvements Section 7. Fi-
nally in Section 8 we summarise our conclusions. All Figures
and relevant measurements are available at the public repository:
https://github.com/lgarma/MWA_pattern_speed
2 DATA
Mapping Nearby Galaxies at Apache Point Observatory (MaNGA)
(Bundy et al. 2015) is the largest galaxy survey using the Integral
Field Spectroscopy (IFS) technique. It observed a sample of over
10, 000 galaxies. Its main sample comprises galaxies of all mor-
phological types (e.g. Vázquez-Mata et al. 2022), within the redshift
interval 0.1 < I < 0.15 and stellar masses between 109 and 1012 "⊙
(for details about the sample see: Wake et al. 2017). The final sample
also includes objects selected as part of 20 ancillary programs driven
by specific scientific objectives (for details refer to: Abdurro’uf et al.
2021), so a small fraction contains galaxies that may lie outside the
aforementioned characteristics.
The MaNGA survey is one of the three major programs of the
SDSS-IV international collaboration (Blanton et al. 2017). Its obser-
vations are carried out by the Sloan 2.5 meter telescope (Gunn et al.
2006), and uses the BOSS spectrograph, which provides a spectral
resolution of ' ∼ 2000 in the wave-length range 3600 Å - 10300 Å
(Smee et al. 2013).
Its Integral Field Units (IFUs) consist of a set of 17 hexagonal
fiber-bundles of five different sizes ranging from 19 to 127 fibers
(Drory et al. 2015). Each fiber has a diameter of 2 arcsec. Data taking
requires that these fiber bundles be plugged into previously drilled
plates, whose positions correspond to the targets to be observed per
night. Finally, a 3-point dithering observing strategy is used to ensure
total coverage of the required field of view (Law et al. 2016a), which
for the MaNGA targets vary from ∼1.5 to ∼2.5 '4.
In this work, we use the MaNGA reduced data (see Law et al.
2016b, for details) as well as the following data products provided
by the Data Analysis Pipeline (DAP) (Westfall et al. 2019): Mean
flux in r-band (as a proxy of the stellar flux), stellar velocity, the HU
velocity and their respective dispersion maps. The data was accessed
via the Python package Marvin (Cherinka et al. 2019). All DAP
datacubes are Voronoi-binned based on the g-band weighted signal-
to-noise ratio (S/N) with a target S/N of 10 per Voronoi bin. The
reconstruction method minimises the spatial covariance, making the
number of independent measurements similar to the number of fibers
(Liu et al. 2020).
Additionally, we usePipe3D to recover some integrated properties,
including the stellar mass, molecular gas, stellar spin parameter,
stellar surface density and the velocity-sigma ratio. (Sánchez et al.
2016a,b, 2018).
We will refer to the galaxies using their MaNGA identification
number, which consist of two sets of numbers (for example 8596-
12704). The first set (8596) corresponds to plate id, while the second
set combines the number of fibers (127) and the fiber-bundle used
(04).
3 TREMAINE WEINBERG METHOD
Using a tracer that follows the continuity equation and assuming a
perfectly flat disc, the Tremaine & Weinberg (1984b) method esti-
mates Ω10A as the ratio between two integrals:
Ω10A sin 8 =
〈E〉
〈G〉
(1)
where 8 is the galaxy inclination, and 〈E〉 and 〈G〉 are the intensity-
weighted means of the line-of-sight velocity and position of the tracer
respectively (Merrifield & Kuijken 1995). In equations this is:
〈E〉 =
∫ +∞
−∞
ℎ(H)3H
∫ +∞
−∞
+!$( (G, H)Σ(G, H)3G
∫ +∞
−∞
ℎ(H)3H
∫ +∞
−∞
Σ(G, H)3G
(2)
〈G〉 =
∫ +∞
−∞
ℎ(H)3H
∫ +∞
−∞
GΣ(G, H)3G
∫ +∞
−∞
ℎ(H)3H
∫ +∞
−∞
Σ(G, H)3G
(3)
where (G, H) is the Cartesian coordinate system of the sky plane,
with G aligned with the line of nodes (i.e the disc position angle). Σ is
the surface brightness and +!$( is the velocity in the line-of-sight,
of the tracer. In this work we are using the old stellar component for
weighting the TW integrals, with the r-band mean flux datacube for
Σ and the stellar velocity map for +!$( (Westfall et al. 2019).
ℎ(H) is an arbitrary odd weight function (ℎ(−H) = −ℎ(H)). To
simulate the position of 2< + 1 slits offset by a separation H0 we use
a sum of Dirac deltas ℎ(H) =
∑<
==0 X(H − =H0) − X(H + =H0)
The separation between slits H0 should be wide enough to prevent
the use of repeated information. This separation depends on the
relative orientation between the slits and the Cartesian grid:
H0 = ?/cos(\) (4)
where ? is the pixel separation and \ is lowest the angle between
the grid and the disc position angle (see equation 6 in G20). We
choose to set the pixel separation to ? = 1 between consecutive slits.
The symmetry in G and H in both 〈E〉 and 〈G〉, makes all axisym-
metric features to cancel out in the integration. Thus, in practice,
the integration limits can be set to cover only the non-axisymmetric
structure. In some galaxies, however, the IFU coverage is not enough
to reach the axisymmetric disc. It is possible a systematic error could
be happening in those galaxies. We cannot estimate the value of
Ω10A using a larger spatial coverage, but we can reduce the length
of the slits to do the opposite. To get an estimate of this systematic
error, we include a slits length error in our measurements.
The symmetry property also makes the TW method extremely
sensible to the disc PA, as a wrongful estimation introduces a false
signal to the integrals (Debattista 2003). It is common that different
estimates of the disc PA are not consistent within their uncertainties.
This introduces another systematic error that is heavily dependent on
the author’s criteria. In G20, we compared the measurements of 10
galaxies in common with Guo et al. (2019) and discussed differences
in procedures to estimateΩ10A . The average difference in the disc PA
was 3°, enough to change most measurements significantly. In G20,
our measurements of Ω10A differed on average 10 km s−1 kpc−1
when using a photometric or kinematic disc position angle. In this
work, we weight the PA of three independent measurements (see
section 5.1).
The TW method cannot be applied to all barred galaxies. If the bar
is oriented towards the disc major or minor axis, the TW integrals will
MNRAS 000, 1–20 (2022)
4 L. Garma-Oehmichen et al.
Figure 1. Image processing of the example galaxy 8596-12704. From left to right: (i) SDSS 6A8 composite colour image, (ii) DESI 6A I composite colour
image, (iii) DESI Residual image after subtracting the best Bulge/Disc 2D model of the galaxy brightness, (iv) Filter-enhanced DESI r-band image. All images
include the MaNGA FoV.
10.0 10.5 11.0 11.5
NSA Stellar Mass log (M* / M )
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
De
ns
ity
0.0 2.5 5.0 7.5 10.0 12.5 15.0
Deprojected bar radius [kpc]
0.00
0.05
0.10
0.15
0.20
0.25
0.30
De
ns
ity
20 40 60 80
Photometric inclination [ ]
0.00
0.01
0.02
0.03
0.04
0.05
De
ns
ity
Figure 2. Statistical distribution of our sample. A kernel density estimate is used to smooth the distribution. Left panel: Logarithmic stellar mass from the
NASA-Sloan Atlas catalogue. Middle panel: Deprojected bar radius, estimated from an isophotal analysis. Right panel: Disc inclination, estimated from the
isophotal analysis.
become symmetric and cancel the non-axisymmetric contributions.
Thus, galaxies with mid inclinations are preferable.
4 SAMPLE SELECTION
By assuming that the Milky Way is not extraordinary among its peers,
we can circumvent the limitation of observing our galaxy by studying
galaxies with similar properties in the extra-galactic context (Board-
man et al. 2020b). Throughout the literature, there have been many
definitions of what makes a good Milky Way Analogue (MWA), usu-
ally based on the science goals of the study. Common limits include
the stellar mass, the bulge-to-total ratio (Boardman et al. 2020a),
the star formation rate (Licquia et al. 2015), galactic companions
(Robotham et al. 2012; Boardman et al. 2020b) and the presence of
galactic structures like the bar, spiral arms (Fraser-McKelvie et al.
2019) or an X-shaped pseudo-bulge (Georgiev et al. 2019; Kormendy
& Bender 2019).
Due to the nature of this work, we tailor our selection criteria to
the stellar mass and the Hubble morphological type. The MW most
prominent features include a strong peanut-shaped bar and four ma-
jor spiral arms. Thus, its Hubble classification may lie somewhere
between SBb and SBc (Shen & Zheng 2020). We used the visual
morphological classification reported in Vázquez-Mata et al. (2022).
Figure 1 shows a mosaic containing a set of post-processed images
from the SDSS and DESI Legacy Image Surveys used for morpho-
logical evaluation.
The stellar mass of the MW is estimated to be ;>6("∗/"⊙) =
10.75 ± 0.2 (Licquia & Newman 2016). We choose to extend this
range to cover a complete dex in stellar mass log " ∼ [10.3 −
11.3]. Stellar masses come from a cross-match with the NASA-Sloan
Atlas (NSA) catalogue (Blanton et al. 2011), using ℎ = 0.71 Hubble
constant and a Chabrier initial mass function (Chabrier 2003).
Using these cuts, we end up with a sample of 233 galaxies. Still,
not all barred galaxies are good candidates for the TW method. The
TW methods weights both the kinematic and positional data, so
the inclination becomes an important selection parameter. Face-on
galaxies tend to have poor kinematic data but great positional data.
The opposite occurs in edge-on galaxies. To filter these galaxies, we
include cuts in inclinations within the range 20° < 8 < 70°.
As mentioned in Section 3, the bar and the disc should not be
oriented parallel nor perpendicular to each other. We included an ad-
ditional cut in their relative orientation: 10° < |%0A −%8B2 | <
80°.
Finally, a good measurement with the TW method should pro-
duce a clear linear relation in between both TW-integrals. However,
some galaxies do not display this trend, possibly because a lack of
symmetry. We choose to remove those galaxies as well.
All together, we discarded ∼ 60% of the galaxies, bringing our
final sample to 97 MWA galaxies. In Figure 2 we show the stellar
mass, bar radius and disc inclination distributions of the final sample.
In Section 6.4 we discuss a sub-sample where we further increase
the selection criteria to include the bar size and disc circular velocity
to be more similar to the MW. Divided by the number of fibers, our
sample contains 1, 14, 18, 12 and 52 galaxies with 19, 37, 61, 91 and
127 fibers respectively.
MNRAS 000, 1–20 (2022)
Bar pattern speed in MWA galaxies 5
Figure 3. Ellipticity and position angle profiles of 8596-12704. The bar radius
is shown with a green vertical stripe, limited by 'n and '%. A sudden jump
in ellipticity near the bar end occurs because of the inner ring. We estimate
the disc PA using the isophotes between 22 and 40 (arcsec) shown as a blue
region. For comparison, we also present the estimates after symmetrizing the
stellar velocity field (PABH<) and from the HU kinematic model (PA<>3). In
this galaxy, the three measurements of PA do not agree within uncertainties.
5 MEASUREMENTS
Throughout this section, we will present our procedure using the
galaxy 8596-12704 shown in Figure 1. This galaxy possesses a strong
bar with an inner ring near the end of the bar. It is a grand design
galaxy with two strong symmetrical arms that extend to the outer
region. The MaNGA field of view only covers the bar and the inner
ring.
In Figure 3 we show the corresponding isophotal profiles per-
formed over the DESI r-band images (Dey et al. 2019). We used the
ELLIPSE routine (Jedrzejewski 1987) within the IRAF environment
(Tody 1993). To increase the signal of the outer disc, we sample the
isophotes using a logarithmic step with the centre fixed.
The isophotes provide a photometric estimate of the disc PA,
inclination, bar radius and bar PA. We discuss different biases in
these measurements in sections 5.1 and 5.3.
In Figure 4 we show the r-band surface brightness and stellar
velocity maps, obtained from Marvin. We compute the TW integrals
over these two maps.
5.1 Disc Position Angle
The disc PA is the most important parameter for making reliable
measurements of Ω10A with the TW method. It is well documented
that few degrees of error in PA can lead to errors in Ω10A of tens of
percent. It has been extensively studied in simulated galaxies (De-
battista 2003; Guo et al. 2019; Zou et al. 2019) and in observations
(Garma-Oehmichen et al. 2020). In G20, we measured the disc PA
by using isophote profiles and by fitting a kinematic model to the HU
velocity map. After discussing the limitations in both methodolo-
gies, we concluded that the photometric estimate was more reliable
as it produced more measurements that made physical sense. How-
ever, our sample was small, and our conclusion based on empirical
arguments.
In this work, we use 3 independent methods for estimating the disc
PA with different tracers. (i) The classical photometric approach,
using the galaxy isophotes. (ii) Symmetrizing the stellar velocity map
using the PAFit package (Krajnović et al. 2006). (iii) Modelling the
U velocity maps using a modified version the package DiskFit
(see Section 5.4) (Spekkens & Sellwood 2007; Sellwood & Sánchez
2010, Aquino-Ortíz et al. in prep.). We will refer to them as PA?ℎ ,
PABH< and PA<>3 , respectively.
Each method has biases and systematic errors that cannot be esti-
mated from the mathematical formal errors. For example, to estimate
the disc PA from isophotes, we have to assume a thin and circular
disc. Strong nearby field stars, light from companion galaxies, disc
warps and strong spiral arms can distort the photometric measure-
ments. In some galaxies, the choice of the outer isophotes heavily
depends on the author’s criteria. In cases of ambiguity, where the PA
profile flattens in multiple sections, we use isophotes that are more
similar with PABH< and PA<>3 .
The symmetrized velocity fields can be affected by local deviations
from axi-symmmetry (Krajnović et al. 2011). In particular, the non-
axisymmetric motions produced by the bar and other structures affect
the stellar kinematics (Stark et al. 2018). In some galaxies, the IFU
data only covers the central region of the galaxy, ignoring the large-
scale movements of the disc.
Modelling the velocity field of the gas has the advantage of us-
ing a tracer that is not affected by the asymmetric drift present in
old stellar populations. However, the U gas can be affected by the
AGN feedback, recent galactic encounters, or episodes of strong star
formation (Tsatsi et al. 2015; Stark et al. 2018). Our kinematic mod-
elling has the additional advantage of disentangling the disc circular
movements from the non-axisymmetric movements produced by the
bar (Spekkens & Sellwood 2007).
In most galaxies of our sample the three measurements are not
consistent within their formal uncertainties. In Figure 5 we show the
distribution of the absolute difference between the three methods for
all galaxies in our sample. It is not surprising that PABH< and PA<>3
are the most similar, as both are based on the galaxy kinematics
(Barrera-Ballesteros et al. 2014). Nonetheless, the average difference
between both is ∼ 4.5°, which is enough to produce significant
differences in Ω10A . The large misalignment between the rotation
of stars and gas is possibly provoked by environmental effects like
mergers, gas accretion and interaction with nearby galaxies (Khim
et al. 2020, 2021; Lu et al. 2021). (See also Jin et al. (2016)).
In order to weight the 3 PA measurements the following proce-
dure was implemented: (i) We assume that the systematic errors are
unknown in all our measurements and that their formal errors can
be increased by using a penalisation factor. (ii) All uncertainties are
increased to be at least 1°. (iii) If all measures agree within their
uncertainty, we proceed to use a traditional variance weighted mean.
(iv) If they do not agree, we check the linearity condition between
〈G〉 versus 〈E〉 at each PA. (v) PAs that do not follow a linear trend
are penalised by increasing their error by a factor of 10. (vi) The
PAs that do present linearity have their errors increased to the point
where they are compatible within their uncertainties. (vii) We use
the variance weighted mean for our final estimation of the disc PA.
Figure 6 shows the estimates of PA for our example galaxy before
and after penalising (in the top and bottom panels respectively). The
differences between PA may be attributed to the strong spirals affect-
ing the estimation of PA?ℎ , or to the small field of view of the IFU
that only covers the barred and ringed region affecting both PABH<
MNRAS 000, 1–20 (2022)
I 
1 
1 
~ r' 1 
..Q 0.4 . " r '--:--
8596-12704 
- DisC E 
- R bar 
- - MaNGA FoV 
+ DESI r-band 
1 1 : 1 ¡ ' 1 I 
.--1 : : I ---¡ I ~ f I I 
11 0.2 ~ : - -~ 
UJ 1
1
I l ' 11 
0.0 1 
: I I - PAph 
o 1:: ~ - i ,,' ~ ~ I ~ l ----t-_¡-------:: : :~ : : 
1 • .1 
.. : 
60 ------~---+----+---~----- I 
\,," : 
40 
o 
1 
I 
20 40 60 80 100 120 
Radius [arcsec] 
6 L. Garma-Oehmichen et al.
Figure 4. The r-band surface brightness and stellar velocity maps of 8596-12704. We display the surface brightness with a logarithmic scale for visualisation
purposes. The bar estimates from the isophotal profiles are shown as black ellipses. We compute the TW integrals over the slits that cover the barred region.
0 10 20 30 40 50 60
0.00
0.02
0.04
0.06
0.08
0.10
0.12
De
ns
ity
Mean(PAsym - PAmod) = 4.59°
|PAph PAsym|
|PAph PAmod|
|PAsym PAmod|
Figure 5. Distribution of the misalignment between three independent PA
methods. Although the differences between PABH< and PA<>3 are smaller,
they are significant enough for the TW method.
and PA<>3 . The strong bar of the system, can also be influencing
the long scale motions of the stars and the gas.
Since these measurements are not compatible in their uncertain-
ties, we proceed to look at the linearity of the TW diagram. Figure 8
shows the TW diagram measured at PABH< (see Section 5.2) where
the lack of a linear relation between 〈G〉 and 〈E〉 is evident. We pe-
nalise the measurement by increasing the PABH< error by a factor 10.
In comparison, both PA<>3 and PA?ℎ satisfy the linearity require-
ment so we make them equal weighted. In Figure 7 we show the TW
diagram measured at the mid-point between both PAs.
In some cases, only one estimate of the disc PA satisfies the lin-
earity condition. In our sample, PABH< tends to produce the better
estimations of 〈G〉 and 〈E〉 (our example galaxy is an exception),
followed closely by PA<>3 .
5.2 Bar pattern speed
In G20, we assumed that each error source was independent and
could be added in quadrature for our final estimation of Ω10A . This
Figure 6. Weighting the 3 PA measurements. Top panel: PA measurements
of 8596-12704 before penalising. The three methods do not agree within their
1-sigma uncertainties: PABH< = 73.0 ± 1.3, PA<>3 = 79.3 ± 0.8, PA?ℎ =
85.9 ± 2.48. Bottom panel: PA measurements after penalisation. PABH< =
73.0 ± 25.0, PA<>3 = 79.3 ± 2.5, PA?ℎ = 85.9 ± 2.5. The TW integrals
at PABH< do not follow the lineal trend (see Figure 8). Thus we choose to
penalise it, and use an equal weight between PA<>3 and PA?ℎ .
MNRAS 000, 1–20 (2022)
Bar pattern speed in MWA galaxies 7
Figure 7. TW integrals 〈G 〉 and 〈E 〉 measured near the midpoint between
PA?ℎ and PA<>3 . Left panel: colour coded by the slit number. The black
dots show the median value for each individual slit. Middle panel: coloured
by the distance of the random centre to the mean centre of mass. Right panel:
coloured by the relative slit length.
assumption was incorrect, as the behaviour of the TW integrals is
non-linear, especially when considering the PA and the slit length.
Instead, in this work, we choose to sample all parameters and error
sources in the same Monte Carlo procedure.
We modelled the PA as a gaussian distribution with parameters
from the weighted measurement of the three PA measurements de-
scribed in the last section. The galaxy centre was estimated from
the centre of mass of the r-band surface-brightness distribution. The
mean values and co-variance matrix between G and H coordinates are
used to model a 2-D Gaussian distribution. Considering the spatial
correlations, the centre distribution looks like an ellipse oriented in
the same direction as the bar.
The relative slits length (relative to the maximum slit length that
fits the data) was modelled using a half-Gaussian distribution with
` = 1 and f = 0.2. We use this distribution to ensure most of
our measurements use large slits, as the TW method suggests, but
also capture some of the systematic error associated with the data
coverage. In total, we draw a sample of ×106 combinations of these
parameters.
The number of slits varies in each galaxy as it depends on the bar
coverage, orientation and the IFU size, but on average we use ∼ 15
slits per galaxy. This results in an average of 15 ×106 TW integrals,
and 1 ×106 measurements of Ω10A × sin 8 for each galaxy.
We can separate the effects of each error source by colouring the
TW integrals as a function of the random parameters. In Figure 7,
we show the TW diagram of our example galaxy measured near the
equal-weighted mean between PA?ℎ and PA<>3 . Each point is a
measurement made by an individual slit, with a random centre and
random slit length. The slope of the plot is Ω10A× sin 8. The three
panels show the same data but are coloured using the slit number
(left panel), the distance to the centre (middle panel) and the relative
slit length (right panel). For the latter two panels, if the colours are
randomly distributed, the measurements do not dependent on the
respective error source, while stratified colours show the opposite.
In this example, 〈G〉 and 〈E〉 have a strong linear relation, and low
dispersion due the random parameters.
Figure 8 shows the same plot, but the integrals are computed
near PABH<. In this orientation, the TW integrals do not follow the
linear trend, and the dispersion of each slit strongly depends on the
slits length, as evidenced by the stratification in colours in the third
panel. When we observe strange behaviours like this, we penalise the
corresponding measurement of PA.
Figure 9 shows the relationship between Ω10A × sin 8 and the disc
PA for our example galaxy. Each point is the slope of a linear fit
between 〈G〉 and 〈E〉 with a random centre and slit length. Again,
we show three panels to disentangle the effects of each error source.
The slit length becomes more important at PAs < 82° where the
shorter slits produce smaller values of Ω10A . In this example, the
Figure 8. Same as Figure 7, but measured near PABH<. At this orientation,
the measurements highly depend on the slit length.
Figure 9. Bar pattern speed versus PA. Left panel: Coloured by the relative
slits length. Middle panel: Coloured by distance to the centre. Third panel:
Coloured by j2 of the fit. Notice how the goodness of fit heavily depends on
the PA.
most important error source is the PA (as usual), as it moves the
distribution of Ω10A× sin 8 between 10 to 25 km s−1 kpc−1. The
third panel shows the j2 of the linear fit. Notice how the linear fit is
getting worse as we move to lower values of PA (closer to PABH<).
To get the final distribution for Ω10A , we divide each measure-
ment by a randomly sampled sin 8. The inclination is recovered from
disc ellipticity using the transformation 8 = arccos(n). We got two
measurements of n : (i) from the isophote analysis and (ii) from the
HU kinematic model. As with the disc PA, we expect similar biases
to be affecting the disc ellipticity estimations, so we use an equally
weighted ellipticity in all our galaxies. The mean and standard devi-
ation of this estimate are used to model n as a Gaussian distribution.
Figure 10 shows the final distribution ofΩ10A for our example galaxy.
The inclination uncertainty smooths the distribution of Ω10A . It
is important to note that a Gaussian distribution in n naturally trans-
forms into a right-skewed distribution in sin 8. This is relevant in
galaxies with high inclination uncertainty (for example Δ8 > 6°), as
this results in a right-skewed distribution in Ω10A . We will discuss
how the inclination affects the determination of R in Section 6.3.
5.3 Bar radius
There is no consensus on what is the best way to estimate the bar
radius. Bars rarely have a well-defined boundary and are often as-
sociated with structures like bulges, spirals, rings and ansae. This
makes different techniques prone to different biases.
Traditionally, the bar radius is recovered from the isophotal el-
lipticity and PA profiles. The barred region is characterised by an
increasing ellipticity that peaks near the end of the bar and remains
constant in PA. The local maximum in ellipticity (hereafter 'n ) is
a measurement that correlates well with visual estimates (Herrera-
Endoqui et al. 2015; Díaz-García et al. 2016a). An upper limit can be
defined as the radius where the PA has changed ∼ 5° from the value
at 'n (hereafter '%) (Aguerri et al. 2009). We can also identify
the bar end with a sharp decline in the ellipticity profile. In galaxies
where we observe a sharp decline in ellipticity before '%, we use
that transition as our upper limit. Figure 3 shows both measurements
MNRAS 000, 1–20 (2022)
- 0.5 0.0 0.5 
(X) [kpe] 
Slits 
- 0.50.0 0,5 
(X) [kpe] 
6 [pixels] 
- 0.5 0.0 0.5 
(Xl [kpc] 
Slits length 
=-25 
"20 
E 
2!.15 
~10 
0 .0 
(X) (kpe] 
Slits 
Sli ts lenq,~8 
. \ , 
0.0 
(X) (kpc] 
82.S 8S.0 
PAlO) 
6 [pixels] 
6 Radius ( pj~els] 
0.0 
(X) [kpe] 
82.S 8S.0 
PAl O) 
Slits lengt h 
x' 
8 L. Garma-Oehmichen et al.
Figure 10. Bar pattern speed distribution of 8596-12704. We used a Monte
Carlo procedure where we modelled the disc PA and disc ellipticity as gaussian
distributions; the slits length as a half-gaussian distribution; and the galaxy
centre as a 2D gaussian distribution.
with a green stripe. In Figure 4 these estimates are drawn as black
ellipses.
However, the isophote technique is prone to several biases. In some
galaxies, the maximum ellipticity is actually produced by open spiral
arms that follow the bar. Ansae structures are relatively common in
late-type galaxies, and it is not clear if they should be considered part
of the bar or the disc (Martinez-Valpuesta et al. 2007). In galaxies
with large bars, the choice between logarithmic or linear isophotal
steps can alter the measurement of 'n by a few arcsec. Finally, the
bulge-to-total ratio (B/T) can affect the estimation in galaxies with
weak exponential bars (Lee et al. 2020).
Other techniques to measure the bar radius include the ratio of
the bar to inter-bar intensity, via Fourier decomposition of the light
(Aguerri et al. 2000), a 2D multi-component surface brightness de-
composition (fitting the light distribution of the disc + bulge + bar)
(Laurikainen et al. 2005; Gadotti 2011; Salo et al. 2015) and the max-
imum bar torque radius (based on the transverse-to-radial force ratio
&) ) (Sanders & Tubbs 1980; Combes & Sanders 1981; Díaz-García
et al. 2016a; Lee et al. 2020).
In G20 we modelled the bar radius using an uniform distribution
with 'n and '% as the lower and upper limits. In this work we
choose to model the bar radius as a log-normal distribution, using
'n as the mean and '% as a 2-sigma upper limit. This is:
` = log ('n ) (5)
f = log
(
√
'%
'n
)
(6)
This distribution let us explore a smaller bar radius, as suggested
by Hilmi et al. (2020) and Cuomo et al. (2021). The logarithmic
distribution also prevents sampling negative bar radius and, when the
f/` ratio is small, the distribution resembles a Gaussian distribution.
We deproject the bar using the analytical procedure described
in Gadotti et al. (2007). The method works well in galaxies with
moderate inclination angles (< 60°) (Zou et al. 2014), which is the
case for most galaxies in our sample . The method requires the relative
orientation of the bar with the disc, as well as the disc inclination
angle.
Figure 11. Rotation model of 8596-12704. Top panel: Rotation curve. The
black dots show the mean circular velocity of the Diskfit model. The blue
and red dots show the fitted radial and tangential non-circular motions. The
cyan and purple curve are the best fit of equation 7 using 60<<0 as a free
parameter or fixing it to 60<<0 = 1 respectively. Bottom panel: Angular
velocity curve. The bar pattern speed distribution is shown in blue. The bar
radius distribution is shown in green. The resulting corotation distribution is
shown in orange. The projections are limited by the 1-sigma errors.
5.4 Corotation radius and the HU rotation curve model
To get a reasonable measurement of corotation, we require a good
estimate of the rotation curve. Several works assume a quick flatten-
ing, and estimate '2A as the ratio +
5 ;0C
2 /Ω10A where +
5 ;0C
2 is the
flat circular velocity of the disc (Aguerri et al. 2015; Cuomo et al.
2019; Guo et al. 2019). Recovering the details in the shape of the
rotation curve requires careful modelling. Massive (brighter) galax-
ies typically display slowly declining rotation curves, while galaxies
with low masses have monotonically rising rotation curves (Persic
et al. 1996; Kalinova et al. 2017). Moreover, bumps and wiggles
are common features of rotation curves, that result from the per-
turbed kinematics of the galactic disc components. The amplitude
of these wiggles varies depending on the galaxy and perturber (bar,
spiral arms, satellites), and takes a proper model to be correctly rep-
resented. In G20, we got a relative difference of 21% in the '2A
estimation between modelling the rotation curve and assuming a flat
disc.
Recovering the rotation curve from 2D velocity maps is straight-
forward when the tracers have small departures from circular orbits
(Begeman 1987). However, stellar bars and other non-axisymmetric
features can drive large non-circular motions that complicate the
determination of the rotation curve (Valenzuela et al. 2007).
