# Exploring stellar and ionized gas non–circular motions in barred galaxies
with MUSE

Carlos López-Cobá Institute of Astronomy and Astrophysics, Academia Sinica,
No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan  Sebastián F. Sánchez
Instituto de Astronomía, Universidad Nacional Autonoma de México, Circuito
Exterior, Ciudad Universitaria, Ciudad de México 04510, Mexico  Lihwai Lin
Institute of Astronomy and Astrophysics, Academia Sinica, No. 1, Section 4,
Roosevelt Road, Taipei 10617, Taiwan  Joseph P. Anderson European Southern
Observatory, Alonso de Córdova 3107, Vitacura, Casilla 190001, Santiago, Chile
Kai-Yang Lin  Institute of Astronomy and Astrophysics, Academia Sinica, No. 1,
Section 4, Roosevelt Road, Taipei 10617, Taiwan  Irene Cruz-González Instituto
de Astronomía, Universidad Nacional Autonoma de México, Circuito Exterior,
Ciudad Universitaria, Ciudad de México 04510, Mexico  L. Galbany Institute of
Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, s/n, E-08193
Barcelona, Spain.  Institut d’Estudis Espacials de Catalunya (IEEC), E-08034
Barcelona, Spain.  Jorge K. Barrera-Ballesteros  Instituto de Astronomía,
Universidad Nacional Autonoma de México, Circuito Exterior, Ciudad
Universitaria, Ciudad de México 04510, Mexico

###### Abstract

We present MUSE integral field stellar and ionized velocity maps for a sample
of 14 barred galaxies. Most of these objects exhibit “S”-shape iso-velocities
in the bar region indicative of the presence of streaming motions in the
velocity fields. By applying circular rotation models we observe that bars
leave symmetric structures in the residual maps of the stellar velocity. We
built non–circular rotation models using the $\mathtt{XookSuut}$ tool to
characterize the observed velocity fields. In particular we adopt bisymmetric
models and a harmonic decomposition for a bar potential for describing the
non-axisymmetric velocities. We find that both models reproduce the observed
kinematic features. The position angle of the oval distortion estimated from
the bisymmetric model correlates with the photometric bar position angle
$(\rho_{pearson}=0.95)$, which suggest that non–circular velocities are caused
by the bar. However because of the low amplitudes of the $s_{3}$ harmonic we
can not rule out radial flows as possible source. Because of the weak
detection of H$\alpha$ in our objects we are not able to compare gas to
stellar non–circular motions in our sample, although we show that when
galaxies are gas rich the oval distortion is also observed but with larger
amplitudes. Finally, we do not find evidence that the amplitude of the
non–circular motions is dependent on the bar size, stellar mass or the global
SFR.

editorials, notices — miscellaneous — catalogs — surveys

##  1 Introduction

Stellar bars are among the multiple non-axisymmetric components observed in
galaxies (de Vaucouleurs et al., 1991), and it is estimated that nearly
$\frac{1}{3}$ of disk galaxies exhibit a stellar bar (e.g., Sellwood &
Wilkinson, 1993), with the number increasing when observing in infrared bands
(e.g., Knapen et al., 2000). Commonly, bars are thought to play multiple roles
in the evolution of galaxies including radial migration of stars (e.g.,
Minchev & Famaey, 2010), triggering the star–formation (SF) and nuclear
activity (e.g., Combes, 2001) and for shaping the metallicity gradients in
galaxies (e.g., Sánchez-Blázquez et al., 2011), among others. All these
processes are intrinsically related with the dynamics of the bar. Therefore,
understand their kinematic properties is crucial to unreveal their real
influence in galaxy evolution. While bars are relatively easy to recognize in
continuum images (Wozniak & Pierce, 1991), their kinematic counterpart is not
evident until detailed examination of their velocity fields. Even so, non-
axisymmetric motions induced by bars have been identified in a wide variety of
objects. These bar–driven motions when projected into the line of sight
manifest in the form of oval distortions which have been observed in the
velocity field of molecular (e.g., Weliachew et al., 1988), neutral (e.g.,
Bosma et al., 1977; Peterson et al., 1978) and ionized gas (e.g., Fathi et
al., 2005; Holmes et al., 2015), as well as in stellar velocity maps (e.g.,
Kormendy, 1983; Bettoni et al., 1988). Such distortion recognized for
producing a S–shaped pattern in the velocity field is less evident when the
bar axis is closely aligned parallel or perpendicular to the line of nodes;
these viewing angles disfavor the detection of bar stream motions (e.g., van
Albada & Roberts, 1981; Pence & Blackman, 1984; Athanassoula & Misiriotis,
2002), making difficult to identify barlike-flows from the line-of-sight
velocities. Hence it is common that bar-flow studies are biased toward
galaxies with bars lying at intermediate orientations from the disk
major/minor axis (e.g., Pence & Blackman, 1984).

With the advent of recent observational techniques such as the integral field
spectroscopy (IFS) we are able to spatially resolve their ionized and stellar
kinematic properties (e.g., Fathi et al., 2005; Barrera-Ballesteros et al.,
2014; Holmes et al., 2015; Fraser-McKelvie et al., 2020; Gadotti et al.,
2020). In the optical, their kinematic counterpart have been identified mostly
using H$\alpha$. This, however, tends to bias the studies towards gas-rich
systems. In addition that H$\alpha$ traces in most cases the location of young
stars. Conversely, stellar bars are dominated by old stellar populations
(e.g., Sánchez-Blázquez et al., 2014), whose ionization is mostly dominated by
old stars (e.g., Stasińska et al., 2008; Gomes et al., 2016; Lacerda et al.,
2018; Sánchez, 2020).

On the other hand, studies of bars using the stellar velocity maps in most
IFS-galaxy surveys are limited by inherent spatial resolution effects,
precluding the identification of bar-driven flows (e.g., Barrera-Ballesteros
et al., 2014).

In the present work, we address the kinematic study of bars by taking full
advantage of integral field spectroscopic data from the multi-unit
spectroscopic explorer instrument to detect bar–like flows on the stellar and
H$\alpha$ velocity maps of 14 galaxies. We built kinematic models to describe
the bar flows and try to relate the amplitude of the bar non–circular motions
with some global properties of galaxies.

The paper is structured as follows. In section 2 we describe the data; in
section 3 the analysis of the velocity maps and the kinematic models adopted;
in section 4 we describe the results and the conclusions are presented in
section 5. Throughout this paper we adopted a $\Lambda$CDM cosmology with
$H_{0}=70$ km s-1 Mpc-1, $\Omega_{m}=0.3$ and $\Omega_{\Lambda}=0.7$.

##  2 Data

In this study we use the dataproducts from the AMUSING++ galaxy compilation
(Galbany et al., 2016; López-Cobá et al., 2020). AMUSING++ is a compilation of
$\sim 600$ nearby galaxies observed with the multi-unit spectroscopic explorer
(MUSE) instrument (e.g., Bacon et al., 2010). With a field–of–view (FoV) of
$1\arcmin\times 1\arcmin$, delivering $\sim$90 K spectra per data–cube, MUSE
is the most advanced integral field spectrocopy (IFS) instrument in the
optical range covering from $4800$Å to $9300$Å. It combines both a moderate
spectral resolution ($R\sim 3000$), and a high spatial resolution limited by
the atmospheric seeing.

The AMUSING++ cubes were analyzed using the pipe3d package (e.g., Sánchez et
al., 2016). This tool has been extensively used on a number of IFS galaxy
surveys such as CALIFA (e.g., Sánchez et al., 2012), MaNGA (e.g., Bundy et
al., 2015) and SAMI (e.g., Allen et al., 2015). pipe3d models the stellar
population using a combination of simple stellar populations (SSP), and
derives the properties of the emission lines once subtracted this model to the
original spectra.

On the other hand, the recovery of the stellar population properties, such as
the stellar kinematics, is performed over a tesselation map that follows the
galaxy light distribution, typically around the V-band. In general, the size
of these tesselations or voxels depend on the target signal-to-noise (S$/$N)
of the continuum, here chosen to reach S$/$N$\geq$ 30\. For the particular
case of the MUSE data, these correspond to $\sim 1\arcsec~{}$ for regions with
high S$/$N, and $\sim 2$″ for regions with low S$/$N, typically for the
outskirt of the disks. We emphasize that this is one of the major advantage of
the current data compared to other IFS-surveys with coarser spatial
resolutions. Then the spectra within each voxel are coadded and over this
spectrum is performed a simple stellar population analysis (SSP) to recover
the velocity, age of the SSPs, metallicity, among other properties. This
procedure is performed over each voxel of the tesselation map.

In addition to the above analysis, the ionized gas properties such as the
flux, velocity and velocity dispersion, are recovered spaxel–by–spaxel by
implementing a moment analysis on a set of $\sim 50$ emission–lines. We refer
the reader to the AMUSING++ presentation paper (e.g., López-Cobá et al., 2020)
for more details of this procedure. The final outputs of the pipe3d analysis
are two dimensional maps of the main properties of ionized gas and the
underlying stellar populations.

For the current analysis, we selected a sample of barred galaxies from the
AMUSING++ dataset. Galaxies were required to exhibit clear stellar bars in the
MUSE $gri$ images. Barred galaxies in interaction were excluded since their
kinematics is dominated by the interaction process and not by internal ones;
in addition, we exclude those objects where the higher S$/$N voxels have sizes
larger than 3 times the seeing of the observing night (based on the ESO DIMM
archive). This condition excludes galaxies observed with non-optimal
atmospheric conditions. Finally the apparent bar length must be resolved in
the stellar velocity map and it must fit in the MUSE FoV. These criteria lead
to a sample of 14 galaxies. Since AMUSING++ is a compilation of objects, our
sample of bars is not an statistical representation of the population of
barred galaxies in the Local Universe. Therefore, the results of this analysis
are not statistical significant.

