# MaNGA galaxy properties – II. A detailed comparison of observed and
simulated spiral galaxy scaling relations

Nikhil Arora,1,2,3 Stéphane Courteau,3 Connor Stone,3,4,5 and Andrea V.
Macciò1,2,6
1New York University Abu Dhabi, PO Box 129188, Abu Dhabi, United Arab Emirates
2Center for Astro, Particle and Planetary Physics (CAP3), New York University
Abu Dhabi
3Department of Physics, Engineering Physics & Astronomy, Queen’s University,
Kingston, ON K7L 3N6, Canada
4Department of Physics, Université de Montréal, Montréal, Québec, Canada
5Mila – Québec Artificial Intelligence Institute, Montréal, Québec, Canada
6Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg,
Germany
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a catalogue of dynamical properties for 2368 late-type galaxies
from the MaNGA survey. The latter complements the catalogue of photometric
properties for the same sample based on deep optical DESI photometry processed
with AutoProf. Rotation curves (RCs), extracted by model fitting H$\alpha$
velocity maps from the MaNGA Data Analysis Pipeline, extend out to 1.4 (1.9)
Re for the primary (secondary) MaNGA samples. The RCs and ancillary MaNGA
Pipe3D data products were used to construct various fundamental galaxy scaling
relations that are also compared uniformly with similar relations from NIHAO
zoom-in simulations. Simulated NIHAO galaxies were found to broadly reproduce
the observed MaNGA galaxy population for $\log(M_{*}/{\rm M_{\odot})>8.5}$.
Some discrepancies remain, such as those pertaining to central stellar
densities and the diversity of RCs due to strong feedback schemes. Also
presented are spatially-resolved scatters for the velocity-size-stellar mass
(VRM∗) structural relations using MaNGA and NIHAO samples. The scatter for
these relations in the galaxian interiors is a consequence of the diversity of
inner RC shapes, while scatter in the outskirts is dictated by the large range
of stellar surface densities which itself is driven by sporadic star
formation. The detailed spatially-resolved scatter analysis highlights the
complex interplay between local and global astrophysical processes and
provides a strong constraint to numerical simulations.

###### keywords:

galaxies: general – galaxies: photometry – galaxies: structure – galaxies:
formation – galaxies: fundamental parameters – galaxies: spiral – methods:
numerical

††pubyear: 2022††pagerange: MaNGA galaxy properties – II. A detailed
comparison of observed and simulated spiral galaxy scaling relations–A

##  1 Introduction

Given their importance for constraining galaxy formation and evolution models,
studies of galaxy scaling relations have enjoyed a rich history (Faber &
Jackson, 1976; Tully & Fisher, 1977; Bender et al., 1992; Mo et al., 1998;
Steinmetz & Navarro, 1999; Courteau et al., 2007; Mo et al., 2010; Kormendy &
Ho, 2013; Lelli et al., 2017; Stone et al., 2021b; D’Onofrio et al., 2021).
The slope, zero-point, and scatter of scaling relations (Courteau et al.,
2007; Kormendy & Ho, 2013; Lange et al., 2015) encode critical information
about the structure of galaxies and provide stringent constraints to galaxy
formation models (Dutton et al., 2007; Brook et al., 2012; Koch et al., 2017;
Dutton et al., 2017; van de Sande et al., 2019; Starkenburg et al., 2019).
Such data-model comparisons require large unbiased multi-band observed galaxy
data (Jarrett et al., 2000; York et al., 2000; Driver et al., 2009) and modern
state-of-the-art galaxy formation simulations (see Vogelsberger et al., 2020,
for a review).

The advent of galaxy surveys with large scale integral field spectroscopy such
as MaNGA (Bundy et al., 2015; Wake et al., 2017), SAMI (Allen et al., 2015),
ATLAS3D (Cappellari et al., 2011) and CALIFA (Walcher et al., 2014) has opened
up new areas of investigations with high quality spatially-resolved dynamical
and chemical properties of galaxies. For instance, the coupling of IFU data
with deep multi-band photometric observations (Courteau, 1996; Hall et al.,
2012; Gilhuly & Courteau, 2018; Ouellette et al., 2017; Arora et al., 2021,
hereafter A21) allows detailed studies of different astrophysical processes
such as star formation (Pandya et al., 2017), baryonic feedback, dynamics,
interactions between baryon and dark matter (Dutton et al., 2011b), impact of
environment (Peng et al., 2012), chemical evolution (Gallazzi et al., 2005;
Sánchez-Menguiano et al., 2020), angular momentum distributions (Romanowsky &
Fall, 2012; Obreschkow & Glazebrook, 2014), and more. On the theoretical
front, modern state-of-the-art cosmological simulations in a cosmological cold
dark matter paradigm allow tracking of the co-evolution of baryons and dark
matter (Wang et al., 2015; Crain et al., 2015; Remus et al., 2017; Pillepich
et al., 2018; Hopkins et al., 2018; Habouzit et al., 2019). While the
distribution (but not the nature) of dark matter in the Universe is fairly
well understood, especially on large-scales, modelling stars, their formation,
and gas distributions within galaxies still challenges most galaxy formation
and evolution theories (Avila-Reese et al., 2011; Sawala et al., 2011;
Weinmann et al., 2012; Agertz & Kravtsov, 2015). Modern large-scale surveys
such as the ones mentioned above provide rich and versatile data that can be
used to constrain the distribution of various baryonic properties in
cosmological simulations.

Providing the most recent, up-to date and uniform comparison between
simulations and observations of galaxies is the motivation for this study.
Such comparisons highlight not only benchmarks for our current understanding
of galaxy formation and evolution, they also identify specific areas of
improvement for galaxy formation models. Different approaches can be taken to
enable uniform data-model comparisons of galaxies. Cosmological simulations
can be post-processed to generate mock images and spectra of galaxies that
include observational characteristics such as atmosphere blur, background sky
noise, photometric bandpasses, etc (Torrey et al., 2015; Bottrell et al.,
2017; Elagali et al., 2018; Deeley et al., 2021; Bottrell & Hani, 2022; Camps
et al., 2022). These can then be compared directly with observations.
Conversely, all known biases and sources of error from observations (such as
extragalactic and Galactic dust extinction, photometric band effects,
inclination effects, sky noise, sample completion, etc.) can be modeled and
removed from the inferred data for direct comparison with intrinsic properties
of simulated galaxies (Arora et al., 2019; Stone et al., 2021a; Frosst et al.,
2022). The consistency of these different data-model comparisons can provide
an internal validation of our methods and results (Stoppa et al., 2023).

Combining the MaNGA photometric catalogue (A21) and auxiliary data from Pipe3D
(Sánchez et al., 2018) with robust rotation curves yields an extensive
catalogue of inferred galaxy structural data which is ideal for comparison
with galaxy formation models. For this paper, the latter is based on the
Numerical Investigation of a Hundred Astrophysical Object (NIHAO) project
which provides $\sim$60 zoom-in late-type galaxy simulations. Data-model
comparisons of galaxies often focus on various informative properties of
galaxies such as star formation rates (Starkenburg et al., 2019), galaxy sizes
(de Graaff et al., 2022), shapes of rotation curves (Santos-Santos et al.,
2018), black hole scaling laws (Çatmabacak et al., 2022), etc. With the
substantial MaNGA data presented in A21 and here, we can create 12 galaxy
scaling relations out of 7 galaxy structural parameters. The multiple scaling
relations (Dutton et al., 2011a; Trujillo-Gomez et al., 2011; Brook et al.,
2012) allow for a multi-dimensional data-model comparison that identifies
specific parameters which simulations struggle to reproduce.

Another important aspect of observation-simulation comparisons is the
quantification of fit parameters of scaling relations, such as their slope and
scatter. While slope comparisons between simulations and observations are
relatively trivial, scatter comparisons require the removal of observed biases
(distance uncertainty, disk thickness, inclination, mass-to-light conversions,
etc.) A common approach to handling such biases consists of removing
observational errors in quadrature to retrieve the intrinsic scatter of galaxy
scaling relations (Strauss & Willick, 1995; Saintonge & Spekkens, 2011; Hall
et al., 2012; Lelli et al., 2017). However, such a method ignores the
correlations between the different biases leading to an underestimation of
intrinsic scatter. A Bayesian formalism to calculate intrinsic scatter (Stone
et al., 2021b) takes correlated observed errors into account and returns
robust estimates of intrinsic scatter. The Bayesian intrinsic scatters for
scaling relations determined here can serve as an accurate robust test for
galaxy formation simulations.

Modern deep multi-band imaging and large IFU surveys also allow the study of
spatially-resolved scaling relations (e.g., Radial Acceleration Relation
(Lelli et al., 2017; Stone & Courteau, 2019) and the Star Formation Main
Sequence (hereafter “SFMS”; Cano-Díaz et al., 2016; Hall et al., 2018)).
Recent studies of spatially-resolved scaling relations have already revealed
new aspects of galaxy formation and evolution. For example, the presence of a
tight spatially-resolved SFMS is comparable to the global SFMS (Wuyts et al.,
2013; González Delgado et al., 2016; Wang et al., 2017; Hall et al., 2018;
Ellison et al., 2018). The existence of a spatially-resolved star formation
law requires a strongly dependent on small-scale local gas and stellar surface
density.

Spatially-resolved scaling relations based on a wide array of structural
parameters are still largely lacking. The variations of slope and scatter as a
function of galactocentric radius for scaling relations (such as size-mass,
size-velocity, and Tully-Fisher relations) can be a powerful tool in
identifying the drivers of diversity in galaxy structure. Spatially-resolved
scatter diagnostics can simultaneously constrain the overall distribution of
galaxy properties and inform us about more advanced tests for cosmological
simulations (especially pertaining to astrophysical processes on sub-galactic
scales). With our deep photometric and dynamical data, for the first time, we
study the variations and drivers of scatter as a function of location in
observed and simulated galaxies.

This paper is organized as follows: Section 2 describes the photometry and
rotation curves (RCs) from the MaNGA galaxy survey and the process of
extracting and correcting various galaxy properties. We also compare our RCs
against other observed rotation curves and scaling relations. Section 3 gives
a brief description of the NIHAO galaxy formation simulations that are
compared against observations. We begin our data-model comparison in Section 4
to show one dimensional comparisons of various galaxy properties. A
simultaneous comparison of twelve major galaxy scaling relations between MaNGA
and NIHAO, including their observed Bayesian intrinsic scatter, is then
presented in Section 5. In Section 6, we examine the radial variations of the
spatially-resolved slopes and scatters for structural galaxy scaling relations
for MaNGA and NIHAO. Concluding remarks are offered in Section 7, with an eye
towards improving numerical simulations of galaxy formation and evolution. All
of our processed data, including the observed light profiles (in A21) and RCs
(Appendix A) are publicly available.

##  2 Observational Data

In this section, we briefly describe our extraction of photometric properties
for the MaNGA galaxies (Bundy et al., 2015; Law et al., 2016; Wake et al.,
2017) using deep imaging from the DESI survey. A detailed description of the
extensive imaging catalogue is found in A21. RCs rotation curves and dynamical
properties are also derived below, and compared against other published
rotation curves and scaling relations.

###  2.1 Photometry

Figure 1:  Surface brightness depth versus maximal spatial extent (in units of
R23.5) for MaNGA galaxies. The surface brightness profiles for LTG (left) and
ETGs (right) were extracted through autoprof in the $z$-band. The text insets
give the number of galaxies, mean maximal spatial extent, and mean surface
brightness depth in the $z$-band.