In this work, we use the code Diskfit that describes the velocity
field with the so-called bi-symmetric model (Spekkens & Sellwood
MNRAS 000, 1–20 (2022)
0.08 
1Il 
§ 0.06 
o 
u 
>. 
.~ 0.04 
e 
QJ 
O 
0.02 
o 5 
Median = 25.59 
+0 = 4.39 
-o = 4.72 
10 15 20 25 30 35 
Bar pattern 5peed [km 5- 1 kpc-1 ] 
40 
250 
200 
...... 
I 
1Il 150 
E 
.::,¿ 
C2 100 
S' 
50 
120 
7
u 
100 
o.. 
~ 80 
I 
1Il 
E 60 
.::,¿ 
40 
ce: 
'3 20 
O 
.,. 
. + , . • .+ . 1 • • • •• • 
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 
Radiu5 [kpc] 
O ~----~ ~~~~~ ----~------~-
O 5 10 15 20 
Radiu5 [kpc] 
Bar pattern speed in MWA galaxies 9
2007; Sellwood & Sánchez 2010). This kinematic model assumes
that most perturbations can be described by an < = 2 order harmonic
decomposition. The code yields the mean circular velocities and
amplitudes of non-circular streaming velocities in the radial and
tangential directions.
The code Diskfit uses a Levenberg Marquardt (LM) algorithm
as a minimisation technique. However, the parameter space may have
several local minimum values, where the LM routine can easily get
trapped. To overcome this issue, we used a modified version that im-
plements a Markov Chain Monte Carlo method (Aquino-Ortíz et al.,
in prep). The input parameters are optimised using the Metropolis-
Hastings algorithm (e.g. Puglielli et al. 2010). This version has been
used in recent kinematic studies (e.g. Aquino-Ortíz et al. 2018, 2020).
In most galaxies of our sample, the kinematic model is not enough
to estimate '2A . Because of the data coverage, the rotation curve only
covers the central region of the galaxy, and additional modelling is
required to extrapolate the outer parts. We use a 3 parameter model
(Bertola et al. 1991):
+ (A) =
+2A
(A2 + :2)W/2
(7)
where +2 controls the amplitude, : the sharpness and W the slope
at large radius. When W > 1 the model corresponds to a declining
curve and W < 1 to a rising curve. It is important to note that there is a
large degeneracy between+2 and W, and in some galaxies, a better fit
can be achieved by fixing W = 1 (a 2 parameter flat rotation model).
Some galaxies require no extrapolation, as '2A can be determined
within the kinematic model. In those cases, we used a spline fit that
recovers small details like wiggles that cannot be modelled with the
parametric functions.
For each measurement of Ω10A we generate a random rotation
curve with the parameters of the best fit. The top panel of Figure
11 shows the rotation curve of the example galaxy. The black dots
correspond to the mean circular velocity from the Diskfit model.
Blue and red dots at the bottom are the m=2 radial and tangential
motions, respectively. The model stops at ∼ 11 kpc due to the data
coverage. The cyan and purple curves show random iterations of
equation 7 where the difference is using W as a free parameter or
fixing it to 1. Here, the 3 parameter model fits a declining rotation
curve, while the flat model remains at +2 = 250 km s−1. At large
radii, the difference between the two curves becomes more important.
The bottom panel of Figure 11 shows the angular velocity curve.
The blue marginalised distribution in the right is the bar pattern speed
we obtained in Figure 10. The green distribution is the bar radius we
modelled in Section 5.3. The corotation radius distribution is shown
in orange, and was obtained form the intersections of Ω10A with the
3 parameter random angular velocity curves.
5.5 Rotation rate R
To estimate R we divide each element of '2A with a random bar
radius modelled with a log-normal distribution. The shape of R
distribution is usually skewed to the right. It is common that galaxies
with large uncertainties (in PA, inclination or bar radius) will show
a large tail in R in the slow bar regime.
In Figure 12 we show the resulting probability distribution of the
example galaxy.We estimate the probability of bar being slow, fast
or ultra-fast by using the area of the probability distribution. In this
particular case, the median of the distribution (R= 1.43) and the
most probable classification (Slow = 0.52) both coincide with a slow
rotating bar.
Figure 12. Rotation rate R of 8596-12704. The distribution is coloured using
the kinematic classifications of the bar. The area under the curve of each
classification is shown in the top right.
Work Ω10A '2A R
(1) (2) (3) (4)
Aguerri et al. (2015) 35% 43% 39%
Guo et al. (2019) 24% 28% 37%
Garma-Oehmichen et al. (2020) 18% 30% 30%
This work 20% 22% 26%
Table 1. Median relative errors in TW based measurements. For all works
we used the light-weighted results at the photometric PA orientation. Col. (1)
Work. Col. (2) Relative error in the bar pattern speed. Col. (3) Relative error
in corotation radius. Col. (4) Relative error in rotation rate.
6 RESULTS
In this section we present the statistical results of our complete sam-
ple. We include measurements of global properties from the Pipe3D
package (Sánchez et al. 2018) including stellar mass, molecular gas
mass, stellar spin parameter and stellar surface density.
6.1 Statistics of the full sample
Figure 13 shows the distribution of Ω10A , '2A and R after adding
the results of the complete sample. The median relative error is
20%, 22% and 26%, respectively. This is slightly smaller than other
TW based measurements. For reference, Table 1 shows the median
relative errors from Aguerri et al. (2015), Guo et al. (2019) and
Garma-Oehmichen et al. (2020). Nonetheless, our treatment is not
free of biases, and we comment on further improvements for future
measurements in Section 7.1.
Most of our sample is near the border between fast and slow bars.
Using the most probable classification from the distribution of R,
our sample is composed of 52 slow, 26 fast and 19 ultra-fast bars.
Figure 14 shows the deprojected bar radius versus corotation radius
of all our sample, coloured by R.
To improve the visualisation of the following figures, we opt to rep-
resent the G axis uncertainty with the dots size and the H axis uncer-
tainty with the opacity. Thus, the more diffuse-transparent (compact-
solid) dots correspond to our most uncertain (certain) measurements.
For reference, we show in red the average error bar. We also include
in the title the Spearman correlation coefficient and the correspond-
ing ?-value. All measurements can be found in the Appendix table
A1 and the public repository (See Data Availability statement).
MNRAS 000, 1–20 (2022)
1.4 
R = 1.43~g : ~~ 
1.2 -P(Slow) = 0.52 -P(Fast) = 0.43 
1.0 -P(Ultra) = 0.06 
g 0.8 
u.. 
O 
c... 0.6 
0.4 
0.2 
0.0 
O 1 2 3 4 5 
Parameter R 
10 L. Garma-Oehmichen et al.
0 10 20 30 40 50 60
Bar [km s 1 kpc 1]
0.000
0.001
0.002
0.003
0.004
0.005
0.006
PD
F(
Ba
r)
Bar=28.14+12.309.55
0 5 10 15 20 25 30
Rcr [kpc]
0.00
0.02
0.04
0.06
0.08
0.10
0.12
PD
F(
CR
)
Rcr = 7.82+3.99
2.96
0 1 2 3 4
Rotation rate 
0.0
0.2
0.4
0.6
0.8
PD
F(
)
= 1.35+0.60
0.40
P(Slow)  = 0.46
P(Fast)  = 0.35
P(Ultra) = 0.19
Figure 13. Distributions of the bar properties for the complete sample of
galaxies. Top panel: Bar pattern speed. Middle panel: Corotation radius.
Bottom panel: Rotation rate R.
6.2 Correlations
Many works have tried to found correlations between the bar pattern
speed and other bar parameters. The strongest of these relations
usually occurs between Ω10A and the bar length '10A (longer bars
rotate with lower pattern speeds). For example in G20 we found
a Spearman correlation coefficient of AB = −0.53 with a sample
0 2 4 6 8 10 12
0
5
10
15
20
1 : 1 1 : 1.4
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
R
ot
at
io
n 
ra
te
 R
r=0.76, p=2.64e-19
Deprojected bar radius [kpc]
C
or
ot
at
io
n 
ra
di
us
 [
kp
c]
Figure 14. Deprojected bar radius versus corotation radius. The dots size and
opacity are used to illustrate the uncertainty in the bar radius and corotation,
respectively. We show the average error bar in red. The ratios 1:1 and 1:1.4
used to separate ultra-fast, fast and slow bars are shown with the segmented
and dotted lines respectively.
2 4 6 8 10 12
10
20
30
40
50
60
70
80
10.4
10.6
10.8
11
11.2
lo
g 
S
te
lla
r 
M
as
s
r=-0.63, p=5.52e-12
Bar Radius [kpc]
Figure 15. Deprojected bar radius versus bar pattern speed coloured by the
logarithmic stellar mass.
of 18 galaxies. Later, using a sample of 77 galaxies with direct
measurements of Ω10A from different pieces of literature, Cuomo
et al. (2020) found a stronger relation with AB = −0.64. In this work,
we found a similar correlation (AB = −0.63) as shown in Figure 15.
The relation is best seen when coloured by the stellar mass, as it
also reveals the well known bar length - mass relation (AB = 0.62 in
our sample). Interestingly, the relation between Ω10A and the stellar
mass is not nearly as strong (we get a weak relation with AB = −0.24).
There is another global parameter that is positively correlated with
Ω10A : the disc flat circular velocity (AB = 0.29). In Figure 16 we show
this relation coloured by R, which is also positively correlated with
+
5 ;0C
2 (AB = 0.29). These two correlations suggest that discs with
large values of circular velocity will host bars with high bar pattern
speed, but will be slower in R. The correlation matrix between all
quoted quantities is shown in Figure 17.
In G20 we reported two strong correlations between the molecular
gas fraction 56 = "60B/("∗ + "60B) with R (AB = 0.54) and with
Ω10A (AB = −0.52). In this work, we found that the relation with the
rotation rate is still present, but with more scatter (AB = 0.23), and
MNRAS 000, 1–20 (2022)
Bar pattern speed in MWA galaxies 11
Figure 16. Bar pattern speed versus disc flat circular velocity coloured by
their rotation rate R. Discs with large values of circular velocity host bars
with high bar pattern speed, but are slower in R
bar RCR Rdep
bar
Vc logM * /M
ba
r
R C
R
Rd
ep ba
r
V c
lo
gM
*/
M
-0.64
-0.12 0.46
-0.63 0.76 -0.18
0.29 0.39 0.29 0.23
-0.24 0.57 -0.01 0.62 0.38
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
Figure 17. Spearman correlation matrix of quantities shown in Figures 15
and 16
that the relation with Ω10A disappeared (AB = 0.07). It is possible
that our previous work was affected by low number statistics, or was
biased by some outliers. Although weak, the relation with R does
suggest that the gas role in the bar evolution is important and should
be explored in more detail.
We also looked for correlations with other global galactic proper-
ties derived from the Pipe3D analysis (Sánchez et al. 2018). Most of
these relations are weak, and have a lot of scatter, but we mention two
that could be of interest: the Sérsic index has a weak anti-correlation
with Ω10A (AB = −0.20), and R (AB = −0.17). Similarly, the rotation
velocity-to-velocity dispersion ratio within 1 effective radius E/f
has a weak correlation with Ω10A (AB = 0.24), and R (AB = 0.12).
These two pairs of correlations suggest that more concentrated mass
distributions (or more pressure supported systems) produce bars that
rotate at lower bar pattern speed, but faster in R. Nonetheless, both =
and E/f are not corrected for the presence of the bar. A more realistic
20 30 40 50 60 70
0.5
1
1.5
2
2.5
3
R= 1 R=1.4
0
10
20
30
40
50
60
70
80
90
D
is
c 
PA
 -
 B
ar
 P
A
Figure 18. Rotation rate R versus disc photometric inclination coloured by
the relative orientation between the disc and the bar. A cluster of ultra-fast
bars is found in the low-inclination region, which we highlight with a green
circle.
study would require such corrections (Gadotti 2008; Weinzirl et al.
2009; Graham & Li 2009).
6.3 Where are the ultra-fast bars?
In recent years, the abundance of fast-rotating bars has attracted at-
tention from the cosmological simulations community. The angular
momentum exchange in simulations produces high rates of bar slow-
down. Roshan et al. (2021b) found mean values for R range between
2.5 to 3 in the cosmological simulations IllustrinsTNG and EAGLE.
Fragkoudi et al. (2021) showed that galaxies more baryon-dominated
can remain fast down to z=0. Moreover, Frankel et al. (2022) showed
that simulated bars are shorter rather than slower compared to their
observed counterparts. Alternatively, Roshan et al. (2021a) showed
that bars remain fast in modified gravity theories.
Our sample has a strong component of ultra-fast bars (using the
probability distributions, they account for ∼ 20% of our sample). Be-
fore attempting to explain their physical origin, we explore other pos-
sible systematic errors. As mentioned in Section 3, the TW method
is susceptible to fictitious signals in galaxies with low and high in-
clinations. Face-on galaxies have poor kinematic information, while
edge-on galaxies do not have positional information. Also, the sym-
metry of the TW-integrals requires the bar not to be oriented towards
the minor or major axis of the disc.
Figure 18 shows the rotation rate versus the disc photometric in-
clination. The figure is coloured by the relative orientation of the disc
and the bar, highlighting in red galaxies where the relative orientation
could be problematic. In particular, we noticed a cluster of ultra-fast
bars in the low inclination region (green circle).
We create a “quality” sub-sample by introducing the following
constrains: inclination within 30° < 8 < 60° and relative orientation
20° < |%38B2 − %10A | < 70°. This quality sub-sample consists
of 55 galaxies, this time with only 6 ultra-fast bars (∼ 10% of the
sample). In other words, 14 of our ultra-fast galaxies were probably
the result of systematic errors related to the geometric limitations of
the TW-method.
Notably, this quality sub-sample also has stronger correlations
compared to the complete sample. In table 2 we compare the corre-
lations discussed in Section 6.2 between both samples.
It is possible that some galaxies in our sample have more than
MNRAS 000, 1–20 (2022)
12 L. Garma-Oehmichen et al.
Relation Complete Sample Quality sub-sample
(1) (2) (3)
Ω10A - log "∗/"⊙ -0.23 -0.46
Ω10A - '4?
10A
-0.63 -0.71
log "∗/"⊙ - '4?
10A
0.58 0.69
Ω10A - '2A -0.64 -0.75
Ω10A - +2 0.29 0.14
R - +2 0.29 0.25
Ω10A - = -0.20 -0.31
R - = -0.17 -0.11
Ω10A - E/f 0.24 0.41
R - E/f 0.12 0.06
Table 2. Spearman correlation coefficients in the complete sample and the
quality cut sub-sample. Col. (1) Pair of variables. Col. (2) Correlation coeffi-
cients in the complete sample Col, (3) Correlation coefficients in the quality
sub-sample.
one pattern speed (for example from the spiral arms or a nested
bar). Although we choose the number of slits to cover only the bar
region, we cannot disentangle the effects that other non-axisymmetric
structures could have in the estimation of R. Some slow bars in our
sample could be affected by the spiral pattern, however it is hard to
quantify.
6.4 Milky Way like galaxies
Because of the restrictive nature of the TW method, the selection
criteria in our sample was based only upon the morphological type
and stellar mass. However, our sample contains galaxies with a great
variety of bars sizes, morphological features (rings, spirals), mass
distribution (with disc circular velocities that range from 150 to 400
km s−1) and galactic environments.
To better reflect the parameters of the MW we build a sub-sample
making an additional cut in the bar length ('10A < 6 kpc) and the
disc circular velocities (190 < +2 < 290 km s−1). We chose these
parameters as they are correlated with Ω10A . We also include the
cuts in inclination and relative orientation of the disc-bar discussed
in the previous section to improve the overall quality. The resulting
MW sub-sample comprises 25 galaxies, that are marked in Table A1
with an asterisk next to the name of the galaxy.
Figure 19 shows the resulting distribution in Ω10A and R. Most
recent estimates of the bar pattern speed in our Galaxy are in the
range Ω10A ∼ 35 − 45 km s−1 kpc−1, which lies within the 1-sigma
upper limit of our distribution.
7 DISCUSSION
7.1 Improving future measurements
Throughout this work, we have discussed many error sources and
biases that appear when measuring the bar pattern speed and de-
rived quantities. The most important of these is the disc PA, where
a few degrees of error can change the measurement dramatically.
This would not be a problem if different measurements agreed on
their uncertainties, but this is usually not the case. In this work, we
choose to do an equal weighting of different PA measurements (by
increasing their errors artificially until they become consistent), and
impose a condition over the linearity of the TW-integrals (see section
5.1). An improvement could be made by using a criterion based on
the goodness of fit j2 (maybe with a threshold condition, or using
0 10 20 30 40 50 60
Bar [km s 1 kpc 1]
0.00
0.01
0.02
0.03
0.04
0.05
0.06
PD
F(
Ba
r)
RDep
Bar  < 6 kpc,  190 < Vc < 290 km/s
Bar = 30.48+10.94
6.57
Portail et al. (2017)
Pérez-Villegas et al. (2017)
Bovy et al. (2019)
Sanders et al. (2019)
Clarke & Gerhard (2021)
Tepper-Garcia et al. (2021)
Kawata et al. (2021)
0 1 2 3 4
Rotation rate 
0.0
0.2
0.4
0.6
0.8
PD
F(
)
RDep
Bar  < 6 kpc,  190 < Vc < 290 km/s
= 1.45+0.57
0.43
P(Slow)  = 0.54
P(Fast)  = 0.31
P(Ultra) = 0.15
Figure 19. Bar pattern speed and R distributions of the MW sub-sample.
This sub-sample is built by adding cuts in the bar lengths ('
4?
10A
< 6kpc)
and the disc circular velocity (190 < +2 < 290 km s−1).
j2 to weight the PA), or by modelling the systemic errors of each
measurement.
Our procedure is biased towards measurements that produce a lin-
ear behaviour in the TW integrals (we are weighting our observations
based on the method). Sometimes this includes galaxies that should
not work with the TW-method, but nonetheless, produce a measure-
ment that has physical sense (see our discussion in sections 6.3 and
6.4).
We estimated the geometric error of Ω10A using a MC procedure
over the inclination, PA, centre and slits length. However, the number
of slits could be an important parameter to consider. In most galaxies
the TW integrals follow a linear trend up to near the bar end. We
tried to use a number of slits to fill the area enclosed by 'n , however,
there are exceptions. In some galaxies the linearity can only be seen
in the innermost slits, while in others, the central slits are clustered
together and only the outermost slits follow a lineal trend. By letting
the number of slits be a free parameter (within the uncertainties of the
bar radius) the measurements would capture some of this behaviour.
Also important is the bar radius estimation, where morphological
features (bulges, rings, ansae, spirals) affect the different measure-
ment techniques. In this work, we choose to model the bar radius
using a log-normal distribution with 'n and '% as the mean and
2-sigma upper limit of the distribution. This could be improved by
MNRAS 000, 1–20 (2022)
Bar pattern speed in MWA galaxies 13
0 2 4 6 8 10 12
0
2
4
6
8
10
12
1:1
1:0.71
Deprojected bar radius [kpc]
M
ax
im
um
 b
ar
 t
or
qu
e 
ra
di
us
 [
kp
c]
Figure 20. Bar radius estimated using the maximum torque versus the depro-
jected bar radius estimated from the maximum ellipticity isophote.
incorporating other measurements like the the 2-D decomposition
model of the surface brightness or the bar maximum torque.
Using hydrodynamical simulations, Hilmi et al. (2020) estimated
that observations could be overestimating the bar radius between
∼ 15 − 55 per cent, depending on the timescale and orientation of
the galactic structures. Cuomo et al. (2021) proposed the maximum
torque radius as a better estimator for the bar radius (hereafter '&1).
However, this method comes other with important limitations and
biases. Most importantly, in various galaxies '&1 does not agree
with the visual estimate of what constitutes the bar. Also, the method
is affected by the bulge-to-total ratio (B/T) as it can dilute &1 and
increase '&1 (Díaz-García et al. 2016a). The B/T ratio also affects
the isophote method, but only in exponential weak bars (Lee et al.
2020).
For comparison purposes, we estimated '&1 by de-projecting the
r-band DESI image and solving the Poisson equation assuming a
constant mass-to-light ratio (Buta & Block 2001). In Figure 20 we
show both bar measuring methods in our sample. We found '&1 to
be ∼ 30% smaller than 'n after deprojection.
We tried using '&1 as the 2-sigma lower limit of the distribution
(with 'n as the mean) but in some galaxies, the difference between
'n and '&1 is so significant, that the distribution had a long tail
towards high bar radii. We got some reasonable results using both
'&1 and '% as the lower and upper 2-sigma limits, but the distri-
bution of R of the complete sample widens significantly, including
some ultra-fast (R∼ 1) and ultra-slow bars (R∼ 4). Thus, we choose
to use the combination that produced more consistent results. Maybe
incorporating a Gaussian Mixture Model to estimate the intrinsic
biases could help improve some of these measurements.
8 CONCLUSIONS
• We have measured the bar pattern speed, bar radius, corotation
radius and rotation rate of a sample of 97 MaNGA galaxies using the
TW method. The sample was constructed to resemble the Milky-Way
by using cuts in the stellar mass and the morphological type. The TW
integrals were computed over the stellar component.
• We used 3 independent measurements of the disc PA from differ-
ent sources: isophote fitting of the surface brightness, symmetrization
of the stellar velocity field and fitting a kinematic model to the HU
velocity field. We assume all measurements are biased and increased
their errors to produce an equal weighted PA. We penalise PAs when
the TW integrals are not linear.
• We used a MC procedure to estimate the Ω10A distribution by
sampling the PA, inclination, centre and slit length. This procedure
let us disentangle the effects of each error source.
• We used the rotation curve from the kinematic bi-symmetric
model of HU velocity field· The model fits non-circular motions pro-
duced by the bar, and does not require an asymmetric drift correction.
• Two measurements of the bar radius were obtained from the
isophotal procedure 'n and '%. We choose to model the bar ra-
dius with a log-normal distribution using these measurements as
parameters (mean and 2-sigma).
• We found two significant correlations within our sample: (i)
Ω10A -'4?
10A
-log "∗/"⊙ which relates the size of the system to the
rotation frequency and (ii) Ω10A -+2-R, that suggest fast rotating
discs tend to host high pattern speed bars, but slow in rotation rate
R.
• We also looked for correlations with various global galactic
properties. We found a weak correlation between R and the gas
fraction. Also, the weak relations between Ω10A and R with the
Sérsic index and E/f ratio within 1 effective radius, suggest that
more concentrated mass distributions produce bars that rotate at
lower pattern speeds, but faster in R.
• We identify the inclination angle and the relative orientation
of the disc-bar as possible sources of ultra-fast rotating bars in our
sample. Using a cut in both parameters reduced the frequency of
ultra-fast from 20% to 10% of the sample.
• We build a sub-sample of MW galaxies using these quality cuts,
and an additional cut in the bar radius and disc circular velocity. The
most recent measurements of the MW bar pattern speed lie within
the upper 1-sigma of our distribution.
• We suggest future measurements to take into account all possible
biases in the procedure or, if possible, model these biases.
DATA AVAILABILITY
All Figures and relevant tables are available in the public reposi-
tory https://github.com/lgarma/MWA_pattern_speed. Other
data that support the findings of this study are available from the
corresponding author, upon reasonable request.
ACKNOWLEDGEMENTS
We thank the referee Isabel Perez for the careful revision and use-
ful comments that significantly improved the quality of the paper.
LGO acknowledge support from CONACyT scholarship. LGO and
LMM acknowledge support from PAPIIT IA101520 and IA104022
grants. OV akcnowledges support from PAPIIT grants: IG101620
and IG10122. EAO acknowledge support from the SECTEI (Secre-
taría de Educación, Ciencia, Tecnología e Innovación de la Ciudad de
México) under the Postdoctoral Fellowship SECTEI/170/2021 and
CM-SECTEI/303/2021.
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APPENDIX A: TABLE
MNRAS 000, 1–20 (2022)
16
L
.
G
a
rm
a
-O
eh
m
ich
en
et
a
l.
Galaxy log("/"⊙) Disc PA (weighted) Bar PA 8 +2 '10A '
4?
