11institutetext: Space Physics and Astronomy Research Unit, University of
Oulu, FI-90014, Finland 22institutetext: Centre for Astrophysics Research,
School of Physics, Astronomy and Mathematics, University of Hertfordshire,
Hatfield AL10 9AB, UK 33institutetext: Astrophysics Research Institute,
Liverpool John Moores University, IC2, Liverpool Science Park, 146 Brownlow
Hill, Liverpool L3 5RF, UK 44institutetext: Finnish Centre of Astronomy with
ESO (FINCA), Vesilinnantie 5, FI-20014 University of Turku, Finland
55institutetext: Specim, Spectral Imaging Ltd., Elektroniikkatie 13, FI-90590
Oulu, Finland 66institutetext: Departamento de Astrofísica, Universidad de La
Laguna, E-38200, La Laguna, Tenerife, Spain 77institutetext: Instituto de
Astrofísica de Canarias E-38205, La Laguna, Tenerife, Spain 88institutetext:
Department for Physics, Engineering Physics and Astrophysics, Queen’s
University, Kingston, ON K7L 3N6, Canada 99institutetext: Department of
Physics and Astronomy, University of Alabama, Box 870324, Tuscaloosa, AL
35487, USA 1010institutetext: Aix Marseille Univ, CNRS, CNES, LAM, Marseille,
France 1111institutetext: Kavli Institute for Astronomy and Astrophysics,
Peking University, Beijing 100871, China 1212institutetext: Department of
Astronomy, School of Physics, Peking University, Beijing 100871, China
1313institutetext: University of Louisville, Department of Physics and
Astronomy, 102 Natural Science Building, 40292 KY Louisville, USA
1414institutetext: Department of Astronomy and Atmospheric Sciences, Kyungpook
National University, Daegu, 41566, Korea 1515institutetext: IPAC, Mail Code
314-6, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA
1616institutetext: Valongo Observatory, Federal University of Rio de Janeiro,
Ladeira Pedro Antônio, 43, Saúde CEP 20080-090 Rio de Janeiro, RJ, Brazil
1717institutetext: Kapteyn Astronomical Institute, University of Groningen, PO
Box 800, NL-9700 AV Groningen, the Netherlands 1818institutetext: NASA
Headquarters Mary W. Jackson Building, 300 E Street SW, Washington DC 20546
1919institutetext: Steward Observatory and Department of Astronomy, University
of Arizona, 933 N. Cherry Ave., Tucson, AZ 85721, USA

# Stellar masses, sizes, and radial profiles for 465 nearby early-type
galaxies: an extension to the _Spitzer_ Survey of Stellar Structure in
Galaxies (S4G)

A. E. Watkins 112233 H. Salo 11 E. Laurikainen 11 S. Díaz-García 667788 S.
Comerón 6677 J. Janz 114455 A. H. Su 11 R. Buta 99 E. Athanassoula 1010 A.
Bosma 1010 L. C. Ho 11111212 B. W. Holwerda 1313 T. Kim 1414 J. H. Knapen 6677
S. Laine 1515 K. Menéndez-Delmestre 1616 R. F. Peletier 1717 K. Sheth 1818 D.
Zaritsky 1919

###### Abstract

Context. The _Spitzer_ Survey of Stellar Structure in Galaxies (S4G) is a
detailed study of over 2300 nearby galaxies in the near-infrared (NIR), which
has been critical to our understanding of the detailed structures of nearby
galaxies. Because the sample galaxies were selected only using radio-derived
velocities, however, the survey favored late-type disk galaxies over
lenticulars and ellipticals.

Aims. A follow-up _Spitzer_ survey was conducted to rectify this bias, adding
465 early-type galaxies (ETGs) to the original sample, to be analyzed in a
manner consistent with the initial survey. We present the data release of this
ETG extension, up to the third data processing pipeline (P3): surface
photometry.

Methods. We produce curves of growth and radial surface brightness profiles
(with and without inclination corrections) using reduced and masked _Spitzer_
IRAC 3.6 $\mu$m and 4.5 $\mu$m images produced through Pipelines 1 and 2,
respectively. From these profiles, we derive the following integrated
quantities: total magnitudes, stellar masses, concentration parameters, and
galaxy size metrics. We showcase NIR scaling relations for ETGs among these
quantities.

Results. We examine general trends across the whole S4G and ETG extension
among our derived parameters, highlighting differences between ETGs and late-
type galaxies (LTGs). ETGs are, on average, more massive and more concentrated
than LTGs, and also show subtle distinctions among ETG morphological sub-
types. We also derive the following scaling relations and compare with
previous results in visible light: mass–size (both half-light and isophotal),
mass–concentration, mass–surface brightness (central, effective, and within 1
kpc), and mass–color.

Conclusions. We find good agreement with previous works, though some relations
(e.g., mass–central surface brightness) will require more careful multi-
component decompositions to be fully understood. The relations between mass
and isophotal radius and between mass and surface brightness within 1 kpc, in
particular, show notably small scatter. The former provides important
constraints on the limits of size growth in galaxies, possibly related to star
formation thresholds, while the latter—particularly when paired with the
similarly tight relation for LTGs—showcases the striking self-similarity of
galaxy cores, suggesting these evolve little over cosmic time. All of the
profiles and parameters described in this paper will be provided to the
community via the NASA/IPAC database on a dedicated website.

###### Key Words.:

galaxies: evolution - galaxies: photometry - galaxies: spiral - galaxies:
structure - galaxies: statistics - galaxies: elliptical and lenticular, cD

##  1 Introduction

Over cosmic time, galaxies acquire and process baryons, turning into stars and
redistributing those stars through interactions with neighboring systems or
through their own internal gravitational processes. Under $\Lambda$CDM
cosmology (White & Rees 1978; Frenk et al. 1985), galaxies form via the
hierarchical merging of dark matter halos, which in the early Universe served
as the potential wells into which baryons initially settled. Given the
hierarchical nature of the merging, the centers of galaxies likely formed
first, and subsequent gas or satellite accretions (e.g., Kauffmann et al.
1993; Bendo & Barnes 2000; Aguerri et al. 2001; Balcells et al. 2003; Bournaud
et al. 2007; Athanassoula et al. 2016) and secular evolution (e.g., Combes &
Sanders 1981; Kormendy 1982; Fathi & Peletier 2003; Kormendy & Kennicutt 2004;
Athanassoula 2005) then conspired to modify both these centers and the
structures built atop them in complex collisional and dissipational
interactions. Nearby galaxies display the present end-results of these
interactions; therefore, detailed study of the stellar structure of nearby
galaxies is critical to understanding galaxy evolution.

It was to this end that the _Spitzer_ Survey of Stellar Structure in Galaxies
(S4G; Sheth et al. 2010) was carried out, providing excellent quality deep
imaging of over 2300 galaxies within $\sim$40 Mpc, in a wavelength regime
(near-infrared, NIR) that suffers little from dust extinction (Draine & Lee
1984; Calzetti 2001) and contains few emission or absorption features (e.g.,
Willner et al. 1977; Tokunaga et al. 1991; Kaneda et al. 2007). Stellar mass-
to-light ratios are also nearly constant across stellar populations in this
wavelength regime, making this imaging a nearly direct tracer of stellar mass
(e.g., Pahre et al. 2004; Jun & Im 2008; Peletier et al. 2012; Meidt et al.
2014; Driver et al. 2016). The survey has been indispensible to the study of
galaxy evolution, having resulted in over one hundred published papers (e.g.,
Buta et al. 2010; Comerón et al. 2011; Martín-Navarro et al. 2012; Zaritsky et
al. 2013; Laine et al. 2014; Herrera-Endoqui et al. 2015; Kim et al. 2016;
Laine et al. 2016; Laurikainen & Salo 2017; Bouquin et al. 2018; Díaz-García
et al. 2019; İkiz et al. 2020). Yet, until now, it has suffered from a serious
bias: the velocity cut in the original survey used only radio measurements,
thereby excluding gas-poor—mostly early-type—galaxies.

Early-type galaxies (ETGs), being typically gas- and dust-poor and dominated
by evolved stellar populations, are often referred to as “red and dead.” This
is an unfortunate term, connoting a lack of interesting properties or
phenomena. The term ETG—referring to the classification scheme of increasing
morphological complexity proposed by Hubble (1926)—covers both the disk-
dominated and rotation-supported lenticular (S0) galaxies, as well as the more
pressure-supported elliptical (E) galaxies. What the exact relationship is
linking these two classes with each other—and with their more gas-rich, late-
type galaxy (LTG) counterparts—is a long-standing question.

For example, due to their many similarities with late-type spirals (including
bulge-to-total mass or light ratios, range of bulge luminosities, bar
properties, and kinematics; e.g., Laurikainen et al. 2011; Cappellari et al.
2011a; Kormendy & Bender 2012), S0 galaxies may have formed from LTGs through
the removal or rapid consumption of gas and dust. S0s also form a parallel
morphological sequence with spirals (Spitzer & Baade 1951; van den Bergh 1976;
Laurikainen et al. 2011; Cappellari et al. 2011a; Kormendy & Bender 2012),
lending further credence to this idea. Detailed photometric analysis has also
provided evidence linking the two disk galaxy classes; Laurikainen et al.
(2010) found that the sizes and masses of bulges in both early- and late-type
disk galaxies correlate strongly with those of the disk, suggesting a similar
kind of concurrent evolution (see also: Kormendy & Kennicutt 2004; Aguerri et
al. 2005), and disk scaling relations between S0s and LTGs also appear similar
(e.g., Eliche-Moral et al. 2015).

Even if the two are evolutionarily linked, however, the precise mechanisms
behind the transformation from LTG to ETG are still open to debate. Because of
the relative abundance of S0s in galaxy clusters, one long-favored such
mechanism is infall into dense environments (e.g., Gunn & Gott 1972; Dressler
1980; Moore et al. 1996; Bekki et al. 2002). This seems to happen quite early;
detailed study of nearby cluster S0 stellar populations frequently shows
quenching epochs over 10 Gyr ago (e.g., Sil’chenko 2013; Comerón et al. 2016;
Katkov et al. 2019). However, isolated field S0s, though rare, do exist
(Karachentsev et al. 2010), ruling out cluster or group infall as the sole
formation pathway. Falcón-Barroso et al. (2019) also argued that simple
quenching and fading of stellar populations is unlikely to be the dominant
spiral–S0 transformation mechanism given differences in angular momentum
between the classes (with simulations suggesting major mergers may instead
explain these S0 kinematics; Querejeta et al. 2015), though such quenching may
still occur in low-density environments (e.g., Rizzo et al. 2018).
Environmental influence may not be necessary, either; Saha & Cortesi (2018)
showed that galaxies with properties very similar to S0s can form in
isolation, via instabilities in gaseous disks. Burstein et al. (2005) also
argued that the relative _K_ -band luminosities of S0s and spirals demonstrate
that direct quenching, while feasible in individual cases, may not be the
dominant transformation mechanism either. Indeed, detailed comparisons between
environments suggest there may be multiple quenching pathways behind the
formation of S0 galaxies (e.g., Wilman & Erwin 2012). Aguerri (2012) provides
a thorough review of different proposed S0 formation pathways.

The origins of elliptical galaxies, despite their relatively simple
appearances, are likely equally complex. Early theories of elliptical galaxy
formation relied on a monolithic collapse (Eggen et al. 1962), either via the
low-rotation, dissipational collapse of gas clouds (e.g., Larson 1974a, b,
1975), or the dissipationless collapse of stars formed prior to the gas
collapse time (e.g., Gott 1973; Thuan 1975; Gott 1977). As $\Lambda$CDM
cosmology came into favor, hierarchical models of elliptical galaxy formation
did alongside (see the review by de Zeeuw & Franx 1991), and the discussion
became more granular. The origin of elliptical galaxy rotation (or lack
thereof), for example, received a lot of focus (Bertola & Capaccioli 1975;
Illingworth 1977; Binney 1978; Davies & Illingworth 1983; Wyse & Jones 1984;
Barnes 1988) given its strong implications for elliptical galaxy merger
histories; indeed, rotation may be critical to understanding ETG formation
pathways as a whole (Cappellari 2016).

From simulations, it became clear that major mergers of gas-poor disk galaxies
can result in remnants that look very much like elliptical galaxies, albeit
with nearly zero net rotation (e.g., Toomre 1977; White 1983; Barnes 1988,
1990; Hernquist 1992). The prevalence of elliptical galaxies in dense
environments (e.g. Oemler 1974; Dressler 1980; Cappellari et al. 2011b; Calvi
et al. 2012) also points toward mergers or interactions as key formation
mechanisms. Elliptical galaxies also show a diversity in their core profiles
that might result from varying merger histories, with some showing continuous
Sérsic profiles, while the most luminous ellipticals show central light
deficits (e.g., King & Minkowski 1966; Gudehus 1973; Davies et al. 1983;
Kormendy & Bender 1996; Faber et al. 1997; Ravindranath et al. 2001; Emsellem
et al. 2007; Kormendy et al. 2009; Dullo & Graham 2012; Krajnović 2015;
Krajnović et al. 2020, and many others). Others still show central light
enhancements (e.g., Kormendy 1999; Côté et al. 2006; Kormendy et al. 2009).
Core light deficits may result from scouring by supermassive black hole (SMBH)
pairs (e.g., Begelman et al. 1980; Ebisuzaki et al. 1991; Milosavljević &
Merritt 2001; Thomas et al. 2014) donated by merged galaxies (implying at
least one major merger event), while core enhancements might occur through
central starbursts induced by dissipative mergers (e.g., Mihos & Hernquist
1994; Kormendy 1999; Kormendy et al. 2009). Mergers may also be necessary to
explain a seeming coupling of baryonic and dark matter kinematics in some ETGs
(if one assumes a constant stellar initial mass function, IMF, and with some
dependence on local environment; Wegner et al. 2012; Corsini et al. 2017);
otherwise, a non-universal IMF is required.

