# A New WISE Calibration of Stellar Mass 111Released on January, 2023

T.H. Jarrett Department of Astronomy, University of Cape Town, Rondebosch,
South Africa  Western Sydney University, Locked Bag 1797, Penrith South DC,
NSW 1797, Australia  M.E. Cluver Centre for Astrophysics and Supercomputing,
Swinburne University of Technology, John Street, Hawthorn, 3122, Australia
Department of Physics and Astronomy, University of the Western Cape, Robert
Sobukwe Road, Bellville, 7535, South Africa  Edward N. Taylor Centre for
Astrophysics and Supercomputing, Swinburne University of Technology, John
Street, Hawthorn, 3122, Australia  Sabine Bellstedt ICRAR, The University of
Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia  A. S. G.
Robotham ICRAR, The University of Western Australia, 35 Stirling Highway,
Crawley WA 6009, Australia  H. F. M. Yao Department of Physics and Astronomy,
University of the Western Cape, Robert Sobukwe Road, Bellville, 7535, Republic
of South Africa

###### Abstract

We derive new empirical scaling relations between WISE mid-infrared galaxy
photometry and well-determined stellar masses from SED modeling of a suite of
optical-infrared photometry provided by the DR4 Catalogue of the GAMA-KiDS-
VIKING survey of the southern G23 field. The mid-infrared source extraction
and characterization are drawn from the WISE Extended Source Catalogue (WXSC)
and the archival ALLWISE catalog, combining both resolved and compact galaxies
in the G23 sample to a redshift of 0.15. Three scaling relations are derived:
W1 3.4 $\mu$m luminosity versus stellar mass, and WISE $\rm W1{-}\rm W2$, $\rm
W1{-}\rm W3$ colors versus mass-to-light ratio (sensitive to a variety of
galaxy types from passive to star-forming). For each galaxy in the sample, we
then derive the combined stellar mass from these scaling relations, producing
M${}_{\scaleto{\star}{5pt}}$ estimates with better than $\sim$25-30% accuracy
for galaxies with $>$109M⊙ and $<$40 - 50% for lower luminosity dwarf
galaxies. We also provide simple prescriptions for rest-frame corrections and
estimating stellar masses using only the W1 flux and the $\rm W1{-}\rm W2$
color, making stellar masses more accessible to users of the WISE data. Given
a redshift or distance, these new scaling relations will enable stellar mass
estimates for any galaxy in the sky detected by WISE with high fidelity across
a range of mass-to-light.

mid-infrared

††facilities: WISE, AAT

##  1 Introduction

Studies of galaxy evolution rely upon accurate estimation of physical
properties, most notably the mass metrics such as halo, gas, and stellar mass.
Since the advent of near-infrared technology, wide-field surveys, and more
recently space-based infrared telescopes, the stellar mass
(M${}_{\scaleto{\star}{5pt}}$) can be effectively estimated from the near-
infrared luminosity and a (typically, single) mass-to-light
($\Upsilon_{\bigstar}$). The fundamental assumption is that the dominant
stellar mass component of a typical galaxy arises from the evolved population
of low mass (e.g., Solar-like) stars that are best traced in the near-infrared
1-2 $\mu$m window where this emission peaks, or at slightly longer wavelengths
in the mid-infrared ($<$5 $\mu$m), where space-based telescopes provide the
additional advantage of a stable PSF and low surface brightnesses.

Galaxies are of course comprised of a variety of stars with differing ages,
masses, histories, dynamics, and metallicity. Attempts to better estimate the
M${}_{\scaleto{\star}{5pt}}$ and $\Upsilon_{\bigstar}$ distribution in
galaxies typically use sophisticated SED modeling across a wide energy
spectrum, but this can be very expensive and limited in areal survey coverage.
A hybrid solution is to create a set of simple scaling functions that connect
or “bootstrap” basic infrared photometric measurements (e.g., color) to well-
determined (e.g., SED modeling) stellar masses, thus extending the
M${}_{\scaleto{\star}{5pt}}$ calibration to much larger surveyed areas without
the benefit of extensive multi-band photometry. Indeed, a single color is
often more than adequate to estimate the global $\Upsilon_{\bigstar}$ for a
galaxy (e.g., Bell & de Jong, 2001; Taylor et al., 2011; Robotham et al.,
2020; Li & Leja, 2022).

For near-infrared (NIR) photometric bands the mass-to-light is optimally near
unity (within 2$\times$; cf. Bell et al., 2003; Taylor et al., 2011), while
the effects of extinction are far less important compared to the shorter wave
bands. This was a key selling point of the 2MASS 1-2 $\mu$m all-sky survey
back in the 1990s: the near-infrared window traces the dominant stellar
backbone of galaxies and is less affected by dust extinction, and hence an
ideal probe of stellar mass. By today’s standards, the 1 mJy flux limits of
2MASS are relatively shallow, limiting studies to very local volumes (e.g.,
redshifts less than 0.05 for the 2MRS redshift survey; Huchra et al., 2012).
More recent infrared surveys (e.g., the VISTA Hemisphere Survey or VHS;
McMahon et al., 2013). are several orders of magnitude deeper and hence enable
studies to extend further in distance, while also revealing the lower surface
brightness emission in nearby galaxies that may be completely hidden in the
shallow surveys.

One such deep survey, covering the near to the mid-infrared window, comes from
the Wide-field Infrared Survey Explorer (WISE) launched in 2009, that fully
surveyed the entire sky in four bands from 3 to 23 $\mu$m (Wright+2010). The
mission was designed to be sensitive to both stellar emission and the
interstellar medium of galaxies. Two near-infrared bands, W1 (3.4 $\mu$m) and
W2 (4.6 $\mu$m, capture the Rayleigh-Jeans (RJ) tail of the stellar continuum
that arises from evolved stars, while the mid-infrared bands at W3 (12 $\mu$m)
– broad enough to capture both atomic and molecular lines and continuum
emission – and W4 (23 $\mu$m) are sensitive to small-grain dust continuum that
arises from the star formation processes that build new mass as galaxies
evolve (Jarrett et al., 2013).

