# Applying the Tremaine–Weinberg Method to Nearby Galaxies: Stellar-mass-based Pattern Speeds and Comparisons with ISM Kinematics

Thomas Williams, Eva Schinnerer, Eric Emsellem, Sharon Meidt, Miguel

Querejeta, Francesco Belfiore, Ivana Bešlić, Frank Bigiel, Mélanie Chevance, Daniel Dale, et al.

► To cite this version:

Thomas Williams, Eva Schinnerer, Eric Emsellem, Sharon Meidt, Miguel Querejeta, et al. Applying the Tremaine–Weinberg Method to Nearby Galaxies: Stellar-mass-based Pattern Speeds and Comparisons with ISM Kinematics. The Astronomical Journal, 2021, 161 (4), pp.185. 10.3847/1538-3881/abe243. hal-03509608

HAL Id: hal-03509608

https://hal.science/hal-03509608v1

Submitted on 2 Aug 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Applying the Tremaine–Weinberg Method to Nearby Galaxies: Stellar-mass-based Pattern Speeds and Comparisons with ISM Kinematics

Thomas G. Williams¹, Eva Schinnerer¹, Eric Emsellem²,³, Sharon Meidt⁴, Miguel Querejeta⁵, Francesco Belfiore⁶, Ivana Bešlic⁷, Frank Bigiel⁷, Mélanie Chevance⁸, Daniel A. Dale⁹, Simon C. O. Glover¹⁰, Kathryn Grasha¹¹, Ralf S. Klessen¹⁰,¹², J. M. Diederik Kruijssen⁸, Adam K. Leroy¹³, Hsi-An Pan¹, Jérôme Pety¹⁴,¹⁵, Ismael Pessa¹, Erik Rosolowsky¹⁶, Toshiki Saito¹, Francesco Santoro¹, Andreas Schruba¹⁷, Mattia C. Sormani¹⁰, Jiayi Sun¹³, and Elizabeth J. Watkins⁸

¹Max Planck Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany; williams@mpia.de

²European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748 Garching, Germany

³Université Lyon 1, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69230 Saint-Genis-Laval, France

⁴Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281 S9, B-9000 Gent, Belgium

⁵Observatorio Astronómico Nacional (IGN), C/Alfonso XII 3, Madrid E-28014, Spain

⁶INAF—Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50157, Firenze, Italy

⁷Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, D-53121 Bonn, Germany

⁸Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstraße 12-14, D-69120 Heidelberg, Germany

⁹Department of Physics & Astronomy, University of Wyoming, 1000 East University Avenue, Laramie, WY 82070, USA

¹⁰Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Str. 2, D-69120 Heidelberg, Germany

¹¹Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia

¹²Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany

¹³Department of Astronomy, The Ohio State University, 140 West 18th Avenue, Columbus, OH 43210, USA

¹⁴IRAM, 300 rue de la Piscine, F-38406 Saint Martin d'Hères, France

¹⁵15 LERMA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, F-75014 Paris, France

¹⁶4-183 CCIS, University of Alberta, Edmonton, Alberta, Canada

¹⁷Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstraße 1, D-85748 Garching bei München, Germany

Received 2020 December 21; revised 2021 January 29; accepted 2021 January 30; published 2021 March 15

## Abstract

We apply the Tremaine–Weinberg method to 19 nearby galaxies using stellar mass surface densities and velocities derived from the PHANGS-MUSE survey, to calculate (primarily bar) pattern speeds ($\Omega_P$). After quality checks, we find that around half (10) of these stellar-mass-based measurements are reliable. For those galaxies, we find good agreement between our results and previously published pattern speeds, and we use rotation curves to calculate major resonance locations (corotation radii and Lindblad resonances). We also compare these stellar-mass-derived pattern speeds with $H\alpha$ (from MUSE) and $CO(J=2-1)$ emission from the PHANGS-ALMA survey. We find that in the case of these clumpy interstellar medium (ISM) tracers, this method erroneously gives a signal that is simply the angular frequency at a representative radius set by the distribution of these clumps ($\Omega_{clump}$), and that this $\Omega_{clump}$ is significantly different from $\Omega_P$ (~20% in the case of $H\alpha$, and ~50% in the case of CO). Thus, we conclude that it is inadvisable to use “pattern speeds” derived from ISM kinematics. Finally, we compare our derived pattern speeds and corotation radii, along with bar properties, to the global parameters of these galaxies. Consistent with previous studies, we find that galaxies with a later Hubble type have a larger ratio of corotation radius to bar length, more molecular-gas-rich galaxies have higher $\Omega_P$, and more bulge-dominated galaxies have lower $\Omega_P$. Unlike earlier works, however, there are no clear trends between the bar strength and $\Omega_P$, nor between the total stellar mass surface density and the pattern speed.

Unified Astronomy Thesaurus concepts: Galaxies (573); Galaxy dynamics (591); Galaxy structure (622)

Supporting material: figure set

## 1. Introduction

A fundamental and currently open question in galaxy formation and evolution is how spiral arms and bars are formed and how they evolve. A natural question to ask is how long-lived these structures are, but the answer to this question remains elusive; even “transient” structures in terms of galaxies may persist for millions of years (e.g., Bournaud et al. 2005; Grand et al. 2012) and so are far beyond the timescales that we are able to observe. The most common theory for the formation of these structures (Lindblad 1963; Lin & Shu 1966; although see Sellwood 2013 and Section 6.4.2 of Binney & Tremaine 2008 for alternative theories) is that density waves propagating through galaxies act on the gas, forming stars along spiral shock lines (forming grand-design spiral arms). Bars can form

from disk instabilities, even in the absence of gas (see Section 6.3 of Binney & Tremaine 2008). A key prediction of this density wave theory is that the waves propagating through these morphological features will have a roughly invariant angular velocity across a large range of galactic radii. These angular velocities are referred to as pattern speeds, $\Omega_P$ (whether this is true in the case of spiral arms is disputed; see the review by Dobbs & Baba 2014).

The pattern speed of a spiral arm or a bar is a key parameter of the structure and is associated with the evolution of the galaxy it is present within. For instance, a bar can only grow self-consistently if it lies within the corotation radius (i.e., where the stars move at the same speed as the density wave; Contopoulos 1980). Density waves driving spiral arms have been shown to trigger star formation (e.g., Rand 1993; Knapen
et al. 1996), and the interface of a bar and a spiral arm can also
trigger massive starburst events (e.g., Beuther et al. 2017), so
galaxy evolution determines Ω<sub>P</sub>. Furthermore, when combined
with the rotation curve of a galaxy, these speeds will set the
location of resonances within a galaxy (the corotation radii and
Lindblad resonances), which in turn maintain and regulate the
density wave (Lin 1970) and can have significant effects on the
distributions of stars (e.g., Fragkoudi et al. 2019).

However, the density wave speed is not directly observable, so we must turn to indirect methods to infer this parameter. There are a number of ways to do this. For example, by identifying resonance locations, one can predict the radius of corotation and thus the pattern speed (e.g., Elmegreen et al. 1989, 1996). This method is limited by an uncertain conversion from resonance radius to corotation radius (e.g., Kranz et al. 2003), and by the fact that, given the particular pattern speed and galaxy rotation curve, certain resonances may not exist. Another option is to match simulations (where the pattern speed is directly known) to galaxies (e.g., Garcia-Burillo et al. 1993; Rautiainen & Salo 1999; Weiner et al. 2001a, 2001b; Sormani et al. 2015). The selection of the “best” model here is somewhat qualitative, as direct comparisons between observations and simulations are difficult. This method has the benefit of being more direct, but it requires a suite of tailored simulations for each individual object (selecting a value of the stellar mass-to-light ratio, the shape of the dark matter halo and its mass, and so on) and so is only feasible for small samples of galaxies.

Due to these difficulties, studies are typically limited to a
single galaxy (or a very small number of them). This means
that the literature contains heterogeneous measures of pattern
speeds using different methods. Given that these different
methods have different systematics (or perhaps are more
sensitive to different pattern speeds within the galaxy), making
direct comparisons between these various literature values is
difficult. Furthermore, applying the same method to different
kinematic tracers may yield differing results (e.g., stars and H I;
see Westpfahl 1998). To draw statistically robust conclusions
about the distributions of pattern speeds in the local galaxy
population, it is important to have homogeneous measures of
these quantities—not only in technique, but in tracer, too.

Ideally, to extend studies of pattern speeds (and resonance
locations) to large samples of galaxies, we desire a method of
pattern speed determination that is widely applicable, which is
data (rather than model or simulation) driven and as
quantitative as possible (i.e., with minimal reliance on by-eye
feature classification). For this, the Tremaine–Weinberg
(Tremaine & Weinberg 1984) method stands out as one of
the most popular approaches, due to its minimal model
assumptions and the fact that it is purely based on the observed
kinematics. We will describe this method in more detail in
Section 3.1. As a brief historical introduction, this method was
originally designed for long-slit spectroscopy and applied to
measure bar pattern speeds of dust-free, early-type galaxies. It
is sensitive to nonaxisymmetry along a number of (ideally
infinitely long) slits positioned along the major axis of the
galaxy. By taking the slope of the intensity-weighted velocity
with respect to the intensity-weighted position for a number of
slits, the velocity component misalignment with the line of
nodes (i.e., the major axis of the galaxy) can be quantified. The
commonly used form of the method was first formulated by
Merrifield & Kuijken (1995) and has since been applied to a

number of other galaxies, including late-type galaxies (e.g.,
Gerssen et al. 1999; Debattista et al. 2002; Aguerri et al. 2003;
Corsini et al. 2003, 2007; Guo et al. 2019, among others). With
the advent of integral field unit spectroscopy, it is now possible
to apply the Tremaine–Weinberg method to derived maps of
the stellar mass and stellar velocities (see, e.g., Debattista &
Williams 2004; Guo et al. 2019), and with the Gaia satellite it
has been applied to stars in the Milky Way (Sanders et al.
2019).

The method has also been used in conjunction with ionized
or neutral gas tracers, with the goal of probing nonaxisym-
metric structure farther out in the disk, either with Hα (e.g.,
Emsellem et al. 2006; Fathi et al. 2009) or occasionally CO
(e.g., Rand & Wallin 2004; Zimmer et al. 2004). The
underlying assumption, in such cases, is that this gas roughly
obeys continuity when there is little chemical transformation
between the gas and other phases of the interstellar medium
(ISM; i.e., molecular to atomic hydrogen, or dust), and that the
chemical abundance of the tracers remains constant. In the case
of CO, Rand & Wallin (2004) argued that it should remain a
valid tracer when the ISM is molecule-dominated (so there is
little conversion from molecular gas to and from atomic gas)
and when the star formation rate is low to moderate (so that
there is little molecular gas lost to star formation or expelled by
feedback; Chevance et al. 2020). However, as discussed in
Rand & Wallin (2004), the clumpiness of ISM tracers like CO
and Hα can pose a further issue: clumpy, highly asymmetric
disks introduce a fake signal in Tremaine–Weinberg integrals.
This effect becomes especially pronounced at high resolution,
when CO and Hα morphology becomes characterized by
clumpiness organized around, for example, H II regions, giant
molecular clouds, or sharp spiral arms (e.g., Kreckel et al.
2018; Schinnerer et al. 2019; S. Meidt et al. 2021, in
preparation) We revisit this in more detail in the context of
our sample in Section 4. Because of these potential short-
comings, fully sampled stellar kinematics are preferable for
applying the Tremaine–Weinberg method.

Recent works have combined a number of literature values of pattern speeds to attempt to perform statistical analyses (Cuomo et al. 2020). Our study complements and builds upon this earlier work by measuring pattern speeds homogeneously for a number of galaxies. In this work, we apply the Tremaine–Weinberg method to two of the Physics at High Angular resolution in Nearby GalaxieS (PHANGS<sup>18</sup>) surveys, namely to observations taken using the Multi Unit Spectroscopic Explorer (MUSE) instrument on the Very Large Telescope (referred to as PHANGS-MUSE; P.I. E. Schinnerer; E. Emsellem et al. 2021, in preparation) and the Atacama Large Millimeter/ Submillimeter Array (ALMA) instrument (referred to as PHANGS-ALMA; P.I. E. Schinnerer, and from pilot proposals with P.I.s G. Blanc and A. K. Leroy; Leroy et al. 2020a). Our work focuses on pattern speeds from the stellar mass surface density (Σ<sub>*</sub>) from MUSE, but we also study the application of this method to ISM tracers: ionized gas (Hα) from MUSE, and cold molecular gas [CO(J = 2 − 1)] emission from ALMA. With 19 galaxies mapped as part of PHANGS-MUSE and 84 with PHANGS-ALMA, this gives us an unprecedented opportunity not only to measure pattern speeds homogeneously for a large sample of galaxies at high (≈100 pc) resolution, but also to perform vital cross-checks

<footnote n="18">¹⁸ phangs.org</footnote>
between different kinematic tracers for many galaxies, which
are currently poorly explored in the literature. This work
derives pattern speeds, along with resonance locations for
these galaxies, which are tabulated in Table 2 and are also
made available online in a machine-readable format.<fnref n="19" /> For
future works that may have improved distance or orientation
measurements, this table also includes these parameters as
used in our work, to allow for simple rescaling in subsequent
works.

The structure of this paper is as follows. We provide an
overview of the PHANGS programs and data products
(Section 2), before summarizing the Tremaine–Weinberg
method, our tests of its efficacy on the data, and our application
to the entire PHANGS-MUSE and PHANGS-ALMA data sets
(Section 3). In Section 4, we present our tests showing that
applying the Tremaine–Weinberg method to ISM tracers can
yield erroneous signals and showing that this is generally an
issue in our sample. We present an overview of our derived
pattern speeds in the context of previous work, and we make a
comparison between our pattern speeds and previously derived
pattern speeds for the same galaxies in the literature
(Section 5). We then calculate the radii of major resonances
for the entire PHANGS sample (Section 6). We study the
potential correlations between these derived dynamical proper-
ties of the galaxy and some of its global parameters (such as
Hubble type and molecular gas fraction; Section 7). We discuss
the implications of these results (Section 8) before presenting a
summary of our work, along with future prospects for large
studies of pattern speeds in the future (Section 9).