Mergers thus appear to be important for elliptical galaxy formation for most
of cosmic history, but the dominant type of merger history remains ambiguous.
Contrary to expectations from a major-merger formation scenario (Nipoti et al.
2003; Hopkins et al. 2008; Bezanson et al. 2009; Naab et al. 2009), mass
evolution of ellipticals has been slow relative to size evolution (e.g.,
Trujillo et al. 2007; Buitrago et al. 2008; van Dokkum et al. 2008; Szomoru et
al. 2012), and metallicity gradients in elliptical outskirts (particularly in
low-luminosity ellipticals) are quite flat (e.g., Peletier et al. 1990; La
Barbera et al. 2004; den Brok et al. 2011). Both indicate that, for most of
cosmic history, elliptical galaxies may have grown via a complex series of
minor mergers (e.g., Bournaud et al. 2007; Naab et al. 2009). Kim et al.
(2012) and Ramos et al. (2015) also showed that tidal debris in their sample
of ellipticals, when present, makes up only a small fraction ($\sim 10$%) of
the hosts’ total mass, again implying a contribution primarily from minor
mergers. Indeed, even the most well-behaved classical ellipticals show
noticeable deviations from simple Sérsic profiles (e.g., Schombert 1986),
suggesting perhaps a more piecemeal evolutionary history (Schombert 2015). The
most massive ellipticals, however, typically show less rotational support
(Cappellari et al. 2007) and more frequently have cored light deficits
compared to low-mass ellipticals (Lauer et al. 2007; Kormendy et al. 2009),
implying that they resulted from larger mass-ratio mergers (though in the case
of disk mergers, dissipational processes may be necessary to produce realistic
kinematics in the resulting remnants; e.g., Thomas et al. 2009). Davari et al.
(2017) also showed that high-redshift progenitors of massive ellipticals (red
nuggets) often have surrounding disks that are mostly destroyed by $z=0.5$,
which should only occur under strong gravitational perturbations. Elliptical
galaxy formation pathways thus seem to have a mass dependence. That said, one
commonality all of these varied scenarios must reproduce is that they must
somehow conspire to yield galaxies that fall on the tight scaling relation
linking their structure and kinematics known as the fundamental plane
(Djorgovski & Davis 1987; Dressler et al. 1987).

Given the unique contribution of ETGs to the overall picture of galaxy
evolution, the S4G Team carried out an additional _Spitzer_ survey to fill in
the initial S4G’s morphological gap (S4G$+$, _Spitzer_ project ID 10043; Sheth
et al. 2013). To avoid confusion, as these new observations are an extension
of the S4G, we hereafter refer to the full survey to date (the original S4G
and the new ETG extension) as S4G+ETG. This new survey echoed the initial
survey’s methodology, now targeting an additional 465 ETGs with the _Spitzer_
Space Telescope’s (Werner et al. 2004) Infrared Array Camera (IRAC; Fazio et
al. 2004) in the 3.6 $\mu$m and 4.5 $\mu$m bands. This new sample selection
used the same criteria as the S4G, but rectified one of its blind spots:
distance estimates based only on radio-derived redshifts, which excluded many
ETGs.

With these new data in hand, we continue the S4G data processing by conducting
the same photometric analyses applied to the previous 2352 S4G galaxies as
part of Pipeline 3 (hereafter, P3; see Muñoz-Mateos et al. 2015, hereafter
MM2015). This includes: intensity and surface brightness profiles, position
angle and ellipticity profiles, total magnitudes, stellar masses, measures of
central concentration, and measures of galaxy size. MM2015 provides in their
Introduction a long list of the applications of this type of structural
analysis, all of which remain applicable to this ETG extension sample (which
includes still both disk and elliptical galaxies). This will lead into future
analyses of this sample as well, including a detailed study of morphology
(morphological classifications already having been completed by R. Buta, which
are used throughout this paper) and multi-component decompositions. In an era
of large-scale deep surveys, most of which are primarily focused on probing
large cosmic volumes (e.g., the Hyper-Suprime Cam Subaru Strategic Program,
the Vera C. Rubin Observatory’s Legacy Survey of Space and Time, the Roman
Space Telescope, and the Euclid Mission; Aihara et al. 2018; Ivezić et al.
2019; Spergel et al. 2015; Laureijs et al. 2010), it is critical to have a
complete census of massive galaxies at redshift $\textrm{z}=0$ for a current
evolutionary end-state comparison. This is ultimately what the S4G hopes to
provide.

This paper is organized as follows. In Section 2, we discuss the S4G+ETG
sample selection, observing strategy, and give a brief overview of the data
reduction steps. In Section 3, we describe our photometric analyses. Section 4
provides an overview of the photometric parameter distributions for the entire
sample, isolating ETGs for comparison. In Section 5, we showcase our results
via a number of different scaling relations derived using the aforementioned
photometric parameters. We discuss what we supply to the community and where
to find it in Section 6, and finally provide a summary of our results in
Section 7.

This paper presents an introduction to the S4G+ETG sample and the quantities
derived from P3. Given the extraordinarily broad scope of the topic, more
detailed investigations into the relations demonstrated in this paper will be
published in future papers.

##  2 S4G ETG extension sample, observations, and data reduction

Figure 1: HyperLEDA $T-$type distribution (_left_ panel) and $T-$type
distribution from 3.6 $\mu$m classifications by Ron Buta (_right_ panel) for
the 465 newly-observed ETGs in the S4G+ETG sample. We use the Buta $T-$types
for the remainder of this paper, given their improved accuracy.

Developed by the original S4G Team, who also carried out the observations, the
S4G ETG extension sample (initially denoted S4G$+$) followed the S4G selection
criteria. This includes all galaxies in the HyperLEDA database (Paturel et al.
2003) with radial velocities $v<3000$ km s-1, total extinction-corrected blue
magnitudes $m_{\rm B,corr}<15.5$, blue isophotal diameters $D_{25}>$
1$\aas@@fstack{\prime}$0, and Galactic latitude $\lvert b\rvert\geq 30$°.
Previously, only radio-derived velocities were used (Sheth et al. 2010),
biasing the sample toward gas-rich LTGs. To resolve this, the ETG extension
included visual band spectroscopic redshifts as well for galaxies with
morphological type $T\leq 0$.

We note, however, that this $T-$type criterion induced another previously
unnoticed bias, as it excludes an additional 425 LTGs lacking radio velocity
measurements. We are currently addressing this bias using ground-based
$i$-band observations, which, due to the vastly different observing strategy
and data reduction procedure, will be the subject of a future paper.

In total, S4G+ETG contains 2817 galaxies. Among these are 690 galaxies with
HyperLEDA Hubble $T<=0$, 225 of which were already present in the initial 2352
galaxy S4G sample (Muñoz-Mateos et al. 2015; Salo et al. 2015). This left an
additional 465 ETGs to be subsequently observed as part of _Spitzer_ proposal
ID 10043 (Sheth et al. 2013). Among these are the following Milky Way
satellites: ESO 351-30 (Sculptor Dwarf), ESO 356-4 (Fornax Dwarf), PGC 88608
(Sextans Dwarf), UGC 5470 (Leo I Dwarf), UGC 6253 (Leo II Dwarf), and UGC
10822 (Draco Dwarf). These galaxies are fully resolved into individual stars
with the _Spitzer_ IRAC resolution, hence cannot be analyzed using the surface
photometry methods we describe here. Therefore, we applied the P3 procedure we
describe below to 459 ETGs total, of which $\sim$67% and $\sim$33% are
classified in HyperLEDA as S0s and Es, respectively. For our results and
discussion, we focus on ETGs as a population, and therefore include all of the
ETGs in the full S4G+ETG sample that are not MW satellites (684 in total).

We show the distribution of the HyperLEDA $T-$types used to select this sample
in the left panel of Fig. 1, demonstrating two peaks at $T=-3$ (S0-) and
$T=-5$ (E). Morphological classifications in HyperLEDA come from a compilation
of sources, including indirect estimates derived from concentration indices,
hence can be somewhat unreliable. We thus derived our own morphological
classifications based on the 3.6 $\mu$m images as in Buta et al. (2015), the
distribution of which is shown in the rightmost panel in Fig. 1 for
comparison. On closer visual inspection, not all of the ETG extension galaxies
are best classified as ETGs ($T>0$ for 75 galaxies in our new
classifications), so, given this discrepancy, we opt to use our own
classifications for the remainder of this paper in lieu of those found in
HyperLEDA. We will make these new classifications available alongside all
other quantities we have derived from this sample; we show a sub-sample of the
full morphology table in the Appendix.

The new observations mirrored the original 2352-galaxy S4G sample observing
strategy. In summary, we observed all 465 new galaxies with the _Spitzer_ IRAC
camera in both 3.6 $\mu$m and 4.5 $\mu$m during the post-cryogenic mission,
with total on-source exposure times of $8\times 23.6$ s px-1 for channel 1
(3.6 $\mu$m) and $8\times 26.8$ s px-1 for channel 2 (4.5 $\mu$m). For
galaxies with diameters $<$ 3$\aas@@fstack{\prime}$3, we split exposures into
two visits separated by at least 30 days to allow the telescope to rotate to a
new orientation, both as an aid in correcting large-scale defects (asteroid or
satellite streaks, scattered light) and to allow for subpixel sampling. For
galaxies with diameters $\geq$ 3$\aas@@fstack{\prime}$3, we used an 8-position
dither (also split between two visits separated by $\geq$ 30 days), with
146$\aas@@fstack{\prime\prime}$6 offsets applied between the maps and frame
times of 30 s at each position. For a more detailed description of the
observation strategy, see Sheth et al. (2010).

The initial data processing procedure is described in detail in MM2015, but we
provide a brief summary here. Pipeline 1 (P1) creates science-ready mosaics by
matching the background levels of individual exposures, then combines the
images using standard dithering and drizzle procedures (Fruchter & Hook 2002).
The final mosaics have a pixel scale of 0$\aas@@fstack{\prime\prime}$75 and
units of MJy sr-1. The S4G uses the AB magnitude system (Oke 1974), hence the
magnitude zeropoint for both 3.6 $\mu$m and 4.5 $\mu$m is 21.0967 mag,
translating to a surface brightness zeropoint of 20.4720 mag arcsec-2.

Pipeline 2 (P2) produces masks of contaminating sources, including foreground
stars, background galaxies, or scattered light artifacts. Proper masking is
crucial for measuring accurate surface photometry of extended sources,
particularly at low surface brightness (see, e.g., the Appendix of Watkins et
al. 2019). For the S4G, we first automatically generate three different masks
using SExtractor (Bertin & Arnouts 1996) on the 3.6 $\mu$m mosaic images, with
high, medium, and low detection thresholds in order to identify sources both
far from the target galaxy and overlapping in projection with the galaxy. The
best of these—for example, high-threshold masks are necessary for objects
embedded within galaxies to avoid masking galaxy light, while low-threshold
masks help reduce the impact of low surface brightness scattered light
artifacts—is then visually identified and subsequently edited by hand (using a
custom IDL editing code; see Salo et al. 2015) to improve masking of diffuse
wings of stellar point spread functions (PSFs) or to remove masks targeting
bright parts of the galaxy such as bar ansae or HII regions. Once completed,
we transfer these masks to the 4.5 $\mu$m images. The masks typically suffice
for both detectors without changes; nonetheless, we visually inspect the 4.5
$\mu$m masks to ensure no such changes are required. These masked mosaic
images are then ready for the third pipeline, described below.

The 2D structural multi-component decomposition analysis (Pipeline 4, P4) for
the S4G+ETG galaxies will be presented in a forthcoming article (Comerón et
al., in preparation). Nevertheless, we make use here of their single-component
GALFIT (Peng et al. 2002, 2010) models where the galaxy light distribution is
fitted with a single Sérsic law. In these decompositions, the same PSF-
function (covering the inner $30$″$\times 30$″) and the same method for
constructing the _sigma-images_ are used as in Salo et al. (2015) for the
original S4G sample. The galaxy centers, and the isophotal ellipticity and
position angles, were fixed to values determined before the decomposition
analysis, leaving the total magnitude, effective radius and Sérsic $n$
parameter as free parameters of the fit. In Sect. 5.1 below we will compare
the concentration derived from the $n$ parameter and the total Sérsic-fit
magnitudes with those obtained directly from light profiles. We also utilize
the Sérsic fits in an approximate deconvolution of the observed light profiles
(Sect. 3.2).

##  3 P3: surface photometry

Pipeline 3 derives galaxy centroids, sky backgrounds, asymptotic magnitudes,
and isophotal properties such as surface brightness profiles, isophotal radii,
and concentration indices. To maintain consistency across the whole S4G, we
adhere closely to the methodology of MM2015, which we briefly summarize here.
To improve the accuracy of our results, however, and to account for necessary
changes in procedure given the steepness of ETG light profiles, we have made
some modifications to this methodology, which we describe in detail below.

###  3.1 Sky Measurement and Uncertainty

Accurate sky subtraction is critical for correctly measuring both surface
brightness profiles (hence isophotal radii) and asymptotic magnitudes and any
derived properties. While the random noise of the S4G imaging is quite low
(Sheth et al. 2010), the image backgrounds themselves are often dominated by
large-scale systematic sources of uncertainty, such as background gradients or
scattered light. Because it affords more control, for the ETG sample we chose
to use the sky subtraction and uncertainty estimation used in the S4G Pipeline
4 analysis (Salo et al. 2015) in lieu of that described by MM2015.