##  3 Analysis

###  3.1 Photometric properties of bars

Figure 1: Top panel: ESO476-16 DES-z image. Yellow contours represent the
isophotes. Bottom panel shows the radial variation of the disk position angle
and ellipticity of the isophotes. The vertical red line shows the approximate
length and position angle of the bar, estimated in
$r^{\prime}_{\mathrm{bar}}\sim 9.5\pm 1.5^{\prime\prime}$ and $\phi=114\pm
3^{\circ}$ respectively. The red shadow region represents the bar length
standard deviation. The green line ontop of the continuum image represents
these values. The galaxy is oriented North-East with North pointing up and
East to the left.

Stellar bars are dominated in general by old stellar populations (e.g.,
Sánchez-Blázquez et al., 2014; Vera et al., 2016; Fraser-McKelvie et al.,
2019; Neumann et al., 2020); therefore, it is common to study these structures
in the reddest bands of the optical spectrum or in infrared bands (e.g., Díaz-
García et al., 2016), where their morphological properties are enhanced. Among
those properties is the bar–length, albeit there is no unambiguous method for
estimating it (see Athanassoula & Misiriotis, 2002, for a thorough description
of methods). Parametric methods often decompose the galaxy light distribution
into axisymmetric and non-axisymmetric components to recover the disk, bulge
and bar properties (e.g., Laurikainen et al., 2018; Méndez-Abreu et al.,
2019). An easier way consists in analyzing the object light distribution by
tracing isophotes. Abrupt changes in the disk position angle
($\phi^{\prime}_{\mathrm{disk}}$) and the axial ratio ($\epsilon$) have been
commonly used to estimate the bar length, as well as its orientation in the
sky (e.g., Kormendy, 1983; Pence & Blackman, 1984; Bettoni & Galletta, 1988;
Wozniak & Pierce, 1991; Pérez et al., 2009). Then, the true length of the bar
along the major axis can be estimated by pure trigonometric relations as
follows:

$r_{\mathrm{bar}}=r^{\prime}_{\mathrm{bar}}(\cos^{2}\phi+\sin^{2}\phi/\sin^{2}i)^{1/2}$
(1)

where $r^{\prime}_{\mathrm{bar}}$ is the apparent length of the bar in the
sky, $\phi=\phi^{\prime}_{\mathrm{disk}}-$
$\phi_{\mathrm{bar,phot}}^{\prime}$, with $\phi_{\mathrm{bar,phot}}^{\prime}$
representing the bar position angle, and $i$ is the disk inclination.

Instead of adopting narrow band images from the MUSE cubes, we use $r-$, $i-$
or $z-$ band images from DES (e.g., Abbott et al., 2018) or Pan-STARRS (e.g.,
Chambers et al., 2016) whenever available. These images have a larger FoV and
they are deeper than our MUSE data. For extracting the isophotes we use the
photoutils Python package (Bradley et al., 2016), which relies on the ellipse
fitting analysis introduced by Jedrzejewski (1987). After background
subtraction, we adopt the last faintest isophote to derive the galaxy position
angle $\phi^{\prime}_{\mathrm{disk}}$ and disk inclination. Then we trace
their radial variations, and measure the bar position angle and bar length
based on the three consecutive isophotes that maximize the difference $\Delta
P=P_{i+1}-P_{i}$, where $P$ is $\phi^{\prime}_{\mathrm{disk}}$ or the
ellipticity ($\epsilon$). Although this analysis is just a proxy for the bar
size and its orientation, it offers a first order estimation of these
properties. As a sanity check, we performed a visual comparison between our
estimations of $\phi_{\mathrm{bar,phot}}^{\prime}$ and
$r^{\prime}_{\mathrm{bar}}$ with their apparent projections in the continuum
images. In all cases a mutual agreement is reached.

Figure 1 shows the implementation of this procedure for one object in our
sample, ESO476-16. The estimated photometric bar position angle for this
object is $\phi_{\mathrm{bar,phot}}^{\prime}$$\sim 114^{\circ}$, which is
consistent with the apparent orientation of the bar in the sky. We finally
applied this analysis to all our galaxies in our sample.

###  3.2 Kinematic signatures of bars

A common and noticeable effect of bars on the velocity field is the change in
orientation of the dynamical major and minor axes (e.g., Barrera-Ballesteros
et al., 2014), a signature of bar stream flows. As mentioned before, oval
distortions on the velocity field of barred galaxies are well known from the
gas velocity fields (CO, H I, H$\alpha$). These kind of perturbations induce
large deviation of the circular motions which can be observed in residual maps
of simple axisymmetric models of circular rotation (e.g., Bettoni et al.,
1988; Bettoni & Galletta, 1988). Thus, we adopt kinematic models to identify
non–circular motions most likely induced by bars.

####  3.2.1 Kinematic models

Multiple algorithms have been developed for describing the velocity field of
galaxies. The vast majority of them are based on the tilted ring model (e.g.,
Begeman, 1989), in which the velocity field is divided in concentric rings
each one rotating with a different velocity around a fixed kinematic centre.
In this work we use $\mathtt{XookSuut}$ (e.g., Lopez-Coba et al., 2021), which
relies on the DiskFit (e.g., Spekkens & Sellwood, 2007) and RESWRI (e.g.,
Schoenmakers et al., 1997) algorithms. $\mathtt{XookSuut}$ performs
non–parametric circular and non–circular rotation models for describing the
line-of-sight velocity (LoS). It adopts Markov Chain Monte Carlo (MCMC)
methods for sampling the posterior distribution of the different kinematic
components and creates an interpolated model based on the highest probability
states.

$\mathtt{XookSuut}$ adopts the following $\log$ likelihood with flat priors on
the parameters:

$\log
p(V_{\mathrm{model}}|\vec{\alpha})=-\frac{1}{2}\sum_{\begin{subarray}{c}xpix\\\
ypix\end{subarray}}^{N}\Big{(}\frac{V_{\mathrm{obs}}-\sum_{k=1}^{K}W_{k}V_{k,\mathrm{model}}}{\sigma}\Big{)}^{2}\\\
-\log\sigma-\frac{N}{2}\log(2\pi)$ (2)

where $\vec{\alpha}$ represents all the parameters describing the considered
model; $V_{\mathrm{obs}}$ is the observed velocity map; $\sigma$ is the error
map of the measured velocities; $W_{k,n}$ is a series of weights computed at
$k$ independent rings and will serve for creating a two-dimensional
interpolated model $V_{\mathrm{model}}$; $N$ is the total number of pixels
included in the model. All $\mathtt{XookSuut}$ models adopt the thin disk
approximation; i.e., it assumes the disk is intrinsically flat, with a
constant disk ellipticity, constant position angle and fixed kinematic center.
These conditions make it possible to represent kinematic models on two-
dimensional maps (hereafter 2D model).

The circular model is the simplest model included in $\mathtt{XookSuut}$,
which is described by the following equation:

$V_{\mathrm{circ,model}}=V_{\mathrm{sys}}+\sin i~{}V_{t}\cos\theta$ (3)

In this expression $V_{t}$ is the circular rotation, which is function of the
radius. $V_{\mathrm{sys}}$ is the systemic velocity assumed constant for all
points in the galaxy. The azimuthal angle $\theta$ is measured on the galaxy
plane and is related to the sky coordinates through the inclination angle
$(i)$, kinematic position angle ($\phi_{\mathrm{disk}}^{\prime}$) measured
from North to East from the galaxy receding side, and the kinematic centre
($x_{c},~{}y_{c}$).

In order to characterize the non–circular motions in our sample, we adopt the
following non–circular models included in $\mathtt{XookSuut}$. The first one
is the bisymmetric model proposed by Spekkens & Sellwood (2007), and is
suitable for describing non–circular motions induced by a bisymmetric
distortion to an axisymmetric potential; for instance that caused by a stellar
bar. This model fits elliptical stream lines around a fixed angle to the LoS
velocities; the equation describing this model is given by:

$V_{\mathrm{bis,model}}=V_{\mathrm{sys}}+\sin i\Big{(}V_{t}\cos\theta-
V_{2,t}\cos 2(\theta-\phi_{\mathrm{bar}})\cos\theta\\\ -V_{2,r}\sin
2(\theta-\phi_{\mathrm{bar}})\sin\theta\Big{)}$ (4)

In this expression the $V_{2,r}$ and $V_{2,t}$ terms represent the bisymmetric
deviations (radial and tangential respectively) from the circular velocity
represented by the $V_{t}$ term. The phase $\phi_{\mathrm{bar}}$, represents
the position angle of the oval distortion with respect to the azimuthal angle.
Its sky projection is given by the following expression (e.g., Bettoni &
Galletta, 1997; Spekkens & Sellwood, 2007) :

$\phi_{\mathrm{bar,kin}}^{\prime}=\phi_{disk}^{\prime}+\arctan(\tan\phi_{\mathrm{bar}}\cos
i)$ (5)

where $\phi_{disk}^{\prime}$ is the kinematic position angle major axis of the
receding side, and $i$ is the galaxy inclination.

