11institutetext: Argelander-Institut für Astronomie, Universität Bonn, Auf dem
Hügel 71, 53121 Bonn, Germany 22institutetext: National Radio Astronomy
Observatory, 520 Edgemont Road, Charlottesville, VA 22903, USA
33institutetext: Department of Astronomy, The Ohio State University, 140 West
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55institutetext: Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan
281 S9, B-9000 Gent, Belgium 66institutetext: Max Planck Institute for
Astronomy, Königstuhl 17, 69117 Heidelberg, Germany 77institutetext:
Netherlands Institute for Radio Astronomy (ASTRON), Oude Hoogeveensedijk 4,
7991 PD Dwingeloo, Netherlands 88institutetext: Kapteyn Astronomical
Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The
Netherlands 99institutetext: Department of Astronomy, University of Cape Town,
Private Bag X3, 7701 Rondebosch, South Africa 1010institutetext: Observatorio
Astronómico Nacional (IGN), C/Alfonso XII 3, Madrid E-28014, Spain
1111institutetext: University of Alberta, Edmonton, Alberta, T6G 2E1, Canada
1212institutetext: Steward Observatory, University of Arizona, Tucson, AZ
85721, USA 1313institutetext: Universität Heidelberg, Zentrum für Astronomie,
Albert-Ueberle-Str. 2, 69120 Heidelberg, Germany 1414institutetext: Max-
Planck-Institut für extraterrestrische Physik, Giessenbachstraße 1, D-85748
Garching, Germany 1515institutetext: Center for Astrophysics $\mid$ Harvard &
Smithsonian, 60 Garden St., 02138 Cambridge, MA, USA 1616institutetext:
Departamento de Fisica de la Tierra y Astrofisica, Facultad de CC Fisicas,
Universidad Complutense de Madrid, 28040, Madrid, Spain 1717institutetext:
Department of Physics, Tamkang University, No.151, Yingzhuan Road, Tamsui
District, New Taipei City 251301, Taiwan 1818institutetext: Sub-department of
Astrophysics, Department of Physics, University of Oxford, Keble Road, Oxford
OX1 3RH, UK

# Neutral atomic and molecular gas dynamics in the nearby spiral galaxies NGC
1512, NGC 4535, and NGC 7496

Sebastian Laudage  11<EMAIL_ADDRESS>Cosima Eibensteiner  Jansky Fellow
of the National Radio Astronomy Observatory, 11 2 2<EMAIL_ADDRESS>Frank
Bigiel 11 Adam K. Leroy 33 4 4 Sharon Meidt 55
Eva Schinnerer 66 W.J.G. de Blok 778899 Miguele Querejeta 1010 Sophia Stuber
66 Dario Colombo  11
Erik Rosolowsky 1111 D.J. Pisano 99 Dyas Utomo 33 Rebecca C. Levy 1212 Ralf
Klessen 1313 Yixian Cao 1414
Eric W. Koch 1515 Sushma Kurapati 99 Patricia Sanchez-Blazquez 1616 Justus
Neumann 66 Lukas Neumann 11 Hsi-An Pan 1515 Thomas G. Williams 1818

(Received ; accepted)

Neutral atomic gas ($\rm H\,\textsc{i}$) effectively traces galactic dynamics
across mid to large galactocentric radii. However, its limitations in
observing small-scale changes within the central few kiloparsecs, coupled with
the often observed H i deficit in galactic centers, necessitates the use of
molecular gas emission as a preferred tracer in these regions. Understanding
the dynamics of both neutral atomic and molecular gas is crucial for a more
complete understanding of how galaxies evolve, funnel gas from the outer disk
into their central parts, and eventually form stars. In this work we aim to
quantify the dynamics of both, the neutral atomic and molecular gas, in the
nearby spiral galaxies NGC 1512, NGC 4535, and NGC 7496 using new MeerKAT H i
observations together with ALMA CO (2-1) observations from the PHANGS
collaboration. We use the analysis tool ${}^{\text{3D}}$Barolo to fit tilted
ring models to the H i and CO observations. A combined approach of using the H
i to constrain the true disk orientation parameters before applying these to
the CO datasets is tested. This paper sets expectations for the results of the
upcoming high-resolution H i coverage of many galaxies in the PHANGS-ALMA
sample using MeerKAT or VLA, to establish a robust methodology for
characterizing galaxy orientations and deriving dynamics from combing new H i
with existing CO data.

###### Key Words.:

Galaxy: kinematics and dynamics, ISM: atoms, ISM: molecules

##  1 Introduction

Different emitting constituents (neutral gas, ionized gas, molecular gas, or
stars) of star-forming disk galaxies and their individual rotation curves
(RCs) can be used to trace galactic dynamics. Each of these galactic
components offers a unique perspective on the dynamics caused by their
distinct and characteristic radial and vertical distribution (Colombo et al.,
2014; Martinez-Medina et al., 2020). The ubiquitous neutral atomic gas ($\rm
H\,\textsc{i}$) is typically the most extended component in galaxies reaching
out to $2-4\times\text{r}_{25}$ (Wang et al. 2016; where r25 is the radial
position of the isophote at 25 mag arcsec-2).

H i kinematics have long been used to trace the galactic potential and its
division into baryonic and dark matter mass components (e.g. de Blok et al.,
2008; de Blok & Bosma, 2002; Gentile et al., 2007; Lelli et al., 2016). The
extended atomic gas disks in particular, provide a key view on the dark matter
halos of galaxies (e.g. Bosma, 1981; Sofue & Rubin, 2001) and the accretion of
new material that supplies the fuel for star formation (see e.g. Schmidt et
al., 2016; Di Teodoro & Peek, 2021; Eibensteiner et al., 2023).

To trace small-scale kinematics in the centers of galaxies that are rich in
dynamical structure (e.g. having bars or rings), CO becomes preferred over
$\rm H\,\textsc{i}$. This is because the ISM tends to be molecular-dominated
towards the center and CO observations can achieve better resolutions (Bigiel
& Blitz, 2012; Levy et al., 2018; Su et al., 2022; Walter et al., 2008). One
of the issues often faced when modeling the kinematics in the centers is that
the gravitational potential and gas kinematics show signs of non-axisymmetry
(i.e due to stellar bars and spiral arms), which can make it difficult to
accurately determine the disk orientation and RC (Lang et al., 2020; Kim &
Ostriker, 2006) at these radii. Additionally, the molecular gas traced by CO
tends to be more clumpy in comparison to H i , making CO velocity fields
sometimes sparsely sampled, presenting a further challenge for kinematic
modeling (Lang et al., 2020).

Given these difficulties, a solution would be to use the H i velocity field
sampling further out in the disk to constrain the true disk orientation,
thereby improving the accuracy in RC determination from either the CO or H i
information. This paper examines the validity and usefulness of this approach,
studying an initial small sample of galaxies with both existing CO and new H i
data. The hypothesis is that H i can provide a good probe of the gas disk’s
intrinsic orientation, as long as additional factors that affect H i and CO
are taken into consideration on a galaxy-by-galaxy basis. These factors
include the fact that outer H i disks of galaxies often exhibit genuine warps
and twists as well as spiral arms (e.g. Wong et al., 2004; de Blok et al.,
2008; Schmidt et al., 2016; Trachternach et al., 2008). H i kinematics may
also trace radial inflows, as newly accreted material in the outskirts of the
disk makes its way inward (e.g. Wong et al., 2004; Di Teodoro & Peek, 2021;
Eibensteiner et al., 2023). All of these factors can impact the derivation of
the disk parameters. Consequently, both the orientation must be allowed to
vary when modeling H i velocity fields, and the validity of that orientation
in the inner CO disk is not guaranteed.