10A
Ω10A '2A R
[°] [°] [°] [km s−1] [arcsec] [kpc] [km s−1 kpc−1] [kpc]
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
7495-12704* 10.8 172.8 ± 1.1 (s+m) 147.8 ± 3.5 55.4+2.5
−2.6 223.1 ± 1.2 5.1 ± 2.0 3.9+1.3
−1.0 26.9+4.1
−5.0 7.6+2.1
−1.3 2.0+0.9
−0.6
7958-6101 10.5 153.9 ± 0.7 (p+s+m) 158.6 ± 0.9 40.7+1.1
−1.2 182.3 ± 1.1 5.2 ± 0.3 2.6+0.1
−0.1 47.9+13.1
−16.5 3.5+1.5
−0.9 1.4+0.6
−0.4
7958-3702 10.7 144.9 ± 1.0 (p+s) 68.2 ± 4.2 50.0+2.5
−2.6 246.8 ± 1.7 2.3 ± 0.2 2.8+0.3
−0.3 74.3+11.2
−11.8 3.3+0.6
−0.5 1.2+0.3
−0.2
7977-9102 10.8 113.0 ± 1.0 (m) 138.2 ± 2.5 49.1+3.2
−3.4 311.9 ± 1.5 2.3 ± 0.5 3.7+0.8
−0.6 41.1+5.2
−5.5 7.4+1.2
−0.9 2.0+0.5
−0.4
7993-12704 11.2 127.6 ± 1.5 (p+s) 26.2 ± 2.2 30.7+3.0
−3.3 252.7 ± 5.5 11.1 ± 1.8 11.2+1.7
−1.6 24.1+6.9
−6.2 10.6+3.5
−2.4 0.9+0.4
−0.2
8078-12703 10.9 4.5 ± 0.9 (p+s) 68.0 ± 1.1 38.5+3.2
−3.4 294.3 ± 2.5 8.2 ± 0.9 5.9+0.6
−0.6 42.6+8.5
−8.0 6.5+1.8
−1.4 1.1+0.3
−0.3
8085-3704 10.7 104.0 ± 3.0 (s) 108.1 ± 3.0 31.8+7.0
−8.8 196.4 ± 12.6 5.1 ± 1.2 4.0+0.8
−0.7 34.8+10.8
−8.0 4.5+1.8
−1.4 1.1+0.5
−0.4
8088-3701* 10.5 44.0 ± 2.0 (s) 169.8 ± 3.1 30.3+5.7
−6.9 244.7 ± 3.0 4.0 ± 0.7 4.2+0.6
−0.6 31.4+13.4
−10.7 7.9+2.6
−1.9 1.9+0.7
−0.5
8091-6101 11.2 85.8 ± 0.6 (p+s+m) 42.8 ± 2.7 46.1+1.8
−1.9 259.3 ± 3.2 7.3 ± 0.8 7.6+0.8
−0.7 23.3+2.6
−3.0 11.1+1.6
−1.2 1.5+0.3
−0.2
8091-12701 10.8 162.1 ± 0.8 (s+m) 111.0 ± 0.4 37.6+4.2
−4.7 224.7 ± 11.9 8.2 ± 0.9 11.7+1.2
−1.2 11.8+1.7
−1.5 18.4+2.9
−2.4 1.6+0.3
−0.3
8135-6103 11.0 34.9 ± 2.4 (m) 8.3 ± 2.2 37.4+0.5
−0.5 239.8 ± 4.4 7.5 ± 1.2 8.3+1.2
−1.1 20.2+4.1
−5.1 11.0+3.0
−2.0 1.3+0.4
−0.3
8146-9102 10.6 17.0 ± 1.0 (s) 157.4 ± 4.2 54.4+2.7
−2.8 209.5 ± 2.4 3.9 ± 0.6 6.2+1.0
−0.9 23.1+1.8
−1.7 8.7+0.5
−0.4 1.4+0.2
−0.2
8241-6102 10.6 117.0 ± 1.5 (s) 174.3 ± 5.3 21.0+1.7
−1.8 297.6 ± 2.3 4.0 ± 0.2 3.3+0.2
−0.2 45.2+10.1
−8.5 5.9+1.3
−1.3 1.8+0.4
−0.4
8245-12702 11.1 17.7 ± 0.7 (p+s) 170.3 ± 3.8 51.8+2.7
−2.8 351.2 ± 3.5 10.8 ± 1.8 9.4+1.5
−1.3 26.3+1.9
−2.8 11.6+1.0
−0.7 1.3+0.2
−0.2
8257-6103 11.2 135.9 ± 0.9 (s+m) 54.6 ± 4.4 25.8+2.6
−2.9 426.1 ± 5.9 4.2 ± 0.4 6.3+0.6
−0.6 32.1+7.0
−5.9 11.9+2.2
−2.1 1.9+0.4
−0.4
8312-12702* 10.6 92.5 ± 1.4 (p+s) 123.6 ± 2.5 38.9+4.2
−4.7 230.2 ± 1.1 6.8 ± 1.6 5.0+1.0
−0.9 29.2+4.1
−3.3 7.5+0.8
−0.8 1.5+0.3
−0.3
8312-12705 10.5 92.4 ± 5.6 (m) 130.8 ± 6.5 36.7+5.6
−6.4 108.2 ± 11.9 7.4 ± 1.7 5.4+1.1
−1.0 14.7+6.3
−5.1 5.6+4.8
−2.8 1.0+0.9
−0.5
8319-12704 10.7 109.2 ± 1.0 (p) 146.0 ± 1.9 42.8+1.6
−1.6 176.9 ± 1.2 6.8 ± 0.7 4.8+0.5
−0.4 30.9+2.5
−2.3 5.7+0.4
−0.4 1.2+0.1
−0.1
8320-6101* 10.4 4.1 ± 1.0 (p+s) 60.2 ± 7.0 47.4+2.0
−2.0 203.2 ± 12.2 4.2 ± 0.7 3.3+0.5
−0.4 54.1+9.8
−10.5 3.1+1.0
−0.9 0.9+0.3
−0.3
8324-12702* 11.0 68.5 ± 0.8 (p+m) 23.9 ± 7.5 35.9+1.2
−1.3 252.4 ± 1.1 7.5 ± 0.4 5.4+0.3
−0.3 33.3+7.0
−7.4 7.2+2.1
−1.5 1.3+0.4
−0.3
8341-12704 10.7 55.5 ± 1.9 (p+m) 80.9 ± 1.2 25.2+3.3
−3.7 116.7 ± 1.4 7.5 ± 0.4 4.9+0.2
−0.2 18.5+4.6
−4.1 5.4+1.8
−1.6 1.1+0.4
−0.3
8444-12703* 10.9 4.9 ± 0.6 (s+m) 31.7 ± 9.5 37.2+1.2
−1.2 237.7 ± 1.3 5.7 ± 0.9 5.0+0.7
−0.7 40.9+4.0
−3.9 4.5+0.8
−0.8 0.9+0.2
−0.2
8450-9102* 10.4 14.7 ± 1.1 (p+s+m) −6.6 ± 4.4 48.6+1.4
−1.4 199.0 ± 2.2 3.9 ± 0.4 4.6+0.4
−0.4 29.5+1.7
−1.9 6.6+0.4
−0.4 1.4+0.2
−0.2
8453-12701 10.4 109.0 ± 1.3 (s) 45.6 ± 2.8 41.6+2.3
−2.4 149.5 ± 4.0 5.6 ± 0.3 3.8+0.2
−0.2 23.4+4.5
−5.4 4.8+2.0
−1.5 1.3+0.5
−0.4
8454-12702 11.0 66.0 ± 1.7 (s) 93.1 ± 2.8 45.4+2.7
−2.8 250.9 ± 2.5 4.4 ± 0.5 8.3+0.8
−0.8 26.1+2.1
−1.8 9.9+0.7
−1.0 1.2+0.2
−0.2
8465-12705 10.9 49.3 ± 0.7 (s+m) 34.9 ± 4.5 53.5+4.8
−5.1 290.6 ± 2.1 5.6 ± 0.6 3.8+0.5
−0.4 65.7+10.2
−10.2 4.9+0.3
−0.3 1.3+0.2
−0.2
8486-6101* 10.8 114.9 ± 1.1 (s+m) 77.6 ± 3.0 40.4+2.8
−3.0 192.7 ± 3.9 2.9 ± 0.9 4.2+1.1
−0.9 27.5+3.0
−2.9 6.8+0.8
−0.7 1.6+0.5
−0.4
8552-9101* 11.0 110.2 ± 0.9 (s+m) 135.4 ± 1.7 32.7+1.3
−1.3 203.8 ± 1.7 3.9 ± 0.4 5.8+0.5
−0.5 26.7+2.7
−3.0 7.1+0.7
−0.6 1.2+0.2
−0.2
8561-3704 10.4 169.2 ± 1.4 (p+s) 156.5 ± 3.3 39.5+2.0
−2.0 234.3 ± 23.3 5.6 ± 1.3 5.7+1.1
−1.0 27.0+3.6
−4.8 4.9+2.6
−2.2 0.8+0.5
−0.4
M
N
R
A
S
0
0
0,1–20
(2022)
B
a
r
p
a
ttern
sp
eed
in
M
W
A
g
a
la
xies
17
8589-12705 11.1 167.8 ± 1.3 (p+m) 34.7 ± 2.7 25.0+2.3
−2.5 309.2 ± 1.3 7.4 ± 1.2 4.8+0.7
−0.7 28.9+9.1
−8.8 10.1+3.4
−2.3 2.1+0.8
−0.5
8596-12704 10.9 82.6 ± 1.8 (p+m) 58.4 ± 1.9 45.0+3.0
−3.2 258.2 ± 1.5 7.6 ± 1.3 6.7+1.0
−0.9 25.6+4.4
−4.7 9.5+1.8
−1.4 1.4+0.3
−0.3
8597-12703* 10.8 170.4 ± 1.9 (p+s+m) 137.3 ± 1.6 42.6+1.1
−1.1 228.2 ± 0.9 8.2 ± 0.9 5.1+0.5
−0.5 28.4+5.5
−6.8 7.4+1.9
−1.2 1.5+0.4
−0.3
8602-3701 10.3 153.1 ± 0.7 (p+s) 186.3 ± 3.6 50.0+1.6
−1.7 175.4 ± 3.1 4.0 ± 0.4 3.2+0.3
−0.3 26.9+2.9
−3.5 5.8+0.7
−0.6 1.8+0.3
−0.3
8602-12701 11.3 156.2 ± 2.2 (p) 193.7 ± 0.7 39.8+1.8
−1.9 256.7 ± 7.3 12.3 ± 0.6 7.8+0.4
−0.4 24.7+6.8
−8.0 9.4+4.3
−2.6 1.2+0.5
−0.3
8602-12705 11.2 143.5 ± 1.5 (m) 185.5 ± 0.7 37.9+2.5
−2.6 184.4 ± 4.7 11.1 ± 1.2 8.4+0.8
−0.8 15.9+3.1
−2.8 11.4+2.5
−1.9 1.4+0.3
−0.3
8612-12702 11.2 47.8 ± 2.1 (p+s) 69.5 ± 2.5 47.5+3.9
−4.2 287.8 ± 0.7 5.1 ± 0.3 7.6+0.5
−0.4 20.6+5.8
−6.7 12.6+5.5
−3.9 1.6+0.7
−0.5
8615-3701 10.8 106.7 ± 1.7 (s+m) 72.2 ± 3.4 33.8+4.6
−5.2 137.9 ± 3.5 2.5 ± 1.0 3.6+1.2
−0.9 29.7+5.5
−4.3 3.7+0.9
−0.9 1.0+0.4
−0.3
8616-6104 10.7 32.0 ± 0.7 (p+s) −0.6 ± 6.4 61.6+4.5
−4.7 251.7 ± 6.7 3.9 ± 0.4 7.0+1.3
−1.0 24.1+3.4
−4.1 9.8+1.9
−1.5 1.4+0.4
−0.3
8622-12704 11.1 76.6 ± 1.3 (p+m) 62.8 ± 0.2 42.8+1.9
−1.9 265.3 ± 4.8 3.8 ± 0.2 5.9+0.3
−0.3 27.3+6.3
−8.9 8.5+3.6
−2.0 1.4+0.6
−0.3
8624-9102 10.6 142.0 ± 1.3 (s) 132.4 ± 1.9 46.6+2.0
−2.1 198.4 ± 3.5 8.9 ± 0.9 5.4+0.5
−0.5 25.3+5.1
−6.8 7.1+1.8
−1.1 1.3+0.4
−0.2
8625-12703 11.0 88.0 ± 1.0 (s) 129.2 ± 5.8 69.1+0.4
−0.4 266.8 ± 3.7 5.0 ± 0.5 7.4+0.7
−0.7 27.6+6.3
−8.3 7.7+3.5
−1.9 1.0+0.5
−0.3
8655-3701 11.0 148.0 ± 1.7 (s) 160.0 ± 2.0 36.0+6.7
−8.1 140.8 ± 3.2 5.9 ± 1.0 9.2+1.4
−1.2 15.4+5.2
−4.9 8.7+3.4
−2.4 1.0+0.4
−0.3
8656-6103* 11.2 14.6 ± 1.6 (s+m) −39.6 ± 2.6 27.4+5.0
−6.1 254.3 ± 3.7 3.3 ± 0.6 4.8+0.7
−0.6 32.7+11.0
−8.6 7.9+2.5
−1.8 1.7+0.6
−0.4
8713-9102 10.6 151.5 ± 1.1 (s+m) 187.7 ± 3.1 34.6+4.2
−4.7 131.9 ± 26.2 6.8 ± 0.7 5.3+0.5
−0.5 21.1+4.0
−3.3 5.3+1.6
−1.3 1.0+0.3
−0.3
8715-12701* 10.9 134.0 ± 1.3 (s) 166.5 ± 1.6 40.2+1.1
−1.1 257.7 ± 4.2 10.3 ± 1.7 5.7+0.8
−0.8 25.7+3.8
−5.4 7.8+3.4
−1.9 1.4+0.6
−0.4
8718-12701 10.7 72.1 ± 2.4 (m) 88.2 ± 0.8 37.0+1.9
−1.9 132.3 ± 3.3 7.5 ± 0.8 8.2+0.8
−0.7 10.0+1.6
−2.2 12.3+3.6
−2.0 1.5+0.5
−0.3
8721-6103 11.2 50.8 ± 0.8 (s+m) 136.3 ± 3.0 32.7+5.4
−6.4 352.8 ± 3.5 4.5 ± 0.7 5.1+0.8
−0.7 40.6+13.1
−9.3 8.5+2.3
−2.2 1.6+0.5
−0.5
8938-12702 11.1 53.4 ± 1.4 (s+m) 32.1 ± 0.3 43.9+1.4
−1.4 303.4 ± 2.5 7.6 ± 0.4 6.9+0.3
−0.3 25.9+8.2
−9.2 10.8+5.2
−3.3 1.5+0.7
−0.5
8940-12702* 11.0 140.2 ± 0.9 (p+m) 112.2 ± 1.3 44.7+1.0
−1.1 240.1 ± 3.2 8.9 ± 2.1 5.7+1.1
−1.0 36.7+2.2
−2.4 5.7+0.4
−0.4 1.0+0.2
−0.2
8948-12702 11.0 1.0 ± 1.3 (s) 49.7 ± 0.6 19.8+1.1
−1.2 246.5 ± 5.7 6.0 ± 1.0 3.4+0.5
−0.5 46.5+28.5
−21.1 4.1+1.2
−0.5 1.3+0.4
−0.3
8978-9101* 10.7 93.6 ± 0.7 (s+m) 251.5 ± 0.8 35.1+2.7
−2.9 202.1 ± 2.3 5.8 ± 0.6 3.8+0.4
−0.3 26.6+3.6
−4.0 6.4+1.2
−0.9 1.7+0.4
−0.3
8978-3701 10.4 19.6 ± 1.4 (p+s) 126.0 ± 0.5 46.4+5.1
−5.6 245.1 ± 47.7 5.6 ± 0.3 4.8+0.5
−0.5 32.6+13.0
−11.2 5.8+2.3
−1.9 1.2+0.5
−0.4
8979-12701 11.2 121.0 ± 1.1 (s+m) 161.9 ± 4.4 50.6+0.7
−0.7 286.5 ± 1.2 4.2 ± 1.0 9.3+1.9
−1.6 24.7+2.8
−2.5 10.4+1.3
−1.3 1.1+0.3
−0.2
8983-12701* 10.5 70.4 ± 0.8 (p+s+m) 125.1 ± 8.7 51.6+4.1
−4.4 245.6 ± 7.4 3.8 ± 0.6 3.1+0.6
−0.5 46.6+9.0
−9.2 4.8+1.0
−1.0 1.5+0.4
−0.4
8983-3703* 10.8 174.9 ± 0.6 (p+s+m) 147.3 ± 3.5 57.9+3.3
−3.5 205.1 ± 2.7 4.9 ± 1.9 4.1+1.3
−1.1 29.2+4.9
−6.6 6.9+2.0
−1.5 1.7+0.8
−0.5
8984-12704* 10.4 114.0 ± 1.7 (s) 73.7 ± 0.8 35.1+6.0
−7.1 255.3 ± 5.0 9.3 ± 1.0 5.8+0.6
−0.6 21.3+5.2
−4.0 11.4+1.9
−1.6 2.0+0.4
−0.3
8985-9102 11.0 127.0 ± 1.7 (m) 45.4 ± 3.4 57.5+1.6
−1.6 295.0 ± 7.6 3.2 ± 0.3 8.4+0.9
−0.8 23.1+5.5
−9.1 10.3+7.2
−4.2 1.2+0.9
−0.5
8989-3703 10.6 36.1 ± 0.9 (s+m) 47.3 ± 0.5 46.3+1.2
−1.3 363.2 ± 20.4 8.2 ± 1.4 4.7+0.7
−0.6 46.3+4.7
−5.4 4.8+1.5
−1.2 1.0+0.4
−0.3
8993-12701 11.1 138.0 ± 0.9 (s+m) 70.6 ± 1.4 54.9+7.4
−8.2 307.3 ± 1.1 8.3 ± 0.9 6.5+1.6
−1.1 53.0+14.3
−15.5 5.7+1.5
−0.9 0.9+0.3
−0.2
9025-3703 11.1 52.0 ± 0.7 (s+m) 163.1 ± 2.2 32.6+2.6
−2.8 303.5 ± 17.6 4.3 ± 0.2 5.6+0.3
−0.3 32.2+3.3
−3.0 7.5+0.6
−0.8 1.3+0.1
−0.1
9028-12704* 10.7 55.6 ± 0.9 (p+s+m) 1.2 ± 0.3 49.3+1.9
−2.0 239.9 ± 0.9 4.7 ± 0.2 4.6+0.3
−0.3 35.7+10.2
−9.7 6.5+2.3
−1.6 1.4+0.5
−0.3
9029-12704* 10.8 116.0 ± 1.0 (s) 85.0 ± 2.0 51.2+2.5
−2.6 227.6 ± 0.8 6.3 ± 1.0 5.1+0.8
−0.7 27.7+3.2
−3.6 8.0+1.1
−1.1 1.5+0.3
−0.3
M
N
R
A
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0
0
0,1–20
(2022)
18
L
.
G
a
rm
a
-O
eh
m
ich
en
et
a
l.
9042-12703 10.8 147.0 ± 1.1 (s+m) 105.5 ± 2.4 40.1+8.8
−10.8 334.6 ± 12.8 8.6 ± 0.4 6.9+0.8
−0.6 36.9+10.0
−5.8 9.1+1.9
−1.7 1.3+0.3
−0.3
9046-12702* 10.9 41.1 ± 1.2 (p+s+m) 89.5 ± 1.7 43.2+0.8
−0.8 245.2 ± 1.5 6.1 ± 0.6 5.1+0.5
−0.5 31.0+3.8
−4.0 7.4+0.8
−1.1 1.4+0.2
−0.2
9047-12703 11.3 104.7 ± 0.7 (p+s) 133.9 ± 1.6 54.1+0.6
−0.6 310.9 ± 2.5 9.3 ± 0.5 11.8+0.6
−0.6 21.5+1.6
−1.8 13.5+1.2
−0.9 1.1+0.1
−0.1
9182-12703 11.1 121.5 ± 0.9 (s+m) 133.4 ± 1.4 31.5+5.0
−5.8 278.0 ± 3.1 5.8 ± 1.3 8.7+1.8
−1.5 32.0+6.5
−4.5 7.8+1.8
−1.2 0.9+0.3
−0.2
9187-3701 11.2 143.9 ± 2.1 (s+m) 99.9 ± 0.5 42.0+5.5
−6.2 105.2 ± 1.5 5.3 ± 0.3 9.5+0.8
−0.7 8.2+2.3
−1.9 11.4+2.5
−2.1 1.2+0.3
−0.2
9187-12704 10.9 78.2 ± 0.9 (s+m) 93.7 ± 1.7 42.0+1.8
−1.8 214.5 ± 1.3 8.4 ± 0.9 5.2+0.5
−0.5 12.4+3.5
−3.7 14.3+3.7
−2.7 2.7+0.8
−0.5
9196-12701 11.0 123.9 ± 2.2 (s+m) 178.7 ± 0.7 29.9+0.5
−0.5 305.9 ± 1.1 12.6 ± 0.6 9.1+0.4
−0.4 23.8+6.9
−7.0 11.7+3.2
−2.2 1.3+0.4
−0.3
9484-12703 10.7 118.0 ± 1.7 (s) 101.5 ± 5.9 27.4+2.0
−2.1 277.4 ± 5.8 9.5 ± 0.5 6.4+0.3
−0.3 32.6+5.5
−5.7 5.4+2.5
−2.1 0.8+0.4
−0.3
9490-6102 10.4 165.4 ± 1.0 (p) 149.5 ± 3.0 58.3+4.1
−4.3 212.1 ± 4.2 8.0 ± 1.3 5.8+1.1
−0.9 27.9+4.9
−6.5 6.7+2.2
−1.3 1.2+0.5
−0.3
9492-6101 10.4 6.0 ± 1.5 (s) 158.4 ± 1.9 35.5+5.1
−5.8 138.6 ± 4.4 4.6 ± 0.8 2.9+0.4
−0.4 24.9+5.2
−4.4 3.7+1.5
−1.4 1.2+0.6
−0.5
9502-12703* 11.2 22.0 ± 1.3 (s) 166.1 ± 3.3 42.0+7.9
−9.4 245.2 ± 2.3 4.9 ± 0.5 3.4+0.5
−0.4 29.3+7.7
−5.6 7.9+1.5
−1.4 2.3+0.5
−0.5
9867-12704* 10.5 90.1 ± 1.6 (p+s) 56.2 ± 2.7 51.0+3.0
−3.1 202.5 ± 1.8 3.5 ± 0.6 2.9+0.5
−0.4 43.5+6.4
−5.9 3.4+1.6
−1.6 1.2+0.6
−0.5
9881-12705 10.8 49.2 ± 1.3 (p+s+m) 62.3 ± 0.9 36.0+3.3
−3.6 225.4 ± 3.6 11.4 ± 0.6 7.5+0.3
−0.4 22.2+2.9
−3.0 9.6+1.5
−1.2 1.3+0.2
−0.2
9890-12702 11.2 139.3 ± 0.7 (s+m) 121.6 ± 1.8 42.2+0.7
−0.7 240.6 ± 0.9 5.3 ± 1.2 7.7+1.5
−1.3 27.0+1.8
−2.0 8.5+0.8
−0.7 1.1+0.2
−0.2
9894-12702 10.7 6.9 ± 1.1 (p+s) 142.3 ± 3.1 43.9+10.2
−12.6 421.7 ± 35.4 3.2 ± 0.3 4.7+1.0
−0.7 39.6+10.7
−6.2 10.2+2.2
−2.0 2.1+0.6
−0.5
10001-6102 10.7 55.2 ± 1.1 (p) 72.9 ± 2.5 55.3+7.0
−7.7 296.9 ± 27.4 6.8 ± 1.6 4.3+1.0
−0.8 38.7+6.4
−7.0 8.4+3.9
−3.5 1.9+1.0
−0.8
10213-12705* 10.7 152.2 ± 2.2 (p+s) 14.1 ± 1.4 40.6+4.2
−4.6 281.7 ± 4.3 7.5 ± 0.4 5.8+0.4
−0.4 24.1+4.4
−4.4 10.6+1.6
−1.3 1.8+0.3
−0.2
10222-12704 11.1 148.7 ± 0.7 (s+m) 38.7 ± 14.7 23.8+2.3
−2.6 346.4 ± 2.7 4.4 ± 0.7 7.2+1.1
−0.9 31.3+4.5
−3.9 10.5+1.6
−1.4 1.4+0.3
−0.3
10510-6101 10.7 36.1 ± 2.4 (p+s) 73.1 ± 3.4 43.2+7.8
−9.2 314.4 ± 6.4 3.8 ± 1.5 4.9+1.7
−1.3 42.3+11.9
−9.1 7.0+2.0
−1.7 1.4+0.7
−0.5
10518-9102 10.5 87.0 ± 1.0 (s) 73.5 ± 0.5 41.8+2.6
−2.7 197.9 ± 9.1 8.2 ± 0.4 5.0+0.2
−0.2 28.1+2.5
−2.6 6.1+0.7
−0.7 1.2+0.1
−0.1
10520-6101 10.4 177.3 ± 2.9 (p+s+m) 59.0 ± 3.4 23.5+2.9
−3.2 217.4 ± 14.5 5.6 ± 0.9 3.8+0.6
−0.5 42.6+10.2
−7.4 4.1+1.5
−1.6 1.1+0.5
−0.4
11016-12703 11.2 92.7 ± 0.9 (p+s+m) 64.4 ± 5.3 47.3+4.6
−5.0 309.1 ± 28.9 6.6 ± 1.1 7.6+1.2
−1.1 31.7+3.6
−3.4 9.4+1.4
−1.3 1.2+0.3
−0.2
11017-12704* 10.6 66.6 ± 1.4 (p+s) 88.7 ± 6.1 47.2+7.2
−8.2 198.9 ± 1.0 6.2 ± 1.0 3.7+0.6
−0.5 44.7+11.2
−10.1 3.8+1.4
−1.1 1.0+0.4
−0.3
11754-12705 11.1 66.0 ± 1.7 (s) 21.2 ± 0.6 44.9+8.4
−9.9 131.5 ± 5.3 12.2 ± 1.3 9.2+1.5
−1.2 9.2+3.1
−2.4 13.0+5.1
−3.7 1.4+0.6
−0.4
11872-12702 10.9 163.3 ± 1.7 (p+s+m) 115.0 ± 4.0 53.3+2.6
−2.6 228.4 ± 5.2 5.0 ± 0.5 9.5+1.1
−1.0 13.1+2.0
−2.2 14.1+3.8
−3.2 1.5+0.4
−0.4
11957-9101 11.0 4.8 ± 1.1 (m) 130.7 ± 20.2 49.7+1.4
−1.4 338.0 ± 20.5 7.3 ± 3.5 8.1+3.1
−2.3 26.0+7.5
−6.7 11.3+4.7
−3.4 1.4+0.8
−0.6
11958-6104 10.9 77.6 ± 1.5 (s+m) 89.6 ± 79.3 26.5+4.6
−5.6 403.0 ± 5.5 6.6 ± 1.1 9.8+1.5
−1.4 23.8+6.8
−5.1 15.6+3.4
−2.9 1.6+0.4
−0.4
11963-9102* 10.4 65.3 ± 0.9 (s+m) 115.6 ± 0.8 43.4+3.0
−3.2 223.1 ± 2.1 5.8 ± 1.0 5.1+0.8
−0.7 36.8+3.4
−3.2 5.8+0.6
−0.6 1.1+0.2
−0.2
11970-12704 10.9 89.9 ± 1.4 (p+s) 42.7 ± 5.1 46.5+5.3
−5.8 192.6 ± 5.4 8.9 ± 1.5 7.7+1.4
−1.2 20.1+5.2
−5.1 9.3+3.1
−2.1 1.2+0.5
−0.3
11970-6102 10.9 168.2 ± 0.8 (s+m) 13.0 ± 2.9 33.6+7.8
−9.9 367.7 ± 15.5 5.2 ± 0.9 5.3+0.8
−0.7 26.2+8.2
−4.7 12.5+2.6
−2.5 2.3+0.6
−0.6
11977-1902 10.7 77.9 ± 1.2 (s+m) 51.6 ± 4.5 52.6+7.6
−8.5 334.0 ± 9.2 4.4 ± 0.7 5.6+1.3
−0.9 29.8+5.8
−5.9 8.7+1.7
−1.4 1.5+0.5
−0.4
12084-3702 10.3 48.5 ± 1.9 (s+m) 108.7 ± 4.6 26.2+2.2
−2.4 244.1 ± 3.0 4.5 ± 0.5 3.0+0.3
−0.3 27.8+12.0
−11.4 7.5+2.9
−2.0 2.5+1.0
−0.7
12487-9101 10.5 179.0 ± 1.3 (p+s+m) 155.5 ± 0.8 51.2+1.8
−1.9 155.2 ± 2.4 3.4 ± 0.8 4.0+0.8
−0.7 40.9+2.7
−2.9 3.4+0.6
−0.6 0.8+0.2
−0.2
M
N
R
A
S
0
0
0,1–20
(2022)
B
a
r
p
a
ttern
sp
eed
in
M
W
A
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a
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xies
19
12490-3703 10.5 −2.9 ± 2.8 (p+s) 103.8 ± 3.0 21.2+1.9
−2.1 247.9 ± 14.5 5.6 ± 1.3 5.1+1.1
−0.9 20.9+9.4
−6.6 9.2+3.4
−2.9 1.8+0.8
−0.6
12700-6102 10.4 169.3 ± 3.3 (p) 131.3 ± 3.0 36.6+9.1
−11.7 333.7 ± 9.5 3.9 ± 0.4 4.1+0.5
−0.4 39.4+14.4
−11.1 8.3+2.5
−1.9 2.0+0.7
−0.5
Table A1: Col. (1): Galaxy MaNGA name. Col. (2): Logarithmic Stellar Mass. Col. (3): Weighted Disc Position Angle (PA that are equally weighted, p:photometric, s:symetric, m:model).
Col. (4): Bar photometric position angle. Col. (5): Weighted disc inclination (photometric + model). Col. (6): Disc circular velocity fitted from equation 7 with W = 1. Col. (7): Bar radius
estimated from the maximum isophote ellipticity method. Col. (8): Deprojected bar radius. Col. (9): Bar pattern speed. Col. (10): Corotation radius. Col. (11): Rotation rate.
M
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(2022)
20 L. Garma-Oehmichen et al.
This paper has been typeset from a TEX/LATEX file prepared by the author.
MNRAS 000, 1–20 (2022)
Chapter 6
Conclusions
6.1 Main conclusions:
In the first paper presented in this thesis, we used numerical simulations in analytical
potentials to perform a statistical study on how the bar properties affect the spiral arms.
Our parameter space included 4 bar properties: (i) length, (ii) ellipticity, (iii) mass and
(iv) rotation rate. These experiments were run in three models with the same total mass
but different mass distribution resulting in : (i) rising, (ii) flat and (iii) declining rotation
curves.
To estimate the properties of the resulting spirals we used a clustering algorithm based
on density called DBSCAN. We used a random forest regressor to quantify the importance
of the relations found. The main results were the following:
• The bar pattern speed and rotation rate are strongly correlated with the spiral pitch
angle.
• The bar strength is strongly correlated with the spiral amplitude.
• Galaxies with declining rotation curves have a stronger response to the bar pertur-
bation
• Radially heated discs produce lower amplitude, more diffuse, and more open spirals.
61
CHAPTER 6. CONCLUSIONS 62
In the second paper we measured the bar pattern speed of 97 Milky Way Analogue
galaxies from the MaNGA survey using the Tremaine-Weinberg method. We included
three independent measurements of the disc PA and introduce a new strategy to weight
them based on the how well they adapt to the method. Our main conclusions are the
following:
• A significant fraction of ultra-fast bars in our sample occurred in galaxies where the
geometrical limitations of the TW method become troublesome. After constraining
the inclination and relative orientation of the bar and disc, we reduced the ultra-fast
fraction from ∼ 20% to 10% of the sample.
• We confirm a well known relation between the bar pattern speed, bar length and
stellar mass was observed.
• Another set of correlations between ΩBar, R and V flat
c , suggest that disc rotating
with high values of circular velocity, will host bars rotating with high values of Ωbar
but will be slower in R. However, the scatter is large and the correlation is weak.
• We also observe a weak correlation between Ωbar and R with the global Sérsic in-
dex and the velocity-to-velocity dispersion ratio within 1 effective radius. These
relations suggest more concentrated mass distributions produce bars that rotate at
lower pattern speeds but faster in R.
• Our measurements agree within 1-sigma with the current estimates of the bar pattern
speed in our Galaxy.
6.2 Future work
As a result from this thesis project, I have developed tools for the analysis of the bar pattern
speed using IFU data. However, the code still requires major refactoring before being
usable by the astronomical community. More specifically, the Monte Carlo computations
and TW integrals are performed in Fortran code, while the data analysis is done in Phyton.
CHAPTER 6. CONCLUSIONS 63
Integrating the tools, with clear documentation, with a new set of measurements would be
ideal.
Cuomo et al. (2021) and Hilmi et al. (2020) have pointed out the bar radius as a possible
explanation for the fast bar problem. Numerous biases related to the position of the spiral
arms can change the measurements substantially. However, no work has tried to quantify
the PA biases. It is well known that the disc orientation is the most important parameter
for the TW method, where an error of few degrees results in dramatically different results.
How much does the spiral arms can affect the PA estimation? What about companion
galaxies or warped discs? How much the bar itself is biasing the kinematic methods?
Performing measurements in numerical simulations could lead to better estimations of the
disc PA, and with that, bar pattern speeds.
In Garma-Oehmichen et al. (2021) we predicted that there is a correlation between the
spiral pitch angle and the bar pattern speed. To tie together the two works presented in
this thesis, we could measure both quantities in a sample of galaxies where the bar could
be driving the spiral structure.