The description of the non–circular motions in our second model is based on
the epicycle theory, where the radial and tangential components of the
non–circular motions are taken into account by inducing small perturbations to
circular orbits. As shown by Schoenmakers et al. (1997), the LoS velocity
$V_{\mathrm{los}}$, can be expressed as a Fourier series as follows:

$V_{\mathrm{los}}=c_{0}+\sum_{m^{\prime}=1}^{\infty}\sin
i\big{[}c_{m^{\prime}}\cos m^{\prime}\theta+s_{m^{\prime}}\sin
m^{\prime}\theta\big{]}$ (6)

where $c_{0}$ is the $0^{\mathrm{th}}$ harmonic term, assumed to be constant
here ($\sim V_{\mathrm{sys}}$). The harmonic coefficients $c_{m^{\prime}}$ and
$s_{m^{\prime}}$ describe different velocity components as a function of the
galactocentric radius, and $\theta$ is the azimuthal angle measured in the
galaxy plane.

Schoenmakers et al. (1997) showed that a perturbation to the potential of
order $m$ affects only the $m^{\prime}=m\pm 1$ harmonics in the Fourier
expansion (Eq. 6). For the case of a bar $m=2$ and the harmonics describing
the LoS velocity adopt the following expression (e.g., Franx et al., 1994):

$V_{\mathrm{los}}=c_{0}+\sin i\big{[}c_{1}\cos\theta+c_{3}\cos
3\theta+s_{1}\sin\theta+s_{3}\sin 3\theta\big{]}$ (7)

Thus, this expression represents the expected behavior of the LoS velocity for
an elongated potential. Note that a bisymmetric perturbation also includes the
$m^{\prime}=3$ and $m^{\prime}=1$ harmonics (e.g., Spekkens & Sellwood, 2007;
Oman et al., 2019); however, as we will see in further sections the amplitude
of the harmonic coefficients allow us to assess the source of the non-circular
motions.

Figure 2: Top row figures: (i) stellar velocity map; (ii) best circular
rotation model; (iii) residual map of the model. The black rectangle on top of
(iii) shows the region with the largest residuals. Iso-velocity contours
spaced each 50 km s-1 are superimposed on each map. Bottom panels have the
same meaning but this time for the H$\alpha$ velocity map. Note the zero
velocity line shows an “S”–shape distortion.

We use $\mathtt{XookSuut}$ to built circular and non–circular flow models of
the stellar and H$\alpha$ velocity maps of our sample of galaxies. Unlike
RESWRI (Schoenmakers et al., 1997), $\mathtt{XookSuut}$ does not allow the
kinematic center and projection angles to change across radii. In fact, this
behavior is not desired in our analysis since we expect that non–circular
motions arise due to the bar potential, rather than being induced by the
presence of a twisted disk. $\mathtt{XookSuut}$ requires guess values for the
disk position angle, inclination and kinematic center. For the first two we
use the values estimated from the isophotal analysis described in Sec. 3.1,
while the kinematic center was estimated by eye in each velocity map. Since
interpolated models are created over rings on the disk plane, we choose the
rings to be spaced each $2\arcsec$. This value is grater than the spatial
resolution of our objects ($\mathrm{FWHM}_{\mathrm{DIMM}}\sim 1.5\arcsec$ on
average). Finally, we use the corresponding error maps for discarding spaxels
with low S$/$N; that is, we remove spaxels with errors larger than 10 km s-1.

In the following we use the same galaxy ESO476-16 as a showcase to go through
the different kinematic models. Figure 2 shows the circular rotation model on
the stellar and H$\alpha$ velocity maps. This figure highlights the
differences in spatial resolution in both maps. Given that the recovery of the
gas kinematics does not involve the binning procedure described in Sec. 2, the
H$\alpha$ velocity map shows a better spatial resolution than the stellar
velocity.

The stellar velocity reveals a central distortion induced most probably by the
presence of the bar, while the iso-velocity contours appear twisted near the
minor axis. The stellar circular model shows a typical rotating disk with
orthogonal major–minor axis. The superimposed iso–velocities show that the
outermost regions in the galaxy are compatible with a pure rotating disk. The
residual map in both cases (stars and H$\alpha$) show several important
features in the inner parts of the disk. A close-up around the minor axis
shows symmetric structures with large residuals of the order $\pm 40$ km s-1.
On the other side, the H$\alpha$ velocity field shows a central ring with no
data because of the low S$/$N of H$\alpha$ in this region. Despite of that,
$\mathtt{XookSuut}$ is able to predict values in these regions by linear
interpolation. We notice that the previous structure with high residuals is
also observed in the gas velocity field, albeit is affected by the low S$/$N
data. This is more evident in the close-up to this region in the third panel.
Such symmetric structures with blueshifted and redshifted components have been
observed in residual maps of barred galaxies with similar amplitudes (e.g.,
Fathi et al., 2005; Castillo-Morales et al., 2007). Thus, we may conclude from
figure 2 that a non-axisymmetric component is present in the residual
velocities from both the ionized gas and stars.

Figure 3: Obtaining of the best fit parameters in $\mathtt{XookSuut}$. Example
adopting the bisymmetric model for the stellar velocity map of ESO 476-16.
Mean values of the projection angles, kinematic center, systemic velocity and
bar position angle are shown at the top of each histogram. Figure 4: Top
figures: bisymmetric model applied to the stellar velocity field of ESO
476-16. From left to right: (i) stellar velocity map: (ii) 2D representation
of the best fit bisymmetric model; (iii) residual map defined as observed $-$
model; (iv) radial variation of the tangential velocity (black), $V_{2,r}$
(red) and $V_{2,t}$ velocities as function of the galactocentric distance.
Bottom figures: harmonic decomposition for an $m=2$ perturbation in the
potential. From left to right: (i) input stellar velocity map; (ii) best 2D
kinematic model; (iii) residual map; (iv) radial variation of the inclination
corrected $c_{1}$, $s_{1}/c_{1}$ and $s_{3}/c_{3}$ harmonics; (v)
$s_{1}/c_{1}$ vs. $s_{s}/c_{1}$ harmonic ratios. The continuous line in this
panel represents a linear fit to the data points with slope $(m)$ pointed in
the upper–left corner (not to be confused with the harmonic coefficient).

Whether the velocity field of a barred galaxy is better described by a
bisymmetric flow or by a pure axisymmetric radial flow is in general not
straightforward to determine (e.g., Wong et al., 2004). In this work we will
not make any a priori assumption on the kinematic model. Instead, we first
adopt a bisymmetric model constraining the radial extension of the
non–circular motions up to the the deprojected bar size plus an additional
value ranging from 1″–2.5″, which takes into account the spatial resolution of
the data. Additionally, we decompose the LoS velocity with a Fourier analysis
including only the $m^{\prime}=1$ and $m^{\prime}=3$ harmonics (Eq. 7), since
the ratio of the $s_{3}$ and $s_{1}$ coefficients can be used to distinguish
between bar–like flows and radial flows. (e.g., Wong et al., 2004; Fathi et
al., 2005; Elson et al., 2011).

The inclusion of complex models such as the bisymmetric model adds extra
variables in the fitting procedure that may reduce the residuals, but at the
expense of overfiting the data. In order to assess the bisymmetric model we
compute the Bayesian information criterion (BIC, Kass & Raftery, 1995),
defined as $\mathrm{BIC}=N\ln(\chi^{2}/N)+\ln(N)N_{params}$, where $\chi^{2}$
is the quadratic sum of the residuals and $N_{params}$ the number of
parameters to estimate from the model . Unlike $\chi^{2}$, BIC is more
sensitive to the number of parameters to estimate from the model, favoring
less complex ones.

We use the outputs of the circular rotation model (namely $x_{c}$, $y_{c}$,
$i$, $\phi_{\mathrm{disk}}^{\prime}$, $V_{\mathrm{sys}}$), as input for the
harmonic and bisymmetric models. We note that there is no significant
differences in the results if we fix these parameters or we let them vary
during fitting. In any case, we let $\mathtt{XookSuut}$ to derive the best set
of parameters. Results adopting the bisymmetric model on the stellar velocity
map of the showcase object is shown in Figure 3. This figure is a corner plot
for the parameters describing the disk geometry and the bar orientation. The
bisymmetric model finds an oval distortion oriented at
$\phi_{\mathrm{bar}}$$=43^{\circ}$, with its corresponding sky projection (Eq.
5) oriented at $\phi_{\mathrm{bar,kin}}^{\prime}$$=289\pm 1^{\circ}$. The
position angle of the oval distortion is in agreement with our previous
estimation for the photometric bar position angle ($114^{\circ}\pm
3^{\circ}$)111 Position angles are measured from the North to the East. The
photometric P.A. goes from $0^{\circ}-180^{\circ}$, while the kinematic P.A.
goes from $0^{\circ}-360^{\circ}$ and is measured from the receding side of
the galaxy; Therefore there could be differences of $180^{\circ}$ between both
angles.

The 2D representation of the bisymmetric and harmonic models together with the
radial profile of the different velocity components are shown in Figure 4. In
the top panels, the bisymmetric model reproduce successfully the “S”shape
distortion observed in the inner regions, furthermore the residual map no
longer exhibits the symmetrical patterns observed in Figure 2. On the other
hand, the non–circular velocities, $V_{2,r}$ and $V_{2,t}$, show maximum
amplitudes of the same of order as the residuals in the circular rotation
model, i.e., $\sim 40$ km s-1, with smooth profiles as those observed in gas
velocity fields (e.g., Sellwood & Sánchez, 2010; Holmes et al., 2015).