## 2. Data

This work uses data from two of the PHANGS large
programs: PHANGS-MUSE and PHANGS-ALMA. For details
on the data reduction and product creation, we refer readers to
E. Emsellem et al. (2021, in preparation) for PHANGS-MUSE,
and Leroy et al. (2020b) and Leroy et al. (2020a) for the
PHANGS-ALMA data processing pipeline and survey descrip-
tion. We provide here only a brief description of the data
products we use in the subsequent sections, which are the
surface brightness and velocity maps for each tracer (MUSE
stellar mass and velocity, MUSE-Hα, and ALMA-CO,
collectively referred to as “kinematic maps”), along with
associated error maps.

Along with these kinematic maps, we use orientation
parameters (the galaxy position angle, inclination, and systemic
velocities) from Lang et al. (2020). We also make use of
distances from Anand et al. (2021, and references therein) to
calculate physical pattern speeds in km s⁻¹kpc⁻¹, rather than in
km s⁻¹arcsec⁻¹. Finally, we make use of catalogs from the
Spitzer Survey of Stellar Structure in Galaxies (S⁴G; Sheth et al.
2010), in particular, bar strengths (Díaz-García et al. 2016), bar
orientations (Herrera-Endoqui et al. 2015), and bulge-to-total
flux ratios and disk scale lengths (Salo et al. 2015).

### 2.1. PHANGS-MUSE

Our primary sample in this study consists of MUSE optical
spectroscopy for 19 galaxies as part of the PHANGS-MUSE
data release (DR) 2.0 (E. Emsellem et al., 2021, in preparation).
The reduction is performed using standard MUSE recipes (e.g.,

wavelength and flux calibration, cosmic-ray rejection, mosaicking) run through PYMUSEPIPE.<fnref n="20" /> These reduced products are then run through the MUSE data analysis pipeline (DAP), which is run in three stages: first, stellar kinematics are measured (the stellar velocity and higher order moments); next, the properties of stellar populations are estimated (e.g., age, stellar metallicity, stellar mass). Both of these stages are performed on Voronoi binned data to a stellar continuum signal-to-noise ratio (S/N) of 35 to maximize reliability. The fit is performed via pPXF (Cappellari & Emsellem 2004; Cappellari 2017) and makes use of E-MILES (Vazdekis et al. 2016) simple stellar population models of eight ages (0.15–14 Gyr) and four metallicities ([Z/H] = [-1.5, -0.35, 0.06, 0.4]). Only the wavelength range 4850–7000 Å is used in the fit, in order to avoid strong sky residuals in the reddest part of the MUSE spectral range. Finally, for individual spaxels, the properties of emission lines are measured (fluxes and kinematics) via a simultaneous fit of continuum and emission lines also performed via pPXF. The DAP fits only a single stellar population, so the stellar masses and velocities are an average of young and older stars. We primarily want to apply the method to the old stellar population, as we do not expect young stars to obey continuity. As the stellar mass is dominated by old stars (typically, the mass-weighted age per spaxel is of the order of gigayears), and the velocities of young and old stars are typically similar (Rosado-Belza et al. 2020; Shetty et al. 2020), averaging these values over stellar populations will not bias our results. Finally, for individual spaxels, the properties of emission lines are measured (fluxes and kinematics). We make use of the stellar mass surface density and velocity maps (from the first two processing stages) and the Hα flux and velocity maps (from the final processing stage).

### 2.2. PHANGS-ALMA

We make use of ¹²CO(J = 2 – 1) maps (hereafter “CO” for
brevity) from the PHANGS-ALMA survey. While the
observations and data reduction are detailed separately (Leroy
et al. 2020a, 2020b), we provide a brief summary of the
products used in this study.

These CO maps are provided as part of the internal PHANGS-ALMA DR v3.4. This includes a total of 84 galaxies (of which two are omitted because no emission was detected in the cubes). We use cubes combined with all available data from the 12 m and 7 m arrays and total power (TP) observations. This means that we maximize the resolution of the observations, while still being sensitive to extended emission on all scales. In total, 69 of our galaxies have 12 m+7 m+TP data, seven have 12 m+7 m, seven have 7 m+TP, and one has 7 m only. Typically, the data without 12 m antenna configurations are for the nearest galaxies, so the spatial resolution remains reasonably consistent. As shown in the right-hand panel of Figure 4, we do not expect the particular antenna configuration to have a strong impact on our results.

The 12 m+7 m data are imaged simultaneously using a
multiscale clean followed by a single-scale clean. This
combined cube is then corrected for primary beam pickup
and convolved to have a round synthesized beam. The TP data
are imaged separately using the method described in Herrera
et al. (2020), and these two cubes are then combined in Fourier
space (“feathered”). The cubes are then collapsed into standard

<footnote n="19">¹⁹ The code used in this work, as well as a `.fits` table, are available online at
https://github.com/thomaswilliamsastro/phangs_pattern speeds.</footnote>

<footnote n="20">²⁰ https://github.com/emsellem/pymusepipe</footnote>
moment 0 (integrated intensity) and moment 1 (intensity-weighted velocity) and error maps. Two types of masks are used to create these maps: first, strict masks that grow out any voxels with $S/N > 4$ in two or more successive channels down to voxels with $S/N > 2$ in two or more successive channels. Second, there are broad masks that take the union of the highest resolution strict mask from a strict mask generated at a spatial resolution of 500 pc, which captures lower-level, extended emission. For more details of this masking, see Leroy et al. (2020b). We opt to use broad rather than strict maps to maximize our completeness (at the cost of increased noise). We find a negligible impact on our results using these different masking schemes in most cases. This choice only matters for the low-surface-brightness galaxies in the sample, for which we typically find a poor fit in any case.

We use maps at their highest resolution ($1.3^{+0.4}_{-0.2}$ arcsec, corresponding to $100^{+31}_{-35}$ pc), whereby the native-resolution maps are smoothed to a common circular beam for each galaxy. Combined with the MUSE stellar and Hα maps, we have three different kinematic tracers, which afford to us useful cross-checks between derived pattern speeds for the sample of 19 galaxies where the observations overlap.

### 3. Calculating Pattern Speeds

In this section, we present a brief overview of the Tremaine-Weinberg method (Section 3.1). We detail our treatments of the various uncertainties associated with the method (Section 3.2), and then investigate the effect of slit lengths and widths on the recovered pattern speeds (Section 3.3). We then apply the method to our data set (Section 3.4) and perform a posteriori checks on the pattern speeds (Section 3.5). We stress that we only provide one pattern speed; for barred galaxies, we expect this to be the pattern speed of the bar, and for nonbarred galaxies this will be a pattern speed for the spiral arms. As an umbrella term, we will refer to these measured pattern speeds as “primary” pattern speeds throughout this work, as these structures will dominate the nonaxisymmetry the Tremaine-Weinberg method is sensitive to when we are in a regime with multiple pattern speeds present (we discuss this further in Section 3.1).

#### 3.1. Tremaine-Weinberg Method

The Tremaine-Weinberg method is a model-independent method of calculating a pattern speed within a galaxy. It has three minimal assumptions:

1. The galaxy disk is flat;

2. The disk contains a single, well-defined pattern speed;

3. The tracer obeys the continuity equation.

Assumption (1) is justified as our observations cover mainly the inner disks of galaxies (typically around $1 R_{25}$; Lang et al. 2020), and the detected emission is less radially extended than, for example, H I. Thus, we do not expect these observations will be sensitive to disk warping. We therefore assume that this assumption holds true across all of our sample.

On point (2), prior studies have shown that many galaxies host multiple pattern speeds, corresponding to different morphological features (e.g., bars and spiral arms within a galaxy; Meidt et al. 2008b; Beckman et al. 2018). In this case, assumption (2) may be invalid. We would expect two different (but linked) phenomena here. For slits that contain no

information from the bar, we would expect to measure a different pattern speed than for slits with bar information if the pattern speeds are different between spiral arms and bars. This is reflected in our quality flagging (Section 3.5), and we believe this only affects a small subset of our sample.

Furthermore, as we make our slits as long as possible, some slits may pass through both bars and spiral arms. If these structures have different pattern speeds, then any pattern speed we measure will be some weighted average of these two distinct values. We expect this measurement to strongly deviate from the bar pattern speed only when the spatial and kinematic nonaxisymmetry of the secondary pattern (i.e., spiral arms) is strong relative to the primary pattern speed, and also when this secondary pattern extends over a considerable portion of the slits (for examples of this, see scenarios considered by Meidt et al. 2008a). For most of the galaxies in our sample, this does not appear to be the case. Outer spirals tend to be both substantially dimmer and exhibit weaker streaming motions than the bars in our sample: whereas residual velocities from the fitted rotation curves by Lang et al. (2020) are typically $20 \text{ km s}^{-1}$ or greater in our sample for bars, they reach a maximum of only $\sim 10 \text{ km s}^{-1}$ in the spiral arms. Attempting to quantify the bias would require detailed simulations of galaxies where we know a ground truth for the pattern speed. This is beyond the scope of this work, but it will be revisited in later studies. For the present work, we assume that the secondary pattern is weak compared to the bar pattern, so the bias is small, much like in earlier works applying the Tremaine-Weinberg method to stellar kinematics (e.g., Aguerri et al. 2015; Guo et al. 2019; Garma-Oehmichen et al. 2020).

While assumption (3) (the tracer obeys the continuity equation) is approximately valid for the old stellar mass surface density distribution that we are primarily sensitive to with our MUSE observations, it may not be formally valid given the clumpy nature of CO and Hα emission, the phase transitions from H I to H$_2$ to H II, and the way these phases participate in the star formation process. We investigate this further in Section 4, and we find that this is generally an issue within our sample. For this reason, the bulk of our analysis is performed on pattern speeds measured from stellar kinematics.

If the continuity equation is valid, then

$$ \frac{\partial \Sigma}{\partial t} + \frac{\partial}{\partial x}(\Sigma v_x) + \frac{\partial}{\partial y}(\Sigma v_y) = 0, \quad (1) $$

where $(v_x, v_y)$ is the mean velocity of the tracer at $(x, y)$, and $\Sigma$ is the surface brightness at $(x, y)$. In polar coordinates $(r, \phi)$, we assume that the surface brightness ($\tilde{\Sigma}$) in a frame rotating with an angular pattern speed $\Omega_P$ is

$$ \Sigma = \tilde{\Sigma}(r, \phi - \Omega_P t), \quad (2) $$

and from this we can write

$$ \frac{\partial \Sigma}{\partial t} = -\Omega_P \frac{\partial \tilde{\Sigma}}{\partial \phi} = \Omega_P \left( y \frac{\partial \Sigma}{\partial x} - x \frac{\partial \Sigma}{\partial y} \right). \quad (3) $$

Substituting this into Equation (1) and integrating over both x and y (see Tremaine & Weinberg 1984), we can obtain

$$ \Omega_P \sin(i) = \frac{\int_{-\infty}^{\infty} h(y) \int_{-\infty}^{\infty} \Sigma v_{\text{LOS}}(x, y) dxdy}{\int_{-\infty}^{\infty} h(y) \int_{-\infty}^{\infty} \Sigma x dxdy}, \quad (4) $$
where *i* is the inclination of the galaxy, $v_{LOS}$ the line-of-sight velocity, and $h(y)$ a weight function. In this work, we take $h(y)$ to be a boxcar function from $y - dy$ to $y + dy$, to represent a pseudo-slit parallel to the line of nodes. Throughout this work, we will refer to “slits” and “integrals” interchangeably in this context. This equation can be simplified to

$$ \Omega_{\mathrm{P}} \sin(i) = \frac{\langle v \rangle}{\langle x \rangle}, \qquad (5) $$

by recognizing that the integrals in the numerator and denominator are simply the intensity-weighted velocity and position along a slit, respectively. In this formalism, nonzero values of $\langle v \rangle / \langle x \rangle$ are caused by nonaxisymmetric structure within the slit. Thus, taking a number of slits and plotting $\langle v \rangle$ versus $\langle x \rangle$ yields a straight line with a slope equivalent to $\Omega_{\mathrm{P}} \sin(i)$.

There are some limitations to this method given the data. First, the integrals formally should extend from $-\infty$ to $\infty$, whereas in reality this is not the case because of the limited field of view of the observations. However, assuming the disk is axisymmetric at large $x$, we can instead integrate from $-x_0$ to $x_0$ (where this is set by the extent of the field of view of the observations). Second, this method is less effective for galaxies that are face-on (due to loss of kinematic information) or edge-on (due to loss of photometric information). Finally, for galaxies with bars, we expect our primary pattern speed to be the bar pattern speed. If the bar is oriented along the galaxy minor axis, the integrals will tend to cancel out, and no pattern speed will be measured.

## 3.2. Uncertainties on the Pattern Speeds

In order to provide robust estimates on the pattern speed uncertainties, we take multiple sources of error into account when calculating them. First, we account for errors in both $\langle x \rangle$ and $\langle v \rangle$ separately, based on the error in each pixel along the slit and summed in quadrature. These are then propagated into the fitting routine using orthogonal distance regression (ODR), which allows us to effectively account for errors in both $\langle x \rangle$ and $\langle v \rangle$. We use the `scipy` implementation of ODR (`scipy.odr`). We find that, given the high S/N of our observations and the large numbers of pixels along each slit, the formal errors on these ODR fits are very small. As an example, one such fit is shown in Figure 1 for NGC 3351, and these visualizations are shown for all MUSE galaxies in Figure Set 1. Even our highest-quality fits have some points that lie off the fitted line. These can be caused by a multitude of reasons, including unmasked foreground stars, small variations in the galaxy position angle with galactocentric radius, or (particularly at the edges of the observations) insufficiently long slits. However, particularly for the fits with a quality flag $Q=1$ (see Section 3.5), the fits tend to look excellent.

We also account for errors in the position angle of the galaxy and the galaxy center. We do this via a Monte Carlo method, perturbing the line of nodes and the galaxy center by the measured errors (recorded in Lang et al. 2020). We use 1000 bootstraps, measure the pattern speed for each of these iterations, and quote the pattern speed as the median of this distribution, with the associated errors as the 16th and 84th percentiles, as we find these errors tend to be asymmetric.

These are listed in Table 2. We find the errors on our pattern speeds to be $11_{-3}^{+15}$% (the median percentage error and 16th and 84th percentiles), and that it is the error in the position angle that dominates, as has been shown in previous work (Debattista 2003; Garma-Oehmichen et al. 2020). We do not include uncertainties from the inclination or distance in our uncertainty for the pattern speeds. This is because many of our comparisons are to other quantities that are inclination- and distance-dependent (see Section 7).

