# Rotation curves and dynamical masses of MaNGA barred galaxies

Eduardo O. Schmidt  ID ,1,2,3 Damián Mast  ID , 1,2 Gaia Gaspar  ID , 1,2 and
Walter Weidmann  ID , 1,2
1Observatorio Astronómico de Córdoba, Universidad Nacional de Córdoba, Laprida
854, Córdoba, Argentina.
2Consejo de Investigaciones Científicas y Técnicas de la República Argentina,
Buenos Aires, Argentina.
3Instituto de Astronomía Teórica y Experimental (IATE), CONICET-UNC, Laprida
854, Córdoba, Argentina E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

In this paper we analyze a sample of 46 barred galaxies of MaNGA. Our goal is
to investigate the stellar kinematics of these galaxies and obtain their
rotation curves. Additionally, we aim to derive the total stellar and
dynamical masses, as well as the maximum rotation velocity, in order to
examine their distributions and scaling relations.

Using the Pipe3D dataproducts publicly available we obtained the rotation
curves, which were fitted considering two components of an axisymmetric
Miyamoto–Nagai gravitational potential.

We found a wide range of the maximum rotation velocities (117-340 ${\rm
km\,s^{-1}}$), with a mean value of 200 ${\rm km\,s^{-1}}$. In addition we
found that the total stellar and dynamical masses are in the range of
log(Mstar/M⊙)$=$ 10.1$-$11.5, with a mean value of log(Mstar/M⊙)$=$ 10.8, and
log(Mdyn/M⊙)$=$ 10.4$-$12.0, with a mean value of log(Mdyn/M⊙)$=$ 11.1,
respectively. We found a strong correlation between dynamical mass and maximum
velocity, between maximum velocity and magnitude, and between stellar mass and
maximum velocity. According to these results, barred galaxies exhibit similar
behaviour to that of normal spiral galaxies with respect to these relations,
as well as in terms of the distribution of their dynamical mass and maximum
rotation velocity. However, we found that the distribution of stellar masses
of barred galaxies is statistically different from other samples including
non-barred galaxies. Finally, analyzing the galaxies that show nuclear
activity, we find no difference with the rotation curves of normal galaxies.

###### keywords:

galaxies: kinematics and dynamics – galaxies: spiral – galaxies: structure

††pubyear: 2022††pagerange: Rotation curves and dynamical masses of MaNGA
barred galaxies–B

##  1 Introduction

Rotation curves (RCs) are the main tracers of the distribution of mass in
spiral galaxies (e.g., Ashman, 1992; Persic & Salucci, 1995; Sofue & Rubin,
2001; Kalinova et al., 2017a). As such, they represent the major tool for
determining the mass distribution, providing fundamental information for
understanding the evolution, dynamics, and formation of spiral galaxies (e.g.,
Persic & Salucci, 1995; Sofue et al., 1999; Sofue & Rubin, 2001; Martinez-
Medina et al., 2020).

There are typical kinematic tracers for galaxies which include stars (e.g.,
Bekeraité et al., 2016a, b; Kalinova et al., 2017a, b; Yoon et al., 2021) and
ionised, atomic or molecular gas (e.g., Levy et al., 2018). In this context,
astronomers have at their disposal several observing techniques and methods
for determining rotation curves and velocity fields for both gas and stars
(Falcón-Barroso et al., 2017; García-Lorenzo et al., 2015). For example,
traditional long slit spectroscopy, Fabry-Perot spectrographs, or integral
field instruments are often used for these purposes (see for example the
review of Sofue & Rubin, 2001). With the advent of large integral-field
spectroscopic (IFS) surveys such as CALIFA (Alto Legacy Integral Field Area,
Sánchez et al., 2012), SAMI (Sydney-AAO (Australian Astronomical Observatory)
Multi-object IFS, Croom et al., 2012), or MaNGA (Mapping Nearby Galaxies at
Apache Point Observatory, Bundy et al., 2015), the possibility of carrying out
detailed studies with samples covering wide ranges of parameters has become a
reality. Rotation curves can be obtained by measuring spectroscopic emission
lines such as H$\alpha$, [OIII], [NII], [SII], HI, and CO lines and stellar
absorption lines such as the generated by Ca, H, He, K, Mg, Na, among other
chemical elements, and techniques vary from single line to full spectrum
fitting (Cappellari & Emsellem, 2004; Cappellari, 2017; Sánchez et al.,
2016a).

Each of the different tracers has its difficulties when studying and analyzing
them (e.g., Leung et al., 2018; Martinez-Medina et al., 2020), in addition to
preferentially tracing different components in the galaxy. For example, the
gas kinematic has an intrinsic turbulent nature and therefore gas velocity
fields are complex (e.g., Barrera-Ballesteros et al., 2014). Particularly, the
CO molecule emission is considered to be a good tracer of the circular
velocity of disc galaxies, however the typical presence of bars and spiral
arms clearly influences the molecular gas orbits, making them turbulent (e.g.,
Laine et al., 1999; Shetty et al., 2007). In this context, although stellar
kinematics can present higher velocity dispersion, stellar velocity fields are
less distorted than gas fields (e.g., Adams et al., 2012; Bekeraité et al.,
2016a). However, stellar kinematics presents its complications when it comes
to deriving properties. Leung et al. (2018) studied and analyzed three
dynamical models used to calculate masses from stellar dynamics, which are the
Jeans method, the asymmetric drift correction method, and the Schwarzschild
model method. These authors found that the three models can reproduce the CO
circular velocity at 1 effective radius to within 10%. The scatter in the
inner regions can be of $\sim$20%, because some assumptions may break down
(see Leung et al., 2018, for more details). These authors found that all three
models can recover the dynamical mass at 1 effective radius better than 20%.
For their part, Martinsson et al. (2013) studied [OIII]$\lambda$5007 and
stellar kinematics of 30 nearby spiral galaxies and found that the [OIII] and
stellar rotation curves exhibit signatures of asymmetric drift with a rotation
difference that is 11% of the maximum rotation speed of the galaxy disk.

It is clear that a sufficiently complete view of the mass distribution in a
galaxy can only be achieved through the use of different tracers. Moreover,
understanding the scope and drawbacks, not only of the tracers but also of the
limitations of the observational techniques used for each of the measurements,
is a fundamental step in our understanding of the mass distribution in
galaxies. Thus, having independent mass determinations for homogeneous samples
of galaxies that allow us to compare analysis methods and even observational
regimes (such as low spatial resolution and spectroscopic observations) is a
fundamental building block in the study of galaxy dynamics and evolution in
this new era of large integral field spectroscopy galaxy surveys.

In this work, we study the stellar kinematics of barred galaxies to obtain
stellar rotation curves that are less affected by the turbulent motions more
characteristic of gas kinematics. We intend to avoid the volatility inherent
to gas curves that are generated by any perturbation. This way, we aim to
analyze the stellar kinematic and derive the stellar rotation curves of a
sample of MaNGA barred galaxies.

Several papers in the literature investigate and present RCs using different
methods. For example, Mathewson et al. (1992) study the RCs of 965 southern
spiral galaxies through H$\alpha$ and HI emission. Makarov et al. (2001)
present RCs of 135 edge-on galaxies through H$\alpha$ and Vogt et al. (2004a,
b) study rotation curves of 329 spiral galaxies using H$\alpha$ and HI
emission. In addition, Márquez et al. (2002) derive rotation curves of 111
galaxies through H$\alpha$ and Spano et al. (2008) study rotation curves of 36
galaxies through HI emission. For their part, Lang et al. (2020) present RCs
of 67 galaxies (including 45 barred galaxies) through CO emission. Moreover,
Martinsson et al. (2013) determined the ionized-gas ([OIII$\lambda$5007]) and
stellar rotation curves of a sample of 30 nearby spiral galaxies and Hernandez
et al. (2005) present the H$\alpha$ gas kinematics of 21 barred spiral
galaxies. In addition, Bekeraité et al. (2016a) study the stellar rotation
curves of 199 rotating galaxies and Kalinova et al. (2017b) present the
stellar rotation curves of 238 galaxies. Moreover, Di Teodoro et al. (2023)
present the HI rotation curves of 15 massive spiral of the local universe and
Kalinova et al. (2017a) study the stellar rotation curves of 18 late type
spirals. Yoon et al. (2021) analysed the shape of the RCs of the MaNGA data
release. In general, the aims of those works vary from testing the cold dark
matter (CDM) models of galaxy formation to analyzing scaling relations. In any
case, it is important to have homogeneously observed and analyzed dataset. In
this context, one of the main goals of this work is to study the stellar
kinematic and obtain the stellar rotation curves of a sample of 46 barred
galaxies, and then, by fitting these RCs, derive the dynamical masses of an
homogeneous galaxy sample.

There are different methods for determining dynamical masses of spiral
galaxies. For example, Aquino-Ortíz et al. (2020) use the stellar kinematics
for galaxies from MaNGA to explore a universal fundamental plane and the
relation between the dynamical and stellar masses. In this scenario, RCs
consitute an independent method for determining the dynamical masses of spiral
galaxies (e.g., Burbidge et al., 1959, 1961; Rubin et al., 1964; Krumm &
Salpeter, 1977; Pence, 1981; Agüero et al., 2004; Jałocha et al., 2010; Daod &
Zeki, 2019). As we mentioned above, obtaining the total dynamical mass of the
sample of barred spiral galaxies by analyzing their stellar rotation curves,
represents a valuable contribution to the study of galaxy dynamics.

Even though barred galaxies constitute $\sim$50% of all disk galaxies (e.g.,
Marinova & Jogee, 2007; Sheth et al., 2008; Aguerri et al., 2009), they have
been observed and studied less than non-barred galaxies in part due to their
kinematic complexity. However, although the kinematic study of barred galaxies
can be more difficult than for non-barred galaxies because their gas tracers
are very in-homogeneously distributed on small scales (e.g., Bosma, 1996), the
large-scale properties of the rotation curves of barred galaxies are generally
similar to those of non-barred galaxies (e.g., Sofue & Rubin, 2001).

This paper is organized as follows. Sect. 2 presents the sample of barred
spiral galaxies and the data we worked with. Sect. 3 describes the procedure
by which the velocity fields and radial velocity curves were obtained. In
Sect. 4 we present our results, i.e., the process of obtaining and fitting the
rotation curves, the derivation of the dynamical and stellar masses of the
galaxies, and the study of the scalling relations. In Sect. 5 we make a
comparison between our results and those of the literature. Finally, in Sect.
6 we provide a brief summary of the work and make final remarks.

##  2 Sample

For our study, we selected the galaxy sample analysed by Guo et al. (2019),
which consists of barred galaxies from MaNGA. This survey investigates the
internal structure and composition of gas and stars in a sample of $\sim$
10000 galaxies of the nearby universe (for more details of the project see
Bundy et al., 2015). We used the integral field unit (IFU) spectroscopic data
taken at the 2.5 m Sloan Telescope (Gunn et al., 2006), which has 1423 fibres
with 2″ core diameters over a 3º diameter field of view.