The bottom panels of figure 4 show the results of the harmonic decomposition.
Again the 2D model seems to be a good representation of the LoS velocities,
and no residual structures are observed in the central regions. The similar
behavior of the $c_{1}$ harmonic with $V_{t}$ in the bisymmetric model,
suggesting that the disk circular rotation is indistinguishable between both
models and that the differences must reside only in the amplitudes of the
non–circular components. The inclusion of the $m^{\prime}=3$ terms reproduce
much better the features observed in the velocity field, however the residual
map is, at first sight, indistinguishable from the bisymmetric model
indicating that the harmonic decomposition produces results as good as the
bisymmetric one. Nevertheless, the interpretation of the harmonic coefficients
is not straightforward. Based on the epicyclic theory, Franx et al. (1994) and
Wong et al. (2004) computed the expected behavior of the $m^{\prime}=1$ and
$m^{\prime}=3$ harmonics for an elliptical potential (i.e., an $m=2$
perturbation). They found that for an elliptical potential, the slope between
the $s_{3}$ and $s_{1}$ coefficients (hereafter $ds_{3}/ds_{1}$), is found to
be negative, i.e. $ds_{3}/ds_{1}<0$, meanwhile for an axisymmetric radial flow
$|ds_{3}/ds_{1}|<0.1$, which means that $s_{3}\ll s_{1}$. The bottom rightmost
panel of figure 4 shows the $s_{1}/c_{1}$ vs. $s_{3}/c_{1}$ ratio together
with the best fit line to the points. For the considered galaxy, the slope of
the non–circular velocities is found in $-0.11$; therefore, the $s_{1}$ and
$s_{3}$ harmonics are compatible with being produced by a bar potential.

##  4 Results

So far we have described our methodology for a single object. In the following
we describe our results when applied to our sample of galaxies. Prior the
kinematic modeling we removed spaxels with errors larger than $10$ km s-1 to
exclude low signal-to-noise data that could affect the final models. For all
objects we first create circular models with initial values taken from the
isophotal analysis. The 2D models for each individual object are shown in
Appendix A. Then we use the best fit values of the geometric parameters
(namely $\phi^{\prime}$, $i$, $x_{c}$, $y_{c}$ and also $V_{\mathrm{sys}}$) as
inputs for the bisymmetric and harmonic decomposition models.

It is expected that bar-like flows are not present across the entire disk, but
only in the region influenced by the bar as observed from the residuals of
Figure 3. Therefore, we fit non–circular motions up to the size of the
photometric bar, plus a constant value that goes from $1\arcsec-2.5\arcsec$ to
compensate for the binning on the stellar velocity maps. The two-dimensional
non–circular models for the stellar and gas velocity maps are shown in Figs.
10 and 11 from Appendix B.

Table 1 shows $\mathtt{XookSuut}$ results for the constant parameters namely
$\phi_{\mathrm{disk}}^{\prime}$, $i$ and $V_{\mathrm{sys}}$. The phometric
estimations of the bar position angle and true bar length (i.e., Eq. 1) are
shown in the third and fourth columns, respectively. In general, we do not
find large differences in the constant parameters, when adopting a bisymmetric
or an harmonic decomposition model. This means that the flat disk assumption
is adequate for our objects, besides that both fits are numerically stable
despite the underlying assumptions. Exception to this might be IC 0004 where a
constant ellipticity by $\mathtt{XookSuut}$ does not seem to reproduce the
kinematic orientation of the disk. Table 1 also shows the BIC value of the
bisymmetric model, but normalized to the BIC circular model. Thus, models with
BIC $>1$ would not favor the bisymmetric model. We note that only in one
object (PGC 055442) BIC definitely favors the circular model over the
bisymmetric one.

###  4.1 Circular rotation models

Figure 9 shows the 2D circular models of the stellar and H$\alpha$ velocity
maps, as well as the residual maps of the models. We notice that in the vast
majority of cases H$\alpha$ is not detected in the inner disks; however, a
careful analysis of the residual velocities in both stellar and H$\alpha$,
large scale residuals are observed around the bar region in most cases, for
instance in NGC 692. Furthermore when H$\alpha$ is detected, symmetric
residual structures with opposite velocities are observed in the inner
regions; for instance IC 2160, IC 0004, PGC 055442 and ESO 18-18. Other
objects although show high residual towards the center, they appear to be
affected by other sources of non–circular motions, such as spiral arms as in
NGC 289 and NGC 3464.

On the other hand, the residuals of the stellar velocity maps show many
examples of galaxies with “S”-shaped iso-velocities together with symmetric
residual structures around the bar. Such residuals are not compatible with an
axisymmetric rotating disk and are most probably induced by the presence of
the stellar bar. The fact we are clearly detecting oval distortions in most of
our stellar velocity maps is likely a consequence of the high spatial
resolution of our data, since similar IFS studies with coarser resolution do
not observe such structures (e.g., Barrera-Ballesteros et al., 2014).

###  4.2 Stellar non–circular motions

Figure 10 shows the bisymmetric and harmonic decomposition models for the
stellar velocity maps. Bisymmetric models tend to reproduce much better the
observed velocity map than circular rotation models do. For most objects,
$V_{2,r}$ and $V_{2,t}$ show smooth behaviors, with velocities decaying at
large radius, a possible signal of the kinematic ending of the bar.

NGC 289 is the only object where the bisymmetric model, for both gas and
stars, does not seem to be a reasonable physical model given the erratic
behavior of the circular and non–circular velocities. As observed in figure 10
and table 1, the bar in NGC 289 is closely aligned with the major axis, being
separated by less than $10^{\circ}$; and it is well known that bar stream
lines in the velocity field become less evident when the bar lies close the
galaxy major-axis (e.g., van Albada & Roberts, 1981). Moreover, when
$\phi_{\mathrm{bar}}$$=90^{\circ}$ or $0^{\circ}$, $V_{2r}$ and $V_{2t}$ from
Eq. 4 become degenerate with the disk circular rotation, making it impossible
to separate the non-circular velocities from the circular rotation. As a
result, $V_{t}$, $V_{2r}$ and $V_{2t}$ can lead to unphysical absurd values as
is probably the case in NGC 289; for instance, note that for $r<40\arcsec$ the
three velocities shows the same uncommon radial profiles. On the other hand
the harmonic model does not depend on the bar phase; thus, the disk circular
rotation related to $c_{1}$ seems to be offer a better representation of the
rotation curve in this case.

When comparing the BIC ratio from the bisymmetric to circular models we note
that, in the case of the stellar maps, these values are mostly close to 1,
with the aforementioned exception. We expect BIC ratios lower than 1 would
favour the bisymmetric model. While this is observed in our sample, in some
cases the differences with the circular model are only marginal. It should be
noted that because of the binning procedure during the SSP analysis, spaxels
are correlated in the stellar maps, affecting the estimation of the BIC value.
Therefore, we favour the bisymmetric models in those cases where we detect
kinematic oval distortions, regardless of the BIC ratio.

The bar position angle is the main parameter in the bisymmetric model since it
defines the axis where the non–circular motions are expected. To visualize
better this parameter, Figure 5 shows the photometric bar position angle vs.
the sky projected kinematic bar position angle for the objects in the sample.
It can be seen that there is a good agreement between both angles; in deed,
the Pearson correlation coefficient for this relation is extremely high, being
${\rho}_{pearson}=0.95$.

On the other side, the harmonic decomposition models show good representations
of the LoS velocities. Table 1 shows that at least in one velocity map
(stellar or ionized), the slopes in the harmonic coefficients $ds_{3}/ds_{1}$
are negative, which is in favor of the hypothesis that the non–circular
motions are induced by the bar. However there are objects where
$|ds_{3}/ds_{1}|<0.1$, which means that radial flows could be contributing to
the observed non–circular motions.

Figure 5: Comparison between the sky projected photometric ($x$–axis) and
kinematic bar position angle for the stellar velocity maps ($y$–axis).
Photometric position angles were shifted $\pm 180^{\circ}$ when required to
match with the definition of the kinematic position angle. The continuous line
represents the 1:1 relation. Each galaxy is observed with a different color.
The error bars are of the size of the points.

###  4.3 Ionized gas non–circular motions

As bar-like flows are expected along the bar, the lack of gas in the inner
regions affects directly the estimation of $\phi_{\mathrm{bar}}$, and hence
the amplitude of the non–circular motions. Because of the weak emission of
H$\alpha$ in the inner disks in our objects, the bar region provides little or
null information of the behavior of the non–circular motions. Only in 6
objects is observed plenty of gas in the inner disk to be able to perform
reliable non-circular flow models. These objects are shown in Fig. 11.
However, the bars in NGC 289 and NGC 3464 are closely aligned parallel to the
disk position angle, preventing the bar streams from being separated from the
disk circular rotation. Even so, the large residuals observed in fig. 9
indicate the presence of strong non–circular motions, although a large
fraction of them could be attributed to the prominent spiral arms in NGC 289
(e.g., Pence & Blackman, 1984).

We notice however that oval distortions appear in the H$\alpha$ velocity maps
when galaxies are gas rich; and such distortions are pronounced when the bar
is elongated at an intermediate position angle from
$\phi_{\mathrm{disk}}^{\prime}$. A remarkable example is observed in IC 2160
where the bar is oriented $\sim 40^{\circ}$ away the kinematic position angle.
As consequence, a strong twist in the kinematic major and minor axes is
observed.

Figure 6: Comparison between ionized gas and stellar bisymmetric models for IC
2160. Black, blue and red curves represent the $V_{t}$, $V_{2,r}$ and
$V_{2,t}$ components for the H$\alpha$ (continuous line) and stellar (dashed
line) velocity maps.

###  4.4 Stellar and ionized gas bar-like flows

As to whether the ionized gas and stars have similar amplitudes of the
non–circular motions, the lack of ionized gas in the bar regions precludes a
direct comparison between them. Figure 6 shows bisymmetric models for IC 2160,
where oval distortions are observed simultaneously in the H$\alpha$ and
stellar velocity maps. We notice similar behaviors between the circular and
non–circular motions of gas and stars, albeit ionized gas have much larger
amplitudes. Such differences in amplitudes could be related to the dynamical
origin of the tracers. Since stars are dynamically hot, they are more affected
by the asymmetric drift (AD) than the gas. Although the AD is more important
in pressure support systems, like dwarf galaxies (e.g., Oh et al., 2011), in
spiral galaxies the bulge and stellar bars, composed largely by old stellar
populations, dominate the AD contribution (e.g., Shetty et al., 2020). Thus,
the lower amplitudes observed in the stellar non–circular motions could be
explained by the AD.