Appendix A
Photometric Signature of Ultraharmonic
Resonances in Barred Galaxies
In this paper, led and written by Dhanesh Krishnarao, we have revisit the “dark gaps”
method, to estimate the bar pattern speed. Dark gaps are regions located along the bar
minor axis that are dimmer in surface brightness than the bar major axis at the equivalent
radius.
Using the GALAKOS high-resolution N-body simulation, we show an alternate inter-
pretation of the dark gaps method. Instead of being the location of the bar corrotation
they seem to be related to the 4:1 ultraharmonic resonance.
My contribution:
I had access to an early draft of the paper thanks to the SDSS collaboration. I contributed
by discussing some of the following ideas:
• Clarifying some misinterpretations of the dark gaps method introduced in Buta
(2017b).
• Discussion on why increasing the physical resolution resulted in lower values for the
ultraharmonic resonance.
64
APPENDIX A. PHOTOMETRIC SIGNATUREOF ULTRAHARMONIC RESONANCES IN BARREDGAL
• Discussion on possible errors and biases. Particularly, on how this method could be
a better alternative in face-on galaxies, where the TW method fails.
Photometric Signature of Ultraharmonic Resonances in Barred Galaxies
Dhanesh Krishnarao
1,2,3
, Zachary J. Pace
1
, Elena D’Onghia
1
, J. Alfonso L. Aguerri
4,5
, Rachel L. McClure
1
,
Thomas Peterken
6
, José G. Fernández-Trincado
7
, Michael Merrifield
6
, Karen L. Masters
8
, Luis Garma-Oehmichen
9
,
Nicholas Fraser Boardman
10
, Matthew Bershady
1,11,12
, Niv Drory
13
, and Richard R. Lane
14
1
Department of Astronomy, University of Wisconsin–Madison, Madison, WI 53706, USA; krishnarao@astro.wisc.edu
2
NSF Astronomy & Astrophysics Postdoctoral Fellow, Johns Hopkins University, Baltimore, MD 21218, USA
3
Department of Physics, Colorado College, Colorado Springs, CO 80903, USA
4
Instituto de Astrofísica de Canarias, Tenerife, Spain
5
Departamento de Astrofísica, Universidad de La Laguna, Tenerife, Spain
6
School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, UK
7
Instituto de Astronomía, Universidad Católica del Norte, Av. Angamos 0610, Antofagasta, Chile
8
Department of Physics & Astronomy, Haverford College, Haverford, PA 19041, USA
9
Instituto de Astronomía, Universidad Nacional Autónoma de México, Apartado Postal 70-264, México D.F., 04510, México
10
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA
11
South African Astronomical Observatory, Cape Town, South Africa
12
Department of Astronomy, University of Cape Town, Private Bag X3, Rondebosch 7701, South Africa
13
McDonald Observatory, The University of Texas at Austin, Austin, TX 78712, USA
14
Instituto de Astronomiá y Ciencias Planetarias de Atacama, Universidad de Atacama, Copayapu 485, Copiapó, Chile
Received 2021 March 31; revised 2022 February 28; accepted 2022 March 10; published 2022 April 19
Abstract
Bars may induce morphological features, such as rings, through their resonances. Previous studies suggested that
the presence of “dark gaps,” or regions of a galaxy where the difference between the surface brightness along the
bar major axis and that along the bar minor axis is maximal, can be attributed to the location of bar corotation.
Here, using GALAKOS, a high-resolution N-body simulation of a barred galaxy, we test this photometric method’s
ability to identify the bar corotation resonance. Contrary to previous work, our results indicate that “dark gaps” are
a clear sign of the location of the 4:1 ultraharmonic resonance instead of bar corotation. Measurements of the bar
corotation can indirectly be inferred using kinematic information, e.g., by measuring the shape of the rotation
curve. We demonstrate our concept on a sample of 578 face-on barred galaxies with both imaging and integral field
observations and find that the sample likely consists primarily of fast bars.
Unified Astronomy Thesaurus concepts: Barred spiral galaxies (136); Galaxy structure (622); Galaxy evolution
(594); Disk galaxies (391); Galaxy kinematics (602); Galaxy dynamics (591)
1. Introduction
Observations reveal that 50% to over 70% of nearby disk
galaxies host a bar (e.g., Eskridge et al. 2000; Aguerri et al.
2009; Nair & Abraham 2010; Masters et al. 2011), including
our own Milky Way (Binney et al. 1991; Weiland et al. 1994;
Hammersley et al. 2001; Benjamin et al. 2005). Bars are long-
lived features with significant effects on the stellar and gas
distribution and kinematics throughout galaxies (Athanassoula
et al. 2005; Barazza et al. 2008; Sheth et al. 2008). To inform a
complete picture of galaxy evolution, we must understand the
formation of bars and their impact on a galaxy’s dynamics and
evolutionary track. Bars form both as a result of interactions
(Noguchi 1987; Elmegreen et al. 1991; Romano-Díaz et al.
2008; Martinez-Valpuesta et al. 2016) and spontaneously from
gravitational instabilities (Toomre 1964; Polyachenko 2013).
The strong nonaxisymmetric nature of bars allows for the
transport of angular momentum, energy, and mass across large
radial scales within a galaxy (e.g., Debattista & Sellwood 2000;
Athanassoula 2003).
Additionally, bars are thought to rotate as a solid body with
an angular velocity called the pattern speed (Ωp) and can thus
drive resonances with the rotating matter in the disk beyond the
physical extent of the bar itself. The pattern speed of bars is one
of the fundamental parameters necessary to understand the
dynamics of disk galaxies. Corotation is defined as the radius at
which the galaxy’s circular angular rotation is equal to the bar
pattern speed. The bar pattern speed also often determines the
locations of the 2:1 Lindblad resonance (Lindblad 1941) and
the 4:1 ultraharmonic resonance (UHR), which can both induce
the formation of structures, such as rings (e.g., Buta 1986).
These dynamical parameters and bar resonance locations
typically require stellar kinematics measurements to derive,
with Tremaine & Weinberg (1984) outlining the most accurate
and only model-independent method. Some attempts to tie
morphological features, such as rings, to the locations of
resonances have been made (e.g., Buta 1986, 2017). The
locations of such features could then be used to infer resonance
locations and pattern speeds but are model dependent. In
particular, Buta (2017) proposed that the location of “dark
gaps,” or regions of a galaxy along the bar minor axis that are
dimmer in surface brightness than the equivalent radius along
the bar major axis, corresponds to the location of bar
corotation.
Buta (2017) used multiband images of ringed, barred
galaxies to identify these “dark gaps” and compared the results
with an N-body simulation from Schwarz (1984). “Dark gaps”
were attributed to a deficiency of material toward L4,5, the often
stable Lagrangian points arising from the gravitational potential
of a bar that may become unstable in the presence of a strong
The Astrophysical Journal, 929:112 (13pp), 2022 April 20 https://doi.org/10.3847/1538-4357/ac5d55
© 2022. The Author(s). Published by the American Astronomical Society.
Original content from this work may be used under the terms
of the Creative Commons Attribution 4.0 licence. Any further
distribution of this work must maintain attribution to the author(s) and the title
of the work, journal citation and DOI.
1
lA ~LJ 
bar (Binney & Tremaine 1987). Buta (2017) argues that
identifying gaps in the galaxy images along the bar minor axis
should correspond to the L4,5 Lagrange points, where unstable
“banana” orbits of the stars result in a decrease in the number
density of stars surrounding these points. Kim et al. (2016)
analyzed the presence of dark gaps with other bar properties
and discovered that the light deficit produced by dark gaps was
larger in longer and stronger bars. They interpreted these
findings as evidence for dark gaps produced by the redistribu-
tion of mass due to bar evolution processes.
Here we use state-of-the-art Milky Way–like N-body
simulation results to test this proposed method directly on a
more general case of a barred galaxy. Using the simulations, we
construct synthetic, face-on maps of the stellar surface density
and identify bar resonance locations with these gaps. We show
that the location of “dark gaps” in this nonringed galaxy does
not correspond to the L4,5 Lagrangian points but instead occurs
close to the location of the 4:1 ultraharmonic resonance.
In this work, an alternative to the Buta (2017) corotation
interpretation of dark gaps is suggested based on results of
high-resolution N-body simulations (GALAKOS; D’Onghia &
Aguerri 2020). This paper is structured as follows. Section 2
briefly describes the numerical simulation and observational
data used to calibrate and apply this method. Section 3
describes our new method and tests its accuracy across
different physical resolutions and epochs of the simulation. In
Section 4, we apply this method to a sample of 578 nearby
galaxies to analyze the bar dynamics in this population of
barred galaxies. Lastly, we discuss the broad implications of
our results in comparison to other methods in Section 5, and we
summarize in Section 6.
2. Data
We use results of an N-body simulation of a Milky Way–like
galaxy described in D’Onghia & Aguerri (2020) and data
associated with the Sloan Digital Sky Survey (SDSS; York
et al. 2000). The details of these data are described below.
Throughout this work, we assume a standard cosmology of
WMAP9 (Hinshaw et al. 2013) implemented via astropy15
(Robitaille et al. 2013).
2.1. N-body Simulations
GALAKOS is a ∼90 million N-body simulation run with
GADGET3. The numerical experiment follows the motion of
disk stars in a Milky Way–like galaxy. The stellar disk has
structural parameters configured to correspond with the current
Milky Way (Bland-Hawthorn & Gerhard 2016). The simula-
tion develops self-consistent spiral patterns and a stellar bar that
arises spontaneously from the stars themselves (D’Onghia
et al. 2013). After 2.5 Gyr of evolution, the simulated galaxy
develops prominent spiral structure and a bar with 4.5 kpc
length but does not develop a ring.
The GADGET3 cosmological code allows for fast computa-
tion of the long-range gravitational field on a particle mesh,
with short-range forces calculated on a tree-based hierarchical
multipole expansion. A spline kernel with scale length, hs,
softens pairwise particle interactions so that interactions
beyond that scale are strictly Newtonian. This is similar to
Plummer softening with a scale length ò= hs/2.8. In this
simulation, hs= 40, 28, and 80 pc for the dark matter halo,
stellar disk, and bulge, respectively. Full details of this
simulation and the code used can be found in D’Onghia &
Aguerri (2020, and their Appendix). Our interpretation of the
bar and spiral structure properties is based on the density wave
theory and is supported by our numerical simulations (see
D’Onghia et al. 2013; D’Onghia & Aguerri 2020).
2.2. SDSS-IV MaNGA
The SDSS-IV (Blanton et al. 2017) Mapping Nearby
Galaxies at Apache Point Observatory (MaNGA) survey
observes nearly 10,000 galaxies with integral field spectrosc-
opy (Bundy et al. 2015). MaNGA observations employ the
BOSS spectrograph on the 2.5 m telescope at Apache Point
Observatory (Gunn et al. 2006), achieving a spectral resolution
of R∼ 2000 for 3600Å< λ< 10300Å and a typical (S/N)
level of 36 (20) at a fiducial fiber magnitude of 21 (22) in the
red (blue) spectrograph arm (Bundy et al. 2015). Optical fibers
subtend 2″ on the sky (Smee et al. 2013) and are bundled into
integral field units (IFUs) with sizes in the range 12″–32″ in
diameter (19–127 fibers; Drory et al. 2015). Sky subtraction
and flux calibration are accomplished using simultaneous
observations of the sky and standard stars (Yan et al. 2016).
The median point-spread function of the resulting data cubes is
2 5 and roughly corresponds to kiloparsec physical scales at
the targeted redshift range (0.01< z< 0.15). Observations are
dithered and mapped onto 0 5 spectroscopic pixels (or
spaxels).
The MaNGA sample is selected to have a flat distribution of
i-band absolute magnitude and uniform radial coverage. This
parent sample is composed of three main components, a
primary sample where 80% of galaxies are covered out to
1.5 Re, a secondary sample where 80% of galaxies are covered
out to 2.5 Re, and a color-enhanced supplement to improve
coverage of poorly sampled regions of the near-UV− i versus
Mi color–magnitude plane (Wake et al. 2017). All MaNGA
data in this work are reduced using v2_5_3 of the MaNGA
Data Reduction Pipeline (Law et al. 2016) and employ the Data
Analysis Pipeline (Belfiore et al. 2019; Westfall et al. 2019)
from the internal eighth MaNGA product launch (MPL-8),
containing 6507 galaxies.
2.2.1. Barred Galaxy Sample—Galaxy Zoo:3D
Galaxy Zoo:3D (Masters et al. 2021) is a citizen science
project in which masks were drawn onto galaxy images to
outline morphological features such as spiral arms, bars, galaxy
centers, and foreground stars. After at least 15 citizen scientists
have classified a galaxy, the combined masks allow for quick
identification of the stellar bar and estimates of its length and
position angle. These masks effectively separate MaNGA
spaxels dominated by bar light from the rest of the galaxy.
Spaxels are considered to be in the bar if so flagged by 40% of
participants. This threshold is stricter than those used in
previous work (Fraser-McKelvie et al. 2019; Krishnarao et al.
2020b) but provides more accurate bar length measurements
(see Krishnarao et al. 2020b, their Appendix A). Details of this
citizen science project can be found on their website.16
Our barred galaxy sample is composed of 578 face-on,
barred galaxies identified in Galaxy Zoo:3D, with minor-to-
15
https://www.astropy.org/
16
https://www.zooniverse.org/projects/klmasters/galaxy-zoo-3d
2
The Astrophysical Journal, 929:112 (13pp), 2022 April 20 Krishnarao et al.
" 
11 
major-axis ratios b/a> 0.6 as estimated in the NASA-Sloan
Atlas (NSA; Blanton et al. 2011).
The bar’s orientation is determined by finding a minimum
area bounding box around bar spaxels in the deprojected galaxy
plane. Each galaxy is also recentered based on this bounding
box, and the bar length is determined using the length of this
box. Additionally, classifications from Galaxy Zoo (Lintott
et al. 2011) are used to match the orientation of spiral arms in
these galaxies such that the counterclockwise direction is the
direction of rotation of the spiral arm pattern, assuming that all
galaxies have trailing spiral arms. Figure 1 shows a polar plot
of all spaxels within the bar of the entire sample, with the bar
axis oriented horizontally.
2.2.2. PCA Resolved Galaxy Parameters
We use i-band stellar mass-to-light ratio maps for MPL-8
galaxies derived using a principal component analysis (PCA)
method of stellar continuum fitting (Pace et al. 2019a, 2019b).
These maps are found to be robust across a wide range of
signal-to-noise ratios (2„ S/N„ 30) and the full range of
realistic stellar metallicities and foreground dust attenuations
(τ 4). These mass-to-light maps allow maps of the resolved
stellar mass surface density to be created after deprojecting the
MaNGA spaxels using the minor-to-major-axis ratio from the
NSA (Blanton et al. 2011) and have been previously used in
Krishnarao et al. (2020b) and Schaefer et al. (2019).
3. Identifying Bar Resonances
Buta (2017) attributed “dark gaps” along the bar minor axis
to the location of bar corotation. These “dark gaps” are defined
as where the difference in the surface brightness along the bar
major axis and that along the minor axis is maximal. In what
follows, we use an N-body simulation to test the validity of this
proposed method in identifying the bar corotation, assuming
that the GALAKOS model is a realistic simulation of a barred
spiral having dark gaps.
3.1. Resonances in Simulated Barred Galaxies
In the simulated galaxy, we measure the bar pattern speed,
Ωp, and locations of resonances using a spectrogram as a
function of radius and frequencies for the m= 2 Fourier
harmonic in the stellar disk (see D’Onghia & Aguerri 2020,
their Figure 11). The spectrogram is constructed using
snapshots sampled at 5 Myr intervals, with the bar pattern
speed corresponding to the greatest power frequency.
Figure 1. 2D Gaussian kernel density estimate of all bar masks in polar
coordinates, normalized by bar radius.
Figure 2. m = 2 Fourier harmonic spectrogram from the GALAKOS
simulation as a function of radius and frequencies between 2.1 <
t/Gyr < 2.6, showing the bar pattern speed Ωp = 39.4 ± 0.4 km s−1 kpc−1.
The solid line shows Ω = Vc/R, used to identify corotation, and the dashed and
dotted lines show similar tracks to identify the Lindblad and ultraharmonic
resonances. In this interval, RCR = 6.40 ± 0.08 kpc and the bar patter speed,
Ωp, is marked in red along the y-axis label.
Figure 3. Polar plot with the same orientation as in Figure 1 showing increases
(red) and decreases (blue) of the stellar surface density between 2.1 <
t/Gyr < 2.6 in the GALAKOS simulation. The differences from the mean
values are scaled by their standard deviation, σ. A decrease in the stellar surface
density is seen in the simulation along the bar minor axis, similar to the dark
gaps used in Buta (2017).
3
The Astrophysical Journal, 929:112 (13pp), 2022 April 20 Krishnarao et al.
2.1 - 2.6 Gyr 
90° 
0° 
5 10 15 
R (kpc) 
~ ~i--~----~ i ~ 
-1.0 -0.5 0 .0 0 .5 1.0 
Deviation from Mean of 
Stellar Surface Density (a) 
Corotation corresponds to the radius at which the bar pattern
speed is equal to the circular angular frequency Ω= Vc/R.
Similarly, the Lindblad and ultraharmonic resonances occur at
the radii at which the bar pattern speed is equal to Ω± κ/2 and
Ω± κ/4, respectively, where κ is the epicyclic frequency.
Figure 2 displays the spectrogram for m= 2 applied to
Figure 4. Left: stellar surface density deviation as a function of radius between 2.1 < t/Gyr < 2.6 and within 30° of the bar major (blue) and minor (pink) axis from
the GALAKOS simulation. Right: the difference between the major- and minor-axis surface density deviations as a function of radius. The radii with a maximal
difference and a null difference are marked with an X and O, respectively. In both panels, the solid, dotted, and dashed black vertical lines mark the locations of
corotation, the Lindblad resonances, and the ultraharmonic resonances, respectively.
Figure 5. The estimated vs. true UHR radius across 23 time intervals of 09.25 Gyr in the range of 1.1 < t/Gyr < 4.1 in the GALAKOS simulation, with points color-
coded by the bar pattern speed at each snapshot. Each panel corresponds to a different physical spaxel resolution (labeled in the upper left corner of each panel) of the
stellar density images used in the measurements, with red dashed outlines highlighting resolution scales corresponding to MaNGA galaxies. The red dotted line shows
a one-to-one relation.
4
The Astrophysical Journal, 929:112 (13pp), 2022 April 20 Krishnarao et al.
1.5 Majar-Axis 
Minar-Axis 
2.0 
I 
-S I CR Vl 
1.0 I 1.5 x e LR (2: 1) 4: ro I 
<l! I UHR (4:1) ~ :;;: I I I 1.0 o 
e 
E 0.5 I X Peak Difference :;;: o I 
~ O First Zera Intersectian 0.5 ..... I 
e : I I o ~ 
·z 0 .0 .. .•. ..... ... . l I o 
ro I ..... •.... ... ···1···· 0.0 ·ro 
.;; I I :;;: 
<l! I I O I I -0.5 
-0.5 I I 
I 
I -1.0 
O 5 10 15 O 5 10 15 
R (kpc) R (kpc) 
Pattern Speed (km S-l kpC' ) 
30 35 40 45 50 
6 0.50 kpc 1.25 kpc 6 
5 • 5 
• 
4 4 
u 
-ª"3 3 
ti 
~ 
~2 2 
cf6 1.50 kpc 1. 75 kpc 2.25 kpc 6 
5 • +t 
5 ..... 
4 liC iIIt- 4 
3 3 
2 2 
2 4 6 2 4 6 2 4 
RUHR• true (kpc) 
6 2 4 6 
GALAKOS that shows the presence of a long bar with a pattern
speed of 39.4± 0.4 km s−1 kpc−1 as measured in the time
interval between 2.1< t/Gyr< 2.6.
The radius of maximal power is identified using a quadratic
fitting approach implemented using the Python package
bettermoments17 (Teague & Foreman-Mackey 2018) after
smoothing the data using a Savitsky–Golay filter (Savitzky &
Golay 1964) with a width of 3 radial bins. This allows for
accurate identification, with uncertainties, of the bar pattern
speed (Ωp), radius of corotation (RCR), radii of inner and outer
ultraharmonic resonance (RiUHR, RoUHR), and radii of inner and
outer Lindblad resonance (RILR, ROLR). We perform this
analysis on 23 time intervals of 0.25 Gyr between 1.1<
t/Gyr< 4.1.
3.2. Simulation Stellar Surface Density
To search for the presence of dark gaps toward the L4,5
Lagrange points and verify their correspondence with corota-
tion, we compute the stellar surface density in the simulation,
averaged across the same time intervals used to measure the
pattern speeds and resonances above. We degrade the
simulation resolution to simulate the effects of dithered
MaNGA IFU observations by first taking three mock
observations within dithered “beams” that sample a specified
physical scale. Then, the resulting mock observations are
interpolated to finer “spaxels” that are 1/4 the scale of the
“beams,” similar to the 2″ fibers and 0 5 spaxels of MaNGA.
The stellar surface density snapshots are converted to bar-
centered polar coordinates, with the bar aligned along 0°–180°.
The bar position angle is determined using a bounding box fit
around spaxels with a stellar surface density greater than
10−0.5 pc−2. Next, the mock observations are binned across a
radial range of 15 kpc, with 76 overlapping radial bins of width
0.5 kpc. Similarly, the data are binned azimuthally with a width
of 2°.5, assuming rotational symmetry. We normalize these
maps for each snapshot by first considering the mean value
across all azimuth in a radial bin and finding the deviation from
this mean value for each azimuthal bin relative to that bin’s
standard deviation. The snapshot maps are then averaged
together using 10,000 bootstrap samples, with a resulting
azimuthal variation map displayed in Figure 3 for the same
time interval as in Figure 2.
Figure 4 shows the relative increase and decrease of the
stellar surface density within 30° of the bar major and minor
axes, respectively, again for the same time interval. The
difference between the major and minor axes’ radial stellar
mass profiles and the peak difference location is shown in this
figure’s right panel. Buta (2017) attributed this effect to the L4,5
and L1,2 Lagrange points, and thus approximately at the radius
of corotation. However, this radius does not correspond to
corotation in the simulation but instead is the inner ultra-
harmonic bar resonance location. In this example, corotation is
located at a larger galactocentric radius, near the innermost
radius where the differences between the major and minor axes
are zero, which we call RCross,CR. Note that this metric is not
Figure 6. Our estimated vs. true corotation radius assuming a range of rotation curve shapes across 23 time intervals of 09.25 Gyr in the range of 1.1 < t/Gyr < 4.1 in
the GALAKOS simulation, with points color-coded by the bar pattern speed at each snapshot. Each panel corresponds to a different physical resolution (labeled in the
upper left corner of each panel) of the stellar density images used in the measurements, with red dashed outlines highlighting resolution scales corresponding to
MaNGA galaxies. The red dotted line shows a one-to-one relation.
17
https://github.com/richteague/bettermoments
5
The Astrophysical Journal, 929:112 (13pp), 2022 April 20 Krishnarao et al.
10 
8 
_ 6 
u 
c. 
6 
g 4 
• '" oi 
.,¿' 10 1.50 kpc 
8 
6 
4 
4 6 
30 
8 4 
Pattern Speed (km 5-1 kpC ' ) 
35 40 
1.75 kpc 
6 
1I 
I 
8 
2.00 kpc 
4 6 
RCR• true (kpc) 
8 
45 50 
1.25 kpc 10 
8 
6 
4 
2.25 kpc 10 
8 
6 
4 
4 6 8 10 
" 
always accurate and tends to underestimate the correct
corotation radius slightly.
These synthetic observations of a simulated barred disk
galaxy provide an interesting alternative to the dark-gap/
corotation interpretation described by Buta (2017). The residual
gap in the GALAKOS model is found to coincide with the
model’s inner ultraharmonic, not its corotation. Athanassoula
et al. (1982) showed that the ratio of the radii of corotation and
the ultraharmonic resonance can be written as
=
- D
d
( )⎜ ⎟⎛
⎝
⎞
⎠
R
R
1
1
, 1
CR
UHR
1
2
where dD = -( )1
1
2
1
2
and δ describes the galaxy rotation
curve where V∼ r− δ+1. δ= 1 corresponds to a flat rotation
curve, and Athanassoula et al. (1982) notes that 0.7„ δ„ 1.0 is
representative of most barred galaxies. We therefore can
estimate the radius of corotation based on the ultraharmonic
resonance, which we call RRatio,CR, as
=  ( )
R
R
1.8 0.3. 2
Ratio,CR
UHR
Alternatively, corotation may also correspond with the inner-
most radius where the major-to-minor-axis differences are zero
(RCross,CR; labeled as “First Zero Intersection” in Figure 4),
though we later show that this method is not always accurate.
We then test the ability of our dark-gap-based method to
identify the UHR throughout the dynamical evolution of our
simulated galaxy. Figure 5 shows the estimated and true radii
of the UHR across 23 time intervals of 0.25 Gyr that overlap in
the range of 1.1< t/Gyr< 4.1 and for different physical
resolutions of the stellar density images. Over this time
interval, the bar evolves and decreases its pattern speed
through angular momentum transport. Our method tends to
work best when considered at the approximate physical
resolution of MaNGA spaxels, where, at higher resolution,
finer density variations add significant noise and local extrema.
Figure 6 shows the same methodology but when applied for
corotation as estimated using Equation (2). The uncertainties
associated with RRatio,CR are large here because we assume a
large range in the δ parameter that describes the rotation curve
shape.
Figure 7 shows the same again, but for an estimate of
corotation using RCross,CR, the innermost radius where the
major-to-minor-axis differences are zero. This method does not
adequately predict the corotation radius’s valid location across
different evolutionary phases of the galaxy. Instead, the
location of this cross point is strongly affected by the bar
pattern speed.
Figure 8 shows all three of these metrics in terms of a
fractional error (True/Estimated) across different time intervals
and physical resolution scales. Within the range of resolutions
corresponding with MaNGA galaxies, we can describe the
accuracy and precision for identifying the UHR as
= -
+1.00
R
R 0.12
0.10UHR,True
UHR,Est
. Similarly, by assuming a rotation curve
shape, we find = -
+0.95
R
R 0.13
0.18CR,True
Ratio,CR
. Although RCross,CR is not an
Figure 7. The estimated vs. true corotation radius using the innermost radius where the major-to-minor-axis differences are zero across 23 time intervals of 09.25 Gyr
in the range of 1.1 < t/Gyr < 4.1 in the GALAKOS simulation, with points color-coded by the bar pattern speed at each snapshot. Each panel corresponds to a
different physical resolution (labeled in the upper left corner of each panel) of the stellar density images used in the measurements, with red dashed outlines
highlighting resolution scales corresponding to MaNGA galaxies. The red dotted line shows a one-to-one relation.
6
The Astrophysical Journal, 929:112 (13pp), 2022 April 20 Krishnarao et al.
Pattern Speed (km 5 - 1 kpC ' ) 
30 35 40 45 50 
10 10 
0.50 kpe 0.75 kpe 1.00 kpe 1.25 kpe 
8 e. ~ 
,. 
~ y"':! • ~ 
8 
T • e ,. 
t y • 
"U 6 • 6 
Q. ,. 
T . .' ::-:. 
• • E 
u. lO 
'" 
10 
u 
1.50 kpe 1.75 kpe 2.00 kpe 2.25 kpe o:: 
8 
f* " 
x .; •• . ~ V +J 8 
* • . f + 
* x + 
** 6 6 
4 +----r ---- ~ -- ~ -- ~----~ -- ~ --_,----,_--~--_,,_--._--_+4 
4 6 8 4 6 8 4 6 8 4 6 8 10 
ReR. true (kpe) 
ideal estimator, it returns an averaged accuracy of
= -
+1.01
R
R 0.19
0.18CR,True
Cross,CR
. We adopt these simulation-derived ratios
as a correction factor, with uncertainties, to adjust the measured
locations and improve the accuracy of the predicted resonance
locations. With this method validated on simulation data and
with an understanding of the corrections needed to be made and
uncertainties involved, we can apply it to a sample of observed
galaxies, both ringed and nonringed, to better understand its
validity.
4. Observational Results
The 578 barred galaxies with bar masks and resolved galaxy
parameters provide a unique statistical view into the impact of a
bar on galaxies’ distribution of matter. To consider our sample
as a whole and understand the significance of rings in our
barred galaxies, we split our sample into two subsamples of
211 ringed and 367 nonringed galaxies using a 50% unbiased
vote fraction on the Galaxy Zoo question “t08_odd_feature_-
a19_ring” from Hart et al. (2016). We then project all ringed
and nonringed galaxies onto a bar-oriented cylindrical
coordinate system, with the radius in units of the bar radius
(Rbar) and azimuth angle oriented such that the bar major axis
spans from 0° to 180°. We use visual classifications from
Galaxy Zoo (Lintott et al. 2011) to orient all spiral arms in a
counterclockwise spiral pattern. In this bar reference frame, we
assume a 180° rotational symmetry so that spaxels at an
azimuth of 30° and 210° are equivalent. To ensure an equal
sampling of different radii in galaxies, we restrict each galaxy
to only extend out to the maximum bar-oriented radius found in
all azimuthal directions. Additionally, we remove spaxels with
a g-band S/N < 2 to only consider spaxels that lie on the
galaxy disk. We can then search for azimuthal and radial
surface density variations in the context of bar-driven
resonances.