Briefly, we measure the sky in each image using 10–20 40 px $\times$ 40 px
(30″$\times$30″) boxes, placed far from the galaxy by hand to avoid scattered
light, image artifacts, and heavily masked areas. We take the median of the
median values of unmasked pixels within each box as the image sky value
(hereafter, SKY, to use the nomenclature from Salo et al. 2015) and the
standard deviation of these medians as its associated uncertainty (hereafter,
DSKY). Finally, we measure the sky root mean square (hereafter, RMS) as the
median of the standard deviations within each box. Although this method is an
improvement due to its enhanced control over sky box placement, the derived
sky values deviate only slightly (showing a median difference in sky
background of 0.003 MJy sr-1 between the P3 and P4 methods) from those of
MM2015 (see Fig. 6 of Salo et al. 2015).

###  3.2 Radial Profiles

Figure 2: Showcasing examples of radial profiles, as described in Section 3.2,
using NGC 720. The _top left_ panel shows the 3.6 $\mu$m image, with masks
shown via the red contours. The _bottom left_ panel shows the same image, with
2″ radial bin width, variable PA and $\epsilon$ isophotal ellipses overlaid in
black. The blue ellipse shows _B_ -band $R_{25}$ from HyperLEDA, while the two
red ellipses show our values of $R_{25.5,3.6}$ and $R_{26.5,3.6}$ (solid and
dashed, respectively). The _rightmost_ panels show radial profiles of, from
top to bottom: surface brightness, ellipticity, and position angle. The 3.6
$\mu$m and 4.5 $\mu$m profiles are offset from each other by the amounts shown
in the legends to avoid significant overlap. Profiles shown here are truncated
in radius compared to those available online, to avoid clutter from poor-
quality fits in the far outskirts. Vertical dotted red lines show
$R_{25.5,3.6}$.

The P3 analysis relies on radial surface brightness profiles and curves of
growth. Again, we generally follow MM2015 in constructing these radial
profiles, with some deviations. We first use a custom IDL routine to locate
the centroids of the galaxies (which we use to fix the galaxy’s center
coordinates), then construct profiles using the IRAF _ellipse_ task
(Jedrzejewski 1987; Busko 1996), varying or holding fixed the ellipticities
($\epsilon$) and position angles (PA). We initially create three different
kinds of profiles, as MM2015:

  * •

Fixed center, free $\epsilon$ and PA, radial bin width $\Delta\textrm{r}=6$″:
these profiles use wide radial bins for high S/N. We thus use them to robustly
measure the outermost structures of each galaxy. From these profiles, we
provide values of $\epsilon$ and PA at both the $\mu_{\lambda}=25.5$ and 26.5
mag arcsec-2 isophotes. If one or both of those isophotes was not found
(generally due to high background noise), we record their values as -999.

  * •

Fixed center, free $\epsilon$ and PA, radial bin width $\Delta\textrm{r}=2$″:
these profiles have lower S/N and are less reliable in the galaxies’
outskirts, but provide more detailed information on isophote shapes in the
galaxies’ inner regions. The value 2″ is chosen to match the IRAC PSF at 3.6
$\mu$m and 4.5 $\mu$m, hence reflects one resolution element. These profiles
are most useful for assessing the properties of bars and other internal
structures. We show an example of these profiles in Fig. 2.

  * •

Fixed center, fixed $\epsilon$ and PA, radial bin width $\Delta\textrm{r}=2$″:
these profiles have parameters fixed to those of the $\mu_{\lambda}=25.5$ mag
arcsec-2 isophotes measured using the $\Delta r=6$″profiles described above.
With fixed isophote shapes, these profiles are most useful for measuring disk
scalelengths, break radii, and for performing 1D bulge-disk decompositions.
With these profiles we measure the curves of growth, from which we derive
asymptotic magnitudes (Sect. 3.3) and derivative values thereof. We note that
the use of these profiles relies on the assumption that the
$\mu_{\lambda}=25.5$ mag arcsec-2 isophote shapes accurately reflect those of
the disk’s underlying structures, such as spiral arms; more detailed analysis
may require a more careful outer isophote selection to ensure this is true.

Generating these profiles required multiple steps. First, IRAF’s _ellipse_
task simply ignores masked pixels, biasing the curves of growth. While there
are options to estimate masked flux within IRAF, because many of the sample
galaxies have spiral arms, bars, and other such symmetrical structures, we opt
instead to estimate masked flux through our own interpolation algorithm,
demonstrated in Fig. 3. For this ETG sample only, we interpolate across masks
using the galaxy’s azimuthal light profile, measured in one-pixel-wide radial
bins from the center outward.

We demonstrate this using the galaxy IC 3381, which is contaminated by a
bright nearby star and so has been heavily masked. First, in each radial bin
we measured the mean and standard deviations within 10-degree-wide azimuthal
bins. As a first pass, we linearly interpolated this azimuthal profile across
the masked pixels and recorded the median standard deviation among azimuthal
bins (hereafter, $\sigma_{\rm rad}$). In order to preserve any periodicity
induced by, e.g., bars, we then decomposed the resulting interpolated
azimuthal profiles into Fourier modes and used the first $m=20$ Fourier modes
to more accurately reconstruct the flux across masked pixels. We then added
Gaussian noise to the interpolated values, distributed as
$\sim\mathcal{N}(0,\sigma_{\rm rad})$. We show an example of this
interpolation for the same example galaxy, IC 3381, in the bottom panel of
Fig. 3.

Figure 3: Demonstrating interpolation across masks for a heavily masked galaxy
(IC 3381). The top panel shows the masked image, while the bottom panel shows
the same image but with the galaxy’s light interpolated azimuthally across the
masked pixels (see Section 3.2).

With the missing flux now preserved, we constructed radial profiles by running
_ellipse_ twice: first, using the software described by Salo et al. (2015) to
select sensible starting values for semi-major axis length (SMA), PA, and
$\epsilon$, then automatically running unconstrained _ellipse_ fits, in the
three manners described above, using these starting parameters. We chose not
to constrain the automated fits in order to measure profiles to as low a
surface brightness level as possible, but the outskirts of these unconstrained
profiles evidently showed large swings in $\epsilon$ and PA where the fits
were dominated by noise. Therefore, we derived maximum radii for the fits
using the point at which the values of $\epsilon$ or PA in the $\Delta r=6$″
profiles began to show regular 3$\sigma$ divergences from their mean values as
measured beyond the $\mu_{3.6}=25.5$ mag arcsec-2 isophote (or, failing this,
where the _ellipse_ uncertainty in these values, beyond $\mu_{3.6}=25.5$ mag
arcsec-2, first was recorded as INDEF). We chose as the maximum fitting radius
1.2 times this value, which ensures in most cases that profiles reach at least
the $\mu_{3.6}=25.5$ mag arcsec-2 isophote if the image’s noise limit is below
this value, even if the isophotal fits there are poor. We then re-fit all
profiles out to these maximum radii for each galaxy, checking the robustness
of each by overplotting the fitted isophotes on the 3.6 $\mu$m images of each
galaxy. Where the fitting failed (evident when the isophotal shapes were
constant even at small radii), we simply re-ran the fits with different
starting SMA until they succeeded. Due to the ETG sample galaxies’ often
simple morphology, we needed only re-run in this manner one or two times.

Figure 4: _Left_ panels show the effect of the aperture correction described
by MM2015. Black shows the unaltered surface brightness profile for NGC 720,
while the gray line shows the same profile after the aperture correction is
applied. The _bottom left_ panel shows the difference between the two
(unaltered$-$corrected). _Right_ panels are the same as the left panels, but
show our more robust aperture correction using GALFIT models (see Section
3.2). We derived all magnitudes and related quantities using the MM2015
correction, to maintain consistency with that study, however for any future
analysis using our datasets we recommend instead using the revised correction.

As in MM2015, we provide these fits as ASCII tables (Sect. 6). We provide the
same parameters as in MM2015, including harmonic deviations from perfect
ellipses and uncertainty estimates. We also include photometric profiles
corrected for PSF scattering in the same manner as in MM2015, who used
correction factors taken from the IRAC Instrument
Handbook111https://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/iracinstrumenthandbook/.
Briefly, the light from any extended source (e.g., galaxies) in IRAC is a
convolution of the astrophysical light distribution and the extended IRAC PSF.
This means that, in a given photometric aperture, light from the astrophysical
source is both scattered out of the aperture into the PSF wings, and scattered
into the aperture from astrophysical sources outside, including other regions
of the same galaxy.

Given the complexity of the IRAC PSFs, modeling this effect requires using
model galaxies convolved with the measured PSF for each IRAC band. From the
IRAC Instrument Handbook, PSF correction factors were measured using single-
Sérsic profile model ETGs with sizes ranging up to 200″. For the total flux
$F$ within an aperture, these are provided in the following form:

$F_{\rm corr}(r_{\rm eq})=F_{\rm obs}(r_{\rm eq})\times(Ae^{-r_{\rm
eq}^{B}}+C)$ (1)

where $r_{\rm eq}=\sqrt{ab}$ is the equivalent elliptical aperture radius in
arcseconds, and the constants $A$, $B$, and $C$ are equal to 0.82, 0.37, and
0.91 for 3.6 $\mu$m and 1.16, 0.433, and 0.94 for 4.5 $\mu$m. Likewise, as
shown by MM2015, the correction for the surface brightness within an
elliptical aperture $I_{\rm obs}$ can be made via a series expansion of Eq. 1:

$I_{\rm corr}(r_{\rm eq})=I_{\rm obs}(r_{\rm eq})\times(Ae^{-r_{\rm
eq}^{B}}+C)-ABr_{\rm eq}^{B-2}e^{-r_{\rm eq}^{B}}F_{\rm obs}(r_{\rm
eq})/(2\pi)$ (2)

These correction factors are nominally accurate to within $\sim$10%, however
they may also contain systematic errors given the limited kinds of models used
to derive them. We therefore now also include a second correction for each
galaxy in our tables. Using the GALFIT Sérsic fit parameters from the
fortcoming P4 analysis, we construct both the theoretical non-convolved and
the PSF-convolved 2D model images, MODEL and PSF $\otimes$ MODEL,
respectively. The convolution is made with the same symmetrized PSF as was
used in Comerón et al. (2018) obtained from the data in Hora et al. (2012). To
make sure the PSF has sufficient coverage, we continued the PSF tails so that
the final coverage was $450$″$\times 450$″. We then performed IRAF _ellipse_
fits to both the non-convolved and convolved model images, using the same
ellipticity and PA as for the observed profile. We denote these profiles as
$I_{\rm non-convolved}$ and $I_{\rm convolved}$, giving an estimate

$I_{\rm corr}(r_{\rm eq})=fI_{\rm obs}(r_{\rm eq})\times(I_{\rm non-
convolved}(r_{\rm eq})/I_{\rm convolved}(r_{\rm eq}))$ (3)

Here $f$ is the infinite aperture correction used to correct flux of extended
sources ($f=0.944$ for 3.6 $\mu$m and 0.937 for 4.5 $\mu$m; Reach et al.
2005). The correction for the total flux $F$ has the same form.

Note that in this analysis we ignore the small difference in the orientation
parameters used in P4 (Salo et al. 2015) compared to our current ones: they
were also estimated from IRAF _ellipse_ profiles, but instead of using a fixed
isophotal level, correspond to the visually selected outer isophote.

Fig. 4 shows example profiles using each kind of aperture correction, for
comparison. We note here that, for consistency across the full S4G+ETG sample,
we derive all P3 values using the aperture correction from MM2015; however,
because it is tailored to specific galaxies, we recommend any future analyses
using these profiles are done with the revised corrections. The choice has no
noticeable impacts on the results presented in this paper; it primarily
results in very slight deviations in our derived magnitudes, well below the
photometric uncertainty. It does not impact any of our broader scientific
conclusions. We make these new corrections available along with our data
release.

Finally, we correct all magnitudes and surface brightnesses for Galactic
extinction using the extinction tables from Schlafly & Finkbeiner (2011).
These corrections are typically extremely small ($<0.01$ mag), but we include
them in the interest of accuracy.

###  3.3 Asymptotic Magnitudes

Figure 5: Example curves of growth for two galaxies, one bright with a large
angular size (NGC 4564, _left_), the other faint with a small angular size (IC
271, _right_). Black points in the main panels show the total magnitudes
enclosed within each radius (bin width 2″). The inset panels show the local
gradients in the outskirts of the curves of growth. Red dashed lines show fits
to the linear parts of these gradients chosen by eye; the $y$-intercepts of
these red lines are the asymptotic magnitudes. Gray shaded regions show the
$1\sigma$ (dark gray) and $2\sigma$ (light gray) perturbations to the curves
of growth from adjusting the sky subtraction (see Section 3.3).

To derive asymptotic magnitudes, we follow MM2015, using each galaxy’s curve
of growth (hereafter, c.o.g.). We derive a given galaxy’s c.o.g. as the
cumulative sum of the galaxy’s flux within elliptical apertures of constant
$\epsilon$ and PA, grown in semi-major axis in steps of 2″ and corrected for
PSF effects using Eq. 1. Given the depth of the S4G+ETG, for most galaxies in
our sample the relationship between the local gradient and the magnitude
enclosed within each elliptical aperture becomes roughly linear near the
outskirts of the c.o.g. We therefore fit a line between the local gradient and
the magnitude in this linear regime; the $y$-intercept of this line is, by
definition, the asymptotic magnitude.

Fig. 5 shows examples of the c.o.g (top panels) and linear fits (inset panels)
for two galaxies with different radial extents and luminosities. Particularly
for galaxies with small angular sizes (hence few points in their surface
brightness profiles) and low luminosities (low S/N at all radii), it is not
always clear whether we are indeed reaching the flat part of the c.o.g.
Additionally, because many galaxies in our sample are ellipticals with steep
luminosity profiles (Sérsic $n>2$), the slopes become approximately linear
only at very large radii. For each galaxy, we therefore hand-picked the limits
within which to fit these lines.