In order to lay the foundations for our study of the photometric and dynamical
properties of MaNGA galaxies, we first take advantage of the extensive
catalogue of non-parametric multi-band photometric and environmental
properties for $\sim$4500 MaNGA galaxies, as presented in A21. This
photometric catalogue relied on the deep optical grz-band imaging from the
Dark Energy Sky Instrument Legacy Imaging Survey
111https://www.legacysurvey.org (DESI Collaboration et al., 2016; Dey et al.,
2019, hereafter DESI) and the WISE Large Galaxy Atlas and the Extended Source
catalogue (WXSC; Jarrett et al. 2019). Non-parametric azimuthally-averaged
surface brightness (SB) profiles were extracted using the automated
astronomical image analysis tool, autoprof (Stone et al., 2021a). autoprof
performs a robust background subtraction and finds an accurate galaxian center
for each galaxy image before fitting elliptical isophotes.

The center, position angle (PA), and ellipticities of each isophote is fit to
the DESI r-band images for MaNGA galaxies. The r-band imaging is chosen for
the extraction of isophotal profiles due to its high signal-to-noise ratio and
low dust extinction relative to the other bands (A21). To obtain multi-band
photometry, the r-band isophotes are applied to the other gz-band images via
forced photometry, resulting in complimentary grz surface brightness profiles.
The “forced photometry” component of autoprof ensures uniformity in
measurement of surface brightness, ellipticities, fluxes, and color gradients.
For a more comprehensive description of the autoprof software package and its
capabilities, the reader is referred to Stone et al. (2021a).

Because of their lowest sensitivity to dust extinction, all photometric
properties were calculated using the DESI z-band surface brightness profiles.
This also ensures uniformity with other studies (Stone et al., 2021b; Arora et
al., 2021; Frosst et al., 2022; Stone et al., 2022).

While the photometric data presented in A21 were based on the galaxies
selected using the public SDSS-DR16 catalogue (Ahumada et al., 2020), the
MaNGA photometric data used in this study are based on the SDSS-DR17
(Abdurro’uf et al., 2022). The cross-correlation between MaNGA data release
from SDSS-DR17 and the DESI sky survey yielded 7864 galaxies; 5166 of which
were classified as LTGs (Domínguez Sánchez et al., 2018), and used in our
comparison between simulations and observations. Fig. 1 presents the maximal
SB depth in the z-band of our photometric LTG (left) and ETG (right) samples.
Thanks to the deep DESI imaging and the versatility of autoprof, we can probe
SBs down to (on average) $\sim$26.3 mag arcsec-2 and $\sim$2
R/R${}_{23.5}^{z}$ for MaNGA LTGs. Our robust non-parametric photometry
through DESI imaging and autoprof results in gathering 0.1 dex more light and
0.3 dex more stellar mass for the same galaxies than presented in the NSA
catalogue as shown in A21.

Any differences between the photometric data presented here and in A21 is due
to sample size (using the updated SDSS-DR17). The method of extracting and
analysing surface brightness profiles is the same in both studies.

###  2.2 Rotation Curves

Figure 2: MaNGA H$\alpha$ velocity maps and extracted RCs. For each row,
corresponding to a different MaNGA target, panel a) shows the composite SDSS-
gri image with the MaNGA IFU footprint (in magenta); panel b) shows the
matching H$\alpha$ velocity map from the MaNGA IFU; and panels c), d) and e)
present the tanh model velocity maps, residual maps ($V_{\rm obs}-V_{\rm
model}$), and the corresponding rotation curve. The inset in panel e) also
gives the MaNGA-ID and average data-model residual $\Delta
V\equiv\sqrt{\langle\delta V^{2}\rangle}$ (see Eq. 5).

A key aspect of this study is the coupling of kinematic data, presented here,
with the photometric information from A21 for the same galaxies. Our
spectroscopic analysis takes advantage of the H$\alpha$ emission line-of-sight
(LOS) velocity maps (hereafter H$\alpha$ velocity maps) provided through the
MaNGA Data Analysis Pipeline (MaNGA-dap; Westfall et al., 2019; Belfiore et
al., 2018) and accessed via the “Marvin” toolkit (Cherinka et al., 2019) for
the MaNGA-DR17. Voronoi binned data are used for the H$\alpha$ emission LOS
velocity maps.

####  2.2.1 Velocity model

RCs were obtained by fitting the following model to the H$\alpha$ velocity map

$V_{\rm LOS}(R,\theta)=V_{\rm sys}+V(R)\cos(\theta)\sin(i),$ (1)

where $V_{\rm LOS}(R)$ is the observed LOS velocity at radius R, $V_{\rm sys}$
is the systematic (heliocentric) velocity, $\theta$ is the azimuthal angle
relative to the positions angle, $i$ is the inclination of the galaxy on the
sky, and $V(R)$ is the adopted rotational velocity model. In Eq. 1,
$R=\sqrt{x_{\rm g}^{2}+y_{\rm g}^{2}}$ and $\theta=\tan^{-1}(y_{\rm g}/x_{\rm
g})$ are calculated using Cartesian coordinates within the disk of the galaxy.
These are defined as:

$x_{\rm g}=x^{\prime}_{\rm s}\cos(-{\rm PA})+y^{\prime}_{\rm s}\sin(-{\rm
PA}),$ (2)

and

$y_{\rm g}=\frac{y^{\prime}_{\rm s}\cos(-{\rm PA})-x^{\prime}_{\rm
s}\sin(-{\rm PA})}{\cos(i)},$ (3)

where $x^{\prime}_{\rm s}=x_{\rm s}-x_{\rm c}$ and $y^{\prime}_{\rm s}=y_{\rm
s}-y_{\rm c}$ are centred projected coordinates on the sky with the observed
galaxy center given by $(x_{\rm c},y_{\rm c})$, and PA is the observed
position angle. For this study, and for uniformity with other studies
(Courteau, 1997; Aquino-Ortíz et al., 2020; Brownson et al., 2022), we write:

$V(R)=V_{\rm max}\times\tanh(R/R_{\rm t}),$ (4)

for the rotational velocity model in Eq. 1, where $V_{\rm max}$ is the maximum
velocity, and $R_{\rm t}$ is a turnover radius between the rising and flat
portion of the RC. Therefore, the kinematic modeling of MaNGA galaxies
requires fitting a dynamical model with seven parameters; $(x_{\rm c},y_{\rm
c})$, $i$, PA, $V_{\rm sys}$, $V_{\rm max}$, and $R_{\rm t}$. The fit
parameters are presented in Appendix A.

For the fit initialization, $(x_{\rm c},y_{\rm c})$, and $V_{\rm sys}$ are
first set to zero, while the inclination and PA use values calculated through
the DESI photometry (A21). $V_{\rm max}$, and $R_{\rm t}$ are initialized by
fitting Eq. 4 to a “mock rotation curve”, calculated by interpolating observed
H$\alpha$ velocity map along to the photometric major axis. These initialized
values are then run through an optimization routine to find the best fit
values for the rotation curves. The quality of the dynamical model is
evaluated as the “mean square error”,

$\langle\delta V^{2}\rangle=\langle(V_{\rm obs}-V_{\rm model})^{2}\rangle.$
(5)

. We calculate robust fit parameters of our dynamical model by running our
fitting algorithm 20 times with initialization values randomly chosen within
$\pm 50$ per cent of the first fit evaluated with the initial values mentioned
above. The final fit is calculate with the set of fit parameters which return
the lowest $\langle\delta V^{2}\rangle$ out of the random 20 runs. Fig. 2
shows the composite SDSS gri image, the corresponding H$\alpha$ velocity map,
the model velocity map, the residual and the rotation curves for six late-type
MaNGA galaxies. The top row of Fig. 2 shows a galaxy with a foreground object;
in general, we find that our fitting routine is robust against
background/foreground objects within the MaNGA IFU.

Figure 3: Probability Density Functions normalized using area for the fit
quality, $\Delta V$ (see Eq. 5; left panel), as well as the model maximum
(asymptotic) velocity, $V_{\rm max}$ (middle), and turnover radius, $R_{\rm
t}$ (right).

Our dynamical models fit the observed MaNGA H$\alpha$ velocity maps very well.
The average $\langle\delta V\rangle$ for all the MaNGA galaxies is 7 km s-1
(see Fig. 3). However, the distribution of $\langle\delta V\rangle$ has a non-
negligible tail at high $\langle\delta V\rangle$. Indeed, objects with
$\langle\delta V\rangle\geq 10\,{\rm km\,s^{-1}}$ present signatures of non-
circular motions (caused by bars and bulges) that are unaccounted for within
our dynamical model. These non-circular motions (caused by e.g., bars and
bulges) are mainly dominant in the central parts of galaxies and have an
amplitude of $\sim$15 km -1. Given the stellar mass distribution of MaNGA
galaxies ($\log(M_{*}/\,{\rm M_{\odot})>8.5}$); such non-circular motions
would not make a large impact on the results presented here. Given that the
galaxy scaling relations that we primarily study trace the outer galaxy
regions, the central non-circular motions should be insignificant in our
analysis (Sellwood et al., 2021).

Furthermore, a slight trend is observed between photometric and kinematic
inclinations as a function of inclination. For more edge-on systems, this
offset is $\sim 5-7^{\circ}$. This is a result of the strong degeneracy
between the Vmax and the inclination in the tanh model. However, the
inclination offset is negligible for tilts below $\sim 65^{\circ}$.

Figure 4: Probability Density Functions for the MaNGA RC extents in terms of
various normalised photometric radii. $R_{\rm max}$, $R_{\rm e,z}$, and
$R_{\rm 23.5,z}$ are, respectively, the luminosity-weighted maximum radius for
the MaNGA IFU, the effective radius, and the isophotal radius measured at a
surface brightness level of 23.5 mag arcsec-2, all reported in the z-band.
$R_{\rm e,z}$, and $R_{\rm 23.5,z}$ were obtained from DESI photometry of
MaNGA galaxies (A21). The left and right panels show primary (+ colour
enhanced) and secondary samples, separately. The insets give the median RC
extents with an error of 1$\sigma$.

The coupling of the MaNGA spectroscopic data with our photometry from A21
enables us to quantify the physical extent of the MaNGA spectral data. The
MaNGA primary and color-enhanced samples (primary+) have indeed been claimed
to probe regions out to 1.5Re, and the secondary sample would extend out to
2.5Re(Bundy et al., 2015). The effective radii came from the NASA Sloan Atlas
(NSA) catalog (Blanton & Roweis, 2007). The distribution of the extent of the
spectroscopic data scaled with various characteristic radii extracted from A21
is shown in Fig. 4. The extent of both primary+ and secondary sample is found
to be lower than the nominal 1.5 and 2.5Re. This is because our non-parametric
approach and more sensitive DESI imaging yield $\sim$0.3 mag more light for
MaNGA-LTGs (A21) than conventional SDSS imaging. Indeed, A21’s effective radii
are larger than those from the NSA catalog. As a result, the MaNGA primary+
and secondary samples extend out to 1.43 $\pm$ 0.35 Re and 1.92 $\pm$ 0.41 Re,
respectively.