4.1. Surface Density Variations
Maps of the stellar mass surface density from Pace et al.
(2019a, 2019b) provide a resolved measure of the distribution
of stars in MaNGA galaxies. The maps are binned in a similar
radial and azimuthal schema as the N-body simulations: radial
bins are computed relative to the bar radius, overlap with a
width of 0.2R/Rbar, and have a step size of 0.04R/Rbar, out to a
maximum radius R= 3Rbar. We normalize these maps for each
galaxy by first considering the mean value across all azimuth in
a radial bin and finding the deviation from this mean value for
each azimuthal bin in terms of the standard deviation computed
across all azimuth. We also do the same using SDSS g-band
imaging to test the ability to perform this analysis using only
imaging data. This process produces maps for all 578 galaxies,
showing azimuthal variations across different radial bins,
scaled by each radial bin’s standard deviation. We combine
the resulting maps using 10,000 bootstrap resamples to create
Figure 9. The stellar mass surface density and g-band
Figure 8. Left: true/estimated radii of the ultraharmonic resonance (RUHR; top panel) and corotation as measured using a ratio (RRatio,CR; middle panel) and using the
first zero crossing point (RCross,CR; bottom panel) at different physical resolution scales in 23 intervals of 0.25 Gyr between 1.1 < t/Gyr < 4.1. The pattern speed of
the bar colors the points in the corresponding simulation snapshot. The approximate physical scale of the ¢¢2 MaNGA fibers spans the resolution scales in bold. A
precise and accurate estimate would be narrowly distributed and centered on one (dashed red lines). Right: histograms of the true/estimated radii for the resolutions
comparable to MaNGA galaxies. The median ratios and uncertainties estimated using the 16th and 84th percentiles of the distributions are displayed for each metric.
These will be used as a correction factor.
7
The Astrophysical Journal, 929:112 (13pp), 2022 April 20 Krishnarao et al.
1.50 
1.25 
Vl 1.00 
::> 
u 0 .75 ro 
a: 
Q) 1.50 u 
" ro 
1.25 " o 
Vl 
Q) 
1.00 a: 
u 
Q) 0.75 ~ 
ro 
E 
:¡:¡ 1.50 
Vl 
w - 1.25 Q) 
::> 
F 1.00 
0 .75 
0.50 
, 
' : . ,, , , 
~ • •• 1 
• • .... . .. 
30 
•• ,
•• 
•• 
If ........... 
• 
Pattern Speed (km 5 - 1 kpC' ) 
• 
I 
• 
35 40 45 
• 
• 
, , , 
...... ~ ... ¡ .. . 
, , , 
, , , ...... ¡ ... ¡ ... 
, , , , 
50 
. ... . .• ... 
J 
.. !I ... ( ......... .• .. ····Ift · .... t!' .... .. • .. + .., ...... , .. . 
I l. . . I ... ' : ..... '- : ..... . l . . . . i • • , 
0 .50 0.75 1.00 1.25 1.50 1.75 2.00 2 .25 
Physical Resolution (kpc) 
'" u 
Ó ., 
ro 
'" o:: 
'" u 
::i 
e 
u 
o:: 
magnitude show similar behavior in the simulations and
Figure 3.
Figure 10 follows the same method used for Figure 4, but
with the observed stellar mass surface density and g-band
magnitudes of MaNGA galaxies. The peak difference is
identified in the same manner, and the correction factors are
applied to adjust for the expected offsets based on the
simulation (see Figure 8). We find the average location of the
inner ultraharmonic resonance to be = -
+
R R0.89UHR 0.13
0.11
bar,
with estimated corotation radii with both methods of
= -
+
R R1.60Cross,CR 0.30
0.29
bar and = -
+
R R1.52Ratio,CR 0.35
0.40
bar. Bars
with RCR/Rbar… 1.4 are considered “slow,” while bars with
RCR/Rbar< 1.4 are considered “fast” (Debattista & Sell-
wood 2000). Bars with RCR/Rbar< 1 are often referred to as
“ultrafast,” but it is unclear whether these bars are a real feature
or a result of imperfect estimates of either RCR or Rbar (e.g.,
Buta 2017). Our results suggest that slow bars are the most
likely scenario for this sample of galaxies as an ensemble, since
RCR/Rbar> 1.4. The sample of galaxies considered in Buta
(2017) are also primarily slow, with RCR/Rbar… 1.58. How-
ever, the relatively large uncertainties in this population-level
diagnostic do not categorically exclude the fast rotator
alternative.
We also perform the same analysis on the 578 individual
galaxies considered here, with 185 galaxies returning reason-
able estimates of RUHR and RCR using the PCA stellar mass
surface density as shown in Figure 11. Four example galaxy
images, with their inferred ultraharmonic resonance location
marked with shaded rings, are shown in Figure 12. While many
galaxies seem to be classified as slow, the large uncertainties
with the corotation radius estimates leave no galaxies with a
2σ estimate of RCR/Rbar that is slow. Combined with the fact
that the peak of the RCR/Rbar distribution for these 185 galaxies
is within the fast threshold, it seems likely that most bars in
MaNGA galaxies in our sample are, in fact, fast rotators. This
is in general agreement with an independent study using a
smaller sample of MaNGA galaxies with the Tremaine &
Weinberg (1984) method, which found = -
+
R R 1.17CR bar 0.41
0.5
(Garma-Oehmichen et al. 2020).
5. Discussion
Rings are often suggested to be linked to resonances with the
pattern speed of bars. However, measurements of the pattern
speed and location of resonances are difficult to observe in
large samples of galaxies, especially for face-on galaxies.
Zhang & Buta (2007) described a phase shift method to locate
corotation radii in face-on barred and spiral galaxies, but this
method assumes that spirals and bars are quasi-steady modes in
galactic disks. Later, Buta (2017) described a novel method for
detecting resonances, attributing “dark gaps” in galaxies to the
location of corotation. However, if we accept that GALAKOS
is a reliable model of a barred galaxy, then it provides an
interesting alternative interpretation of the gaps. These same
“dark gaps” may indicate the location of the 4:1 ultraharmonic
resonance with the bar, in both ringed and nonringed galaxies.
This finding now opens up a new avenue to explore the effects
of bar resonances on galaxies’ structure using large samples of
galaxy images.
Kim et al. (2016) analyzed the relation of the dark gaps with
the bar properties for a sample of barred galaxies. They found
that larger and stronger bars produced stronger dark gaps,
Figure 9. Same as in Figure 3, but for the 578 barred MaNGA galaxies using the PCA stellar mass surface density of Pace et al. (2019a, 2019b) (top half; 0°–180°)
and SDSS G-band imaging (bottom half; 180°–360°), separated into 211 ringed (left) and 367 nonringed (right) galaxies. The radial axis is normalized by the bar
radius.
8
The Astrophysical Journal, 929:112 (13pp), 2022 April 20 Krishnarao et al.
Deviation from Mean of 
Stellar Mass Surface Density (a) 
-1.0 -0.5 0.0 0.5 1.0 
; ; 
Ringed MaNGA Galaxies Nonringed MaNGA Galaxies 
90 0 90 0 
270 0 270 0 
~¡ "~ I ==~ I ==~I ~ ~ ¡ 
-1.0 -0.5 0.0 0.5 1.0 
Deviation from Mean of 
g-band magnitude (a) 
/ 
Figure 10. Same as in Figure 4, but for the 578 barred MaNGA galaxies, separated into 211 ringed (top panel) and 367 nonringed (bottom panel) galaxies using the
PCA stellar mass surface density of Pace et al. (2019a, 2019b) (solid lines) and SDSS g-band imaging (dotted lines). The inner ultraharmonic resonance (RUHR) is
approximated at the location of maximal difference between the major and minor axes, and corotation is approximated as RRatio,CR/RUHR = 1.8 ± 0.3 (solid shading)
and as the first zero intersection (RCross,CR; hatched shading). All estimates have also been adjusted to include the correction factors found in the simulations (see
Figure 8).
Figure 11. Estimated radius of the inner ultraharmonic resonance (RUHR) and radius of corotation (RCR) for individual ringed (left) and nonringed (right) MaNGA
galaxies. Data are limited to galaxies where measurement errors are under 0.2Rbar for identifying the peak difference. Error bars are shown for one-fifth of the data
points. The dotted horizontal lines mark the boundaries between ultrafast, fast, and slow bars. Marginalized Gaussian kernel density estimates are shown along both
axes, with the dashed lines including the associated errors. A total of 74 ringed and 111 nonringed galaxies are plotted, with 5 ultrafast, 12 fast, and 57 slow bars in the
ringed sample and 22 ultrafast, 30 fast, and 59 slow bars in the nonringed sample as diagnosed using RCross,CR. When considering the errors, the number of galaxies
with ultrafast and slow bars drops significantly.
9
The Astrophysical Journal, 929:112 (13pp), 2022 April 20 Krishnarao et al.
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indicating that the growth process of the bar is linked with the
formation or the dark gaps. Similar to our cases, their dark gaps
are located at radius smaller than the bar radius (see their
Figure 5). In addition, they run a galaxy numerical model to
understand the formation of the dark gaps. In this simulation
the dark gaps formed were also located at a radius smaller than
the bar radius. This is similar to what we obtain in our
GALAKOS simulation. All these facts imply that if the dark
gaps would be located at the corotation radius (Buta 2017), all
bars would be ultrafast rotators. This would be in contradiction
with the classical view of bars supported by the orbits of the
x1-family (e.g., Contopoulos & Papayannopoulos 1980;
Athanassoula 1992).
The most accurate method for measuring the bar pattern
speed, which is used to derive the corotation radius, is using the
kinematic and model-independent method proposed by Tre-
maine & Weinberg (1984). The method is, however, affected by
multiple error sources, notably the determination of geometric
parameters and modeling of the rotation curve (Garma-
Oehmichen et al. 2020). Additionally, it is often best suited to
work with early-type galaxies. Guo et al. (2019) used this
method to measure the bar pattern speed and corotation radii of a
sample of 53 MaNGA galaxies. Of these, 10 galaxies are also in
our barred sample and have reliable estimates from our new
method without considering any kinematic information.
Figure 13 shows the difference between our estimated corotation
radius and the measurements from Guo et al. (2019) using their
kinematic and photometric methods. While large discrepancies
are present in some individual galaxies, the uncertainties
combined with the kinematic method cannot rule out an overall
agreement. The median value of the ratio between our estimate
Figure 12. Example SDSS gri images with the inferred location of the inner ultraharmonic resonance and corotation shown in the right-hand image pair in blue and
red, respectively, for ringed (left) and nonringed (right) galaxies. The image axes are scaled based on the measured bar radius. The MANGA-ID of each galaxy is
displayed above the left-hand image pair. The purple hexagons mark the footprint of the MaNGA survey. These images are exactly what are used in the citizen science
classifications of Masters et al. (2021).
Figure 13. The ratio of the estimated corotation radii and the estimates from the
mass-weighted (cyan circle) and light-weighted (pink cross) measurements
from Guo et al. (2019) for galaxies overlapping in both samples. The top and
middle panels are for the kinematic- and photometric-based estimates from Guo
et al. (2019), respectively, and shading encompasses both measurements’
errors. Bottom panel: the ratio between our bar length estimate (Galaxy
Zoo:3D) and the estimates in Guo et al. (2019; units of arcsec). The “Galaxy
Index” is an arbitrary index describing each overlapping galaxy in our sample
and that of Guo et al. (2019).
10
The Astrophysical Journal, 929:112 (13pp), 2022 April 20 Krishnarao et al.
Ringed Examples 
1-29901 3 Inferred 
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Galaxy Index 
using RRatio,CR and the Guo et al. (2019) mass-weighted
kinematic method is = -
+
R R 1.7Cross,CR kin,CR,m 0.2
0.4, showing that
our method tends to estimate the corotation radius to be farther
out than using the Tremaine & Weinberg (1984) method.
Figure 13 also shows differences between our bar length
estimates from Galaxy Zoo:3D and the estimates derived in Guo
et al. (2019). In the future, applying the Tremaine & Weinberg
(1984) method to the full sample of MaNGA galaxies will allow
for more statistically robust validation of our method. The
kinematic approach to identifying the corotation radius is likely
still the most accurate method, but our new method explored in
this work may be preferable for more face-on galaxies, whose
kinematics are less reliable in projection and whose morpholo-
gical features can be better characterized.
Though the method used to measure bar length can impact
estimates of resonance location, it is difficult to gauge which
method is best. We note that the Galaxy Zoo:3D bar length
estimates are generally consistent with estimates using other
methods, including in our overlap with the work of Guo et al.
(2019) with a median of Rbar,GZ:3D/Rbar,Guo= 0.9± 0.1. Guo
et al. (2019) used a combination of three methods to estimate
bar lengths, including the ellipticity radial profile, the ellipticity
position angle profile, and the Fourier decomposition. In
Krishnarao et al. (2020, their Appendix), the Galaxy Zoo:3D
bar lengths were tested more extensively with a Fourier-based
method of Kraljic et al. (2012), showing consistent results.
In addition, we can explore the correlation between different
galaxy morphological types and their bar speeds. Using the
most likely morphological types as diagnosed in Galaxy Zoo 2
(Hart et al. 2016), we compare differences in the corotation
radii. Figure 14 shows the distribution of measured resonance
radii separated by their morphological types. Several galaxies
have a likely morphology from Galaxy Zoo 2 as SA, or not
containing a bar. These galaxies likely have weaker bars than
those with a likely morphology of SB, which would have bars
more clearly visible in SDSS imaging. Here we use this
difference as an approximate proxy for bar strength. With our
current sample, we cannot robustly diagnose any correlation
with morphology or bar strength and bar speeds. Understanding
the impact of the bar strength on the bar speed could be studied
in the future with more detailed analysis, but it is beyond the
scope of this paper.
In our analysis of individual galaxies, only 32% of our full
sample of 578 galaxies showed a clear “dark-gap” signature in
the PCA stellar mass surface density maps. While this is
relatively low, it may still be possible to improve the success
rate using deeper imaging data and to vary the amount of
smoothing. In particular, it may be possible to identify “dark
gaps” using the deep multiband imaging from the DESI legacy
imaging survey (Dey et al. 2019). This approach also does not
involve the more complicated effects of spatial covariances
resulting from dithered IFU observations.
In this paper, we only consider the stellar mass and light
concerning bar-driven resonances because the GALAKOS
simulation lacks gas. It has been shown that gas can
significantly impact the stellar dynamics of bars, which our
test simulations do not account for (e.g., Athanassoula 2003;
Combes 2008). However, cosmological simulations with the
same level of detail as GALAKOS that include gas are not yet
available but in the future will provide an ideal way to calibrate
this gap method. In addition, including simulations of ringed
galaxies and other types will help determine the universality of
the gap method on all barred galaxies. Initial examination of
the MaNGA emission-line data suggests that the gas distribu-
tion may also closely track the azimuthal variations of the
stellar density shown in Figure 9 based on a similar map using
the Hα equivalent width. However, the precise location of the
“dark gaps” is slightly shifted from the stellar matter. Fraser-
McKelvie et al. (2020) investigated the correlation between the
Hα morphology of bars and their star formation rates, finding
correspondence between different gas morphologies and the
amount of quenching in bars. To fully understand the nature of
the gas with resonances, we must, in the future, similarly
calibrate our metrics using hydrodynamic simulations.
6. Conclusions
This paper has proposed a new method to measure the radius
of both corotation and the ultraharmonic resonance in galaxies
without measurements of their internal kinematics. Our method
is very similar to the “dark-gap” method of Buta (2017), but
Figure 14. Same as in Figure 11, but separated by morphological types from Galaxy Zoo 2 (Hart et al. 2016), with SB vs. SA (left panel; proxy for strong vs. weak
bar) and spirals vs. lenticulars (right panel).
11
The Astrophysical Journal, 929:112 (13pp), 2022 April 20 Krishnarao et al.
3.0 
25 
~ 2.0 
a:: 1.5 
LO 
0.5 
I 
1> 
.. .. ~ . 
~~ .. 
'~ .. ;f .. " . ............ ~ . ...f:i;" t ························· 
. ...... ~ . I .. , 
-- ~ ~ - ~ ---¡----------------
: Rbllr 
. ' I S8 
: SAB 
.2 M U U U U U U ,. 2. 
RUHRIRblJr 
/ 
3.' 
2.5 
~ 2.0 
ce 15 
, .• 
'.5 
. " 
•• ~ 4 • 
,',' 
.,,:;¡;..: 
........... ~ . ::c.;;: ~ ........................ . 
~ '. " :- I 
... .,. I 
-- : ~~ ~ ----:----------------
., • I R b~r 
.... . I Spirals 
• • : Lenticular 
~ M U U ,. U U U U U 
RUHRIRbM 
with a significant change in the interpretation, prompted by
tests on the N-body simulation, GALAKOS (D’Onghia &
Aguerri 2020). We summarize our main findings below:
1. In the GALAKOS simulation, dark gaps are found in the
stellar mass surface density but appear not to be related to
the L4,5 Lagrangian points as was proposed by Buta
(2017). Instead, the model favors the gaps to be linked to
the inner 4:1 ultraharmonic resonance.
2. If the rotation of the galaxy hosting the “dark gap” is
known, then the corotation resonance can be precisely
identified using Equation (1).
3. Alternatively, assuming a generally flat rotation curve,
the radius of corotation can be predicted as
RRatio,CR/RUHR= 1.8± 0.3 but with decreased precision.
4. This method works best for galaxies with imaging, or
IFU spaxels, at ∼0.2–0.4 kpc scales, similar to MaNGA
galaxies, so that the strong nonaxisymmetric variations
from the bar in the innermost regions do not dominate.
5. Applying this method to a sample of 578 barred MaNGA
galaxies reveals that, on average, the MaNGA sample
represents a population of bars that are slow rotators, but
it cannot be ruled out that the bars are fast rotators given
the uncertainties, especially on the determination of the
bar radius.
6. About 32% of the members of our barred galaxy sample
show clear signatures of a “dark gap” in the PCA-based
resolved stellar mass surface density maps of Pace et al.
(2019a, 2019b).
7. Our results for these individual galaxies are generally
consistent with estimates derived using the Tremaine &
Weinberg (1984) method in previous work (Guo et al.
2019), though both methods have large systematic errors
in our sample.
8. None of the individual bars can be confidently deter-
mined to be slow rotators when considering their
2σ uncertainties.
In particular, our method does not require kinematic
information to determine the fundamental parameters describ-
ing the dynamics of bars. It can be used on large samples of
barred galaxies from imaging surveys, such as the Legacy
Survey (Dey et al. 2019). This method allows for the nature of
rings and other structures in barred galaxies to be reexamined
in terms of bar-driven resonances, paving the way for better
diagnostics of galaxies’ internal dynamical structures. With the
continued use of citizen science projects like Galaxy Zoo to
build larger training samples, it may soon be possible to
identify bar resonances with automated machine-vision tech-
niques or neural networks.
D.K. and Z.J.P. acknowledge support from the NSF
CAREER award AST-1554877. D.K. is supported by an
NSF Astronomy and Astrophysics Postdoctoral Fellowship
under award AST-2102490. J.A.L.A. is supported by the
Spanish MINECO grant AYA2017-83204-P.
J.G.F.-T. gratefully acknowledges the grant support provided
by Proyecto Fondecyt Iniciación No. 11220340, and also from
ANID Concurso de Fomento a la Vinculación Internacional
para Instituciones de Investigación Regionales (Modalidad
corta duración) Proyecto No. FOVI210020, and from the Joint
Committee ESO-Government of Chile 2021 (ORP 023/2021).
Funding for the Sloan Digital Sky Survey IV has been
provided by the Alfred P. Sloan Foundation, the U.S.
Department of Energy Office of Science, and the Participating
Institutions. SDSS acknowledges support and resources from
the Center for High-Performance Computing at the University
of Utah. The SDSS website is www.sdss.org.
SDSS is managed by the Astrophysical Research Con-
sortium for the Participating Institutions of the SDSS
Collaboration including the Brazilian Participation Group, the
Carnegie Institution for Science, Carnegie Mellon University,
the Chilean Participation Group, the French Participation
Group, Harvard-Smithsonian Center for Astrophysics, Instituto
de Astrofísica de Canarias, The Johns Hopkins University,
Kavli Institute for the Physics and Mathematics of the Universe
(IPMU)/University of Tokyo, the Korean Participation Group,
Lawrence Berkeley National Laboratory, Leibniz Institut für
Astrophysik Potsdam (AIP), Max-Planck-Institut für Astro-
nomie (MPIA Heidelberg), Max-Planck-Institut für Astrophy-
sik (MPA Garching), Max-Planck-Institut für Extraterrestrische
Physik (MPE), National Astronomical Observatories of China,
New Mexico State University, New York University, Uni-
versity of Notre Dame, Observatório Nacional/MCTI, The
Ohio State University, Pennsylvania State University, Shanghai
Astronomical Observatory, United Kingdom Participation
Group, Universidad Nacional Autónoma de México, Univer-
sity of Arizona, University of Colorado Boulder, University of
Oxford, University of Portsmouth, University of Utah,
University of Virginia, University of Washington, University
of Wisconsin, Vanderbilt University, and Yale University.
Facility: Sloan.
Software: astropy (Robitaille et al. 2013; Astropy Collabora-
tion et al. 2018), matplotlib (Hunter 2007), seaborn (Waskom
et al. 2020), scipy (Virtanen et al. 2020), sdss-marvin (Cherinka
et al. 2019), bettermoments (Teague & Foreman-Mackey
2018).
ORCID iDs
Dhanesh Krishnarao https://orcid.org/0000-0002-
7955-7359
Zachary J. Pace https://orcid.org/0000-0003-4843-4185
Rachel L. McClure https://orcid.org/0000-0001-5928-7155
Thomas Peterken https://orcid.org/0000-0003-3217-7778
José G. Fernández-Trincado https://orcid.org/0000-0003-
3526-5052
Michael Merrifield https://orcid.org/0000-0002-4202-4727
Karen L. Masters https://orcid.org/0000-0003-0846-9578
Matthew Bershady https://orcid.org/0000-0002-3131-4374
Niv Drory https://orcid.org/0000-0002-7339-3170
Richard R. Lane https://orcid.org/0000-0003-1805-0316
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The Astrophysical Journal, 929:112 (13pp), 2022 April 20 Krishnarao et al.
Appendix B
Galaxy Zoo: Kinematics of strongly and
weakly barred galaxies
In this work led and written by Tobias Géron, we have measure the bar pattern speed of
a sample of 225 barred galaxies from the MaNGA project with the Tremaine-Weinberg
method. The objective of the study is to identify differences between strongly and weakly
barred galaxies. This is currently the largest sample of galaxies with measurements of bar
pattern speed, and posses a lower fraction of ultrafast bars compared to other studies.
The paper is currently submitted and in review process. With permission of the first
author, here I include last version of the paper, with the latest corrections highlighted in
red.
My contribution:
I helped in the implementation and usage of the Tremaine-Weinberg method in early
versions of the draft. More specifically, we discussed and exchanged ideas on:
• Placement of pseudo-slits. The importance of the symmetry.
• Measurement of the corotation radius from the rotation curve.
• Discussion on how which PA measurement should be used.
79
APPENDIX B. GALAXY ZOO: KINEMATICS OF STRONGLY ANDWEAKLY BARREDGALAXIES80
• Discussion on the error treatment and biases induced by the manual measurements.
MNRAS 000, 1–19 (2022) Preprint 18 November 2022 Compiled using MNRAS LATEX style file v3.0
Galaxy Zoo: Kinematics of strongly and weakly barred galaxies
Tobias Géron1⋆, Rebecca J. Smethurst1, Chris Lintott1, Sandor Kruk2, Karen L. Masters3,
Brooke Simmons4, Kameswara Bharadwaj Mantha5,6, Mike Walmsley7,
L. Garma-Oehmichen8, Niv Drory9, Richard R. Lane10
1 Oxford Astrophysics, Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK
2 Max-Planck-Institut für extraterrestrische Physik (MPE), Giessenbachstrasse 1, D-85748 Garching bei München, Germany
3 Haverford College, Department of Physics and Astronomy, 370 Lancaster Avenue, Haverford, Pennsylvania 19041, USA
4 Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK
5 Minnesota Institute for Astrophysics, University of Minnesota, 116 Church St SE, Minneapolis, MN 55455, USA
6 Department of Physics and Astronomy, University of Minnesota, 116 Church St SE, Minneapolis, MN 55455, USA
7 Jodrell Bank Centre for Astrophysics, Department of Physics & Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, UK
8 Instituto de Astronomía, Universidad Nacional Autónoma de México, Apartado Postal 70-264, CDMX, 04510, México
9 McDonald Observatory, The University of Texas at Austin, 1 University Station, Austin, TX 78712, USA
10 Centro de Investigación en Astronomía, Universidad Bernardo O’Higgins, Avenida Viel 1497, Santiago, Chile
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We study the bar pattern speeds (Ωb) and corotation radii (RCR) of a sample of 225 barred
galaxies, using IFU data from MaNGA and the Tremaine-Weinberg method. Our sample,
which is divided between strongly and weakly barred galaxies identified via Galaxy Zoo, is the
largest that this method has been applied to. We find lower pattern speeds for strongly barred
galaxies than in weakly barred galaxies. As simulations show that the pattern speed decreases
as the bar exchanges angular momentum with its host, these results suggest that strong bars
are more evolved than weak bars. Interestingly, the corotation radius is not different between
weakly and strongly barred galaxies, despite being proportional to bar length. We also find that
the corotation radius is significantly different between quenching and star forming galaxies.
Additionally, R, the ratio between the corotation radius and the bar radius, is not signifi-
cantly different between the two bar types for our sample. Additionally, we could not find
a significant difference between the two bar types for R, the ratio between the corotation
radius and the bar radius. This ratio classifies bars into ultrafast bars (R < 1.0; 21% of
our sample), fast bars (1.0 < R < 1.4; 32%) and slow bars (R > 1.4; 47%). We know from
simulations that R is correlated with the bar formation mechanism, so our results suggest that
weak and strong bars are triggered by similar mechanisms. Finally, we discuss our results in
the context of the recently claimed tension with ΛCDM. Finally, we find a lower fraction
of ultrafast bars than most other studies, which somewhat decreases the recently claimed
tension with ΛCDM. However, the median value of R is still lower than what is predicted
by simulations.
Key words: galaxies: general – galaxies: bar – galaxies: evolution – galaxies: structure –
galaxies: kinematics and dynamics
1 INTRODUCTION
Bars are a relatively common structure in galaxies, with about
30%-60% of nearby galaxies hosting a bar, depending on the red-
shift and wavelength range of the study (Marinova & Jogee 2007;
Menéndez-Delmestre et al. 2007; Barazza et al. 2008; Sheth et al.
⋆ E-mail: tobias.geron@physics.ox.ac.uk (TG)
2008; Nair & Abraham 2010b; Masters et al. 2011). Bars can drive
angular momentum outwards and funnel gas to the centre of the
galaxy (Athanassoula 1992b; Davoust & Contini 2004; Rodriguez-
Fernandez & Combes 2008; Athanassoula et al. 2013; Villa-Vargas
et al. 2010; Fragkoudi et al. 2016; Vera et al. 2016; Spinoso et al.
2017; George et al. 2019; Seo et al. 2019). This will result in sig-
nificant “secular” evolution of the host, caused directly by its bar
(Kormendy & Kennicutt 2004; Cheung et al. 2013; Sellwood 2014;
© 2022 The Authors
2 Tobias Géron
Díaz-García et al. 2016b; Kruk et al. 2018). Moreover, multiple
studies have found that bars appear more often in massive, red and
gas-poor galaxies (Masters et al. 2012; Cervantes Sodi 2017; Vera
et al. 2016; Fraser-McKelvie et al. 2020). For example, Kruk et al.
(2018) found that bars are redder than disks and that the disks of
barred galaxies are redder than the disks in unbarred galaxies. These
results suggest that bars might be linked to the quenching of their
host. This can be either the result of triggering a starburst in the
centre of the galaxy, after extensive inflows of gas (Alonso-Herrero
& Knapen 2001; Sheth et al. 2005; Jogee et al. 2005; Hunt et al.
2008; Carles et al. 2016), or by making the gas too dynamically hot
for star formation (Zurita et al. 2004; Haywood et al. 2016; Khop-
erskov et al. 2018; Athanassoula 1992b; Reynaud & Downes 1998;
Sheth et al. 2000). In any case, a bar is a very common and impor-
tant structure in a galaxy, so understanding bars is fundamental to
understanding galaxy evolution.