Since the launch of WISE, many studies have used the mid-IR to probe the
current and past star formation history properties of galaxies. Similar to the
Spitzer-IRAC 3.6 $\mu$m band in which a number of studies have shown to be an
effective metric for the mass-to-light and hence stellar mass (e.g. Eskew et
al., 2012; Meidt et al., 2014; Querejeta et al., 2015), the WISE W1 3.4 $\mu$m
band luminosity (LW1) may be used to estimate the global stellar mass using a
single mass-to-light (M/L) value.

Studies using IRAC 3.6 $\mu$m point to a M${}_{\scaleto{\star}{5pt}}$/L3.6μm
($\Upsilon_{\bigstar}^{3.6\mu m}$) with a value of 0.4 to 0.6 (McGaugh, 2012;
Schombert & McGaugh, 2014; Meidt et al., 2014). Using a single M/L is only an
approximation; galaxies are not monolithic populations but contain stars in
different stages of evolution and with a variety of physical properties (e.g.,
metallicity). A single M/L is best used with early type or spheroidal
galaxies, whose emission tends to be dominated by older evolved stars,
comprising a more homogeneous class that may be described by simple stellar
populations. Conversely, it is less effective with late-type or SF disk
galaxies whose composition is a mix of old and young populations with distinct
dynamical distribution (e.g., disk, bulge, bars, etc).

To allow for variation in the $\Upsilon_{\bigstar}$ across galaxy types,
photometric colors are typically used both globally and internally. The
assumption is that the range in color (or set of colors) will indicate the
dominant populations (e.g., disk vs spheroidal) and map to a mass-to-light
that is more appropriate to its aggregate M${}_{\scaleto{\star}{5pt}}$. In
this way, studies have shown correlations of color with $\Upsilon_{\bigstar}$
for normal galaxies, including those using the IRAC [3.6 $\mu$m]-[4.5 $\mu$m]
color (e.g., Meidt et al., 2014; Querejeta et al., 2015; Ponomareva et al.,
2018) and the WISE $\rm W1{-}\rm W2$ color (e.g. Cluver et al., 2014; Norris
et al., 2014). At these wavelengths, $\Upsilon_{\bigstar}$ dependencies with
metallicity is likely small, although there are conflicting studies, see for
example Norris et al. (2014) and Röck et al. (2015).

In addition to dust absorption mitigation, the great advantage of using these
3-5 $\mu$m mid-IR bands is superior detection sensitivity (relative to ground
NIR observations). But as pointed out by Meidt et al. (2012), they are also
subject to warm dust continuum emission from star formation and AGN activity,
and additionally, the 3.3 $\mu$m PAH emission line that lies within the IRAC-1
and W1 bands. These non-stellar mechanisms act as contaminants to the
luminosity and colors, and caution is in order when working with extrema
galaxies (e.g., infrared-bright galaxies). See Ponomareva et al. (2018) for a
comprehensive review of the efforts to measure the $\Upsilon_{\bigstar}$ using
the 3-5 $\mu$m bands of Spitzer-IRAC.

The WISE study from Cluver et al. (2014) showed that the $\rm W1{-}\rm W2$
color trends such that higher $\Upsilon_{\bigstar}^{3.4\mu m}$ tracks with
bluer (passive, early-types) colors, while redder colors track to lower
$\Upsilon_{\bigstar}^{3.4\mu m}$, consistent with SF disks and late-type
galaxies. The full range in $\Upsilon_{\bigstar}^{3.4\mu m}$ is from 0.8 to
0.2, respectively from early to late-types.

More recently, a novel “star formation history” approach was used to estimate
the mid-IR mass-to-light. Salim et al. (2016) and Leroy et al. (2019) have
shown that the $\Upsilon_{\bigstar}^{3.4\mu m}$ trends with the specific star
formation rate (sSFR) as determined through WISE photometry and SED fitting,
with a $\Upsilon_{\bigstar}^{3.4\mu m}$ ranging from 0.2 and 0.5,
corresponding to actively building galaxies and passive galaxies,
respectively. Moreover, combining several methods, Schombert et al. (2022)
demonstrated that the baryonic Tully-Fisher Relation is improved if the
$\Upsilon_{\bigstar}$ range is determined using a wide suite of optical-
infrared bands, in conjunction with stellar structure (bulge vs disk)
modeling, with the expected behavior of lower values for disk populations
($\Upsilon_{\bigstar}$ ranging from 0.4 to 0.8) and roughly constant, large-
values (0.8-0.9) for bulge-dominated systems.

We see that in a range of methods from single M/L, to colors, SF information,
and population-kinematic structure, both the aggregate and local mass density
of a galaxy may be estimated from optical, near-IR, mid-IR (and combinations
thereof) imaging surveys.

In this study, our goal is to update and improve upon the early WISE work by
Jarrett et al. (2013) and Cluver et al. (2014) with a new set of scaling
relations that use our latest galaxy measurements with WISE (see below),
coupled to a new and rigorous stellar mass catalog derived from SED modeling
of a complete set of optical-infrared photometry and spectroscopic redshifts
from the GAMA-KIDS-VIKING DR4 Catalog of the G23 field in the southern
hemisphere (Driver et al., 2022).

Since the launch of WISE in 2009, our team has been building a resolved galaxy
database based on WISE imaging and source extraction called the WISE Extended
Source Catalogue (WXSC). The WXSC improves upon both the imaging and source
measurements, compared to the published ALLWISE catalogs that were optimized
for point sources, through custom mosaic construction (Jarrett et al., 2012)
and characterization that includes careful foreground star (and background
galaxy) removal, local background estimation, and a full suite of
size/orientation, photometry, surface brightness, and radial profile
measurements (see Jarrett et al., 2013, 2019)

In this new study, we deploy the “total” fluxes (details next section) to
derive the W1 3.4 $\mu$m luminosity, which will be the fundamental scaling
tracer of stellar mass. However, in order to be sensitive to
M${}_{\scaleto{\star}{5pt}}$/LW1 differences from one galaxy to the next, we
also derive scaling relations for the colors that WISE provides, namely $\rm
W1{-}\rm W2$ and $\rm W1{-}\rm W3$, which are both sensitive to population
differences and SF activity (Wright et al., 2010; Jarrett et al., 2013; Cluver
et al., 2014).