The sample studied by Guo et al. (2019) is made up of 53 plate-IFU
observations. Note that plates IFU 8256$-$6101 and 8274$-$6101 correspond to
the galaxy SDSS J105456.35+412954.3 and plates 8588-3701 and 8603-12701
correspond to SDSS J163233.73+390751.7. In addition, we excluded from the
analysis the galaxies SDSS J154702.69+542027.1, SDSS J162200.73$+$493114.8,
SDSS J164029.54$+$391406.3 and SDSS J162907.83$+$403954.1 (plate-IFU:
8481-12701, 8482-12703, 8601-12705, and 8603-12703, respectively) which have a
noisy velocity map. Moreover, as we will see below (Sect. 4) one galaxy (SDSS
J090641.14$+$412154.3, plate-IFU 8247$-$3701) was rejected from the analysis
because the dispersion of velocities dominates its entire curve compared with
the rotation velocity. This way, our final sample consists of 46 barred
galaxies. Table 1 lists the main parameters of the galaxy sample, such as the
MaNGA ID, the galaxy name, right ascension (J2000), declination (J2000),
morphological type, absolute r-band magnitude from SDSS-DR9 (Ahn et al.,
2012), effective radius in the r-band from SDSS-DR9, redshift, galaxy
inclination, galaxy kinematic PA and systemic velocity. All data were taken
from Guo et al. (2019), except the galaxy name (column 2), which has been
taken from the Sloan Digital Sky Survey (SDSS) IV DATA Release 13 (Albareti et
al., 2017) and the systemic velocity (column 11), which has been obtained in
this work (Sect. 3). The galaxy inclination was measured from the ellipticity
radial profile using ellipse fitting of r-band SDSS image, while the galaxy
kinematic PA was measured from the velocity map using
fit_kinematic_pa.py111https://www-astro.physics.ox.ac.uk/ cappellari/software/
code (see Guo et al., 2019, for more details). In addition, we have made use
of the Nasa Extragalactic Database (NED)222The NASA/IPAC Extragalactic
Database (NED) is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration. to verify galaxy parameters such us the radial velocity.

In order to make the characterisation of the sample more comprehensive, in
Fig. 1 we present the redshift and r-band magnitude distributions for our
sample of galaxies (left and right panels, respectively). The redshift of the
entire sample ranges from 0.0228 to 0.1303 and has a distribution with a mean
value of z $=$ 0.0399, a standard deviation of 0.019 and an interquartile
range (IQR) of 0.021. There is one object with $z\sim$ 0.13, while the
remaining galaxies (98 % of the sample) have redshifts in the range 0.0228 $-$
0.0753. By removing this object from the analysis, the redshift distribution
would have a standard deviation of 0.0138. Regarding the magnitude, it ranges
from $-$22.89 to $-$19.77, with a mean value of $-$21.33 and a distribution
with a standard deviation of 0.82 and an IQR of 1.16. Both distributions show
that the galaxy sample is homogeneous in both redshift and magnitude.

Figure 1: Redshift and r-band magnitude distributions for the galaxies of our sample are shown in the left and right panels, respectively. The data of both plots were taken from Guo et al. (2019). Table 1:  The sample of 46 barred galaxies. Columms: (1) MaNGA ID; (2) galaxy name; (3) right ascension (J2000); (4) declination (J2000); (5) morphological type; (6) absolute r-band magnitude from SDSS-DR9; (7) effective radius in the r-band from SDSS-DR9; (8) redshift; (9) galaxy inclination; (10) galaxy kinematic PA; (11) systemic velocity. All data were taken from Guo et al. (2019), except the galaxy name (from the SDSS DR13) and the systemic velocity (Sect. 3). Plate IFU | Galaxy name | RA | DEC | Morph. | Mr | $R_{e}$ | $z$ | $i$ | $PA_{kin}$ | $V_{syst}$
---|---|---|---|---|---|---|---|---|---|---
|  | (°) | (°) |  | (mag) | (“) |  | (°) | (°) | (${\rm km\,s^{-1}}$)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11)
7495$-$12704 | SDSS J134145.21$+$270016.9 | 205.4384 | 27.0048 | SBbc |  $-$21.40 | 8.65 | 0.0289 | 52.2$\pm$0.6 | 173.0 $\pm$0.6 | 8768
7962$-$12703 | SDSS J172452.14$+$280441.8 | 261.2173 | 28.0783 | SBab |  $-$22.33 | 8.34 | 0.0477 | 61.2$\pm$1.2 | 37.0 $\pm$0.9 | 14452
7990$-$3704 | SDSS J172817.97$+$564629.2 | 262.0749 | 56.7748 | SB0 |  $-$20.15 | 3.83 | 0.0291 | 39.4$\pm$1.4 | 15.2 $\pm$3.4 | 8800
7990$-$9101 | SDSS J171901.32$+$571024.6 | 259.7555 | 57.1735 | SBc |  $-$19.77 | 4.51 | 0.0280 | 71.8$\pm$0.2 | 20.0 $\pm$3.8 | 8488
7992$-$6104 | SDSS J170107.08$+$644036.6 | 255.2795 | 64.6769 | SBc |  $-$20.31 | 8.78 | 0.0271 | 46.7$\pm$1.8 | 6.0 $\pm$2.8 | 8188
8082$-$6102 | SDSS J031947.01$+$003504.4 | 49.9459 | 0.5846 | SB0 |  $-$21.46 | 6.91 | 0.0242 | 41.3$\pm$0.5 | 99.0 $\pm$0.9 | 7346
8083$-$6102 | SDSS J032427.59$-$000510.8 | 51.1150 |  $-$0.0863 | SBa |  $-$21.62 | 4.70 | 0.0365 | 70.4$\pm$0.2 | 62.8 $\pm$0.9 | 11028
8083$-$12704 | SDSS J032247.22$+$000857.6 | 50.6968 | 0.1494 | SBbc |  $-$21.03 | 13.32 | 0.0228 | 41.7$\pm$0.9 | 167.0 $\pm$1.4 | 6946
8133$-$3701 | SDSS J072819.02$+$431807.5 | 112.0793 | 43.3021 | SBb |  $-$20.10 | 2.39 | 0.0437 | 44.6$\pm$1.1 | 102.8 $\pm$3.5 | 13159
8134$-$6102 | SDSS J073941.88$+$455445.4 | 114.9245 | 45.9126 | SB0a |  $-$21.40 | 5.98 | 0.0320 | 53.8$\pm$0.9 | 93.0 $\pm$0.8 | 9670
8137$-$9102 | SDSS J074809.26$+$433526.4 | 117.0386 | 43.5907 | SBb |  $-$21.07 | 6.68 | 0.0311 | 43.3$\pm$2.2 | 132.8 $\pm$1.9 | 9417
8140$-$12701 | SDSS J074743.27$+$412310.9 | 116.9303 | 41.3864 | SBa |  $-$20.61 | 5.69 | 0.0286 | 37.8$\pm$1.3 | 62.8 $\pm$1.8 | 8674
8140$-$12703 | SDSS J075135.62$+$425248.3 | 117.8985 | 42.8801 | SBb |  $-$21.87 | 9.85 | 0.0320 | 55.0$\pm$0.6 | 28.0 $\pm$1.1 | 9652
8243$-$6103 | SDSS J083641.96$+$534338.0 | 129.1749 | 53.7272 | SB0 |  $-$21.65 | 4.75 | 0.0315 | 59.1$\pm$0.6 | 9.8 $\pm$0.6 | 9526
8244$-$3703 | SDSS J084758.26$+$513603.6 | 131.9928 | 51.6010 | SB0 |  $-$21.03 | 2.50 | 0.0483 | 46.1$\pm$1.1 | 71.5 $\pm$1.6 | 14561
8249$-$6101 | SDSS J091014.98$+$461735.8 | 137.5625 | 46.2933 | SBc |  $-$20.27 | 4.64 | 0.0267 | 48.7$\pm$1.4 | 63.5 $\pm$1.6 | 8092
8254$-$9101 | SDSS J104502.80$+$434217.0 | 161.2617 | 43.7048 | SBa |  $-$21.78 | 8.00 | 0.0253 | 44.1$\pm$1.6 | 27.2 $\pm$0.8 | 7698
8256$-$6101 | SDSS J105456.35$+$412954.3 | 163.7348 | 41.4985 | SBa |  $-$20.79 | 6.06 | 0.0246 | 51.4$\pm$2.6 | 134.0 $\pm$0.9 | 7492
8257$-$3703 | SDSS J110637.35$+$460219.6 | 166.6557 | 46.0388 | SBb |  $-$20.34 | 4.03 | 0.0250 | 58.3$\pm$0.6 | 155.2 $\pm$1.2 | 7543
8257$-$6101 | SDSS J110102.71$+$445317.7 | 165.2613 | 44.8882 | SBc |  $-$20.86 | 5.77 | 0.0294 | 45.0$\pm$2.2 | 159.2 $\pm$1.5 | 8904
8312$-$12702 | SDSS J162105.00$+$395502.6 | 245.2709 | 39.9174 | SBc |  $-$21.24 | 7.23 | 0.0320 | 42.9$\pm$1.1 | 95.2 $\pm$1.8 | 9698
8312$-$12704 | SDSS J162912.98$+$410903.1 | 247.3041 | 41.1509 | SBb |  $-$21.00 | 7.47 | 0.0296 | 46.1$\pm$0.7 | 34.0 $\pm$1.8 | 8977
8313$-$9101 | SDSS J155847.40$+$415617.1 | 239.6975 | 41.9381 | SBb |  $-$21.87 | 6.76 | 0.0387 | 38.6$\pm$0.7 | 110.5 $\pm$1.0 | 11685
8317$-$12704 | SDSS J125448.95$+$440920.1 | 193.7040 | 44.1556 | SBa |  $-$22.68 | 7.14 | 0.0543 | 69.2$\pm$0.3 | 101.8 $\pm$0.9 | 16355
8318$-$12703 | SDSS J130455.76$+$473013.0 | 196.2324 | 47.5036 | SBb |  $-$22.21 | 9.09 | 0.0393 | 61.8$\pm$0.9 | 53.8 $\pm$0.9 | 11869
8320$-$6101 | SDSS J134630.60$+$224221.6 | 206.6275 | 22.7060 | SBb |  $-$20.37 | 5.22 | 0.0266 | 50.0$\pm$0.6 | 5.0 $\pm$1.1 | 8046
8326$-$3704 | SDSS J141924.05$+$455402.8 | 214.8502 | 45.9008 | SBa |  $-$20.25 | 3.83 | 0.0265 | 50.4$\pm$1.1 | 159.8 $\pm$3.0 | 8039
8326$-$6102 | SDSS J142004.29$+$470716.8 | 215.0179 | 47.1213 | SBb |  $-$22.06 | 2.95 | 0.0704 | 51.9$\pm$0.9 | 145.8 $\pm$1.6 | 21156
8330$-$12703 | SDSS J133329.90$+$403146.8 | 203.3746 | 40.5297 | SBbc |  $-$20.67 | 7.51 | 0.0269 | 45.0$\pm$0.5 | 68.5 $\pm$1.9 | 8143
8335$-$12701 | SDSS J142134.86$+$402129.1 | 215.3953 | 40.3581 | SBb |  $-$21.66 | 4.39 | 0.0633 | 67.0$\pm$0.5 | 78.2 $\pm$1.4 | 19096
8439$-$6102 | SDSS J093106.75$+$490447.1 | 142.7782 | 49.0797 | SBab |  $-$21.64 | 4.54 | 0.0339 | 49.3$\pm$0.5 | 45.5 $\pm$1.1 | 10229
8439$-$12702 | SDSS J092609.43$+$491836.7 | 141.5393 | 49.3102 | SBa |  $-$21.57 | 8.10 | 0.0269 | 55.1$\pm$0.4 | 31.5 $\pm$0.5 | 8167
8440$-$12704 | SDSS J090434.15$+$412352.1 | 136.1423 | 41.3978 | SBb |  $-$21.12 | 4.56 | 0.0270 | 57.9$\pm$0.4 | 150.0 $\pm$0.8 | 8201
8447$-$6101 | SDSS J134431.98$+$401424.0 | 206.1333 | 40.2400 | SBb |  $-$22.89 | 4.48 | 0.0753 | 63.9$\pm$0.8 | 178.2 $\pm$1.2 | 22661
8452$-$3704 | SDSS J103009.35$+$471642.2 | 157.5390 | 47.2784 | SBc |  $-$19.97 | 4.34 | 0.0251 | 59.7$\pm$0.3 | 72.0 $\pm$2.5 | 7597
8452$-$12703 | SDSS J102713.35$+$481441.2 | 156.8057 | 48.2448 | SBb |  $-$22.83 | 8.13 | 0.0610 | 45.7$\pm$2.4 | 65.0 $\pm$1.2 | 18356
8482$-$9102 | SDSS J102713.35$+$481441.2 | 242.9559 | 49.2287 | SBb |  $-$21.59 | 3.54 | 0.0580 | 62.6$\pm$0.6 | 63.8 $\pm$1.9 | 17493
8482$-$12705 | SDSS J161652.00$+$501655.8 | 244.2167 | 50.2822 | SBb |  $-$22.06 | 7.39 | 0.0417 | 63.0$\pm$1.0 | 117.0 $\pm$1.0 | 12583
8486$-$6101 | SDSS J155209.48$+$461911.2 | 238.0396 | 46.3198 | SBc |  $-$21.57 | 3.56 | 0.0589 | 40.4$\pm$1.2 | 113.5 $\pm$1.4 | 17734
8548$-$6102 | SDSS J162205.36$+$463726.9 | 245.5224 | 46.6242 | SBc |  $-$20.83 | 3.85 | 0.0478 | 54.1$\pm$0.4 | 58.8 $\pm$3.6 | 14386
8548$-$6104 | SDSS J162259.35$+$464031.2 | 245.7474 | 46.6753 | SBc |  $-$20.47 | 2.73 | 0.0480 | 62.2$\pm$1.6 | 120.2 $\pm$5.0 | 14465
8549$-$12702 | SDSS J160505.14$+$452634.7 | 241.2714 | 45.4430 | SBb |  $-$22.03 | 6.72 | 0.0433 | 54.3$\pm$2.6 | 100.8 $\pm$1.0 | 13084
8588$-$3701 | SDSS J163233.73$+$390751.7 | 248.1406 | 39.1310 | SBb |  $-$22.88 | 4.43 | 0.1303 | 40.4$\pm$1.7 | 136.2 $\pm$1.9 | 39164
8604$-$12703 | SDSS J163103.40$+$395018.5 | 247.7642 | 39.8385 | SBab |  $-$21.67 | 9.08 | 0.0305 | 48.8$\pm$1.0 | 97.8 $\pm$1.0 | 9254
8612$-$6104 | SDSS J170001.67$+$384857.7 | 255.0069 | 38.8160 | SBb |  $-$21.83 | 8.60 | 0.0356 | 42.4$\pm$2.3 | 153.5 $\pm$1.8 | 10749
8612$-$12702 | SDSS J165547.13$+$391837.8 | 253.9464 | 39.3105 | SBc |  $-$22.60 | 8.26 | 0.0631 | 52.3$\pm$1.0 | 44.0 $\pm$1.4 | 19004