Bars are historically classified into weak or strong. Since de
Vaucouleurs (1959, 1963), three subclasses are recognised: un-
barred (SA), strongly barred (SB) and weakly barred (SAB). Weakly
barred galaxies were thought to be an intermediate class between
unbarred and strongly barred, having lengths and contrast in be-
tween SA and SB bars. Bars classified as weak were usually small
and faint, whereas bars classified as strong were long and obvious
de Vaucouleurs (1959, 1963). Morphological arguments are still
used to determine bar type. Nair & Abraham (2010a) produced
a catalogue of detailed visual morphological classifications and
they distinguished between weak and strong by looking at whether
the bar dominated the light distribution. The bar strength can also
be estimated using the maximum ellipticity and boxiness of the
isophotes (Athanassoula 1992a; Laurikainen & Salo 2002; Erwin
2004; Gadotti 2011). One can also look at the surface brightness
profiles of bars. Previous work has shown that stronger bars have flat
profiles, while weaker bars have exponential profiles (Elmegreen &
Elmegreen 1985; Elmegreen et al. 1996; Kim et al. 2015; Kruk et al.
2018). Clearly, there are many ways to characterise bars and their
strength. However, the community has yet to reach a consensus on
how to best define weak and strong bars and on which detection
method is superior.
This problem was addressed more recently by the Galaxy Zoo
(GZ) team, who combined the efforts of citizen scientists and ma-
chine learning to provide morphological classifications of galaxies
(Lintott et al. 2008; Walmsley et al. 2021). These morphological
classifications included a distinction between weak and strong bars
based on visual morphology. Volunteers are shown examples of
weak and strong bars prior to classification. The strong bars
are typically large and obvious structures, whereas the weak
bars can be smaller and fainter. Géron et al. (2021) used the mor-
phological classifications from GZ based on images from the Dark
Energy Camera Legacy Survey (DECaLS, Dey et al. 2019) to study
weak and strong bars, and found that around 28% of all disk galax-
ies have a weak bar, while 16% had a strong bar. They also found
that, when correcting for bar length, any difference they observed
between weak and strong bars disappeared. Thus, they suggested
that weak and strong bars are not fundamentally different physical
phenomena. Instead, they proposed the existence of a continuum of
bar types, which varies from ‘weakest’ to ‘strongest’. Most research
on bars has traditionally been focussed on stronger bars, as they are
more obvious and clearer structures. However, weaker bars are still
very common structures in galaxies and need to be included in more
studies to obtain a more complete picture.
The bar pattern speed (Ωbar), or the rotational frequency of
the bar, is one of the most important dynamical parameters that
describe a bar. It is intrinsically linked to the evolution of the bar
and its host. It is typically found that, as the bar the exchanges
angular momentum with its host, the bar grows and the pattern
speed decreases (Debattista & Sellwood 2000; Athanassoula 2003;
Martinez-Valpuesta et al. 2006; Okamoto et al. 2015).
If the bar pattern speed and galaxy kinematics are known, one
can calculate the corotation radius (RCR), which is the radius at
which the angular speed of the stars in the disc is equal to the
pattern speed of the bar. Additionally, one can also calculate the
dimensionless corotation radius-to-bar radius ratio,R. A large value
forR implies that the point of corotation is far outside the bar region.
This ratio is typically used to separate bars into ‘fast’ (1.0 < R < 1.4)
and ‘slow’ (R > 1.4) bars (Debattista & Sellwood 2000; Rautiainen
et al. 2008; Aguerri et al. 2015). There is a known correlation
between the formation of the bar and R. Bars that are triggered by
tidal interactions tend to be in the slow regime for a longer time and
have higher values for R than bars formed by global bar instabilities
(Sellwood 1981; Miwa & Noguchi 1998; Martinez-Valpuesta et al.
2016, 2017).
There is also a known tension between simulations and obser-
vations on the distribution of the ratio R. Cosmological simulations
predict that bars slow down significantly due to dynamical friction
with their dark matter halo, which results in high values for R. How-
ever, observations typically find lower values of R, which has been
highlighted as a challenge for the ΛCDM cosmology used in these
cosmological simulations (Algorry et al. 2017; Peschken & Łokas
2019; Roshan et al. 2021b).
It is suggested that bars cannot extend beyond their corotation
radius (Contopoulos 1980, 1981; Athanassoula 1992b). This implies
that bars with R < 1 should not exist. However, these so-called
‘ultrafast’ bars have been repeatedly observed (Buta & Zhang 2009;
Aguerri et al. 2015; Cuomo et al. 2019; Guo et al. 2019; Garma-
Oehmichen et al. 2020; Krishnarao et al. 2022). This discrepancy
between observation and theory remains an open question, although
some suggest that the cause for this problem is rooted at incorrect
estimates of the bar radius (Cuomo et al. 2021; Roshan et al. 2021a).
It is becoming clear that the pattern speed and the parameters
derived from it (such as corotation radius and R) are important to
understand. However, it is also quite challenging to correctly esti-
mate the bar pattern speed. Nevertheless, various methods exist to
measure this dynamical parameter. For example, one can match the
observed surface gas distribution or gas velocity field with simula-
tions whereΩbar is a free parameter (Sanders & Tubbs 1980; Hunter
et al. 1988; Lindblad & Kristen 1996; Weiner et al. 2001; Rauti-
ainen et al. 2008; Treuthardt et al. 2008). Alternatively, one can
subtract a rotation model from the gas velocity field and look at the
morphology of the residuals to estimate the pattern speed (Sempere
et al. 1995; Font et al. 2011, 2017). Other morphological features
are helpful to determine the bar pattern speed, such as rings (Buta
1986; Rautiainen & Salo 2000; Muñoz-Tuñón et al. 2004; Pérez
et al. 2012), the shape and offset of dust lanes (Athanassoula 1992b;
Sánchez-Menguiano et al. 2015) and the morphology of spiral arms
(Puerari & Dottori 1997; Aguerri et al. 1998; Buta & Zhang 2009;
Sierra et al. 2015).
However, all these methods require some sort of modelling.
The only reliable direct and model-independent method to deter-
mine the bar pattern speed is the Tremaine-Weinberg (TW) method
(Tremaine & Weinberg 1984). It has been used extensively in the
past to study bar pattern speeds (Aguerri et al. 2015; Cuomo et al.
2019; Guo et al. 2019; Garma-Oehmichen et al. 2020). The TW
method uses surface brightness and line-of-sight (LOS) velocity
data to estimate the pattern speed.
MNRAS 000, 1–19 (2022)
Kinematics of strong and weak bars 3
Figure 1. DECaLS postage stamps (64x64 arcsec) of a strongly (left) and
weakly barred galaxy (right), on which we will apply the Tremaine-Weinberg
method.
In this paper, we use the TW method on integral-field spec-
troscopy data from the Mapping Nearby Galaxies at Apache Point
Observatory (MaNGA) survey (Bundy et al. 2015) to estimate bar
pattern speeds, corotation radii and the dimensionless ratio R for a
sample of 225 galaxies. This is the largest sample to date measured
with the TW method and includes both weakly and strongly barred
galaxies, identified using GZ.
The structure of the paper is as follows: in Section 2, we explain
the Tremaine-Weinberg method in detail. The data and sample se-
lection is explained in Section 3. Section 4 shows our results, which
are discussed in Section 5. Finally, our conclusions are summarised
in Section 6. Where necessary, we assumed a standard flat cosmo-
logical model with H0 = 70 km s−1 Mpc−1, Ωm = 0.3 and ΩΛ =
0.7.
2 THE TREMAINE-WEINBERG METHOD
2.1 Theory
The Tremaine-Weinberg (TW) method is a model-independent
method to determine the pattern speed of a galaxy (Tremaine &
Weinberg 1984). The main assumptions of the TW method are that
there is a well-defined pattern speed and that the tracer used (i.e.
stars or gas) satisfies the continuity equation. To illustrate the dif-
ferent steps of the TW method, we will apply the TW method to
one strongly barred galaxy and one weakly barred galaxy, shown in
Figure 1.
Take a Cartesian coordinate system (X,Y ) in the sky plane with
the origin in the centre of the galaxy and the X-axis aligned with
the line of nodes (LON), which is defined as the intersection of the
sky plane and the disc plane, so it is effectively the major axis of the
galaxy. Then, the Tremaine-Weinberg method can be formulated as:
Ωb sin (i) =
∫
+∞
−∞
h(Y )
∫
+∞
−∞
Σ(X,Y ) VLOS(X,Y ) dXdY
∫
+∞
−∞
h(Y )
∫
+∞
−∞
X Σ(X,Y ) dXdY
, (1)
where Ωb is the bar pattern speed, i is the inclination of the
galaxy, VLOS is the line of sight velocity, Σ is the surface bright-
ness of the galaxy and h(Y ) is a weight function. A delta function
like h(Y ) = δ (Y − Y0) is typically used here, so that the integration
happens in pseudo-slits across the IFU parallel to the LON. Multi-
ple integrations are usually done with different offset distances Y0
Figure 2. The stellar flux (top row) and stellar velocity (bottom row) for
our strongly barred (left column) and weakly barred (right column) example
galaxies. The different pseudo-slits, over which the kinematic and photo-
metric integrals are calculated, are visualised on top of the maps in white
outlines.
to ensure reliable measurement of the pattern speed (Tremaine &
Weinberg 1984). In this case, Equation 1 can be simplified to:
Ωb sin (i) =
〈V〉
〈X〉
, (2)
where 〈X〉 is called the photometric integral and 〈V〉 the kine-
matic integral. They are defined as:
〈X〉 =
∫
+∞
−∞
XΣdΣ
∫
+∞
−∞
ΣdΣ
; 〈V〉 =
∫
+∞
−∞
VLOSΣdΣ
∫
+∞
−∞
ΣdΣ
. (3)
〈X〉 is effectively the luminosity-weighted mean position and
〈V〉 is the luminosity-weighted mean line of sight velocity. These
photometric and kinematic integrals are calculated for the multiple
different pseudo-slits across the IFU. These pseudo-slits are visu-
alised on top of the MaNGA stellar flux and velocity maps in Figure
2. Every pseudo-slit has a width of 1 arcsec and is carefully placed
next to each other so that no pixel is in two different slits. We make
the pseudo-slits as long as the data allows, however each slit should
be symmetrical in length centred on the disc minor axis, i.e. both
sides of the slit should be the same length on either side of the
disc minor axis. We place as many slits as we can fit within the bar,
but impose a minimum of three slits. The maximum amount of slits
placed on one galaxy was 23, and the median is 7.
We do not calculate the ratio of 〈V〉 and 〈X〉 directly. Instead,
we plot 〈V〉 against 〈X〉 for the different pseudo-slits and the slope
of the best-fit line going through these points will then be equal to
Ωb sin (i). This is done to help avoid centering errors and account
for incorrect estimates of the systemic velocity (Guo et al. 2019).
An example of such a plot can be found in Figure 3.
It is important to note that for an axisymmetric disc, the
weighted mean position and velocity integrals will equal zero. This
MNRAS 000, 1–19 (2022)
Strongly barred: 11956-12702 Weakly barred: 8323-610 1 ,. • w. • .,-
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• ", 
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4 Tobias Géron
Figure 3. The kinematic integral, 〈V 〉, is plotted against the photometric
integral, 〈X 〉, for all the pseudo-slits for a strongly barred galaxy (left) and
a weakly barred galaxy (right). The equation of the best-fit line going
through these points is shown in each plot. The equation of the best-fit
line going through these points is written in the top-right corner of each
plot. The slope of these lines is equal to Ωb sin (i).
means that any non-zero values will be due to additional structures
such as the bar (if it is not aligned or perpendicular to the LON).
The code used in this work to calculate the bar pattern speed
is publicly available here1.
2.2 Concerns and limitations
As alluded to in Section 2.1, the TW method requires that the chosen
tracer satisfies the continuity equation. Multiple studies have used
gas as the tracer and have been successful in determining the pattern
speed using the TW method (Zimmer et al. 2004; Hernandez et al.
2005; Emsellem et al. 2006; Fathi et al. 2009; Gabbasov et al. 2009).
Many studies have also successfully determined pattern speeds
by using stars as the tracer (Merrifield & Kuijken 1995; Debattista
et al. 2002; Aguerri et al. 2003; Corsini et al. 2007), although they
usually limited their sample to early-type barred galaxies. This is
because there were concerns that dust obscuration and star formation
in late-type galaxies could cause the surface brightness to not trace
the mass distribution properly. However, other papers show that it
is possible to use the TW method on late-type galaxies, despite
initial concerns (Gerssen et al. 2003; Gerssen & Debattista 2007;
Treuthardt et al. 2007; Aguerri et al. 2015; Cuomo et al. 2019;
Guo et al. 2019; Garma-Oehmichen et al. 2020). More recently,
Williams et al. (2021) applied the TW method to stellar and gaseous
tracers (using both CO and Hα) and found significantly different
results. They attributed this inconsistency to the clumpy nature of
the gaseous tracers they used, which resulted in incorrect pattern
speed measurements. Thus, in this work we decided to use stars as
our tracer.
As Garma-Oehmichen et al. (2020) show, centering issues are
not negligible and it is crucial that the LON goes through the centre
of the galaxy. In this work, we find the centre by smoothing the
stellar flux data with a Gaussian filter and finding the brightest pixel
in the smoothened data.
The TW method is also only applicable to galaxies with regu-
lar kinematics and on galaxies with intermediate inclinations (20◦
< i < 70◦) (Tremaine & Weinberg 1984; Aguerri et al. 2015;
Cuomo et al. 2019; Garma-Oehmichen et al. 2020). Edge-on galax-
ies do not have enough spatial data, while face-on galaxies have
1 github.com/tobiasgeron/Tremaine_Weinberg
low quality kinematic data.while the stellar velocity is not well
constrained in face-on galaxies. Additionally, detecting bars in
edge-on galaxies is very difficult and unreliable. It is known that
the TW method is very sensitive to incorrect estimates of the PA of
the galaxy (Debattista 2003; Zou et al. 2019; Garma-Oehmichen
et al. 2020). Thus, a correct and reliable estimate of the position
angle is crucial. We try to account for this sensitivity by performing
a Monte Carlo (MC) simulation over the uncertainty of the PA (see
Section 3.3 for more details). For the TW method to work, it is also
important that the bar is not aligned with the major or minor axis
of the galaxy, as otherwise the integrals will cancel out. We also
need to be able to place a sufficient amount of slits, otherwise the
straight line in the 〈V〉 over 〈X〉 plot that is used to determine Ωb is
not well constrained. As the slits have to be placed on top of the bar,
the TW method is not ideal for the shortest of bars, where only very
few slits can be placed. This will disproportionally affect weak bars,
which should be kept in mind. The specific thresholds we impose
are detailed in Section 3.5.
2.3 Calculation of corotation radius and R
After the bar pattern speed is obtained using the TW method, one can
calculate the corotation radius (RCR). This is where the centrifugal
and gravitational forces balance each other in the rest frame of the
bar, which means that the stars in the disc will have the same angular
velocity as the bar pattern speed at the corotation radius (Cuomo
et al. 2019; Guo et al. 2019). Various papers calculate this by doing
RCR = Vc/Ωb, where Vc is the circular velocity in the flat part of the
rotation curve (Aguerri et al. 2015; Cuomo et al. 2019; Guo et al.
2019). However, this assumes that the corotation radius lies in the
region where the rotation curve has flattened. This is not necessarily
the case and can lead to incorrect estimates of RCR and R.
Instead, we will use the rotation curve of the galaxy to calculate
the corotation radius. The rotation curve can be obtained by using
the MaNGA stellar velocity data (see Section 3.4). The bar pattern
speed is multiplied by a radius range, which effectively shows how
fast the tracer moves at any radius for that particular pattern speed.
The radius at which this curve intersects with the galaxy rotation
curve, is the corotation radius. An example of this can be found in
Figure 4.
Garma-Oehmichen et al. (2020) used a similar approach to
ours and compared their results to results obtained by using the
RCR = Vc/Ωb method. On average, they found a relative differ-
ence of ∼15%, indicating that the simplified approach introduces a
significant bias.
The corotation radius can be used to calculate the dimension-
less parameter R, defined as R = RCR/Rbar, where Rbar is the
deprojected bar radius. We obtain estimates for the uncertainty on
Ωb, RCR and R by performing a Monte Carlo simulation using the
errors on the input variables and assuming Gaussianity (see Section
3.3 for more details). The posterior distributions of the final pattern
speed, corotation radius and R for our example galaxies are shown
in Figure 5.
R can be used to classify bars into fast (1.0 < R < 1.4) and
slow (R > 1.4) bars (Debattista & Sellwood 2000; Rautiainen et al.
2008; Aguerri et al. 2015). Thus, slow bars have bar lengths that
are shorter that the cororation radius, whereas fast bars end near
the corotation radius. It is suggested that bars cannot extend beyond
corotation (Contopoulos 1980, 1981; Athanassoula 1992b), which
means that bars with R < 1.0 are not expected. However, multiple
studies have observed these so-called ultrafast bars (Buta & Zhang
MNRAS 000, 1–19 (2022)
->O .' 
->O L~_" -,,_,C----¡;--,,------,---,---' 
< )( :> Larcsec[ 
Weakly barred: 8323·6101 
<'f>.··121U ... ~.·189 
.. 1)( -0 .1 o» 02 " < )( > 11'rtsecJ 
Kinematics of strong and weak bars 5
Figure 4. Visualisation of how the corotation radius is obtained, for a
strongly barred galaxy (left) and a weakly barred galaxy (right). The blue
dots are stellar velocity measurement from MaNGA in a 5 arcsec aperture
along the major axis of the galaxy. The black line is the best-fit rotation
curve, please refer to Section 3.4 for more details on how the rotation curve
is calculated. The black dashed line is obtained by multiplying Ωb with a
radius range. The distance where this line and the rotation curve intersect
(indicated by the orange cross), defines the corotation radius. The dotted
vertical line is the deprojected bar radius.
2009; Aguerri et al. 2015; Cuomo et al. 2019; Guo et al. 2019;
Garma-Oehmichen et al. 2020).
3 DATA
3.1 MaNGA survey
We need resolved stellar velocity and stellar flux data in order to
implement the TW method, which we obtain from the Mapping
Nearby Galaxies at Apache Point Observatory (MaNGA) survey
(Bundy et al. 2015). MaNGA is part of the Sloan Digital Sky Survey
IV (SDSS-IV) collaboration (Blanton et al. 2017). More specifically,
we used data from the seventeenth data release of SDSS (Abdurro’uf
et al. 2022). MaNGA used the Baryon Oscillation Spectroscopic
Survey (BOSS) Spectrograph, which has a resolution of R ∼ 2000
and a wavelength coverage of 3600 - 10,000 Å (Smee et al. 2013), on
the 2.5m Sloan Telescope at Apache Point Observatory (Gunn et al.
2006). Every integral field unit (IFU) consists of 19-127 optical
fibers, stacked hexagonally (Drory et al. 2015). Most galaxies are
covered out to 1.5 effective radii (Re), while a third are covered
out to 2.5 Re. We make use of the maps that are binned to S/N
∼ 10 using the Voronoi binning algorithm (Westfall et al. 2019).
For more information on the observing strategy, survey design, data
reduction process, sample selection and the data analysis pipeline,
please refer to Law et al. (2015); Yan et al. (2016); Law et al. (2016);
Wake et al. (2017); Belfiore et al. (2019); Westfall et al. (2019). All
the stellar masses and SFRs in this paper come from the Pipe3D
value added catalog (Sánchez et al. 2016a,b). The SFRs in Pipe3D
are estimated from the Hα flux and is dust and aperture corrected.
For more details, please refer to (Sánchez et al. 2016b). Finally, this
paper made extensive use of the Marvin software in order to access
MaNGA data (Cherinka et al. 2019).
3.2 Galaxy Zoo and the Legacy Survey
We have used the Galaxy Zoo (GZ) project to obtain morphological
classifications and find weak and strong bars. Here, citizen scientists
classify galaxies according to a decision tree (Lintott et al. 2008,
2011). We made use of the latest iteration of Galaxy Zoo, namely
Galaxy Zoo DESI (GZ DESI, Walmsley et al. in prep.). GZ DESI
sources images from the DESI Legacy Imaging Surveys2 (Dey et al.
2019), which consists of three individual projects: the Dark Energy
Camera Legacy Survey (DECaLS), the Beijing-Arizona Sky Sur-
vey (BASS) and the Mayall z-band Legacy Survey (MzLS), which
covers ∼ 14,000 deg2 of sky. As shown by Géron et al. (2021), the
DESI Legacy Imaging Surveys are sufficiently deep so that weak
bars are visible and can be identified by the volunteers (the median
5σ point source depth of DECaLS is r = 23.6, Dey et al. 2019).
GZ DESI uses classifications from citizen scientists to train ma-
chine classifications based on the Bayesian convolutional neural
networks described in Walmsley et al. (2021), which we rely on for
our morphology measurements. The decision tree of GZ DESI, up
to the bar question, is shown in Figure 6. Note that volunteers will
only reach the bar question after they identified the target as being
a disk that is not edge-on.
3.3 Inclination, bar length and position angles
In order to perform the TW method, we need multiple additional
parameters. We need the inclination of the galaxy, the kinematic
position angle of the galaxy, the length of the bar and the position
angle of the bar.
3.3.1 Position angle
Various papers using the TW method often use photometric ap-
proaches to obtain the position angle of the galaxy, such as fitting
ellipses in the sky plane (Aguerri et al. 2015; Cuomo et al. 2019;
Guo et al. 2019). The outermost isophotes are then used to estimate
the position angle. However, multiple issues are associated with
this method. First of all, there is no systematic way to determine
which and how many outer isophotes to use to determine the po-
sition angle. Secondly, the presence of a bar, especially a strong
bar, will influence these estimates. Additionally, spiral arms, rings,
companion galaxies and foreground stars will all also affect these
measurements significantly.
However, MaNGA allows us to use kinematic position angles
rather than photometric ones. We obtained the global kinematic po-
sition angle and its uncertainty by using the Python package PaFit
on the stellar velocity IFU data from MaNGA. This package is based
on the method detailed in Appendix C of Krajnović et al. (2006)3.
It constructs a bi-anti-symmetric map based on the original input.
The position angle that minimises the difference between the origi-
nal velocity map and the symmetrised map is considered to be the
best-fit global kinematic position angle. The error on the best-fit
position angle is defined as the range of angles for which the dif-
ference in χ2 with the best-fit angle is less than 9 (Krajnović et al.
2006). This corresponds to a 3σ confidence limit, which is then
converted to a 1σ limit. Multiple other papers have successfully
used this code to study galaxy kinematics before (Cappellari et al.
2007; Krajnović et al. 2011). The kinematic position angle does not
suffer from the issues that plague the photometric position angle,
which is why we used the kinematic one in this work. However, the
kinematic position angle is not infallible. The bar can twist the inner
parts of the disc velocity field, which can affect the measurement
of the kinematic position angle. Additionally, the coverage of the
IFU, inclination of the galaxy and the difference in position angle
2 www.legacysurvey.org/
3 www-astro.physics.ox.ac.uk/~cappellari/software#pafit
MNRAS 000, 1–19 (2022)
Strongly barred: 11956·12702 
SO 1.5 10.0 11. S 15.0 
Ulstance [arcsec J 
Weakly barred: 8323·6101 
. / 
. , , 
UIstanc2 [arcsec J 
6 Tobias Géron
Figure 5. The posterior distributions of the pattern speed (left column), corotation radius (middle column) and R (right column) for a strongly barred galaxy
(top row) and weakly barred galaxy (bottom row), obtained from performing Monte Carlo (MC) simulations of a 1,000 iterations in order to characterise the
uncertainty on each measurement. The median value is indicated in every plot by a black vertical line, while the 16th and 84th percentile are shown by the
dashed vertical lines. These values are also printed in each subplot.
Is the galaxy simply smooth and 
rounded, with no sign of a disk?
Features or 
Disk
Smooth
Star or 
Artifact
Could this be a disk viewed edge-on?
No - something else Yes - Edge On Disk
Is there a bar feature through the 
centre of the galaxy?
No bar Weak bar Strong bar
Galaxy Zoo DECaLS
Figure 6. The decision tree of GZ DESI up to the bar question. It is worth
noting that volunteers will only reach the bar question after they said the
target is a disk galaxy that is not viewed edge-on. The full decision tree is
shown in Walmsley et al. (2021).
between the bar and disc will have some influence as well. These
effects are described in more detail in Appendix A3 of Guo et al.
(2019).
3.3.2 Inclination
Estimating the inclination of a barred galaxy, especially a strongly
barred one, is not straight forward. As a strong bar is a very obvious
component, it will make the galaxy appear more inclined. There-
fore, we carefully measured the inclinations of our barred galaxies
ourselves, using the elliptical isophote analysis technique described
by Jedrzejewski (1987) using the Python package photutils4 on
the r-band images from the Legacy Survey. We typically averaged
the ellipticity of the outermost 5% of fitted isophotes, which
usually corresponded to 5 isophotes. However, to guarantee the
bar does not affect our measurement, we excluded any isophotes
that are within the bar region. This meant that we used less than
5 isophotes for some targets that had long bars. The ellipticity
profiles and r-band images of all our targets were inspected
individually to make sure that the final value was correct. We
averaged the ellipticity of the outermost isophotes. To guaran-
tee the bar does not affect the measurement, it is made sure
that the isophotes used are outside the bar region by inspecting
every galaxy individually.However, spiral arms, foreground stars
and rings will all bias this measurement to some degree. To estimate
the error on this ellipticity measurement, we correctly combine the
errors associated with the isophotes used to calculate the ellipticity.
3.3.3 Bar length and bar position angle
There are multiple ways to determine the length of the bar. One
possibility involves ellipse fitting again (Laine et al. 2002; Erwin
2005; Marinova & Jogee 2007; Aguerri et al. 2009), but as bars are
associated with spiral arms, rings and ansae, this method is prone
to inconsistencies. Other methods include Fourier decomposition
(Aguerri et al. 2000) and using explainable artificial intelligence
and saliency mapping techniques (Bhambra et al. 2022). Addition-
ally, the Galaxy Zoo:3D project provides bar masks for galaxies in
MaNGA, based on SDSS images, which can be used to estimate bar
length (Masters et al. 2021).
4 https://photutils.readthedocs.io/en/stable/index.html
MNRAS 000, 1–19 (2022)
_ 0.17 
J 
", 
---
Kinematics of strong and weak bars 7
A more straight-forward approach is to manually mea-
sure bar lengths. Manual bar length measurements have been
successfully used in various studies (Erwin 2019; Géron et al.
2021). Additionally, Hoyle et al. (2011) have found that manual
bar length measurements between different volunteers agree
within 10% of each other and they show that manual measure-
ments can be unbiased and robust against systematic effects.
Additionally, Díaz-García et al. (2016a) have found that their
manual bar lengths agree with bar lengths determined by vari-
ous other automated techniques.
However, in this work, the bar lengths were measured man-
ually by one of the authors (TG) on images from the Legacy
Survey. Thus, manual bar length measurements are used in this
work. The bar lengths were measured manually by one of the
authors (TG) on grz-images obtained from the Legacy Survey. A
number of measures were put in place to make sure these mea-
surements were done as consistently and correctly as possible.
For example, the order of the measurements was completely ran-
domised, so that it was was not known whether the bar that was being
measured was classified as a strong or weak bar by GZ. The mea-
surements themselves were done in DS9 (Joye & Mandel 2003)
with a measurement tool that automatically records the dis-
tance measured. Additionally, every bar was measured twice and
the final bar length distribution is modelled by a Gaussian centered
around the average of the two measurements and with an uncertainty
equal to half the difference between the two measurements. Finally,
all measurements were inspected again afterwards to make sure
no mistakes were made. These bar lengths were successfully used
before in Géron et al. (2021), where they were compared to an-
other bar length catalog (Hoyle et al. 2011). The bar lengths are
deprojected using the method described by Gadotti et al. (2007):
Rb,deproj = Rb,obs
√
cos2 φ + sin2 φ/cos2 i , (4)
where i is the inclination of the galaxy, φ is the difference be-
tween the position angle of the bar and of the galaxy and Rb,obs and
Rb,deproj are the observed and deprojected bar lengths, respectively.
The position angles of the bar were also obtained from these manual
measurements and are similarly modelled by a Gaussian centered
around the average of the measurements and with an uncertainty
equal to half the difference between the two measurements.
The uncertainties on the inclination, disk PA, bar length and
bar PA are used to estimate the uncertainty on the bar pattern speed,
corotation radius and R. This is achieved by assuming Gaussianity
over these input parameters and performing a Monte Carlo simula-
tion with 1,000 iterations.
An overview of all the input parameters for 50 randomly se-
lected targets is given in Table 1. The full table can be found online
here5.
3.4 Rotation curve
As mentioned in Section 2.3, we need the rotation curve of the
galaxy in order to obtain the corotation radius. The rotation curve
can be determined from the stellar velocity IFU data from MaNGA.