Fig. 4 also shows the MaNGA RC extents scaled by isophotal radius. The
isophotal (or mass density) measurements for galaxy sizes are more robust,
accurate, and reproducible. Furthermore, the use of isophotal radii (either
calibrated by surface brightness and/or stellar mass surface density) to
measure structural parameters reduces the scatter of related structural
scaling relations (Hall et al. 2012; Trujillo et al. 2020, A21). In term of
isophotal size, measured at the 23.5 mag arcsec-2 isophote, the MaNGA
spectroscopic data extends out to 0.78 $\pm$ 0.19 (1.02 $\pm$ 0.17) R23.5 for
the primary+ (secondary) samples, respectively.

####  2.2.2 Error estimation

We use “Jackknife resampling” (Quenouille, 1949) to estimate the fit parameter
errors of our dynamical models (Eq. 1). A spaxel is randomly selected and all
spaxels within 2.5$\scriptscriptstyle\prime$ $\scriptscriptstyle\prime$ radius
are removed to produce a new truncated velocity map. The new velocity map is
then fit again with Eq. 1 to calculate truncated fit parameters. The Jackknife
resampling process is repeated 100 times. The error on the fit parameters is
the 16-84 per centile range for the 100 Jackknife runs. These errors are also
reported in the (public) catalogue of dynamical properties presented in
Appendix A. No velocity data were rejected during the fitting process; rather,
galaxies were excised from the sample (if at all) during the error estimation
process.

####  2.2.3 Quality cuts

Our MaNGA LTG sample (Domínguez Sánchez et al., 2018) consists of 5166
galaxies for which DESI-grz imaging and robust photometry through AUTOPROF is
available . To obtain RCs for MaNGA LTGs, we only use galaxies which achieve
photometric inclinations between 30-80 degrees. This inclination cut ensures
robust inclination corrections for the RC. For galaxies with DESI photometric
data and favourable inclinations, we run our velocity model to extract RCs.
For the galaxies with both photometry and RCs, we only use systems where the
seven fits parameters for the velocity models have errors $<20\%$. In the end,
we are left for the remainder of this study with a final set of 2368 galaxies
which have robust photometry and RCs.

###  2.3 Parameter Extraction and Correction

While the photometric (A21) and dynamical catalogues presented here are highly
versatile, we are primarily concerned with the sizes, stellar mass, velocity
metrics, dynamical masses, and stellar mass surface density within a physical
radius (e.g., 1 kpc).

The photometric properties were extracted and corrected using methods
described in A21. These are briefly described below; see A21 for more details.

####  2.3.1 Corrections

All SB profiles used in this study were corrected for Galactic and internal
extinction, as well as for cosmological K-corrections. The Galactic extinction
corrections were retrieved from the NSA catalogue (Blanton et al., 2011),
originally taken from Schlegel et al. (1998). These corrections are available
for the optical grz photometric bands. The cosmological K-corrections were
calculated using templates from Blanton & Roweis (2007).

Our corrections for internal dust extinction and geometry/inclination were
applied after extracting the structural parameters. The inclination
corrections were calculated using a linear fit between each structural
parameter and the log of the cosine of the inclination (Giovanelli et al.,
1994). The inclination correction can be written as:

$\log X_{0}=\log X_{i}+\gamma\log(\cos i),$ (6)

where $X_{0}$ is the galaxy property corrected to face-on view, $X_{i}$ is the
observed galaxy property, $\gamma$ is the inclination correction factor, and
$i$ is the observed inclination of the galaxy disk on the sky. The photometric
inclination, $i$, is related to the projected axis ratio ($b/a$) of the object
as

$\cos^{2}i=\frac{(b/a)^{2}-q_{0}^{2}}{1-q_{0}^{2}},\qquad(q_{0}=0.13)$ (7)

where $q_{0}$ is the assumed stellar disk thickness (Hall et al., 2012). For
more details about the various corrections to the MaNGA photometric data, see
A21.

The RCs are also corrected for inclination by dividing the sin of the disk
inclination (see Eq. 1). It should be noted that the kinematic inclinations
are independent of the photometry as the former is a fit parameter in our
dynamical model. The difference between kinematic and photometric inclinations
was found to be $\sim 5\pm 9^{\circ}$; where the error presents the standard
deviation.

####  2.3.2 Parameter extraction

The sizes, stellar masses, and surface densities were estimated using the
photometric data. For greater compatibility with simulations (which predict
mass rather than light), we convert all of our light-weighted properties into
mass-weighted properties. Stellar mass estimates for the MaNGA galaxies were
calculated using various stellar mass-to-light color relations (MLCRs Courteau
et al., 2014). To achieve the maximum robustness for our stellar mass
estimates, five different MLCRs were averaged from Roediger & Courteau
(2015a), Zhang et al. (2017), and García-Benito et al. (2019). The stellar
mass estimates use $g-r$ and $g-z$ optical colours and luminosities from all
three DESI bands. As a result, the stellar masses are calculated as the
average of 30 different measurements. The average scatter on stellar mass
measurements from different MLCRs is $\sim 0.2\,{\rm dex}$. This is consistent
with systematic errors in stellar mass based on extensive MLCRs calibrations
reported in Courteau et al. (2014) and Roediger & Courteau (2015b).

The physically-motivated galaxy sizes used in this study represent the semi-
major axis of various isophotal levels, measured in mag arcsec-2 or M⊙ pc-2.
As shown by Trujillo et al. (2020), such a representation of sizes yields a
tighter scatter for the size–mass relation.

We use the quantity, $\Sigma_{1}=M_{\rm*,1\,kpc}/\pi$ [$M_{\odot}$ pc-2], as a
representation of stellar mass surface density within a projected radius;
specifically 1 kpc.

Velocity metrics were calculated by interpolating/extrapolating the
inclination-corrected RCs at various radii. Finally dynamical masses, $M_{\rm
dyn}(R)$, are calculated using the velocity, $V(R)$, at projected radius, R,
assuming a spherical halo:

$M_{\rm dyn}(R)=\sqrt{\frac{V^{2}(R)R}{G}}.$ (8)

The error in the dynamical mass at radius R is calculated in quadrature as

$\delta M_{\rm dyn}(R)=\Bigg{[}\Bigg{(}\frac{2V(R)R\delta
V(R)}{G}\Bigg{)}^{2}+\Bigg{(}\frac{V(R)^{2}\delta
R}{G}\Bigg{)}^{2}\Bigg{]}^{1/2}.$ (9)

In Eq. 9, $\delta V(R)$ and $\delta R$ are the measured uncertainties.

###  2.4 Structural Scaling Relations

Figure 5: Comparison of inclination-corrected RCs for the ten galaxies in
common between MaNGA and the PROBES sample (Stone et al., 2021b). Each panel
compares the long slit H$\alpha$ RC (purple points with error bars) from
PROBES with the RC (pink solid lines with shaded regions) extracted from the
MaNGA H$\alpha$ velocity map. The inset text gives the MaNGA object ID and the
RMS between the MaNGA/PROBES data sets.

We confirm the validity of our MaNGA RCs by comparing them with similar
independent observations for the same object. Fig. 5 shows such a comparison
for the ten galaxies overlapping between MaNGA and the PROBES (“Photometry and
Rotation Curve Observations from Extragalactic Surveys”) sample (hereafter
PROBES; Stone & Courteau, 2019; Stone et al., 2021b; Stone et al., 2022).
MaNGA’s spectroscopic data were taken with an IFU, while most PROBES RCs were
obtained via H$\alpha$ long-slit spectra (except 1-135468 taken in HI). While
the MaNGA RCs can be extrapolated to infinity (as a result of the tanh fit),
the extent of the models shown in Fig. 5 is limited to the IFU spatial
coverage. The inclinations for the PROBES-MaNGA comparisons use the same
photometric ellipticities obtained via AutoProf (A21). This ensures a uniform
(“apples-to-apples”) comparison for the RCs. The pair of 10 RCs between PROBES
and MaNGA trace each other quite well; on average the RCs from both samples
differ by ${\sim}24$ km s-1 or ${\sim}0.1\,V_{\rm max}$. The match is nearly
perfect for galaxies (MaNGA-ID) 1-446131 and 1-605088. Galaxy (MaNGA-ID)
1-135468 shows a great match between the MaNGA IFU and HI observations (which
naturally extends further). While PROBES RCs have typically greater spatial
extent, MaNGA velocity maps provide better coverage in the central parts of
disk galaxies and take advantage of the two dimensional velocity fields.

At this point, it is worth mentioning that our velocity models are not
corrected for any beam-smearing correction. Our decision is two-fold. Firstly,
the excellent agreement between the MaNGA and PROBES rotation curves presented
in Fig. 5 confirms that the effect of beam smearing is minimal for our galaxy
sample. Secondly, for the spatially resolved analysis, we ignore the inner two
most data points (corresponding roughly to 2 arcseconds). This lessens any
beam smearing effect in the centrals parts of galaxies. Beam smearing ought to
be more pronounced for stellar velocity dispersion measurements, an effect
that we shall revisit elsewhere.

Figure 6: MaNGA VRM∗ scaling relations measured at an isophote of 23.5 mag arcsec-2 in the $z$-band. The three panels show the stellar TFR (top-left), R–V relation (bottom-left), and the R–M∗ relation (bottom-right). In each panel, the density contours show the underlying MaNGA data fitted with a linear orthogonal distance regression (orange solid line; the 1$\sigma$ scatter is shown as an orange dotted line); the points are presented in the low density regions. Each scaling relation is compared with various literature sources. The legend (top-right) highlights those sources with relevant scaling relations in parenthesis. Scaling Relation | N | $y$ | $x$ | $\alpha$ | $\beta$ | $\sigma_{\rm obs}$ | $\sigma_{\perp}$
---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
Stellar Tully-Fisher | 2368 | $\log M_{\rm*,23.5}$ | $\log V_{23.5}$ | $3.59\pm 0.04$ |  -$2.21\pm 0.08$ | $0.24\pm 0.01$ | 0.06$\pm 0.01$
Projected size–velocity | 2368 | $\log V_{23.5}$ | $\log R_{23.5,z}$ | $1.28\pm 0.02$ | $-1.85\pm 0.05$ | $0.14\pm 0.01$ | 0.09$\pm 0.01$
Projected size–stellar mass | 2368 | $\log R_{\rm 23.5,z}$ | $\log M_{\rm*,23.5}$ | $0.35\pm 0.01$ | $-2.57\pm 0.03$ | $0.10\pm 0.01$ | 0.09$\pm 0.01$
Table 1: VRM∗ scaling relations for the 2368 MaNGA galaxies. Each scaling
relation is modelled as $y=\alpha x+\beta$ through orthogonal distance
regression. Column (1) gives the scaling relation, column (2) shows the number
of points fit; columns (3-4) present the $x$ and $y$ variables for the fit;
columns (5-6) show the slope ($\alpha$), zero-point ($\beta$), and the
observed scatter ($\sigma_{\rm obs}$), respectively; and, column (8) gives the
orthogonal scatter ($\sigma_{\perp}$) for the scaling relation.

The full catalogue enables us construct scaling relations which can also be
compared with the literature. Fig. 6 presents the VRM∗ scaling relations for
$\sim$2300 LTGs compared with other sources. The orthogonal distance
regression (hereafter ODR) linear fits and scatters for each relation are
given in Table 1. In general, our photometry and RC modelling yield scaling
relations that match other studies. For the sake of uniformity and unless
otherwise stated, the selected comparisons in Fig. 6 use isophotal radii
measured at the 23.5 mag arcsec-2 isophote. In what follows, we briefly
discuss and compare each scaling relation.