This contributes some uncertainty into the final derived magnitudes, but this
uncertainty is quite small compared to the systematic uncertainty induced by
the sky subtraction. Because the c.o.g. relies on summing flux in concentric
apertures, a slight under- (over-)subtraction of the sky will result in a
positive (negative) gradient in the outermost areas of the galaxy disk, where
the sky level becomes comparable to the galaxy disk brightness, thus
potentially altering the $y$-intercept used to derive the magnitude. To derive
the uncertainties associated with the sky subtraction, we subtract from each
c.o.g. 1000 different sky levels randomly sampled from a normal distribution
$\sim\mathcal{N}($SKY, DSKY$)$ (see Section 3.1). We take the $1\sigma$
amplitude of the variation in the final derived asymptotic magnitudes as this
sky uncertainty, using our initial hand-picked fitting limits to derive
$y$-intercepts for every perturbation. The gray shaded regions in the sub-
panels of Fig. 5 show the effects of these perturbations, with the dark and
light gray regions showing the $1\sigma$ and $2\sigma$ variations on the
c.o.g., respectively. We provide these systematic uncertainties in addition to
the photometric uncertainties.

To measure absolute magnitudes, we derived distances (again following MM2015)
by either taking the average of all redshift-independent distance estimates
for each galaxy from the NASA Extragalactic Database (NED)222The NASA/IPAC
Extragalactic Database (NED) is funded by the National Aeronautics and Space
Administration and operated by the California Institute of Technology., if
available (60.4% of the new sample), or else by taking the average of all
redshift distances available from HyperLEDA (Paturel et al. 2003).

We derive stellar masses using these distances and asymptotic magnitudes
following the dust-corrected (Meidt et al. 2012) expression given by Eq. 8 of
Querejeta et al. (2015):

$\mathcal{M}_{*}=10^{8.35}\times F_{3.6}^{1.85}\times F_{4.5}^{-0.85}\times
D^{2}$ (4)

where $\mathcal{M}_{*}$ is in solar masses, $F_{\lambda}$ are the total fluxes
in Jy, and $D$ is the distance to the galaxy in Mpc. This expression was
derived using the independent component analysis technique demonstrated by
Meidt et al. (2012). This differs from MM2015, who used the value
$\mathcal{M}_{*}/L_{3.6}=0.56$ from Eskew et al. (2012). We found the impact
of this choice to be almost unnoticeable. One might expect some noticeable
differences given that Querejeta et al. (2015) had few ETGs available for
their calibration, but ETGs are also relatively dust-poor compared to LTGs
(e.g., Skibba et al. 2011), hence any dust corrections should be, and
seemingly are, slight.

In the following cases, the target galaxy overlaps strongly in projection with
another, much brighter object: ESO 400-30, NGC 3636, NGC 4269, NGC 5846A, and
PGC 93119. The c.o.g. method is incapable of correctly measuring asymptotic
magnitudes for such galaxies, as their flux cannot be disentangled from that
of their brighter neighbors. We will therefore obtain more accurate magnitudes
for these galaxies at a future date through 2D decompositions (Pipeline 4),
taking into account both the galaxy and its neighbor simultaneously. For now,
we urge caution regarding the magnitudes we provide for these particular
galaxies.

###  3.4 Concentration Indices

The radial distribution of light—the light concentration—is easily measurable
and, when used in conjunction with galaxy asymmetry and clumpiness (Conselice
2003), can serve as a quantitative alternative to visual morphological class
(e.g., Hubble 1926; Morgan & Mayall 1957; Morgan 1958; Okamura et al. 1984;
Kent 1985; Abraham et al. 1994; Bershady et al. 2000; Conselice 2003, 2006;
Taylor-Mager et al. 2007; Holwerda et al. 2014). In elliptical galaxies, the
light concentration—being explicitly related to the Sérsic index $n$ (e.g.,
Bershady et al. 2000; Trujillo et al. 2001)—correlates with galaxy luminosity
(e.g., Caon et al. 1993; Conselice 2003; Ferrarese et al. 2006; Mahajan et al.
2015; Graham 2019) and thereby with stellar mass, suggesting that the light
profile is somehow indicative of an elliptical galaxy’s evolutionary history.
It is thus a rather useful first measure of a given galaxy’s intrinsic
properties.

Again following MM2015, we measured the following two concentration
parameters: $C_{31}$ (de Vaucouleurs 1977) and $C_{82}$ (Kent 1985), defined
as:

$C_{31}=R_{75}/R_{25}$ (5) $C_{82}=5\log\,(R_{80}/R_{20})$ (6)

where $R_{x}$ is the radius containing $x\%$ of the total luminosity of the
galaxy (see also Fraser 1972). As in MM2015 and Bershady et al. (2000), to
avoid assumptions about the shapes of the light profiles, we measure $R_{x}$
from the c.o.g. Also, in constrast with, e.g., Bershady et al. (2000), we
extrapolate the galaxies’ total luminosities out to infinity rather than
measuring them within set apertures.

###  3.5 Galaxy Sizes

ETG mass–size relations are of considerable interest, particularly given their
connection to ETG formation and evolution mechanisms (e.g., Daddi et al. 2005;
Trujillo et al. 2007; van Dokkum et al. 2008; van der Wel et al. 2014).
Recently, the mass–size relation for galaxies generally has come under
additional scrutiny due to the recent rediscovery of sometimes abundant
populations of extremely extended but low-mass galaxies in clusters,
originally identified by Binggeli et al. (1985) in the Virgo Cluster, now
broadly referred to as ultra-diffuse galaxies (e.g., van Dokkum et al. 2015;
Koda et al. 2015; Mihos et al. 2015; Iodice et al. 2020). Depending on the
size metric used, such galaxies either appear as extreme outliers in the
mass–size relation, or show little difference from other dwarf galaxies
(Chamba et al. 2020), leading to the argument that a more physically motivated
definition of galaxy size (based on, e.g., star formation thresholds) is
called for (Trujillo et al. 2020, and references therein).

For our ETGs, we provide 3.6 $\mu$m and 4.5 $\mu$m half-light (or effective)
radii, as well as two isophotal radii: $R_{25.5}$ and $R_{26.5}$, denoting the
radii at which the galaxy’s surface brightness profiles reach 25.5 and 26.5 AB
mag arcsec-2, respectively. As NIR imaging is such a close tracer of stellar
mass, the effective radius is nearly equivalent to the half-mass radius, while
$R_{25.5,3.6}$ and $R_{26.5,3.6}$ denote the $\sim
3.6\,{\mathcal{M}}_{\odot}\,{\rm pc}^{-2}$ and $\sim
1.5\,\mathcal{M}_{\odot}\,{\rm pc}^{-2}$ surface mass density contours,
respectively (using the $\mathcal{M}/L$ calibration from Eskew et al. 2012).
We measure all radii along the galaxies’ major axes, i.e., all radii are
deprojected. We also derive isophotal radii both from the raw and inclination-
corrected surface brightness profiles.

We measure the half-light radii in the same manner as the concentration
parameters described above, taking them as the radii at which the c.o.g.
reaches 50% of the galaxies’ total flux (converted from the asymptotic
magnitudes). We measure the isophotal radii directly from the $\Delta
r=6$″surface brightness profiles described in Section 3.2 (i.e., using
elliptical apertures), to take advantage of the improved S/N. Given these
profiles’ rather coarse sampling, we interpolate the exact radii between the
two bins that enclose the desired isophote. We record as well the full
isophotal parameters at each radius (PA and $\epsilon$), interpolated in the
same manner. If the background noise precluded our ability to find either a
25.5 or 26.5 mag arcsec-2 isophote for a given galaxy, we recorded these
values as -999 (for 3.6 $\mu$m, 20 cases and 177 cases, respectively; for 4.5
$\mu$m, 178 and 204 cases, respectively). For the remainder of this paper we
focus on the $\mu_{3.6}=25.5$ isophotal radii, given their higher S/N.

All of these values—magnitudes, concentrations, sizes, and associated
uncertainties (including SKY, DSKY, and RMS)—will be made available in table
form on the NASA/IPAC Infrared Science Archive (IRSA). We now showcase them,
starting with a summary of observed parameters across the S4G+ETG population,
then moving to a variety of NIR scaling relations.

##  4 Overview: general trends

Figure 6: Distributions of the photometric parameters discussed in Section 3
(_top_), and parameter values plotted against morphological $T-$type
(_bottom_). From left to right, the parameters shown are: decimal logarithm of
stellar mass (in solar units); concentration parameter $C_{82}$ (equation 6);
concentration parameter $C_{31}$ (equation 5); logarithm of the
$\mu_{3.6,\textrm{AB}}=25.5$ mag arcsec-2 isophotal radius in kiloparsecs; and
logarithm of the half-light ($\sim$half-mass) radius in kiloparsecs. Solid red
lines show the median value of each parameter measured in bins of $\Delta
T=1$, while dashed red lines show the associated first and third quartiles.
Unfilled histograms and gray points show distributions for all the galaxies,
while filled histograms and black points show distributions for galaxies with
$T\leq 0$ and $T=11$ (ETGs).  Table 1: Numerical $T-$types summary

Stage | Numerical index ($T-$type)
---|---
cE | -6
E | -5
E+ | -4
S0- | -3
S00 | -2
S0+ | -1
S0/a | 0
Sa | 1
Sab | 2
Sb | 3
Sbc | 4
Sc | 5
Scd | 6
Sd | 7
Sdm | 8
Sm | 9
Im | 10
dE, dS0, dSph | 11
333References: de Vaucouleurs et al. (1991), Buta et al. (2015)

The top panels in Fig. 6 show the distributions of stellar mass (luminosity),
concentration, and size for the entire S4G+ETG sample, with ETGs ($T\leq 0$
and $T=11$; filled histograms) also displayed separately from the whole sample
(unfilled histograms). The bottom panels show individual galaxies’ parameter
values as a function of morphological $T-$type. ETGs are plotted as black
points, with the full population plotted underneath in gray. Red solid lines
show the median trends, with red dashed lines showing the first and third
quartiles.

We use $T-$types from Buta et al. (2015) for the original S4G sample, and new
classifications by R. Buta for the ETG extension galaxies. These
classifications follow the Comprehensive de Vaucouleurs revised Hubble-Sandage
(CVHRS; Buta et al. 2007) system, which follows the de Vaucouleurs revised
Hubble-Sandage system (de Vaucouleurs 1959) but is modified to include many
additional morphological features. For quick reference, we reproduce the
integer $T-$types associated with each broad morphological classification from
Buta et al. (2015) in Table 1. A sample of the full classification table can
be found in Table 2.

We focus here on the ETG population, in its context within the general
population of nearby galaxies. We replicate a few previous observations. For
example, the high-mass end of the galaxy stellar mass function tends to be
dominated by ETGs at low redshift (e.g. Baldry et al. 2004; Pozzetti et al.
2010; Calvi et al. 2012), and in our sample the bootstrap median mass
(measured as the median of the medians of 10000 randomly resampled mass
distributions, with replacement) of ETGs is indeed higher than that of either
LTGs or the total galaxy population (ETGs and LTGs combined):
$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})=10.15\pm 0.03$ for ETGs only, vs.
$9.63\pm 0.03$ for LTGs only, and $9.78\pm 0.02$ for all the galaxies, where
uncertainties are the $1\sigma$ bootstrap uncertainty (measured as the
standard deviation of the bootstrapped median values described above) on the
median (we use the bootstrap medians and uncertainties in all following
comparisons). Bright ETGs also have the highest concentration ($C_{82}=4.40\pm
0.04$ vs. $3.12\pm 0.02$ for the whole sample, and $C_{31}=5.13\pm 0.09$ vs.
$3.13\pm 0.02$). Nearby ETGs are also larger in isophotal size compared to
either LTGs or the whole galaxy population (median $R_{25.5}=12.74\pm 0.36$
kpc, vs. $R_{25.5}=9.91\pm 0.16$ kpc for LTGs, and $R_{25.5}=10.47\pm 0.15$
kpc for all the galaxies); combined with the previous point, this results in
them having smaller effective radii on average (median $R_{\rm eff}=2.30\pm
0.11$ kpc vs. $R_{\rm eff}=4.47\pm 0.08$ kpc for all the galaxies; a similar
trend was also noted by Jarrett et al. 2003).

More granular trends are also visible. S0s ($-3\leq T\leq 0$) are undermassive
compared to early-type spirals (Sa, Sab, Sb), with median
$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})=10.23\pm 0.03$ compared to
$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})=10.39\pm 0.01$. S0s are closer in
mass to Sc galaxies ($\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})=10.10\pm
0.01$). This too was shown previously, by van den Bergh (1998). Most of this
low-mass tendency is driven by early-type S0s (S0- and S00), which have median
mass $\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})=10.09\pm 0.05$, compared to
$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})=10.36\pm 0.01$ for S0+ and S0a
galaxies, making Sc and early-type S0s the most similar in mass. S0s are also
smaller on average in isophotal radius ($R_{25.5}=12.65\pm 0.38$ kpc) than all
but Sc ($R_{25.5}=12.62\pm 0.35$ kpc) and dwarf galaxies ($T\geq 8$, with
median $R_{25.5}=6.49\pm 0.16$ kpc), although S0s are more concentrated (hence
have smaller $R_{\rm eff}$) than such dwarf disks. Again, this tendency toward
small size among S0s is driven by the early-type S0s ($R_{25.5}=11.30\pm 0.72$
kpc, compared to $R_{25.5}=13.36\pm 0.63$ kpc for late-type S0s).