Table 1: Main properties

AMUSING++ | $\log~{}\mathrm{M_{\star}/M_{\odot}}$ | $\phi_{\mathrm{bar,phot}}^{\prime}$ | $r_{\mathrm{bar}}$ | model | $\phi_{\mathrm{disk}}^{\prime}$ | $i$ | $V_{sys}$ | $\phi_{\mathrm{bar}}$ | $\phi_{\mathrm{bar,kin}}^{\prime}$ | BIC | slope
---|---|---|---|---|---|---|---|---|---|---|---
|  | $(^{\circ})$ | (kpc) |  | (∘) | (∘) | (km s-1) | (∘) | (∘) |  | $(ds_{3}/ds_{1})$
ESO476-16 stars | 10.8 | 114.4 $\pm$ 3.2 | 4.0 $\pm$ 0.6 | bis | 321.6 $\pm$ 0.8 | 53.6 $\pm$ 0.3 | 5960.9 $\pm$ 0.3 | 42.8 $\pm$ 0.5 | 288.9 $\pm$ 0.9 | 0.89 |  $\cdots$
ESO476-16 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 321.0 $\pm$ 0.4 | 54.0 $\pm$ 0.7 | 5960.9 $\pm$ 0.6 |  $\cdots$ |  $\cdots$ |  $\cdots$ | -0.04
NGC6947 stars | 10.9 | 4.3 $\pm$ 1.9 | 6.6 $\pm$ 1.0 | bis | 238.8 $\pm$ 1 | 47.8 $\pm$ 1 | 5602.3 $\pm$ 0.2 | 35.4 $\pm$ 0.7 | 195.4 $\pm$ 1.2 | 0.93 |  $\cdots$
NGC6947 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 235.9 $\pm$ 0.4 | 48.1 $\pm$ 0.7 | 5602.3 $\pm$ 0.6 |  $\cdots$ |  $\cdots$ |  $\cdots$ | -0.25
NGC692 stars | 11.2 | 117.6 $\pm$ 3.3 | 7.3 $\pm$ 1.1 | bis | 259.4 $\pm$ 0.2 | 27.0 $\pm$ 0.3 | 6343.4 $\pm$ 0.7 | 38.8 $\pm$ 0.3 | 295.0 $\pm$ 0.4 | 1.01 |  $\cdots$
NGC692 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 259.0 $\pm$ 0.7 | 28.0 $\pm$ 0.5 | 6343.4 $\pm$ 0.7 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.13
IC2160 stars | 10.6 | 144.2 $\pm$ 1.1 | 5.8 $\pm$ 0.5 | bis | 106.0 $\pm$ 0.0 | 47.6 $\pm$ 0.2 | 4729.6 $\pm$ 0.5 | 33.1 $\pm$ 1.1 | 129.7 $\pm$ 0.9 | 0.96 |  $\cdots$
IC2160 gas |  $\cdots$ |  $\cdots$ |  $\cdots$ | bis | 110.4 $\pm$ 1.2 | 47.2 $\pm$ 0.0 | 4730.5 $\pm$ 10.0 | 38.9 $\pm$ 0.7 | 139.2 $\pm$ 1.3 | 0.97 |  $\cdots$
IC2160 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 108.9 $\pm$ 0.5 | 44.9 $\pm$ 0.7 | 4728.5 $\pm$ 0.6 |  $\cdots$ |  $\cdots$ |  $\cdots$ | -0.07
IC2160 gas |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 113.3 $\pm$ 0.4 | 43.4 $\pm$ 0.7 | 4727.9 $\pm$ 0.6 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.02
UGC3634 stars | 11.0 | 1.1 $\pm$ 1.5 | 6.0 $\pm$ 0.9 | bis | 120.5 $\pm$ 0.2 | 39.1 $\pm$ 1.1 | 7876.0 $\pm$ 5.0 | 41.4 $\pm$ 0.9 | 154.9 $\pm$ 1.0 | 1.0 |  $\cdots$
UGC3634 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 120.5 $\pm$ 0.4 | 48.9 $\pm$ 2.7 | 7876.5 $\pm$ 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ | -0.24
NGC5339 stars | 10.4 | 83.1 $\pm$ 1.0 | 5.7 $\pm$ 0.8 | bis | 208.9 $\pm$ 0.1 | 35.5 $\pm$ 0.3 | 2736.2 $\pm$ 0.1 | 44.9 $\pm$ 0.1 | 247.9 $\pm$ 0.2 | 1.03 |  $\cdots$
NGC5339 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 208.5 $\pm$ 1.2 | 33.6 $\pm$ 0.8 | 2736.2 $\pm$ 0.6 |  $\cdots$ |  $\cdots$ |  $\cdots$ | -0.23
ESO325-43 stars | 11.1 | 28.2 $\pm$ 1.3 | 4.3 $\pm$ 0.5 | bis | 109.2 $\pm$ 1 | 36.6 $\pm$ 1 | 10460.2 $\pm$ 0.9 | 29.9 $\pm$ 0.7 | 54.8 $\pm$ 1.3 | 0.99 |  $\cdots$
ESO325-43 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 109.2 $\pm$ 0.4 | 36.8 $\pm$ 1.0 | 10460.4 $\pm$ 0.6 |  $\cdots$ |  $\cdots$ |  $\cdots$ | -0.21
IC0004 stars | 10.5 | 150.0 $\pm$ 4.5 | 4.7 $\pm$ 0.5 | bis | 178.3 $\pm$ 0.7 | 52.5 $\pm$ 1.4 | 4991.2 $\pm$ 1.1 | 9.9 $\pm$ 0.9 | 104.4 $\pm$ 0.9 | 1.0 |  $\cdots$
IC0004 gas |  $\cdots$ |  $\cdots$ |  $\cdots$ | bis | 178.4 $\pm$ 1 | 45.8 $\pm$ 1 | 5002.3 $\pm$ 16.8 | 61.3 $\pm$ 1.7 | 157.5 $\pm$ 2.3 | 0.97 |  $\cdots$
IC0004 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 177.6 $\pm$ 0.6 | 45.8 $\pm$ 1.3 | 4996.1 $\pm$ 0.7 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.12
IC0004 gas |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 176.3 $\pm$ 1.3 | 49.7 $\pm$ 1.3 | 4996.9 $\pm$ 0.8 |  $\cdots$ |  $\cdots$ |  $\cdots$ | -0.37
PGC055442 stars | 10.9 | 39.3 $\pm$ 1.9 | 2.5 $\pm$ 0.4 | bis | 193.8 $\pm$ 0.1 | 37.7 $\pm$ 0.6 | 7036.3 $\pm$ 0.9 | 104.4 $\pm$ 1.1 | 205.3 $\pm$ 1.4 | 1.06 |  $\cdots$
PGC055442 gas |  $\cdots$ |  $\cdots$ |  $\cdots$ | bis | 194.7 $\pm$ 1 | 46.7 $\pm$ 1 | 7031.8 $\pm$ 21.7 | 87.2 $\pm$ 3.1 | 280.7 $\pm$ 4.6 | 1.16 |  $\cdots$
PGC055442 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 193.9 $\pm$ 0.5 | 38.9 $\pm$ 1.6 | 7036.5 $\pm$ 0.8 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.05
PGC055442 gas |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 194.1 $\pm$ 1.1 | 38.3 $\pm$ 1.9 | 7028.8 $\pm$ 1.0 |  $\cdots$ |  $\cdots$ |  $\cdots$ | -0.49
NGC7780 stars | 10.5 | 160.2 $\pm$ 2.2 | 3.9 $\pm$ 0.6 | bis | 188.2 $\pm$ 1 | 63.3 $\pm$ 1 | 5185.7 $\pm$ 0.9 | 37.4 $\pm$ 0.0 | 157.7 $\pm$ 1.0 | 1.0 |  $\cdots$
NGC7780 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 189.1 $\pm$ 0.5 | 61.2 $\pm$ 0.5 | 5185.0 $\pm$ 0.7 |  $\cdots$ |  $\cdots$ |  $\cdots$ | -0.14
NGC3464 stars | 10.6 | 112.0 $\pm$ 2.5 | 3.9 $\pm$ 0.6 | bis | 109.4 $\pm$ 1 | 45.6 $\pm$ 1 | 3740.0 $\pm$ 1.1 | 19.2 $\pm$ 0.7 | 123.2 $\pm$ 1.1 | 1.0 |  $\cdots$
NGC3464 gas |  $\cdots$ |  $\cdots$ |  $\cdots$ | bis | 109.4 $\pm$ 1 | 45.6 $\pm$ 1 | 3740.0 $\pm$ 1.1 | 19.2 $\pm$ 0.7 | 123.2 $\pm$ 1.1 | 1.03 |  $\cdots$
NGC3464 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 109.6 $\pm$ 0.5 | 47.6 $\pm$ 0.6 | 3740.3 $\pm$ 0.6 |  $\cdots$ |  $\cdots$ |  $\cdots$ | -0.26
NGC3464 gas |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 109.2 $\pm$ 0.4 | 54.8 $\pm$ 0.7 | 3730.6 $\pm$ 0.5 |  $\cdots$ |  $\cdots$ |  $\cdots$ | -0.49
NGC1591 stars | 10.4 | 47.2 $\pm$ 2.0 | 2.6 $\pm$ 0.4 | bis | 24.0 $\pm$ 1.2 | 56.8 $\pm$ 0.4 | 4120.7 $\pm$ 1.9 | 46.3 $\pm$ 3.0 | 53.8 $\pm$ 2.9 | 0.68 |  $\cdots$
NGC1591 gas |  $\cdots$ |  $\cdots$ |  $\cdots$ | bis | 25.4 $\pm$ 0.0 | 50.8 $\pm$ 0.2 | 4129.6 $\pm$ 5.6 | 26.8 $\pm$ 1.0 | 43.1 $\pm$ 0.8 | 0.97 |  $\cdots$
NGC1591 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 29.2 $\pm$ 0.6 | 60.5 $\pm$ 1.0 | 4117.1 $\pm$ 0.6 |  $\cdots$ |  $\cdots$ |  $\cdots$ | -0.04
NGC1591 gas |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 25.8 $\pm$ 0.5 | 47.4 $\pm$ 2.8 | 4128.0 $\pm$ 0.7 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.06
ESO18-18 stars | 10.5 | 53.6 $\pm$ 2.5 | 5.7 $\pm$ 0.6 | bis | 281.4 $\pm$ 0.0 | 38.9 $\pm$ 0.9 | 5117.2 $\pm$ 4.8 | 12.0 $\pm$ 3.5 | 206.7 $\pm$ 2.8 | 0.53 |  $\cdots$
ESO18-18 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 277.3 $\pm$ 0.6 | 18.1 $\pm$ 0.4 | 5113.2 $\pm$ 0.7 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.01
NGC289 stars | 10.5 | 120.7 $\pm$ 1.7 | 1.9 $\pm$ 0.1 | bis | 128.3 $\pm$ 1 | 45.1 $\pm$ 1 | 1623.8 $\pm$ 0.0 | 85.9 $\pm$ 0.3 | 125.4 $\pm$ 1.1 | 0.93 |  $\cdots$
NGC289 gas |  $\cdots$ |  $\cdots$ |  $\cdots$ | bis | 125.0 $\pm$ 0.1 | 43.6 $\pm$ 0.1 | 1629.3 $\pm$ 14.3 | 61.9 $\pm$ 6.1 | 178.7 $\pm$ 7.0 | 0.98 |  $\cdots$
NGC289 stars |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 130.0 $\pm$ 1.0 | 37.7 $\pm$ 0.7 | 1624.0 $\pm$ 0.6 |  $\cdots$ |  $\cdots$ |  $\cdots$ | -0.24
NGC289 gas |  $\cdots$ |  $\cdots$ |  $\cdots$ | hrm | 126.2 $\pm$ 0.9 | 43.8 $\pm$ 0.6 | 1624.7 $\pm$ 0.7 |  $\cdots$ |  $\cdots$ |  $\cdots$ | 0.1