####  2.4.1 Projected size – stellar mass relation

The MaNGA size-mass (R – M∗) relation, shown in the bottom-right panel of Fig.
6, has an ODR slope of $\log(R_{23.5,z}/{\rm kpc})=(0.35\pm
0.01)\log(M_{*,23.5}/{\rm M_{\odot}})+(-2.57\pm 0.03)$ and an observed scatter
of $0.10\pm 0.01$ dex. That relation, the same as that presented in A21, can
be compared with similar ones presented by Ouellette et al. (2017), Trujillo
et al. (2020), and Stone et al. (2021b) for different samples. For instance,
Ouellette et al. (2017) used SDSS i-band photometry for the SHIVir survey of
Virgo cluster galaxies to derive a R – M∗ relation with a similar ODR slope of
$0.34\pm 0.02$, shown as the dark orange solid line, though with a larger
scatter of 0.15 dex. That larger scatter is due to a much lower sample size
(69 galaxies) in the SHIVir survey. Stone et al. (2021b) also constructed a R
– M∗ relation for 1152 PROBES galaxies with structural parameters measured
using similar methods as ours. Both analyses of the PROBES and MaNGA data sets
used DESI imaging and SB profiles extracted through the Autoprof. It is
therefore not surprising that their ODR slope of $0.334^{0.009}_{0.004}$ and
scatter of $0.099^{0.002}_{0.003}$ should be so close to ours for MaNGA. Any
(very small) difference ought to be explained by the use of two different
samples.

Studying 464 LTGs from the SDSS IAC Stripe82 survey, Trujillo et al. (2020)
found an isophotal ODR slope of $0.318\pm 0.014$ (light green solid line) and
a scatter of $0.087\pm 0.005$. Isophotal sizes are sensitive to the choice of
photometric bands; while we use z-band, Trujillo et al. (2020) used the SDSS-i
band. The difference in the fit parameters are a result of choice of
photometric band for our respective analysis.

####  2.4.2 Projected size – velocity relation

The size – velocity (RV) relation for MaNGA LTGs, shown in the bottom-left
panel in Fig. 6, has an ODR fit of $\log(R_{23.5,z}/{\rm kpc})=(1.28\pm
0.02)\log(V_{23.5}/{\rm km\,s^{-1}})+(-1.85\pm 0.05)$ with an observed scatter
of $0.14\pm 0.01$. The PROBES galaxies also benefit from long-slit H$\alpha$
RCs and yield an RV relation whose parameters are closely matched with
MaNGA’s; $\alpha=1.325^{0.074}_{0.034}$ and $\sigma_{\rm obs}=0.128\pm 0.004$.
The RV relation for the SHIVir galaxies (Ouellette et al., 2017) has a much
smaller slope of $1.1\pm 0.1$. The difference could be due to environmental
effects, or small number statistics. The latter actually thwarts a conclusive
answer. Hall et al. (2012) used SDSS photometry and rotational velocities from
integrated HI linewidths from Springob et al. (2005, 2007) to find a steeper
RV slope of $1.518\pm 0.065$ and an observed scatter of 0.152 dex. While our
scatters agree within the error, the slopes for our respective RV relations
differ significantly. While integrated HI linewidths are often shown to
reproduce various well-defined spatially-resolved rotational velocities
(Courteau, 1997), an exhaustive study should be conducted to characterize the
spatial coverage of the HI linewidths used by Hall et al. (2012). The latter
is beyond the scope of this paper.

Different velocity measurements can definitely result in different slopes for
our respective R – V relation. Other sources of uncertainties include distance
measurements.

####  2.4.3 Stellar Tully – Fisher relation

The combination of our photometry (A21) and RC analysis yields a stellar
Tully-Fisher (STFR), or M∗–V relation, which informs us about the cross-talk
between the stellar and dark matter in MaNGA galaxies (Dutton et al., 2010;
Hall et al., 2012; Ouellette et al., 2017; Stone et al., 2021b; Aquino-Ortíz
et al., 2020). The STFR for MaNGA LTGs, measured at R23.5, shown in the top-
left panel of Fig. 6, is described by $\log(M_{*,23.5}/{\rm
M_{\odot}})=(3.59\pm 0.04)\log(V_{23.5}/{\rm km\,s^{-1}})+(2.21\pm 0.08)$ with
an observed scatter of $0.24\pm 0.01$ dex. This ODR fit slope agrees with
theoretical derivations of $\alpha\sim 3-4$ (Courteau, 1997). Moreover, of the
three VRM scaling relations studied here, the STFR is the tightest with a
small orthogonal scatter ($\sigma_{\perp}=\sigma_{\rm obs}/(1+m^{2})^{0.5}$),
as shown in Table 1. The study of 1152 PROBES galaxies by Stone et al. (2021b)
yielded a STFR slope of $3.997^{0.144}_{0.039}$ and an observed scatter of
$0.248^{0.009}_{0.007}$ dex. Similarly, Ouellette et al. (2017) found a STFR
slope of $3.99\pm 0.18$ for SHIVir galaxies. These two STFRs are actually
steeper than the one found here for MaNGA galaxies. This discrepancy can be
explained by the sample differences; SHIVir and PROBES sample consists of more
dwarf galaxies than MaNGA. As is seen in the Baryonic Tully-Fisher relation,
dwarfs have a different stellar TFR slope until corrected for gas mass.

Brownson et al. (2022) also studied the correlation between dynamical tracers
and star formation quenching in galaxies with MaNGA galaxies. Their modelling
of stellar velocities used a formalism similar to ours with Eq. 1, Brownson et
al. (2022) yiedling a STFR slope of 3.62$\pm$0.13. The stellar velocity used
by Brownson et al. (2022) would further suffer from asymmetric drift and would
also be a source of differences between our respective STFR. The latter is
certainly a closer match to our H$\alpha$ velocity-based STFR slope than those
measured by Ouellette et al. (2017) or Stone et al. (2021b). However, the
scatters differ, 0.24 dex in this study versus 0.09 dex in Brownson et al.
(2022), as a result of the definition of scatter interquartile range versus
root mean square error. The zero-points also differ by $\sim$0.2 dex, due to
the different stellar measurements and various assumptions such as the choice
of IMFs and the location of stellar mass measurements.

###  2.5 Selection Functions

A discussion about the selection functions of the observed and simulated
galaxy sample is warranted. For the sake of uniformity within our data-model
comparison, we down-sample the complete observed MaNGA sample to a
morphologically-limited massive LTG sample (Domínguez Sánchez et al., 2018).
While the conclusions of the comparison presented in this paper remain valid,
they are only limited to the structure of massive disk systems. Our aim is not
to achieve a volume-complete or bias-corrected observed sample for comparison
with simulations but rather to examine the ability of current simulations to
reproduce various observed properties and scaling relations for a segment of
the galaxy population. Similar data-model comparisons have indeed already
served as a guide for cosmological simulations (Starkenburg et al., 2019;
Nanni et al., 2022; Cannarozzo et al., 2022; Frankel et al., 2022; Goddy et
al., 2023).

##  3 NIHAO Galaxy Formation Simulations

Figure 7: Stellar mass surface density profiles for NIHAO-AGN (left panel) and
NIHAO-LG (right panel) galaxies. The colour of each line represents the
stellar mass of each galaxy at $z=0$. The radial axis is normalized by $R_{\rm
200}\,(=600H_{0}^{2}/8\pi G)$. The stellar mass density profiles are
calculated for the halo aligned face-on.

Figure 8: RCs for NIHAO-AGN (left panel) and NIHAO-LG (right panel) galaxies.
The colour of each line represents the stellar mass of each galaxy. The radial
axis is normalized by $R_{\rm 200}$. RCs are calculated for halos viewed edge-
on.

Having established a solid baseline of observed disk galaxy scaling relations
for MaNGA sample, we are now poised to make direct comparisons with numerical
simulations of galaxy formation in order to tease out any differences that
could either indicate limitations of our theoretical formalism or biases in
the way that structural parameters are being measured. For the sake of this
comparison, we use galaxies from the “Numerical Investigation of a Hundred
Astrophysical Objects" (NIHAO) cosmological zoom-in simulations (Wang et al.,
2015). The NIHAO simulations were run with a flat $\rm\Lambda$CDM cosmology
with parameters from the Planck Satellite Investigation: $\rm
H_{0}=100h\,km\,s^{-1}\,Mpc^{-1}$ with $\rm
h=0.671,\,\Omega_{m}=0.3175,\,\Omega_{\Lambda}=0.6824,\,\Omega_{b}=0.049,\,\sigma_{8}=0.8344$
and $\rm n=0.9624$ (Planck Collaboration et al., 2014). The hydrodynamics used
an updated N-body SPH solver GASOLINE (Wadsley et al., 2017). All NIHAO
galaxies were allowed to form stars following the Kennicutt-Schmidt law
(Kennicutt, 1998) with suitable density and temperature thresholds, $\rm
T<15000\,K$ and $\rm n_{th}>10.3\,cm^{-3}$. Energy is re-injected back into
the interstellar medium (ISM) from stars through blast wave supernova
feedback. Massive stars also ionize the ISM before their supernova explosion
referred to as “early stellar feedback" (Stinson et al., 2006; Wang et al.,
2015). The “early stellar feedback" (ESF) mode allows 13% of the total stellar
flux to be injected into the ISM. This differs from the original prescription
of Stinson et al. (2013) to account for increased mixing and conform to the
abundance matching presented in Behroozi et al. (2013). For the supernova
feedback, massive stars with $\rm 8\,M_{\odot}<M_{star}<40\,M_{\odot}$ inject
energy and metals into the the ISM. The energy is injected into high density
gas and is radiated away due to efficient cooling. For gas particles inside
the blast radius, cooling is delayed by $30\,Myr$ (Stinson et al., 2013).

Supermassive black holes (SMBH) and their associated feedback are also
included in the latest version of the NIHAO simulation (Blank et al., 2019).
All NIHAO galaxies with halo mass $M_{200}>5\times 10^{10}\rm{M_{\odot}}$ were
seeded with a SMBH with mass 10${}^{5}\rm{M_{\odot}}$. SMBH accretion in NIHAO
galaxies follows the Bondi-Hoyle-Lyttleton parametrization (Bondi, 1952) where
the accretion is limited by the Eddington rate. The feedback from SMBH is a
result of the accretion where 15 per cent of the generated luminosity is
deposited into the nearby gas thermal energy. Simulated NIHAO galaxies have
been shown to match various observed galaxy properties and scaling relations
(Macciò et al., 2016; Obreja et al., 2016; Buck et al., 2017; Dutton et al.,
2017). However, our detailed analysis below will shed some contrasting light
on these assertions. Some genuine challenges, such as matching the galaxy
population diversity, do indeed exist (Frosst et al., 2022).

For the present analysis, we use all NIHAO galaxies with AGN feedback, which
are centrals, and that have $\log(M_{*}/\rm{M_{\odot})>8.5}$. The latter
stellar mass restriction is made to match the MaNGA stellar mass distribution.
Out of the 91 galaxies from the NIHAO project, 45 NIHAO objects satisfy the
stellar mass criteria. All NIHAO-AGN galaxies were simulated with a dark
matter particle mass $1.38\times 10^{7}\,\rm{M_{\odot}}$ with a softening
length of 1.86 kpc and gas softening of 782 pc. To further increase the size
of our simulated sample, we use massive galaxies from the NIHAO-LG simulations
performed with the same hydrodynamics (no AGN feedback) but constrained to
reproduce the Local Group environment (Arora et al., 2022). For the NIHAO-LG
simulations, we also utilize all galaxies with
$\log(M_{*}/\rm{M_{\odot})>8.5}$. NIHAO-LG galaxies were run with a dark
matter particle mass of $1.62\times 10^{6}\,\rm{M_{\odot}}$ with a softening
length of 860.3 pc and a gas softening length of 487 pc. Figs. 7 and 8 show
the individual stellar mass surface density profile and rotation curves for
NIHAO-AGN and NIHAO-LG galaxies. Both NIHAO simulations span a large range of
stellar mass surface densities and velocities, similar to the MaNGA
observations.