In this paper, we perform a detailed modeling of the CO and H i observations
to determine where their RCs agree and under which circumstances they differ.
Our initial target consists of three galaxies, but it should be sufficient for
reaching our goal of determining a best-practice recommendation for fitting
the disk orientation and RC when both CO and H i observations are available in
future works.

For our dynamic analysis and comparison of atomic and molecular rotation
curves, we use observations from the PHANGS collaboration (Physics at High
Angular resolution in Nearby GalaxieS111www.phangs.org) (Schinnerer et al.,
2019). The PHANGS-ALMA survey presents 90 nearby
($d\leq$20\text{\,}\mathrm{Mpc}$$ ) galaxies targeting CO J=2$\rightarrow$1
emission (Leroy et al., 2021). By using the excellent capabilities of the
MeerKAT telescope it is possible to complement these data with a unique view
in H i . Eibensteiner et al. (2024) present a refined view of the H i in eight
nearby star-forming galaxies observed with the MeerKAT telescope. Three of
these galaxies are part of the PHANGS-MeerKAT sample (cycle0; PI: D. Utomo)
which is growing in the near future (cycle1; PI: D.J. Pisano). The focus of
this paper is on the first three galaxies in the PHANGS-MeerKAT sample: NGC
1512, NGC 4535, and NGC 7496.

In addition to pure circular velocities, we also analyze radial velocities in
H i of the three galaxies. Studying radial motions of matter in galaxies
contributes to the explanation of their current star formation rates and
evolution (Wong et al., 2004; Spekkens & Sellwood, 2007). Many galaxies,
including the Milky Way, have been forming stars at a nearly constant or
slightly decreasing rate throughout cosmic time (Bigiel et al., 2011; Panter
et al., 2007). This raises the question of why the star-forming gas is not yet
depleted. A prominent explanation for the current star formation rates (SFRs)
is radially transported neutral atomic gas (see e.g. Schmidt et al., 2016). We
analyze radial mass flow rates even though our sample size is currently not
large enough to formulate conclusions on this topic. However, with upcoming H
i observations the presented methodology in this paper can be applied to a
much larger sample.

Figure 1: Top row: Moment 1 maps from PHANGS MeerKAT H i observations of NGC
1512, NGC 4535, and NGC 7496 built with the strict masking procedure described
in Section 3. The maps are overplotted with isovelocity-lines (grey) in
intervals of $30\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. Middle row:
Integrated intensity or Moment 0 maps from the MeerKAT H i observations built
with the same masking procedure. The location of the companion galaxy NGC 1510
is indicated by the white circle. Bottom row: Integrated intensity or Moment 0
maps of the three galaxies built from CO observations taken from PHANGS-ALMA.
The extent of the CO maps is indicated in comparison to the H i maps.

We describe our observations in Section 2. Section 3 outlines our methodology
for implementing the tilted ring model. We present our results that encompass
the best-fit parameters and rotation curves, in Section 4. Section 5, is
dedicated to the implications and trustworthiness of our models and we further
quantify radial motions in our models.

Table 1: Physical Properties of the MeerKAT galaxies NGC 1512, NGC 4535 and
NGC 7496. The columns from left to right contain the morphological type,
central right ascension (RA) and declination (Dec.), the distance, the
systemic velocity, the stellar effective radius, and the bar radius. The
values are taken from: (1) (Makarov et al., 2014), (2) (Salo et al., 2015),
(3) distanced are curated by (Anand et al., 2021)(with original references for
NGC 1512: Scheuermann et al. (2022); for NGC 4535: Freedman et al. (2001), and
for NGC 7496: Shaya et al. (2017) and Kourkchi et al. (2020))

, (4) (Lang et al., 2020),(5) (Leroy et al., 2021), (6) (Querejeta et al.,
2021).  Galaxy ID Morphology $\alpha_{\text{J2000}}$ $\delta_{\text{J2000}}$
Distance ($\text{\,}\mathrm{Mpc}$) V${}_{\text{sys}}$
($\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$) R${}_{\text{eff}}$
($\text{\,}\mathrm{kpc}$) R${}_{\text{bar}}$ ($\text{\,}\mathrm{kpc}$)
(Reference) (1) (2) (2) (3) (4) (5) (6) NGC 1512 SB(r)ab
$4^{\text{h}}03^{\text{m}}54.1^{\text{s}}$
$-$43\text{\,}\mathrm{\SIUnitSymbolDegree}$$20\text{\,}\mathrm{\SIUnitSymbolArcminute}$$55\text{\,}\mathrm{\SIUnitSymbolArcsecond}$$
$17.92\pm 0.88$ $871.4\pm 5$ 4.8 6.5 NGC 4535 SAB(s)c
$12^{\text{h}}34^{\text{m}}20.31^{\text{s}}$
$+$8\text{\,}\mathrm{\SIUnitSymbolDegree}$$11\text{\,}\mathrm{\SIUnitSymbolArcminute}$$56\text{\,}\mathrm{\SIUnitSymbolArcsecond}$$
$15.77\pm 0.37$ $1953.6\pm 5$ 6.3 2.9 NGC 7496 SBb
$23^{\text{h}}09^{\text{m}}47.3^{\text{s}}$
$-$43\text{\,}\mathrm{\SIUnitSymbolDegree}$$25\text{\,}\mathrm{\SIUnitSymbolArcminute}$$40\text{\,}\mathrm{\SIUnitSymbolArcsecond}$$
$18.72\pm 2.82$ $1639.2\pm 5$ 3.8 3.4

##  2 Sample and Observations

###  2.1 Sample

The galaxies NGC 1512, NGC 4535, and NGC 7496 are all nearby (d$\lesssim$ 20
Mpc), star-forming, barred spiral galaxies (Table 1). Targets were selected
from the parent sample of galaxies that have PHANGS-ALMA CO observations and
where MeerKAT has an opportunity to substantially improve on the quality of
data available in the literature because of its excellent $uv$ coverage,
sensitivity and resolution capabilities. Previous H i studies of nearby
galaxies that are part of the PHANGS-ALMA sample (e.g Koribalski et al., 2018;
de Blok et al., 2008; Murphy et al., 2010; de Blok et al., 2024; Chung et al.,
2009) have led to a detailed view of the atomic gas distribution already. NGC
1512 for example was covered with the Australia Telescope Compact Array (ATCA)
(Koribalski & López-Sánchez, 2009) and additionally observed at a coarser
resolution with MeerKAT (Elson et al., 2023). NGC 4535 has been mapped as part
of the Imaging survey of Virgo galaxies in Atomic gas (VIVA) (Chung et al.,
2009) before.