Note. — In some cases $V_{2t}$ and $V_{2r}$ have both negative values. When
this occurs, $\phi_{\mathrm{bar}}$ no longer represents the bar major axis,
but the bar minor axis in the galaxy plane. This is taken into account when we
compute position angle of the bar in the sky plane by shifting
$\phi_{\mathrm{bar}}$ by $90^{\circ}$. This guarantees that
$\phi_{\mathrm{bar,kin}}^{\prime}$ is measuring the bar major axis in column
10. The reported photometric bar position angle takes into account the
possible shift in $180^{\circ}$ due to the different definitions between
kinematic and photometric position angle. The BIC value is normalized to the
BIC value from the circular rotation model.

##  5 Discussion

###  5.1 Bar-like or radial flows?

Wong et al. (2004) categorized different source of non–circular motions based
on the slope of the $s_{1}$ and $s_{3}$ coefficients. For radial flow
dominated systems, they found that $|ds_{3}/ds_{1}|<0.1$. Table 1 show objects
that fall in this category; however, for describing barred galaxies the
bisymmetric model is usually preferred over radial flows. In this model the
perturbation is driven along a fixed axis ($\phi_{\mathrm{bar}}$) and the
non–circular velocities correspond to those of elliptical orbits. In the
epicyclic theory, radial stellar motions are not expected to contribute
significantly to the non–circular motions (e.g., Sellwood & Binney, 2002). In
addition, the large speeds on the observed non–circular velocities could not
persist for a very long time without rearranging the mass distribution of the
disk (e.g., Wong et al., 2004; Spekkens & Sellwood, 2007); therefore, invoking
pure axisymmetric radial flows (inflows/outflows) brings considerable
consequences in the stability of stellar disks.

In addition, numerical simulations also show that inflows in bars are expected
to be $<5$ km s-1(e.g., Athanassoula, 1992), but not of tens of kilometers per
second as observed in most objects. Recent studies however, observe radial
motions in spirals of the order of 10-30 km s-1(e.g., Di Teodoro & Peek,
2021). Such large speeds of gas inflow/outflow would invoke necessary
efficient mechanisms of gas depletion through star formation processes or
galactic scale winds to remove gas out of the galaxy. The former scenario,
although it has been widely observed (e.g., López-Cobá et al., 2019, 2020),
the link of axisymmetric radial flows with out of plane outflows is a subject
that has not yet been addressed. For all the above reasons pure axisymmetric
radial flows are infrequently considered for describing the non–circular flows
in barred galaxies.

###  5.2 Deficit of ionized gas in bars

The absence of ionized gas in the central regions observed in halve of our
sample, suggests it could be related with the presence of the bar itself. In
fact, in most barred galaxies there is observed a lack of gas in the inner
disk (e.g., Erroz-Ferrer et al., 2015; Fraser-McKelvie et al., 2020). However
there is no clear explanations in the literature for why bars tend to not
exhibit ionized gas. Two scenarios are related with the absence of molecular
gas. If there is absence of molecular gas along the bars, no new stars are
formed. Closely related might be the presence of, indeed, radial flows that
could have transported cold gas towards the center. In such case one should
expect a higher star formation rate (SFR) towards the center, which is not yet
observed (e.g., Erroz-Ferrer et al., 2015). It is therefore important to trace
the cold gas abundance in bars to help to distinguish between both scenarios.

###  5.3 Connection between non–circular motions and galaxy properties

Figure 7: Stellar mass versus the deprojected bar semi–major axis for our
sample. The overplotted discontinuous line shows the relation observed by
Erwin (2019) with more than $1000$ barred galaxies from the $\mathrm{S^{4}G}$
survey (e.g., Díaz-García et al., 2016).

As noted in Figure 10 and 11, the amplitudes of the non–circular velocities
$V_{2,r}$ and $V_{2,t}$ vary widely from galaxy to galaxy. Thus, we ask
whether the amplitude of the non–circular motions depend on some galaxy
properties or they are local processes. For instance recent studies show a
two-slope relation between the stellar mass and the bar length Erwin (2019).
Since bars are composed largely of old stellar populations, the bar must
contribute to a large fraction of the total stellar mass, as observed in some
barred galaxies (e.g, Sánchez-Blázquez et al., 2011). Thus the relation
between $r_{bar}$ and the host galaxy stellar mass, should be expected as well
with the bar-mass.

Figure 8: Stellar mass, bar length and global SFR versus some descriptions of the non–circular motions. The top set of figures show the maximum value of the non–circular to circular motions ratio (Eq. 9). Middle and bottom figures show maximum and average value of the amplitude of the non–circular motions (Eq. 8) | $\rho_{\mathrm{pearson}}$ | $\rho_{\mathrm{spearman}}$
---|---|---
|  $\log~{}\mathrm{M_{\star}/M_{\odot}}$ |  $r_{\mathrm{bar}}$ |  $\log$ SFR |  $\log~{}\mathrm{M_{\star}/M_{\odot}}$ |  $r_{\mathrm{bar}}$ |  $\log$ SFR
$f_{\mathrm{nc}}$ |  -0.42 (0.13) |  -0.25 (0.39) |  0.24 (0.44) |  -0.58 (0.03) |  -0.31 (0.28) |  0.36 (0.22)
$\max~{}A_{\mathrm{bis}}$ |  0.37 (0.19) |  0.11 (0.71) |  0.01 (0.98) |  0.39 (0.17) |  0.3 (0.3) |  0.01 (0.99)
$<A_{\mathrm{bis}}>$ |  0.37 (0.19) |  0.37 (0.19) |  -0.39 (0.18) |  0.40 (0.16) |  0.45 (0.11) |  -0.38 (0.2)
Table 2: Pearson and Spearem correlation coefficients between non–circular
motions and galaxy global properties. The first column from top to bottom
correspond to the fraction of non–circular over the local circular rotation,
the maximum amplitude of the non–circular motions and the average velocity of
non–circular motions. These parameters are compared with the galaxy stellar
mass, the size of the bar and the SFR integrated across the MUSE FoV. For each
set of parameters we have computed the Pearson correlation coefficient and the
Spearman rank correlation with p–values shown in parenthesis.

Figure 7 shows the deprojected stellar bar length (Eq. 1) versus the stellar
mass obtained from the SSP analysis. We notice that our objects fall around
the steeper slope of this relation where galaxies with larger bars are
located. Indeed most of our objects have deprojected bar sizes larger than
$2.5$ kpc.