Many ETGs classified as $T=11$, which are morphologically dwarf galaxies, have
masses of $\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})>10.0$, suggesting that
they are not best classified as dwarfs in terms of their masses. The reason
for this is unclear; either their morphological classifications do not match
their high stellar masses, or the distances we use to derive their masses are
incorrect. The median distance to these $T=11$ galaxies is 17.6 Mpc; at this
distance, a dwarf galaxy with a diameter $D_{25}<1$ kpc would have a angular
size of $<12$″, hence should have been excluded from the sample via the
criterion of $D_{25}>1^{\prime}$ (Sec. 2), which biases the sample against
dwarf galaxies even within the selected distance constraints ($D\lesssim 40$
Mpc). However, because these $T=11$ galaxies tend to follow the rest of the
ETGs in each scaling relation we examine, we will defer a detailed discussion
of them for future work.

On average, LTGs and ETGs show clear disparities in concentration, as noted
already by de Vaucouleurs (1948). Specifically, this disparity sets in for
galaxies with $T\leq 3$ (Sb), with both concentration parameters rising
monotonically with decreasing T below this value (this trend is visible
already with the smaller ETG sample from Muñoz-Mateos et al. 2015, e.g., their
Fig. 9) save for a slight dip among early-type S0s ($-3\leq T\leq-2$). Díaz-
García et al. (2016) found similar morphological separation using stacked
profiles, which was echoed in the central stellar contribution to the circular
velocity curves. This likely reflects the similar steady increase of the
bulge-to-total luminosity ratio (B$/$T) with decreasing morphological type
observed by, e.g., Laurikainen et al. (2010).

##  5 Scaling relations

Here we showcase several ETG scaling relations derived from the P3 isophotal
parameters. We begin with concentration, then move to size, surface
brightness, and finally discuss integrated $m_{3.6}-m_{4.5}$ colors. At the
end of each sub-section, we briefly summarize our results.

###  5.1 Concentration

Figure 7: Correlation between concentration and stellar mass. Here and in all
subsequent figures with this color scheme, ETGs ($T\leq 0$) are color-coded by
$T-$type, while all S4G galaxies are shown in grey. Dwarf ETGs ($T=11$) are
shown in cyan. The dashed black line shows a linear fit between
$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})$ and $\log\,(n)$, converted from
$C_{82}$, for ETGs (Eq. 7). The red dashed line shows a similar fit, but here
$n$ was estimated for each ETG via single-component 2D decompositions (Eq. 8).
Figure 8: Investigating the divergence between the Sérsic $n$ derived from
$C_{82}$ and that measured from single-component 2D decompositions. The _top-
left_ panel shows concentration parameter vs. Sérsic index, the latter derived
from single-component 2D decompositions for all S4G galaxies. The solid black
line shows the theoretical behavior of $C_{82}$ for Sérsic profiles of varying
$n$. Here and in subsequent panels, the dashed black line shows the median of
points in equal-width bins (bins with fewer than five points are ignored). In
the _bottom-left_ panel, we show the difference between our asymptotic
magnitudes and those derived from decompositions (negative numbers mean the
asymptotic magnitudes are fainter). The solid black line here and in the
following panels shows a value of 0 (no difference). The _top-right_ panel
shows the difference between $R_{20}$ derived from single-component
decompositions and those derived from our curves of growth for 3.6 $\mu$m
imaging. The vertical dotted line shows the size of three IRAC 3.6 $\mu$m
resolution elements. The _bottom-right_ panel is the same as the _top-right_
panel, but shows the behavior of $R_{80}$. ETG data points are multi-colored
(with dwarfs in cyan), while LTGs are shown in gray.  Figure 9: Left panel:
difference between single-component decomposition total magnitudes and
asymptotic magnitudes (P3; MM2015), as a function of curve of growth
concentration parameters (P3) for the original S4G sample. Right panel: as the
left panel, but showing the difference between multi-component decomposition
total magnitudes (P4; Salo et al. 2015) and asymptotic magnitudes. While both
decomposition types show slight offsets from the asymptotic magnitudes on
average, as well as downward trends (brighter magnitudes) with increasing
concentration, the downward trend in the single-component decompositions is
much stronger than that in the multi-component decompositions. Also, at least
$\sim 0.06$ magnitudes of offset in the multi-component magnitudes arise
because we applied no infinite aperture correction to the total flux (see Eq.
3, and Reach et al. 2005).

Particularly for high-redshift galaxies, where individual morphological
features are unresolvable, light concentration is useful for separating ETGs
and LTGs (e.g., Morgan 1958; de Vaucouleurs & Agüero 1973; Okamura et al.
1984; Kent 1985; Conselice 2003). Concentration also scales with luminosity,
showing an almost log-linear relationship for ETGs (e.g., Schombert 1986;
Binggeli & Cameron 1991; Graham et al. 2001; Conselice 2003; Kauffmann et al.
2004), which we demonstrate as well in Fig. 7. Structural parameters such as
concentration in turn show little dependence on local environment, but
environment correlates well with star formation rate (e.g., Kauffmann et al.
2004; van der Wel 2008), suggesting that the structure of the bright, visible
parts of galaxies was in place early on in most galaxies (Kauffmann et al.
2004). Still, secular evolution and subsequent star formation episodes can
modify galaxy structure over time, so even under this early formation scenario
one might expect that galaxies with bars, spiral arms, and other dynamical
structures capable of moving mass (e.g., Sellwood & Binney 2002), as well as
those with ongoing, wide-spread star formation, might show the largest scatter
in the concentration–stellar mass relation. Indeed, Méndez-Abreu et al. (2021)
found that disk galaxy bulges evolve little over cosmic time, while their
disks can change dramatically (depending on the central concentration of the
galaxy, e.g., Athanassoula et al. 2005). Additionally, Sheth et al. (2008)
showed that the distribution of bars among massive S0s at $z=0.84$ was quite
similar to that of massive S0s at $z=0$, suggesting that the disk structures
were in place early on in these galaxies; lower-mass LTGs, by contrast, appear
to have formed their bars much more gradually during this time, but evidently
continued forming stars in their disks much longer than their S0 counterparts.
We see evidence for this in Fig. 7. Ellipticals (navy blue points) show a
fairly tight correlation; S0s (yellow and gold points) of the same stellar
mass scatter toward low $C_{82}$ values, with the tightest scatter found at
the high-mass end; and the scatter and distribution of $C_{82}$ values among
LTGs is quite similar to that of S0s, with the highest scatter at the lowest
stellar masses.

Though this correlation appears roughly log-linear, it does show some mild
curvature, suggesting the true log-linear correlation is between stellar mass
and Sérsic index (e.g., Caon et al. 1993; Young & Currie 1994; Graham 2001;
Ferrarese et al. 2006; Savorgnan et al. 2013; Graham 2019). We demonstrate
this via the black dashed curve in Fig. 7, which we derived by first
converting our measured values of $C_{82}$ for ETGs (colored points) to their
corresponding values of $n$ (see, e.g., Graham & Driver 2005; Janz et al.
2014), then fitting the resulting
$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})$–$n$ relationship as a line:

$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})=2.47\log\,(n)+9.03$ (7)

We then converted this linear fit back to a relation between
$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})$ and $C_{82}$ using the relation
between $n$ and $C_{82}$, which produces the aforementioned black dashed
curve. Using the single-component decompositions described in Sect. 2, we also
derived empirical values of $n$ for all ETGs for comparison. These too showed
a linear relationship with stellar mass, which we overplot in Fig. 7 via the
red dashed line (again, converted to $C_{82}$). This has the form:

$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})=1.52\log\,(n)+9.21$ (8)

which has a similar zeropoint but a slope different by nearly a factor of
$\sim 1.5$. Even though we are using the PSF-corrected radial profiles to
derive $C_{82}$, the values of $n$ implied by our concentration parameters
thus agree poorly with those derived via single-component 2D fits.

We expand on this in the top-left panel of Fig. 8, which shows our $C_{82}$
values plotted against the values of $n$ for all our ETGs derived from single-
component GALFIT decompositions. The solid black line here shows the
theoretical relationship between the $C_{82}$ and $n$, while the dashed line
(here and in all subsequent panels) shows the median trend for all the
galaxies (again shown in grey) as measured within consecutive equal-width
$C_{82}$ bins. Clearly, at the high concentration (high mass) end, the
measured values of $C_{82}$ are too low compared to the theoretical
expectation from the measured $n$.

Venhola et al. (2018) found superficially similar behavior for galaxies in the
Fornax Cluster region using Fornax Deep Survey (Peletier et al. 2020)
$u^{\prime}$-, $g^{\prime}$-, $r^{\prime}$-, and $i^{\prime}$-band imaging
(see their Fig. 13). Like us, Venhola et al. (2018) observed that the Sérsic
$n$ values from single component GALFIT decompositions were systematically
higher than the theoretical values of $n$ estimated from $C_{82}$. In their
case, the aperture magnitudes and the corresponding $C_{82}$ values were
calculated using a region within one Petrosian radius, which was also taken
into account in their theoretical $n$ vs. $C_{82}$ relation (the $n$
corresponding to a given $C_{82}$ is larger than if infinite extent is
assumed). They concluded that the offset, seen for all values of $n$, was due
to resolution effects: the GALFIT decompositions included PSF-convolution,
whereas the aperture photometry did not444Note that the galaxies shown in Fig.
13 in Venhola et al. (2018) are mainly background galaxies at distances much
larger than the 20 Mpc distance of the Fornax Cluster, making the resolution a
more pronounced problem than for our sample, even if the FWHM of the FDS
observations was smaller (about 1″ instead our _Spitzer_ 2″)..

We investigate this explicitly in the right two panels of Fig. 8. In these two
panels, we show the values
$\Delta\log\,(R_{x})=\log\,(R_{x,\leavevmode\nobreak\ 1\leavevmode\nobreak\
\textrm{comp.}})-\log\,(R_{x,\leavevmode\nobreak\ \textrm{c.o.g.}})$, where
$x$ is the fraction of light contained within the radius, “1 comp.” represents
values derived from single-component decompositions, and “c.o.g.” represents
values we derive from the radial curves of growth. We show both components
used for $C_{82}$, the 20% (top-right panel) and 80% (bottom-right panel) flux
radii. Vertical dotted lines in each panel show three times the 3.6 $\mu$m
FWHM; if resolution was purely at fault for the diverging $C_{82}$–$n$
profiles, because $R_{20}$ often falls within this limit, it might show the
strongest method-to-method difference. However, while $\Delta\log\,(R_{20})$
is systematically slightly positive (albeit with large scatter), the offset is
small compared to that of $\Delta\log\,(R_{80})$, for which resolution’s
impact should be mild. Resolution does not thus appear to be the driving
factor here.

Typically, we find $\Delta\log\,(R_{80})>0$, indicating that single-component-
fit $R_{80}$ are systematically $\sim 1.3$ times larger on average (albeit
with large scatter) than those derived from the curves of growth. In the
bottom-left panel of Fig. 8, we also show the difference between the
magnitudes estimated from single-component Sérsic fits and our asymptotic
magnitudes. These tend to be negative, meaning single-component fit magnitudes
are systematically $\sim 0.2$ mag brighter than asymptotic magnitudes. This,
combined with the larger values of $R_{80}$, suggests that single-component
fitting often overestimates the flux in the galaxies’ outskirts. This is
expected for disk galaxies, which often show down-bending breaks in their
surface brightness profiles (e.g., Pohlen & Trujillo 2006; Erwin et al. 2008).
However, from the color distribution of the points, it is clear that the
magnitude of the difference is similar for all ETG morphological types.
Various studies have shown that elliptical galaxies, much like disks, are
poorly characterized by single Sérsic component models (e.g., Schombert 1986;
Hopkins et al. 2009), which may result in the behavior we see here.

We provide further evidence for this in Fig. 9. In each panel, we show the
difference between decomposition magnitudes—both single- and multi-
component—and asymptotic, c.o.g-derived magnitudes, as a function of $C_{82}$,
for the original S4G sample. Asymptotic magnitudes and $C_{82}$ here come from
MM2015, while multi-component decomposition magnitudes come from Salo et al.
(2015). Both decomposition magnitudes show slight systematic offsets from the
asymptotic magnitudes, as well as trends with $C_{82}$: the higher the
concentration, the brighter the estimated decomposition magnitude relative to
the asymptotic value. However, this trend is much stronger for the single-
component decompositions than for the multi-component decompositions.
Additionally, $\sim 0.06$ magnitudes of the offset in the multi-component
magnitudes arises because these magnitudes were derived directly from the
images, without applying the $3.6\mu$m infinite aperture correction of $0.944$
(see Eq. 3, and Reach et al. 2005). This confirms our suspicion that single-
component fits are not as reliable as either multi-component fits or the curve
of growth method for estimating total galaxy magnitudes. Combined with Fig. 8,
it is clear that most of the systematic errors in single-component fits occur
in the galaxy outskirts.

In summary, we reproduce well the known trend between $C_{82}$ and stellar
mass for ETGs, thereby reproducing the trend between $n$ and stellar mass
(Fig. 7). In comparing our values of $C_{82}$ derived from radial profiles to
those derived from single-component GALFIT decompositions, we find that such
decompositions tend to overestimate the concentration and the total flux of
these galaxies (Fig. 8). Most of this over-estimate of flux occurs in the
galaxy outskirts, in a manner that correlates with the galaxies’
concentration. Multi-component decompositions fare much better—total magnitude
estimates from these agree more closely with radial profile–derived asymptotic
magnitudes, with only small systematic differences due to galaxy concentration
(Fig. 9).