##  3 Procedure

The data included in this paper are from Sloan Digital Sky Survey (SDSS) IV
DATA Release 13 333https://data.sdss.org/sas/dr13/manga/spectro/redux/v1_5_4/
(Albareti et al., 2017). We used the publicly available dataproducts obtained
by Pipe3D (Sánchez et al., 2016b, 2018a). From these cubes we extracted the
V-band pseudo-image, the stellar radial velocity field, and the error in the
stellar radial velocity determination for each spaxel. Using MARVIN software
(Cherinka et al., 2019) we also obtained the SDSS color image with the
indicated IFU MaNGA FoV for comparison with the cube maps. The pseudo V-image
was used to determine the peak emission and the systemic velocity. From the
kinematic position angle (PAKin) determined by Guo et al. (2019) we carried
out an extraction of the radial velocities in a 1 spaxel width pseudo-slit.
This allowed us to construct the stellar radial velocity curves. Errors were
estimated from the error plane in MaNGA datacubes. We used the value of the
radial velocity error indicated in the Pipe3D maps for each spaxel. A typical
error bar is indicated in the plots. In the upper panels of Fig. 15 we show
the SDSS images, the pseudo V-images, and the stellar radial velocity fields
for each galaxy.

For the choice of the systemic velocity ($V_{syst}$) for each galaxy, the
velocities corresponding to the peak of the pseudo-image obtained in the
V-band were compared with the velocity that best symmetrizes the radial
velocity curve. This comparison is performed to analyze possible off-centering
of the velocity fields with the luminosity distribution, but in all cases the
displacements were less than 2 arcsec, which implies that they are within the
spatial resolution element. Thus the $V_{syst}$ adopted are those that best
symmetrize the radial velocity curves. In all cases these $V_{syst}$ are in
agreement with the values found in the literature and in the NED.

##  4 Results

###  4.1 The rotation curves

Once the radial velocity curves are determined, the rotation velocity (Vrot)
can be calculated as:

$V_{rot}=\frac{V_{obs}-V_{syst}}{sin(i)}$ (1)

where Vobs is the observed radial velocity, Vsyst is the systemic radial
velocity and $i$ is the galaxy inclination.

In order to fit all the RCs, we made use of
galpy444http://github.com/jobovy/galpy (Bovy, 2015), which is a Python package
for galactic dynamics. For this, we considered two mass components
corresponding to axisymetric Miyamoto$-$Nagai gravitational potentials
(Miyamoto & Nagai, 1975):

$\Phi~{}(R,z)=-\frac{GM}{\sqrt{R^{2}+[a+\sqrt{z^{2}+b^{2}}]^{2}}}$ (2)

where $\Phi(R,z)$ is the Miyamoto$-$Nagai potential at $(R,z)$ position, $a$
and $b$ are shape parameters and $M$ is the total mass. By way of example, the
ratio b/a $\sim$ 0.4 corresponds to a flattened disk distribution, b/a $=$ 1
to an ellipsoidal distribution and b/a $\sim$ 5 to a spherical distribution
(e.g., Binney & Tremaine, 1987).

Given its simplicity, the Miyamoto$-$Nagai potential is one of the most widely
used models for fitting rotation curves (e.g., Binney & Tremaine, 1987; Sofue,
2017; Granados et al., 2017; Ciotti, 2021, 2022). It is a generalisation of
the Plummer’s spherical potential (when a $=$ 0) and the Kuzmin’s potential of
a razor-thin disk (when b $=$ 0). This way, depending on the values of the two
parameters a and b, the Miyamoto$-$Nagai model can represent the potential of
anything from an infinitesimally thin disk to a spherical distribution (see
for example Binney & Tremaine, 1987, for more details).

From the centrifugal-equilibrium condition, the rotation velocity in the
equatorial plane is related to the gravitational potential $\Phi$(R, z$=$0) in
the form:

$V_{rot}~{}(R)=\sqrt{-R(\partial\Phi/\partial R)}$ (3)

Taking into account Eq. 2 and Eq. 3, the rotation velocity in the galactic
plane at z $=$ 0 is given by:

$V_{rot}~{}(R)=R\sqrt{\frac{GM}{[R^{2}+(a+b)^{2}]^{3/2}}}$ (4)

The fitting of the rotation curves was carried out in a two-stage procedure.
The first thing we did was to perform a least squares fit to the data in order
to have a first approximation. However, an automatic fit may consider regions
where there are non-circular motions generated by nonaxisymmetric disturbances
and thus affect the global fit that takes into account rotation velocities.
Because of this, we carefully checked each curve individually, and using the
least square fit as a reference, we made used of galpy improving each fit by
making it take into account only the observed rotation motions, which are
representative of the gravitational potential. This procedure was carried out
varying the input parameters mass, a, and b and considering each galaxy
individually to ensure the best fit, which has been carefully inspected by
eye. To determine the uncertainties of the fits, we repeated the measurement
procedure 15 times for all galaxies. We assumed that any errors are given by
the dispersion in the distribution of these measurements, considering them at
1$\sigma$. Concerning the fits, relative errors in the velocity are in the
range 3% $-$ 8%. These errors take into account only our measurements
procedures, and do not involve other factors such as the observation process,
reduction or calibrations.

All galaxies of the sample were fitted considering the presence of two
galactic subsystems: a central ellipsoidal component and a flattened disk, in
agreement with previous works that also have fitted rotation curves of barred
spiral galaxies (e.g., Díaz et al., 1999; Agüero et al., 2004; Schmidt et al.,
2019).

Given the morphological characteristics of this sample, a significant fraction
of the galaxies are late type objects (see Table 1). Related to this and
considering the rotation curve fitting, the central component is almost absent
or negligible compared to the disk component in all but a few galaxies (see
for example the fits of 7495$-$12704 and 7962$-$12703, among others, in Fig.
15). Although for consistency in all the fittings the two components were
kept. In none of the fits a third component corresponding to the halos was
used, on the one hand because the fit was satisfactory with two components and
on the other hand because the scale radius of the halo is larger than the
radii involved in our data (e.g., Sofue, 2016). Related to this, it has to be
noted that the possible gravitational influence of a dark matter halo may be
implied by the disc component. However, in this work we do not intend to
physically interpret the different components separately, but consider the
total fit of the rotation curve that will allow us to robustly calculate the
dynamical masses of the galaxies of the sample (see Sect. 4.2).

In order to check for the reliability of the rotation curves as dynamical mass
estimators, we follow the procedure used by Kalinova et al. (2017a) and
Bekeraité et al. (2016a). This procedure considers that when the ordered-over-
random motion V$\phi/\sigma_{R}$ < 1.5, the rotation curve might not be a
reliable tracer. To do this, we made use of the dispersion maps available as
part of the Pipe3D dataproducts to construct the dispersion velocity radial
profiles to evaluate the V$\phi/\sigma_{R}$ ratio. We performed this test for
all galaxies in our sample. As we said in Sect. 2, we rejected one galaxy
(SDSS J090641.14+412154.3, plate-IFU 8247-3701) whose entire extension
presents an important contribution of the stellar velocity dispersion compared
with the rotation velocity. This way, our whole sample is constituted by
galaxies whose dynamic is dominated by rotation. Considering that practically
all the galaxies in our sample present a negligible bulge component, and in
order to make an homogeneous treatment to our data and to avoid any
potentially incorrect assumption, we will study the stellar kinematic of the
galaxies neglecting the correction methods mentioned in Sect.1 (e.g., Bureau &
Carignan, 2002; Di Teodoro et al., 2021; Di Teodoro et al., 2023).

Table 4 lists the obtained parameters of our fits, such as the scale
parameters, the mass of each galactic subsystem and the total mass. Fig. 15
presents all the fitted rotation curves of the galaxy sample. In addition,
Fig. 2 shows the distribution of the maximun rotation velocity, i.e., the
highest rotation velocity measured over each rotation curve. According to our
results, the maximun rotation velocities show a wide range, with values from
117 to 340 ${\rm km\,s^{-1}}$, with most of the objects showing a maximun
rotation velocity in the range $\sim$120$-$250 ${\rm km\,s^{-1}}$. The
distribution has a mean value of 200 ${\rm km\,s^{-1}}$, a standard deviation
of 50 ${\rm km\,s^{-1}}$ and an IQR of 57. These values are in agreement with
previous results on spiral galaxies (e.g., Persic & Salucci, 1995; Sofue &
Rubin, 2001; Agüero et al., 2004; Sofue, 2017; Schmidt et al., 2019; Martinez-
Medina et al., 2020, among others). There are 11 galaxies from our sample
(7495-12704, 8137-9102, 8257-6101, 8318-12703, 8330-12703, 8335-12701,
8439-12702, 8452-12703, 8482-12705, 8549-12702 and 8601-12705) that were also
studied by Pilyugin et al. (2019), who obtained rotation curves through the
analysis of the H$\alpha$ velocity field. In general, the rotation velocities
determined in this work are in good agreement with the results reported in the
mentioned paper. There are, however, some differences in the rotation
velocities of a few galaxies. These differences may probably be due to the
procedure used to construct the rotation curves, the adopted galaxy
inclination, and to the fact that these authors used H$\alpha$ kinematics
instead of the stellar velocity fields.