Moment maps in H i and CO and the moment 1 map in H i are presented in Figure
1. The most unique appearance is seen in NGC 1512. It has a double-ring
structure, a nuclear ring around the galactic center, and an additional ring
further out in the main disk. The galaxy presents a very extended disk in H i
(going out to $\sim$120\text{\,}\mathrm{\SIUnitSymbolArcsecond}$$) with the
dwarf galaxy NGC 1510 located about $26.5\text{\,}\mathrm{kpc}$ away (assuming
the distance presented in table Table 1). A clear peak in H i emissions is
seen at the position of NGC 1510 from our integrated intensity map (compare
Figure 1. The galaxy pair is believed to be strongly interacting because of
the proximity, creating an asymmetric distribution of H i across the disk of
NGC 1512 (Koribalski & López-Sánchez, 2009). A similar but, not as strong
unwinding of spiral arms can be seen in NGC 4535. Such unwinding of spiral
arms is typical for galaxies influenced by ram pressure inside galaxy clusters
(Bellhouse et al., 2021). NGC 4535 is part of the Virgo Cluster and the
peculiar structure at large radii might be caused by this effect.

###  2.2 MeerKAT 21cm observations

The H i observations used in this work were taken during the first observing
cycle (Cycle 0) of the new MeerKAT (Meer Karoo Array Telescope) radio
interferometer (PI: D. Utomo). It is the most sensitive centimeter-wavelength
interferometer in the southern hemisphere. The special design of the MeerKAT
antennas, results in low system temperatures and an antenna gain of $\sim$2.8
K/Jy (Jonas, 2016). These properties make it ideally suited to detect low
column density H i emission (see e.g. Koribalski et al., 2020; de Blok et al.,
2016).

The galaxies (see Table 1) were each observed for 6 hours in a frequency range
from $1.411\text{\,}\mathrm{GHz}$ to $1.414\text{\,}\mathrm{GHz}$. The
observations were imaged to a resulting angular resolution of
$15\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ and a spectral resolution of
$5.5\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ per channel (see
Eibensteiner et al., 2024, for more details). We used the strict masking
procedure described in Leroy et al. (2021) that was initially used for the
PHANGS-ALMA CO(2-1) observations. This high-confidence masking includes only
voxels with a high signal-to-noise ratio and few or no noise-dominated sight
lines. This is achieved by creating a core mask that includes all voxels with
a signal-to-noise (S/N) larger than 4 in two consecutive velocity channels and
a lower S/N outer mask that includes voxels with a S/N larger than 2 in two
consecutive velocity channels. The strict mask then consists of all contiguous
regions in the outer mask that contain pixels from the high-significance inner
mask.

###  2.3 PHANGS-ALMA CO observations

To trace the molecular component we used observations from the PHANGS-ALMA
survey (Leroy et al., 2021). The survey maps the CO $\text{J}=2\rightarrow 1$
line emission in these galaxies at an angular resolution of
${\sim}1^{\prime\prime}$ which corresponds to linear scales of
${\sim}$100\text{\,}\mathrm{pc}$$. The spectral resolution is
$2.5\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. For CO we used a
different masking procedure than for the H i described above. As CO emission
is generally less smoothly distributed, and good spatial coverage is needed
for reliable results, we used another product offered by the PHANGS-ALMA
pipeline, the ”moment1 wprior” map. These maps include all moment 1 values
calculated from the strict mask (procedure described above), but add line-of-
sights that are in agreement with a prior guess (the prior is created from a
strictly masked $15\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ resolution cube).
This process can result in increased spatial coverage of the moment-1 map at
high resolution compared to the strictly masked moment-1 maps, while still
being reliable at rejecting severe outliers in the velocity space (see Leroy
et al., 2021, for more details).

##  3 Methods

RCs are commonly derived by fitting tilted ring models to the velocity field
of the line emission and extracting the rotational velocity parameter from the
best-fit model. Several available software packages exist that support this
approach such as ROTCUR (Begemann, 1987) or DISKFIT (Spekkens & Sellwood,
2007). These two-dimensional approaches provide reliable RCs for highly
resolved data but have the disadvantage of being dependent on assumptions made
during the extraction of the velocity fields.

The beam-smearing effect is another severe problem in deriving kinematics from
the velocity field (Bosma, 1981). The effect is caused by the finite size of a
telescope’s beam which causes es the line emission to be smeared in the
adjacent regions. As a typical result, the rising part of the extracted
rotation curve can be underestimated.

A solution to take both mentioned effects into account is to use a
3-dimensional tilted ring modeling algorithm such as TiRiFiC (Jozsa et al.,
2007) or ${}^{\text{3D}}$Barolo (Di Teodoro & Fraternali, 2015). In this
paper, we compute rotation curves using ${}^{\text{3D}}$Barolo which fits
tilted ring models to emission line data cubes in three dimensions. It does
not depend on the methodology of the velocity field derivation and takes beam
smearing into account by introducing the instrumental effect to the model
during the convolution.

###  3.1 Tilted Ring Model

Figure 2: Top row: Line-of-sight velocities along the major axis position from
the H i observations of NGC 1512 (left), NGC 4535 (middle), and NGC 7496
(right). Bottom row: Line-of-sight velocities along the minor axis. The x-axis
represents the offset of the galactic center and the y-axis represents the
observed velocities with (left) and without (right) subtracting the systemic
velocity of the galaxy.

${}^{\text{3D}}$Barolo shares the basic scheme adopted by any tilted-ring
model, modeling the geometry and dynamics of a galaxy by subdividing the
galactic disk radially into several circular rings. Each ring is described by
a set of parameters that (at first) do not depend on parameters from other
rings. The ring parameters are connected to line-of-sight velocities via a
harmonic expansion:

$\displaystyle
V_{\text{los}}(R,\theta)=V_{\text{sys}}+V_{\text{rot}}(R)\,\text{cos}\,\theta\,\text{sin}\,i+V_{\text{rad}}(R)\,\text{sin}\,\theta\,\text{sin}\,i$
(1)

with $i$ being the inclination angle and $\theta$ the azimuthal angle measured
inside the galactic disk (Iorio et al., 2017). The ring radius $R$ is defined
as the median distance of each ring to the center. The orientation of the
projection of the inclined galactic disk is described by the position angle
$\phi$. It is an azimuthal angle between the north direction and the receding
major axis of the projection on the celestial sphere measured eastwards. The
connection between $\theta$ and $\phi$ is given by the following relations:

$\displaystyle\cos{\theta}=\frac{-(x-x_{0})\,\sin{\phi}+(y-y_{0})\,\cos{\phi}}{R}$
(2)
$\displaystyle\sin{\theta}=\frac{-(x-x_{0})\,\cos{\phi}+(y-y_{0})\,\sin{\phi}}{R\,\cos{i}}$
(3)

with $x_{0}$ and $y_{0}$ describing the coordinates of the center of the
rings. This description yields line-of-sight velocities based on a set of
parameters for an arbitrarily chosen coordinate in the disk. With the
equation, it is possible to fit these parameters to an observed velocity
field.

###  3.2 Initial Parameters and Assumptions with ${}^{\text{3D}}$Barolo

Finding the best initial parameters for the fitting procedure is one of the
most important steps when working with algorithms like ${}^{\text{3D}}$Barolo.
The two-dimensional tilted ring model already involves seven fitting
parameters (see Equation 1 and Equation 2), namely
$V_{\text{sys}},x_{0},y_{0},i,\phi,V_{\text{rad}}$, and $V_{\text{rot}}$. In
the 3D version, three additional parameters are needed to apply a model
directly to the emission line data cube, which are $z_{0}$ (the thickness of
the disk), $\Sigma_{\text{gas}}$ (gas surface density), and
$\sigma_{\text{gas}}$ (the velocity dispersion in the gas disk). The used
initial parameters were taken from previous works on these galaxies and are
listed with sources in Table 2.