##  4 One-Dimensional Comparisons

Figure 9: Comparison of the probability density function for eight galaxy
properties between MaNGA observations (cyan line) and NIHAO simulations (pink
line). The text inset on each panel gives the p-value for the KS test applied
to the MaNGA and NIHAO samples. All properties for both observed and simulated
galaxies were calculated at a radius corresponding to a stellar surface
density of 10 M${}_{\rm\odot}\,$pc-2 (with the exception of the gas mass, used
for $\log(f_{\rm g})$ and $\log(M_{\rm bar}/{\rm M_{\odot}})$, and SFR
obtained by Pipe3D).

With our observed and simulated galaxy samples in place, we can now perform a
multi-dimensional comparison. The simplest data-model comparisons use one
dimensional probability density functions (PDF). In general, the comparison of
data-model PDFs highlights clear areas of agreement/disagreement between
galaxy simulations and observations. Fig. 9 presents such a comparison for
eight galaxy properties extracted for MaNGA and NIHAO galaxies measured at a
radius corresponding to $\Sigma_{*}=10\,\rm{M_{\odot}\,pc^{-2}}$.
Qualitatively, certain differences between the MaNGA and NIHAO galaxy
properties stand out, with the largest differences being found for $M_{*}$,
$\Sigma_{\rm 1}$, $f_{\rm g}(\equiv M_{\rm g}/M_{*})$, and star formation
rate. For instance, the NIHAO sample has smaller values of $M_{*}$,
$\Sigma_{1}$ and SFR than MaNGA observations. NIHAO zoom-in simulations were
designed with a roughly uniform stellar mass distribution, which explains the
discrepancies for the PDFs in $M_{*}$. On the other hand the gas fractions,
$f_{\rm g}$, in NIHAO have values higher relative to observed galaxies.

For a quantitative comparison, we perform a Kolmogorov–Smirnov (KS) test on
the simulated and observed distributions of galaxy properties. The text inset
in each panel of Fig. 9 shows the p-value for the KS test applied to the MaNGA
and NIHAO samples. Generally, a value of $p_{\rm KS}<0.05$ is indicative of
two samples being drawn from statistically different samples. This is indeed
what we find for all eight galaxy properties shown in Fig. 9. However, the
disagreement between observations and simulations is largely due to the
incompleteness of our samples. The NIHAO and MaNGA galaxy samples were
designed to have a roughly uniform stellar mass distribution with a non-
trivial selection function; both samples are also statistically incomplete in
different ways. While volume corrected statistics exist for the MaNGA sample
(Wake et al., 2017), these are not applied here. Furthermore, most zoom-in
simulations involves non-trivial, complex stellar mass distribution functions
and large scale observations are often volume incomplete. Therefore, caution
is advised in the calculation and interpretation of such comparisons.

##  5 Galaxy Scaling Relations

The next level of complexity in data-model comparisons is to use galaxy
scaling relations. Our approach is to construct multiple galaxy scaling
relations to build a more complete picture. This can capture the multi-faceted
nature of galaxy formation and avenues of progress for hydrodynamical
simulations. Typically, structural parameters would be measured at a light-
based radius such as R23.5,z, however isophotal levels are challenging to
evaluate for numerical simulations. To uniformly compare simulated and
observed galaxies, we measure all structural properties at a physically-
motivated radius defined by a stellar mass surface density. Through the DESI
photometry of MaNGA galaxies, we find that R23.5,z corresponds to a median
$\Sigma_{*}\sim 13\,\rm{M_{\odot}\,pc^{-2}}$ with 16-84 per cent quartile
range of 9.5-16.7 $\rm{M_{\odot}\,pc^{-2}}$. Therefore, to match closely with
previous studies based on R23.5,z, we measure all properties at the
$\Sigma_{*}=10\,\rm{M_{\odot}\,pc^{-2}}$ radius.

###  5.1 Qualitative Comparisons

Figure 10: Comparison of 12 scaling relations between MaNGA and simulated
NIHAO galaxies. The density contours match the underlying MaNGA observations;
the cyan solid and dotted lines present the ODR best fit and scatter
respectively. The pink stars represent the simulated NIHAO galaxies. All
properties for both observed and simulated galaxies are calculated at a radius
which corresponds to a stellar surface density of 10 M${}_{\rm\odot}\,$pc-2
(with the exception of the gas mass (Panel j) and SFR (Panel l) which are
obtained using Pipe3D). The inset label identifies the scaling relation.

Fig. 10 presents a qualitative comparison of various structural scaling
relations between MaNGA and NIHAO. The corresponding fitting parameters are
reported in Table 2 below. The top row of Fig. 10 shows the structural VRM∗
scaling relations; size-stellar mass, size-velocity, and STFR. Overall, the
simulated NIHAO galaxies follow similar relations as the observed galaxies, as
the pink stars scatter more or less evenly about the best fit cyan line, but
some differences exist. Most conspicuously, it is seen that simulated disk
galaxies, with large sizes, high stellar masses and circular velocities, are
underrepresented. The under-representation of such galaxies is primarily due
to NIHAO’s selection function.

For the parameter spaces where NIHAO and MaNGA galaxies overlap, both
$R-M_{*}$ and $R-V$ relations are in broad agreement between simulations and
observations. With respect to the STFR, BTFR, $M_{\rm*}-M_{\rm dyn}$, and
$R-M_{\rm dyn}$, all NIHAO galaxies scatter within observed 1$\sigma$ region.
The comparison of scaling relations involving $\Sigma_{1}$ also shows that
NIHAO galaxies with high stellar mass ($M_{*}\gtrsim 10^{10}$) have higher
central stellar densities than observed. All massive NIHAO galaxies
consistently lie at or above the upper 1$\sigma$ observed line. These high
central stellar densities for massive spiral galaxies are indicative of the
weaker baryonic feedback which fails to remove material to prevent over-
cooling (see also McCarthy et al., 2012). At the low mass end, simulated NIHAO
galaxies have slightly lower $\Sigma_{1}$ than the observed MaNGA sample (see
Panels d, e, g, and i of Fig. 10). For low mass galaxies, the strong stellar
feedback removes baryons perhaps too efficiently from the central parts to
depress $\Sigma_{1}$ relative to observations. The strong stellar feedback in
NIHAO also leads to an overall expansion of the halo (Dutton et al., 2016)
causing slightly larger sizes compared to MaNGA systems for low mass systems.

Finally, the bottom panel of Fig. 10 shows comparison of scaling relations for
gas masses (Panel j) and star formation rates (SFRs; Panel l). Gas masses and
SFRs were taken from the Pipe3D outputs for the MaNGA IFU data (Sánchez et
al., 2016). The observed gas masses are estimated from the dust attenuation
while taking the oxygen abundance into account, while SFRs are derived from
Pipe3D’s simple stellar population (SSP) template fitting for the last 100
Myr. 222For a subset of MaNGA galaxies, neutral hydrogen gas measurements were
made through the ALFALFA survey (Haynes et al., 2018) and the Robert C. Bryd
Green Bank Telescope. Details of the MaNGA-HI catalogue are found in Masters
et al. (2019) and Stark et al. (2021). For simulated NIHAO galaxies, gas mass
estimates correspond to the total mass of gas particles within a radius where
the stellar surface density is 10 M⊙ pc-2. SFR estimates are an average SFR
within the last 100 Myr. We note that the gas mass estimates between
simulations and observations are not uniformly assessed, since Pipe3D uses
MaNGA data that are limited to the extent of the IFU (1.5/2.5 Re).
Furthermore, the Pipe3D outputs do not provide errors for the gas mass and SFR
estimates, and therefore intrinsic scatter for scaling relations involving
these properties cannot be robustly calculated.

Panel j) in Fig. 10 shows the fraction of gas ($f_{g}\equiv M_{g}/M_{*}$) as a
function of stellar mass for the MaNGA and NIHAO galaxies. As expected, low
stellar mass galaxies have a larger fraction of gas relative to high stellar
mass galaxies (Catinella et al., 2010; Cortese et al., 2011; Huang et al.,
2012). In general, and especially for the lower mass systems, the NIHAO
simulated galaxies retain more gas than observed MaNGA distributions; the
opposite is true at higher masses. As stated above, the low and intermediate
stellar mass NIHAO galaxies show signatures of excess cooling (Arora et al.
2022). However, the excess gas could also be a result of the mis-match between
the observed and simulated gas measurements. A few massive NIHAO galaxies have
reduced gas content, however consistent with MaNGA observations.

Panel k) of Fig. 10 shows the baryonic Tully-Fisher relation (BTFR) for MaNGA
and NIHAO galaxies. Unlike the STFR, which showed a great simulation-
observation match (Panel c), the simulated galaxies rotate too slow for a
given baryonic mass. This is a systematic trend at most masses, but especially
at the low mass end, possibly as a result of over-cooling. The significant
discrepancy at the low-mass end of the BTFR has also been observed in other
simulation-observation comparisons (Brook et al., 2012; McQuinn et al., 2022).

Finally, panel l) features the SFR$-M_{*}$ relation for MaNGA and NIHAO. Along
with the star formation main sequence (SFMS; Whitaker et al., 2012; Cano-Díaz
et al., 2016; Hall et al., 2018), some of the MaNGA LTGs are also found in the
“green valley” (Salim, 2014). As with many other scaling relations presented
here, NIHAO galaxies lie on the distribution of observed MaNGA galaxies.
However, for a given mass, some NIHAO galaxies have much lower SFRs than
observed. Again, this is likely cause by the strong stellar feedback within
NIHAO galaxies, leading to lower SFRs than observations. Massive NIHAO
galaxies with high star formation rates are also missing.

Qualitatively, NIHAO galaxies reproduce the broad observed MaNGA spiral galaxy
properties. However, some discrepancies remain. In the next section, we
quantify our model-observation comparisons by measuring ODR slopes and
intrinsic scatters for the 12 scaling relations presented in Fig. 10.

###  5.2 Quantitative Comparisons

Figure 11: Comparison of the slopes and orthogonal scatters for the MaNGA and
NIHAO scaling relations presented in Fig. 10. The top panel shows the slopes,
$\alpha$, of the scaling relations, while the bottom panel gives the
orthogonal scatters, $\sigma_{\perp}$. All properties for both observed and
simulated galaxies are calculated at a radius which corresponds to a stellar
surface density of 10 M${}_{\rm\odot}\,$pc-2 (with the exception of the gas
mass and SFR which are obtained using Pipe3D). The pink and teal dots
represent simulated NIHAO and observed fits, respectively. The x-axis shows
each scaling relation. The black dashed line in the bottom panel shows scaling
relations based on galaxy properties from Pipe3D.