In this work we, therefore, construct three stellar-mass scaling relations
based on the luminosity and colors that can be accessed using WISE, harnessing
the fidelity of the WXSC to provide the necessary photometric accuracy. We
find that our new WISE-derived stellar masses are accurate to $\simeq$ 25-30%,
or 0.1-0.2 in dex for Log M${}_{\scaleto{\star}{5pt}}$. Since WISE is a whole-
sky survey, and consequently is an effective means to estimate stellar masses
for galaxies anywhere in the sky, it is notably powerful in this day and age
of wide-field radio to X-ray surveys, including the southern hemisphere next-
generation SKA (radio) and LSST (optical) missions. We note that one can
always derive more accurate stellar masses using additional photometry,
spectroscopy, kinematic and sophisticated SED modeling, if high-quality data
is available. Yet WISE provides a unique data set that is both uniform in
properties, sensitive to the dominant mass populations of most galaxies and
covers the entire sky. Our aim is to produce uniform and consistent stellar
masses (and SFRs) for all nearby galaxies and thus provide a point of entry to
galaxy evolution studies that rely upon stellar mass and star formation
activity metrics.

The paper is organized using the following sections: (2) data and
measurements, including the M${}_{\scaleto{\star}{5pt}}$ catalog and WXSC
source characterization, (3) procedure for how the $\Upsilon_{\bigstar}$
scaling relations are constructed and used to derive a final
M${}_{\scaleto{\star}{5pt}}$ per galaxy, (4) the primary results that
including the scaling relations for 3.4 $\mu$m luminosity, $\rm W1{-}\rm W2$
and $\rm W1{-}\rm W3$ colors, (5) simple pragmatic methods computing your own
masses, (6) implications of these new relations and (7) summary.

Throughout this paper, we frequently interchange M${}_{\scaleto{\star}{5pt}}$
to mean Mstellar, the global stellar mass for individual galaxies. M/L refers
to the mass-to-light ratio (sometimes denoted using $\Upsilon_{\bigstar}$).
The adopted cosmology is H0 = 70 km s-1 Mpc-1, $\Omega_{M}$ = 0.3 and
$\Omega_{\Lambda}$= 0.7. All magnitudes are in the Vega system (WISE
photometric calibration described in Jarrett et al. (2011).

##  2 Data

Our aim is to construct scaling relations between WISE mid-IR galaxy
properties (LW1 luminosity and colors) and ‘fiducial’ stellar masses that have
been recently constructed from the GAMA redshift survey. In this section, we
describe the GAMA measurements and stellar masses, and the WISE source
characterization measurements. For WISE, we use both the WXSC measurements and
those coming from the ALLWISE catalog to include both resolved and compact
sources that have been targeted by the GAMA survey. The redshift range is
restricted to less than 0.15 to minimize systematic effects from observational
biases and errors that grow with (1+z) (e.g., the k-correction), while also
providing a large enough sample to cover all galaxy types,
$\Upsilon_{\bigstar}$, and stellar mass.

###  2.1 M${}_{\scaleto{\star}{5pt}}$ Sample

The primary M${}_{\scaleto{\star}{5pt}}$ sample is drawn from the Galaxy and
Mass Assembly (GAMA) Survey (Driver et al., 2011), a spectroscopic survey
highly complete down to 19.8 mag in the r-band in its three equatorial bands
and to 19.2 mag in $i$-band in the G23 field (see Liske et al., 2015; Baldry
et al., 2018). For this M${}_{\scaleto{\star}{5pt}}$ study, we utilize the
GAMA-KiDS-VIKING DR4 catalog (Driver et al., 2022) of the G23 field, located
between right ascension 339∘ to 351∘ and declination from -35∘ to -30∘and
covering 50 deg2 in area. A total of $\sim$47,000 sources have high-quality
redshifts in the G23 field.

The catalog also includes a full suite of multi-band ProFound-derived
photometry, notably the optical-infrared (ugrizjhk) bands, as detailed in
Bellstedt et al. (2020a). From this rich spectro-photometric data set, the
GAMA team produced high-quality stellar masses222StellarMassesv24 GAMA catalog
through SED modeling with a fixed grid of relatively simple stellar population
models (based on Bruzual & Charlot, 2003) that includes exponential star
formation histories, uniform metallicity, Calzetti (2001) dust law and
Chabrier (2003) IMF. The models are constrained by Bayesian minimum variance
parameter estimation marginalized over uniform priors in population age, dust
opacity, and extinction. Physical parameters are in the co-moving frame
system, with the same cosmology used in this study (H0 = 70 km s-1, etc).
Formal errors are approximately 30% for the aggregate stellar masses. More
details on this method may be found in Driver et al. (2022) and Taylor et al.
(2011).

Independent comparison comes from a study using the same data with a more
sophisticated ProSpect modeling treatment (see Robotham et al., 2020;
Bellstedt et al., 2020b) that reveals a 0.1 dex offset with 0.11 dex scatter,
but essentially no differential systematics as a function of color (see Sec 5
and Fig 34 of Robotham et al., 2020). Given that ProSpect retrieves more
massive values (by 0.1 $\pm$11% systematic) compared to color-based masses
that this current WISE study uses for its scaling relations, this should be
noted for the resulting WISE stellar masses. Nevertheless, the stellar mass is
surprisingly robust against the well-known degeneracies between age,
metallicity, and dust; and indeed, using a single optical color like (g-i)
gives nearly as accurate a mass (Bell & de Jong, 2001; Taylor et al., 2011;
Robotham et al., 2020).

Hence we believe these GAMA-G23 stellar masses are a reliable and accurate
calibration and will serve well as the “fiducial” to scale with the WISE mid-
IR luminosities and colors. Not only will this significantly improve the
scaling relations since the Cluver et al. (2014) work (where masses relied on
a scaling factor to convert aperture-derived values to total values), but also
places the WISE “stellar masses” on the same (co-moving H0 = 70 km s-1 frame
and Chabrier IMF) calibration as SDSS, GAMA, DEVILS, WAVES, SAMI, Hector,
MAGPI and other major surveys that use SDSS/GAMA data. In total, over 16000
sources have high-quality global M${}_{\scaleto{\star}{5pt}}$ values for
redshifts $<$ 0.15 that will be used in this study.

###  2.2 WISE Measurements

Resolved galaxies are cataloged in the WXSC (Jarrett et al. 2013, 2019), while
compact sources come from the ALLWISE (point-source) catalog (Cutri et al.,
2012) served by the NASA/IPAC Infrared Science Archive (IRSA). In the rest
frame, we will be using aperture-matched colors; $\rm W1{-}\rm W2$ and $\rm
W1{-}\rm W3$ in magnitudes, and the LW1 in-band luminosity derived from the W1
3.4 $\mu$m total flux density.