Table 2: A total of 9 parameters are used during the fitting: The systemic
velocity $V_{\text{sys}}$, the coordinates of the galactic center given in
right ascension and declination, the rotational velocity $V_{\text{rot}}$, the
inclination angle (Inc.), the position angle (PA), the dispersion velocity
$\sigma_{\text{gas}}$, the scale height (z0) and the radial velocity
($V_{\text{rad}}$). Here we list the initial values for the fitting process.

Parameter | NGC 1512 | NGC 4535 | NGC 7496
---|---|---|---
$V_{\text{sys}}$ [$\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$](2) | 871.4 | 1953.6 | 1639.2
R.A. [deg] (2) | 60.97557 | 188.58459 | 347.44703
Decl. [deg] (2) | -43.34872 | 8.19797 | -43.42785
$V_{\text{rot}}[$\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$]$ (1) | 210 | 200 | 175
Inc. [deg] (2) | 42.5 | 44.7 | 35.9
PA [deg] (2) | 261.9 | 179.7 | 193.7
$\sigma_{\text{gas}}$ [$\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$] (4) | 8 | 8 | 8
Z0 [$\mathrm{kpc}$] (3) | 0.68 | 0.68 | 0.68
Z0 [arcsec] (3) | 7.5 | 8.9 | 7.5
$V_{\text{rad}}$ [$\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$] (4) | 0 | 0 | 0

Notes:(1) Lang et al. (2020); (2) Galaxy centers taken from Leroy et al.
(2021) ; (3) Randriamampandry et al. (2021); (4) Default value of
${}^{\text{3D}}$Barolo.

Ten free parameters lead to degeneracy in the parameter space, which can cause
discontinuities within the results. We deal with this issue by dividing the
modeling process into a series of runs, reducing the number of parameters to
fit each run.

  * -

First run: The set of initial parameters (Table 2) is applied in a first run
where all parameters except for $V_{\text{rad}}$ are allowed to vary. Radial
velocities are non-dominant for most of the regions in galactic disks and we
are interested in pure circular motion, which is why we set radial velocities
to zero. By using a weighting of $w(\theta)=|\cos{\theta}|^{2}$, we made sure
that radial motions do not severely influence the fits (compare Equation 1).
The ring width is set to the resolution of the observation
($15\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ for H i ;
$\sim$1\text{\,}\mathrm{\SIUnitSymbolArcsecond}$$ for CO).

  * -

Second run: For the second fitting stage, the central position and systemic
velocity of the rings are fixed to the median value of the results for all
rings from the first run, which is a common method for galactic modeling (e.g.
Iorio et al., 2017; Di Teodoro, 2015). If the rings, especially at large
radii, become asymmetric, the fitting process is not reliable anymore which
will be seen in a jump in the size of the error bars on all parameters. Rings
that have large error bars ($>$\pm
30\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$) on the V${}_{\text{rot}}$
values are neglected from here on. A flagging of rings after the final fitting
run is not advisable since the regularization of the fitting parameters used
in the third run might be influenced by badly fitted rings.

  * -

Regularization: The results of the second run can contain numerical scatter,
caused by the aforementioned degeneracy in the parameter space. All geometric
parameters are further regularized to a function or a constant value to
suppress this. The scale height $Z_{0}$ is regularized by a constant median
value. The inclination $i$ and the position angle $\phi$ are regularized to a
functional form that represents the fitting results of the second run. To
differentiate between real shifts in geometry and numerical scatter we
investigated the position velocity diagrams (Figure 2) and moment maps (Figure
1) of the observations, which are a good indicator for warped structures.
Warps will affect the structure of the major axis position velocity diagram,
while radial motions influence the minor axis position velocity diagram (Di
Teodoro & Peek, 2021, for a detailed description of this technique). All three
major axis position velocity diagrams show variations in the velocities which
can be an indicator for warps. Therefore we regularize the inclination and
position angle by using Bezier functions and do not use more restrictive
functional forms.

  * -

Final run: Once the geometry of the galaxy is fixed, the kinematics are
fitted. The only free parameters in this run are the rotation velocity
$V_{\text{rot}}$ and the velocity dispersion $\sigma_{\text{gas}}$. This run
yields the final rotation curve. The last two runs are repeated several times
to get separate fits for the approaching side, the receding side, and both
sides at once.

The errors of the parameters for the final model are calculated during the
second (for $i$ and $\phi$) and third run (for $V_{\text{rot}}$) based on
${}^{\text{3D}}$Barolo’s default error calculation method.

For the CO rotation curves, we use the best-fit results of our H i fits as
starting parameters but adapt the fitting algorithm slightly to account for
the different properties of CO disks. Two stages are used in this case. The
galaxy’s systemic velocity and central position are fixed from the beginning
to the values found by our H i models and a $w(\theta)=|\cos{\theta}|^{2}$
weighting is applied. The scale height $Z_{0}$ of the disk is expected to be
much thinner when observed in CO (compared to $\rm H\,\textsc{i}$). Therefore
a thin disk approximation of $100\text{\,}\mathrm{pc}$ is assumed (similar to
Roper et al. (2022) and Meidt et al. (2023)) and held fixed throughout the
runs. After a first initial fitting stage where $i$, $\phi$,
V${}_{\text{rot}}$, and $\sigma_{\text{gas}}$ are kept free to vary, the
geometry ($i$ and $\phi$) is fixed to the median value of the best-fit values
of all rings. The second run only fits the kinematics $V_{\text{rot}}$ and
$V_{\text{disp}}$ for the final model.

Figure 3: Top row: Final H i rotation curves (black circles) for NGC 1512
(left), NGC 4535 (middle) and NGC 7496 (right) overplotted with rotation
curves derived by only considering the approaching (blue) and the receding
(red) side of the galaxy. The bar radius is indicated with a dashed grey line
and the distance of NGC 1510 is indicated with the green dashed line. Bottom
row: The panels show the adopted geometry in our fitting procedure for NGC
1512 (left), NGC 4535 (middle) and NGC 7496 (right) for the H i observations.
The grey squares and circles represent the inclination $i$ and position angle
$\phi$ results for each ring after the second fitting stage, respectively. The
blue (inclination) and yellow (position angle) lines represent the
regularization function that was used to smooth out the geometry in the final
stage of the fitting procedure.

###  3.3 Radial Velocities

For a more complete picture of the atomic gas dynamics, we additionally
modeled radial motions inside the galactic disks. For that, the third term of
Equation 1 is now included and $V_{\text{rad}}$ is free to vary instead of
being fixed to zero as before. As shown in Eibensteiner et al. (2023), radial
velocities are notoriously difficult to measure in galaxies, since only small
changes in their geometry can lead to drastic changes of V${}_{\text{rad}}$.
The most reliable radial velocities are acquired using the most robust
geometric predictions of the galaxies. Therefore, following an approach
similar to Di Teodoro & Peek (2021), we fit the radial velocities in a
singular run using ${}^{\text{3D}}$Barolo with all the other parameters fixed
to the previously determined values from our fitting routine. As mentioned
before, radial motions dominantly leave a mark near the minor kinematic axis
of galaxies (Section 3.2), which is why for this fitting run a
$\sin{\phi}^{2}$ weighting is applied.