Figure 2: Distribution of the maximun rotation velocity in units of ${\rm
km\,s^{-1}}$ for our sample of barred galaxies.

There are some galaxies in our sample that present nuclear activity:
8257$-$6101 (Edelson et al., 1987; Toba et al., 2014), 8312$-$12704 (Toba et
al., 2014; Rembold et al., 2017), 8317$-$12704 (e.g., Rembold et al., 2017;
Bing et al., 2019), 8320$-$6101 (e.g., Stern & Laor, 2012), 8326$-$6102 (e.g.,
Greene & Ho, 2007; Stern & Laor, 2012), 8452$-$12703 (e.g., Rembold et al.,
2017), 8549$-$12702 (e.g., Greene & Ho, 2007; Stern & Laor, 2012), 8588$-$3701
(e.g., Rembold et al., 2017), 8604$-$12703 (e.g., Rembold et al., 2017) and
8612$-$12702 (e.g., Rembold et al., 2017; Ilha et al., 2019). We did not find
any specific peculiarity in the rotation curves of these galaxies, which is to
be expected due to the distances involved. Related to this, nuclear activity
should not affect the kinematics at large distances from the center (e.g.,
Sofue et al., 1999; Schmidt et al., 2016; Sofue, 2017; Yu et al., 2022). This
would be in accordance with the hypothesis that nuclear activity is generated
by a more local cause rather than by a global dynamical mass distribution
(e.g., Sofue et al., 1999; Sofue, 2017), although a study with a larger sample
size would be necessary to delve deeper into this aspect (Sánchez et al.,
2018b).

On the other hand, as mentioned above, there are galaxies that present
evidence of non-circular motions which are manifested through strong
departures from the fit. These galaxies are 7962-12703, 7992-6104, 8312-12704,
8317-12704, 8330-12703, 8452-12703, 8482-9102, 8486-6101, 8548-6102 and
8549-12702. This behaviour is expected in this kind of galaxies (e.g.,
Vaughan, 1989; Regan & Vogel, 1994; Schmidt et al., 2019) and may be caused by
the presence of nonaxisymmetric structures such as spiral arms and bars (e.g.,
Binney et al., 1991; Athanassoula, 1992; Burton & Liszt, 1993; Sofue, 2017).

###  4.2 Dynamical masses

The derived dynamical masses for all barred galaxies of the sample are
presented in column 6 of Table 4. In addition, Fig. 3 shows the distribution
of this parameter. The masses are in the range of log(Mdyn/M⊙) $=$
10.4$-$12.0, with a mean value of log(Mdyn/M⊙) $=$ 11.1. These values are in
agreement with previous results considering masses of spiral galaxies (e.g.,
Burbidge et al., 1959, 1961; Rubin et al., 1964; Krumm & Salpeter, 1977;
Pence, 1981; Agüero et al., 2004; Salucci et al., 2008; Jałocha et al., 2010;
Daod & Zeki, 2019; Schmidt et al., 2019, among others). The distribution has a
standard deviation of 0.36 and an IQR of 0.47. This result would suggest that
barred galaxies present dynamical masses similar to those of the general
spiral galaxies population.

Considering the uncertainties given by the fitting process of the rotation
curves (see Sect. 4.1), the typical relative error in the determination of the
dynamical mass is in the range $\sim$7% $-$ 15% (Table 4). As mentioned above,
these errors take into account only our measurement process.

There are 11 galaxies in our sample which were also studied by Garma-Oehmichen
et al. (2020). These objects are 7495-12704, 7962-12703, 8256-6101, 8257-3703,
8312-12704, 8313-9101, 8317-12704, 8318-12703, 8439-6102, 8439-12702 and
8453-12701. These authors present the stellar mass and the molecular gaseous
mass derived from the Pipe3D analysis. In the mentioned work, the molecular
gas mass was estimated adopting the dust-to-gas ratio based on the dust
attenuation obtained by Pipe3D. As expected, the dynamic masses obtained in
this work are higher than the sum of the masses obtained in Garma-Oehmichen et
al. (2020). This difference is in general $\sim$0.3 dex. The observed
difference between our dynamically determined mass and the sum of stellar and
molecular mass estimates may be attributed to the contribution of other
components, such as ionized gas and dark matter (e.g., Sofue & Rubin, 2001).
Indeed, it is well-known that the dynamical mass estimates of galaxies account
for all gravitationally bound matter within their extent, including components
beyond the visible stellar and molecular matter (e.g., Lelli et al., 2016). In
this context, the contribution of ionized gas, which can comprise a
substantial fraction of the total gas content in galaxies, can be particularly
relevant. Moreover, the presence of dark matter, which is thought to
constitute an important fraction of matter in the universe, is expected to
impact the dynamical mass of galaxies and, thus, the observed discrepancy
between different mass estimates (e.g., Persic & Salucci, 1995; Sofue & Rubin,
2001; Lelli et al., 2016). Therefore, our finding of a greater dynamical mass
relative to the sum of stellar and molecular mass estimates, while expected,
is noteworthy as it underscores the importance of knowing the dynamical masses
of these objects as they consider the contribution of all elements to the
gravitational potential. To our best knowledge, this is the first direct
determination of dynamical masses through rotation curves in this sample of
galaxies.

Figure 3: Total dynamical mass distribution for our sample of barred galaxies.

###  4.3 Stellar masses

The relation between the dynamical and stellar mass of galaxies gives us clues
to understand their structure and evolution, since it links the mass already
contained in stars with the total mass of the galaxy, i.e. gas, dust, and dark
matter. The integrated stellar masses reported in Table 4 were determined from
the absorption-corrected stellar mass density maps as indicated in Sánchez et
al. (2016a, c). The errors of stellar masses are of the order of $\sim$ 1% and
were obtained through error propagation considering the mass determination of
each spaxel. These errors refer to our measurement process and do not include
errors in observations, calibrations, etc. Fig. 4 shows the distribution of
stellar mass for the whole sample, which is in the range of log(Mstar/M⊙) $=$
10.13 $-$ 11.53, with a mean value of log(Mstar/M⊙) $=$ 10.8. According to our
results, this distribution has a standard deviation of 0.35 and an IQR of
0.54. These values are in agreement with previous results on spiral galaxies
(e.g., Williams et al., 2010; Davis et al., 2018). As mentioned in Sec.4.2,
Garma-Oehmichen et al. (2020) studied the stellar mass of 11 galaxies that are
also part of our sample and as expected the results are in good agreement.

Figure 4: Stellar mass distribution for our sample of barred galaxies.

###  4.4 Scaling relations

Scaling relations are empirical relationships between different parameters
that can be used to study the properties of galaxies (e.g., Di Teodoro et al.,
2021; Krut et al., 2023). They are a fundamental tool for understanding the
structure, dynamics, and evolution of galaxies (e.g., Aquino-Ortíz et al.,
2018; Ferrero et al., 2021). For example, in the context of rotation curves,
scaling relations provide insights into the underlying physical processes that
govern the dynamical properties of galaxies. These relations quantify the
relationships between different observable properties and are widely used to
study the formation and evolution of galaxies (e.g., Lapi et al., 2018; Di
Teodoro et al., 2023) . As such, scaling relations provide a powerful
framework for connecting observations with theoretical models and for testing
our understanding of galaxy formation and evolution. In this context, we
analyse and study the scaling relations taking into account the magnitude and
the parameters we have measured namely dynamical mass, stellar mass, and
maximum rotation velocity.

Firstly, we study the relation between the dynamical mass and the stellar
mass. In Fig. 5 we show this correlation, which has a Pearson coefficient
r${}_{p}=$ 0.70 and a p-value of 1.4 $\times$10-7. As expected, this result
implies that as the stellar mass of a galaxy increases, so too does its total
dynamical mass. Although the relation is clear, it can be seen that there is
some dispersion in the data. It is worth noting that there are many different
factors that can influence the dynamics of spiral galaxies, including the
distribution of dark matter, the presence of gas and dust, and the possible
effects of mergers and interactions with other galaxies. These factors may
interact in complex ways and can be different in each galaxy. We performed an
OLS (Ordinary Least Squares) bisector fit for our data, which has a slope of
1.0 $\pm$ 0.1 and an offset of -0.80 $\pm$ 1\. Due to the deviation, the
galaxies 8082$-$6102 and 8318$-$12703 are not shown in the plot and are not
considered for this particular analysis. These objects, however, are
considered for the rest of the analysis regarding the other parameters.
Additionally, it has to be noted that a few objects present higher stellar
mass compared to the dynamical mass. These galaxies are: 8083-6102, 8249-6101,
8257-3703, 8326-6102, 8548-6104 and 8612-12702 (see Fig. 5 and Table 4). While
these differences fall within the errors, caution should be exercised when
dealing with these particular values.

Figure 5: Relation between the dynamical mass and the stellar mass. The
galaxies 8082-6102 and 8318-12703 are not shown in the plot and are not
considered in the analysis. The black solid line represents the OLS bisector
fit for our data.

In addition, we study the correlation between dynamical mass and maximum
rotation velocity, which is a well-known relationship in the study of spiral
galaxies (e.g., Rubin et al., 1978). This correlation implies that as the mass
of a galaxy increases, so does its maximum rotation velocity. Physically, this
can be understood as a result of the balance between gravity and centrifugal
forces within the galaxy (e.g., Binney & Tremaine, 1987). Fig. 6 shows the
relation between the dynamical mass and the maximun rotation velocity. This
correlation has a Pearson coefficient r${}_{p}=$ 0.90 and a p-value of 7.9
$\times$10-17, indicating the close relationship between the two parameters.
As in the previous case, the galaxies 8082$-$6102 and 8318$-$12703 are not
shown in the plot and are not considered for this particular analysis. These
objects, however, are considered for the rest of the analysis. We performed an
OLS bisector fit for our data, which has a slope of 0.33 $\pm$ 0.02 and an
offset of -1.36 $\pm$ 0.27.

Figure 6: Relation between the dynamical mass and the maximun rotation
velocity (km s-1). The galaxies 8082-6102 and 8318-12703 are not shown in the
plot and are not considered in the analysis. The black solid line represents
the OLS bisector fit for our data.

Moreover, in Fig. 7 we show the relation between the stellar mass and the
magnitude Mr. As expected, a close correlation is seen, with a Pearson
coefficient r${}_{p}=$ -0.93 and a p-value of 3.4 $\times$10-21. The OLS
bisector fit has a slope of -2.36 $\pm$ 0.12 and an offset of 4.2 $\pm$ 1.4.
As expected, galaxies with higher stellar masses tend to have lower
magnitudes, indicating that they are brighter.

Figure 7: Relation between stellar mass and magnitude Mr. The black solid line
represents the OLS bisector fit for our data.

In the context of scaling relations, the Tully-Fisher (TF) relation is a
powerful tool for understanding the intrinsic properties of rotationally
supported galaxies (Tully & Fisher, 1977; Courteau, 1997a). While the primary
use of the TF relation has been to determine distances in extragalactic
studies, it also provides fundamental insights into the mechanisms of disk
assembly and evolution (e.g., Bekeraitė, 2017). In one of its best-known
forms, this relation provides a link between two key properties of galaxies:
their circular rotation velocities and their luminosities (e.g., Pizagno et
al., 2007; Bekeraité et al., 2016a). We study the TF relation in our sample
through the maximun rotation velocity and the magnitude Mr (Fig. 8).
Considering our data, this relation has a Pearson coefficient r${}_{p}=$ -0.76
and a p-value of 1.1 $\times$10-9. An OLS bisector fit yields a slope of -7.3
$\pm$ 0.7 and an offset of -4.6 $\pm$ 1.6. Our fitting results were compared
with Courteau (1997b), Pizagno et al. (2007) and Bekeraité et al. (2016a)
findings. The first two studies investigated the TF relation through rotation
curves using H$\alpha$ emission and considering log(v) as independent
variable. On the other hand, Bekeraité et al. (2016a) explored the TF relation
in a sample of galaxies from CALIFA using stellar dynamics and considering the
magnitude as the independent variable. Our bisector fit’s slope and offset are
presented in Table 2, along with the corresponding values reported by Courteau
(1997b), Pizagno et al. (2007) and Bekeraité et al. (2016a) (see Table 1 of
this last work). As can be seen, the slope and offset reported in this work
are within the range obtained by Bekeraité et al. (2016a) and Courteau
(1997b). This suggests that the behaviour of the sample of barred galaxies
considering this relation does not differ from that of other studies involving
galaxies of different morphological types including non-barred galaxies.