The WXSC includes full source characterization in the four WISE bands: 3.4,
4.6, 12, and 23 $\mu$m, extracted from custom-made mosaics of WISE single
frames using the ICORE software developed by the WISE team (details in Jarrett
et al., 2012). The frames database also includes images from follow-up NEOWISE
mission observations (W1 and W2 only), and hence creating very deep (typically
confusion-limited) mosaics, 1-sigma depths fainter than 23 mag arcsec-2 (26.7
mag arcsec-2 in AB mags). Native angular resolution is preserved in all four
bands.

Source characterization consists of four primary steps for the target galaxy:
(1) identify and remove foreground stars and background galaxies, (2) local
background estimation, (3) 3-$\sigma$ shape and orientation, (4) aperture and
1-$\sigma$ isophotal photometry, (5) surface brightness and radial profile
fitting to derive total fluxes, and other miscellaneous characterization;
further details in Jarrett et al. (2013, 2019). For well-resolved galaxies,
axi-symmetric radial profiles were constructed and fitted with a double-Sersic
function, thus attempting to model the spheroidal and disk population
distributions, as well as extrapolate to larger radii to determine total
fluxes down to several disk scale lengths.

Figure 1: Mid-IR brightness and color properties of the G23 galaxy sample used
to derive the stellar mass scaling relations. Resolved sources were measured
and cataloged through the WXSC, while fainter (see LEFT panel) compact sources
were extracted from the ALLWISE catalog. All measurements have been rest-frame
corrected (ranging to $z<$ 0.15); see the Appendix for details and
prescriptions for simplifying the process. For colors (Vega system
magnitudes), RIGHT panel, a general trend from left to right is observed, the
sequence (cyan dashed line; Jarrett et al. (2019) shows how the colors
indicate increasing SF activity from blue to red colors. A small fraction of
the sample has warm $\rm W1{-}\rm W2$ colors indicative of AGN emission
significantly contributing to the overall host galaxy emission (and hence
eliminated from the analysis). Table 1 summarizes the number of optical-
infrared cross-match detections in these plots.

Typically the W1 1-$\sigma$ isophote captures 95% of the total light (Jarrett
et al., 2019), but some (e.g., low surface brightness) cases it can be more
hidden in the background noise (e.g., 50% of the light from such a galaxy may
be unaccounted). We, therefore, use total fluxes, when available, to derive
the W1 luminosities for the M${}_{\scaleto{\star}{5pt}}$ scaling relation of
this study. In the case of compact sources where little emission is detected
beyond the 6″ FWHM beam, and hence radial profiles are not possible, we
instead apply an aperture correction that at the very least accounts for the
PSF emission that extends beyond the small, compact aperture.

All fluxes and colors are rest-frame corrected, accounting for the (1+z)
wavelength stretching, using a suite of composite SED templates from Brown et
al. (2014) and Spitzer-SWIRE/GRASIL Polletta et al. (2006, 2007); Silva et al.
(1998) that range from early-type spheroidals to late-type spirals, Seyferts,
and AGNs of differing types. For redshifts $<$0.15, we can expect $<$10-15%
scatter from this k-correction operation based on a recent analysis in Yao et
al. (2022). Further details of the SED template fitting are given in the
Appendix, including prescriptions to estimate the corrections for the flux and
$\rm W1{-}\rm W2$ color.

Using the redshift information, transforming to the CMB frame, and using Ho=70
km/s/Mpc cosmology to derive luminosity distances, the rest-frame W1 total
fluxes are converted to the in-band luminosity, LW1,

$L_{\rm W1}[L_{\odot}]=10^{-0.4(M-M_{\rm SUN})}$ (1)

where M is the absolute magnitude of the source in W1 (determined from the
distance modulus) and 3.24 mag is the W1 in-band solar value (Jarrett et al.,
2013). It is important to note that this luminosity is not a spectral
luminosity ($\nu$Lν), but instead is bolometric in nature (normalizing the in-
band flux by the solar value in the same band). This has implications for the
k-correction (see Appendix) and means the in-band luminosity is much larger
than the spectral luminosity (because the latter is normalized by the full
bolometric luminosity of the Sun).

A final note with regard to the 3-5 $\mu$m light, we do not attempt to model
and correct for any non-stellar emission in the W1 and W2 bands. It is likely
to be negligible, or small relative to the total propagating errors, for most
galaxies. For those cases where the mid-infrared is subject to additional AGB
dust emission (old stars in the asymptotic giant branch phase), ISM molecular
emission (strong PAH emission in starbursts), or AGN-accretion emission (e.g.,
luminous infrared galaxies), the W1 luminosity (and hence stellar masses) may
be over-estimated. We have taken measures (see below when discussing the WISE
colors) to eliminate ‘warm’ galaxies (i.e., AGN and powerful starbursts) from
the scaling relation analysis to follow; see below.

In the case of compact and point sources, we use the ALLWISE standard aperture
magnitudes. Nearby galaxies, which tend to be resolved even in the W1 imaging,
are not well approximated by PSF profile fitting. Compared to the default
(PSF-fit) ALLWISE magnitudes, we have determined that the ALLWISE cataloged
standard fixed aperture – small circular aperture, 8.25″ in radius, corrected
to capture the total flux of a point source, is better suited to
characterizing nearby, compact galaxies. As discussed in Cluver et al. (2014,
2020), the fixed circular aperture tends to capture more ‘extended’ flux that
compact (or nearby galaxies) galaxies will possess. Nevertheless, we have
found that these standard apertures require further correction to better
capture the total extended flux and bring them onto a consistent system with
the WXSC isophotal magnitudes; hence, we have applied a further correction to
the ALLWISE measurements to capture the total flux of the compact
extragalactic source. We can expect 10-20% uncertainty in the resulting fluxes
both from aperture corrections and due to the fainter and hence lower S/N
sample relative to the well-resolved galaxies of the WXSC.