The value of radial velocities derived by ${}^{\text{3D}}$Barolo alone does
not inform whether the motion is directed inwards (inflow) or outwards
(outflow). The direction of the galaxy’s rotation is needed to correctly
interpret the nature of the radial motion. By definition of
${}^{\text{3D}}$Barolo, a positive (negative) $V_{\text{rad}}$ is tracking
outflow (inflow) if the galaxy rotates clockwise and inflow (outflow) when it
rotates counterclockwise. Assuming that spiral galaxies spin with their arms
trailing the rotation movement it is possible to determine the rotation
velocities from optical images. We present the results, such that positive
values are tracking outflow and negative values inflow. The errors on the
radial velocities are determined by ${}^{\text{3D}}$Barolo’s error calculation
method.

###  3.4 Mass flow rates

Detecting radial motions in galaxies can potentially indicate mass transport
along the radial direction of galaxies, which is needed to fuel ongoing star
formation. To estimate how efficient the transport is, we take the H i surface
density profiles $\Sigma_{\text{HI}}\text{(R)}$ into account. Average mass
flow rates can be calculated using a simplified approach similar to
Eibensteiner et al. (2023) or Di Teodoro & Peek (2021). For that H i intensity
profiles $I(R)$ are taken from the ${}^{\text{3D}}$Barolo task ELLPROF, which
gives the average flux along an elliptical ring defined by the best-fit
geometrical parameters found in our kinematic modeling. For the error of
$I(R)$, we adopt the standard deviation provided by the task for each ring.
The radial intensity profile can be converted to a face-on H i mass surface
density $\Sigma_{\text{HI}}(R)$ when assuming that the H i emission is
optically thin:

$\frac{\Sigma_{\text{HI}}(R)}{[M_{\odot}\,\text{pc}^{-2}]}=8794\frac{I(R)\cos{i(R)}}{[\text{Jy}\,\text{beam}^{-1}\,$\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$]}\left(\frac{\text{B}_{\text{maj}}\text{B}_{\text{min}}}{[\text{arcsec}^{2}]}\right)^{-1}$
(4)

with B${}_{\text{maj, min}}$ being the full width at half maximum of the beam
(here $15\text{\,}\mathrm{\SIUnitSymbolArcsecond}$) and the factor
$\cos{i}(R)$ correcting for the inclination of the disk.
Using this result, it is now possible to calculate the average H i mass flow
$\dot{M}$ for a given radius $R$ by:

$\dot{M}_{\text{HI}}(R)=2\pi R\Sigma_{\text{HI}}(R)V_{\text{rad}}(R)$ (5)

with $R$ being the radial distance of the ring to the center. The error of
$\dot{M}_{\text{HI}}$ is calculated using a Gaussian error propagation of the
errors on $\Sigma_{\text{HI}}(R)$ and $V_{\text{rad}}(R)$. Positive values of
$\dot{M}_{\text{HI}}$ can be understood as mass outflow and negative values of
$\dot{M}_{\text{HI}}$ as mass inflow (accretion). To calculate the total
neutral gas mass flow $\dot{M}$, the H i mass flow $\dot{M}_{\text{HI}}$ is
multiplied by the factor 1.33 which takes the primordial abundance of helium
into account.

##  4 Results

In this section, we present the resulting best-fit parameters derived from our
modeling process. First, we will describe the achieved rotation curves from
the MeerKAT H i data and then present the results achieved using the PHANGS-
ALMA CO data. Values without an error were fixed from the start and not fitted
by ${}^{\text{3D}}$Barolo.

###  4.1 HI rotation curves

Table 3: Comparison of best-fit parameters obtained from MeerKAT-H i and ALMA-CO data. All values in this table are derived from this work. The ring count is given for rings with a thickness of $15\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ width for H i and $1\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ for CO. | NGC 1512 | NGC 4535 | NGC 7496
---|---|---|---
| H i | CO | H i | CO | H i | CO
$V_{\text{sys}}$ [$\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$] | $880.8\pm 6.3$ | $880.8$ | $1951.7\pm 7.7$ | $1951.7$ | $1639.2\pm 5.2$ | $1639.2$
R.A. [deg] | 60.97557 | 60.97557 | 188.58459 | 188.58459 | 347.44703 | 347.44703
Decl. [deg] | -43.34872 | -43.34872 | 8.19797 | 8.19797 | -43.42785 | -43.42785
$z_{0}$ [$\text{\,}\mathrm{kpc}$] | $1.06\pm 0.08$ | 0.1 | $0.79\pm 0.08$ | 0.1 | $0.83\pm 0.1$ | 0.1
$i$ median [$\text{\,}\mathrm{\SIUnitSymbolDegree}$] | $43.47\pm 5.4$ | $43.3\pm 3.03$ | $44.3\pm 2.93$ | $43.7\pm 4.5$ | $37.99\pm 3.59$ | $44.49\pm 2.72$
$\phi$ median [$\text{\,}\mathrm{\SIUnitSymbolDegree}$] | $265.8\pm 10.8$ | $252.53\pm 9.15$ | $179.91\pm 5.75$ | $182.77\pm 7.78$ | $191.8\pm 7.22$ | $201.22\pm 7.17$
V${}_{\text{disp.}}$ [$\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$] | $4.28\pm 4.23$ | $5.97\pm 2.33$ | $6.53\pm 4.52$ | $3.88\pm 2.02$ | $7.32\pm 3.98$ | $5.23\pm 2.54$
V${}_{\text{rot, max}}$ [$\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$] | $177.3\pm 16.4$ | $204.8\pm 9.7$ | $188.8\pm 6.2$ | $181.93\pm 6.09$ | $125\pm 5.8$ | $118.11\pm 6.09$
R${}_{\text{max}}$ [kpc] | 517.5 | 89.93 | 247.5 | 109.33 | 157.5 | 74.54
Ring Count | 32 | 36 | 15 | 55 | 10 | 45

Figure 3 shows the best-fit HI rotation velocities, as well as the results
achieved using one side (receding or approaching) of the galactic disk. The H
i rotation curves reach velocities between ${\sim}90$ and
${\sim}$200\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$. Successful fits
were achieved for 32 rings of $15\text{\,}\mathrm{\SIUnitSymbolArcsecond}$
width for NGC 1512, 15 rings for NGC 4535, and 10 rings for NGC 7496. The
innermost rings of the RCs exhibit velocities close to the maximum of the
curves. A steep rise of the velocities is therefore expected in the central
regions, which are not covered by the H i observations.

The approaching and receding rotation curves are typically close to each other
(within $\approx$20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$) for most
radii in all three galaxies. Differences in the approaching and receding
fitting results are caused by asymmetries in the disk. Especially for NGC
1512, we see very pronounced differences between the approaching and receding
sides at radii larger than $300\text{\,}\mathrm{\SIUnitSymbolArcsecond}$,
which are caused by the influence of the companion galaxy NGC 1510.

Our best-fit values of the central positions and systemic velocities are
provided in Table 3. We took the median for these two parameters for all
successfully fitted rings after the first stage and used these values for the
following stages. The galactic centers agree well (same pixel) with the
previously determined values (our initial parameters). For the systemic
velocity, the values were in very good agreement with the previously found
values for NGC 4535 and NGC 7496. The new value for NGC 1512 of $880.8\pm
6.3\,$$\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ is still within the
errors of the previous $871.4\pm
5\,$$\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$.