## 3.3. Effects of Slit Length and Width on Recovered Pattern Speeds

Finally, we investigate the effects of both the slit length and slit width on recovered pattern speeds. These can affect the pattern speeds in two ways: first, the slits must be long enough to reach a sufficient radius at which the disk is roughly axisymmetric (and the effect from morphological features no longer prominent). If this is not achieved, the measurement will be biased. Second, if the slits are too wide, we may have insufficient slits covering the bar to retrieve a reliable pattern speed measurement. We use a number of slit lengths, from 10" to 150" (~1–5 kpc, dependent on the distance and inclination of the galaxy), and slit widths from 0"/5 to 10" (in general, the resolution of our data is ~1", but the pixels oversample the beam; this corresponds to ~50–1 kpc). Here, 1" corresponds to between around 25 pc for the nearer targets in the sample and 120 pc for the farthest. The results of this experiment for NGC 3351 are shown in Figure 2. We find that the measured pattern speeds tend to converge as the slits become longer (typically, slightly longer than the bar). We also find that the slit width has a minimal impact on the recovered pattern speed (right panel of Figure 2). With larger slit widths, we have fewer points to fit, and typically the uncertainty in the pattern speed becomes slightly larger. Motivated by these results, we opt to make each individual slit as long as possible and 1" wide to approximately match the resolution of the data. For surveys with a larger number of galaxies, but with physical resolutions of kiloparsec scales, rather than ~100 pc scales (e.g., MaNGA; Bundy et al. 2015), efforts to measure pattern speeds in these galaxies should still produce reliable results (right panel of Figure 2; see also work by Guo et al. 2019). We perform these slit-length and slit-width diagnostics for all galaxies, and these form a critical component of our diagnostic assessments (Section 3.5).

## 3.4. Data Preparation

We perform a number of preprocessing steps on the data before calculating the Tremaine–Weinberg integrals:

1. **Mask foreground stars.** For the MUSE data only, foreground stars can be a contaminant. These are clearly recognizable in the data with extreme (positive or negative) velocities (with respect to the systemic velocity of the galaxy). We therefore remove any pixels with $|v| > 300 \text{ km s}^{-1}$. This cut is arbitrary and may need to be tailored for other data sets. However, we find that the recovered pattern speeds with masked or nonmasked stars are not significantly different.

2. **Subtract systemic velocity.** For the ALMA data only, the systemic velocity is not subtracted in the moment 1 maps.
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Top left: stellar mass surface density map of NGC 3351 shown in grayscale, with Tremaine–Weinberg integral slits of 1" width, oriented parallel to the major axis overlaid. Only one in every four slits is shown, due to the slit density, and are colored according to their position along the kinematic minor axis. For this galaxy, the quality flag $Q = 1$ (see Section 3.5). Top right: stellar velocity map for the same galaxy, with a dashed black line showing the kinematic major axis passing through the galaxy center. Bottom: intensity-weighted velocity ($\langle v \rangle$) vs. intensity-weighted position ($\langle x \rangle$) for each of the slits (the color corresponds to the slit color in the above top left panel). The black line shows the best fit, and the gray shaded region the errors on the fit (in this case, this region is extremely small). One point has an extremely large uncertainty in this panel, and the error bar extends across the entire range of $\langle v \rangle$ shown. The complete figure set (19 images) is available in the online journal.</figcaption></figure>

(The complete figure set (19 images) is available.)

We estimate the systemic velocity using PAFIT (Krajnović et al. 2006) and subtract it from the ALMA velocity field (moment 1) map. This has the effect of centering the $\langle v \rangle$ integrals. As we are primarily interested in the slope of $\langle v \rangle$ versus $\langle x \rangle$, this value is not particularly important, but it is included so that the line approximately passes through the origin.

3. *Remove integrals that do not cross the bar.* For galaxies with bars, we attempt to isolate the bar pattern speed from any other pattern speeds present in the galaxy (i.e., to minimize the effects of potentially different pattern speeds in the spiral arms). This is done by removing any slits that do not at least partially cover the bar. We note that these slits may cross multiple features with multiple patterns, but as the slit lengths need to be suitably long to reach the axisymmetric disk, this is unavoidable.

4. *Symmetrize the integral.* Finally, after the other preprocessing steps, we make sure each integral goes from $-x_0$ to $x_0$. To do this, we find the maximum distance from the minor

axis where both sides of each slit still contain data. Given the field of view of the observations and previous bar masking, this may lead to slits that are different lengths from each other. It is critically important that each slit is symmetric, as if they are not, this asymmetry can induce a false signal in the integrals.

Having performed this preprocessing, we place a number of slits across the surface brightness and velocity maps and calculate the intensity-weighted position and velocity of all the pixels within each slit. We do this for all 19 galaxies covered by MUSE and all 84 of our galaxies covered by ALMA. We find that all of the MUSE galaxies can be fitted, but because no CO emission is detected in the cube, the ALMA data for IC 5332 and NGC 3239 cannot be fitted. This leaves us with a total of 82 ALMA galaxies, and 18 of these overlap the MUSE observations. IC 5332 is present in both samples, but as no emission is detected in the ALMA cube, we only present measurements from the MUSE observations.
<figure><img src="image_4.png" /></figure>

**Figure 2.** Left: recovered pattern speed, $\Omega_p$, vs. Tremaine–Weinberg integral slit length, $r$, for MUSE stellar mass surface density observations of NGC 3351 (at this distance, $1''$ corresponds to ~50 pc). The vertical dashed black line indicates the bar extent in this galaxy. Right: recovered pattern speed vs. Tremaine–Weinberg integral slit width for the same galaxy (the resolution of this map is ~1''). As the pattern speed measured for each slit width is not independent from other slit widths (they use the same data, and larger slit widths combine information from narrower slits), the scatter in each measurement is much smaller than the typical uncertainty. In both cases, the horizontal black line indicates our fiducial pattern speed, and the dashed black lines either side of it the uncertainty on this measurement.

### 3.5. Quality Flagging

Having applied the Tremaine–Weinberg method to all of the galaxies (with all available tracers) in our sample, we perform a posteriori quality flagging on these values. We include all measured values in Table 2 along with these quality flags ($Q$). We take a minimalist approach to the quality flagging, to provide clear delineation between the flags, which are defined as follows:

1. Single, well-defined slope: integrals have converged, and pattern speed is stable with decreasing slit width;

2. Clear multiple slopes visible in the $\langle v \rangle/\langle x \rangle$ plot, but otherwise the fit would be a quality flag (1);

3. Poor fit: integral has not converged, and points do not form a clear, well-defined slope;

4. Data of insufficient quality to calculate a reliable slope.

Our flags span this entire range, from 1 to 4 (see Table 2). We note that these quality flags simply represent the quality of the fits and will not take into account the issues present in the ISM tracers discussed in Section 4. We do include these in the table with the caveats mentioned in this later section, both because we will make a comparison between the tracers in Section 5, and because future work will investigate these values in the context of other methods.

This flagging is performed independently on each tracer by three of the authors, and our final flag is the mode of these three flags. In the case of disagreement between all three authors, we instead take the highest flag value of any author, to be as conservative as possible. We opt for the mode to ensure there are no fractional flags that may cause confusion as to their definition. In this work going forward, we will define “well-constrained” Tremaine–Weinberg values as having $Q=1$ or 2 and only use these in our analyses. The only galaxy we use with a quality flag of 2 is NGC 3627, for which points toward the southwest appear to have a different slope (see Figure B.13, which is available only online).

The distribution of well-constrained values is shown as a function of Hubble morphological type (from the HyperLeda database; Makarov et al. 2014) in Figure 3. Many of the galaxies flagged as having poorly measured values are from

<figure><img src="image_5.png" /></figure>

**Figure 3.** Distribution of well-constrained Tremaine–Weinberg values as a function of Hubble morphological types: top panel: MUSE-$M_*$ measured values; middle: MUSE-H$\alpha$ measured values; and bottom: ALMA-CO measured values. In each case, gray indicates all of the values for each tracer (i.e., the underlying population), and the colored bars the well-constrained sample: red for MUSE-$M_*$, cyan for MUSE-H$\alpha$, and blue for ALMA-CO.

earlier Hubble types. This is likely due to their lack of prominent morphological features and thus no strong signal in the Tremaine–Weinberg plots. For the MUSE stellar mass values, we find that the fit is well constrained for 10 of the 19 galaxies. Of these, nine have bars present, and one (NGC 0628) does not. The PHANGS-MUSE sample contains 15 barred galaxies and four nonbarred, so we preferentially measure bar pattern speeds. For some with a quality flag of 3 or 4, we detect no evidence for a pattern (IC 5332 and NGC 5068), but in these cases the velocity field is very irregular (and so the position angles change with radius).
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Left: comparison between the Tremaine–Weinberg signal generated and axisymmetric surface brightness and velocity distribution with incomplete map coverage ($\Omega_{null}$), and from the actual ALMA-CO (blue) and MUSE–H$\alpha$ (cyan) data ($\Omega_{clump}$). This test has been carried out for all ALMA galaxies where a rotation curve has been fitted (65 galaxies, Lang et al. 2020; and to the 13 galaxies in MUSE DR1.0). The 1:1 relation is shown as a dashed black line. Right: comparison between $\Omega_{clump}$ measured for high-resolution data (generally 12 m+7 m+TP) and lower resolution data (7 m+TP).</figcaption></figure>

## 4. Application to ISM Tracers

While we base the majority of this study on pattern speeds derived from stellar kinematics, the PHANGS-ALMA data set in particular offers a much larger sample size. The pattern speed for the Milky Way bar derived from gas dynamics (Sormani et al. 2015) has been found to match well with that derived from stellar dynamics (Sanders et al. 2019), so using the ISM may provide useful, independent pattern speed measurements. We therefore turn to the question of whether ISM tracers are intrinsically compromised by their incomplete coverage and clumpiness. If each slit goes through a single clump, even if there is no pattern present, applying the Tremaine–Weinberg method will give nonzero integrals, and this can lead us to measure a false pattern speed. In reality, in these data, each slit will pass through many of these clumps, which are not positioned randomly (i.e., they cluster on spiral arms or bars). We thus critically examine whether we can disentangle the true signal from the effect of incomplete coverage.

To test this, for each galaxy we take an axisymmetrized velocity and surface brightness profile based on the CO maps. For the velocity field, we project the fits from Lang et al. (2020) into the frame of the galaxy, and for the CO data we take the average surface brightness within a number of annuli. We then mimic the incomplete coverage of these maps by blanking pixels in these axisymmetrized profiles where emission is not detected, and we apply the method described in Section 3.1 to these. In this case, there is no pattern speed present in these maps, and any deviation is simply due to the incomplete CO coverage and the nonaxisymmetry of the CO morphology. We refer to this as $\Omega_{null}$ and compare this to the “pattern speed” measured from the actual CO data, as shown in the left panel of Figure 4. We refer to this as $\Omega_{clump}$. As we can see, the points tend to lie close to the 1:1 relation, indicating that much of the signal we measure using the Tremaine–Weinberg method may simply be due to the clumpy nature of the tracer itself. We also find a similar trend when repeating this test for H$\alpha$. The angular velocity that we measure with this experiment is some function of the underlying rotation curve of the galaxy and the sampling of this rotation curve by the tracer’s distribution of “clumps.” In the right panel of Figure 4, we compare $\Omega_{clump}$ from the higher resolution (~1") and lower resolution (~5")

ALMA data. The values calculated are very similar, and we find that repeating the $\Omega_{null}$ test with these values calculated from lower resolution data yields much the same results as the higher resolution data. This indicates that this effect is endemic to the use of CO and not simply due to the high resolution of our data.

As an initial exploration of this, we take some simple density-wave spiral models following Binney & Tremaine (2008), with a known pattern speed, and blank regions to mimic the clumpiness of the employed tracer. We find that more spatially extended distributions of clumps (probing farther out in galactocentric radius) produce lower $\Omega_{clump}$ than more centrally concentrated arrangements. We believe that this leads to the results in Section 5.2, where $\Omega_{clump}$ and $\Omega_{P}$ appear systematically offset. As CO extends farther out in galaxies hosting larger bars (the outermost CO-emitting radius tabulated by Lang et al. 2020 increases roughly with increasing $R_{bar}$). Second, longer bars tend to have lower pattern speeds than shorter bars in similar-sized galaxies (i.e., $R \sim 1.2$). As applied to PHANGS-ALMA, we will thus find lower $\Omega_{clump}$ in galaxies with lower bar pattern speeds. Note that this may not always hold for other tracers, such as for much more extended H I disks, when the extent of the tracer is not tied to the morphology of the dynamical feature in question. However, it does appear that this is the case for both our ALMA-CO and MUSE–H$\alpha$ data. A full exploration of the links between $\Omega_{clump}$ and $\Omega_{P}$ is beyond the scope of this work and will require detailed simulations of galaxies.