Colors are generally determined using matched apertures that are large enough
to capture most of the galaxy flux (and hence, representative of the global
color). As detailed in Jarrett et al. (2019), matching W1 and W2 bands is
straightforward since they both trace the same emission mechanism (stellar),
while matching with W3 (or W4) is not always optimal because of the combined
mechanisms of stellar and interstellar medium (gas and dust), as well as the
overall (lower) sensitivity of the long wavelength bands of WISE. For stellar-
dominated galaxies, the W3 and W4 detections tend to be weak or undetermined,
resulting in smaller apertures (with the appropriate aperture correction
applied, see above), while SF-dominated galaxies have strong W3 detections and
aperture matching is possible. In either case, the $\rm W1{-}\rm W3$ color is
not as stringent as that of $\rm W1{-}\rm W2$ because of the large dynamic
range – 4 to 5 magnitudes vs. 0.5 mag – so we allow lower S/N detections to
create $\rm W2{-}\rm W3$ and $\rm W1{-}\rm W3$ colors.

Using a typical optical-infrared cross-match cone-radius of 3″ (Cluver et al.,
2014; Jarrett et al., 2017), we positionally matched the G23
M${}_{\scaleto{\star}{5pt}}$ catalog with our rest-wavelength WISE
measurements to draw the M${}_{\scaleto{\star}{5pt}}$-WISE sample of this
study. We have restricted the sample to $z<$ 0.15 in order to maximize
brightness S/N, mitigate luminosity bias and other redshift trends that may
skew the scaling relations, minimize rest-frame k-correction uncertainties
(see also the Appendix for a discussion on k-corrections) and general
completeness and reliability uniformity. Next, we eliminate suspected AGNs
(using WISE colors, see below) and low S/N detections in the W1 (3.4 $\mu$m)
band. The resulting match in total with 6671 WISE sources, of which $\sim$51%
are resolved (with S/N $>$ 5 from the WXSC) and 49% are unresolved or compact
(S/N $>$ 3 from ALLWISE). For the $\rm W1{-}\rm W2$ color, we demand better
accuracy for the scaling relation, and hence we set S/N $>$ 7, resulting in a
lower number of matches, 3305 and 984 for resolved and unresolved
respectively. Finally, for the $\rm W1{-}\rm W3$ color, the scaling relation
can tolerate a lower sensitivity because of the large dynamic range in the
color, and hence the S/N limit is set to 3, resulting in 2692 and 505 matches
respectively (note that the detection sensitivity in the W3 12 $\mu$m band is
considerably less than that of W1). A summary of the cross matches between
GAMA and WISE is given in Table 1.

In Fig. 1, we show the rest-frame W1 fluxes and colors for the WXSC and
ALLWISE measurements that are used in the M${}_{\scaleto{\star}{5pt}}$
calibration (see also Table 1 for a summary of the cross-match statistics).
Most resolved sources peak around 1 mJy, but are as faint as 0.1 mJy, while
the compact sources peak at 0.1 mJy and reach down to 25 $\mu$Jy. In terms of
color, early-types (spheroidal and passive in SF activity) tend to have blue
$\rm W1{-}\rm W3$ colors (left side of the plot), while dusty SF disks
populate the right side (i.e., red colors). There is a clear sequence in
color-color (cyan dashed line, from Jarrett et al., 2019) that shows how the
$\rm W1{-}\rm W2$ color ’warms’ as SF activity increases, due to the W2 4.6
$\mu$m band increasingly sensitive to the warm-dust ISM component. Additional
mid-IR emission may arise from AGN accretion, which tends to further redden
the colors, most notably in the $\rm W1{-}\rm W2$ axis, while for dusty
(AGN+)starbursts the W1 3.4 $\mu$m band may be enhanced due to the 3.3 $\mu$m
molecular PAH molecular (typically very weak for normal galaxies, yet has been
detected in active galaxies; cf Kim et al., 2012).

As determined by Jarrett et al. (2011) and Stern et al. (2012), galaxies with
$\rm W1{-}\rm W2$ colors larger than 0.8 are clearly dominated by AGN,
including QSOs, BLAGNs, and dust-obscured Type-II varieties. However, a number
of more recent studies show that lower power AGN may reside in the range from
$\rm W1{-}\rm W2$ $>$ 0.4 mag (cf Assef et al., 2013; Yao et al., 2020), so
caution is in order when the color is warmer than the expected sequence value
(cyan dashed line in Fig. 1). Fortunately, the sample has only a few obvious
AGNs, and quite a small (statistically insignificant) number of ’warm’ AGNs
and hybrid-AGN+starbursts ($\rm W1{-}\rm W2$ $>$ 0.4 mag). We have eliminated
all sources with $\rm W1{-}\rm W2$ $>$ 0.4 from the analysis to follow.

Table 1: Summary of GAMA-M${}_{\scaleto{\star}{5pt}}$ and WISE Catalogs GAMA
G23 M${}_{\scaleto{\star}{5pt}}$ sources ($z<0.15$): 16838
---
| x-matches | x-matches
WISE | (resolved) | (unresolved)
LW1 (SNR $>$ 3) | 3374 | 3297
$\rm W1{-}\rm W2$ (SNR $>$ 7) | 3305 | 984
$\rm W1{-}\rm W3$ (SNR $>$ 3) | 2692 | 505

##  3 Procedure for Deriving M${}_{\scaleto{\star}{5pt}}$ Relations

In the previous work on deriving the M${}_{\scaleto{\star}{5pt}}$-mid-IR
relation, Cluver et al. (2014) determined that the color $\rm W1{-}\rm W2$ was
correlated with the M${}_{\scaleto{\star}{5pt}}$/L ratio, showing a trend in
which redder colors had lower mass-to-light ratios. The trend may (broadly)
account for population, morphology, and metallicity changes with color, from
passive spheroidal galaxies at one extreme to dusty SF disk galaxies at the
other. The observed scatter in the M${}_{\scaleto{\star}{5pt}}$ vs. $\rm
W1{-}\rm W2$ relation is large ($\sim$50%), and the derived stellar masses
have correspondingly large uncertainties. Since that work, we have
significantly improved the WXSC measurements, both in terms of deriving better
isophotal and total fluxes and better matched-aperture colors, as well as
improving our rest-frame correction pipeline. For this new work, we also
investigate the scaling relation between M${}_{\scaleto{\star}{5pt}}$ and the
$\rm W1{-}\rm W3$ color, which is likely a better tracer compared to W2
because of the improved sensitivity in the W1 measurements and the broader
range in color.