The inferred disk geometry (Figure 3 bottom row) for all three galaxies warps
at large galactocentric radii that entail subtle changes in $\phi$ and $i$
moving outward. For NGC 4535 and NGC 7496, a warp in inclination at large
radii and evolving position angles throughout the disks are modeled. These
warps are also indicated by the deviation from constant velocities at similar
radii in the major axis position velocity diagrams (Figure 2). NGC 1512
exhibits the most complex geometry with a varying behavior of the inclination
at ${\sim}$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$ throughout its radial
extent. Such oscillations are also evident in the position velocity diagram of
the major axis and are likely caused by the presence of the companion galaxy
NGC 1510.

Figure 4: PHANGS-ALMA CO rotation curve of the sample produced by 3D-Barolo
(black circles). The previous fits by Lang et al. (2020) are plotted as grey
squares. The radius of the galactic bar is indicated by the vertical dashed
line. Some bins of the 3D-Barolo rotation curves did not converge and are
therefore not represented. The two different methods lead to very similar
results overall. The rotation curve of NGC 1512 has a large area without data
points, which corresponds to regions with too few emissions for reliable
fitting.

###  4.2 CO Rotation Curves

The circular velocities derived from the CO fitting procedure are presented in
Figure 4 in red. We overplotted the results with previous RCs derived by Lang
et al. (2020), which represent the latest kinematic analysis of these
galaxies. Their method differs from ours as they assumed that the CO emitting
disks are infinitely thin and rotating with a single, global orientation and
thus derived V${}_{\text{sys}}$, $i$, PA, and V${}_{\text{rot}}$ with 2D
tilted ring fitting (with $\phi$, $i$ and V${}_{\text{sys}}$ held fixed). For
a more direct comparison, we adopted the same radial bins as Lang et al.
(2020).

Our RCs based on CO recover the central rising part of the curves well. For
all three galaxies a strong rise within the first
$\approx$1\text{\,}\mathrm{kpc}$$ can be seen with a flatter behavior for
larger radii. The rotation curve of NGC 1512 has a large area without data
points, which correspond to regions with too few emission for reliable
fitting. NGC 4535 shows a pronounced dip around the bar radius, then rises
again and flattens out at around $6\text{\,}\mathrm{kpc}$. NGC 7496 shows
overall lower circular velocities than the other two galaxies and rises slowly
until the largest radii.

The adopted best-fit parameters are listed in Table 3. Overall the circular
velocities derived by ${}^{\text{3D}}$Barolo agree well with the values from
the previous work by Lang et al. (2020) (compare Figure 4). The best-fit
parameters of this work (Table 3) are also in good agreement with the
previously derived values (our starting parameters Table 2). For NGC 7496, our
three-dimensional CO fit produces a higher inclination and position angle.
While the errors of both values are overlapping, we still investigated if the
new methodology is causing the different results in the geometry. Different
values for the scale height for H i did not have a big impact on our fitting
results (1pc and 500pc stay within the errors of the final fit; see Appendix
A) and therefore we trust that they do not have a decisive impact on our CO
fits. Additionally, we fitted the galaxy by fixing the geometry to the
starting parameters, but this produced unrealistic rotation curves with large
jumps in velocity ($\pm$20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$$
between many bins). Because of the better agreement with the H i results we
think that the geometric values from Lang et al. (2020) are more trustworthy
for the inner CO disk of NGC 7496.

##  5 Discussion

Figure 5: Left column: Moment 1 map of the H i observations for NGC 1512
(top), NGC 4535 (middle), and NGC 7496 (bottom). The black ring indicates the
outermost radius that was used for the model. The pink line tracks the
systemic velocity line. Middle column: Moment 1 map of the H i model using the
best-fit parameters found. Right column: Residual of the observation and the
model. Models and residuals larger than R${}_{\text{max}}$ are not shown due
to unphysical results as discussed in Section 3 as part of the second run.
Figure 6: Left column: First-moment map of the CO observations for NGC 1512
(top), NGC 4535 (middle), and NGC 7496 (bottom). The black ring indicates the
outermost radius that was used for the model. The pink line tracks the
systemic velocity line. Middle column: First-moment map of the H i model using
the best-fit parameters found. Right column: Residual velocity field between
the observation and the model. Models and residuals larger than
R${}_{\text{max}}$ are not shown due to unphysical results as discussed in
Section 3 as part of the second run.

In this section, we discuss the reliability and implications of previously
presented results and evaluate how our assumptions affect them.

###  5.1 Reliability of the models

The H i rotation curves (Figure 3) exhibit a slight rise at the inner most
radii, but overall H i seems to track mostly the flat regime for all three
galaxies. The velocities in this regime correspond to very typical values
compared to previous studies. Kinematic H i studies on larger samples such as
THINGS (de Blok et al., 2008) or more recent publications (Di Teodoro & Peek,
2021) establish a typical range of
$100\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ to
$250\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. NGC 4535 and NGC 1512
therefore show normal velocity behaviour with NGC 7496 being at the lower end
of the typical range. Di Teodoro & Peek (2021) derived a H i -based rotation
curve for NGC 4535 using ${}^{\text{3D}}$Barolo, finding a lower inclination
angle ($36\text{\,}\mathrm{\SIUnitSymbolDegree}$ compared to our
$44.3\text{\,}\mathrm{\SIUnitSymbolDegree}$) resulting in higher rotation
velocities. The velocities are especially deviating at very large radii
because of the assumed constant inclination in contrast to the slightly warped
structure of our model. Applying a Tully-Fisher relation similar to McGaugh &
Schombert (2015) implies a velocity of about
$182\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ which is also in good
agreement with our rotation curve. The rotation curve for NGC 1512 published
by Koribalski & López-Sánchez (2009) shows higher velocities overall, as they
assume a constantly increasing inclination in contrast to our varying
inclination angle. The better resolution of the MeerKAT data used for our RC
($15\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ compared to their
$65\text{\,}\mathrm{\SIUnitSymbolArcsecond}$) reveals a more detailed picture
of the geometry, which can explain the differences.

A direct comparison between the observation and the models can help to
understand how the assumptions affect our fits. For that, we created residual
maps by subtracting the modeled velocities from the observations (Figure 5 for
H i and Figure 6 for CO). For NGC 1512 we see overall residual velocities
between $-20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ and
$20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ in H i throughout most of
the disk, but it has a large spike on the receding side of the disk at the
location of NGC 1510. For NGC 4535 and NGC 7496, the residual velocities are
also between $-20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ and
$20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ in H i for most of the
disk, which supports our modeling routine. For radii $\lesssim
R_{\text{Bar}}$, large residual velocities of up to
$60\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ are present and are likely
caused by non-circular motions associated with the bar (Schmidt et al., 2016).
These non-circular motions are not part of our model. Both galaxies further
show spiral patterns in the residual maps of H i that correspond to the
position of their spiral arms implying streaming motions along the spiral
arms. The model tries to account for real radial motions within the
observation, leading to an overestimation of regions without radial motions
and vice versa.

###  5.2 Caveats

Several effects can influence the outcome of a tilted ring fit. The tilted
ring approach relies on assumptions and approximations that might not always
be fully appropriate for real galaxies. Models are always symmetric, while
real H i disks are known to be subject to asymmetries (Espada et al., 2005;
Eibensteiner et al., 2023). The one-sided fitting results (Figure 3) in our
sample all show asymmetries, especially at large radii. Modeling the
kinematics of interacting galaxies (like NGC 1512 and NGC1510) is therefore
challenging and not reliable due to the distortion introduced by tidal
torques.