Figure 8: T-F relation between the maximun rotation velocity (km s-1) and the magnitude Mr. The black solid line represents the OLS bisector fit for our data. For comparison we also show the Courteau (1997b), Pizagno et al. (2007) and Bekeraité et al. (2016a) T-F fits. Table 2: Fit parameters of the Tully-Fisher relation (Vmax and magnitude Mr) and literature values. Work | Slope | Offset
---|---|---
This work | -7.3 $\pm$ 0.7 | -4.6 $\pm$ 1.6
Bekeraité et al. (2016a) | -7.5 $\pm$ 0.5 | -4.0 $\pm$ 1.0
Pizagno et al. (2007) | -5.72 $\pm$ 0.19 | -7.90 $\pm$ 0.03
Courteau (1997b) | -6.99 $\pm$ 0.33 | -5.23 $\pm$ 0.46

A known alternative form of the TF relation is expressed in terms of the
stellar mass and the maximum rotation velocity (e.g., Avila-Reese et al.,
2008). We examine this relation in Fig. 9, which has a Pearson coefficient
r${}_{p}=$ 0.74 and a p-value of 3.3 $\times$10-9. The OLS bisector fit has a
slope of 0.32 $\pm$ 0.03 and an offset of -1.2 $\pm$ 0.3. We compared our
fitting results with those found by Avila-Reese et al. (2008) and Aquino-Ortíz
et al. (2018). Avila-Reese et al. (2008) study the TF relation in a compiled
sample of 76 normal disk galaxies of all morphological types and performed,
among others, a bisector fit (their table 1), while Aquino-Ortíz et al. (2018)
study the TF relation in galaxies from CALIFA performing an orthogonal linear
fit considering, among others, the stellar dynamics of only spiral galaxies
and using the stellar mass as independent variable (their table 1). In Table 3
we present our values of slope and offset for our bisector fit of the TF
relation. This table also shows the parameters provided by Avila-Reese et al.
(2008) and Aquino-Ortíz et al. (2018). The slope value we obtain is higher
than the values obtained by both works, however, within error it is similar to
the one reported by Aquino-Ortíz et al. (2018). In Fig. 9 it can be seen that
our fit is between the fits of these two mentioned works, what indicates that
the behaviour of this sample of barred galaxies does not differ from samples
of varying morphologies.

Table 3: Fit parameters of the Tully-Fisher relation (Mstar/M⊙ and Vmax in km s-1) and literature values. Work | Slope | Offset
---|---|---
This work | 0.32 $\pm$ 0.03 | -1.2 $\pm$ 0.3
Aquino-Ortíz et al. (2018) | 0.30 $\pm$ 0.02 | -1.00 $\pm$ 0.02
Avila-Reese et al. (2008) | 0.284 $\pm$ 0.012 | -0.741 $\pm$ 0.127
Figure 9: T-F relation between the stellar mass and the maximun rotation
velocity (km s-1). The black solid line represents the OLS bisector fit for
our data. For comparison we also show the Aquino-Ortíz et al. (2018) and
Avila-Reese et al. (2008) T-F fits.

##  5 Comparison with literature

Figure 10: Comparison of distributions of stellar mass for our sample and
other three spiral galaxy samples (see text for details). Top panel shows
histograms for each sample with the barred and non-barred samples marked with
"B" and "NB" labels. The bottom panel displays box plots with the same color
scheme. The orange horizontal line within each box plot represents the median
value. The height of the boxes indicate the interquartile range (IQR) for each
sample, while the whiskers extending from the boxes indicate the range of non-
outlier values. Additionally, the notches in the boxes provide a visual
estimate of the statistical significance of differences between groups. If the
notches of two boxes do not overlap, it suggests that there is a significant
difference in the medians of the two groups at a 95% confidence level.

This section presents a comparison of the parameters obtained for our sample
of galaxies with those of other samples found in the literature. The primary
objective is to examine the results in relation to other studies on barred
galaxies and to investigate potential differences between barred and non-
barred galaxies in terms of their stellar mass, total mass, and Vrot. By
conducting this analysis, we aim to gain a deeper understanding of the
properties and behavior of barred galaxies and how they relate to the broader
population of galaxies.

We compared the stellar mass distribution of our sample with three additional
samples of spiral galaxies by generating histograms and box plots (see Fig.
10). Box plots are an effective tool for summarising the distribution of data
based on quartiles and offer several advantages, including the ability to
visualize central tendency, dispersion, and outliers of each sample, even for
small samples (n $>$ 5). In addition, we have included the histograms of the
samples as they provide a quick and direct visualization of their
distributions. Barred and non-barred galaxies were distinguished for each
sample using the morphological type reported in the NED, and we only included
in this analysis sub-samples with a sufficient number of galaxies to conduct a
meaningful comparison.

Of the sub-samples examined, only the sample presented in Davis et al. (2018)
contained a sufficient number of barred galaxies for a valid comparison. The
medians of our sample and the Davis et al. (2018) sample are statistically
similar, as shown in the bottom panel of Fig. 10, taking into account the
uncertainties represented by the notches of the boxes. Furthermore, we used
the Kolmogorov-Smirnov (KS) test to compare the underlying distributions of
the two samples and obtained a statistic of 0.2123 and a p-value of 0.2895. As
the p-value is significantly higher than the limit value of 0.05, there is no
strong evidence to reject the null hypothesis. Therefore, based on the KS
test, it is likely that the two samples were drawn from the same underlying
distribution suggesting that our sample and the sample of Davis et al. (2018)
may follow a similar stellar mass distribution.

The non-barred sub-sample we used to compare our results came from Leung et
al. (2018) where the stellar masses were measured for the EDGE-CALIFA survey
(Bolatto et al., 2017), which mainly includes face-on galaxies. The box plot
in Fig. 10 reveals some overlapping between our sample and the sample from
Leung et al. (2018) but the medians of both samples are clearly different. The
KS test yielded a statistic of 0.3849 and a p-value of 0.0008 for the
comparison between them, indicating that the two samples likely have different
underlying distributions. This suggests that there could be a statistically
significant difference in the stellar mass distributions of barred and non-
barred galaxies.

Finally, we compared our stellar masses with a third sample of edge-on spiral
galaxies compiled by Williams et al. (2010). We have selected only the non-
barred galaxies based on their classification in the NED. However, we note
that despite the NED classification, Williams et al. (2010) report that 75% of
the galaxies in this sample exhibit boxy structures that may be interpreted as
bars viewed side-on. Furthermore, given that all galaxies in this sample are
edge-on and that bars in such galaxies may be harder to observe, it is likely
that this sample is strongly contaminated with barred galaxies. As such, care
must be taken when comparing our results to this sample. The box plot in Fig.
10 shows a significant difference between the median of the stellar mass for
this sample and ours and even more with respect to the non-barred sample from
Leung et al. (2018). For the comparison with our sample the KS test yielded
similar results as the obtained when comparing with the barred sample of Davis
et al. (2018): a statistic of 0.2876 and a p-value=0.1016, implicating that we
can not rule out the null hypothesis where both samples, ours and the one from
Williams et al. (2010), have been drawn from the same stellar mass
distribution. This reinforces the idea that the galaxies in Williams et al.
(2010) sample could be mainly barred.

Figure 11: Same as Fig. 10 but for the total dynamical mass.

Regarding the total dynamical mass we found that Williams et al. (2010) used
the $K_{s}$ total magnitude to estimate the total mass of the galaxies in
their sample. To evaluate the agreement between their result and our own, we
present a comparison in Fig. 11, where we used histograms and box plots to
illustrate the distributions of both samples. Despite the relatively small
sample size in Williams et al. (2010), our analysis indicates significant
agreement between the two datasets. Results from a KS test yield a statistic
of 0.1572 and a corresponding p-value of 0.7364, indicating that the two
distributions are statistically indistinguishable within the limits of these
samples.

In order to compare the $V_{max}$ values of our sample with those of other
studies, we used a sample of spiral galaxies presented by Rubin et al. (1980).
We employed the NED to classify the galaxies into barred and non-barred
subsamples, which resulted in 12 and 9 galaxies, respectively. Although the
sample size is relatively low, we conducted a statistical analysis to
investigate the differences between the samples. The histograms and box-plots
depicted in Fig. 12 suggest that the three samples are statistically
indistinguishable.

Furthermore, we implemented the Kolmogorov-Smirnov test to evaluate the
significance of the similarities and differences between the samples. The test
yielded a statistic of 0.2608 and a p-value of 0.4612 for the comparison
between our sample and the barred sample of Rubin et al. (1980). Additionally,
we obtained a statistic of 0.2777 and a p-value of 0.5121 for the comparison
with their non-barred sample. In both cases, the null hypothesis of the
samples corresponding to the same distribution cannot be rejected. Thus, based
on the KS test, we can conclude that the two samples are statistically similar
to ours in their distribution, even though caution should be exercised due to
the low number of galaxies in each of the sub-samples from Rubin et al.
(1980). A larger sample of non-barred galaxies is necessary in order to study
if bars have some impact in the $V_{max}$ of galaxies.

Figure 12: Same as Fig. 10 but for the maximum velocity.

##  6 Final remarks

We have obtained and presented the rotation curves of 46 barred galaxies of
MaNGA (Fig. 15). By intrinsic quality of the individual curves, size, and
homogeneity, this sample constitutes one of the best samples of RCs available
for barred galaxies to date. As such, it will offer a very good opportunity
for investigating and analyzing the dynamical properties of this kind of
objects.
We have fitted all the RCs considering two components of an axisymmetric
Miyamoto–Nagai gravitational potential, the parameters of which are presented
in Table 4. Our main results can be summarised as follows:

  * •

We have found a wide range of rotation velocities (117 $-$ 340 ${\rm
km\,s^{-1}}$), with most galaxies showing values in the range 120 $-$ 250
${\rm km\,s^{-1}}$. These velocities are consistent with previous results in
barred galaxies and no difference have been found with a sample of non-barred
galaxies. Nevertheless a larger sample of galaxies is necessary to confirm
this trend.

  * •

The total dynamical masses of our sample lie in the range of log(Mdyn/M⊙)$=$
10.1$-$12.0, with a mean value of log(Mdyn/M⊙)$=$ 11.0. These values are in
agreement with previous results considering total masses of spiral galaxies.
According to our results, the distribution has a standard deviation of 0.39
and an IQR of 0.55.

  * •

Stellar masses of our sample of barred galaxies are in the range
log(Mstar/M⊙)$=$ 10.13$-$11.53 with a standard deviation of 0.35 and an IQR of
0.54. The comparison with other sample of barred galaxies yielded no
statistical difference while comparison with a non-barred galaxies sample
yielded that both distributions are statistically different. This difference
between the stellar mass of barred and non-barred galaxies could imply that
the bar plays a role in the assembly of stellar mass of galaxies.