Finally, to account for faint sources where colors (such as $\rm W1{-}\rm W2$)
are poorly determined, we have derived the simplest relation possible,
M${}_{\scaleto{\star}{5pt}}$-to-LW1 in Log-Log space. This relation has the
largest scatter (compared to the color relations) because it does not account
for population, metallicity, or other factors that color is tracking. It does
have a big advantage in that it is the most sensitive band of WISE: the WXSC
is fundamentally based on the detection of the W1 3.4 $\mu$m emission from
galaxies. In effect, the general linear trend between
M${}_{\scaleto{\star}{5pt}}$ and LW1 indicates that a single
$\Upsilon_{\bigstar}^{3.4\mu m}$ (or narrow range thereof) adequately (better
than 40-50%) describes most galaxies.

The procedure, therefore, is to derive three M${}_{\scaleto{\star}{5pt}}$ (or
M/L) relations: (1) W1 luminosity, (2) $\rm W1{-}\rm W2$ color, and (3) $\rm
W1{-}\rm W3$ color. With these three relations 333For completeness, we note
that surface brightness is also correlated with galaxy property types ($cf.$
Jarrett et al., 2019), so we investigated the W1 central surface brightness
compared to the GAMA stellar masses. However, we have determined that the
(large) resulting scatter does not improve the final derived stellar mass. We
do not show any further results for surface brightness. and their
corresponding error models, we then derive a weighted average
M${}_{\scaleto{\star}{5pt}}$ and corresponding uncertainty.

##  4 M${}_{\scaleto{\star}{5pt}}$ Results

In this section, we present the scaling relations between the optically
determined stellar masses and the primary WISE measurements of the W1
luminosity, $\rm W1{-}\rm W2$, and $\rm W1{-}\rm W3$ colors. We will use all
of this information to create a weighted-average combination that represents
the mid-IR stellar calibration, which may be used for any WISE 3.4 $\mu$m
total flux measurements of extragalactic sources.

###  4.1 M${}_{\scaleto{\star}{5pt}}$ vs. 3.4 $\mu$m Light

The W1 3.4 $\mu$m flux is determined for all WISE-cataloged sources (since it
is the most sensitive band in WISE), and hence is the most general of the
scaling relations with M${}_{\scaleto{\star}{5pt}}$. For resolved sources, W1
detections range from 5 to 17.2 mag (3.1 Jy to 40 $\mu$Jy in flux density; see
Fig 1a), and corresponding LW1 luminosities (Eq. 1) that range from 107 to
10${}^{1}1.8$ L⊙. For compact sources, W1 mags range from 13.5 to 17.9 mag
(1.2 mJy to 21 $\mu$Jy), thus pushing down to fainter magnitudes and
luminosities. Although most of the resolved sources are more luminous than 109
L⊙, (compact sources stretching 0.5 dex fainter), the sample does allow
scaling down to the dwarf galaxy regime, which was a key goal of this study to
improve the M${}_{\scaleto{\star}{5pt}}$ estimates for low mass and dwarf
galaxy populations. The scaling results are shown in Fig 2, Log
M${}_{\scaleto{\star}{5pt}}$ versus Log LW1.

Figure 2: W1 scaling relation, comparing the M${}_{\scaleto{\star}{5pt}}$-to-
light global properties of galaxies as measured in the WISE 3.4 $\mu$m band.
Resolved galaxies from the WXSC are shown in black, whose luminosities are
derived from the total integrated flux. Point sources are shown in blue, whose
aperture-corrected measurements are adapted from the ALLWISE catalog. A 3rd-
order polynomial least-squares fit to the distribution (dashed line) accounts
for a small upturn in the M${}_{\scaleto{\star}{5pt}}$/LW1 at the high
luminosity (i.e., massive galaxy) end; see also Eq. 2. Relative to the fit,
the vertical scatter is 0.12 dex for masses $>$ 108.5 M⊙.

As expected in Fig. 2, luminosity (or “light”) scales with stellar mass, with
a range of slope (i.e., mass-to-light) that is highly constrained to values
between 0.30 to 0.40 (as shown in the next Section). The scaling trend is not
perfectly linear, there is a small upturn at the luminous end, where the slope
is steepening, most notably for the compact sources. At higher masses, the
$\Upsilon_{\bigstar}^{3.4\mu m}$ ratio is steeping, reflecting the population
change as older, SF-quenched galaxies dominate at these masses. Based on the
resolved and compact distributions in Fig 2, a least-squares fit using a 3rd-
order cubic polynomial covers the range from 7 to 12 in the Log LW1,
accordingly:

$\begin{split}Log\,M_{\scaleto{\star}{5pt}}=A_{0}+&A_{1}\,(Log\,L_{\rm
W1})+\\\ &A_{2}\,(Log\,L_{\rm W1})^{2}+A_{3}\,(Log\,L_{\rm
W1})^{3}\end{split}$ (2)

where M${}_{\scaleto{\star}{5pt}}$ is the stellar mass in M⊙, LW1 is the WISE
3.4 $\mu$m in-band rest-frame luminosity in L⊙ (see Eq. 1), and the A
coefficients are -12.62185, 5.00155, -0.43857, 0.01593, respectively.

The resulting $\Upsilon_{\bigstar}^{3.4\mu m}$ (slope in Fig. 2) has a mean of
0.35 $\pm$ 0.05. The vertical scatter in M${}_{\scaleto{\star}{5pt}}$ relative
to the quadratic relation is about 0.12 dex (i.e., 32%), but considerably
higher, 0.2 dex, at the low end where the relation is not as well constrained.
This small M/L range essentially means a single value crudely scales the 3.4
$\mu$m luminosity to the expected stellar mass. The small range and random
scatter in the trend demonstrate the limits of only using the W1 band to
estimate masses, it cannot account for population differences or other
properties that may alter the global M${}_{\scaleto{\star}{5pt}}$/LW1. On the
other hand, the cubic polynomial is tracking the change in M/L for the passive
galaxies at the high mass end. As we shall see below, colors do a better job
of tracking $\Upsilon_{\bigstar}^{3.4\mu m}$ differences between galaxies; as
such, we will downgrade the W1-M${}_{\scaleto{\star}{5pt}}$ scaling relation
when color information is available by lowering its weight by a factor of 5
when weight-combined with the other relations. Nevertheless, with this
W1-M${}_{\scaleto{\star}{5pt}}$ scaling relations, there now exists a simple
way to derive the stellar mass for faint sources, such as dwarfs, with an
accuracy that is 0.1-0.2 dex (30-50%) in Log M${}_{\scaleto{\star}{5pt}}$ for
most galaxy masses. In the following section, we give an example of how to do
simple mass estimation using this relation.