While three-dimensional modeling does lessen the effects of beam smearing
(Rathborne et al., 2015), it also comes with its own drawbacks. As described
by Roper et al. (2022), when the model emission of a given ring is compared to
the corresponding region on the sky, it is assumed that the emission comes
from a fixed radial interval within the disk. This would almost be true if the
rings were razor-thin vertically or two-dimensional. For rings that have a
volume, emission from above and below the mid-plane is taken into account,
which results in overlapping emission from other radii that contaminate a
given interval. This effect is boosted in particular for moderately to highly
inclined galaxies. Therefore, an observation from one given location might be
used for setting the parameters of several rings.

In reality, the outer gas disk of galaxies is often flared, which means that
its scale height increases with radius (Dib et al., 2021). In the model, the
average disk thickness of a large galaxy sample was used as an initial
parameter and the gas disk was fixed to have a constant scale height after a
first fitting run. It is therefore expected that the disks of the three
galaxies are generally thinner in the central regions and can be thicker in
the outer regions (depending on how strongly the H i is flared) than the
assumed fixed scale height parameter. We tested how this affects the final
parameters by varying the fixed scale height of our models from
unrealistically small ($Z_{0}=$1\text{\,}\mathrm{pc}$$) to unrealistically
large ($Z_{0}=$2\text{\,}\mathrm{kpc}$$) constant values over several fitting
runs. The outcome of these fits is presented in Appendix A. Most of the
results are still within the error range of our final adopted model. Assuming
a constant average scale height does not seem to be a dominant source of
uncertainty.

###  5.3 Combined fitting of H i and CO data

We compared the circular velocities of the different tracers in Figure 7.
Combining the two, leads to a more complete view of the disk dynamics,
covering the inner regions by CO to the most extended radii probed by H i .
The H i and CO circular velocities are in agreement for most bins where
measurements overlap. For all three galaxies, we find an offset towards higher
velocities in H i at a radius corresponding to the scale of the galactic bar.
Especially in NGC 1512, the velocities derived from the two tracers seem to
track different motions at the largest radii recovered by CO. Overall, the CO
rotation curves draw a more granular and less smooth picture of the
kinematics, which can be partially attributed to the higher resolution in CO.
Colombo et al. (2014) and Frank et al. (2016) reported similar local
differences between the CO and H i rotation velocities in M51. CO is
associated with tracing denser cold gas (compared to H i ), while emission
from atomic gas is more broadly distributed throughout the whole disk. Bars or
spirals cause more streaming motions in comparison to the inter-arm regions
which will contribute to the observed velocities. Additionally, CO emission
arises from a thinner disk compared to H i , which causes CO to trace the
gravitational potential differently than H i (e.g. Levy et al., 2018, 2019;
Zschaechner et al., 2015).

Figure 7: MeerKAT-H i rotation curves (black circles) are directly compared to
PHANGS-ALMA CO rotation curves (pink circles) derived in this work.

The residuals of the CO velocity fields reveal that assuming pure circular
velocities is a very simplified way to model the complex dynamics in the inner
disk. These regions are heavily influenced by non-circular motions. Large
differences especially in the systemic velocity line (pink) can be seen when
comparing observations and models of the first moments. The larger residuals
of the CO models in comparison to H i imply that the parameters determined
from H i are less disturbed by non-axisymmetric influences that are in general
difficult to model correctly. The inner CO disk does not necessarily share the
same orientation as the outer parts of the galaxy, but deriving first
orientations for the H i disk and then using these H i -based orientations as
initial parameters for the CO observations worked well for our observations
(see Section 4 and Figure 7). This is why we recommend using H i to correctly
model the disk orientation parameters before deriving CO-based RCs.

###  5.4 Radial motions

Figure 8: Radial velocities of HI determined by the tilted ring model on the
MeerKAT-H i observations (blue triangles). Mass flow rates show the neutral
atomic gas (purple circles). The bar radii and the radial position of NGC
1510’s center are indicated with the dashed and dotted lines.

In Figure 8 we show the radial motions that were extracted from the tilted
ring model (see Section 3.3). All galaxies show radial in- and outflow
velocities up to $\sim|$20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$|$.
In all of the three galaxies, we note that the bar affects the direction and
magnitude of the radial motions strongly. We also show the mass flow rates
$\dot{M}$ indicated as purple markers in Figure 8. The general direction of
the mass flow traces the direction of the radial velocities, but the magnitude
differs because of its additional dependence on how much gas is actually in
that radial bin (traced by $\Sigma_{\text{HI}}$).

The values for NGC 4535 and NGC 7496 vary between inflow and outflow up to -2
and 2 M⊙yr-1. Stronger mass flow rates can be observed in NGC 1512 where
values of $\dot{M}>10\,{\text{M}}_{\odot}\,\text{yr}^{-1}$ are reached. We see
the strongest changes in NGC 1512 at around
$R\approx$300\text{\,}\mathrm{\SIUnitSymbolArcsecond}$$, where the companion
galaxy NGC 1510 is located. At smaller radii, NGC 1512 shows a dominant
outflow with velocities of less than
$10\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$ while it suddenly changes
to strong inflow at 300 to $400\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ with
a magnitude of about $20\text{\,}\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$. These
shifts could be caused by the gravitational influence both galaxies have on
each other. It is more likely that the fitting tries to account for the large
residuals caused by the presence of NGC 1510 and might therefore overestimate
the radial velocities at this point. This uncertainty makes the radial bins
larger than $\text{R}_{c,1512}$ (Radial distance of NGC 1510 to the center of
NGC 1512) unreliable and we further ignore them in the ongoing analysis.

Existing studies of radial gas flow from HI observations of local galaxies
(e.g., Trachternach et al., 2008; Colombo et al., 2014; Schmidt et al., 2016;
Eibensteiner et al., 2023) have mostly derived radial velocities using 2D
harmonic decomposition. In this work, in contrast, we employ a 3D modeling
approach to gather information about the radial movement of gas.

Overall, the magnitude of $V_{\text{rad}}$ and $\dot{M}$ are in agreement with
typical values for comparable galaxies. Di Teodoro & Peek (2021) recently
published radial motions and mass flow rates for NGC 4535 using
${}^{\text{3D}}$Barolo. The magnitude of the mass flow is very similar to the
results in this work, but the orientation of the flow differs. Deriving radial
velocities based on tilted ring models is very sensitive to the assumed
geometrical parameters (e.g. Eibensteiner et al., 2023). Relatively small
differences in inclination can lead to a complete flip of the inferred flow
directions. Therefore, the differences in the results, even though the same
methods were used, are likely to be caused by the differently assumed
inclination angle ($i=$43.7\pm 4.5\text{\,}\mathrm{\SIUnitSymbolDegree}$$ with
a slight rise at large radii in this work compared to
$i=$36\text{\,}\mathrm{\SIUnitSymbolDegree}$$).