###  5.2 Mass–size

Figure 10: Effective radius mass–size relation. The _left_ panel shows our
effective radius values (derived from 3.6 $\mu$m imaging) plotted against
stellar mass, with a linear fit to ETGs with masses
$>10^{10.7}\mathcal{M}_{\odot}$ shown in black and the $z=0.25$ ETG mass–size
relation from Newman et al. (2012) shown in green (labeled N12). The _right_
panel is the same as the left, but here $R_{\rm eff}$ has been adjusted by a
factor of $\sqrt{b/a}$ to adjust the values to those measured using circular
apertures, following Shen et al. (2003), whose fit is overplotted in blue
(labeled S03). ETG data points are multi-colored (with dwarfs in cyan), while
LTGs are shown in gray.  Figure 11: Isophotal mass–size relation. The _left_
panel shows the relation between $R_{25.5,3.6}$ and stellar mass, in which
$R_{25.5,3.6}$ was derived from the raw surface brightness profiles. The
_right_ panel shows the same but with $R_{25.5,3.6}$ corrected for
inclination, measured from the outermost isophote shapes’ axial ratios. ETG
data points are multi-colored (with dwarfs in cyan), while LTGs are shown in
gray. The arrow indicates the galaxy IC 3413 (see text).  Figure 12: Comparing
$V/\sigma_{e}$ values (denoting those measured at 1 $R_{\rm eff}$) from the
ATLAS3D survey to isophotal axial ratios measured at $R_{25.5}$ for galaxies
matching between the S4G+ETG and ATLAS3D samples. Orange point denote
elliptical galaxies, while black points denote S0s. The red dashed line shows
a linear fit to all points. Points circled in red are edge-on disks (denoted
‘spindle’ by Buta et al. 2015), with unreliable axial ratios.

The mass–size relation for ETGs has been crucial to understanding the growth
of ETGs throughout cosmic history (e.g., Daddi et al. 2005; Longhetti et al.
2007; Toft et al. 2007; Trujillo et al. 2007; Hopkins et al. 2008; van Dokkum
et al. 2008; Hopkins et al. 2009; van der Wel et al. 2009; Szomoru et al.
2012; Andreon et al. 2016). A series of gas-poor, low mass-ratio mergers can
explain elliptical galaxies’ rapid growth in size compared to stellar mass
(e.g., Nipoti et al. 2003; Naab et al. 2009; Szomoru et al. 2012), though
dissipational processes seem necessary to explain specific aspects of their
evolution, e.g., the tilt in the fundamental plane (Hopkins et al. 2008).
Typically, size is measured via the effective radius, given the utility of
Sérsic functions in describing their light profiles (though see Sect. 5.1),
but it has long been known that all galaxies also show correlations between
isophotal size and the absolute magnitude (or, modulo the mass-to-light ratio,
stellar mass; Schombert 1986; Binggeli & Cameron 1991). These correlations are
always positive, suggesting that whatever drives the mass evolution of most
galaxies drives the size evolution in a similar way.

MM2015 showed the stellar mass–$R_{\rm eff}$ relation for the initial
2352-galaxy S4G sample (their Fig. 15). They demonstrated a clear
morphological segregation, with ETGs occupying the lower-right (low-
luminosity, compact) corner of the point distribution and the latest type
galaxies (Sm, Irr) occupying the lower-left (low luminosity, extended). We
broadly reproduce this trend in Fig. 10. Gray points here are LTGs, while ETGs
are color-coded by morphological type. Among ETGs, ellipticals and S0s are
mostly intermingled, though for stellar masses $\gtrsim
10^{10}\mathcal{M}_{\odot}$, S0s show a mild tendency toward higher $R_{\rm
eff}$ than ellipticals at equal stellar mass. ETGs as a population form a
visibly tighter stellar mass–$R_{\rm eff}$ relation than LTGs, even including
the increase in dispersion emerging below stellar masses $\lesssim
10^{10.5}\mathcal{M}_{\odot}$.

Only the high-mass end of the relation is roughly linear: curvature in the
mass–$R_{\rm eff}$ relation is known to result from the dependence of light
profile shapes on stellar mass (Sect. 5.1), alongside structural differences
between dwarf and massive ETGs (e.g., Graham & Guzmán 2003; Boselli et al.
2008; Janz & Lisker 2008; Kormendy et al. 2009). Massive galaxies are also not
well-described by single Sérsic profiles (Figs. 8 and 9), likely leading to
divergences even from the theoretical curved relationship described by Graham
& Guzmán (2003) and others. For comparison with past studies, however, in Fig.
10 we show a linear fit to the S4G mass–size relation via the black lines,
including only those ETGs with stellar masses $>10^{10.7}\mathcal{M}_{\odot}$
(126 galaxies total). This has the following form:

$\log\,(R_{\rm
eff}/\rm{kpc})=0.51\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})-5.02$ (9)

We overplot similar fits from two previous studies of nearby galaxies, derived
above similar stellar mass thresholds: in the left panel, we show the fit for
$z=0.06$ ETGs from Table 1 of Newman et al. (2012); and in the right panel, we
show the ETG sample fit from Eq. 17 of Shen et al. (2003). These have the
following forms:

$\log\,(R_{\rm
eff}/\rm{kpc})=0.57\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})-5.73$ (10)

for the Newman et al. (2012) relation (labeled N12 in Fig. 10), and:

$\log\,(R_{\rm
eff}/\rm{kpc})=0.56\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})-4.46$ (11)

for the Shen et al. (2003) relation (labeled S03 in Fig. 10) . We have
corrected our rightmost panel effective radii by a factor of $\sqrt{b/a}$
(with axial ratios measured at $R_{25.5,3.6}$) for this comparison, as Shen et
al. (2003) used circular apertures for their photometry (see MM2015).

Our fit is very close to those of both Newman et al. (2012) and Shen et al.
(2003). Our derived slope is slightly shallower than those of either study,
and our intercept is very close to that of Shen et al. (2003) but noticeably
smaller than that of Newman et al. (2012). While this may have physical
significance—for example, the change in intercept compared with that of Newman
et al. (2012) could imply a size evolution since $z=0.06$—we note that both
Newman et al. (2012) and Shen et al. (2003) used Sérsic fits (via GALFIT and
from the radial profiles, respectively) to derive their masses and radii, so
the differences could also be explained by differences in methodology (see
Sect. 5.1). For example, van der Wel et al. (2014) found a substantially
different slope of 0.75, which they attribute to differences in sample
selection and their use of Petrosian (Petrosian 1976) half-light radii.
Additionally, the aforementioned studies used visible light photometry to
estimate stellar mass for their $z=0$ samples, which can be more subject to,
e.g., metallicity effects (e.g., Meidt et al. 2014). Within these ambiguities,
therefore, we find our fits broadly consistent with those of previous studies,
confirming the $z=0$ calibration for the observed tendency for the slope in
this relation to remain stable since as early as $z=3$ (e.g., van der Wel et
al. 2014).

We show the stellar mass–isophotal size relation for S4G+ETG galaxies in Fig.
11, using $R_{25.5,3.6}$ for the isophotal radii. This relation looks similar
when using $R_{26.5,3.6}$ or the isophotal radii measured in 4.5 $\mu$m, but
these have higher scatter due to the lower S/N in the latter. The left panel
shows the relation using unmodified surface brightness profiles, while the
right panel shows the relation after correcting the surface brightness
profiles to their face-on equivalents. We do this by multiplying the flux in
each radial bin by the axial ratio, following Kent (1985); we note that, while
such a correction may be an appropriate inclination correction for thin disks,
for ETGs, many of which have spheroidal or triaxial shapes, this correction
should be considered an isophotal circularization rather than a true
inclination correction. The scatter decreases substantially once this
correction is applied. Extreme outlier points in the rightmost panel (mostly
LTGs) are all nearly edge-on galaxies with unreliable axial ratios. Outliers
among the ETGs have a muddier origin; for example, the point at
($\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})=9.657$,
$\log\,(R_{25.5,3.6})=1.129$, shown by the arrow) is IC 3413, a seemingly
undisturbed, mildly inclined, and mostly featureless galaxy, though it is a
Virgo Cluster member. Deeper imaging of some of these outliers could reveal
evidence of tidal interactions, which may move them away from their expected
positions in this relation.

MM2015 show the stellar mass–size relation for the initial S4G sample in their
Fig. 14, again demonstrating clear morphological segregation, which accounts
for much of the scatter in the left panel of Fig. 11. Once an inclination
correction is applied555While MM2015 did apply some manner of inclination
correction to their isophotal radii, the scatter in their relation much more
closely resembles our uncorrected relation’s scatter. Our axial ratio
correction reduces the scatter significantly, so it would appear to be the
more appropriate choice., this morphological segregation reduces, as the wider
array of axial ratios displayed by thin disk galaxies results in stronger
corrections on average compared to the vertically thicker ellipticals. Yet,
with only an axial ratio correction applied, the scatter reduces just as much
for elliptical galaxies as for S0s, surprising given that ellipticals are
usually oblate systems. Possibly this correction is as effective for
elliptical galaxies due to their significant rotational support (e.g.,
Cappellari et al. 2007), which implies that their axial ratios are in some
ways reflective of their instrinsic alignments. We show this explicitly in
Fig. 12, in which we plot the axial ratios of galaxies overlapping the S4G+ETG
and the ATLAS3D (Cappellari et al. 2011a; Emsellem et al. 2011a) surveys
against their rotational velocity–to–velocity dispersion ratios
($V/\sigma_{\rm e}$, measured at one effective radius; Emsellem et al. 2011b).
Even with the ambiguity presented by nearly edge-on galaxies (marked with red
circles), there is a clear downward trend, as expected if more rotationally
supported galaxies are also intrinsically vertically thinner. We show
ellipticals as orange points and S0s as black points—the downward trend is
visible using both classes of galaxies.

The tightness in the stellar mass–isophotal radius relation is noteworthy, and
has been mentioned before in studies utilizing different isophotes and
photometric bands (e.g., Schombert 1986; Binggeli & Cameron 1991; Saintonge &
Spekkens 2011; Hall et al. 2012). Indeed, several papers have recently begun
to explore it in the context of star formation thresholds, assessing the shape
and scatter of the relation for the radius at which the stellar mass surface
density reaches 1 $\mathcal{M}_{\odot}$ pc-2 (Chamba et al. 2020; Trujillo et
al. 2020; Sánchez Almeida 2020), which is comparable to previous estimates of
star-formation thresholds using gas surface density (3–10
$\mathcal{M}_{\odot}$ pc-2, e.g., Schaye 2004). The scatter in the right panel
of Fig. 11 is roughly equivalent to that found by Trujillo et al. (2020), who
used $g$\- and $r$-band imaging from the IAC Stripe82 Legacy Project (Fliri &
Trujillo 2016; Román & Trujillo 2018) to derive stellar masses. Comparing to
their Fig. 4, the curvature we find in our relation is even quite similar, as
is the morphological segregation. This is not particularly surprising, as 3.6
$\mu$m imaging is a very close tracer of stellar mass—the 1
$\mathcal{M}_{\odot}$ pc-2 radius explored by these papers corresponds roughly
to $R_{27,3.6}$. Sánchez Almeida (2020) demonstrated that the likely origin of
this low scatter lies in the anticorrelation between $R_{\rm eff}$ and $n$,
and found that it minimizes when using the 2.4 $\mathcal{M}_{\odot}$ pc-2 mass
surface density radius (very close to our $R_{25.5,3.6}\sim
3.5\mathcal{M}_{\odot}$ pc-2). Still, the physical origin behind such an
optimal value is unknown. More careful analysis of this relation is required
(akin to that done by, e.g., Ouellette et al. 2017), which will be the subject
of a separate paper.

In summary, we find good agreement with previous studies in the stellar-
mass–$R_{\rm eff}$ relation for high-mass ETGs, modulo methodological
differences. We also find that the stellar-mass–isophotal radius relation,
once the surface brightnesses are corrected for inclination, has very little
scatter even among elliptical galaxies, which serves as a secondhand
demonstration of their rotational support.

###  5.3 Surface Brightness

Figure 13: All ETG 3.6 $\mu$m surface brightness profiles (fixed isophotal
parameters, $\Delta r=2$″; see Section 3.2), normalized by effective radius.
The left panel shows profiles color-coded by total stellar mass, while the
right panel shows profiles color-coded by morphological $T-$type. Surface
brightness is converted to mass surface density in the right panel as well,
for convenience. Data quality at very low surface brightness is poor, hence
all profiles are truncated at $\mu_{3.6,\textrm{AB}}=27.5$.  Figure 14: 3.6
$\mu$m central surface brightnesses ($\mu_{0,3.6}$) vs. stellar mass. The
_left panel_ shows $\mu_{0,3.6}$ derived from single-component 2D
decompositions, while the _right panel_ shows $\mu_{0,3.6}$ derived by fitting
generalized single Sérsic functions to each galaxy’s radial surface brightness
profile (fixed isophotal parameter, $\Delta r=2$″, with aperture correction
applied; see Section 3.2). Galaxies with fitted $n>7$ are circled in blue.
Linear fits to the two distributions are shown via the red dashed lines,
excluding galaxies with $\mathcal{M}_{*}/\mathcal{M}_{\odot}<8$ and fitted
$n>7$). Points lying above the axis limits are shown as up-pointing arrows;
each of these had fitted $n>7$. ETG data points are multi-colored (with dwarfs
in cyan), while LTGs are shown in gray.