## 5. Derived Pattern Speeds

With our final selection of well-constrained Tremaine–Weinberg values in hand (Table 1), we show the distribution of the 19 MUSE stellar mass pattern speeds in Figure 5. Note that we use kernel density estimate (KDE) plots throughout this work to improve readability, by avoiding the “steps” we would see in histograms for these small number of data points. However, any statistics are calculated from the data themselves, rather than from the distribution of the KDE. For the values we consider well constrained, the average speed is $30_{-10}^{+10}$ km s$^{-1}$ kpc$^{-1}$ (this is the median of the sample, along with the 16th and 84th percentiles of the distribution). The distributions of all of the pattern speeds derived from the
<table><caption>**Table 1**
Pattern Speeds and Corotation Radii for the 10 Well-constrained Stellar Mass Pattern Speeds</caption><thead><tr><th>Galaxy</th><th>PGC</th><th>D<br>Mpc</th><th>i<br>°</th><th>PA<br>°</th><th>Bar?</th><th>Ω<sub>P</sub><br>km/s</th><th>Q</th><th>R<sub>CR</sub><br>kpc</th></tr></thead><tbody><tr><td>NGC 0628</td><td>5974</td><td>9.84</td><td>8.9</td><td>20.7</td><td>0</td><td>31.1<sup>+4.0</sup><sub>-2.9</sub></td><td>1</td><td>4.5 ± 2.0</td></tr><tr><td>NGC 1087</td><td>10496</td><td>15.85</td><td>42.9</td><td>359.1</td><td>1</td><td>31.9<sup>+3.2</sup><sub>-1.6</sub></td><td>1</td><td>4.3 ± 1.1</td></tr><tr><td>NGC 1433</td><td>13586</td><td>12.11</td><td>28.6</td><td>199.7</td><td>1</td><td>20.0<sup>+2.4</sup><sub>-1.9</sub></td><td>1</td><td>6.3 ± 0.4</td></tr><tr><td>NGC 1512</td><td>14391</td><td>17.13</td><td>42.5</td><td>261.9</td><td>1</td><td>22.3<sup>+5.5</sup><sub>-5.4</sub></td><td>1</td><td>5.4 ± 2.2</td></tr><tr><td>NGC 1672</td><td>15941</td><td>19.4</td><td>42.6</td><td>134.3</td><td>1</td><td>22.7<sup>+0.6</sup><sub>-0.6</sub></td><td>1</td><td>6.1 ± 1.6</td></tr><tr><td>NGC 2835</td><td>26259</td><td>12.38</td><td>41.3</td><td>1.0</td><td>1</td><td>34.6<sup>+4.3</sup><sub>-3.1</sub></td><td>1</td><td>2.8 ± 0.6</td></tr><tr><td>NGC 3351</td><td>32007</td><td>9.96</td><td>45.1</td><td>193.2</td><td>1</td><td>43.6<sup>+11.6</sup><sub>-12.4</sub></td><td>1</td><td>3.4 ± 0.8</td></tr><tr><td>NGC 3627</td><td>34695</td><td>11.32</td><td>57.3</td><td>173.1</td><td>1</td><td>29.1<sup>+20.6</sup><sub>-8.1</sub></td><td>2</td><td>1.9 ± 0.1</td></tr><tr><td>NGC 4303</td><td>40001</td><td>16.99</td><td>23.5</td><td>312.4</td><td>1</td><td>43.5<sup>+5.3</sup><sub>-10.0</sub></td><td>1</td><td>3.8 ± 2.5</td></tr><tr><td>NGC 7496</td><td>70588</td><td>18.72</td><td>35.9</td><td>193.7</td><td>1</td><td>16.8<sup>+6.5</sup><sub>-12.1</sub></td><td>1</td><td>4.3 ± 2.0</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Distribution of pattern speeds derived from MUSE stellar mass maps for 19 galaxies. The red line shows the kernel density estimate distribution for all calculated pattern speeds (using the optimal bandwidth formula from Silverman 1986), and the shaded black line shows the distribution of well-constrained pattern speeds. The KDE is normalized such that the integral of the distribution is equal to one, so the shape of each distribution, rather than the absolute scaling, is what should be considered here.</figcaption></figure>

MUSE stellar mass maps are also shown in this figure, and it can be seen that they occupy a similar range of speeds ($29_{-10}^{+14}$ km s$^{-1}$ kpc$^{-1}$). Thus, poorly constrained pattern speeds are not found at any particular low or high values of $\Omega_P$, so this manual quality flagging is required compared to the simple threshold cut.

## 5.1. Comparison to the Literature

Some of these galaxies have previously published pattern speeds,$^{21}$ and we make a comparison between our values and these earlier values in Figure 6 (scaling to our assumed galaxy distances and inclinations). These pattern speeds come from a variety of methods, but we note that the pattern speeds of NGC 1365, NGC 3627, and NGC 4321 in the literature come from applying the Tremaine–Weinberg method to ISM tracers. For completeness, we show those points with Q-values greater

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Comparison between derived pattern speeds for MUSE stellar mass maps (red symbols), ALMA-CO-based $\Omega_{\text{clump}}$ (blue symbols), and previously published literature values (gray symbols). If the pattern speed is not well measured ($Q = 3$ or 4), the dot is unfilled. For literature pattern speeds from applying Tremaine–Weinberg to ISM tracers, we outline these points in light green.</figcaption></figure>

than 2, but we stress that these do not constitute true measurements of pattern speeds and are not used in any of our analysis here or later in this work. In general, our pattern speeds agree well with previously published values and typically agree within the uncertainties (a median absolute

<footnote>$^{21}$ NGC 0628: Martínez-García & Puerari (2014); NGC 1300: Lindblad & Kristen (1996); NGC 1365: Speights & Rooke (2016); NGC 1433: Treuthardt et al. (2008); NGC 1512: Koribalski & López-Sánchez (2009); NGC 1566: Korchagin et al. (2005); NGC 1672: Díaz et al. (1999); NGC 3627: Rand & Wallin (2004); NGC 4254: Egusa et al. (2004); NGC 4303: Schinnerer et al. (2002); NGC 4321: Hernandez et al. (2005).</footnote>
<figure><img src="image_9.png" /><figcaption>**Figure 7.** KDE plot showing the relative pattern speed difference (Equation (6)) for both CO (blue line) and Hα (cyan line) with respect to the stellar mass pattern speed. The dashed black line indicates where Ωp = Ωclump. The vertical dashed blue and cyan lines show the median of the distributions for CO and Hα, respectively.</figcaption></figure>

deviation of ~10%, compared to the larger differences we see
when comparing to ISM tracers in Section 4). Given that these
come from extremely heterogeneous pattern speed measure-
ments (either in technique or tracer), the fact that we see good
agreement across the board is reassuring. We are therefore
confident that our stellar mass pattern speeds presented in
Table 1 are robust and reliable.

5.2. Comparisons between Tracers

As the PHANGS-MUSE sample overlaps entirely the PHANGS-ALMA sample, we can compare how closely the ISM tracers’ (Hα, CO) measured Ωclump tends to agree with Ωp. We compare values flagged as well constrained in both the stellar mass and the ISM tracer in question (for both Hα and CO, this leaves us with nine galaxies). We then define a “pattern speed difference” as

$$
\text{Pattern Speed Difference} = \frac{\Omega_{\text{clump}} - \Omega_{\text{p}}}{\Omega_{\text{p}}} \qquad (6)
$$

and show the distributions for both ISM tracers in Figure 7.
Typically, we find that Ωclump is higher than Ωp, somewhat
greater than our ~10% errors (~40% in the case of CO, ~20%
in the case of Hα). Therefore, while applying the Tremaine–
Weinberg method to clumpy ISM tracers may yield reasonable
results, it is clear from this exercise that Ωclump and Ωp are
systematically different quantities, and thus Ωclump should not
be used as a proxy for Ωp.

6. Locations of Major Resonances

We next use our pattern speeds to calculate major resonance
locations for each galaxy. In this work we focus on the
corotation radius, R<sub>CR</sub>, but we also provide estimates of the
outer and inner Lindblad radii (OLR and ILR). These
resonance locations are derived with the assumption of
circularity, so we give a single radius for each resonance. To
do this, we make use of the rotation curves derived from the
PHANGS-ALMA data in Lang et al. (2020) and rescaled to our
assumed distances. These rotation curves approximate the
circular velocity, v<sub>circ</sub>, in a number of radial bins, where the
galaxy center, inclination, and position angle are first fit using a

Bayesian Markov Chain Monte Carlo (MCMC) analysis. Following this, least-squares fitting uses a harmonic decomposition to model the rotational velocity within a series of radial annuli. However, as these curves are measured from CO, there may be deviations from this circularity, particularly in the centers of galaxies, or for strong spiral or bar streaming motions. We use this observed velocity profile as a proxy for the true circular velocity of the galaxy and as such is a first-order approximation to calculate the resonance locations. Where nonaxisymmetry of the potential dominates, the observed velocity profile will overestimate the true circular velocity, so this approximation will lead to incorrect resonance locations, but assessing the impact of this requires detailed simulations for each galaxy and as such is beyond the scope of this work. However, observational works have applied this method previously (see, e.g., Schinnerer et al. 2000; Fathi et al. 2009) and find the location of expected structures near these resonance locations, which would indicate this first-order approximation is often close to the true value. As such, we highlight that these resonance locations are approximations but represent our best estimation of the true value. Throughout this work, we will refer to these resonance locations as the corotation radius and ILR/OLR, but bear in mind these caveats.

Typically, the rotation curves derived from the CO maps are
very similar to those derived from Hα and stars in our sample,
so we are confident our choice of using CO-based rotation
curves will not bias our results. By converting the rotation
velocity to an angular velocity, Ω(R), the corotation radius is
simply where the angular velocity is equal to the pattern speed.

To calculate R<sub>CR</sub>, we use the velocities as fitted to each radial bin. This allows us to propagate through the uncertainties both in the rotation curve and the pattern speed. In some cases, we have multiple regions where the angular velocity crosses the pattern speeds, and we report all of these in Table 2. In many cases, due to the shape of the rotation curve, a number of consecutive points are consistent with being corotation, within the errors. For these points, we associate these with the same R<sub>CR</sub> if they are within 2σ, and we take the mean of them as the nominal values, with the range as the uncertainty. An example of this R<sub>CR</sub> measurement is shown in Figure 8 for NGC 3351. As can be seen, our error propagation leads to reliable estimates of R<sub>CR</sub>, but quite large uncertainties, on the order of ~1 kpc (the error is mainly dependent on the shape of the rotation curve). These errors are similar to those obtained using gravitational torque analysis (Querejeta et al. 2016).

We also calculate the OLR and ILR for each galaxy, where
they exist. In this case, to avoid issues with numerical
derivatives, we use smooth splines fitted to the rotation curves.
As these do not account for the errors in the rotation curve
fitting, the errors reported in Table 2 only consider the errors in
our pattern speed derivation. We suggest taking an uncertainty
on these values at least as large as that of R<sub>CR</sub>. The locations of
these resonances occur when

$$
\Omega_p = \Omega(R) \pm \kappa/2 \qquad (7)
$$

where positive is for the OLR, and negative is for the inner
ILR. Here, κ is the epicyclic frequency, which is given by

$$
\kappa^2 = \frac{2\Omega(R)}{R} \frac{d}{dR} (R^2 \Omega(R)). \qquad (8)
$$
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Angular speeds ($\Omega$) vs. galactic radius for NGC 3351. Angular speeds from Lang et al. (2020) are shown as black points. When one of these $\Omega$ values is consistent with the pattern speed (green line, with associated errors in green), it is highlighted in red. These points are then combined to form our estimate of the corotation radius (red line) and associated error (shaded red region). The bar radius is shown as a vertical, black, dotted-dashed line. We also include dashed, black lines to indicate $\Omega \pm \kappa/2$, which we use to calculate ILR and OLR (these values and associated errors are omitted here to maintain readability, but are included in Table 2).</figcaption></figure>

These curves are also highlighted in Figure 8, and we calculate their average radii and uncertainties in the same way as for $R_{CR}$. The locations of the resonances are shown on the MUSE white-light map of NGC 3351 in Figure 9.

With $R_{CR}$ directly measured, we can investigate two commonly assumed ratios for inferring the corotation radius when it is not directly accessible. The first is $R_{CR}/R_{bar}$ (commonly referred to as $\mathcal{R}$), where we calculate the deprojected bar length as

$$R_{bar,deproj} = R_{bar,proj} \times \sqrt{\cos(\Delta PA)^2 + (\sin(\Delta PA) \sec(i))^2}, \quad (9)$$

where $R_{bar,proj}$ is the projected bar length from Herrera-Endoqui et al. (2015), $\Delta PA$ is the relative alignment of the bar with the galaxy major axis, and $\sec(i)$ is the secant function of the inclination (the inverse cosine). We show the distribution of $\mathcal{R}$ for our sample in the left panel of Figure 10. The average value (and spread) for our sample is $1.1_{-0.4}^{+0.6}$ (this spread is calculated from the data, rather than from the KDE plot in Figure 10). Taking into account the error in corotation radius, along with a characteristic uncertainty in the bar length of 20% (Díaz-García et al. 2016), we find that around half of our measured values of $\mathcal{R}$ are inconsistent with the commonly assumed value of 1.2 in the literature (e.g., Elmegreen et al. 1996; Aguerri et al. 1998) to $1\sigma$. This is well within the expectation when including Poisson noise due to small number statistics, so our values of $\mathcal{R}$ are, on average, consistent with the expected value of 1.2.

Another commonly used ratio for inferring $R_{CR}$ is through the disk scale length $h_r$. We use the $S^4G$ scale lengths calculated in Salo et al. (2015) from multiple component image decompositions of Spitzer 3.6µm images, and we show the ratio of these in the right panel of Figure 10. Commonly, $h_r/R_{CR}$ is taken to be 3 (e.g., Kranz et al. 2003), and we find the value for our sample to be $2.6_{-0.4}^{+0.7}$. Again, given the small number statistics, we conclude our sample is consistent with a value for $h_r/R_{CR}$ of 3.

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Resonances highlighted on MUSE white-light map of NGC 3351 Corotation (and associated errors) is shown in red, and the inner Lindblad radius in blue.</figcaption></figure>

## 7. Correlations with Global Galaxy Parameters

With a number of pattern speeds robustly derived, we can statistically study how pattern speeds and resonance locations vary with a number of global parameters of the galaxy. Given that our sample is still small, we see this investigation as exploratory, but it provides a basis for future studies to expand on the results we present here. We expect that these further investigations may come from both an observational perspective (i.e., if we see these trends for a larger sample of galaxies) and simulations (i.e., if we should expect to see these trends in the first place). We first investigate the possibility of any pathological biases arising from the projection of the galaxy to our line of sight, in particular, the inclination and the relative alignment of the bar with the galaxy major axis ($\Delta PA$). In this work, we will focus on $\Omega_P$, $R_{CR}$, $R_{bar}$, $\mathcal{R}$, and the bar strength, $S_{bar}$. We take the bar lengths from Herrera-Endoqui et al. (2015) and the bar strength as the normalized $m=2$ Fourier density amplitude ($A_2^{\text{max}}$) from Díaz-García et al. (2016). Following Cuomo et al. (2020), we only use points that satisfy $\Delta\Omega_P/\Omega_P \le 0.5$, where $\Delta\Omega_P$ is the error in $\Omega_P$, but in this work we find that this removes only one galaxy (NGC 7496). This leaves us with a maximum of nine galaxies, but depending on the available literature data, the number of galaxies compared may vary slightly with parameter (for example, only six galaxies have bar strengths in Díaz-García et al. 2016). We also show the values from the ALMA points in gray, to highlight any misleading conclusions that may be drawn from using $\Omega_{\text{clump}}$ instead of $\Omega_P$ (or, equivalently, the bias of using CO rather than stars).