With the observed MaNGA sample being two orders of magnitude larger than the
simulated NIHAO sample, a fair quantitative comparison of the respective
scaling relations requires similar numbers of observed and simulated galaxies.
The selection of the MaNGA galaxies is performed randomly to match the size of
the smaller NIHAO samples while keeping the NIHAO stellar mass distribution.
Specific scaling relation fits are then carried out. The random selection of
MaNGA sub-samples and measurements of the scaling relation fits were carried
out 500 times; the slope and scatter and their corresponding errors were
calculated as the median and standard deviations of the distribution. It is
noted that the fit parameters (slope, zero-point, and scatter) for the NIHAO-
sized MaNGA sub-samples differ only marginally $(<1\sigma)$ from the fit
parameters based on the complete sample. These results are presented in Fig.
11. However, the dearth of data points in the both the observed and simulated
samples thwarts the assessment of robust Bayesian intrinsic scatters for our
scaling relation comparisons. For observed galaxy scaling relations, the
Bayesian intrinsic scatters were calculated and presented in Table 2.

The results presented in Fig. 11 show decent agreement for slopes and scatter
of scaling relations, though differences exist. Agreements are found for the
derived slopes of various scaling relations, including the $R-M_{*}$,
$\Sigma_{1}-M_{*}$, $M_{*}-M_{\rm dyn}$, and $f_{\rm g}-M_{*}$ relations.
Stellar mass is a common denominator in these scaling relations. Conversely,
disagreements between observed and simulated slopes exist for relations
involving velocity estimates (specifically STFR, $\Sigma_{1}-V$, and BTFR).
Disagreements for scaling relations involving velocity metrics are also
reflective of the simulations inabililty to reproduce the diversity of
observed RCs and central stellar densities (Oman et al., 2015; Frosst et al.,
2022). Indeed, NIHAO galaxies show excess halo expansion due to over-efficient
stellar feedback which results in lower central densities and different RC
slopes than observations (Dutton et al., 2017; Frosst et al., 2022). For STFR
and BTFR, NIHAO galaxies agree with theoretical scaling relation slope
expectations (Courteau et al., 2007) but a $1\sigma$ disagreement with
observations exists. NIHAO galaxies have been shown to qualitatively reproduce
observed slopes for the BTFR (Dutton et al., 2017; McQuinn et al., 2022);
albeit with fewer galaxies.

Along with the slope of scaling relations, the scatter informs us about the
cosmic variations within galaxy formation and evolution processes. The bottom
panel of Fig. 11 shows the observational and simulated orthogonal scatters for
the MaNGA sample. All galaxy structural scaling relations presented in Fig. 11
show reasonable agreement between simulations and observations. Roughly,
$1\sigma$ disagreements exist for STF, BTF, and SFMS relations. A source of
discrepancies between MaNGA and NIHAO, and specifically for the SFMS, owes to
the differences in measured or inferred SFRs. While both simulations and
observations infer SFRs over the last 100 Myr, SFR values retrieved from
Pipe3D (Sánchez et al., 2018) are measured within 1.5$R_{\rm e}$ (the extent
of the MaNGA IFU) while NIHAO properties are measured within a spatial
location corresponding to 10 M⊙ pc-2. Measurement differences also exist for
the BTFR which involves gas masses (retrieved from Pipe3D); however, as gas
contributions to the mass budget for massive LTGs are relatively small
($f_{\rm g}\sim 10\%$), the slope and scatter disagreements for our data-model
comparison remain small.

In conclusion, while NIHAO structural scaling relations agree reasonably well
with MaNGA observations, some differences are observed. The latter can be
attributed to the lack of RC diversity (Santos-Santos et al., 2018; Frosst et
al., 2022) and lower central stellar densities as a result of the strong
feedback prescriptions in NIHAO. However, some of the differences
(specifically scaling relations involving gas masses and SFR) stem from
different measurement techniques. This is evidenced by the large scatters of
the $f_{\rm g}-M_{*}$ and SFMS relations.

Sample | MaNGA | NIHAO
---|---|---
Scaling Relation | $y$ | $x$ | $\alpha$ | $\sigma_{\rm obs}$ | $\sigma_{i,\perp}$ | $\alpha$ | $\sigma_{i}$ | $\sigma_{i,\perp}$
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
Project size-stellar mass | $\log R$ | $\log M_{*}$ | 0.37$\pm$0.01 | 0.09$\pm$0.01 | 0.07$\pm$0.01 | 0.35$\pm$0.01 | 0.08$\pm$0.01 | 0.08$\pm$0.01
Project size-velocity | $\log R$ | $\log V$ | 1.33$\pm$0.02 | 0.14$\pm$0.01 | 0.09$\pm$0.01 | 0.99$\pm$0.05 | 0.08$\pm$0.01 | 0.06$\pm$0.01
Stellar Tully-Fisher | $\log M_{*}$ | $\log V$ | 3.59$\pm$0.04 | 0.24$\pm$0.01 | 0.08$\pm$0.01 | 2.84$\pm$0.15 | 0.15$\pm$0.03 | 0.05$\pm$0.01
$\Sigma_{1}$-stellar mass | $\log\Sigma_{1}$ | $\log M_{*}$ | 0.93$\pm$0.01 | 0.24$\pm$0.01 | 0.17$\pm$0.01 | 0.83$\pm$0.08 | 0.29$\pm$0.05 | 0.22$\pm$0.04
$\Sigma_{1}$-velocity | $\log\Sigma_{1}$ | $\log V$ | 3.59$\pm$0.05 | 0.34$\pm$0.01 | 0.11$\pm$0.01 | 2.62$\pm$0.32 | 0.34$\pm$0.06 | 0.12$\pm$0.04
Stellar mass-halo mass | $\log M_{*}$ | $\log M_{\rm dyn}$ | 1.05$\pm$0.01 | 0.16$\pm$0.01 | 0.13$\pm$0.01 | 0.94$\pm$0.05 | 0.16$\pm$0.02 | 0.12$\pm$0.02
$\Sigma_{1}$-size | $\log\Sigma_{1}$ | $\log R$ | 0.33$\pm$0.01 | 0.16$\pm$0.01 | 0.15$\pm$0.01 | 0.36$\pm$0.06 | 0.18$\pm$0.04 | 0.17$\pm$0.04
Size-halo mass | $\log R$ | $\log M_{\rm dyn}$ | 0.38$\pm$0.01 | 0.10$\pm$0.01 | 0.09$\pm$0.01 | 0.33$\pm$0.01 | 0.05$\pm$0.01 | 0.05$\pm$0.01
$\Sigma_{1}$-halo mass | $\log\Sigma_{1}$ | $\log M_{\rm dyn}$ | 0.97$\pm$0.02 | 0.32$\pm$0.01 | 0.24$\pm$0.01 | 0.75$\pm$0.12 | 0.42$\pm$0.06 | 0.34$\pm$0.05
Gas mass-stellar mass∗ | $\log f_{g}$ | $\log M_{*}$ | -0.65$\pm$0.03 | 0.39$\pm$0.01 | – | -0.37$\pm$0.14 | 0.42$\pm$0.11 | 0.39$\pm$0.13
Baryonic Tully Fisher∗ | $\log M_{\rm bar}$ | $\log V$ | 3.32$\pm$0.04 | 0.24$\pm$0.01 | – | 2.65$\pm$0.08 | 0.13$\pm$0.02 | 0.05$\pm$0.01
Star formation main sequence∗ | $\log SFR$ | $\log M_{*}$ | 0.72$\pm$0.02 | 0.41$\pm$0.02 | – | 0.26$\pm$0.22 | 0.68$\pm$0.14 | 0.66$\pm$0.14
Table 2: ODR linear fits for various scaling relations using MaNGA and NIHAO
galaxies evaluated at the radius corresponding to a stellar surface density of
10 M${}_{\rm\odot}\,$pc-2. Column (1) shows the scaling relation, and columns
(2-3) give the variables for the corresponding relation. Columns (4-6) show
the slope ($\alpha$), observed scatter ($\sigma_{\rm obs}$), and orthogonal
Bayesian intrinsic scatter ($\sigma_{i,\perp}=\sigma_{i}/\sqrt{1+\alpha^{2}})$
of each scaling relation for the observed MaNGA sample. Likewise, columns
(7-9) give the slope, scatter ($\sigma_{i}$) and orthogonal scatter
($\sigma_{i,\perp}$) for the NIHAO galaxies. The errors in each fit parameters
are calculated using 1000 bootstrap runs. The “ – ” indicate the scaling
relations for which Bayesian intrinsic scatter could not be calculated.

##  6 Spatially-Resolved Properties

We now take advantage of our spatially-resolved data to address the notion of
spatially-resolved scaling relations in the fundamental VRM∗ parameter space.
Once again galaxy scaling relations, along with the observational errors,
reflect fluctuations in the different processes that contribute to the
distribution of galaxy properties in the Universe.

To infer the VRM∗ properties, the rotation curves and stellar mass profiles
were interpolated (or in some cases extrapolated) at radii based on stellar
mass surface density. If a certain stellar mass density does not exist, the
object is discarded; i.e. no extrapolation is performed. The process is
applied consistently between MaNGA and NIHAO objects. Ideally, the proper
analysis of spatially resolved galaxy scaling relations should involve a
Bayesian intrinsic fit parameter to analyse the amplitude of such a
slope/scatter curve as a function of radius. For the computation of Bayesian
intrinsic fit parameters, the variation of each galaxy structural property and
its associated error with galactocentric radius must be known (Stone &
Courteau, 2019; Stone et al., 2021b). Major sources of error on most
structural parameters, and their corresponding scaling relations, include
mass-to-light ratios, distance, and intrinsic disk thickness. Some of these
vary with radius (e.g., M*/L, disk thickness), others not (distance). However,
in all cases, we assume the error profile to be constant with galactocentric
radius. As a result, the presentation of such spatially resolved parameter
would only alter the amplitude of the function and not the shape (which is of
interest here).

###  6.1 Slopes

Figure 12: Variation of observed and simulated VRM∗ slopes as a function of
radius scaled using stellar mass surface density. The solid and dashed-dotted
lines represent the MaNGA observations and NIHAO simulations respectively. The
green vertical line and shaded regions correspond to an isophotal level of
23.5 mag arcsec-2 in the z-band for the MaNGA sample.

Fig. 12 shows the spatially-resolved slopes for the VRM∗ as a function of
stellar mass surface density for the MaNGA and NIHAO galaxies. As expected,
all three scaling relations for MaNGA and NIHAO galaxies show significant
local (spatially-resolved) variations in galaxies. The slope of the $R-M_{*}$
relation varies from $0.411\pm 0.003$ in the inner parts to the $0.544\pm
0.017$ in the outskirts. However, at $\Sigma_{*}\sim 6\,{\rm
M_{\odot}\,pc^{-2}}$ the slope of $\sim$ 0.36 is closer to theoretical
expectations, $R\propto M^{\sim 1/3}$ (Courteau et al., 2007). In the
outskirts of MaNGA galaxies ($\Sigma_{*}<2\,{\rm M_{\odot}\,pc^{-2}}$), the
slope of the $R-M_{*}$ relation is less stable with fewer extended galaxies
and larger size errors. On the other hand, NIHAO simulations have the opposite
trends for the spatially-resolved slopes for the $R-M_{*}$. While agreement is
found for $\Sigma_{*}\gtrsim 2\,{\rm M_{\odot}\,pc^{-2}}$ for simulations and
observations, differences are found at low stellar mass surface densities
where scatter in our size metric is high.