We look at the spaxels in a 5 arcsec aperture along the position angle
5 github.com/tobiasgeron/Tremaine_Weinberg
of the galaxy. The true stellar velocity in every spaxel is calculated
from the observed stellar velocity by doing:
Vrot = Vobs/(sin i × cos φ) , (5)
where Vobs and Vrot are the observed and true velocity in that
spaxel, i is the inclination of the galaxy and φ is the azimuthal angle
measured relative to the position angle of the galaxy. The distance
to the centre of the galaxy is deprojected using equation 4. The
corrected velocities and deprojected distances are used to fit a two
parameter arctan function, described in Courteau (1997):
Vrot = Vsys +
2
π
Vc arctan
(
r − r0
rt
)
, (6)
where Vsys is the systemic velocity, Vc is the asymptotic ve-
locity, r0 is the spatial centre of the galaxy and rt is the transition
radius. The rotation curve flattens at rt and goes towards Vc in this
model. For our purposes, Vsys and r0 are assumed to equal zero.
3.5 Sample selection
We use the machine classifications from GZ DESI (Walmsley et al.
in prep.). GZ works based on a decision tree structure. As you can
see in Figure 6, this means that the question “Is there a bar feature
through the centre of the galaxy?” is only answered when the galaxy
is a not edge-on disk galaxy. To guarantee reliable bar classifications,
we must apply additional thresholds on the fraction of people that
would have been asked the bar question (Nbar
6) and the fraction of
people that would have voted for a certain answer (e.g., pstrong bar),
as predicted by the automated classifications. We choose to apply
pfeatures/disk ≥ 0.27, pnot edge-on ≥ 0.68 and Nbar ≥ 0.5. For more
information on these thresholds, please refer to Géron et al. (2021)
and Walmsley et al. (2021). These thresholds resulted in a sample
of 3,140 galaxies that consists of relatively face-on disk galaxies
with reliable bar classifications. The same classifications are also
used to assign a bar type (no bar, weak bar or strong bar) to every
galaxy. The galaxy had no bar if pstrong bar + weak bar < 0.5. If this
was not the case and if pweak bar ≥ pstrong bar, then the galaxy had a
weak bar. Otherwise, it had a strong bar. This classification scheme
was used before in Géron et al. (2021) and is shown in Table 2.
The galaxies that are identified as unbarred were removed from our
sample, which reduced the sample size to 1,696 barred galaxies.
In order to avoid selection effects, we work with a volume-
limited sample by imposing additional thresholds on the redshift
(0.01 < z < 0.05) and absolute r-band magnitude (Mr < −18.96,
values obtained from the NASA-Sloan Atlas), which removed 525
galaxies from our sample. Limitations of the TW method (see Sec-
tion 2.2) also impose a few additional thresholds on our sample
selection. The TW method is also not developed for galaxies with
irregular kinematics, as one of the main assumptions of the TW
method is the existence of a well-defined pattern speed. The stel-
lar velocity field of every galaxy was inspected by eye and 477
irregular galaxies were removed. Additionally, the bar cannot align
with the disc major or minor axis. Thus, galaxies where the PA of
the bar was within 10◦ of the major or minor axis of the galaxy
6 Please note that, as we are using machine classifications, in this context,
Nbar is not the amount of people that have been asked the bar question.
Rather, it is the estimated fraction of people that would have been asked the
bar question.
MNRAS 000, 1–19 (2022)
8 Tobias Géron
Table 1. The plate-ifu number, right ascension, declination, inclination, position angle of the disk, position angle of the bar, the (projected) bar radius, redshift
and the bar type for 50 randomly selected galaxies. The full table can be found online here.
Plate-ifu RA [◦] DEC [◦] Inclination [◦] PAdisk [◦] PAbar [◦] Rbar [arcsec] Rbar,deproj [kpc] Redshift Bar type
8723-12701 126.9739 55.1586 62.95±1.73 1.40±0.20 37.38±0.18 3.93±0.18 4.07±0.19 0.0388 Weak bar
9872-9102 234.1347 41.7993 42.18±0.70 64.90±0.28 114.68±2.91 3.90±0.09 3.48±0.08 0.0414 Weak bar
8616-12702 322.3060 -0.2948 48.54±1.37 38.00±0.73 110.31±3.03 7.67±0.05 5.58±0.03 0.0305 Strong bar
8077-12705 41.6946 -0.7288 40.20±6.21 21.40±1.48 8.45±2.24 6.75±0.02 5.96±0.02 0.0439 Strong bar
8322-1901 198.7842 30.4038 45.30±0.69 99.70±1.45 22.61±0.43 2.30±0.28 1.29±0.16 0.0232 Strong bar
9027-12704 245.3466 32.3490 54.90±1.87 135.70±2.07 87.73±0.54 10.57±0.28 9.83±0.26 0.0347 Strong bar
10220-9101 120.8259 31.7764 39.91±1.08 61.40±0.35 82.59±1.43 5.95±0.06 4.53±0.04 0.0364 Strong bar
9187-9101 311.3127 -5.6227 52.69±1.42 66.70±0.72 103.32±1.54 3.87±0.50 2.38±0.31 0.0273 Weak bar
11758-3701 200.9719 51.4352 56.04±1.06 168.40±1.33 59.36±0.23 4.87±0.54 3.34±0.37 0.0253 Weak bar
11866-6102 150.1015 0.7046 56.50±0.47 177.60±0.63 37.97±2.58 2.80±0.44 2.19±0.34 0.0334 Weak bar
11866-12701 149.8591 -0.2539 50.26±0.42 93.50±0.27 33.79±0.76 4.25±0.18 4.66±0.20 0.0475 Weak bar
9886-12701 236.3470 24.5068 53.22±0.91 90.70±0.37 152.87±0.90 3.78±0.11 2.22±0.06 0.0230 Weak bar
9876-9102 195.8556 28.0140 52.52±1.76 32.10±0.17 75.66±3.69 3.51±0.11 2.39±0.07 0.0265 Strong bar
11836-3702 150.5171 0.6875 32.58±6.07 58.60±0.55 102.45±2.08 4.17±0.38 4.24±0.39 0.0451 Weak bar
9033-12705 224.0182 45.4053 28.90±2.33 71.90±1.03 100.33±1.16 6.60±0.28 5.05±0.22 0.0362 Strong bar
11982-9102 311.3108 -5.4843 52.31±0.36 60.40±0.88 140.15±1.07 7.24±0.18 6.07±0.15 0.0270 Strong bar
8721-6103 133.1213 56.1127 42.22±1.56 51.50±1.03 117.64±0.97 4.19±0.33 4.33±0.34 0.0454 Strong bar
8935-6104 195.5329 27.6483 40.45±2.19 85.10±1.13 53.03±3.60 2.46±0.18 1.38±0.10 0.0230 Strong bar
7979-12702 333.3396 13.4365 46.69±2.22 142.00±0.17 8.43±1.65 5.90±0.36 4.11±0.25 0.0281 Weak bar
12495-6102 160.4608 4.3308 64.23±0.21 45.80±1.97 85.44±2.11 4.05±0.13 3.14±0.10 0.0268 Weak bar
8983-12705 203.9100 25.8752 62.42±1.95 59.30±0.23 72.77±2.71 7.39±0.11 4.06±0.06 0.0256 Weak bar
11747-9102 130.6514 27.2702 63.37±0.81 84.30±0.33 56.86±1.65 2.51±0.01 1.63±0.00 0.0255 Weak bar
9879-6102 197.0069 26.7654 34.80±3.01 68.10±0.95 121.01±2.42 4.68±0.24 3.94±0.20 0.0341 Strong bar
11979-9101 252.3505 22.9414 32.97±1.12 0.90±0.42 39.90±3.83 6.55±0.44 6.30±0.42 0.0443 Weak bar
8932-6104 195.5361 28.3871 38.44±5.51 85.50±0.53 50.64±0.28 7.10±0.86 4.03±0.49 0.0253 Weak bar
11977-12703 255.5133 25.4164 50.92±0.52 123.70±1.05 94.74±0.77 9.42±0.21 6.81±0.15 0.0314 Weak bar
8093-12703 22.2241 14.7207 42.09±2.22 69.40±1.52 83.82±0.67 10.41±0.25 7.91±0.19 0.0367 Strong bar
9499-6102 119.9470 25.3570 38.87±0.59 141.60±1.90 2.96±4.91 2.65±0.90 1.58±0.53 0.0267 Weak bar
11952-12701 252.4987 27.0748 59.42±0.41 132.30±0.92 164.91±5.06 7.28±1.31 8.08±1.46 0.0458 Strong bar
11016-12703 212.5579 51.1136 52.09±0.54 92.80±0.17 63.89±0.03 5.01±0.14 5.07±0.14 0.0461 Weak bar
8312-12704 247.3041 41.1509 45.71±0.78 27.90±0.42 140.83±0.82 6.28±0.19 4.67±0.14 0.0296 Strong bar
8600-12701 245.7265 41.5206 48.84±0.92 77.10±0.45 138.50±0.92 8.56±0.64 6.46±0.48 0.0280 Strong bar
7992-6104 255.2795 64.6769 54.93±0.33 0.80±1.05 122.90±0.51 4.93±0.09 4.20±0.08 0.0271 Strong bar
8249-6101 137.5625 46.2933 50.93±0.69 63.60±0.62 107.99±0.83 5.88±0.47 4.55±0.36 0.0267 Strong bar
8938-12702 120.8144 30.7965 42.90±0.59 55.50±0.22 42.50±0.69 5.98±0.26 4.86±0.21 0.0403 Weak bar
11834-12705 223.3961 0.0104 56.12±0.43 89.40±0.42 68.02±0.18 6.48±0.61 6.00±0.56 0.0424 Strong bar
9026-12705 251.4585 43.8665 41.04±0.75 147.50±0.53 45.09±1.46 6.42±0.36 4.99±0.28 0.0338 Weak bar
8604-12703 247.7642 39.8385 47.61±0.28 95.40±0.75 151.93±0.67 6.56±0.44 4.71±0.32 0.0305 Weak bar
8442-9102 200.2228 32.1908 34.07±7.31 15.30±1.15 90.08±0.29 6.20±0.29 4.23±0.20 0.0230 Strong bar
11751-6102 146.6335 35.3303 48.25±0.57 53.30±0.67 159.29±1.15 4.10±0.44 4.35±0.47 0.0428 Strong bar
8312-12702 245.2709 39.9174 44.29±2.33 93.50±0.60 117.32±2.43 3.45±0.03 2.36±0.02 0.0320 Weak bar
9484-12703 120.6746 34.9446 32.20±1.85 119.20±1.30 91.81±0.42 7.04±0.21 4.74±0.14 0.0318 Strong bar
10515-6104 149.2979 5.2014 57.85±1.47 0.90±1.22 167.64±0.22 6.18±0.49 2.81±0.22 0.0216 Strong bar
8140-12701 116.9303 41.3864 30.66±0.20 59.60±0.68 123.66±1.37 6.88±0.32 4.87±0.23 0.0286 Strong bar
8983-3703 205.4384 27.0047 54.22±1.68 178.20±0.37 135.74±3.01 4.26±0.21 3.11±0.16 0.0289 Weak bar
11956-6103 187.1800 53.3577 59.11±0.23 1.00±0.75 103.59±0.23 6.04±0.03 5.60±0.03 0.0366 Weak bar
8484-12703 248.7883 46.2139 38.90±1.63 65.00±0.57 53.68±0.91 7.51±0.03 4.71±0.02 0.0308 Strong bar
9029-12704 247.2170 42.8120 58.38±0.40 115.70±0.28 78.55±1.51 5.46±0.16 4.21±0.12 0.0316 Strong bar
9192-6101 46.3470 -0.2421 66.39±1.09 86.00±0.17 134.94±0.35 1.37±0.24 1.15±0.20 0.0288 Weak bar
8989-3703 177.4404 50.5270 48.37±0.68 35.50±0.85 59.18±0.29 8.46±0.05 5.02±0.03 0.0264 Strong bar
and 175 more rows...
MNRAS 000, 1–19 (2022)
Kinematics of strong and weak bars 9
Table 2. The vote fractions of the galaxy are used to determine its bar type
(no bar, weak bar or strong bar), according to the following scheme. This
method of classification is identical to the one in Géron et al. (2021).
Condition 1 Condition 2 Result
pstrong bar + weak bar < 0.5 N/A No bar
pstrong bar + weak bar ≥ 0.5 pstrong bar < pweak bar Weak bar
pstrong bar + weak bar ≥ 0.5 pstrong bar ≥ pweak bar Strong bar
were removed from our sample, which affected 206 galaxies. The
TW method only works on galaxies with intermediate inclination,
so we limit our sample to galaxies with inclinations between 20◦
and 70◦, which removed a further 31 galaxies.
As our methodology requires to reliably perform a linear fit in
the 〈V〉 against 〈X〉 plots, we require each galaxy to have at least
three pseudo-slits. Similarly, to ensure the robustness of the linear
fit, we use normalized root mean squared error (NRMSE) to estimate
the fit quality. We only included targets that had a median NRMSE of
all the MC iterations lower than 0.2. Finally, we fit the rotation curve
of the galaxy with a two-parameter arctan function (see Section 3.4).
However, some galaxies are not described correctly by this function,
especially galaxies with highly irregular kinematics (which have
mostly already been removed by this point). Thus, a threshold of
median NRMSE < 0.2 is imposed on this fit as well. These threshold
values for NRMSE were chosen after careful visual inspection of
their fits. Additionally, targets where more than 10% of the MC
iterations were unable to provide a value for pattern speed (e.g.,
due to not being able to place enough pseudo-slits), were excluded
as well. Applying these last thresholds result in a final sample that
contains 225 galaxies, with 125 strongly barred and 100 weakly
barred galaxies.
Reliable bar pattern speed estimates were obtained for all these
targets. However, it was found that for a small subset of these tar-
gets, especially for those with low pattern speeds, estimating the
corotation radius and R is difficult. This was because the corota-
tion radius was so high, that it fell far outside the MaNGA field of
view. It was judged that extrapolating the velocity curves too much
results in unreliable estimates for the corotation radius. Therefore,
we excluded any targets where we had to extrapolate by more than
a factor of two. This affected 28 of our 225 galaxies.
4 RESULTS
4.1 Bar pattern speeds, corotation radii and R
The final bar pattern speeds of all our weakly and strongly barred
galaxies are shown in Figure 7. As mentioned in Section 3.3.2, incli-
nation measurements are prone to biases, so to be cautious we show
both Ωb sin (i) (top row) and Ωb (bottom row). Additionally, the
pattern speed is measured in observational units (km s−1 arcsec−1,
left column), which are then converted to physical units (km s−1
kpc−1, right column).
Despite the distributions for the weakly and strongly barred
samples overlapping considerably, an Anderson Darling test reveals
that they are still significantly different, with strongly barred galax-
ies having lower average pattern speeds than weak barred galaxies.
The p-values for the Ωb sin (i) distributions are 0.004 and 0.006
for the plots using observational units and physical units, respec-
tively. The p-values for the Ωb plots are 0.069 and 0.041. As we
do not expect any intrinsic differences in inclination between the
weakly and strongly barred galaxy samples, this result highlights
the importance of a robust inclination measurement.
We can conclude that strongly barred galaxies have sig-
nificantly lower bar pattern speeds than weakly barred galax-
ies. The median, together with the 16th and 84th percentiles, is
Ωb = 26.32+11.41
−13.22 km s−1 kpc−1 for strongly barred galaxies and
Ωb = 28.05+14.77
−11.54 km s−1 kpc−1 for weakly barred galaxies.
The final corotation radii for our target galaxies are shown in
Figure 8. Strongly and weakly barred galaxies do not seem to differ
significantly in their corotation radii. The median values are RCR =
6.26+3.97
−2.31 kpc and RCR = 6.44+4.0
−2.99 kpc for the strongly barred
and weakly barred sample, respectively. A two-sample Anderson
Darling test reveals, with a p-value of >0.25, that both distributions
are not significantly different.
With a p-value of 0.01 from a two-sample Anderson Dar-
ling test, we could not clearly conclude that there is a significant
difference between strongly and weakly barred galaxies for the
ratio R.The determination of R strongly depends on correctly
estimating the bar length and inclination. Therefore, with a p-
value of 0.01 from a two-sample Anderson Darling test, we do
not have enough evidence to conclude that there is a significant
difference between strongly and weakly barred galaxies for the
ratioR, as shown in Figure 9. The median value for strongly barred
galaxies is R = 1.32+0.87
−0.48 and R = 1.55+1.44
−0.58 for weakly barred
galaxies. Albeit not significantly different, there is a difference in
median values between the weakly and strongly barred sample.
Additionally, we see more ultrafast bars among strong bars than
weak bars (24.0% and 17.0%, respectively). However, we believe
this can mostly be attributed to uncertainties in the bar length and
inclination estimates, which are crucial to correctly estimating
R. Comparisons to other bar length estimates or larger sample sizes
will help to clarify this issue.
As mentioned above, R is used to divide bars into ultrafast
(R < 1.0), fast (1.0 < R < 1.4) and slow (R > 1.4). Most bars in
our sample seem to be slow bars (47% of our sample), as shown
in Table 4. This fraction is higher than what most other studies
find. Conversely, we find less ultrafast and fast bars (21% and 32%,
respectively) than most other studies.
The final values for pattern speeds, corotation radii and R of
50 randomly selected galaxies is shown in Table 3. The full table
can be found online here7.
4.2 Relationship between the parameters
The bar pattern speed and corotation radius should be inversely
proportional to each other. A higher pattern speed will result in a
steeper gradient of the straight line shown in Figure 4, resulting in
it intersecting with the rotation curve at a shorter distance, which
produces a lower corotation radius (see Section 2.3 for more de-
tails). This is shown explicitly in Figure 10, where the inversely
proportional relationship becomes very clear.
We can also see that galaxies with the highest values of R
tend to have lower values for the bar pattern speed. This also makes
sense, as low pattern speeds will result in larger corotation radii,
which increases R. Conversely, galaxies with lower R tend to have
lower values for the corotation radius.
Figure 11 shows the corotation radius plotted against the bar
radius. Lines of equal values of R are found diagonally over this
Figure, as R is defined as the ratio between the corotation radius and
7 github.com/tobiasgeron/Tremaine_Weinberg
MNRAS 000, 1–19 (2022)
10 Tobias Géron
Figure 7. The final median values for Ωb sin (i) (top row) and Ωb (bottom row) for every galaxy, after doing a Monte Carlo simulation of 1,000 iterations.
The sample is divided into strongly barred (orange) and weakly barred (blue). The MC is done over observational units (which include arcsec), which are
afterwards converted to kpc. The left column shows the results for the observational units, while the right column shows the results for the physical units. The
vertical dashed lines show the median values for every histogram. The full lines are kernel density estimates of these histograms, using a Gaussian kernel. The
p-value of a two-sample Anderson-Darling test is shown inside each subplot, with the null hypothesis being that the two samples are drawn from the same
population. We see that, on average, strongly barred galaxies have significantly lower bar pattern speeds, despite there being significant overlap between the
two populations.
Figure 8. The final median values for RCR for every galaxy, after doing a Monte Carlo simulation of 1,000 iterations. The sample is divided into strongly
barred (orange) and weakly barred (blue). The MC is done over observational units (which is in arcsec), which are afterwards converted to kpc. The left column
shows the results for the observational units, while the right column shows the results for the physical units. The vertical dashed lines show the median values
for every histogram. The full lines are kernel density estimates of these histograms, using a Gaussian kernel. The p-value of a two-sample Anderson-Darling
test is shown inside each subplot, with the null hypothesis being that the two samples are drawn from the same population. We see no significant difference
between weak and strong bars in terms of corotation radii.
MNRAS 000, 1–19 (2022)
~ 
e 
Q¡ 0. 10 -, 
~ \lOO -
o •. "• " •• • E 0.02 -
" O 
< •• • 
(ltlar SIn(I) [km s- , arcsec' J 
Obor [km s- , c¡ rcsec'[ 
Observational units 
, " , 
Ru< [ilrcSec] 
~. ~> 0.25 
SO""",, loar 
_ .".., bu 
" 
Otlar sin(i) [km s- ' I::pc'] 
• 
~ r [I::m s- ' kpc '[ 
ROl [kpc] 
Kinematics of strong and weak bars 11
Figure 9. The final median values for R for every galaxy, after doing a
Monte Carlo simulation of 1,000 iterations. The sample is divided into
strongly barred (orange) and weakly barred (blue). The vertical dashed lines
show the median values for every histogram. The full lines are kernel density
estimates of these histograms, using a Gaussian kernel. The p-value of a two-
sample Anderson-Darling test is shown inside each subplot, with the null
hypothesis being that the two samples are drawn from the same population.
We see that there is no difference in terms of R.
Figure 10. The bar pattern speed (Ωb) of all our targets plotted against the
corotation radius (RCR). The colour of the data points is determined by R.
The median error on the x and y axis is shown in the top-right corner. We
can see that Ωb and RCR are clearly inversely proportional, as expected.
Additionally, we see that low R values cluster at lower values for RCR. To
aid visualisation, the colors used to indicate R were capped at the 16th and
84th percentile.
the bar radius. Thus, we can divide this figure into three regions,
one with all the ultrafast bars, one with all the fast bars and one with
all the slow bars. The bar pattern speed is shown with the colour
gradient. The galaxies with the fastest pattern speeds mostly have
low values for the bar radius and corotation radius. The galaxies
with the lowest pattern speeds tend to have higher values for the
corotation radius, as well as higher values for R.
The bar radius is plotted against the bar pattern speed, corota-
tion radius and R in Figure 12. We find that the bar pattern speed
decreases as bar radius increases. Though the Spearman correlation
index is quite small (-0.26) due to the high amounts of scatter, its
significance is high (3.94σ). A more careful look reveals that all the
largest bars have lower values for the bar pattern speeds. Conversely,
all bars with higher values for their pattern speed are relatively short.
Figure 11. The corotation radius is plotted against the deprojected bar length.
As R = RCR/Rbar, this figure is divided into three regions: the region with
slow bars (R > 1.4), the region with fast bars 1 < R < 1.4) and the region
with ultrafast bars R < 1). The colour indicates the bar pattern speed. The
median error on the x and y axis is shown in the top-left corner. To aid
visualisation, the colors used to indicate the bar pattern speed were capped
at the 16th and 84th percentile.
The corotation radius increases with bar radius. This is because
larger bars need to have a larger corotation radii, as a bar can only
grow up to its corotation radius. R is observed to decrease with bar
radius. However, R is very sensitive to correct bar length estimates,
so this trend could merely be a reflection of that. Interestingly, the
median trends for the weakly and strongly barred subsamples are
very similar to each other, except for the trends of the corotation
radii for the larger bars.
Many properties of galaxies vary with stellar mass (Brinch-
mann & Ellis 2000; Brinchmann et al. 2004; Noeske et al. 2007;
Lara-López et al. 2010) and bars, especially stronger bars, are known
to appear more often in massive galaxies (Masters et al. 2012; Cer-
vantes Sodi 2017; Géron et al. 2021). In Figure 13, we plot the
stellar mass against the bar pattern speed, corotation radius and R
to see if any of these parameters are correlated with stellar mass
as well. We see that the pattern speed and R do not correlate with
stellar mass. This shows that the differences we observed in pattern
speed in Figure 7 are not due to differences in stellar mass of our
targets. Interestingly, the corotation radius does increase with stellar
mass. This is because more massive galaxies tend to host stronger
and longer bars, which tend to have larger corotation radii, as shown
in the middle panel of Figure 12.
4.3 Quenching
Our previous results reveal a complicated interplay between bar
pattern speed, corotation radii, R, bar length and bar type. It is
also known that strong bars are more often found in red sequence
galaxies (Masters et al. 2012; Vera et al. 2016; Géron et al. 2021).
This suggest a potential link between these dynamical parameters
and quenching.
Galaxies can be classified as star forming or quiescent based
on their location on the SFR-stellar mass plane. We can use the star
formation main sequence (SFMS) defined in Belfiore et al. (2018):
log
(
SFR/M⊙ yr−1
)
= (0.73 ± 0.03) log (M*/M⊙)−(7.33 ± 0.29) ,
MNRAS 000, 1–19 (2022)
>-
~ 07 
g: 0& 
~ 05 
- O' "C 
~ O] 
70 01 
E 01 
O 
Z O 
;:;IB 
~ 100 
J " 
" 
: 
" 
p.wlllue - O 01 
' 0 
,. 
u 
1.1 ",;" 
,,' 
,. 
u 
•• 
.. /0 • /"> 
• ".... '4 
(" ' /~ . \ 
':. . ,/ . /' ~ . . . 
• • ~ ' , ,-' t . .. .. .' ."' .. .. ... , . 
.. I" / :~ J ... .. .. 
' . .. .. 
/ ~ •. ' .. '. '., .< -, "".. .. .. <,./ .. . .. :¡ ~ . 
• 
,A/, .. : •• ' .. .. 
/ / , 
,'/ .. 
.;,' 
• • u 
Rc~ [kpc] 
• 
12 Tobias Géron
Table 3. The bar pattern speeds, corotation radii and R for 50 randomly selected bars. The full table can be found online here.