  * •

The behaviour of our sample in the scaling relations was investigated. We
found a Pearson coefficient rp = 0.70 and a p-value of 1.4 x 10-7 for the
relation between Mdyn and Mstar while for the relation between Mdyn and Vmax
the statistic yielded rp= 0.9 and a p-value of 7.9 x 10-17. We also explored
the Tully-Fisher relation in two ways: by considering the relationship between
Vmax and Mr (rp= -0.76 and a p-value of 1.1 x 10-9), and separately by
examining Vmax and Mstar (rp= 0.74 and a p-value of 3.3 x 10-9). Our analysis
revealed that the Tully-Fisher relation in barred galaxies is consistent with
reported relations in the literature for samples of varying morphologies,
indicating that barred galaxies do not deviate from this scaling relation
commonly observed in other galaxy types.

Summing up, we studied the dynamical masses, stellar masses and maximun
rotation velocities of barred galaxies, investigating their distributions and
scaling relations and comparing them with other studies. Our results suggest
that stellar masses could be different in barred galaxies compared with non-
barred galaxies. The dynamical mass and maximun rotation velocities as well as
scalling relations such as the Tully-Fisher relation show no difference with
other samples involving galaxies with different morphological types including
non-barred galaxies.

It has to be noted the importance of knowing the dynamical masses of galaxies
as they consider the contribution of all constituent elements to the
gravitational potential. To our best knowledge, this is the first time that
the dynamical masses of these galaxies have been determined in a direct and
independent way through their rotation curves. We hope that the work presented
here will constitute a useful tool to increase our understanding of the
dynamical behaviour of barred spiral galaxies.

## Acknowledgements

We want to thank our referee for constructive comments and suggestions that improved this work. We are also grateful to Rubén Díaz for fruitful discussions. This work was partially supported by grant PICT 2017-3301 awarded by Fondo para la Investigación Científica y Tecnológica (FonCyT). G.G is a postdoctoral fellow of Consejo de Investigaciones Científicas y Técnicas (CONICET), Argentina. W.W., E.S., and D.M. are members of the Carrera del Investigador Científico of CONICET, Argentina. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We made an extensive use of the following Python libraries and packages: galpy, pandas, Matplotlib, NumPy and SciPy. This project makes use of the MaNGA-Pipe3D dataproducts. We thank the IA-UNAM MaNGA team for creating this catalogue, and the ConaCyt-180125 project for supporting them. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard & Smithsonian (CfA), the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatório Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.

## Data Availability

All data used and analysed in this article are available in www.sdss.org/dr17/

## References

  * Adams et al. (2012) Adams J. J., Gebhardt K., Blanc G. A., Fabricius M. H., Hill G. J., Murphy J. D., van den Bosch R. C. E., van de Ven G., 2012, ApJ, 745, 92
  * Agüero et al. (2004) Agüero E. L., Díaz R. J., Bajaja E., 2004, A&A, 414, 453
  * Aguerri et al. (2009) Aguerri J. A. L., Méndez-Abreu J., Corsini E. M., 2009, A&A, 495, 491
  * Ahn et al. (2012) Ahn C. P., et al., 2012, ApJS, 203, 21
  * Albareti et al. (2017) Albareti F. D., et al., 2017, ApJS, 233, 25
  * Aquino-Ortíz et al. (2018) Aquino-Ortíz E., et al., 2018, MNRAS, 479, 2133
  * Aquino-Ortíz et al. (2020) Aquino-Ortíz E., et al., 2020, ApJ, 900, 109
  * Ashman (1992) Ashman K. M., 1992, PASP, 104, 1109
  * Athanassoula (1992) Athanassoula E., 1992, MNRAS, 259, 328
  * Avila-Reese et al. (2008) Avila-Reese V., Zavala J., Firmani C., Hernández-Toledo H. M., 2008, AJ, 136, 1340
  * Barrera-Ballesteros et al. (2014) Barrera-Ballesteros J. K., et al., 2014, A&A, 568, A70
  * Bekeraitė (2017) Bekeraitė S., 2017, PhD thesis, University of Potsdam, Germany
  * Bekeraité et al. (2016a) Bekeraité S., et al., 2016a, A&A, 593, A114
  * Bekeraité et al. (2016b) Bekeraité S., et al., 2016b, ApJ, 827, L36
  * Bing et al. (2019) Bing L., et al., 2019, MNRAS, 482, 194
  * Binney & Tremaine (1987) Binney J., Tremaine S., 1987, Galactic dynamics
  * Binney et al. (1991) Binney J., Gerhard O. E., Stark A. A., Bally J., Uchida K. I., 1991, MNRAS, 252, 210
  * Bolatto et al. (2017) Bolatto A. D., et al., 2017, ApJ, 846, 159
  * Bosma (1996) Bosma A., 1996, in Buta R., Crocker D. A., Elmegreen B. G., eds, Astronomical Society of the Pacific Conference Series Vol. 91, IAU Colloq. 157: Barred Galaxies. p. 132
  * Bovy (2015) Bovy J., 2015, ApJS, 216, 29
  * Bundy et al. (2015) Bundy K., et al., 2015, ApJ, 798, 7
  * Burbidge et al. (1959) Burbidge E. M., Burbidge G. R., Prendergast K. H., 1959, ApJ, 130, 739
  * Burbidge et al. (1961) Burbidge E. M., Burbidge G. R., Prendergast K. H., 1961, ApJ, 134, 874
  * Bureau & Carignan (2002) Bureau M., Carignan C., 2002, AJ, 123, 1316
  * Burton & Liszt (1993) Burton W. B., Liszt H. S., 1993, A&A, 274, 765
  * Cappellari (2017) Cappellari M., 2017, MNRAS, 466, 798
  * Cappellari & Emsellem (2004) Cappellari M., Emsellem E., 2004, PASP, 116, 138
  * Cherinka et al. (2019) Cherinka B., et al., 2019, AJ, 158, 74
  * Ciotti (2021) Ciotti L., 2021, Introduction to Stellar Dynamics, doi:10.1017/9780511736117.
  * Ciotti (2022) Ciotti L., 2022, ApJ, 936, 180
  * Courteau (1997a) Courteau S., 1997a, AJ, 114, 2402
  * Courteau (1997b) Courteau S., 1997b, AJ, 114, 2402
  * Croom et al. (2012) Croom S. M., et al., 2012, MNRAS, 421, 872
  * Daod & Zeki (2019) Daod N. A., Zeki M. K., 2019, ApJ, 870, 107
  * Davis et al. (2018) Davis B. L., Graham A. W., Cameron E., 2018, ApJ, 869, 113
  * Di Teodoro et al. (2021) Di Teodoro E. M., Posti L., Ogle P. M., Fall S. M., Jarrett T., 2021, MNRAS, 507, 5820
  * Di Teodoro et al. (2023) Di Teodoro E. M., et al., 2023, MNRAS, 518, 6340
  * Díaz et al. (1999) Díaz R., Carranza G., Dottori H., Goldes G., 1999, ApJ, 512, 623
  * Edelson et al. (1987) Edelson R. A., Malkan M. A., Rieke G. H., 1987, ApJ, 321, 233
  * Falcón-Barroso et al. (2017) Falcón-Barroso J., et al., 2017, A&A, 597, A48
  * Ferrero et al. (2021) Ferrero I., Navarro J. F., Abadi M. G., Benavides J. A., Mast D., 2021, A&A, 648, A124
  * García-Lorenzo et al. (2015) García-Lorenzo B., et al., 2015, A&A, 573, A59
  * Garma-Oehmichen et al. (2020) Garma-Oehmichen L., Cano-Díaz M., Hernández-Toledo H., Aquino-Ortíz E., Valenzuela O., Aguerri J. A. L., Sánchez S. F., Merrifield M., 2020, MNRAS, 491, 3655
  * Granados et al. (2017) Granados A., Torres D., Castañeda L., Henao-O. J. L., Vanegas S., 2017, arXiv e-prints, p. arXiv:1705.01665
  * Greene & Ho (2007) Greene J. E., Ho L. C., 2007, ApJ, 667, 131
  * Gunn et al. (2006) Gunn J. E., et al., 2006, AJ, 131, 2332
  * Guo et al. (2019) Guo R., Mao S., Athanassoula E., Li H., Ge J., Long R. J., Merrifield M., Masters K., 2019, MNRAS, 482, 1733
  * Hernandez et al. (2005) Hernandez O., Carignan C., Amram P., Chemin L., Daigle O., 2005, MNRAS, 360, 1201
  * Ilha et al. (2019) Ilha G. S., et al., 2019, MNRAS, 484, 252
  * Jałocha et al. (2010) Jałocha J., Bratek Ł., Kutschera M., Skindzier P., 2010, MNRAS, 406, 2805
  * Kalinova et al. (2017a) Kalinova V., van de Ven G., Lyubenova M., Falcón-Barroso J., Colombo D., Rosolowsky E., 2017a, MNRAS, 464, 1903
  * Kalinova et al. (2017b) Kalinova V., et al., 2017b, MNRAS, 469, 2539
  * Krumm & Salpeter (1977) Krumm N., Salpeter E. E., 1977, A&A, 56, 465
  * Krut et al. (2023) Krut A., Argüelles C. R., Chavanis P. H., Rueda J. A., Ruffini R., 2023, ApJ, 945, 1
  * Laine et al. (1999) Laine S., Kenney J. D. P., Yun M. S., Gottesman S. T., 1999, ApJ, 511, 709
  * Lang et al. (2020) Lang P., et al., 2020, ApJ, 897, 122
  * Lapi et al. (2018) Lapi A., Salucci P., Danese L., 2018, ApJ, 859, 2
  * Lelli et al. (2016) Lelli F., McGaugh S. S., Schombert J. M., Pawlowski M. S., 2016, ApJ, 827, L19
  * Leung et al. (2018) Leung G. Y. C., et al., 2018, MNRAS, 477, 254
  * Levy et al. (2018) Levy R. C., et al., 2018, ApJ, 860, 92
  * Makarov et al. (2001) Makarov D. I., Burenkov A. N., Tyurina N. V., 2001, Astronomy Letters, 27, 213
  * Marinova & Jogee (2007) Marinova I., Jogee S., 2007, ApJ, 659, 1176
  * Márquez et al. (2002) Márquez I., Masegosa J., Moles M., Varela J., Bettoni D., Galletta G., 2002, A&A, 393, 389
  * Martinez-Medina et al. (2020) Martinez-Medina L. A., Pichardo B., Peimbert A., 2020, MNRAS, 496, 1845
  * Martinsson et al. (2013) Martinsson T. P. K., Verheijen M. A. W., Westfall K. B., Bershady M. A., Schechtman-Rook A., Andersen D. R., Swaters R. A., 2013, A&A, 557, A130
  * Mathewson et al. (1992) Mathewson D. S., Ford V. L., Buchhorn M., 1992, ApJS, 81, 413
  * Miyamoto & Nagai (1975) Miyamoto M., Nagai R., 1975, Publications of the Astronomical Society of Japan, 27, 533
  * Pence (1981) Pence W. D., 1981, ApJ, 247, 473
  * Persic & Salucci (1995) Persic M., Salucci P., 1995, ApJS, 99, 501
  * Pilyugin et al. (2019) Pilyugin L. S., Grebel E. K., Zinchenko I. A., Nefedyev Y. A., Vílchez J. M., 2019, A&A, 623, A122
  * Pizagno et al. (2007) Pizagno J., et al., 2007, AJ, 134, 945
  * Regan & Vogel (1994) Regan M. W., Vogel S. N., 1994, ApJ, 434, 536
  * Rembold et al. (2017) Rembold S. B., et al., 2017, MNRAS, 472, 4382
  * Rubin et al. (1964) Rubin V. C., Burbidge E. M., Burbidge G. R., Prendergast K. H., 1964, ApJ, 140, 80
  * Rubin et al. (1978) Rubin V. C., Ford W. K. J., Thonnard N., 1978, ApJ, 225, L107
  * Rubin et al. (1980) Rubin V. C., Ford W. K. J., Thonnard N., 1980, ApJ, 238, 471
  * Salucci et al. (2008) Salucci P., Yegorova I. A., Drory N., 2008, MNRAS, 388, 159
  * Sánchez et al. (2012) Sánchez S. F., et al., 2012, A&A, 538, A8
  * Sánchez et al. (2016a) Sánchez S. F., et al., 2016a, Rev. Mex. Astron. Astrofis., 52, 21
  * Sánchez et al. (2016b) Sánchez S. F., et al., 2016b, Rev. Mex. Astron. Astrofis., 52, 171
  * Sánchez et al. (2016c) Sánchez S. F., et al., 2016c, Rev. Mex. Astron. Astrofis., 52, 171
  * Sánchez et al. (2018a) Sánchez S. F., et al., 2018a, Rev. Mex. Astron. Astrofis., 54, 217
  * Sánchez et al. (2018b) Sánchez S. F., et al., 2018b, Rev. Mex. Astron. Astrofis., 54, 217
  * Schmidt et al. (2016) Schmidt E. O., Ferreiro D., Vega Neme L., Oio G. A., 2016, A&A, 596, A95
  * Schmidt et al. (2019) Schmidt E. O., Mast D., Díaz R. J., Agüero M. P., Günthardt G., Gimeno G., Oio G., Gaspar G., 2019, AJ, 158, 60
  * Sheth et al. (2008) Sheth K., et al., 2008, ApJ, 675, 1141
  * Shetty et al. (2007) Shetty R., Vogel S. N., Ostriker E. C., Teuben P. J., 2007, ApJ, 665, 1138
  * Sofue (2016) Sofue Y., 2016, PASJ, 68, 2
  * Sofue (2017) Sofue Y., 2017, PASJ, 69, R1
  * Sofue & Rubin (2001) Sofue Y., Rubin V., 2001, ARA&A, 39, 137
  * Sofue et al. (1999) Sofue Y., Tutui Y., Honma M., Tomita A., Takamiya T., Koda J., Takeda Y., 1999, ApJ, 523, 136
  * Spano et al. (2008) Spano M., Marcelin M., Amram P., Carignan C., Epinat B., Hernandez O., 2008, MNRAS, 383, 297
  * Stern & Laor (2012) Stern J., Laor A., 2012, MNRAS, 423, 600
  * Toba et al. (2014) Toba Y., et al., 2014, ApJ, 788, 45
  * Tully & Fisher (1977) Tully R. B., Fisher J. R., 1977, A&A, 54, 661
  * Vaughan (1989) Vaughan J. M., 1989, The Fabry-Perot interferometer. History, theory, practice and applications
  * Vogt et al. (2004a) Vogt N. P., Haynes M. P., Giovanelli R., Herter T., 2004a, AJ, 127, 3300
  * Vogt et al. (2004b) Vogt N. P., Haynes M. P., Giovanelli R., Herter T., 2004b, AJ, 127, 3325
  * Williams et al. (2010) Williams M. J., Bureau M., Cappellari M., 2010, MNRAS, 409, 1330
  * Yoon et al. (2021) Yoon Y., Park C., Chung H., Zhang K., 2021, ApJ, 922, 249
  * Yu et al. (2022) Yu S.-Y., et al., 2022, A&A, 666, A175