While concentration parameters provide an easy-to-measure approximation of
galaxy structure, surface brightness profile shapes contain much more detailed
information, including the true central surface brightness, $\mu_{0}$. In a
typical hierarchical accretion scenario, a galaxy’s core will form early,
during a cosmological era in which the horizon was small and therefore mergers
were more frequent, whereas the outskirts form later from less frequent
mergers (Frenk et al. 1985) and from reorganization of matter due to secular
processes (e.g., Kormendy & Kennicutt 2004). The former process builds the
very dense innermost structures that $\mu_{0}$ traces (classical bulges;
Sandage 1961). Subsequent secular processes can lead to mass growth in galaxy
cores as well, if the bar induces angular momentum loss in the gas and dust
(e.g., Athanassoula 1992; Maciejewski et al. 2002; Carles et al. 2016), but
the resulting structures are often less dense and more disklike (pseudobulges
or disky bulges, in the nomenclature of Kormendy & Kennicutt 2004;
Athanassoula 2005, respectively) than classical bulges. In the absence of star
formation, the reorganization of mass in the disk then depends on the growth
of bars, whose pattern speeds set the locations of resonances. This in turn
depends on a large number of properties of the galaxy (Athanassoula 2003),
including the mass and size of the central mass concentration (e.g., Sellwood
1980; Norman et al. 1996; Athanassoula et al. 2005; Debattista et al. 2006;
Kataria et al. 2020), which is often small or negligible in S0s (Laurikainen
et al. 2007, 2010). Elliptical galaxies lack large-scale bars and the disks
necessary to form them; they also have the densest cores, which show little
evolution over cosmic time (e.g., Szomoru et al. 2012), implying a two-phase
evolutionary process, with early violent growth in the core followed by a
slower, merger-fueled growth in the outskirts that had little impact on
subsequent core evolution. In the highest-mass ellipticals, however,
subsequent core evolution can occur through scouring effects of SMBH mergers
and the subsequent reorganization of matter while the systems relax (e.g.,
Milosavljević & Merritt 2001). Modulo resolution effects (for example, the
resolution of our profiles is likely too coarse to accurately probe the SMBH’s
sphere of influence), surface brightness profile shapes can reflect all of
these complex histories.

For an overall picture, Fig. 13 shows the radial surface brightness profiles
of every ETG ($T\leq 0$) in the S4G+ETG sample, with radii normalized by
$R_{\rm eff,3.6}$. We show here the fixed isophotal parameter, 2″ radial bin
width curves described in Section 3.2. In the left panel, each curve is color-
coded by the galaxy’s total stellar mass, while in the right panel the curves
are color-coded by $T-$type. Additionally, to reduce noise in the plot, we
truncate each curve at $\mu_{3.6}=27.5$.

Mass segregation is clear in the left panel, with high-mass galaxies typically
showing higher surface brightnesses than low-mass galaxies at all radii. High-
mass galaxies are also larger than low-mass galaxies relative to their
effective radii (see also Section 5.2). This, combined with the tendency for
high-mass ETGs to have more concentrated light profiles (Fig. 7), leads to
high-mass ETGs showing longer profiles in this space. By contrast, the
profiles are well-mixed by morphological $T-$type, suggesting that stellar
mass is the primary driver behind ETG surface brightnesses. A more detailed
examination of individual profiles is evidently necessary to disentangle the
various contributions to the shapes of these profiles, including bars, rings,
disk breaks (e.g., Pohlen & Trujillo 2006), and cored vs. Sérsic ellipticals
(e.g., Kormendy & Bender 1996; Faber et al. 1997).

We thus expand on the stellar-mass–surface brightness relation in Fig. 14, in
which we plot only the central surface brightnesses $\mu_{0,3.6}$ of each
galaxy against their stellar masses. As in Fig. 8, we show $\mu_{0,3.6}$
derived both from single-component 2D decompositions, as well as those derived
through single-component least-squares fitting to the curves shown in Fig. 13.
We correct for PSF effects in the 1D curve fits by using the aperture-
corrected profiles described in Section 3.2.

Either method demonstrates clear stellar mass–$\mu_{0}$ trends, such that
higher mass galaxies have higher central surface brightnesses, in agreement
with, e.g., Graham (2019, and references therein). Linear fits to these
relations result in the following (excluding galaxies with
$\mathcal{M}_{*}/\mathcal{M}_{\odot}<8$ and fitted $n>7$):

$\mu_{0,3.6,\textrm{2D}}=-3.78\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})+52.99$
(12)

for the 2D decompositions, and

$\mu_{0,3.6,\textrm{1D}}=-3.23\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})+47.49$
(13)

for the values derived from curve-fitting. The two fits are somewhat similar
despite the differing methods used to estimate $\mu_{0,3.6}$, though the
profile-fit slope is noticeably shallower than the decomposition slope.

In both cases, many galaxies with masses between $\sim
10^{9.5}\mathcal{M}_{\odot}$ and $\sim 10^{11}\mathcal{M}_{\odot}$ lie far
above the relation, meaning they have large estimated central light
enhancements. Most of these have very high fitted Sérsic indices ($n>7$, which
we have marked using blue circles). Examination of individual profiles shows
that these galaxies typically have strong central peaks indicative of large,
bright bulges (relative to their disks), which thus dominate the profiles’
curvature. The distribution of these galaxies’ $T-$types is also heavily
weighted toward disks, with the strongest peak at $T=0$. Due to the complex
structure of disk galaxies (which can host bulges, bars, disks, lenses, rings,
nuclear bars, nuclear rings, and even active galactic nuclei), the very high
fitted values of $n$ and $\mu_{0,3.6}$ we find for these galaxies may mostly
result from poor fits to the galaxies’ total light profiles.

Among ellipticals, we see some hint of a light deficit for many galaxies with
masses $>10^{10.5}\mathcal{M}_{\odot}$. The distribution of a fair number of
points falls well below the fitted line in this mass regime, seemingly
leveling out at a central surface brightness of $\sim 15$ (this is clearer in
the right panel of Fig. 14). These may be examples of cored ellipticals, which
do tend to be high-mass galaxies (e.g., Kormendy & Bender 1996; Emsellem et
al. 2007; Kormendy et al. 2009; Dullo & Graham 2012), though we note that many
ellipticals in this mass range lie roughly on or above the best-fit relation
as well. Whether the massive galaxies that fall on the relation are truly
uncored or not is unclear given the large uncertainties inherent to our
estimates of $\mu_{0}$, as evidenced by the many poor, $n>7$ fits, described
above; higher resolution profiles for the cores, in conjunction with more
detailed fitting procedures, are needed to verify this.

Thus, while we broadly reproduce the known relation between stellar mass and
$\mu_{0}$, evidently great care is required when estimating the central
surface brightnesses of galaxies. As we demonstrated with the concentration
index in Sect. 5.1, single-Sérsic component fits can be inaccurate and
misleading.

Figure 15: Mean surface brightness within the effective radius (_left panel_)
and within 1 kpc (_right panel_) vs. stellar mass. The solid black lines in
both panels show linear fits to all S4G+ETG galaxies. The dashed red lines in
the right panel show fits to only ETGs, separated at
$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})=10.4$ (see text). ETG data points
are multi-colored, while LTGs are shown in gray.

Given this, we also investigate scaling relations using the mean surface
brightnesses within fixed radii. We show two scaling relations of this nature
in Fig. 15: the mean surface brightness within $R_{\rm eff}$,
$\langle\mu_{3.6,\rm{eff}}\rangle$ (left panel); and the mean surface
brightness within 1 kpc radius, $\langle\mu_{3.6,1\rm{kpc}}\rangle$ (right
panel), both plotted against stellar mass. The former relation has been
demonstrated since photometry of low-mass, low surface brightness galaxies
became possible, and was always known to have a fairly high scatter (e.g.,
Impey et al. 1996). A more recent demonstration of this relation came from
Sedgwick et al. (2019), who found a correlation between effective $r$-band
surface brightness and stellar mass for core-collapse supernova host galaxies
with a slope of $\sim-1.35$. Our regression line is shown in Fig. 15, and has
the following form:

$\mu_{3.6,\rm{eff}}=-1.79\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})+37.90$
(14)

where the steeper slope than that of Sedgwick et al. (2019) likely arises from
the difference in photometric band and galaxy sample (core-collapse SNe hosts
typically being star-forming galaxies). In NIR, ETGs and LTGs both follow
similar relations, although below
$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})\sim 10$ ETGs trend slightly
higher in effective surface brightness than LTGs of the same stellar mass.
This is to be expected, however, as ETGs in this mass range also have smaller
$R_{\rm eff}$ on average than LTGs (Fig. 10) and higher $C_{82}$ (Fig. 7).
Much of the scatter in this relation thus might arise from the large range of
relative physical radii probed by $R_{\rm eff}$ for galaxies with a variety of
photometric profiles.

A more consistent choice of physical radius yields a much tighter correlation,
as we demonstrate in the rightmost panel of Fig. 15 via the
$\langle\mu_{3.6,1\rm{kpc}}\rangle$–$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})$
relation. This, too, is a known feature of ETGs; Saracco et al. (2017), for
example, found a tight relation between stellar mass and the mass surface
density within 1 kpc ($\Sigma^{*}_{1\rm{kpc}}$) among ETGs at a wide range of
redshifts (see also Saracco et al. 2012; Tacchella et al. 2016; Arora et al.
2021), indicating that the average core densities of these galaxies have not
evolved much during the past $\sim$10 Gyr (echoing, in a different manner, the
results found by Szomoru et al. 2012, and references therein).

The linear fits shown by the red dashed lines mirror those done by Saracco et
al. (2017), who found a kink in the
$\Sigma^{*}_{1\rm{kpc}}$–$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})$
relation at $\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})\sim 10.4$. Here we
fit only ETGs, with $T\leq 0$, for better comparison with that study.
Tacchella et al. (2016) found a similar bend using galaxy models from a
hydrodynamical cosmological zoom simulation, although they placed its location
at $\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})\sim 10.2$. Both find a low-
mass-end slope in this relation of nearly 1 and a high-mass-end slope of $\sim
0.6$. If we split our ETG population as Saracco et al. (2017), we find the
following linear fits for the low- and high-mass end, respectively:

$\mu_{3.6,\leavevmode\nobreak\
\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})<10.4}=-2.39\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})+42.19$
(15) $\mu_{3.6,\leavevmode\nobreak\
\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})>10.4}=-1.19\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})+29.65$
(16)

For easier comparison, we convert our surface brightnesses to stellar mass
surface densities using the relation given by Eq. A5 from Muñoz-Mateos et al.
(2013), which yields:

$\log(\Sigma^{*}_{1\rm{kpc},\leavevmode\nobreak\
\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})<10.4})=0.96\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})-6.12$
(17) $\log(\Sigma^{*}_{1\rm{kpc},\leavevmode\nobreak\
\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})>10.4})=0.48\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})-1.10$
(18)

Where $\Sigma^{*}_{\rm{kpc}}$ is in units of
$\mathcal{M}_{\odot}/\rm{kpc}^{2}$. The low-mass-end fit is very close to that
found by Saracco et al. (2017), though we find a slightly shallower high-mass-
end fit. Recently, Arora et al. (2021) compared this relation between many
different studies, and found all of them showed similar behavior: LTGs and
low-mass ETGs show similar slopes, while high-mass ETGs show a much shallower
slope. They replicate this in their own study as well, though they provide a
different location for the high-mass bend than either study cited above:
$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})\sim 10.7$.

The bend does appear to be real, and not just a result of increased scatter at
the high-mass end of the relation: if we do a similar double-fit for LTGs, we
find slopes in the $\mu_{3.6}$–$\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})$
relation of $-2.1$ and $-1.9$ for the low- and high-mass end, respectively,
both very similar to the low-mass end fit for the ETGs despite the higher
scatter. Likewise, a fit to the entire S4G+ETG population, shown in black, has
the form:

$\langle\mu_{3.6,1\rm{kpc}}\rangle=-2.30\log\,(\mathcal{M}_{*}/\mathcal{M}_{\odot})+41.74$
(19)

which, when converted to mass surface density, has a slope of $\sim 0.93$,
very close still to the slope of $\sim 1.02$ given by Saracco et al. (2017)
for the low-mass end of their ETG relation, and very similar to the ETG+LTG
best-fit slope of 0.91 found by Arora et al. (2021). Both S0s and ellipticals
contribute to the bend in the relation at high stellar mass; the relation’s
high-mass-end slope is $-1.23$ including only ellipticals, and $-1.28$
including only S0s.

The range of values in $\langle\mu_{3.6,1\rm{kpc}}\rangle$ for ETGs above
$10^{10.4}\mathcal{M}_{\odot}$ are the same as those for LTGs in the same mass
range. The most evident morphological divide in this relation lies between
ETGs and LTGs below this stellar mass, where ETGs occupy the high-surface-
brightness end of the distribution at a given stellar mass, which is
interesting if ETGs indeed evolve directly from LTGs. In disk galaxies, 1 kpc
is well within the typical sizes of bars and possibly related structures like
rings or lenses (on average, $\sim 4$ kpc; Laurikainen et al. 2011, 2013, and
references therein), hence this metric $\langle\mu_{3.6,1\rm{kpc}}\rangle$
traces the regime of central structure, such as bulges.

If ETGs do evolve directly from LTGs, the bend at high stellar masses could
imply that whatever mechanisms causing the evolution increase central
densities only in lower mass galaxies, leaving high-mass ETG cores untouched
relative to their LTG ancestors. Kim et al. (2012), for example, found
evidence for the immutability of high-mass ETGs by demonstrating that the
cores of ETGs in their sample with clear tidal debris lie on the same Kormendy
(1977) relation as their undisturbed counterparts. This could be related to
bulge formation. The bulge-to-total ratio for S0s increases with increasing
galaxy mass (see Fig. 7 of Laurikainen et al. 2010), and, as stated
previously, large bulges tend to inhibit bar formation (Kataria et al. 2020,
and references therein). This in turn could stabilize gas and dust against
angular momentum loss (e.g., Athanassoula 1992; Maciejewski et al. 2002;
Carles et al. 2016), or inhibit the formation of inner rings (assuming such
structures are related to bar resonances; e.g., Buta & Combes 1996; Rautiainen
& Salo 2000; Athanassoula et al. 2010), which are often sites of intense star
formation even among S0s (e.g., Comerón 2013). Inner structures, given their
deeper gravitational potentials, are less likely to be stripped of gas and
dust than disks, meaning central star formation (and thus central mass growth)
should outlast disk star formation (and thus disk mass growth) during many
quenching scenarios, leading to the higher central surface brightnesses we see
here. Otherwise, some subsequent mechanism in high-mass ETGs (both S0s and
ellipticals) must act to decrease their core densities relative to their lower
mass counterparts.