Plate-ifu Ωb [km s−1 arcsec−1] Ωb [km s−1 kpc−1] RCR [arcsec] RCR [kpc] R [-]
8723-12701 14.10+0.34
−0.31 18.33+0.44
−0.40 16.78+0.31
−0.30 12.91+0.24
−0.23 2.79+0.18
−0.18
9872-9102 9.80+2.42
−1.73 11.98+2.96
−2.11 21.89+5.23
−4.35 17.90+4.28
−3.55 4.75+1.11
−1.14
8616-12702 24.98+2.38
−2.84 40.89+3.89
−4.65 9.48+2.38
−1.57 5.79+1.45
−0.96 0.84+0.21
−0.14
8077-12705 23.23+4.90
−2.88 26.89+5.68
−3.33 6.50+0.78
−0.54 5.61+0.67
−0.47 0.94+0.11
−0.08
8322-1901 35.82+7.26
−4.87 76.63+15.53
−10.43 4.05+1.46
−1.43 1.89+0.68
−0.67 1.25+0.47
−0.45
9027-12704 5.67+0.56
−0.88 8.21+0.80
−1.28 12.60+3.60
−1.97 8.69+2.48
−1.36 0.81+0.26
−0.13
10220-9101 20.61+0.99
−0.88 28.54+1.37
−1.22 11.01+0.58
−0.73 7.95+0.42
−0.53 1.77+0.10
−0.11
9187-9101 13.79+1.90
−1.74 25.13+3.46
−3.18 14.12+2.71
−2.54 7.75+1.49
−1.39 2.80+0.90
−0.56
11758-3701 16.48+4.65
−5.76 32.32+9.12
−11.31 16.91+13.67
−5.42 8.62+6.97
−2.76 2.02+1.73
−0.66
11866-6102 26.26+1.38
−1.46 39.46+2.08
−2.19 8.29+0.67
−0.61 5.52+0.44
−0.41 2.09+0.38
−0.22
11866-12701 49.42+1.92
−2.73 53.08+2.06
−2.93 4.02+0.47
−0.33 3.75+0.43
−0.31 0.66+0.08
−0.06
9886-12701 10.88+0.66
−0.62 23.45+1.41
−1.33 12.73+1.10
−0.95 5.91+0.51
−0.44 2.18+0.20
−0.19
9876-9102 9.77+0.36
−0.36 18.30+0.68
−0.67 15.35+0.63
−0.26 8.19+0.34
−0.14 3.28+0.20
−0.20
11836-3702 24.45+5.41
−3.82 27.54+6.09
−4.30 8.94+0.65
−1.52 7.94+0.58
−1.35 1.94+0.25
−0.29
9033-12705 23.04+2.25
−1.97 32.04+3.13
−2.73 8.90+0.74
−0.79 6.40+0.53
−0.57 1.30+0.11
−0.10
11982-9102 15.87+1.70
−1.68 29.20+3.14
−3.10 6.57+1.10
−0.88 3.57+0.60
−0.48 0.56+0.09
−0.08
8721-6103 17.21+5.94
−3.52 19.27+6.66
−3.94 12.15+3.98
−3.61 10.85+3.55
−3.23 2.25+0.75
−0.66
8935-6104 19.12+1.51
−1.43 41.14+3.26
−3.09 4.62+0.83
−0.46 2.14+0.39
−0.22 1.71+0.33
−0.22
7979-12702 7.35+0.43
−1.29 13.05+0.76
−2.29 28.04+6.34
−0.87 15.78+3.57
−0.49 3.82+1.07
−0.31
12495-6102 17.76+1.81
−10.01 32.97+3.36
−18.58 10.42+18.60
−1.76 5.62+10.02
−0.95 1.55+3.05
−0.30
8983-12705 12.12+3.13
−2.27 23.51+6.08
−4.41 14.99+4.66
−4.71 7.73+2.40
−2.43 1.84+0.61
−0.62
11747-9102 11.91+0.40
−0.29 23.17+0.77
−0.56 8.50+0.31
−0.37 4.37+0.16
−0.19 2.48+0.16
−0.15
9879-6102 18.25+2.52
−2.11 26.89+3.71
−3.10 12.73+2.28
−0.95 8.64+1.55
−0.65 2.47+0.35
−0.35
11979-9101 18.93+1.83
−2.14 21.69+2.10
−2.45 11.37+1.79
−1.24 9.92+1.57
−1.08 1.57+0.39
−0.18
8932-6104 29.44+4.44
−3.27 57.87+8.73
−6.42 5.78+0.99
−0.97 2.94+0.50
−0.49 0.75+0.17
−0.15
11977-12703 11.80+1.29
−1.53 18.80+2.06
−2.45 19.15+3.84
−2.66 12.01+2.41
−1.67 1.74+0.37
−0.25
8093-12703 13.74+2.41
−2.04 18.86+3.31
−2.81 14.01+3.75
−2.79 10.21+2.73
−2.03 1.32+0.37
−0.28
9499-6102 11.26+6.82
−3.76 20.99+12.70
−7.01 14.99+9.68
−8.36 8.04+5.19
−4.49 4.96+4.13
−2.19
11952-12701 25.27+1.08
−0.96 28.05+1.19
−1.06 6.81+0.61
−0.65 6.14+0.55
−0.58 0.69+0.16
−0.12
11016-12703 28.34+0.40
−0.34 31.29+0.44
−0.38 8.53+0.12
−0.14 7.73+0.11
−0.12 1.45+0.05
−0.05
8312-12704 6.18+0.55
−0.41 10.43+0.93
−0.68 21.82+1.64
−1.96 12.94+0.97
−1.16 2.53+0.21
−0.22
8600-12701 7.38+0.64
−0.54 13.14+1.14
−0.96 12.92+1.09
−1.21 7.26+0.61
−0.68 1.06+0.09
−0.08
7992-6104 11.61+0.37
−2.70 21.29+0.67
−4.96 11.66+5.48
−0.90 6.36+2.99
−0.49 1.51+0.72
−0.12
8249-6101 15.95+0.46
−0.51 29.69+0.86
−0.95 7.08+0.36
−0.22 3.81+0.19
−0.12 0.92+0.08
−0.07
8938-12702 23.67+6.14
−1.10 29.69+7.70
−1.38 10.71+0.50
−3.29 8.54+0.40
−2.62 1.70+0.19
−0.48
11834-12705 14.85+0.42
−0.41 17.75+0.50
−0.49 15.47+0.61
−0.51 12.94+0.51
−0.42 2.08+0.27
−0.22
9026-12705 11.80+1.57
−1.64 17.49+2.33
−2.43 20.50+3.56
−2.85 13.82+2.40
−1.92 2.42+0.51
−0.36
8604-12703 25.19+2.23
−1.89 41.22+3.65
−3.09 7.39+0.88
−0.99 4.52+0.54
−0.60 0.83+0.13
−0.13
8442-9102 14.01+3.83
−2.64 30.14+8.23
−5.68 7.34+1.40
−1.42 3.41+0.65
−0.66 0.97+0.26
−0.20
11751-6102 30.97+2.11
−3.56 36.72+2.50
−4.22 8.01+1.34
−0.65 6.76+1.13
−0.55 1.35+0.24
−0.15
8312-12702 13.61+0.73
−0.62 21.28+1.15
−0.98 11.85+0.27
−0.50 7.58+0.17
−0.32 3.19+0.10
−0.14
9484-12703 20.93+1.99
−2.19 32.93+3.14
−3.44 3.95+1.78
−2.14 2.51+1.13
−1.36 0.54+0.24
−0.29
10515-6104 9.49+0.56
−7.14 21.77+1.28
−16.37 14.57+67.09
−1.02 6.35+29.26
−0.44 2.23+10.23
−0.25
8140-12701 30.75+2.53
−1.96 53.64+4.40
−3.41 3.63+0.63
−0.83 2.08+0.36
−0.47 0.47+0.08
−0.11
8983-3703 25.86+0.92
−0.78 44.57+1.59
−1.34 5.89+0.49
−0.35 3.41+0.29
−0.20 1.01+0.09
−0.08
11956-6103 23.05+1.19
−1.16 31.71+1.63
−1.59 9.52+0.80
−0.72 6.92+0.58
−0.52 0.82+0.07
−0.06
8484-12703 9.70+0.96
−0.97 15.75+1.56
−1.58 21.69+3.05
−1.82 13.35+1.88
−1.12 2.86+0.38
−0.26
9029-12704 17.06+0.84
−0.33 27.03+1.33
−0.52 11.12+0.30
−0.99 7.02+0.19
−0.62 1.44+0.09
−0.11
9192-6101 8.35+0.18
−4.51 14.47+0.31
−7.80 14.86+28.59
−0.51 8.58+16.51
−0.29 5.30+16.21
−0.80
8989-3703 26.27+1.89
−2.60 49.40+3.56
−4.90 5.36+1.74
−1.22 2.85+0.93
−0.65 0.58+0.19
−0.14
and 175 more rows...
MNRAS 000, 1–19 (2022)
Kinematics of strong and weak bars 13
Figure 12. The bar pattern speed (left panel), corotation radius (middle panel) and R (right panel) against the bar radius. All strongly barred galaxies are
coloured orange, while all weakly barred galaxies are coloured blue. The median trend for the weakly and strongly barred galaxies is shown with the blue
and orange full lines, respectively. Additionally, the general median trend of all barred galaxies is shown in the dashed black line. The Spearman correlation
coefficient, R, and its significance, σ, are shown in every subplot. The median error on the x and y axis is shown in the top-right corner. We see that the pattern
speed and R decrease with bar length, while the corotation radius increases.
Figure 13. The bar pattern speed (left panel), corotation radius (middle panel) and R (right panel) against the stellar mass. All strongly barred galaxies are
coloured orange, while all weakly barred galaxies are coloured blue. The median trend for the weakly and strongly barred galaxies is shown with the blue
and orange full lines, respectively. Additionally, the general median trend of all barred galaxies is shown in the dashed black line. The Spearman correlation
coefficient, R, and its significance, σ, are shown in every subplot. The median error on the x and y axis is shown in the top-right corner. We see that the pattern
speed and R do not show a significant trend with stellar mass, while the corotation radius seems to increase with stellar mass.
(7)
and assume that all galaxies that are 1σ (= 0.39 dex) below this
line are undergoing quenching and everything else is star forming
(Belfiore et al. 2018). 59% of our barred galaxies are quenching,
whereas 41% are star forming. The bar pattern speeds, corotation
radii and values for R for all the star forming and quenching galaxy
subsamples are shown in Figure 14. An Anderson-Darling test be-
tween the subsamples shows that the pattern speed and R are not
significantly different. However, with a p-value of <0.001, the coro-
tation radii are significantly different. Thus, our results suggest that
quenching galaxies tend to have significantly higher corotation radii
than star forming galaxies.
4.4 Comparison with other work
Various other studies have also tried to measure bar pattern speeds,
corotation radii and R. Rautiainen et al. (2008) determine pattern
speeds, corotation radii and R for a sample of 38 galaxies with
data from the Ohio State University Bright Spiral Galaxy Survey
(Eskridge et al. 2002). Aguerri et al. (2015) used the Calar Alto
Legacy Integral Field Area (CALIFA, Sánchez et al. 2012) survey
on 15 galaxies and found that all of their bars were consistent with
being fast. Font et al. (2017) combined Spitzer images of 68 barred
galaxies with previously determined corotation radii to estimate
values for R. Cuomo et al. (2019) looked at 16 weakly barred
galaxies using data from CALIFA. Guo et al. (2019) used MaNGA
data to obtain estimates for pattern speeds, corotation radii and R
for a total of 53 barred galaxies. Garma-Oehmichen et al. (2020)
combined data from MaNGA and CALIFA to study a sample of 18
MNRAS 000, 1–19 (2022)
• R - <0 .26, a - 3 94 r Ro - 0.44, o - 6.55 
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lu~(M. [Mol) loy(M. [M"jI luy(M. [Mol) 
14 Tobias Géron
Figure 14. The final median values for bar pattern speed (left panel), corotation radius (middle panel) and R (right panel) after doing a Monte Carlo simulation
of 1,000 iterations. The sample is divided into quenching galaxies (red) and star forming galaxies (blue). The vertical dashed lines show the median values for
every histogram, while the full lines are kernel density estimates of these histograms, using a Gaussian kernel. The p-value of a two-sample Anderson-Darling
test is shown in the top-right corner of every subplot. The null hypothesis is that the two samples in each subplot are drawn from the same population. We can
see that the quenching and star forming subsamples are not significantly different in terms of pattern speed and R, but are in terms of corotation radius.
galaxies. Finally, Garma-Oehmichen et al. (in prep.) used MaNGA
to study 97 barred galaxies.
We compare our results with these studies in Figure 15. Our
distribution of the bar pattern speed, corotation radius and R falls
well within the range that is usually observed and we see no obvious
deviations. Our bar pattern speed distribution agrees especially well
with the other studies that have larger sample sizes (n > 50). An
interesting trend is observed when looking at the various distribu-
tions of R. The median value of R seems to be moving upwards as
the sample sizes increase. This could be attributed to larger samples
typically being more representative of large variety of galaxy and
bar types, which could have an effect on the observed distribution
of R.
Please refer to Appendix A for a similar comparison to
various other studies, but distinguishing between weakly and
strongly barred galaxies as well.
5 DISCUSSION
5.1 Are strong bars older than weak bars?
We have found in Figure 7 that strongly barred galaxies have sig-
nificantly lower bar pattern speeds than weakly barred galaxies,
especially in terms of Ωb sin (i). However, there is still a large over-
lap between the two samples. It is worth noting that this difference
is not due to differences in stellar mass, as shown in Figure 13.
Additionally, we found that pattern speed is negatively correlated
to bar length in Figure 12. This is in agreement with Cuomo et al.
(2020), who used the CALIFA and MaNGA surveys and also found
that stronger bars have lower bar pattern speeds. Font et al. (2017)
also found that the largest bars have the lowest pattern speeds, while
the bars with the largest pattern speeds are all very small. Using
CALIFA, MaNGA and Pan-STARRS DR1 (PS1), Lee et al. (2022)
also found that the bar pattern speed is negatively correlated to bar
length and strength.
Simulations suggest that the bar grows in size and the pattern
speed slows down as the bar exchanges angular momentum with its
host (Debattista & Sellwood 2000; Athanassoula 2003; Martinez-
Valpuesta et al. 2006; Okamoto et al. 2015). Thus, our results suggest
that strong bars are older and more evolved structures than weakly
barred galaxies. Alternatively, stronger bars could be simply more
efficient at redistributing angular momentum, as this depends on
various parameters, such as the velocity dispersion and mass dis-
tribution of the emitting and absorbing components (Athanassoula
2003).
5.2 How are strong and weak bars triggered?
We could not find a significant difference between weakly and
strongly barred galaxies in terms of R, which was defined as
R = RCR/Rbar, in Figure 9. Interestingly, we do find a rela-
tionship between R and bar length in Figure 12. However, R
is very dependent on the bar length and inclination estimates
(Cuomo et al. 2021; Roshan et al. 2021a), so this trend could
be primarily caused by differences in bar length between weak
and strong bars. A more detailed study using different metrics
to measure bar length could help clarify this issue.
We know from simulations that R depends on the formation of
the bar. Bars triggered by tidal interactions tend to have higher val-
ues for R than bars formed by global bar instabilities. Additionally,
tidally induced bars stay in the slow regime for a longer time (Sell-
wood 1981; Miwa & Noguchi 1998; Martinez-Valpuesta et al. 2016,
2017). Thus, our results seem to suggest that strong and weak
bars are triggered by similar mechanisms. Thus, not being able
to find a significant difference between weak and strong bars
suggests that they might be triggered by similar mechanisms.
This is in agreement with Cuomo et al. (2019), who also found that
weak and strong bars have similar values for R. Guo et al. (2019)
found no relationship between bar strength and R. This statement
could possibly be tested observationally by looking at the envi-
ronment of a large mass-matched sample of strongly and weakly
barred galaxies.
Interestingly, we do find a relationship between R and bar
length. However, R is very dependent on the bar length esti-
mates, so this trend could be primarily caused by differences
MNRAS 000, 1–19 (2022)
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O"",[km s - 1 lepel ] Roo [lepe] R [·1 
Kinematics of strong and weak bars 15
Figure 15. A comparison of estimates of the bar pattern speeds (left), corotation radii (middle) and R (right) found in various works. All the histograms are
normalised and offset from each other vertically to facilitate comparison. The median, 25th and 75th percentile for every distribution are indicated by the short
full and dashed lines. The studies are ordered by sample size, which is also shown in the left panel. The values from Rautiainen et al. (2008) and Aguerri et al.
(2015) were converted from the observational units cited in their papers to physical units using redshifts obtained from the NASA/IPAC Extragalactic Database
(NED). The values obtained from Guo et al. (2019) were converted using their own cited redshifts. Similarly, the values from Font et al. (2017) were converted
using the distances cited. The values for the bar pattern speeds from Rautiainen et al. (2008) are not publicly available, hence the empty histogram.
in bar length between weak and strong bars. A more detailed
study using different
5.3 Why and where do we see ultrafast bars?
Ultrafast bars, which are bars that have R < 1, should not exist
according to our current theoretical understanding. This is because
bars are thought to not be able to extend beyond the corotation
rate of the galaxy (Contopoulos 1980, 1981; Athanassoula 1992b).
Nevertheless, they have been found observationally. Multiple studies
find that 26 - 67% of their bars have R < 1, while 7-40% have a 1σ
upper limit that has R < 1 (Aguerri et al. 2015; Cuomo et al. 2019;
Guo et al. 2019; Garma-Oehmichen et al. 2020). In our sample,
∼21% of galaxies have R < 1, while ∼11% have a 1σ upper limit
that has R < 1. A more detailed breakdown can be found in Table
4.
Given thatR is defined asR = RCR/Rbar, a low value forR can
arise either because RCR is underestimated or Rbar is overestimated.
Both seem to happen simultaneously: even though most ultrafast
bars have a lower corotation radius (see Figure 10), we see in Figure
11 that ultrafast bars still have a relatively broad range of bar and
corotation radii.
Bars are often associated with spiral arms and rings, so it is of-
ten not straight-forward to measure the bar length correctly (Hilmi
et al. 2020; Cuomo et al. 2021; Roshan et al. 2021a). An underes-
timation of the corotation radius can be a consequence of either an
overestimation of the pattern speed, or because the rotation curve
is not fitted properly. Additionally, if the inclination of the galaxy
is not measured properly, the line of sight velocities in the rotation
curve will be corrected incorrectly, which will affect the corota-
tion radius as well. Interestingly, we see slightly more ultrafast bars
among strong bars than weak bars (24.0% and 17.0%, respectively).
5.4 Slow bars
As noted in Table 4, we find that ∼47% of galaxies have a slow
bar. Our fraction of slow bars is higher than what other studies have
typically found. This can be due to multiple factors, such as our
larger and more representative sample, which includes weak and
strong bars from a volume-limited sample, from a wide range of
magnitudes. Additionally, a correct measurement of the bar length
is crucial to a correct estimate of R (Cuomo et al. 2021). Differ-
ent authors use different methods of measuring bar length, which
will change the final distribution of R (for more details on our bar
length measurements, see Section 3.3). However, perhaps most im-
portantly, Guo et al. (2019) have shown that estimates of the pattern
speed will be systematically lower when using kinematic position
angles, compared to photometric position angles. This is because the
method to calculate the kinematic integrals position angle works
by minimising asymmetry in the velocity field, which will reduce
the values for the kinematic integrals. This will lower the estimates
for the patterns speed, which will, in turn, increase the estimates for
R and produce more slow bars. We used kinematic position angles
in this work, which could partially explain the higher fraction of
slow bars. Finally, if we take the errors into account, we find that
∼31% of our targets have a 1σ lower limit that is greater than
1.4. Thus, we can confidently exclude the fast regime for only
these targets, which is more consistent with other studies.
5.5 Strong and weak: part of a continuum
We have found that strong bars tend to have lower bar pattern speeds.
However, as Figure 7 shows, their distributions overlap significantly.
There is no clear threshold in pattern speed, corotation radius or R
that separates weak and strong bars. A closer look at Figure 12,
where we plot these parameters against bar radius, reveals that the
differences in pattern speed are driven by the smallest and largest
bars. At intermediate bar radii, the distributions of the two popu-
lations overlap. Additionally, the median trend of the weakly and
strongly barred subsamples are almost identical in these figures. Fig-
ure 8 shows that weakly and strongly barred galaxies do not have
significantly different corotation radii. As bars are able to grow up
until their corotation radius (Contopoulos 1980, 1981; Athanassoula
1992b), this result suggests that weak bars still have the possibility
MNRAS 000, 1–19 (2022)
I.H ,1015 n _ 15 
16 Tobias Géron
Table 4. Summary of how many ultrafast bars (R < 1), fast bars (1 < R < 1.4) and slow bars (R > 1, 4) are found in various works. Note that Aguerri et al.
(2015) and Guo et al. (2019) have multiple different samples, hence the range.
Sample size % Ultrafast % Fast % Slow
Rautiainen et al. (2008) 38 16 34 50
Aguerri et al. (2015) 15 46-67 20-40 7-13
Font et al. (2017) 68 1 59 40
Cuomo et al. (2019) 16 44 50 6
Guo et al. (2019) 53 26-47 13-34 38-43
Garma-Oehmichen et al. (2020) 18 39 22 39
Garma-Oehmichen et al., in prep. 97 11 43 45
This work 225 21 32 47
to grow up to the same length as strong bars. Either they have not
had to time to do so yet or something else is preventing them.
These results are consistent with the idea of a bar continuum,
proposed by Géron et al. (2021), who found that any distinction
in fibre SFR, gas mass and depletion timescale between weak and
strong bars disappeared when correcting for bar length. They sug-
gested that weak and strong bars are not fundamentally distinct
physical phenomena. Instead, bar types are continuous, and vary
from ‘weakest’ to ‘strongest’. Our measurements of the dynamical
parameters of weak and strong bars support this conclusion as well.
As mentioned in Section 2.2, we require that at least three
pseudo-slits can be placed on the bar. This means that this
method does not work for the shortest of bars, which are typ-
ically weak bars. Thus, it should be kept in mind when inter-
preting our results that the shortest and weakest bars are not
included in our analysis. This manifasts itself as well in the er-
rors associated with the measured pattern speeds. The highest
errors are associated with bars with the fewest pseudo-slits. This
means that weak bars typically have higher errors for the bar
pattern speed than strong bars: 4.55 km s−1 kpc−1 and 3.38 km
s−1 kpc−1, respectively.
5.6 Dark matter haloes and tension with ΛCDM
The fraction of ultrafast and fast bars found in this work is somewhat
less than what some other studies tend to find, but combined they
are still more than half of our sample. Observing a high fraction of
ultrafast and fast bars has been raised as a tension for the ΛCDM
cosmological paradigm, as cosmological simulations predict that
bars should slow down significantly (Algorry et al. 2017; Peschken
& Łokas 2019; Fragkoudi et al. 2021; Roshan et al. 2021a,b; Frankel
et al. 2022). This slowdown of the bar and increase of R is typically
attributed to the dynamical friction applied to the bar by the DM
halo (Debattista & Sellwood 1998, 2000; Fragkoudi et al. 2021).
As we find fewer ultrafast and fast bars than other studies, this
tension is decreased somewhat, but the median value of R in our
sample (R = 1.45+1.07
−0.56) is still significantly lower than that predicted
from simulations, whose average values at z∼0 are typically R > 2.5
(Algorry et al. 2017; Peschken & Łokas 2019; Roshan et al. 2021b).
Other studies have tried to relieve the tension as well. Frankel
et al. (2022) has recently shown that simulations obtain higher
values of R than observations, mostly because simulations predict
shorter bars, rather than slower bars. Additionally, Fragkoudi et al.
(2021) actually do find fast bars in their cosmological simulations
in baryon-dominated discs and claim that the DM fraction is too
high in other simulations. A lower DM fraction or lower central
DM density will lower the dynamical friction, and thus, allow fast
bars to exist. Finally, Beane et al. (2022) have shown that the gas
phase of the disk can help to stablise the bar pattern speed and
prevent it from slowing down.
As mentioned above, we observe no differences in terms of R
between weak and strong bars. This suggests that the DM halo is
similar between weakly and strongly barred galaxies as well. How-
ever, we do see that weak bars have significantly higher pattern
speeds than strongly barred galaxies. Studying the relationship be-
tween the DM halo and bars should clear up some of these issues
and help us understand the evolution of bars in general. This can
be done, for example, with Jeans Anisotropic Modelling (JAM) of
these galaxies (Cappellari 2008). This would provide estimates for
the DM fraction and allow us to study the intrinsic connection be-
tween the DM halo and the dynamical parameters of bars in greater
detail.
5.7 Effect on quenching
It is also known that strong bars are more often found in red sequence
galaxies (Masters et al. 2012; Vera et al. 2016; Cervantes Sodi
2017; Fraser-McKelvie et al. 2020). In addition, Géron et al. (2021)
showed that stronger bars have the ability to facilitate quenching,
whereas weaker bars do not. Figure 14 shows that the bar pattern
speed and R are not significantly different between star forming
and quenching galaxies. These results suggest that how fast bars
rotate, both in terms of pattern speed and R, has no significant or
measurable impact on quenching. This seems odd, but one way to
look at this is by looking at the timescales involved. A bar with a
pattern speed of ∼25 km s−1 kpc−1 will make a full rotation once
every ∼250 Myr. As (secular) quenching usually happens on ∼Gyr
timescales (Smethurst et al. 2015), this means that a bar will usually
have made multiple full rotations before the galaxy is quenched.
Interestingly, the corotation radius is significantly higher for
quenching galaxies. It is known that a bar can grow up to its corota-
tion radius (Contopoulos 1980, 1981; Athanassoula 1992b), so this
result relates back to longer and stronger bars having more of an
effect on quenching, which agrees with the findings of Géron et al.
(2021).
6 CONCLUSION
We have used the TW method on MaNGA IFU data for a sample
of 225 barred galaxies, which is the largest sample this method
has been applied to so far. The TW method produces bar pattern
speeds, which is used to calculate corotation radii and R, the ratio
between the corotation radius and bar radius. We have used Galaxy
Zoo morphological classifications to distinguish between strongly
MNRAS 000, 1–19 (2022)
Kinematics of strong and weak bars 17
and weakly barred galaxies. This allows us to study the bar pattern
speed, corotation radius and R for a statistically significant sample
of strongly and weakly barred galaxies, and compare them with
each other. We have found that:
• Though there is significant overlap, we find that the bar pattern
speeds between weakly and strongly barred galaxies are significantly
different. The median bar pattern speed of strongly barred galaxies
(Ωb = 26.32+11.41
−13.22 km s−1 kpc−1) is lower than that of weakly
barred galaxies (Ωb = 28.05+14.77
−11.54 km s−1 kpc−1). Additionally, we
find that bar pattern speed is inversely proportional to bar length.
We also show that this difference is not due to differences in stellar
mass in our targets. Simulations suggest that the pattern speed goes
down as the bar evolves and exchanges angular momentum, so our
results suggest that strong bars are older and more evolved structures
than weak bars.
• The corotation radius is not significantly different between
weakly (RCR = 6.44+4.0
−2.99 kpc) and strongly barred galaxies (RCR =
6.26+3.97
−2.31 kpc). As bars can grow up until their corotation radius, this
result suggests that weak bars still have the possibility of becoming
as long as strong bars.
• We could not find a significant difference for the ratio R be-
tween strongly and weakly barred galaxies (R = 1.32+0.87
−0.48 and
R = 1.55+1.44
−0.58, respectively). As R is related to the formation of
bars, this could suggest that strong and weak bars are formed by
similar mechanisms. Nevertheless, we do see more ultrafast bars
among strong bars than weak bars. However, it would be beneficial
to study the effect of the bar length measurement on R to reach a
more robust conclusion.
• We do not see a distinct cutoff or threshold in pattern speed,
corotation radius orR that separates strong and weak bars from each
other. In fact, the overlap is still quite significant. This is consistent
with Géron et al. (2021), who stated that strong and weak bars are
not distinct physical phenomena, but rather lie on a continuum of
bar types that vary from ‘weakest’ to ‘strongest’.
• 21% of our sample host ultrafast bars, 32% host fast bars and
47% of all bars are slow. We have a slightly higher fraction of slow
bars and a lower fraction of ultrafast bars than most other studies.
This can be attributed to various factors, such as the bigger and
more representative sample used in this study.
• However, only ∼11% of our targets have a 1σ upper limit
that has R < 1. Similarly, we can only confidently exclude the
(ultra)fast regime for ∼31% of our galaxies (i.e. they have a 1σ
lower limit that has R > 1.4)
• The lower fraction of ultrafast bars among our sample de-
creases the recent tension with ΛCDM. However, the median value
of R in our sample is still significantly lower than what is predicted
from simulations.
• We do not see any significant difference between between the
star forming and quenching subsamples in terms of pattern speed
or R. However, quenching galaxies do have significantly higher
corotation radii than star forming galaxies.
ACKNOWLEDGEMENTS
The data in this paper are the result of the efforts of the Galaxy Zoo
volunteers, without whom none of this work would be possible.
Their efforts are individually acknowledged at http://authors.
galaxyzoo.org.
Funding for the Sloan Digital Sky Survey IV has been pro-
vided by the Alfred P. Sloan Foundation, the U.S. Department of
Energy Office of Science, and the Participating Institutions. SDSS
acknowledges support and resources from the Center for High-
Performance Computing at the University of Utah. The SDSS web
site is www.sdss.org.
Funding for the Sloan Digital Sky Survey IV has been provided
by the Alfred P. Sloan Foundation, the U.S. Department of Energy
Office of Science, and the Participating Institutions.
SDSS-IV acknowledges support and resources from the Center
for High Performance Computing at the University of Utah. The
SDSS website is www.sdss.org.
SDSS-IV is managed by the Astrophysical Research Consor-
tium for the Participating Institutions of the SDSS Collaboration
including the Brazilian Participation Group, the Carnegie Institu-
tion for Science, Carnegie Mellon University, Center for Astro-
physics | Harvard & Smithsonian, the Chilean Participation Group,
the French Participation Group, Instituto de Astrofísica de Canarias,
The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, the
Korean Participation Group, Lawrence Berkeley National Labora-
tory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-
Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut
für Astrophysik (MPA Garching), Max-Planck-Institut für Extrater-
restrische Physik (MPE), National Astronomical Observatories of
China, New Mexico State University, New York University, Uni-
versity of Notre Dame, Observatário Nacional / MCTI, The Ohio
State University, Pennsylvania State University, Shanghai Astro-
nomical Observatory, United Kingdom Participation Group, Uni-
versidad Nacional Autónoma de México, University of Arizona,
University of Colorado Boulder, University of Oxford, University
of Portsmouth, University of Utah, University of Virginia, Univer-
sity of Washington, University of Wisconsin, Vanderbilt University,
and Yale University.
TG gratefully acknowledges funding from the University of
Oxford Department of Physics and the Saven Scholarship.
RJS gratefully acknowledges funding from Christ Church, Uni-
versity of Oxford.
This project makes use of the MaNGA-Pipe3D dataproducts.
We thank the IA-UNAM MaNGA team for creating this catalogue,
and the Conacyt Project CB-285080 for supporting them.
DATA AVAILABILITY
The Galaxy Zoo data used in this article will be made publicly
available soon. The data from MaNGA is publicly available and can
be found at http://www.sdss.org/dr17/manga/. The Legacy survey
can be accessed at https://www.legacysurvey.org/.
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APPENDIX A: COMPARISON TO OTHER WORK
In Section 4.4, we compared our results to that of other studies.
However, we did not make a distinction between weak and strong
bars. In Figure A1, we compare our results to that of various
other works that clearly mentioned whether the bars studied
were weak or strong. We see that, in all the other studies, the
bar pattern speed for weakly barred galaxies tends to be higher
than that of strongly barred galaxies, which corresponds to our
findings. Contrary to our findings, strongly barred galaxies in
the other studies do seem to have higher corotation radii than
weakly barred galaxies. However, as mentioned in Section 2.3,
Aguerri et al. (2015), Cuomo et al. (2019) and Guo et al. (2019)
calculate the corotation radius assuming that it lies in the region
where the rotation curve has flattened, which is not always the
case and will bias the final values. Finally, weakly and strongly
barred galaxies seem to have comparable values for R in the
other studies, similarly to what we have found.
This paper has been typeset from a TEX/LATEX file prepared by the author.
MNRAS 000, 1–19 (2022)
20 Tobias Géron
Figure A1. A comparison of estimates of the bar pattern speeds (left), corotation radii (middle) and R (right) found in various works. The data is split
between weakly barred galaxies (blue) and strongly barred galaxies (orange). Font et al. (2017) have both weak and strong bars, whereas the other
studies focus only on either weak or strong bars. All the histograms are normalised and offset from each other vertically to facilitate comparison. The
median, 25th and 75th percentile for every distribution are indicated by the short full and dashed lines. The studies are ordered by sample size, which
is also shown in the left panel. The values from Aguerri et al. (2015) were converted from the observational units cited in the papers to physical units
using redshifts obtained from the NASA/IPAC Extragalactic Database (NED). The values obtained from Guo et al. (2019) were converted using their
own cited redshifts. Similarly, the values from Font et al. (2017) were converted using the distances cited in the paper.
MNRAS 000, 1–19 (2022)
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