We first consider any pathological effects that may arise from the orientation of the galaxy. Our ability to measure morphological properties of bars may be affected by the inclination or relative misalignment of the bar with the galaxy itself, and it is important to consider whether this may bias our results. Figure 11 shows the relationships between $\Omega_P$, $R_{CR}$, $R_{bar}$, $\mathcal{R}$, and $S_{bar}$ with $\Delta PA$ and the galaxy inclination, where we also indicate the Kendall (1938) $\tau$ correlation coefficient. As we are correlating a number of variables against each other, by random chance we may see some “significant” correlations
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Left: KDE plot showing the distribution of *R* (the ratio of corotation radius to bar length) for our nine well-constrained stellar mass bar pattern speeds. Right: KDE plot showing the distribution of the ratio of the corotation radius to the galaxy scale length for the same sample of galaxies. In each case, we highlight the median value as a solid black vertical line, and the 16th and 84th percentiles as dashed black lines (these percentiles are calculated directly from the data, rather than from the KDE). The commonly assumed literature values are shown as red, vertical lines.</figcaption></figure>

in random data, based simply on using *p*-values (i.e., for 100 random distributions, five would be expected to have *p* < 0.05; see, e.g., Holm 1979, as well as Kruijssen et al. 2019 for a recent discussion of this). The *p*-value also assumes all errors are equal, which is not the case here. Instead, we take a Monte Carlo approach to assign a statistical error to the significance of the correlation. For 10,000 realizations, we perturb each value by its associated errors in both *x* and *y* and repeat the *τ* calculation. We then quote the median value of the correlation and the 1σ spread in these values. If the correlation is larger than this 1σ error, then the correlation is significant.

Ideally, we should see no correlations with projection parameters. Reassuringly, the only significant correlations we see are between $S_{\text{bar}}$ and $\Delta\text{PA}$, and between $\mathcal{R}$ and $\Delta\text{PA}$. These may give some artificially more significant correlation going forward, but otherwise we are confident that the projection of the galaxy will not systematically bias our results. If we consider the PHANGS-ALMA points we flag as well constrained, which cover a wider range of galaxy inclinations, there is clearly an anticorrelation between bar length and inclination and a correlation between bar length and $\Delta\text{PA}$. As these bars are typically defined by eye, at high inclination recovering the true bar length becomes increasingly difficult, and this tends to lead to the length being underestimated. For studies that attempt to measure such quantities in highly inclined systems, it is therefore important to take this bias into account.

We next correlate these quantities with a number of global galaxy parameters. These are the Hubble morphological type (from HyperLeda), the bulge-to-total flux ratio at 3.6μm (B/T) and galaxy disk scale length ($h_r$; both from Salo et al. 2015), and the total stellar mass ($M_*$) from the $z=0$ Multiwavelength Galaxy Synthesis (z0MGS; Leroy et al. 2019) normalized by $R_{25}$ for a measure of the average stellar surface density of the galaxy. We also take the asymptotic velocity of the galaxy rotation curve ($v_{r,\text{inf}}$) and turnover radius between the rising and flat part of the rotation curve ($R_t$) from Lang et al. (2020, see their Equation (10)) and the molecular gas fraction ($f_{\text{mol}}$; Leroy et al. 2020a). These are shown in Figure 12, where we also highlight the number of points in each combination of these parameters, along with the Kendall $\tau$. We color each point by

inclination to highlight any potential dependence of these correlations on galaxy orientation (see Figure 11).

Even given our relatively small number of galaxies, we find a number of significant correlations. In terms of the pattern speeds themselves, we find higher $\Omega_{\text{P}}$ in later-type galaxies and galaxies that are more molecular gas-dominated. We also find lower pattern speeds in galaxies that are more bulge-dominated (with a higher bulge-to-total flux ratio) and in those with a larger scale length. We find the corotation radius to be larger in more bulge-dominated galaxies, and also galaxies with a larger scale length. We see no significant correlations with the $R_{25}$ normalized bar length. We find that $\mathcal{R}$ tends to be higher in later-type galaxies. This appears to be contradictory to numerical simulations, showing that the bar pattern speed slows down over time (Wu et al. 2018). However, as we do not know the age of the bars in these galaxies, we cannot draw this conclusion from these results. Further work to calculate the bar ages in these galaxies may help to answer this question. We also find that $\mathcal{R}$ tends to be lower in galaxies with a larger scale length. We also see that bars tend to be stronger in galaxies with a higher $R_t$ and a higher $f_{\text{mol}}$. While we do see correlations between other parameters in this figure, the small number of galaxies means that we cannot robustly draw conclusions about the strength of the correlations between them. We see no strong dependence with inclination in any of these correlations, and we therefore are confident any biases seen in Figure 11 do not strongly affect the results presented here. Across all of these parameters, our single nonbarred galaxy (NGC 0628) occupies the same parameter space as the barred galaxies, as do galaxies with clear spiral arms versus more flocculent morphologies.

The gray points indicate PHANGS-ALMA values that we consider well constrained (those for which the straight-line fits to $\langle v \rangle$ versus $\langle x \rangle$ look reasonable, although these are $\Omega_{\text{clump}}$ values, rather than $\Omega_{\text{P}}$). Notably, these values—even if they do not measure the true pattern speed of the structure present in the disk—appear to occupy very similar regions in parameter space as our more robust stellar pattern speed estimates. This serves as a warning that similar measurements made with ISM tracers may give the appearance of meaningful correlations between dynamical structure and global galaxy properties, even when none are present. Indeed, $\Omega_{\text{clump}}$ exhibits a trend with stellar
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Pattern speeds measured from MUSE stellar mass data and global parameters with galaxy orientation parameters. Left: from top to bottom, the pattern speed, corotation radius, bar radius, $R$, and bar strength vs. $\Delta PA$ (the difference between bar and galaxy position angles). Points are colored by the inclination of the galaxy. Right: same as the left column, but now vs. inclination, with points colored by $\Delta PA$. For galaxies without a bar, we show the pattern speed and corotation radius as an unfilled circle. In each case, the number of points, $N$, and the Kendall $\tau$ correlation are given. We also show the well-constrained PHANGS-ALMA sample as gray crosses.</figcaption></figure>

mass surface density, while we see no significant trends with our stellar-mass-based pattern speeds. Based on our studies described in Section 4 (which suggest that $\Omega_{\text{clump}}$ is a function of both the shape of the rotation curve of the galaxy and the radial extent of the ISM clumps), the trends in Figure 12 arise as a result of a relation between the galaxy gravitational potential and the way the ISM is distributed within it.

Finally, we also correlate the y-axis quantities of Figure 12 with each other, and this is shown in Figure 13. The only significant correlation we see here is unsurprising: galaxies with higher pattern speeds have shorter corotation radii. We also see that there is variation both in $R_{\text{CR}}$ and $R_{\text{bar}}/R_{25}$ with $\Omega_{\text{P}}$ (although we do not find these correlations individually to be significant). The decrease in $R_{\text{bar}}/R_{25}$ appears to be driven
<figure><img src="image_14.png" /><figcaption>**Figure 12.** From top to bottom: pattern speed, corotation radius, bar radius, $R$, and bar strength vs., from left to right, the Hubble morphological type, bulge-to-total flux ratio, galaxy disk scale length, stellar mass divided by $R_{25}^2$, asymptotic rotation velocity, galaxy rotation turnover radius, and molecular gas fraction. Points are colored by the inclination of the galaxy, galaxies with bars are shown as circles and galaxies without as squares, and galaxies with noted spiral arms are highlighted in red. In each subplot, the number of points *N* and the Kendall *τ* correlation are given, along with its associated uncertainty. We also plot well-constrained PHANGS-ALMA values as gray crosses.</figcaption></figure>

mainly by the decrease in $R_{bar}$, and this means we do not see a
strong variation in $R$ across our galaxies.

8. Discussion

This work presents homogeneous measurements of pattern speeds and resonance locations for 10 galaxies observed as part of the PHANGS-MUSE survey. This sample size means that we can draw exploratory conclusions about trends in these parameters with the global properties of the galaxy, and, as our methodology for the application of the Tremaine–Weinberg method is generally data-driven, with minimal qualitative checks (except for the quality flagging at the end), we believe that future studies can build on the present work in a homogeneous way.

We have first investigated the scatter in $R$, the ratio of corotation to bar length. Theoretical arguments by Contopoulos (1980) have shown that this value should be a little larger than 1, and measurements of $R$ in simulation (Athanas-soula 1992; Debattista & Sellwood 2000) and in observation (e.g., Merrifield & Kuijken 1995; Gerssen et al. 1999; Font et al. 2017) have shown $1 < R < 1.4$. This has led to, when the corotation radius cannot be measured from the pattern speed, a common expectation that $R_{CR} \sim 1.2R_{bar}$. We find for our sample that the median and 16th–84th percentile spread in the sample is $R = 1.3_{-0.5}^{+0.4}$. The scatter on this distribution is consistent with the typical uncertainty in the measurement of $R_{CR}$. Thus, we find that our sample is compatible with a value of $R = 1.2$. We also find this to be the case for the ratio of the corotation radius, $R_{CR}$, to the disk scale length, $h_r$. This ratio is commonly assumed to be 3 (Kranz et al. 2003), with little

scatter. We find a large scatter, with a median and 16th–84th
percentile spread in the sample of $R_{CR}/h_r = 2.7_{-0.1}^{+0.7}$. There is
some scatter here, but given the small number statistics of this
study, we find our measurement to be in agreement with the
commonly assumed $R_{CR}/h_r = 3$.

8.1. Relationships between Pattern Speeds, Corotation Radii, and Bar Parameters with Global Galaxy Properties

Next, we discuss the relationships between the pattern
speeds, corotation radii, and bar parameters in the context of
global galaxy parameters (Figure 12). Even given our small
number of galaxies, we find some significant correlations here.
For galaxies with a later Hubble type, we find a larger *R*
(driven by shorter bar lengths). This is in agreement with, for
example, Erwin (2005), so this finding is not surprising. We
also find that galaxies with a longer disk scale length tend to
have a lower *R*. Given that we expect galaxies with higher
scale lengths to have larger R<sub>CR</sub> (Kranz et al. 2003) but not
necessarily larger bars, this naturally follows. More molecular
gas-rich systems tend to have a higher pattern speed. This has
been observed in simulations by Ghosh & Jog (2016) and also
in some observational studies (Aguerri et al. 2015; Guo et al.
2019; Garma-Oehmichen et al. 2020). Galaxies that have larger
bulges tend to have somewhat lower pattern speeds. This is
seen in simulations (Kataria & Das 2019), and our results
appear to confirm that. Finally, unlike Cuomo et al. (2020), we
do not find a significant correlation between Ω<sub>P</sub> and the stellar
mass surface density.
<figure><img src="image_15.png" /><figcaption>**Figure 13.** Corner plot of the y-axis quantities from Figures 11 and 12. Galaxies without bars are indicated as squares (i.e., NGC 0628); galaxies with bars are circles. In each subplot, the number of points *N* and the Kendall τ correlation are given, along with its associated uncertainty. We observe a weak anticorrelation between corotation radius and pattern speed, while all other quantities appear uncorrelated.</figcaption></figure>

## 8.2. Relationships between Pattern Speeds, Corotation Radii, and Bar Parameters

Finally, we turn to the interrelation between our derived parameters and various bar parameters (Figure 13). We find only one significant correlation: galaxies with higher pattern speeds tend to have smaller corotation radii. Despite this, our sample of galaxies has a roughly constant *R*. Although, it is also informative to look at parameters where we find no significant correlations; as in Cuomo et al. (2020), we find no correlation between *R* and $R_{bar}$. Their galaxy sample tends to focus on earlier-type galaxies than the PHANGS sample does, so this may be a trend throughout the entire galaxy population. Unlike previous works (e.g., Erwin 2005; Díaz-García et al. 2016; Kruk et al. 2018; Guo et al. 2019), we find no trend between bar length and bar strength. We also find no correlation between *R* and $S_{bar}$, much like in Cuomo et al. (2020). Finally, there is a theoretical prediction that there should be an anticorrelation between $S_{bar}$ and $\Omega_{P}$ (e.g., Debattista & Sellwood 2000; Athanassoula 2003). As a bar loses angular momentum, it slows down and hence the pattern speed is reduced. However, we do not see this in our sample. This may be due to low number statistics, or the variety of initial conditions in these galaxies washing out any correlation we may see between the bar strength and the pattern speed. It

may be useful to study this in the context of more similar galaxies (e.g., age, size, morphology), but our sample has insufficient numbers to allow for this sort of binning.

## 9. Conclusions

In this work, we have applied the Tremaine–Weinberg method to stellar mass maps obtained as part of the PHANGS-MUSE survey, to measure pattern speeds for a sample of galaxies. This work improves on previous studies in a number of ways. These pattern speeds are calculated homogeneously, with a consistent methodology on a consistent kinematic tracer. We rigorously account for various uncertainties in our measurements of $\Omega_{P}$, and we perform a number of tests that allow us to determine whether these pattern speeds are reliable. We find that of our sample of 19 galaxies, 10 have well-measured pattern speeds. For the nine galaxies that do not have well-measured pattern speeds, in two we detect no evidence of a pattern (IC 5332 and NGC 5068), and for the others we find the velocity field to be extremely messy, meaning the galaxy position angle changes strongly with radius.

Leveraging $H\alpha$ maps from PHANGS-MUSE, as well as CO maps from PHANGS-ALMA, we also critically examine the use of clumpy ISM tracers in determining pattern speeds. We find that the incomplete coverage can lead to a false signal in
Tremaine–Weinberg integrals (which we refer to as $\Omega_{\text{clump}}$), and that this $\Omega_{\text{clump}}$ is systematically different from $\Omega_{\text{P}}$ (~40% higher for CO, ~20% higher for H$\alpha$). An important result from this paper, therefore, is that these tracers produce a compromised measure of the pattern speed. We thus view with caution the reliability of pattern speeds in the literature derived from ISM tracers.