Table 4: Star formation rates taken from Leroy et al. (2019) compared to the average radial velocities and mass flow rates derived in this paper. Negative (positive) values represent inflow (outflow). | SFR | $\langle V_{\text{rad}}\rangle$ | $\langle\dot{M}\rangle$
---|---|---|---
| [M${}_{\odot}\,\text{yr}^{-1}]$ | [$\mathrm{km}\text{\,}{\mathrm{s}}^{-1}$] | [M${}_{\odot}\,\text{yr}^{-1}]$
NGC 1512 | $0.75\pm 0.75$ | $4.47\pm 8.1$ | $2.02\pm 1.85$
NGC 4535 | $2.04\pm 1.6$ | $-0.55\pm 5.57$ | $-0.13\pm 1.79$
NGC 7496 | $2.09\pm 1.6$ | $-0.27\pm 5.39$ | $-0.03\pm 1.49$

We aim to know whether radial motions in neutral atomic gas of our three
galaxies can sustain current star formation rates (SFRs). Table 4 compares the
star formation rates taken from Leroy et al. (2019) to our derived average
radial velocities and average mass flow rates. The tilted ring model is
susceptible to underlying non-axisymmetric dynamical features (bars or
spirals), which are very difficult to incorporate and mainly influence the
inner regions of galactic disks. To prevent a strong influence of this effect
we calculated an average in radial velocity and mass flow rates only for radii
larger than $1.2\,\text{R}_{\text{bar}}$, where the influence of the bars is
almost negligible (Aguerri et al., 1998).

The averaged mass flow rates and their uncertainties for NGC 4535 and NGC 7496
are largely consistent with no mass flow at all. The average mass flow in NGC
1512 is directed radially outwards, which might again be a sign of the
gravitational influence of its companion NGC 1510. The limited sample size and
the presence of substantial uncertainties in our data preclude us from making
definitive conclusions. However, other studies, such as Di Teodoro & Peek
(2021) and Radburn-Smith et al. (2014), have also indicated that a significant
portion of the observed SFRs cannot be ascribed to radial accretion of neutral
atomic gas. Such results suggest that observed SFRs may not solely stem from
the radial accretion of neutral atomic gas. Additional factors like high-
velocity clouds, episodic accretion, or gravitational interactions could
contribute (Kerp et al., 2016; Radburn-Smith et al., 2014). This study
provides a methodological framework for future investigations when a larger
sample of similar observations is available despite the need for a cautious
interpretation of our specific results.

##  6 Conclusions

In conclusion, our study delved into the dynamics of both the neutral atomic
and CO-traced molecular components within the galactic disks of the three
nearby spiral galaxies NGC 1512, NGC 4535, and NGC 7496. Using MeerKAT H i and
PHANGS-ALMA CO observations at resolutions of
$15\text{\,}\mathrm{\SIUnitSymbolArcsecond}$ and
$\sim$1\text{\,}\mathrm{\SIUnitSymbolArcsecond}$$, respectively, we employed
the ${}^{\text{3D}}$Barolo, tilted ring fitting code to derive robust tilted
ring models. Our key findings and insights include:

  1. 1.

We established reliable H i tilted ring models and utilized them to constrain
the geometry of the CO observations, deriving corresponding tilted ring models
for molecular components.

  2. 2.

Signatures of warps were identified in all three galactic disks through
position-velocity diagrams and kinematic fitting results. NGC 4535 and NGC
7496 exhibited slight warps at large H i radii, while NGC 1512 displayed a
more intricate structure, likely influenced by the proximity of NGC 1510.

  3. 3.

Comparison with the approach used in Lang et al. (2020) affirmed the validity
of our approach. Similar orientations and dynamics in both analyses provided
confidence in the robustness of our assumptions.

  4. 4.

Rotation velocities based on neutral atomic and molecular gas were combined
and compared. The agreement between the two differently traced rotation curves
was observed for most overlapping radii, demonstrating the effectiveness of
combining these tracers to derive RCs across the entire disk.

  5. 5.

Investigation into radial motions and mass flow rates revealed minimal mass
flow in all galaxies, with a small outflow in NGC 1512 potentially attributed
to the presence of NGC 1510. The methodology employed for deriving radial
motions and mass flow rates serves as a viable approach for future studies
when expanded datasets become available.

With the ongoing H i coverage of the PHANGS-ALMA sample, it will be possible
to apply the approach presented here to the whole sample. With a larger number
of galaxies, we will be able to gain new insights and formulate reliable
conclusions on how the dynamics and the interplay of H i and molecular gas
determine the structure of spiral galaxies. It will be possible to investigate
the fueling of star formation via radial infall of neutral atomic gas similar
to the approach presented in this paper and in Di Teodoro & Peek (2021). With
the possibilities of fitting the scale height, it would be interesting to
investigate how and if it is related to the rotation velocity, as indications
for such a relation were already found by O’Brien et al. (2010).

###### Acknowledgements.

The work of AKL is partially supported by the National Science Foundation
under Grants No. 1615105, 1615109, and 1653300.
ES acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 694343).
MQ acknowledges support from the Spanish grant PID2019-106027GA-C44, funded by
MCIN/AEI/10.13039/501100011033.
SKS acknowledges financial support from the German Research Foundation (DFG)
via Sino-German research grant SCHI 536/11-1.
ER acknowledges the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC), funding reference number RGPIN-2022-03499.
RCL acknowledges support provided by a National Science Foundation (NSF)
Astronomy and Astrophysics Postdoctoral Fellowship under award AST-2102625.
RSK acknowledges financial support from the German Research Foundation (DFG)
via the collaborative research center (SFB 881, Project-ID 138713538) “The
Milky Way System” (subprojects A1, B1, B2, and B8). He also thanks for funding
from the Heidelberg Cluster of Excellence “STRUCTURES” in the framework of
Germany’s Excellence Strategy (grant EXC-2181/1, Project-ID 390900948) and for
funding from the European Research Council via the ERC Synergy Grant “ECOGAL”
(grant 855130).
EWK acknowledges support from the Smithsonian Institution as a Submillimeter
Array (SMA) Fellow and the Natural Sciences and Engineering Research Council
of Canada.
This work has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 882793 “MeerGas”). LN acknowledges funding from the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) - 516405419.
HAP acknowledges support by the National Science and Technology Council of
Taiwan under grant 110-2112-M-032-020-MY3.
TGW acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 694343).

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##  Appendix A Influence of the scale height parameter

Since the true height of the disks is not known for our sample we made
assumptions regarding the scale height parameter. We took a simple approach,
by using a constant value for all radii. The initial parameter is based on an
average value for H i scale heights by Randriamampandry et al. (2021), based
on a large H i survey. By using a constant value we cannot take any variations
of scale height into account, although large H i disks are known to flare at
large radii (Dib et al. 2021; Randriamampandry et al. 2021).

To justify the simple approach, we were interested in how much the scale
height affects our final rotation curves. We conducted eight additional
fitting runs, which fixed the constant scale height parameter to a variety of
values. All other parameters were left unchanged from our original best-fit
model. The scale heights range from unrealistically small
($1\text{\,}\mathrm{pc}$) to unrealistically large
($2000\text{\,}\mathrm{pc}$) values.

The results of this procedure are presented in Figure 9. Most of the RCs
obtained in the additional runs are within the errors of our best-fit model
(black circles), which supports the simple approach we have used. A trend to
how the thickness of the disk affects the rotation velocities cannot be
identified.

Figure 9: Fitting results achieved for varying constant values of scale height
for 9 different values for NGC 1512 (left), NGC 4535 (middle), and NGC 7496
(right). The scale height was chosen from unrealistically small (green) to
unrealistically large (blue) values. The final rotation curve adopted in this
paper is plotted in black.