We should point out that these new stellar masses are lower by 0.15 dex than
what we derived in Cluver et al. (2014) (notably for early-type galaxies, see
below for the colors analysis), reflecting the new GAMA
M${}_{\scaleto{\star}{5pt}}$ fiducials and improved WISE measurements of the
total global flux. Likewise, the ‘single’ $\Upsilon_{\bigstar}^{3.4\mu m}$ =
0.35 is considerably lower than the value adopted in the early S4G work with
Spitzer IRAC-1 at 3.6 $\mu$m (Meidt et al., 2014). Conversely, we note that
our lower M/L (and hence, lower M${}_{\scaleto{\star}{5pt}}$ estimates) is
more in line with the recent work of Ponomareva et al. (2018) and Leroy et al.
(2019).

###  4.2 Colors

####  4.2.1 $\rm W1{-}\rm W2$

Applying a S/N limit of 7 to the $\rm W1{-}\rm W2$ colors restricts the sample
to 3305 and 984 WXSC and ALLWISE sources, respectively. The limit was chosen
to avoid undue scatter in the relation given the small range or tolerance in
color, while also maximizing the useful range as demonstrated in Fig. 1b for
normal galaxies. The resulting scaling with $\Upsilon_{\bigstar}^{3.4\mu m}$
is shown in Fig 3, where we have fit the distribution using a simple linear
function that ranges across -0.2 $<$ $\rm W1{-}\rm W2$ $<$ 0.4 mag, and
corresponding M/L range of $\sim$0.2 to 0.6.

Figure 3: Mass-to-light trend with $\rm W1{-}\rm W2$ color (Vega magnitude).
Black and blue points indicate resolved and point-like measurements,
respectively. The $\rm W1{-}\rm W2$ color is sensitive to warm dust associated
with SF activity (and AGN, if present), see Fig. 1b, and hence the linear
trend (red dashed line) from higher to lower $\Upsilon_{\bigstar}^{3.4\mu m}$
follows from early-type (bluer colors) to later-types (redder colors).
Relative to the fit, the vertical scatter is 0.11 dex. For comparison, we also
show the relation from Cluver et al. (2014), indicating previously larger
overall M${}_{\scaleto{\star}{5pt}}$ at the high mass (low color value) end.

The $\rm W1{-}\rm W2$ color shows a wide range in $\Upsilon_{\bigstar}^{3.4\mu
m}$, trending from high M/L with blue (stellar-dominated) colors to low M/L
with red (dusty ISM) colors. Least-square fitting to the distribution, we
derive the simple scaling relation:

$\text{Log}\Upsilon_{\bigstar}^{3.4\mu m}=A_{0}+A_{1}\,C12$ (3)

where C12 = $\rm W1{-}\rm W2$ color is in Vega magnitudes, and the A
coefficients are (-0.376, -1.053, respectively). For more extreme colors, the
floor/limits should be -0.2 and 0.4 mag, respectively, in the $\rm W1{-}\rm
W2$ color, corresponding to $\Upsilon_{\bigstar}^{3.4\mu m}$ range 0.68 to
0.16, respectively. Compared to the linear least-squares fit, the scatter is
about 0.11 dex (in Log $\Upsilon_{\bigstar}^{3.4\mu m}$) for the entire range.
In terms of the associated error model, in addition to this vertical scatter,
we have propagated the formal uncertainties in the color and W1 luminosity.

For comparison in Fig. 3, we show the relation from Cluver et al. (2014)
(orange dashed line), demonstrating that the stellar masses and WISE-GAMA
photometry from that study formed a steeper and higher offset M/L relation.
Galaxies with bluer colors, typically passive or spheroidal types, will thus
have larger masses in the previous color scaling. For example, at a color of
0.0 mag, the older relation gives a $\Upsilon_{\bigstar}^{3.4\mu m}$ $\sim$
0.68, while the new relation indicates a lower value of 0.42; hence, in terms
of Log M${}_{\scaleto{\star}{5pt}}$, the difference is about 0.21 in dex. At
the most extreme blue color limit for the 2014 relation, $\rm W1{-}\rm W2$ =
-0.05, the difference in Log M${}_{\scaleto{\star}{5pt}}$ is 0.26 dex. On the
other hand, the two relations are in close agreement for warmer colors
corresponding to late-type and dusty SF galaxies, $\rm W1{-}\rm W2$ $\sim$0.14
mag; while for reddest colors, the steeper 2014 relation (which is limited to
a color of 0.2 mag) will give lower $\Upsilon_{\bigstar}^{3.4\mu m}$ values:
0.21 versus 0.26 for the new scaling (equivalent to 0.09 dex in the Log
M${}_{\scaleto{\star}{5pt}}$).

####  4.2.2 $\rm W1{-}\rm W3$

Next, we considered the ISM-sensitive (i.e., star-formation activity) color,
$\rm W1{-}\rm W3$. Here we relaxed the S/N criteria to 3-$\sigma$ color
uncertainty limit to include as many sources as possible. Accordingly, 2692
resolved (and 505 compact) sources were used in the scaling relation; plotted
in Fig 4, showing the Log $\Upsilon_{\bigstar}^{3.4\mu m}$-[$\rm W1{-}\rm W3$]
distribution.

Figure 4: Mass-to-light ($\Upsilon_{\bigstar}^{3.4\mu m}$) trends with $\rm
W1{-}\rm W3$ color. This color is notably sensitive to dust content and SF
activity (see also Fig. 1b). As expected, the $\Upsilon_{\bigstar}^{3.4\mu m}$
tends to decrease from blue to red colors (i.e., passive to SF active) but has
a non-linear distribution that is suitably modeled by a Schechter Function
(red dashed line; best-fit parameters indicated in the lower left). Relative
to the fit, the vertical scatter is 0.09-0.15 dex, increasing to larger values
for redder colors.