##  Appendix A Fitted rotation curves of the 46 barred galaxies

|
---|---
|
|
Figure 13: Each panel shows from left to right and top to bottom: MaNGA FoV depicted over $gri$ false-color images from SDSS, pseudo-V-band image extracted from the datacubes, the velocity field with red color representing positive radial velocities and blue representing negative radial velocities with respect to the systemic velocity (see text for details), and the rotation curve fitted with two mass components corresponding to an axisymetric Miyamoto$-$Nagai gravitational potential. In all the maps and images, North is up and East is to the left. Note that the maps are not at the same scale as the false color image. Ticksmarks on the maps are separated each 0.5 arcsec. Circles and stars represent different sides of the velocity field. A typical error bar is shown in the top left corner. |
---|---
|
|
Figure 14: continued |
---|---
|
|
Figure 15: continued |
---|---
|
|
Figure 16: continued |
---|---
|
|
Figure 17: continued |
---|---
|
|
Figure 18: continued |
---|---
|
|
Figure 19: continued |
---|---
|
Figure 20: continued

##  Appendix B Dynamical parameters of the galaxies

Table 4:  Dynamical parameters of the galaxies. Columns: (1) MaNGA ID; (2) stellar mass; (3) and (4) shape parameters and dynamical mass of the component 1 and 2 of the fitted Miyamoto–Nagai model, respectively; (5) maximun rotation velocity measured over each rotation curve; (6) total dynamical mass given by the sum of M1 and M2. All data were obtained in Sect. 4. As mentioned in the text, errors in stellar masses are of $\sim$ 1 %. These errors, Vmax, and Mdyn errors only include uncertainties in our measurement process (see Sect. 4.3, 4.1, and 4.2 respectively for details). Plate-IFU | Mstar/ M⊙ | Component 1 | Component 2 | Vmax (km s-1) | Mdyn/ M⊙
---|---|---|---|---|---
7495-12704 | 8.04 $\times$ 1010 | a${}_{1}=$ 2.0 kpc | a${}_{2}=$ 4.3 kpc | 205 $\pm$ 7 | (1.45 $\pm$ 0.07) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$0.39 |  |
|  | M${}_{1}=$ 5.0 $\times$ 109 M⊙ | M${}_{2}=$ 1.4 $\times$ 1011 M⊙ |  |
7962-12703 | 2.25 $\times$ 1011 | a${}_{1}=$ 2.0 kpc | a${}_{2}=$ 6.4 kpc | 255 $\pm$ 13 | (3.4 $\pm$ 0.3) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$0.38 |  |
|  | M${}_{1}=$ 3.0 $\times$ 109 M⊙ | M${}_{2}=$ 3.4 $\times$ 1011 M⊙ |  |
7990-3704 | 2.08 $\times$ 1010 | a${}_{1}=$ 1.0 kpc | a${}_{2}=$ 2.4 kpc | 117 $\pm$ 7 | (2.70 $\pm$ 0.24) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$0.37 |  |
|  | M${}_{1}=$ 1.5 $\times$ 109 M⊙ | M${}_{2}=$ 2.5 $\times$ 1010 M⊙ |  |
7990-9101 | 2.47 $\times$ 1010 | a${}_{1}=$ 1.0 kpc | a${}_{2}=$ 4.3 kpc | 132 $\pm$ 8 | (7.10 $\pm$ 0.64) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$0.58 |  |
|  | M${}_{1}=$ 1.0 $\times$ 109 M⊙ | M${}_{2}=$ 7.0 $\times$ 1010 M⊙ |  |
7992-6104 | 3.59 $\times$ 1010 | a${}_{1}=$ 0.1 kpc | a${}_{2}=$ 6.1 kpc | 221 $\pm$ 11 | (3.60 $\pm$ 0.22) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$0.57 |  |
|  | M${}_{1}=$ 3.0 $\times$ 108 M⊙ | M${}_{2}=$ 3.6 $\times$ 1011 M⊙ |  |
8082-6102 | 8.02 $\times$ 1010 | a${}_{1}=$ 1.0 kpc | a${}_{2}=$ 4.4 kpc | 221 $\pm$ 9 | (2.7 $\pm$ 0.1) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$0.36 |  |
|  | M${}_{1}=$ 8.0 $\times$ 109 M⊙ | M${}_{2}=$ 1.7 $\times$ 1011 M⊙ |  |
8083-6102 | 1.29 $\times$ 1011 | a${}_{1}=$ 1.5 kpc | a${}_{2}=$ 3.3 kpc | 220 $\pm$ 9 | (1.20 $\pm$ 0.05) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.32 |  |
|  | M${}_{1}=$ 5.0 $\times$ 109 M⊙ | M${}_{2}=$ 1.2 $\times$ 1011 M⊙ |  |
8083-12704 | 3.66 $\times$ 1010 | a${}_{1}=$ 2.0 kpc | a${}_{2}=$ 2.2 kpc | 134 $\pm$ 8 | (3.8 $\pm$ 0.3) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.59 |  |
|  | M${}_{1}=$ 1.0 $\times$ 109 M⊙ | M${}_{2}=$ 3.7 $\times$ 1010 M⊙ |  |
8133-3701 | 2.44 $\times$ 1010 | a${}_{1}=$ 0.8 kpc | a${}_{2}=$ 6.6 kpc | 173 $\pm$ 12 | (2.0 $\pm$ 0.1) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.48 |  |
|  | M${}_{1}=$ 4.0 $\times$ 109 M⊙ | M${}_{2}=$ 2.0 $\times$ 1011 M⊙ |  |
8134-6102 | 6.06 $\times$ 1010 | a${}_{1}=$ 3.0 kpc | a${}_{2}=$ 4.5 kpc | 266 $\pm$ 8 | (2.50 $\pm$ 0.07) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.33 |  |
|  | M${}_{1}=$ 6.0 $\times$ 109 M⊙ | M${}_{2}=$ 2.5 $\times$ 1011 M⊙ |  |
8137-9102 | 4.34 $\times$ 1010 | a${}_{1}=$ 2.0 kpc | a${}_{2}=$ 5.7 kpc | 178 $\pm$ 5 | (1.5 $\pm$ 0.5) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.40 |  |
|  | M${}_{1}=$ 6.0 $\times$ 109 M⊙ | M${}_{2}=$ 1.5 $\times$ 1011 M⊙ |  |
8140-12701 | 4.53 $\times$ 1010 | a${}_{1}=$ 2.0 kpc | a${}_{2}=$ 4.0 kpc | 144 $\pm$ 5 | (6.6 $\pm$ 0.6) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.35 |  |
|  | M${}_{1}=$ 6.0 $\times$ 109 M⊙ | M${}_{2}=$ 6.0 $\times$ 1010 M⊙ |  |
8140-12703 | 1.07 $\times$ 1011 | a${}_{1}=$ 1.0 kpc | a${}_{2}=$ 4.3 kpc | 199 $\pm$ 6 | (1.5 $\pm$ 0.5) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.53 |  |
|  | M${}_{1}=$ 3.0 $\times$ 109 M⊙ | M${}_{2}=$ 1.5 $\times$ 1011 M⊙ |  |
8243-6103 | 5.83 $\times$ 1010 | a${}_{1}=$ 3.0 kpc | a${}_{2}=$ 3.5 kpc | 269 $\pm$ 8 | (2.2 $\pm$ 0.1) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.42 |  |
|  | M${}_{1}=$ 4.0 $\times$ 1010 M⊙ | M${}_{2}=$ 1.8 $\times$ 1011 M⊙ |  |
8244-3703 | 4.98 $\times$ 1010 | a${}_{1}=$ 2.0 kpc | a${}_{2}=$ 5.4 kpc | 205 $\pm$ 9 | (1.90 $\pm$ 0.06) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.44 |  |
|  | M${}_{1}=$ 8.0 $\times$ 109 M⊙ | M${}_{2}=$ 1.9 $\times$ 1011 M⊙ |  |
8249-6101 | 3.16 $\times$ 1010 | a${}_{1}=$ 1.5 kpc | a${}_{2}=$ 2.5 kpc | 122 $\pm$ 8 | (2.7 $\pm$ 0.2) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.20 |  |
|  | M${}_{1}=$ 2.0 $\times$ 109 M⊙ | M${}_{2}=$ 2.5 $\times$ 1010 M⊙ |  |
8254-9101 | 1.31 $\times$ 1011 | a${}_{1}=$ 0.5 kpc | a${}_{2}=$ 4.8 kpc | 340 $\pm$ 12 | (4.8 $\pm$ 0.1) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.42 |  |
|  | M${}_{1}=$ 5.0 $\times$ 109 M⊙ | M${}_{2}=$ 4.8 $\times$ 1011 M⊙ |  |
Table 5: continued 8256-6101 | 3.40 $\times$ 1010 | a${}_{1}=$ 1.5 kpc | a${}_{2}=$ 3.0 kpc | 188 $\pm$ 6 | (8.3 $\pm$ 0.2) $\times$ 1010
---|---|---|---|---|---
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.33 |  |
|  | M${}_{1}=$ 7.0 $\times$ 109 M⊙ | M${}_{2}=$ 7.6 $\times$ 1010 M⊙ |  |
8257-3703 | 5.92 $\times$ 1010 | a${}_{1}=$ 1.5 kpc | a${}_{2}=$ 2.4 kpc | 163 $\pm$ 5 | (5.1 $\pm$ 0.2) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.33 |  |
|  | M${}_{1}=$ 3.0 $\times$ 109 M⊙ | M${}_{2}=$ 4.8 $\times$ 1010 M⊙ |  |