With these pattern speeds, combined with measured CO-based rotation curves for PHANGS galaxies by Lang et al. (2020), we calculate a number of resonances for the sample: the corotation radius, and the outer and inner Lindblad radii. The full list of pattern speeds, quality flags, and resonance locations is given in Table 2, along with our adopted orientation parameters. We have compared the corotation radius both to the bar length and disk scale length, and we find that these ratios are consistent with commonly assumed literature values of 1.2 and 3, respectively. Given the spread in our values, we would suggest taking an uncertainty in these ratios of ~30% and ~15%, respectively.

We have investigated how our derived parameters depend on a number of global galaxy parameters. We confirm a number of previous findings in our work: that later-type galaxies have higher pattern speeds and larger values of $R$, that more molecular gas-dominated galaxies have higher pattern speeds, and that more bulge-dominated galaxies have lower pattern speeds. However, we also find an absence of correlations where we may expect them. In particular, we find no correlation between bar strength and pattern speed, nor between total stellar mass surface density and pattern speed. With larger number statistics, the small correlations we see here may become more significant, and future work will be able to rule out whether these correlations truly are absent or are simply due to small number statistics.

There are a number of future studies that follow on naturally from this work. First, it is important to note that the Tremaine–Weinberg method can only recover a single pattern speed in a galaxy. We may expect different morphological features to have different pattern speeds, which we are unable to recover in this work. We have flagged those we believe to have strong signals of multiple pattern speeds. Applying a method that allows for radial variation in the pattern speed has been beyond the scope of this work, but it will be the focus of future work. Second, our work has been applied to data with cloud-scale resolution (1", corresponding to ~100 pc). We have performed tests for a number of slit widths ranging from ~100 pc to ~1 kpc, and we find that the width of the slit has little effect on the pattern speed measurement. Thus, it should be possible to extend this study to much larger samples of galaxies with surveys such as MaNGA, which have kiloparsec resolution. Given that we are only confident in pattern speeds measured using stellar masses and velocities, this work is hindered by low number statistics. The possibility of having reliable pattern speeds for a factor of 100 more galaxies will present a significant increase in statistical power for studying the links between the properties of galaxies and pattern speeds.

This work has been carried out as part of the PHANGS collaboration.

The authors would like to thank the anonymous peer reviewer for their constructive comments that have improved the quality of the paper. T.G.W. would also like to personally thank P. Woolford for discussions and support during the

preparation of this manuscript. T.G.W., E.S., H.-A.P., T.S., and F.S. acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 694343). I.B. and F.B. acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 726384/Empire). S.C.O.G., R.S.K., M.C.S., and E.J.W. acknowledge financial support from the German Research Foundation (DFG) via the Collaborative Research Center (SFB 881, Project-ID 138713538) "The Milky Way System" (subprojects A1, B1, B2, B8, and P2). S.C.O.G. and R.S.K. also acknowledge financial support from the Heidelberg Cluster of Excellence STRUCTURES in the framework of Germany's Excellence Strategy (grant EXC-2181/1-390900948), and from the ERC via the ERC Synergy Grant ECOGAL (grant 855130). J.M.D.K. and M.C. gratefully acknowledge funding from the Deutsche Forschungs-gemeinschaft (DFG) through an Emmy Noether Research Group, grant No. KR4801/1-1, and the DFG Sachbeihilfe, grant No. KR4801/2-1. J.M.D.K. gratefully acknowledges funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program via the ERC Starting Grant MUSTANG (grant agreement No. 714907). The work of A.K.L. and J.S. is partially supported by the National Science Foundation under grant Nos. 1615105, 1615109, and 1653300. E.R. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (NSERC), funding reference number RGPIN-2017-03987.

This paper makes use of the following ALMA data, which have been processed as part of the PHANGS-ALMA survey: ADS/JAO.ALMA#2012.1.00650.S, ADS/JAO.ALMA#2013.1.00803.S, ADS/JAO.ALMA#2013.1.01161.S, ADS/JAO.ALMA#2015.1.00121.S, ADS/JAO.ALMA#2015.1.00782.S, ADS/JAO.ALMA#2015.1.00925.S, ADS/JAO.ALMA#2015.1.00956.S, ADS/JAO.ALMA#2016.1.00386.S, ADS/JAO.ALMA#2017.1.00392.S, ADS/JAO.ALMA#2017.1.00766.S, ADS/JAO.ALMA#2017.1.00886.L, ADS/JAO.ALMA#2018.1.01321.S, ADS/JAO.ALMA#2018.1.01651.S, ADS/JAO.ALMA#2018.1.00062.S. ALMA is a partnership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.

This work is based on observations collected at the European Southern Observatory under ESO programs 1100.B-0651, 095. C-0473, and 094.C-0623.

This work has made use of NumPy (Oliphant 2006), AstroPy (Astropy Collaboration et al. 2013; Price-Whelan et al. 2018), SciPy (Virtanen et al. 2020), Matplotlib (Hunter 2007), and Seaborn (Waskom et al. 2017). This work has also made use of the HyperLeda<fnref n="22" /> database.

## Appendix

### Derived Pattern Speeds and Resonance Locations

Table 2 contains $\Omega_{\text{p}}$ and $\Omega_{\text{clump}}$ values, along with resonance locations, for the full PHANGS sample considered in this work.