We investigate if the stellar mass, size of the bar and the star formation
rate (SFR) are coupled with the amplitude of the non–circular motions. Instead
of adopting the average value of the circular rotation residuals, we
characterize the non–circular motions with the amplitude of the bisymmetric
components $V_{2,r}$ and $V_{2,t}$. As these terms are function of radius we
define their amplitud, $A_{\mathrm{bis}}$, as the quadratic sum of both
components:

$A_{\mathrm{bis}}(r)=\sqrt{V_{2,t}^{2}(r)+V_{2,r}^{2}(r)}$ (8)

and the fraction of non–circular over circular motions as:

$f_{\mathrm{nc}}(r)=A_{\mathrm{bis}}(r)/V_{t}(r)~{}~{}\mathrm{for}~{}r\leq
r_{bar}$ (9)

where $V_{t}$ is the tangential velocity component of the bisymmetric model,
which gives a better description of the circular rotation than if we consider
the pure circular model. Note again that these two parameters depend on the
galactocentric distance. Thus in the second expression we are comparing at
each distance the non–circular motions with the local circular velocity.

Figure 8 shows $A_{bis}$ and $f_{nc}$ computed for the stellar velocity maps
against the stellar mass and the bar lengths of our objects. In this figure we
also include the integrated SFR derived with H$\alpha$, with the dust
attenuation correction using the Cardelli et al. (1989) extinction law and
case B of recombination (e.g., Osterbrock, 1989), and use the distances
reported in the AMUSING++ paper (e.g., López-Cobá et al., 2020). The first
thing to notice is the strength of non–circular motions ranges between 10–50%
of the local circular rotation on average, which is similar to what other
studies have found (e.g., Bettoni & Galletta, 1997).

In figure 8 we tried to find possible correlations between kinematic
properties and global properties, while Table 2 shows the Pearson and Spearman
coefficients for those relations. We find however only weak correlations
between these parameters. Even more, the large p-values show that such
correlations are not statistically significant. These results are in line with
recent studies (e.g., Erroz-Ferrer et al., 2015), where no obvious relation is
observed with the residuals of circular rotation and morphological properties
of bars. However, we are aware of the low statistic provided by this sample.

##  6 Conclusions

In this study we analyzed the incidence of non–circular flows in the stellar
and H$\alpha$ velocity field of a sample of non-interacting barred galaxies
observed with the MUSE spectrograph. The exquisite resolution of the data
allowed us to detect oval distortions in the stellar velocity maps most
probably associated to the presence of the bar. Such perturbations are
recognized in the residual maps of pure circular rotation models as symmetric
structures with blueshifted and redshifted velocities, clearly evidencing the
presence of a non-axisymmetric potential affecting the expected disk rotation.
Despite not observing ionized gas in the inner disk in most of the objects,
the bar leave imprints in the circular residual velocities in regions close to
the bar. When ionized gas is observed along the bar, the oval distortion is
also revealed in the H$\alpha$ velocity field. This evidence that both stars
and gas are affected by the bar potential.

We characterize the kinematics of these galaxies with models that include
non–circular motions, in particular the bisymmetric model and a harmonic
decomposition for a $m=2$ potential perturbation. We use the slope of the
harmonic coefficients, the $ds_{3}/ds_{1}$ ratio, to determine the source of
the observed non–circular motions.

Based on this parameter we find that the harmonic decomposition on the LoS
velocity is compatible in most cases with being produced by an elongated
potential. However, some radial flows could be present in some objects since
$|ds_{3}/ds_{1}|<0.1$ is observed in 4/14 cases. In these cases, we attribute
to the dominance of other sources of non–circular motions such as spiral arms.

We find that the bisymmetric model produces successful fittings in the
velocity maps within the bar region. We also find that the position angle of
the oval distortion correlates with the photometric position angle of the bar.
This supports the scenario of a stellar bar potential inducing deviations of
the circular rotation in the observed velocity fields. We find that the lack
of ionized gas along the bars restricts the robustness of a modeling of the
non–circular motions, in which case the stellar velocity could be considered
for studying bar-like flows when no gas is detected. However we notice that
not only the stellar circular rotation is affected by the asymmetric drift,
but also the non–circular motions since gas velocities appear rotate faster.

We find the average amplitude of the non–circular motions in our sample is
$\sim 30$ km s-1 for stars and ionized gas, while the strength of the
non–circular motions reaches values of up to 50% of the local circular
velocity, although this fraction varies among galaxies.

When trying to relate the non–circular motions with galaxy properties we do
not find any clear correlation with the stellar mass, the bar size or the
global SFR. These results point that bar-flows are rather local process;
however, we stress that larger statistical samples of barred galaxies with
high spatial resolution are required to reveal possible correlations between
kinematic properties of bars and global properties of galaxies.

We would like to thank the anonymous referee for their comments and
suggestions that contribute to improve the quality of this paper.

L. L and C. L. C. thank the supports by the Academia Sinica under Career
Development Award CDA-107-M03 and the Ministry of Science & Technology of
Taiwan under the grant MOST 108-2628-M-001-001-MY3. ICG acknowledges support
from DGAPA-UNAM grant IN113320. L.G. acknowledges financial support from the
Spanish Ministry of Science, Innovation and Universities (MICIU) under the
2019 Ramón y Cajal program RYC2019-027683 and from the Spanish MICIU project
HOSTFLOWS PID2020-115253GA-I00.

##  Appendix A Circular rotation models

Following are shown the maps of the circular and non–circular kinematic models
for the sample of galaxies shown in Table 1. In each figure, only one object
is shown as an example, the remaining figures are available in the online
journal. Figure 9 shows the circular rotation models derived by
$\mathtt{XookSuut}$ for the ionized-gas and stellar velocity maps of each
object. The three figures on the left (right) correspond to the observed
velocity map, best kinematic model and the residual map for the ionized
(stellar) velocity maps.

Fig. Set9. Circular rotation models for the H$\alpha$ and stellar velocity
maps.

Figure 9: NGC 6947 galaxy. Each block of figures refers to the results of the
kinematic modelling in each galaxy. The three leftmost (rightmost) figures
show the observed velocity field; the best circular rotation model obtained by
$\mathtt{XookSuut}$; and the residual map for the stellar (H$\alpha$) velocity
map. Overlayed contours are spaced each $\pm 50$ km s-1.

##  Appendix B Noncircular rotation models

Figure 10 shows the results from the isophotal analysis together with the
kinematic modeling of $\mathtt{XookSuut}$ adopting bisymmetric and harmonic
decomposition models. In this set of figures, only one object is shown as an
example, the remaining figures are available in the online journal. This
figure shows results only for the stellar velocity maps.

Fig. Set10. Non–circular rotation models for the stellar velocity maps.

Figure 10: The left-most figure shows the reconstructed continuum image from
the MUSE datacube using $gri$ filters. The photometric position angle of the
bar is shown with a green line, while the kinematic one derived by
$\mathtt{XookSuut}$ is shown with a red dashed line; the size of these lines
represent the aparent size of the bar. Overlayed yellow lines represent
isophotes estimated with DES or Pan-STARRS images. The right top (bottom)
figures show the bisymmetric (harmonic) model for the stellar velocity map.
From left to right: (i) observed stellar velocity field; (ii) best kinematic
model; (iii) residual map; (iv) radial distribution of the different kinematic
components describing the considered model. Iso-velocity contours are spaced
by $\pm 50$ km s-1. Objects are oriented North-East with North pointing up and
East to the left.

Similarly, Fig 11 shows results for the H$\alpha$ velocity maps, but only for
those objects with H$\alpha$ emission detected along the bars. The complete
figures set is available in the online journal.

Fig. Set11. Non–circular rotation models for the H$\alpha$ velocity maps.

Figure 11: Same as Fig. 10 but this time for the H$\alpha$ velocity maps. We
show results only for those galaxies with H$\alpha$ emission detected along
the bars.