In summary, no matter the metric used, ETG stellar mass scales positively with
ETG core surface brightnesses, with little dependence on galaxy morphology. We
reproduce well the low-scatter log-linear relationship between total stellar
mass and the mean stellar mass surface density within 1 kpc, providing further
evidence that the cores of ETGs have evolved little over cosmic time. This is
also true of LTGs, albeit with subtle differences from ETGs: we find that low-
mass ETGs have typically higher core surface brightnesses than LTGs of the
same stellar mass, suggesting that whatever mechanism might transform the
latter to the former typically halts star formation from the outside-in: disks
quench first, followed by bulges.

###  5.4 m${}_{3.6}-$m4.5 Color

Figure 16: Relationships between $m_{3.6}-m_{4.5}$ color and stellar mass
(_left_ panel) or concentration (_right_ panel). The dashed red lines show
linear fits to the ETG populations only, while dotted black lines shows fits
to the LTGs. ETG data points are multi-colored, while LTGs are shown in gray.

The 3.6 $\mu$m and 4.5 $\mu$m bands are fairly insensitive to star formation
history, making the $m_{3.6}-m_{4.5}$ color nearly constant across stellar
populations (e.g., Meidt et al. 2012; Bonanos et al. 2010; Meidt et al. 2014;
Querejeta et al. 2015; Williams & Bonanos 2016). Indeed, given how they trace
only the Rayleigh-Jeans tail in normal stellar populations, these colors are
most often employed in either very low-mass stellar or high-mass planetary
studies (e.g., Allers et al. 2006; Luhman et al. 2008), or else are used to
trace highly evolved stars (e.g., Mould et al. 2008; Bonanos et al. 2009,
2010; Williams & Bonanos 2016).

Due to a CO absorption feature present in the 4.5 $\mu$m band (Cohen et al.
1996), $m_{3.6}-m_{4.5}$ can be a tracer of metallicity in integrated stellar
populations. In a detailed study of the NIR colors of elliptical galaxies,
Peletier et al. (2012) demonstrated that the mass-metallicity relation is
traceable in these bands via a slight blueward $m_{3.6}-m_{4.5}$ color trend
with increasing mass, as lower metallicities yield higher average stellar
temperatures, which weaken the absorption in the 4.5 $\mu$m band (see also
Meidt et al. 2014). Young stellar populations are also hotter. Hence, this
relation should be clearer for galaxies with predominantly old stellar
populations than for mixed young and old populations. For Peletier et al.
(2012), this manifested as a higher scatter in the color–velocity dispersion
relation for colors measured within $R_{\rm eff}/8$ than for those measured
within $R_{\rm eff}$, as many galaxies in their sample had centrally
concentrated star formation and dust. However, from synthesized stellar
population models calibrated in the NIR using the $m_{3.6}-m_{4.5}$ colors of
giant stars, Meidt et al. (2014) found that stellar age has an extremely
limited impact on these colors (their Figs. 2 and 3), suggesting that dust and
other non-stellar emission and absorption processes constitute the primary
factor behind this increased dispersion. Stellar age does influence this color
when dust emission is taken into account, however, as dust heated by young
stars contributes to both 3.6 $\mu$m and 4.5 $\mu$m luminosities (e.g., Zhu et
al. 2010; Meidt et al. 2012; Jarrett et al. 2013; Querejeta et al. 2015),
making the exact correlation between $m_{3.6}-m_{4.5}$ and stellar population
age quite complicated.

We show this correlation for integrated colors in the left panel of Fig. 16,
using our 3.6 $\mu$m and 4.5 $\mu$m asymptotic magnitudes (in the AB system,
hence they show an offset from the Vega magnitudes used by Peletier et al.
2012). Despite using asymptotic rather than central colors, we also find a
blueward trend with increasing stellar mass for ETGs (shown via the red dashed
line, a linear fit to all ETG points; the corresponding Pearson’s correlation
coefficient is $\rho=-0.136$, with $p=0.00015$). While Peletier et al. (2012)
studied only elliptical galaxies, we find that S0s follow the same trend. LTGs
do not; the linear fit to LTGs is shown as a black dotted line, and is nearly
flat, with a Pearson’s $\rho$ correlation coefficient that is not
statistically significant ($p\textrm{-value}=0.86$, meaning the null
hypothesis of no correlation cannot be rejected).

In the right panel of Fig. 16, we show the correlation between integrated
$m_{3.6}-m_{4.5}$ color and the concentration parameter $C_{82}$ (Eq. 6). This
shows a similar blueward trend (red dashed line; $\rho=-0.120$, $p=2.67\times
10^{-10}$) as the stellar mass correlation, as expected given that $C_{82}$
correlates already with stellar mass (Fig. 7). Again, LTGs show no significant
correlation. They do, however, fall at the tail end of the ETG correlation,
appearing as a cloud of points near $C_{82}\sim 2.6$ (corresponding to $n=1$).
Indeed, a fit to the entire population of S4G+ETG galaxies gives a very
similar slope to the ETG fit alone ($-0.024$ and $-0.025$, respectively).
Meidt et al. (2014) also found that metallicity had the strongest impact on
dust-free synthetic stellar population colors, due to CO absorption in the
photospheres of giant stars (echoing Peletier et al. 2012, and references
therein). Given their lower masses (Fig. 6), LTGs are also on average lower in
metallicity (e.g., Tremonti et al. 2004; Gallazzi et al. 2005), so this may be
primarily what is driving this behavior. A simple calculation provides
evidence for this. From our linear fit:

$m_{3.6}-m_{4.5}=-0.025C_{82}-0.365$ (20)

we can predict the expected color difference between the two populations. The
median value of $C_{82}$ for ETGs is 4.40 (Section 4), while that of LTGs is
2.96, yielding a difference in $C_{82}$ of 1.44. The expected color difference
from Eq. 20 is thus $-0.025\times 1.44\approx-0.04$ mag. The median color of
the ETG sample is -0.48 mag, while that of the LTG sample is -0.42 mag,
yielding a median color difference of -0.06 mag, slightly redward from the
expectation from the linear relation. If age has little impact on these colors
in uncontaminated stellar photosphere emission, the remaining $\sim 0.02$ mag
color difference could be attributable to non-stellar emission, which Meidt et
al. (2012) showed has a fairly small impact on integrated quantities (of order
10% on stellar mass estimates, for example).

In summary, we replicate the blue-ward $m_{3.6}-m_{4.5}$ color trend with
stellar mass among ETGs using integrated magnitudes, in contrast to the LTG
population, which shows no color trend with mass. This trend leads to a color-
concentration relation as well, with LTGs occupying the low-concentration, red
color end of the entire S4G+ETG population, when metallicity and dust effects
are taken into consideration.

##  6 Data online access

As for the initial S4G results (MM2015; Salo et al. 2015; Querejeta et al.
2015), all of our data will be made available online at a dedicated webpage at
the NASA/IPAC Infrared Science Archive. We will provide, for each galaxy, the
P1 images and weight maps, the P2 masks, and the P3 radial profiles (as ASCII
tables). Once they become available, we will also upload the P4 multi-
component decompositions and any additional pipeline products to this same
location.

Additionally, we will make all of our derived parameters available in table
format: asymptotic magnitudes in both photometric bands (and associated
uncertainties), both apparent and absolute; concentration parameters in both
photometric bands; stellar masses; distances; effective radii in both
photometric bands; local sky estimates in both photometric bands (and
associated uncertainties); isophotal radii ($R_{25.5}$ and $R_{26.5}$) in both
photometric bands; isophotal parameters associated with these radii (position
angles and ellipticities); and revised morphological classifications based on
the 3.6 $\mu$m images. For the sake of reproducibility, we will also include
the limits within which we made our linear fits for deriving our asymptotic
magnitudes (drawn from the radial profiles). Also following MM2015, we are
including ancillary values from NED and HyperLEDA666These values are not
dynamically linked to either catalogue, hence changes to either will not be
reflected in these tables., including optical colors, gas content, internal
kinematics, etc., if the user wishes to define specific subsamples selected
according to specific criteria.

Finally, we will also make all of the above tables and FITS images available
at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsarc.u-strasbg.fr/viz-bin/cat/J/A+A.

##  7 Summary

We have presented the data release of the early-type galaxy extension to the
S4G, an NIR survey of nearby galaxies in a wavelength regime (3.6 $\mu$m and
4.5 $\mu$m) that is a nearly direct tracer of stellar mass. This extension
includes newly calibrated images of 465 nearby ETGs not observed during the
original S4G. We have now processed these images up through Pipeline 3, which
derives surface photometry and associated parameters for each galaxy. This
includes radial surface brightness profiles, curves of growth, asymptotic
magnitudes (converted as well to stellar masses), effective radii, isophotal
radii, and concentration parameters. We also include updated CVHRS
morphological classifications for these ETGs.

Using these parameters, we have conducted a preliminary investigation into the
following scaling relations:

  * •

Stellar mass–concentration, in which we show that S0 and elliptical galaxies
follow roughly the same correlation, though disk galaxies tend to scatter to
lower concentrations at a given stellar mass (which is also true of LTGs). We
also demonstrate that single-component fits to ETGs tend to underestimate
concentration by overestimating the luminosity of the galaxies’ outskirts.
This results in single-component magnitudes that are slightly too bright
compared to their empirical counterparts; multi-component fits are necessary
even for these otherwise visually simple elliptical galaxies.

  * •

Stellar mass–size, including both effective and isophotal radius. We find good
agreement with past measures of the mass–effective radius relation for ETGs at
$z=0$, taking into account methodological differences. We find that the
corresponding isophotal radius relation has surprisingly low scatter once
galaxy inclination is corrected for, including among the more pressure-
supported, triaxial ellipticals.

  * •

Surface brightness profile shape, including relations between stellar mass and
central surface brightness, median surface brightness within the effective
radius, and median surface brightness within 1 kpc radius. Surface brightness
profiles of high-mass ETGs are brighter and, when normalized by effective
radius, more extended than those of low-mass ETGs, but there is no noticeable
trend with ETG morphological class. We reproduce the log-linear mass–central
surface brightness trend found by other studies, but we note that, as with
concentration parameters, single-component fits used to estimate this tend to
fall short in galaxies with complex structure, often leading to extreme fitted
Sérsic indices. The tightest of these surface brightness scaling relations is
between stellar mass and surface brightness within 1 kpc, with little
variation even between ETGs and LTGs, suggesting that the central 1 kpc of
most galaxies is only mildly affected by whatever processes lead to
morphological differentiation over cosmic time.

  * •

Stellar mass–NIR color, in which we reproduce the tendency for
$m_{3.6}-m_{4.5}$ to become bluer for higher mass galaxies, a result of a CO
absorption feature in the 4.5 $\mu$m band. This trend is echoed in the
concentration-color relation, where LTGs form roughly the low-concentration,
reddest color tail of the ETG trend.

Altogether, our preliminary look into these scaling relations shows good
agreement with those of previous studies, but also includes some details that
are worthy of further investigations. All of these images, profiles, and
derived parameters will be made available via the NASA/IPAC Infrared Science
Archive.

###### Acknowledgements.

AW acknowledges support from the STFC [ST/S00615X/1]. SC acknowledges funding
from the State Research Agency (AEI-MCINN) of the Spanish Ministry of Science
and Innovation under the grants “The structure and evolution of galaxies and
their central regions” with reference
PID2019-105602GB-I00/10.13039/501100011033, and “Thick discs, relics of the
infancy of galaxies” with reference PID2020-113213GA-I00. This project has
received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No.
893673 (SDG), as well as grant agreement No. 721463 to the SUNDIAL ITN network
(AHS, HS, JHK), from the State Research Agency (AEI-MCINN) of the Spanish
Ministry of Science and Innovation under the grant “The structure and
evolution of galaxies and their central regions” with reference
PID2019-105602GB-I00/10.13039/501100011033, and from the IAC project P/300724
which is financed by the Ministry of Science and Innovation, through the State
Budget and by the Canary Islands Department of Economy, Knowledge and
Employment, through the Regional Budget of the Autonomous Community. EA and AB
gratefully acknowledge financial support from CNES (Centre National d’Études
Spatiales, France). LCH is supported by the National Science Foundation of
China (11721303 and 11991052) and the National Key R&D Program of China
(2016YFA0400702). TK acknowledges support from the Basic Science Research
Program through the National Research Foundation of Korea (NRF) funded by the
Ministry of Education (NRF-2019R1I1A3A02062242). KMD thanks the support of the
Serrapilheira Institute (grant Serra-1709-17357) as well as that of the
Brazilian National Council for Scientific and Technological Development (CNPq
grant 312702/2017-5) and of the Carlos Chagas Filho Foundation for Supporting
Research in the State of Rio de Janeiro (FAPERJ grant E-26/203.184/2017),
Brazil. This work is based on observations made with the Spitzer Space
Telescope, which was operated by the Jet Propulsion Laboratory, California
Institute of Technology under a contract with NASA. Support for this work was
provided by NASA through an award issued by JPL/Caltech. This research has
made use of the NASA/IPAC Extragalactic Database (NED), which is funded by the
National Aeronautics and Space Administration and operated by the California
Institute of Technology. This work made use of the Python packages Numpy
(Harris et al. 2020), Scipy (Virtanen et al. 2020), and Matplotlib (Hunter
2007). We thank Jarkko Laine for providing us with some of the IRAF scripts
used to produce the radial profiles discussed in MM2015, which were critical
to us for keeping the methodology consistent across such a gap of time.
Finally, we thank the anonymous referee for their valuable feedback.

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