<footnote n="22">²² http://leda.univ-lyon1.fr</footnote>
<table><caption>Table 2
Pattern Speeds (Ω<sub>p</sub> for Stellar Mass, Ω<sub>clump</sub> for ISM Tracers) and Inferred Resonance Locations for 83 Galaxies</caption><thead><tr><th>Galaxy</th><th>PGC</th><th>D<br>Mpc</th><th>i<br>°</th><th>PA<br>°</th><th>Bar?</th><th>Ω<sub>clump, A</sub><br>kpc s</th><th>Q<sub>A</sub></th><th>R<sub>CR1, A</sub><br>kpc</th><th>Ω<sub>P, MM</sub><br>kpc s</th><th>Q<sub>MM</sub></th><th>R<sub>CR1, MM</sub><br>kpc</th><th>Ω<sub>clump, MHα</sub><br>kpc s</th><th>Q<sub>MHα</sub></th><th>R<sub>CR1, MHα</sub><br>kpc</th></tr></thead><tbody><tr><td>ES 0097-013</td><td>50779</td><td>4.2</td><td>64.3</td><td>36.74</td><td>0</td><td>-12.8<sup>+68.7</sup><sub>-124.8</sub></td><td>3</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>IC 1954</td><td>13090</td><td>12.0</td><td>57.1</td><td>63.4</td><td>1</td><td>69.3<sup>+1.4</sup><sub>-1.8</sub></td><td>4</td><td>0.1 ± 0.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>IC 5273</td><td>70184</td><td>14.18</td><td>52.0</td><td>234.1</td><td>1</td><td>40.6<sup>+2.8</sup><sub>-1.9</sub></td><td>1</td><td>2.9 ± 0.8</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>IC 5332</td><td>71775</td><td>8.18</td><td>26.9</td><td>74.4</td><td>0</td><td>...</td><td>...</td><td>...</td><td>-9.2<sup>+16.5</sup><sub>-4.4</sub></td><td>3</td><td>...</td><td>31.4<sup>+1.2</sup><sub>-1.0</sub></td><td>3</td><td>...</td></tr><tr><td>NGC 0253</td><td>2789</td><td>3.7</td><td>75.0</td><td>52.48</td><td>0</td><td>51.4<sup>+32.0</sup><sub>-7.1</sub></td><td>3</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 0300</td><td>3238</td><td>2.09</td><td>39.8</td><td>114.3</td><td>0</td><td>31.6<sup>+4.7</sup><sub>-4.2</sub></td><td>3</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 0628</td><td>5974</td><td>9.84</td><td>8.9</td><td>20.7</td><td>0</td><td>32.9<sup>+0.2</sup><sub>-0.3</sub></td><td>1</td><td>4.4 ± 1.0</td><td>31.1<sup>+4.0</sup><sub>-2.9</sub></td><td>1</td><td>4.5 ± 2.0</td><td>35.4<sup>+0.4</sup><sub>-0.5</sub></td><td>1</td><td>3.8 ± 1.2</td></tr><tr><td>NGC 0685</td><td>6581</td><td>19.94</td><td>23.0</td><td>100.9</td><td>1</td><td>36.7<sup>+2.9</sup><sub>-2.3</sub></td><td>1</td><td>0.1 ± 0.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 1087</td><td>10496</td><td>15.85</td><td>42.9</td><td>359.1</td><td>1</td><td>46.7<sup>+0.6</sup><sub>-0.8</sub></td><td>1</td><td>3.4 ± 0.7</td><td>31.9<sup>+3.2</sup><sub>-1.6</sub></td><td>1</td><td>4.3 ± 1.1</td><td>49.7<sup>+6.9</sup><sub>-7.4</sub></td><td>1</td><td>3.7 ± 1.7</td></tr><tr><td>NGC 1097</td><td>10488</td><td>13.58</td><td>48.6</td><td>122.4</td><td>1</td><td>39.9<sup>+2.6</sup><sub>-0.8</sub></td><td>2</td><td>6.4 ± 2.0</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 1300</td><td>12412</td><td>18.99</td><td>31.8</td><td>278.0</td><td>1</td><td>24.9<sup>+1.7</sup><sub>-0.2</sub></td><td>1</td><td>6.1 ± 2.5</td><td>18.9<sup>+9.4</sup><sub>-8.9</sub></td><td>3</td><td>0.1 ± 0.1</td><td>25.1<sup>+0.7</sup><sub>-0.6</sub></td><td>1</td><td>6.1 ± 2.5</td></tr><tr><td>NGC 1317</td><td>12653</td><td>19.11</td><td>23.2</td><td>221.5</td><td>1</td><td>94.6<sup>+43.7</sup><sub>-14.9</sub></td><td>3</td><td>0.2 ± 0.2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 1365</td><td>13179</td><td>19.57</td><td>55.4</td><td>201.1</td><td>1</td><td>109.1<sup>+22.0</sup><sub>-36.8</sub></td><td>3</td><td>2.3 ± 1.6</td><td>38.1<sup>+20.1</sup><sub>-21.2</sub></td><td>3</td><td>2.8 ± 1.1</td><td>26.8<sup>+5.6</sup><sub>-20.7</sub></td><td>2</td><td>...</td></tr><tr><td>NGC 1385</td><td>13368</td><td>17.22</td><td>44.0</td><td>181.3</td><td>0</td><td>27.4<sup>+7.1</sup><sub>-7.1</sub></td><td>3</td><td>0.1 ± 0.1</td><td>25.0<sup>+5.6</sup><sub>-19.3</sub></td><td>3</td><td>0.1 ± 0.1</td><td>21.9<sup>+10.4</sup><sub>-5.9</sub></td><td>3</td><td>0.1 ± 0.1</td></tr><tr><td>NGC 1433</td><td>13586</td><td>12.11</td><td>28.6</td><td>199.7</td><td>1</td><td>105.6<sup>+4.6</sup><sub>-8.1</sub></td><td>3</td><td>0.1 ± 0.1</td><td>20.0<sup>+2.4</sup><sub>-1.9</sub></td><td>1</td><td>6.3 ± 0.4</td><td>41.2<sup>+0.7</sup><sub>-0.7</sub></td><td>1</td><td>0.1 ± 0.1</td></tr><tr><td>NGC 1511</td><td>14236</td><td>15.28</td><td>72.7</td><td>297.0</td><td>0</td><td>44.3<sup>+1.6</sup><sub>-1.0</sub></td><td>1</td><td>0.3 ± 0.3</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 1512</td><td>14391</td><td>17.13</td><td>42.5</td><td>261.9</td><td>1</td><td>27.6<sup>+2.3</sup><sub>-2.0</sub></td><td>1</td><td>5.4 ± 2.2</td><td>22.3<sup>+5.1</sup><sub>-5.5</sub></td><td>1</td><td>5.4 ± 2.2</td><td>25.0<sup>+2.3</sup><sub>-2.2</sub></td><td>1</td><td>5.4 ± 2.2</td></tr><tr><td>NGC 1546</td><td>14723</td><td>17.69</td><td>70.3</td><td>147.8</td><td>0</td><td>74.5<sup>+1.8</sup><sub>-2.1</sub></td><td>3</td><td>2.3 ± 0.4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 1559</td><td>14814</td><td>19.44</td><td>65.4</td><td>244.5</td><td>1</td><td>21.2<sup>+3.1</sup><sub>-1.2</sub></td><td>1</td><td>0.2 ± 0.2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 1566</td><td>14897</td><td>17.69</td><td>29.5</td><td>214.7</td><td>1</td><td>58.5<sup>+7.2</sup><sub>-5.7</sub></td><td>1</td><td>4.0 ± 2.5</td><td>29.4<sup>+9.1</sup><sub>-7.2</sub></td><td>3</td><td>6.8 ± 2.7</td><td>64.7<sup>+6.4</sup><sub>-7.1</sub></td><td>1</td><td>3.8 ± 2.3</td></tr><tr><td>NGC 1637</td><td>15821</td><td>11.7</td><td>31.1</td><td>20.61</td><td>1</td><td>90.5<sup>+10.2</sup><sub>-29.5</sub></td><td>3</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 1672</td><td>15941</td><td>19.4</td><td>42.6</td><td>134.3</td><td>1</td><td>32.5<sup>+2.0</sup><sub>-1.3</sub></td><td>2</td><td>3.3 ± 1.0</td><td>22.7<sup>+0.6</sup><sub>-0.6</sub></td><td>1</td><td>6.1 ± 1.6</td><td>28.1<sup>+0.1</sup><sub>-0.1</sub></td><td>1</td><td>4.0 ± 0.5</td></tr><tr><td>NGC 1792</td><td>16709</td><td>16.2</td><td>65.1</td><td>318.9</td><td>0</td><td>57.7<sup>+2.8</sup><sub>-4.2</sub></td><td>1</td><td>1.9 ± 0.5</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 1809</td><td>16599</td><td>19.95</td><td>57.6</td><td>138.2</td><td>0</td><td>47.4<sup>+3.5</sup><sub>-6.1</sub></td><td>4</td><td>1.4 ± 1.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 2090</td><td>17819</td><td>11.75</td><td>64.5</td><td>192.46</td><td>0</td><td>58.1<sup>+4.0</sup><sub>-4.6</sub></td><td>1</td><td>0.1 ± 0.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 2283</td><td>19562</td><td>13.68</td><td>43.7</td><td>-4.1</td><td>1</td><td>41.0<sup>+1.9</sup><sub>-3.1</sub></td><td>1</td><td>0.1 ± 0.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 2566</td><td>23303</td><td>23.44</td><td>48.5</td><td>312.0</td><td>1</td><td>34.6<sup>+2.0</sup><sub>-4.8</sub></td><td>1</td><td>3.8 ± 2.2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 2775</td><td>25861</td><td>23.15</td><td>41.2</td><td>156.5</td><td>0</td><td>54.9<sup>+0.1</sup><sub>-0.1</sub></td><td>1</td><td>5.6 ± 0.2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 2835</td><td>26259</td><td>12.38</td><td>41.3</td><td>1.0</td><td>1</td><td>41.3<sup>+0.3</sup><sub>-0.8</sub></td><td>1</td><td>2.8 ± 0.6</td><td>34.6<sup>+4.3</sup><sub>-3.1</sub></td><td>1</td><td>2.8 ± 0.6</td><td>37.6<sup>+0.4</sup><sub>-0.4</sub></td><td>1</td><td>3.2 ± 0.2</td></tr><tr><td>NGC 2903</td><td>27077</td><td>10.0</td><td>66.8</td><td>203.7</td><td>1</td><td>25.0<sup>+1.8</sup><sub>-0.5</sub></td><td>1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 2997</td><td>27978</td><td>14.06</td><td>33.0</td><td>108.1</td><td>1</td><td>36.0<sup>+0.7</sup><sub>-0.3</sub></td><td>4</td><td>7.2 ± 3.8</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr></tbody></table>
<table><caption>Table 2
(Continued)</caption><thead><tr><th>Galaxy</th><th>PGC</th><th>D<br>Mpc</th><th>i<br>°</th><th>PA<br>°</th><th>Bar?</th><th>Ω<sub>clump, A</sub><br>km<sup>2</sup><br>kpc s</th><th>Q<sub>A</sub></th><th>R<sub>CR1, A</sub><br>kpc</th><th>Ω<sub>P, MM</sub><br>km<sup>2</sup><br>kpc s</th><th>Q<sub>MM</sub></th><th>R<sub>CR1, MM</sub><br>kpc</th><th>Ω<sub>clump, MHα</sub><br>km<sup>2</sup><br>kpc s</th><th>Q<sub>MHα</sub></th><th>R<sub>CR1, MHα</sub><br>kpc</th></tr></thead><tbody><tr><td>NGC 3059</td><td>28298</td><td>20.23</td><td>29.4</td><td>-14.8</td><td>1</td><td>36.8<sup>+5.2</sup><sub>-7.8</sub></td><td>1</td><td>0.1 ± 0.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 3137</td><td>29530</td><td>16.37</td><td>70.3</td><td>-0.3</td><td>0</td><td>21.6<sup>+0.6</sup><sub>-0.7</sub></td><td>1</td><td>5.0 ± 0.5</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 3351</td><td>32007</td><td>9.96</td><td>45.1</td><td>193.2</td><td>1</td><td>60.5<sup>+7.2</sup><sub>-12.6</sub></td><td>1</td><td>2.9 ± 1.2</td><td>43.6<sup>+11.6</sup><sub>-12.4</sub></td><td>1</td><td>3.4 ± 0.8</td><td>89.7<sup>+12.6</sup><sub>-16.3</sub></td><td>3</td><td>2.5 ± 1.7</td></tr><tr><td>NGC 3489</td><td>33160</td><td>11.86</td><td>63.68</td><td>70.0</td><td>0</td><td>193.9<sup>+78.9</sup><sub>-56.8</sub></td><td>4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 3507</td><td>33390</td><td>23.55</td><td>21.7</td><td>55.8</td><td>1</td><td>33.9<sup>+2.3</sup><sub>-1.6</sub></td><td>1</td><td>0.1 ± 0.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 3511</td><td>33385</td><td>13.94</td><td>75.1</td><td>256.8</td><td>1</td><td>22.6<sup>+1.8</sup><sub>-0.9</sub></td><td>1</td><td>5.7 ± 1.5</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 3521</td><td>33550</td><td>13.24</td><td>68.8</td><td>343.0</td><td>0</td><td>35.7<sup>+0.4</sup><sub>-0.7</sub></td><td>1</td><td>6.4 ± 0.4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 3596</td><td>34298</td><td>11.0</td><td>25.1</td><td>78.4</td><td>0</td><td>67.8<sup>+0.6</sup><sub>-1.1</sub></td><td>1</td><td>1.9 ± 1.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 3599</td><td>34326</td><td>19.86</td><td>23.0</td><td>41.9</td><td>0</td><td>55.6<sup>+16.0</sup><sub>-13.0</sub></td><td>4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 3621</td><td>34554</td><td>7.06</td><td>65.8</td><td>343.8</td><td>0</td><td>36.3<sup>+0.4</sup><sub>-0.4</sub></td><td>2</td><td>3.4 ± 0.2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 3626</td><td>34684</td><td>20.05</td><td>46.6</td><td>165.2</td><td>1</td><td>170.2<sup>+11.6</sup><sub>-4.1</sub></td><td>1</td><td>1.5 ± 0.4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 3627</td><td>34695</td><td>11.32</td><td>57.3</td><td>173.1</td><td>1</td><td>45.8<sup>+1.0</sup><sub>-2.5</sub></td><td>1</td><td>4.7 ± 0.5</td><td>29.1<sup>+20.6</sup><sub>-8.1</sub></td><td>2</td><td>1.9 ± 0.1</td><td>50.7<sup>+5.6</sup><sub>-6.4</sub></td><td>1</td><td>4.2 ± 1.4</td></tr><tr><td>NGC 4207</td><td>39206</td><td>15.78</td><td>64.5</td><td>121.9</td><td>0</td><td>64.6<sup>+4.6</sup><sub>-14.8</sub></td><td>4</td><td>1.2 ± 1.2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4254</td><td>39578</td><td>13.0</td><td>34.4</td><td>68.1</td><td>0</td><td>47.3<sup>+1.8</sup><sub>-1.7</sub></td><td>1</td><td>3.4 ± 0.3</td><td>52.4<sup>+6.2</sup><sub>-4.5</sub></td><td>3</td><td>3.1 ± 0.6</td><td>36.1<sup>+0.5</sup><sub>-0.3</sub></td><td>3</td><td>4.4 ± 0.6</td></tr><tr><td>NGC 4293</td><td>39907</td><td>15.76</td><td>65.0</td><td>48.3</td><td>1</td><td>185.2<sup>+6.3</sup><sub>-4.8</sub></td><td>4</td><td>0.2 ± 0.2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4298</td><td>39950</td><td>13.0</td><td>59.2</td><td>313.9</td><td>0</td><td>27.2<sup>+0.7</sup><sub>-0.2</sub></td><td>1</td><td>4.8 ± 0.6</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4303</td><td>40001</td><td>16.99</td><td>23.5</td><td>312.4</td><td>1</td><td>46.7<sup>+2.2</sup><sub>-8.2</sub></td><td>1</td><td>4.0 ± 2.3</td><td>43.5<sup>+5.3</sup><sub>-10.0</sub></td><td>1</td><td>3.8 ± 2.5</td><td>48.4<sup>+2.5</sup><sub>-6.7</sub></td><td>1</td><td>3.9 ± 2.4</td></tr><tr><td>NGC 4321</td><td>40153</td><td>15.21</td><td>38.5</td><td>156.2</td><td>1</td><td>26.6<sup>+5.7</sup><sub>-9.6</sub></td><td>3</td><td>6.3 ± 2.3</td><td>43.4<sup>+3.1</sup><sub>-9.1</sub></td><td>4</td><td>5.6 ± 3.0</td><td>34.7<sup>+8.1</sup><sub>-3.4</sub></td><td>3</td><td>5.1 ± 2.6</td></tr><tr><td>NGC 4424</td><td>40809</td><td>16.2</td><td>58.2</td><td>88.3</td><td>0</td><td>17.2<sup>+1.4</sup><sub>-2.2</sub></td><td>4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4457</td><td>41101</td><td>15.0</td><td>17.4</td><td>78.7</td><td>0</td><td>122.4<sup>+1.0</sup><sub>-0.4</sub></td><td>3</td><td>0.1 ± 0.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4459</td><td>41104</td><td>15.85</td><td>46.95</td><td>108.75</td><td>0</td><td>89.1<sup>+366.8</sup><sub>-40.6</sub></td><td>3</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4476</td><td>41255</td><td>17.54</td><td>60.14</td><td>27.38</td><td>0</td><td>174.6<sup>+33.9</sup><sub>-25.1</sub></td><td>4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4477</td><td>41260</td><td>15.76</td><td>33.51</td><td>25.68</td><td>0</td><td>1440.3<sup>+43.92</sup><sub>-623.3</sub></td><td>4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4496A</td><td>41471</td><td>14.86</td><td>53.8</td><td>51.1</td><td>1</td><td>23.9<sup>+2.3</sup><sub>-1.2</sub></td><td>1</td><td>1.8 ± 1.8</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4535</td><td>41812</td><td>15.77</td><td>44.7</td><td>179.7</td><td>1</td><td>30.9<sup>+0.5</sup><sub>-0.3</sub></td><td>1</td><td>5.4 ± 2.8</td><td>21.0<sup>+7.7</sup><sub>-1.3</sub></td><td>3</td><td>5.4 ± 3.0</td><td>30.7<sup>+0.9</sup><sub>-0.3</sub></td><td>1</td><td>5.4 ± 2.8</td></tr><tr><td>NGC 4536</td><td>41823</td><td>16.25</td><td>66.0</td><td>305.6</td><td>1</td><td>17.9<sup>+2.6</sup><sub>-4.1</sub></td><td>1</td><td>9.0 ± 2.2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4540</td><td>41876</td><td>15.76</td><td>28.7</td><td>12.8</td><td>1</td><td>55.6<sup>+11.9</sup><sub>-9.0</sub></td><td>1</td><td>1.5 ± 1.5</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4548</td><td>41934</td><td>16.22</td><td>38.3</td><td>138.0</td><td>1</td><td>26.2<sup>+2.7</sup><sub>-2.5</sub></td><td>2</td><td>5.7 ± 0.7</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4569</td><td>42089</td><td>15.76</td><td>70.0</td><td>18.0</td><td>1</td><td>31.9<sup>+10.5</sup><sub>-11.9</sub></td><td>1</td><td>0.1 ± 0.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4571</td><td>42100</td><td>14.0</td><td>32.7</td><td>217.5</td><td>0</td><td>30.8<sup>+1.2</sup><sub>-1.0</sub></td><td>1</td><td>4.1 ± 0.7</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4579</td><td>42168</td><td>21.0</td><td>40.22</td><td>91.3</td><td>1</td><td>24.0<sup>+5.5</sup><sub>-6.5</sub></td><td>2</td><td>8.2 ± 1.8</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr></tbody></table>
<table><caption>**Table 2**
(Continued)</caption><thead><tr><th>Galaxy</th><th>PGC</th><th>D<br>Mpc</th><th>i<br>°</th><th>PA<br>°</th><th>Bar?</th><th>Ω<sub>clump, A</sub><br>kpc s</th><th>Q<sub>A</sub></th><th>R<sub>CR1, A</sub><br>kpc</th><th>Ω<sub>P, MM</sub><br>kpc s</th><th>Q<sub>MM</sub></th><th>R<sub>CR1, MM</sub><br>kpc</th><th>Ω<sub>clump, MHα</sub><br>kpc s</th><th>Q<sub>MHα</sub></th><th>R<sub>CR1, MHα</sub><br>kpc</th></tr></thead><tbody><tr><td>NGC 4596</td><td>42401</td><td>15.76</td><td>36.56</td><td>120.0</td><td>0</td><td>1290.0<sup>+604.7</sup><sub>-754.1</sub></td><td>4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4654</td><td>42857</td><td>21.98</td><td>55.6</td><td>123.2</td><td>1</td><td>19.0<sup>+0.8</sup><sub>-0.2</sub></td><td>1</td><td>8.3 ± 1.8</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4689</td><td>43186</td><td>15.0</td><td>38.7</td><td>164.1</td><td>0</td><td>35.1<sup>+0.7</sup><sub>-0.8</sub></td><td>1</td><td>3.8 ± 0.5</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4694</td><td>43241</td><td>15.76</td><td>60.7</td><td>143.3</td><td>0</td><td>95.9<sup>+12.0</sup><sub>-7.3</sub></td><td>4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4731</td><td>43507</td><td>13.28</td><td>64.0</td><td>255.4</td><td>1</td><td>22.7<sup>+5.0</sup><sub>-10.6</sub></td><td>4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4781</td><td>43902</td><td>11.31</td><td>59.0</td><td>290.0</td><td>1</td><td>32.8<sup>+0.9</sup><sub>-0.8</sub></td><td>1</td><td>3.9 ± 0.7</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4826</td><td>44182</td><td>4.41</td><td>59.1</td><td>293.6</td><td>0</td><td>236.9<sup>+48.6</sup><sub>-26.0</sub></td><td>3</td><td>0.8 ± 0.3</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4941</td><td>45165</td><td>15.0</td><td>53.4</td><td>202.2</td><td>1</td><td>46.2<sup>+0.2</sup><sub>-0.2</sub></td><td>1</td><td>3.7 ± 0.2</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 4951</td><td>45246</td><td>15.0</td><td>70.2</td><td>91.2</td><td>0</td><td>64.0<sup>+1.6</sup><sub>-1.3</sub></td><td>3</td><td>0.1 ± 0.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 5042</td><td>46126</td><td>16.78</td><td>49.4</td><td>190.6</td><td>0</td><td>28.1<sup>+0.5</sup><sub>-0.5</sub></td><td>1</td><td>4.2 ± 0.9</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 5068</td><td>46400</td><td>5.16</td><td>35.7</td><td>342.4</td><td>1</td><td>18.5<sup>+0.3</sup><sub>-3.3</sub></td><td>1</td><td>0.2 ± 0.2</td><td>16.3<sup>+2.0</sup><sub>-9.4</sub></td><td>3</td><td>1.1 ± 1.1</td><td>15.8<sup>+4.5</sup><sub>-2.0</sub></td><td>1</td><td>0.2 ± 0.2</td></tr><tr><td>NGC 5128</td><td>46957</td><td>3.69</td><td>45.33</td><td>32.17</td><td>0</td><td>644.1<sup>+376.2</sup><sub>-1416.9</sub></td><td>3</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 5134</td><td>46938</td><td>19.92</td><td>22.7</td><td>311.6</td><td>1</td><td>31.1<sup>+4.0</sup><sub>-1.3</sub></td><td>1</td><td>0.3 ± 0.3</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 5248</td><td>48130</td><td>14.87</td><td>47.4</td><td>109.2</td><td>1</td><td>48.0<sup>+6.6</sup><sub>-10.8</sub></td><td>3</td><td>4.7 ± 2.4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 5530</td><td>51106</td><td>12.27</td><td>61.9</td><td>305.4</td><td>0</td><td>32.8<sup>+0.8</sup><sub>-0.3</sub></td><td>1</td><td>0.1 ± 0.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 5643</td><td>51969</td><td>12.68</td><td>29.9</td><td>318.7</td><td>1</td><td>54.7<sup>+5.6</sup><sub>-5.3</sub></td><td>1</td><td>0.1 ± 0.1</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 6300</td><td>60001</td><td>11.58</td><td>49.6</td><td>105.4</td><td>1</td><td>41.7<sup>+4.1</sup><sub>-4.8</sub></td><td>3</td><td>4.4 ± 1.3</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 6744</td><td>62836</td><td>9.39</td><td>52.7</td><td>14.0</td><td>1</td><td>27.7<sup>+0.0</sup><sub>-0.1</sub></td><td>1</td><td>6.1 ± 0.4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 7456</td><td>70304</td><td>15.7</td><td>67.3</td><td>16.0</td><td>0</td><td>23.9<sup>+0.6</sup><sub>-0.8</sub></td><td>1</td><td>4.0 ± 0.9</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>NGC 7496</td><td>70588</td><td>18.72</td><td>35.9</td><td>193.7</td><td>1</td><td>24.4<sup>+3.1</sup><sub>-3.0</sub></td><td>1</td><td>4.3 ± 2.0</td><td>16.8<sup>+6.5</sup><sub>-12.1</sub></td><td>1</td><td>4.3 ± 2.0</td><td>23.0<sup>+7.5</sup><sub>-4.3</sub></td><td>3</td><td>4.0 ± 2.2</td></tr><tr><td>NGC 7743</td><td>72263</td><td>20.32</td><td>37.1</td><td>86.24</td><td>0</td><td>-13.9<sup>+31.0</sup><sub>-17.8</sub></td><td>4</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr></tbody></table>