The slope of the $R-V$ relation shows great variation, monotonically
increasing from $\alpha_{\rm obs}\sim 0.7$ in the inner parts of galaxies to
$\alpha_{\rm obs}\sim 2.4$ in the outskirts. The $R-V$ relation has a constant
slope of $\sim\,$1.2 in regions corresponding to $\Sigma_{*}\sim 2-10\,{\rm
M_{\odot}\,pc^{-2}}$ which represents the plateau in galaxy RCs. In very low
stellar surface density regimes, the $R-V$ slope increases greatly and shows
larger errors. This once again corresponds to the large diversity in low
stellar mass surface densities in the outer regions of galaxies due to
sporadic star formation. The NIHAO simulated galaxies show a roughly constant
slope for the $R-V$ relation as a function of stellar mass surface density.
While in tension with the observations, the constant slope for the NIHAO $R-V$
relation agrees with theoretical expectations, $R\propto V^{\sim 1.1}$
(Courteau et al., 2007).

Finally for the STFR, the observed slopes increase with radius to reach an
agreement with the theoretical expectations of $\alpha\sim 3-4$ for values of
$\Sigma_{*}\sim 10\,{\rm M_{\odot}\,pc^{-2}}$ or less. Beyond this
characteristic stellar surface density, the observed STFR slope remains
constant. This is explained by a combination of the cumulative stellar mass
profile and RC flattening leading to a constant contribution of baryons. NIHAO
simulations show slightly increasing slopes for the STFR up to a threshold
surface density of $\Sigma_{*}\sim 2\,{\rm M_{\odot}\,pc^{-2}}$, beyond which
the slope sharply rises. Such behaviour is indicative of rising RC and/or
stellar mass profiles (Figs. 7 and 8) for NIHAO galaxies (also Frosst et al.,
2022).

In conclusion, similar to the spatially-resolved scatters, the range of VRM∗
slopes changes greatly with the adopted size metrics (calculated here using
stellar mass surface density). The simulated NIHAO galaxies result in smaller
VRM∗ slopes than inferred observationally; the disagreement is at the
$2-\sigma$ level and so represents only weak tension. Similar to the scatter,
the spatially-resolved slopes of scaling relations as a function radius can
effectively constrain realistic numerical models of galaxies.

###  6.2 Scatters

The variation of scatter as a function of spatial position within galaxies
traces the degree to which the universe creates a uniform (or variable) set of
mass (baryon and dark matter) distributions. The variation of scatter as a
function of spatial positions within galaxies is a powerful tool to connect
astrophysics at the different scales of galaxy formation. To understand the
variation of the scatter for a scaling relation, we study the variation of the
“normalized scatter” ($\sigma_{\perp}/\sigma_{\parallel}$) as a function of a
physically-motivated size metric (measured here using projected stellar mass
surface densities). If a scaling relation is constructed using galaxy
properties $X$ and $Y$ with a slope, $\alpha$, and observed scatter,
$\sigma_{\rm obs}$, then the orthogonal scatter, $\sigma_{\perp}=\sigma_{\rm
obs}/\sqrt{1+\alpha^{2}}$, informs us about the scatter normal to that scaling
relation. The choice of orthogonal scatter enables a uniform comparison of
spatially-resolved scatters while taking into account the variations in the
slope as well. $\sigma_{\parallel}$ represents the scatter parallel to a
scaling relation; in other words, it corresponds to the dynamical range of a
scaling relation in a two-dimensional space. Mathematically, the parallel
scatter is the standard deviation calculated as:

$\sigma_{\parallel}=std\Bigg{[}\frac{-X}{\sqrt{\alpha^{2}+1}}+\frac{-\alpha
Y}{\sqrt{\alpha^{2}+1}}\Bigg{]}.$ (10)

The ratio, $\sigma_{\perp}/\sigma_{\parallel}$, is therefore, able to
represent how informative a scaling relation is. Since for the same orthogonal
scatter, if a relation was spread over a greater dynamic range
($\sigma_{\parallel}$) then there is more meaningful diversity in that
population. This can be seen by considering the opposite limit of
$\sigma_{\perp}=\sigma_{\parallel}$ in which the data is an uninformative
circle.

Figure 13: Variation of the scatter as a function of stellar surface density
for the VRM∗ scaling relations. (Left) Showing observational MaNGA data.
(Right) Showing the simulated NIHAO data. The dashed-dotted and solid lines
represent the orthogonal scatter ($\sigma_{\perp}$)and the normalized scatter
($\sigma_{\perp}/\sigma_{\parallel}$), respectively. The shaded regions show
the error calculated for the scatter using 1000 bootstrap runs. For clarity,
the shaded region represents twice the error. For the left column (all three
windows), the dashed black vertical line shows the isophotal radius where
$\sigma_{\perp}/\sigma_{\parallel}$ is minimized. The green vertical line and
shaded regions correspond to an isophotal level of 23.5 mag arcsec-2 in the
z-band for the MaNGA sample. The x-axis baseline is the same in all windows to
facilitate a direct comparison.

Fig. 13 shows variations of the scatter metric as a function of stellar mass
surface density for the VRM∗ scaling relations for MaNGA and NIHAO galaxies.
The green vertical line represents the normalized and orthogonal scatter
calculated for a stellar surface density range of $9.5-16.7\,{\rm
M_{\odot}\,pc^{-2}}$, which itself corresponds to an isophotal level of
approximately 23.5 mag arcsec-2 in the z-band for MaNGA LTGs. Independent of
the scatter metric, the VRM∗ scaling relations show a significant variation of
$\sim$0.2 dex for different sub-galactic regions within the MaNGA sample.
Thus, regardless of the size metric, the scatter variations of VRM∗ scaling
relations can inform us about the radial dependence of processes controlling
the evolution of galaxy structures. Analysing spatially-resolved scatter for
scaling relations (e.g., for the radial acceleration relation (Lelli et al.,
2017) and the SFMS (Wuyts2013; Cano-Díaz et al., 2016; González Delgado et
al., 2016; Wang et al., 2017; Hall et al., 2018; Ellison et al., 2018))
provides a more complete understanding of the physics shaping galaxy
properties in the Universe. As with the spatially resolved SFMS, this
spatially resolved scatter can allow us to connect the physics on the local
scale to the global astrophysics.

The top panel in the left-hand column of Fig. 13 shows variations of the
scatter metrics for the observed MaNGA size-mass relation. The dashed vertical
black line represents the density where the lowest normalized scatter is
recorded. The minimized normalized scatter shows the region where a particular
scaling relation provides the most information; this is a region where the
dynamic range ($\sigma_{\parallel}$) is maximized and the orthogonal scatter
($\sigma_{\perp}$) is minimized. For the $R-M_{*}$ relation, the minimum
normalized scatter is found at $\Sigma_{*}\sim 444\,{\rm M_{\odot}\,pc^{-2}}$
or $19.6\pm 0.3\,{\rm mag\,arcsec^{-2}}$ in the z-band. The minimum density
and SB level are found in the inner parts of the LTGs where the bulge
component dominates the light and mass budget. From this analysis, it is
evident that the stellar contents of galaxy bulges are quite uniform while
disk components show greater diversity. The orthogonal scatter (dashed-dotted
line) for the $R-M_{*}$ relation decreases with increasing stellar surface
densities. The larger stellar surface densities probe inner/bulge dominated
regions within LTGs; these regions behave similarly to the tighter (smaller
scatter) size-mass projections for ETGs (Lange et al., 2015; Trujillo et al.,
2020; Arora et al., 2021). The smaller scatter is the result of bulge growth
via repeated wet/dry mergers (Shen et al., 2003; Huertas-Company et al.,
2013).

The correlation between galaxy sizes and velocity informs us about the growth
of angular momentum in galaxies, via the connection between luminous and dark
content in galaxies. The lowest normalized scatter in the $R-V$ relation is
found at $\Sigma_{*}\sim 196\,{\rm M_{\odot}\,pc^{-2}}$. This corresponds to a
z-band isophotal level of $20.4\pm 0.04\,{\rm mag\,arcsec^{-2}}$, which is
typically found in the inner parts of LTGs. The larger normalized scatter for
higher stellar surface densities results from the wide variations in the
rising parts of LTG RCs (Oman et al., 2015; Oman et al., 2019; Frosst et al.,
2022).

We contrast the diversity of inner RCs with the larger inner scatter of the
size-velocity and stellar TFR. The larger diversity of the low stellar surface
density ($\Sigma_{*}<196\,{\rm M_{\odot}\,pc^{-2}}$) is the result of a
broader distribution of galaxy sizes. The quartile range of isophotal radius
increases as lower stellar surface density are examined. This behaviour is
similar to the $R-M_{*}$ relation where larger normalized scatters are
obtained in the disk-dominated regions of LTGs. The larger scatters on both
sides of the minimum density are indeed a result of the parameter diversities
for LTGs. In the inner parts of galaxies, velocity metrics show significant
diversity (Oman et al., 2015; Oman et al., 2019; Frosst et al., 2022), while
physically-motivated galaxy sizes (corresponding to stellar mass surface
densities) show larger diversities in the galaxies outskirts. Such diversity
in physically-motivated galaxy sizes are a result sporadic star formation in
the outer regions of galaxies (Rodríguez-Baras et al., 2018).

The combination of $R-M_{*}$ and $R-V$ relations also yields the STFR which
directly relates the luminous and dark matter content in galaxies. The minimum
scatter of the STFR is found at $\Sigma_{*}\sim 17{\rm M_{\odot}\,pc^{-2}}$,
which corresponds to an isophotal level of $23.2\pm 0.7\,{\rm
mag\,arcsec^{-2}}$ (in the $z$-band). This result agrees with other findings
that the Tully-Fisher relation has its scatter minimized at an isophotal size
of 23.5 ${\rm mag\,arcsec^{-2}}$ (in the $i$-band) (Giovanelli et al., 1994;
Courteau, 1996; Hall et al., 2012). Slight differences between specific
isophotal levels emerge from our measure of scatter based on normalized
scatter. The larger scatters for ($\Sigma_{*}>17\,{\rm M_{\odot}\,pc^{-2}}$)
are due to the inner shapes of the stellar mass and velocity profiles. The
larger diversity in the shapes of the stellar mass and velocity profiles leads
to a large inner normalized and orthogonal scatter. The smaller, near
constant, normalized and orthogonal scatter are a result of the flat nature of
stellar mass profiles and rotation curves in the outer/disk dominated regions
of the MaNGA galaxies.

The study of spatially-resolved scatters (normalized and orthogonal) has shown
that the inner scatters of the combined VRM∗ scaling relations are dominated
by the great diversity of galaxy RCs; only reflected in the scaling relation
involving a velocity metric. Conversely, scaling relations built on parameters
measured in the galaxy’s outskirts are controlled by the non-uniformity in
stellar surface density, likely driven by stochastic star formation and
feedback. It is evident from the left-hand panel of Fig. 13 that the scatter
varies significantly as a function of radius. Therefore, for a more complete
understanding of the physics of galaxy scaling relations, we caution against
the use of scatter based on a single radial metric. Given the availability of
IFU surveys, such as MaNGA(Bundy et al., 2015; Wake et al., 2017), CALIFA
(Walcher et al., 2014), and SAMI (Allen et al., 2015); and deep imaging such
as provided by DESI (Dey et al., 2019) and LSST (Ivezić et al., 2019);
spatially-resolved scaling relations must be investigated in order to achieve
a thorough understanding of structure evolution on local and global scales.

We note that the interpretation of the spatially-resolved scatter for VRM∗
scaling relations in NIHAO simulations depends greatly on sample definition
and the size metric. For example, the trends in Fig. 13 can differ for dwarf
only samples and if size definitions are based on fractions of total light
(see A21 for more details). However, with MaNGA’s target selection being
restricted to $\log(M_{*}/{\rm M_{\odot})>8.5}$, our conclusions apply mostly
to massive LTGs.