## References

  * Abbott et al. (2018) Abbott, T. M. C., Abdalla, F. B., Allam, S., et al. 2018, ApJS, 239, 18, doi: 10.3847/1538-4365/aae9f0
  * Allen et al. (2015) Allen, J. T., Croom, S. M., Konstantopoulos, I. S., et al. 2015, MNRAS, 446, 1567, doi: 10.1093/mnras/stu2057
  * Athanassoula (1992) Athanassoula, E. 1992, MNRAS, 259, 345
  * Athanassoula & Misiriotis (2002) Athanassoula, E., & Misiriotis, A. 2002, MNRAS, 330, 35, doi: 10.1046/j.1365-8711.2002.05028.x
  * Bacon et al. (2010) Bacon, R., Accardo, M., Adjali, L., et al. 2010, in Proc. SPIE, Vol. 7735, Ground-based and Airborne Instrumentation for Astronomy III, 773508, doi: 10.1117/12.856027
  * Barrera-Ballesteros et al. (2014) Barrera-Ballesteros, J. K., Falcón-Barroso, J., García-Lorenzo, B., et al. 2014, A&A, 568, A70, doi: 10.1051/0004-6361/201423488
  * Begeman (1989) Begeman, K. G. 1989, A&A, 223, 47
  * Bettoni & Galletta (1988) Bettoni, D., & Galletta, G. 1988, A&A, 190, 52
  * Bettoni & Galletta (1997) —. 1997, A&AS, 124, 61, doi: 10.1051/aas:1997180
  * Bettoni et al. (1988) Bettoni, D., Galletta, G., & Vallenari, A. 1988, A&A, 197, 69
  * Bosma et al. (1977) Bosma, A., Ekers, R. D., & Lequeux, J. 1977, A&A, 57, 97
  * Bradley et al. (2016) Bradley, L., Sipocz, B., Robitaille, T., et al. 2016, Photutils: Photometry tools, Astrophysics Source Code Library.  http://ascl.net/1609.011
  * Bundy et al. (2015) Bundy, K., Bershady, M. A., Law, D. R., et al. 2015, ApJ, 798, 7, doi: 10.1088/0004-637X/798/1/7
  * Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245, doi: 10.1086/167900
  * Castillo-Morales et al. (2007) Castillo-Morales, A., Jiménez-Vicente, J., Mediavilla, E., & Battaner, E. 2007, MNRAS, 380, 489, doi: 10.1111/j.1365-2966.2007.12104.x
  * Chambers et al. (2016) Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints, arXiv:1612.05560.  https://arxiv.org/abs/1612.05560
  * Combes (2001) Combes, F. 2001, in Advanced Lectures on the Starburst-AGN, ed. I. Aretxaga, D. Kunth, & R. Mújica, 223, doi: 10.1142/9789812811318_0006
  * de Vaucouleurs et al. (1991) de Vaucouleurs, G., de Vaucouleurs, A., Corwin, Herold G., J., et al. 1991, Third Reference Catalogue of Bright Galaxies
  * Di Teodoro & Peek (2021) Di Teodoro, E. M., & Peek, J. E. G. 2021, ApJ, 923, 220, doi: 10.3847/1538-4357/ac2cbd
  * Díaz-García et al. (2016) Díaz-García, S., Salo, H., Laurikainen, E., & Herrera-Endoqui, M. 2016, A&A, 587, A160, doi: 10.1051/0004-6361/201526161
  * Elson et al. (2011) Elson, E. C., de Blok, W. J. G., & Kraan-Korteweg, R. C. 2011, MNRAS, 411, 200, doi: 10.1111/j.1365-2966.2010.17672.x
  * Erroz-Ferrer et al. (2015) Erroz-Ferrer, S., Knapen, J. H., Leaman, R., et al. 2015, MNRAS, 451, 1004, doi: 10.1093/mnras/stv924
  * Erwin (2019) Erwin, P. 2019, MNRAS, 489, 3553, doi: 10.1093/mnras/stz2363
  * Fathi et al. (2005) Fathi, K., van de Ven, G., Peletier, R. F., et al. 2005, MNRAS, 364, 773, doi: 10.1111/j.1365-2966.2005.09648.x
  * Franx et al. (1994) Franx, M., van Gorkom, J. H., & de Zeeuw, T. 1994, ApJ, 436, 642, doi: 10.1086/174939
  * Fraser-McKelvie et al. (2019) Fraser-McKelvie, A., Merrifield, M., Aragón-Salamanca, A., et al. 2019, MNRAS, 488, L6, doi: 10.1093/mnrasl/slz085
  * Fraser-McKelvie et al. (2020) Fraser-McKelvie, A., Aragón-Salamanca, A., Merrifield, M., et al. 2020, MNRAS, 495, 4158, doi: 10.1093/mnras/staa1416
  * Gadotti et al. (2020) Gadotti, D. A., Bittner, A., Falcón-Barroso, J., et al. 2020, A&A, 643, A14, doi: 10.1051/0004-6361/202038448
  * Galbany et al. (2016) Galbany, L., Anderson, J. P., Rosales-Ortega, F. F., et al. 2016, MNRAS, 455, 4087, doi: 10.1093/mnras/stv2620
  * Gomes et al. (2016) Gomes, J. M., Papaderos, P., Kehrig, C., et al. 2016, A&A, 588, A68, doi: 10.1051/0004-6361/201525976
  * Holmes et al. (2015) Holmes, L., Spekkens, K., Sánchez, S. F., et al. 2015, MNRAS, 451, 4397, doi: 10.1093/mnras/stv1254
  * Jedrzejewski (1987) Jedrzejewski, R. I. 1987, MNRAS, 226, 747, doi: 10.1093/mnras/226.4.747
  * Kass & Raftery (1995) Kass, R. E., & Raftery, A. E. 1995, Journal of the American Statistical Association, 90, 773, doi: 10.1080/01621459.1995.10476572
  * Knapen et al. (2000) Knapen, J. H., Shlosman, I., & Peletier, R. F. 2000, ApJ, 529, 93, doi: 10.1086/308266
  * Kormendy (1983) Kormendy, J. 1983, ApJ, 275, 529, doi: 10.1086/161552
  * Lacerda et al. (2018) Lacerda, E. A. D., Cid Fernandes, R., Couto, G. S., et al. 2018, MNRAS, 474, 3727, doi: 10.1093/mnras/stx3022
  * Laurikainen et al. (2018) Laurikainen, E., Salo, H., Laine, J., & Janz, J. 2018, A&A, 618, A34, doi: 10.1051/0004-6361/201833138
  * Lopez-Coba et al. (2021) Lopez-Coba, C., Lin, L.-H., & Sanchez Sanchez, S. F. 2021, arXiv e-prints, arXiv:2110.05095.  https://arxiv.org/abs/2110.05095
  * López-Cobá et al. (2019) López-Cobá, C., Sánchez, S. F., Bland -Hawthorn, J., et al. 2019, MNRAS, 482, 4032, doi: 10.1093/mnras/sty2960
  * López-Cobá et al. (2020) López-Cobá, C., Sánchez, S. F., Anderson, J. P., et al. 2020, AJ, 159, 167, doi: 10.3847/1538-3881/ab7848
  * Méndez-Abreu et al. (2019) Méndez-Abreu, J., Sánchez, S. F., & de Lorenzo-Cáceres, A. 2019, MNRAS, 484, 4298, doi: 10.1093/mnras/stz276
  * Minchev & Famaey (2010) Minchev, I., & Famaey, B. 2010, ApJ, 722, 112, doi: 10.1088/0004-637X/722/1/112
  * Neumann et al. (2020) Neumann, J., Fragkoudi, F., Pérez, I., et al. 2020, A&A, 637, A56, doi: 10.1051/0004-6361/202037604
  * Oh et al. (2011) Oh, S.-H., de Blok, W. J. G., Brinks, E., Walter, F., & Kennicutt, Robert C., J. 2011, AJ, 141, 193, doi: 10.1088/0004-6256/141/6/193
  * Oman et al. (2019) Oman, K. A., Marasco, A., Navarro, J. F., et al. 2019, MNRAS, 482, 821, doi: 10.1093/mnras/sty2687
  * Osterbrock (1989) Osterbrock, D. E. 1989, Astrophysics of gaseous nebulae and active galactic nuclei
  * Pence & Blackman (1984) Pence, W. D., & Blackman, C. P. 1984, MNRAS, 210, 547, doi: 10.1093/mnras/210.3.547
  * Pérez et al. (2009) Pérez, I., Sánchez-Blázquez, P., & Zurita, A. 2009, A&A, 495, 775, doi: 10.1051/0004-6361:200811084
  * Peterson et al. (1978) Peterson, C. J., Rubin, V. C., Ford, W. K., J., & Thonnard, N. 1978, ApJ, 219, 31, doi: 10.1086/155752
  * Sánchez (2020) Sánchez, S. F. 2020, ARA&A, 58, 99, doi: 10.1146/annurev-astro-012120-013326
  * Sánchez et al. (2012) Sánchez, S. F., Kennicutt, R. C., Gil de Paz, A., et al. 2012, A&A, 538, A8, doi: 10.1051/0004-6361/201117353
  * Sánchez et al. (2016) Sánchez, S. F., Pérez, E., Sánchez-Blázquez, P., et al. 2016, Rev. Mexicana Astron. Astrofis., 52, 21.  https://arxiv.org/abs/1509.08552
  * Sánchez-Blázquez et al. (2011) Sánchez-Blázquez, P., Ocvirk, P., Gibson, B. K., Pérez, I., & Peletier, R. F. 2011, MNRAS, 415, 709, doi: 10.1111/j.1365-2966.2011.18749.x
  * Sánchez-Blázquez et al. (2014) Sánchez-Blázquez, P., Rosales-Ortega, F. F., Méndez-Abreu, J., et al. 2014, A&A, 570, A6, doi: 10.1051/0004-6361/201423635
  * Schoenmakers et al. (1997) Schoenmakers, R. H. M., Franx, M., & de Zeeuw, P. T. 1997, MNRAS, 292, 349, doi: 10.1093/mnras/292.2.349
  * Sellwood & Binney (2002) Sellwood, J. A., & Binney, J. J. 2002, MNRAS, 336, 785, doi: 10.1046/j.1365-8711.2002.05806.x
  * Sellwood & Sánchez (2010) Sellwood, J. A., & Sánchez, R. Z. 2010, MNRAS, 404, 1733, doi: 10.1111/j.1365-2966.2010.16430.x
  * Sellwood & Wilkinson (1993) Sellwood, J. A., & Wilkinson, A. 1993, Reports on Progress in Physics, 56, 173, doi: 10.1088/0034-4885/56/2/001
  * Shetty et al. (2020) Shetty, S., Bershady, M. A., Westfall, K. B., et al. 2020, ApJ, 901, 101, doi: 10.3847/1538-4357/ab9b8e
  * Spekkens & Sellwood (2007) Spekkens, K., & Sellwood, J. A. 2007, ApJ, 664, 204, doi: 10.1086/518471
  * Stasińska et al. (2008) Stasińska, G., Vale Asari, N., Cid Fernandes, R., et al. 2008, MNRAS, 391, L29, doi: 10.1111/j.1745-3933.2008.00550.x
  * van Albada & Roberts (1981) van Albada, G. D., & Roberts, W. W., J. 1981, ApJ, 246, 740, doi: 10.1086/158969
  * Vera et al. (2016) Vera, M., Alonso, S., & Coldwell, G. 2016, A&A, 595, A63, doi: 10.1051/0004-6361/201628750
  * Weliachew et al. (1988) Weliachew, L., Casoli, F., & Combes, F. 1988, A&A, 199, 29
  * Wong et al. (2004) Wong, T., Blitz, L., & Bosma, A. 2004, ApJ, 605, 183, doi: 10.1086/382215
  * Wozniak & Pierce (1991) Wozniak, H., & Pierce, M. J. 1991, A&AS, 88, 325