Note. For each galaxy, we list its NGC number, PGC number, distance, inclination, position angle, and whether it has a bar (1 for yes, 0 for no). We use a subscript to refer to the tracer in question: MM for MUSE stellar mass, MHα for MUSE-Hα measurements, and A for ALMA CO measurements. We also show the quality flags (indicated as Q), corotation radii (R<sub>CR</sub>), and outer/inner Lindblad radii (R<sub>O/ILR</sub>). For brevity, only the first corotation radius is shown, and Lindblad radii are hidden.
ORCID iDs

Thomas G. Williams https://orcid.org/0000-0002-0012-2142

Eva Schinnerer https://orcid.org/0000-0002-3933-7677

Eric Emsellem https://orcid.org/0000-0002-6155-7166

Sharon Meidt https://orcid.org/0000-0002-6118-4048

Miguel Querejeta https://orcid.org/0000-0002-0472-1011

Francesco Belfiore https://orcid.org/0000-0002-2545-5752

Frank Bigiel https://orcid.org/0000-0003-0166-9745

Mélanie Chevance https://orcid.org/0000-0002-5635-5180

Daniel A. Dale https://orcid.org/0000-0002-5782-9093

Simon C. O. Glover https://orcid.org/0000-0001-6708-1317

Kathryn Grasha https://orcid.org/0000-0002-3247-5321

Ralf S. Klessen https://orcid.org/0000-0002-0560-3172

J. M. Diederik Kruijssen https://orcid.org/0000-0002-8804-0212

Adam K. Leroy https://orcid.org/0000-0002-2545-1700

Hsi-An Pan https://orcid.org/0000-0002-1370-6964

Jérôme Pety https://orcid.org/0000-0003-3061-6546

Erik Rosolowsky https://orcid.org/0000-0002-5204-2259

Toshiki Saito https://orcid.org/0000-0002-2501-9328

Francesco Santoro https://orcid.org/0000-0002-6363-9851

Mattia C. Sormani https://orcid.org/0000-0001-6113-6241

Jiayi Sun https://orcid.org/0000-0003-0378-4667

Elizabeth J. Watkins https://orcid.org/0000-0002-7365-5791

References

Aguerri, J. A. L., Beckman, J. E., & Prieto, M. 1998, AJ, 116, 2136

Aguerri, J. A. L., Debattista, V. P., & Corsini, E. M. 2003, MNRAS, 338, 465

Aguerri, J. A. L., Méndez-Abreu, J., Falcón-Barroso, J., et al. 2015, A&A, 576, A102

Anand, G. S., Lee, J. C., Van Dyk, S. D., et al. 2021, MNRAS, 501, 3621

Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33

Athanassoula, E. 1992, MNRAS, 259, 345

Athanassoula, E. 2003, MNRAS, 341, 1179

Beckman, J. E., Font, J., Borlaff, A., & García-Lorenzo, B. 2018, ApJ, 854, 182

Beuther, H., Meidt, S., Schinnerer, E., Paladino, R., & Leroy, A. 2017, A&A, 597, A85

Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition (2nd ed.; Princeton, NJ: Princeton Univ. Press)

Bournaud, F., Combes, F., & Semelin, B. 2005, MNRAS, 364, L18

Bundy, K., Bershady, M. A., Law, D. R., et al. 2015, ApJ, 798, 7

Cappellari, M. 2017, MNRAS, 466, 798

Cappellari, M., & Emsellem, E. 2004, PASP, 116, 138

Chevance, M., Kruijssen, J. M. D., Hygate, A. P. S., et al. 2020, MNRAS, 493, 2872

Contopoulos, G. 1980, A&A, 81, 198

Corsini, E. M., Aguerri, J. A. L., Debattista, V. P., et al. 2007, ApJL, 659, L121

Corsini, E. M., Debattista, V. P., & Aguerri, J. A. L. 2003, ApJL, 599, L29

Cuomo, V., Aguerri, J. A. L., Corsini, E. M., & Debattista, V. P. 2020, A&A, 641, A111

Debattista, V. P. 2003, MNRAS, 342, 1194

Debattista, V. P., Gerhard, O., & Sevenster, M. N. 2002, MNRAS, 334, 355

Debattista, V. P., & Sellwood, J. A. 2000, ApJ, 543, 704

Debattista, V. P., & Williams, T. B. 2004, ApJ, 605, 714

Díaz, R., Carranza, G., Dottori, H., & Goldes, G. 1999, ApJ, 512, 623

Díaz-García, S., Salo, H., & Laurikainen, E. 2016, A&A, 596, A84

Dobbs, C., & Baba, J. 2014, PASA, 31, e035

Egusa, F., Sofue, Y., & Nakanishi, H. 2004, PASJ, 56, L45

Elmegreen, B. G., Elmegreen, D. M., Chromey, F. R., Hasselbacher, D. A., & Bissell, B. A. 1996, AJ, 111, 2233

Elmegreen, B. G., Elmegreen, D. M., & Seiden, P. E. 1989, ApJ, 343, 602

Emsellem, E., Fathi, K., Wozniak, H., et al. 2006, MNRAS, 365, 367

Erwin, P. 2005, MNRAS, 364, 283

Fathi, K., Beckman, J. E., Piñol-Ferrer, N., et al. 2009, ApJ, 704, 1657

Font, J., Beckman, J. E., Martínez-Valpuesta, I., et al. 2017, ApJ, 835, 279

Fragkoudi, F., Katz, D., Trick, W., et al. 2019, MNRAS, 488, 3324

Garcia-Burillo, S., Combes, F., & Gerin, M. 1993, A&A, 274, 148

Garma-Oehmichen, L., Cano-Díaz, M., Hernández-Toledo, H., et al. 2020, MNRAS, 491, 3655

Gerssen, J., Kuijken, K., & Merrifield, M. R. 1999, MNRAS, 306, 926

Ghosh, S., & Jog, C. J. 2016, MNRAS, 459, 4057

Grand, R. J. J., Kawata, D., & Cropper, M. 2012, MNRAS, 421, 1529

Guo, R., Mao, S., Athanassoula, E., et al. 2019, MNRAS, 482, 1733

Hernandez, O., Wozniak, H., Carignan, C., et al. 2005, ApJ, 632, 253

Herrera, C. N., Pety, J., Hughes, A., et al. 2020, A&A, 634, A121

Herrera-Endoqui, M., Díaz-García, S., Laurikainen, E., & Salo, H. 2015, A&A, 582, A86

Holm, S. 1979, Scandinavian Journal of Statistics, 6, 65

Hunter, J. D. 2007, CSE, 9, 90

Kataria, S. K., & Das, M. 2019, ApJ, 886, 43

Kendall, M. G. 1938, Biometrika, 30, 81

Knapen, J. H., Beckman, J. E., Cepa, J., & Nakai, N. 1996, A&A, 308, 27

Korchagin, V., Orlova, N., Kikuchi, N., Miyama, S. M., & Moiseev, A. V. 2005, arXiv:astro-ph/0509708

Koribalski, B. S., & López-Sánchez, Á. R. 2009, MNRAS, 400, 1749

Krajnović, D., Cappellari, M., de Zeeuw, P. T., & Copin, Y. 2006, MNRAS, 366, 787

Kranz, T., Slyz, A., & Rix, H.-W. 2003, ApJ, 586, 143

Kreckel, K., Faesi, K., Kruijssen, J. M. D., et al. 2018, ApJL, 863, L21

Kruijssen, J. M. D., Pfeffer, J. L., Crain, R. A., & Bastian, N. 2019, MNRAS, 486, 3134

Kruk, S. J., Lintott, C. J., Bamford, S. P., et al. 2018, MNRAS, 473, 4731

Lang, P., Meidt, S. E., Rosolowsky, E., et al. 2020, ApJ, 897, 122

Leroy, A., Lee, J., Van Dyk, S., et al. 2020a, submitted

Leroy, A., Lee, J., Van Dyk, S., et al. 2020b, submitted

Leroy, A. K., Sandstrom, K. M., Lang, D., et al. 2019, ApJS, 244, 24

Lin, C. C. 1970, in IAU Symp. 38, The Spiral Structure of our Galaxy, ed. W. Becker & G. I. Kontopoulos (Dordrecht: Reidel), 377

Lin, C. C., & Shu, F. H. 1966, PNAS, 55, 229

Lindblad, B. 1963, StoAn, 5, 5

Lindblad, P. A. B., & Kristen, H. 1996, A&A, 313, 733

Makarov, D., Prugniel, P., Terekhova, N., Courtois, H., & Vauglin, I. 2014, A&A, 570, A13

Martínez-García, E. E., & Puerari, I. 2014, ApJ, 790, 118

Meidt, S. E., Rand, R. J., Merrifield, M. R., Debattista, V. P., & Shen, J. 2008a, ApJ, 676, 899

Meidt, S. E., Rand, R. J., Merrifield, M. R., Shetty, R., & Vogel, S. N. 2008b, ApJ, 688, 224

Merrifield, M. R., & Kuijken, K. 1995, MNRAS, 274, 933

Oliphant, T. E. 2006, A Guide to NumPy (Spanish Fork, UT: Trelgol Publishing)

Price-Welan, A. M., Sipőcz, B. M., Günther, H. M., et al. 2018, AJ, 156, 123

Querejeta, M., Meidt, S. E., Schinnerer, E., et al. 2016, A&A, 588, A33

Rand, R. J. 1993, ApJ, 410, 68

Rand, R. J., & Wallin, J. F. 2004, ApJ, 614, 142

Rautiainen, P., & Salo, H. 1999, A&A, 348, 737

Rosado-Belza, D., Falcón-Barroso, J., Knapen, J. H., et al. 2020, A&A, 644, A116

Salo, H., Laurikainen, E., Laine, J., et al. 2015, ApJS, 219, 4

Sanders, J. L., Smith, L., & Evans, N. W. 2019, MNRAS, 488, 4552

Schinnerer, E., Eckart, A., Tacconi, L. J., Genzel, R., & Downes, D. 2000, ApJ, 533, 850

Schinnerer, E., Hughes, A., Leroy, A., et al. 2019, ApJ, 887, 49

Schinnerer, E., Maciejewski, W., Scoville, N., & Moustakas, L. A. 2002, ApJ, 575, 826

Sellwood, J. A. 2013, in Planets, Stars and Stellar Systems, ed. T. Oswalt & G. Gilmore, Vol. 5 (Dordrecht: Springer), 923

Sheth, K., Regan, M., Hinz, J. L., et al. 2010, PASP, 122, 1397

Shetty, S., Bershady, M. A., Westfall, K. B., et al. 2020, ApJ, 901, 101

Silverman, B. W. 1986, Density Estimation for Statistics and Data Analysis (London: Chapman Hall)

Sormani, M. C., Binney, J., & Magorrian, J. 2015, MNRAS, 454, 1818

Speights, J. C., & Rooke, P. C. 2016, ApJ, 826, 2

Tremaine, S., & Weinberg, M. D. 1984, ApJL, 282, L5
Treuthardt, P., Salo, H., Rautiainen, P., & Buta, R. 2008, AJ, 136, 300

Vazdekis, A., Koleva, M., Ricciardelli, E., Röck, B., & Falcón-Barroso, J. 2016, MNRAS, 463, 3409

Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, NatMe, 17, 261

Waskom, M., Botvinnik, O., O'Kane, D., et al. 2017, mwaskom/seaborn: v0.8.1, Zenodo, doi:10.5281/zenodo.883859

Weiner, B. J., Sellwood, J. A., & Williams, T. B. 2001a, ApJ, 546, 931

Weiner, B. J., Williams, T. B., van Gorkom, J. H., & Sellwood, J. A. 2001b, ApJ, 546, 916

Westpfahl, D. J. 1998, ApJS, 115, 203

Wu, Y.-T., Pfenniger, D., & Taam, R. E. 2018, ApJ, 860, 152

Zimmer, P., Rand, R. J., & McGraw, J. T. 2004, ApJ, 607, 285