The right-hand column in Fig. 13 shows spatially-resolved scatters for the
VRM∗ scaling relations using the NIHAO simulations. Given the (small) central
densities observed in Fig. 7 for NIHAO galaxies, we cannot achieve the large
stellar surface densities seen for MaNGA galaxies in Fig. 13. We can only
study resolved scatters for $\Sigma_{*}<50\,{\rm M_{\odot}\,pc^{-2}}$ using
NIHAO galaxies. The x-axis baseline for the left and right panels in Fig. 13
were made equal to enable uniform model-observation comparisons. The lower
number of zoom-in simulations slightly increases the random errors within our
simulated fits for the VRM∗ relations.

Qualitatively, for all the VRM∗ relations, NIHAO and MaNGA galaxies have
similar spatially resolved scatter trends. For example, the spatially resolved
scatter (orthogonal and normalized) for NIHAO and observed galaxies increases
with decreasing stellar surface density. Similarly, the STFR normalized
spatially resolved scatters for both MaNGA and NIHAO stay roughly constant for
$\Sigma_{*}\leq 50\,{\rm M_{\odot}\,pc^{-2}}$. The same is broadly true for
the $R-V$ relations.

However, though these scatter trends are similar, their absolute values differ
($\sim\,$0.05 dex) due to the contribution of observed errors in MaNGA
galaxies. Furthermore, NIHAO simulations struggle to produce the high stellar
surface densities found in the observed MaNGA galaxies. While we have
demonstrated the usefulness of spatially-resolved scatters, a larger,
morphologically-diverse observed sample would enable a less-biased study.

##  7 Conclusions

We have presented a dynamical catalogue of galaxy RCs for MaNGA LTGs, as a
complement to the extensive photometric and environmental catalogue of A21.
Based on these data and the NIHAO zoom-in simulations, an extensive comparison
between galaxy observations and simulations of galaxies was established to
highlight similarities and differences.

The MaNGA H$\alpha$ velocity maps were fitted with an inclined rotating disk
model assuming a hyperbolic tangent model for the circular velocities as a
function galactocentric radius. The fitting procedure allowed us to extract
inclination-corrected RCs for MaNGA galaxies which provided us with accurate
dynamical properties. The tanh velocities fit the MaNGA H$\alpha$ velocity
cubes very well, with an average error of $\sim$7$\,{\rm km\,s^{-1}}$.
However, errors as high as $\sim$15-20$\,{\rm km\,s^{-1}}$ were recorded due
to non-circular flows and PSF effects in the galaxy’s central parts. While
non-circular motions affect our study only weakly, their investigations would
greatly benefit the comprehensive and rigorous analysis of galaxy velocity
fields.

The combination of RCs with robust photometry for the MaNGA galaxy reveals the
full extent of the MaNGA spectral data. With robust photometry based on deep
imaging ($\mu_{z}\sim$26 ${\rm mag\,arcsec^{-2}}$), we establish that the
primary+ (secondary) MaNGA samples extend out to $1.4\pm 0.4$ ($1.9\pm 0.4$)
Re. In general, our DESI photometry collects 0.3 mag more light than the NSA
SDSS photometry which explains the lesser extent of the MaNGA velocity fields
in units of effective radii. On the other hand, the use of isophotal radius is
independent of total light and provides a more accurate extent for the MaNGA
IFU. In units of R23.5 ($z$-band), the primary+ (secondary) MaNGA
spectroscopic data extend out to $0.8\pm 0.2$ ($1.0\pm 0.2$).

The ten RCs overlapping the MaNGA and PROBES surveys (the former using
H$\alpha$ velocity maps, the latter long-slit H$\alpha$ RCs) show excellent
agreement, with average differences no greater than $\sim$20 km s-1. MaNGA
velocity maps and extracted RCs are indeed well suited for analyzing the
dynamics of a galaxy’s interiors. The RCs for MaNGA allow for the construction
of spatially-resolved structural scaling relations. ODR fit parameters for the
VRM∗ scaling relations (measured at isophotal levels) are provided in Table 1
and Fig. 6. The VRM∗ scaling relations from MaNGA agree well with other
similar relations presented in the literature (see Stone et al., 2021b, for
more comparisons), including studies using MaNGA galaxies.

For uniform comparisons between simulations and observations, the metric at
which structural parameters are measured is no longer isophotal but rather
corresponding to a stellar mass surface density threshold. A stellar mass
density of 10 M⊙ pc-2 was chosen for this comparison as it closely relates to
the common isophotal R23.5. The versatility and richness of the MaNGA data
enable comparisons of observations and simulations through multiple scaling
relations. Using dynamical information, DESI photometry (A21), gas and SF
properties, we construct 12 galaxy scaling relations to compare observed MaNGA
and simulated NIHAO galaxies. NIHAO simulations broadly reproduce the observed
LTG populations ($\log(M_{*}/{\rm M_{\odot})>8.5}$) seen in the MaNGA data.
Qualitatively, most NIHAO galaxies lie within the observed 1$\sigma$ region
(see Fig. 10). Good matches are indeed found for the $R-M_{*}$,
$\Sigma_{1}-M_{*}$, $M_{*}-M_{\rm dyn}$, and $f_{\rm g}-M_{*}$ relations. The
simultaneous comparison using multiple scaling relations is beneficial in
constraining various astrophysical phenomena (star formations, dynamics,
stellar populations, etc.)

However, as a result of their complex selection function, NIHAO simulations
also lack galaxies with high stellar mass, high circular velocity, and high
star formation rates. The largest discrepancies between NIHAO and MaNGA are
found for scaling relations involving velocity metrics. Both of these are
controlled by the feedback prescription and halo response to feedback
(McCarthy et al., 2012; Dutton et al., 2016). For scaling relations involving
gas masses, NIHAO galaxies produce a larger gas fraction than observed in low
stellar mass galaxies ($\sim$0.3 dex) due to excess cooling. However, part of
that discrepancy may be due heterogeneous measurements for gas masses. Any
excess gas does not result in high SFR for NIHAO galaxies; rather, the NIHAO
SFMS show broad agreement with the observed one (see also Blank et al., 2021).

The spatially-resolved scatter for VRM∗ scaling relations for MaNGA and NIHAO
systems is also studied. The scatter of scaling relations encodes
astrophysical processes that cause cosmic variation in galaxy formation and
evolution. We study the variations of the normalized scatter, which provides
information about the orthogonal dispersion as well as the dynamical range of
a scaling relation, as a function of stellar mass surface densities. In the
inner parts, the larger scatter of the $R-V$ and STFR are dominated by the
diversity of galaxy RCs (Frosst et al., 2022). Conversely, for the $R-M_{*}$
relation, the normalized scatter in the inner regions remains low likely due
to uniform bulge populations and formation mechanisms (Courteau et al., 1996;
Kormendy & Kennicutt, 2004). The scatter for the VRM∗ scaling relations at
large galactocentric radii is dominated by the diverse stellar mass surface
densities. The effect of the diverse stellar mass surface density is most
evident in the $R-M_{*}$ relation where the scatters, both orthogonal and
normalized, continue to increase with decreasing stellar surface density. The
larger scatter in stellar mass surface density could result from sporadic star
formation at larger galactic radii due to mergers and interactions.

We also compare the spatially-resolved scatters of observed MaNGA and
simulated NIHAO galaxies. For the surface densities available in NIHAO, the
simulated $R-M_{*}$ relation is broadly consistent with MaNGA data.
Furthermore, both the $R-V$ and STFR relations are broadly consistent with a
constant scatter as a function of stellar surface density, albeit for lower
surface stellar density. The spatially-resolved scatters also show that NIHAO
fails to produce the diverse range of observed RCs (Frosst et al., 2022). Most
conspicuously, NIHAO simulations do not produce the high stellar masses and
central densities that are seen in MaNGA data. Other similar numerical
simulations fail in similar fashion (Oman et al., 2015; Oman et al., 2019;
Roper et al., 2022).

While NIHAO can broadly reproduce properties of the observed LTG galaxy
population, some discrepancies remain such as those pertaining to central
stellar densities and the diversity of RCs due to AGN feedback. With the
spatially-resolved data presented here, we may now constrain simulations at
all possible physically-motivated radii which represent different sub-galactic
environments. Not only should simulations be able to reproduce global galaxy
scaling relations (size-mass, SFMS, TFR), they should also match the observed
spatially-resolved properties and relations (such as the diversity of inner RC
slopes, spatially-resolved SFMS, RAR, metallicity gradients, etc.) These
detailed comparisons bring together the complex interplay between local and
global astrophysical processes.

## Acknowledgements

NA, SC, and CS are grateful to the Natural Sciences and Engineering Research
Council of Canada, the Ontario Government, and Queen’s University for generous
support through various scholarships and grants. This material is based upon
work supported by Tamkeen under the New York University Abu Dhabi Research
Institute grant CAP3. NA also acknowledges support from the Arthur B. McDonald
Canadian Astroparticle Physics Research Institute. The research was performed
using the PYNBODY package (Pontzen et al., 2013), SCIPY (Virtanen et al.,
2020), and NUMPY (Harris et al., 2020) and used MATPLOTLIB (Hunter, 2007) for
all graphical representation. The authors gratefully acknowledge the Gauss
Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project
by providing computing time on the GCS Supercomputer SuperMUC at Leibniz
Supercomputing Centre (www.lrz.de). This research was carried out on the High
Performance Computing resources at New York University Abu Dhabi. We greatly
appreciate the contributions of all these computing allocations.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions.

SDSS-IV acknowledges support and resources from the Center for High
Performance Computing at the University of Utah. The SDSS website is
https://www.sdss.org.

SDSS-IV 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, 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, Observatário 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 observed data for this paper have been incorporated into the online
supplementary material. The simulated data underlying this article will be
shared upon reasonable request to the corresponding author.

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##  Appendix A The Kinematic Catalogue

This section presents the headers for the kinematic catalogue based on the
2368 MaNGA LTGs used in this study (Table 3). The extracted fit parameters for
our velocity model are provided in this supplementary material.

Column Name | Description | Unit | Data Type
---|---|---|---
(1) | (2) | (3) | (4)
MaNGA-ID | MaNGA Identification | — | string
RA | Right Ascension of the object (J2000) | ∘ | float
DEC | Declination of the object (J2000) | ∘ | float
Z | NSA or SDSS redshift | — | float
TType | Morphological T-Type (from MDLM-VAC) | — | float
XC | X coordinate for the centre of the observed galaxy | arcsec | float
dXC | Error in XC | arcsec | float
YC | Y coordinate for the centre of the observed galaxy | arcsec | float
dYC | Error in YC | arcsec | float
Vsys | Systemic (heliocentric) velocity of the object | km s-1 | float
dVsys | Error in V_sys | km s-1 | float
PA | Fit position angle | rad | float
dPA | Error in PA | rad | float
I | Fit inclination | rad | float
dI | Error in inclination | rad | float
Vm | Fit Vmax for the tanh function (Eq. 4) | km s-1 | float
dVm | Error in Vm | km s-1 | float
Rt | Fit Rt for the tanh function (Eq. 4) | arcsec | float
dRt | Error in rt | arcsec | float
Table 3: Kinematic quantities for the MaNGA galaxies. Columns (1), (2) and (3)
present the column name, description, units, and type for a given kinematic
parameter, respectively. The table is found in the supplementary material in a
comma separated values file format.