8257-6101 | 3.67 $\times$ 1010 | a${}_{1}=$ 1.5 kpc | a${}_{2}=$ 3.2 kpc | 155 $\pm$ 9 | (6.0 $\pm$ 0.3) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.31 |  |
|  | M${}_{1}=$ 3.0 $\times$ 109 M⊙ | M${}_{2}=$ 5.7 $\times$ 1010 M⊙ |  |
8312-12702 | 6.59 $\times$ 1010 | a${}_{1}=$ 2.5 kpc | a${}_{2}=$ 4.5 kpc | 180 $\pm$ 6 | (1.20 $\pm$ 0.04) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.40 |  |
|  | M${}_{1}=$ 3.0 $\times$ 109 M⊙ | M${}_{2}=$ 1.2 $\times$ 1011 M⊙ |  |
8312-12704 | 3.58 $\times$ 1010 | a${}_{1}=$ 1.1 kpc | a${}_{2}=$ 8.0 kpc | 149 $\pm$ 10 | (9.8 $\pm$ 0.9) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.50 |  |
|  | M${}_{1}=$ 2.8 $\times$ 1010 M⊙ | M${}_{2}=$ 7.0 $\times$ 1011 M⊙ |  |
8313-9101 | 1.28 $\times$ 1011 | a${}_{1}=$ 1.7 kpc | a${}_{2}=$ 4.5 kpc | 236 $\pm$ 7 | (1.90 $\pm$ 0.05) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.40 |  |
|  | M${}_{1}=$ 4.0 $\times$ 1010 M⊙ | M${}_{2}=$ 1.5 $\times$ 1011 M⊙ |  |
8317-12704 | 2.33 $\times$ 1011 | a${}_{1}=$ 2.0 kpc | a${}_{2}=$ 6.0 kpc | 229 $\pm$ 10 | (2.7 $\pm$ 0.2) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.46 |  |
|  | M${}_{1}=$ 1.0 $\times$ 1010 M⊙ | M${}_{2}=$ 2.6 $\times$ 1011 M⊙ |  |
8318-12703 | 1.3 $\times$ 1011 | a${}_{1}=$ 2.0 kpc | a${}_{2}=$ 8.3 kpc | 214 $\pm$ 9 | (2.90 $\pm$ 0.09) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.44 |  |
|  | M${}_{1}=$ 5.0 $\times$ 1010 M⊙ | M${}_{2}=$ 2.4 $\times$ 1011 M⊙ |  |
8320-6101 | 3.79 $\times$ 1010 | a${}_{1}=$ 1.1 kpc | a${}_{2}=$ 3.2 kpc | 183 $\pm$ 5 | (7.6 $\pm$ 0.3) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.25 |  |
|  | M${}_{1}=$ 1.0 $\times$ 1010 M⊙ | M${}_{2}=$ 6.6 $\times$ 1010 M⊙ |  |
8326-3704 | 1.34 $\times$ 1010 | a${}_{1}=$ 2.5 kpc | a${}_{2}=$ 4.0 kpc | 128 $\pm$ 4 | (7.3 $\pm$ 0.4) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.50 |  |
|  | M${}_{1}=$ 3.0 $\times$ 109 M⊙ | M${}_{2}=$ 7.0 $\times$ 1010 M⊙ |  |
8326-6102 | 1.72 $\times$ 1011 | a${}_{1}=$ 1.5 kpc | a${}_{2}=$ 5.0 kpc | 175 $\pm$ 12 | (1.00 $\pm$ 0.07) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.20 |  |
|  | M${}_{1}=$ 7.0 $\times$ 109 M⊙ | M${}_{2}=$ 1.0 $\times$ 1011 M⊙ |  |
8330-12703 | 3.53 $\times$ 1010 | a${}_{1}=$ 1.5 kpc | a${}_{2}=$ 5.2 kpc | 165 $\pm$ 13 | (9.7 $\pm$ 0.9) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.19 |  |
|  | M${}_{1}=$ 7.0 $\times$ 109 M⊙ | M${}_{2}=$ 9.0 $\times$ 1010 M⊙ |  |
8335-12701 | 1.25 $\times$ 1011 | a${}_{1}=$ 3.5 kpc | a${}_{2}=$ 6.0 kpc | 218 $\pm$ 12 | (2.2 $\pm$ 0.1) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.33 |  |
|  | M${}_{1}=$ 9.0 $\times$ 109 M⊙ | M${}_{2}=$ 2.2 $\times$ 1011 M⊙ |  |
8439-6102 | 1.06 $\times$ 1011 | a${}_{1}=$ 2.7 kpc | a${}_{2}=$ 4.0 kpc | 202 $\pm$ 6 | (1.40 $\pm$ 0.04) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.42 |  |
|  | M${}_{1}=$ 1.0 $\times$ 1010 M⊙ | M${}_{2}=$ 1.3 $\times$ 1011 M⊙ |  |
8439-12702 | 6.05 $\times$ 1010 | a${}_{1}=$ 2.5 kpc | a${}_{2}=$ 4.0 kpc | 235 $\pm$ 7 | (1.60 $\pm$ 0.05) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.25 |  |
|  | M${}_{1}=$ 6.0 $\times$ 109 M⊙ | M${}_{2}=$ 1.6 $\times$ 1011 M⊙ |  |
8440-12704 | 5.17 $\times$ 1010 | a${}_{1}=$ 2.0 kpc | a${}_{2}=$ 2.8 kpc | 205 $\pm$ 6 | (9.7 $\pm$ 0.3) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.35 |  |
|  | M${}_{1}=$ 7.0 $\times$ 109 M⊙ | M${}_{2}=$ 9.0 $\times$ 1010 M⊙ |  |
8447-6101 | 3.66 $\times$ 1011 | a${}_{1}=$ 3.0 kpc | a${}_{2}=$ 7.2 kpc | 312 $\pm$ 14 | (5.7 $\pm$ 0.3) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.39 |  |
|  | M${}_{1}=$ 9.0 $\times$ 109 M⊙ | M${}_{2}=$ 5.7 $\times$ 1011 M⊙ |  |
8452-3704 | 1.84 $\times$ 1010 | a${}_{1}=$ 0.6 kpc | a${}_{2}=$ 3.0 kpc | 142 $\pm$ 4 | (4.9 $\pm$ 0.2) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.33 |  |
|  | M${}_{1}=$ 1.0 $\times$ 109 M⊙ | M${}_{2}=$ 4.8 $\times$ 1010 M⊙ |  |
Table 6: continued 8452-12703 | 2.25 $\times$ 1011 | a${}_{1}=$ 3.5 kpc | a${}_{2}=$ 6.1 kpc | 216 $\pm$ 11 | (2.4 $\pm$ 0.1) $\times$ 1011
---|---|---|---|---|---
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.44 |  |
|  | M${}_{1}=$ 5.0 $\times$ 109 M⊙ | M${}_{2}=$ 2.4 $\times$ 1011 M⊙ |  |
8482-9102 | 9.51 $\times$ 1010 | a${}_{1}=$ 4.0 kpc | a${}_{2}=$ 5.5 kpc | 208 $\pm$ 15 | (2.1 $\pm$ 0.2) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.45 |  |
|  | M${}_{1}=$ 1.0 $\times$ 1010 M⊙ | M${}_{2}=$ 2.0 $\times$ 1011 M⊙ |  |
8482-12705 | 1.26 $\times$ 1011 | a${}_{1}=$ 1.0 kpc | a${}_{2}=$ 5.0 kpc | 227 $\pm$ 7 | (2.20 $\pm$ 0.05) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.42 |  |
|  | M${}_{1}=$ 1.0 $\times$ 109 M⊙ | M${}_{2}=$ 2.2 $\times$ 1011 M⊙ |  |
8486-6101 | 1.12 $\times$ 1011 | a${}_{1}=$ 2.5 kpc | a${}_{2}=$ 5.0 kpc | 189 $\pm$ 11 | (1.40 $\pm$ 0.07) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.30 |  |
|  | M${}_{1}=$ 1.0 $\times$ 109 M⊙ | M${}_{2}=$ 1.4 $\times$ 1011 M⊙ |  |
8548-6102 | 2.69 $\times$ 1010 | a${}_{1}=$ 1.0 kpc | a${}_{2}=$ 4.4 kpc | 155 $\pm$ 9 | (9.6 $\pm$ 0.5) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.54 |  |
|  | M${}_{1}=$ 6.0 $\times$ 109 M⊙ | M${}_{2}=$ 9.0 $\times$ 1010 M⊙ |  |
8548-6104 | 6.04 $\times$ 1010 | a${}_{1}=$ 1.0 kpc | a${}_{2}=$ 3.0 kpc | 136 $\pm$ 6 | (4.10 $\pm$ 0.15) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.33 |  |
|  | M${}_{1}=$ 6.0 $\times$ 109 M⊙ | M${}_{2}=$ 3.5 $\times$ 1010 M⊙ |  |
8549-12702 | 1.61 $\times$ 1011 | a${}_{1}=$ 2.6 kpc | a${}_{2}=$ 4.5 kpc | 249 $\pm$ 7 | (2.20 $\pm$ 0.07) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.36 |  |
|  | M${}_{1}=$ 6.0 $\times$ 1010 M⊙ | M${}_{2}=$ 1.6 $\times$ 1011 M⊙ |  |
8588-3701 | 2.86 $\times$ 1011 | a${}_{1}=$ 2.1 kpc | a${}_{2}=$ 12.9 kpc | 303 $\pm$ 9 | (1.10 $\pm$ 0.04) $\times$ 1012
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.53 |  |
|  | M${}_{1}=$ 1.4 $\times$ 109 M⊙ | M${}_{2}=$ 1.1 $\times$ 1012 M⊙ |  |
8604-12703 | 1.10 $\times$ 1011 | a${}_{1}=$ 3.0 kpc | a${}_{2}=$ 3.6 kpc | 220 $\pm$ 9 | (1.50 $\pm$ 0.06) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.24 |  |
|  | M${}_{1}=$ 3.4 $\times$ 109 M⊙ | M${}_{2}=$ 1.5 $\times$ 1011 M⊙ |  |
8612-6104 | 7.90 $\times$ 1010 | a${}_{1}=$ 2.0 kpc | a${}_{2}=$ 3.2 kpc | 190 $\pm$ 8 | (9.6 $\pm$ 0.4) $\times$ 1010
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.37 |  |
|  | M${}_{1}=$ 9.0 $\times$ 109 M⊙ | M${}_{2}=$ 8.7 $\times$ 1010 M⊙ |  |
8612-12702 | 3.06 $\times$ 1011 | a${}_{1}=$ 2.0 kpc | a${}_{2}=$ 7.0 kpc | 214 $\pm$ 11 | (2.9 $\pm$ 0.2) $\times$ 1011
|  | b1/a${}_{1}=$ 1.00 | b2/a${}_{2}=$ 0.50 |  |
|  | M${}_{1}=$ 1.0 $\times$ 109 M⊙ | M${}_{2}=$ 2.9 $\times$ 1011 M⊙ |  |