Unlike the $\rm W1{-}\rm W2$ relation, here the $\rm W1{-}\rm W3$ comparison
to the $\Upsilon_{\bigstar}^{3.4\mu m}$ indicates two distinct trends:
somewhat flat M/L for blue colors and a steeper trend for red colors.
Specifically at the blue end, Fig. 4 shows a flat (single
$\Upsilon_{\bigstar}^{3.4\mu m}$ $\sim$ 0.45) for a range in color from 0 to 2
mag in the $\rm W1{-}\rm W3$ color, essentially covering the range that
stellar-dominated galaxies occupy, including early-type and bulge-dominated
spirals. Note that this M/L is higher than the overall (all-galaxies) value
determined with the simpler M${}_{\scaleto{\star}{5pt}}$-LW1 relation,
consistent with these types of galaxies having a more massive bulge population
that is reflected in the WISE color. For redder colors, $>$3 mag, the
distribution in Fig. 4 then rolls off to lower M/L, the vast majority between
0.2 to 0.4, thereafter flooring around 0.15.

To create a $\rm W1{-}\rm W3$ scaling relation with
$\Upsilon_{\bigstar}^{3.4\mu m}$, the two trends may be fit with corresponding
linear functions, joining (in discontinuity) at a color of 3.1 mag or so. It
may also fit (somewhat poorly) with a simple exponential function. However, we
also note that the overall Log-Log distribution conveniently resembles a
Schechter Function; hence, we have carried out a least squares fit using a
model with variables $\phi_{\star}$ (vertical scaling factor), $\alpha$
(controls the slope of the flat trend at the blue end), and ($\rm W1{-}\rm
W3$)⋆ which controls the ‘knee’ bend; accordingly:

$\begin{split}\text{Log}\Upsilon_{\bigstar}^{3.4\mu
m}=\phi_{\star}\>\gamma^{(\alpha+1)}exp(-\gamma)\\\
\gamma=10^{[0.4(C13-C13^{\star})]}\\\ C13=W1{-}W3\end{split}$ (4)

where we have fit for $\phi_{\star}$, $\alpha$ and color ($\rm W1{-}\rm W3$)⋆;
the resulting best fit values are, respectively 0.454, -1.00, 4.690, also
indicated in Fig. 4. For more extreme colors, the floor/ceiling color limits
should be -1.0 and 5.0 mag, respectively, in the $\rm W1{-}\rm W3$ color,
which corresponds to $\Upsilon_{\bigstar}^{3.4\mu m}$ values of 0.45 and 0.12,
respectively. Compared to the best fit function, the vertical scatter is about
0.09-0.15 dex (in Log $\Upsilon_{\bigstar}^{3.4\mu m}$ ) for the full range,
with increasing uncertainty at the red color extreme where it is poorly
constrained. As with the $\rm W1{-}\rm W2$ error model, we propagate the
vertical scatter, color uncertainty (horizontal scatter), and W1 luminosity
uncertainty.

Figure 5: WISE mid-IR stellar masses derived from the combined WISE W1
luminosity and mid-IR color scaling relations. (left) shows the newly
estimated masses compared to the “fiducial’ masses from which the scaling
relations were derived. Resolved WXSC (black points) sources have a residual
scatter around the zero line of $\sim$0.1-0.12 dex (red dashed line) in the
LW1 range from 109 to 1012 L⊙; aperture-corrected point sources (blue) have a
larger scatter of 0.16-0.17 dex for a luminosity range between 108.5 to 1010.5
L⊙ (magenta dashed line). (right) Corresponding mass-to-light histograms for
the complete WXSC distribution (dark blue line), which may be compared with
fiducial $\Upsilon_{\bigstar}^{3.4\mu m}$ (green line). The mean
$\Upsilon_{\bigstar}^{3.4\mu m}$ is 0.34, and a full-width half-maximum range
0.23 to 0.44.

###  4.3 Combined M${}_{\scaleto{\star}{5pt}}$ Relations: WISE Stellar Masses

Combining the estimated stellar masses from the three scaling relations and
their corresponding error models, using an inverse-variance weighting scheme,
the mid-IRbased stellar mass is derived. The color relations have a smaller
total scatter compared to the LW1 relation, and hence will have a higher
weighting in general. However as noted earlier, our aim is to cover as wide a
M/L range as possible to better model real galaxies, and consequently the LW1
relation (Eq. 2) is further de-weighted (by a factor of 5) in order to
emphasize the color relations (Eqs. 3 and 4) which better trace the variation
in $\Upsilon_{\bigstar}^{3.4\mu m}$ that is exhibited in the fiducial ‘truth’
set. The corresponding weighted mean in the global stellar mass is shown in
Fig. 5 for the WXSC and ALLWISE sub-samples, where we have compared the mid-IR
masses and M/L with the fiducial counterparts to assess the new scaling
relations performance.

The resulting scatter seen in the Log M${}_{\scaleto{\star}{5pt}}$ residual is
shown in Fig. 5a (left panel). For resolved sources (black points), most of
the points are in the luminosity range from 109 to 1012 L⊙, scatter around the
zero residual line with a 0.1 to 0.12 standard deviation, but may also have
extremes as deviant as 0.5 in dex. This empirical result is mostly consistent
with the propagated error models which predict most sources will have Log
M${}_{\scaleto{\star}{5pt}}$ errors between 0.09-0.11 dex (25-30%). Since
there is some cross-correlation between the three scaling relations (they do
share information, notably the W1 flux), we can expect the standard error of
the weighted mean to have a slightly lower value than the observed residual
scatter. Yet, the residual is still remarkably tight considering that we are
only using three pieces of mid-IR characterization information to derive Log
M, all of which are cataloged in the WXSC.

Remember the formal errors in the GAMA masses is 0.12 dex (30%), and hence
appear to be the dominant uncertainty in these new WISE scaling relations (as
opposed to the photometry). Moreover, given the relatively flat, zero
systematic between the fiducial and the derived, and the 0.1-0.12 dex residual
scatter in Log M${}_{\scaleto{\star}{5pt}}$, it is reassuring that our mid-
infrared bands are not obviously contaminated by non-stellar emission or
observational (e.g., redshift) biases.

For compact source from ALLWISE (blue points) the scatter is considerably
larger, 0.16-0.17 dex, and some fairly large deviations of $\pm$0.75 dex, but
also extending to lower luminosities (and hence, lower masses), $\sim$108.5
L⊙. The error models predict Log M${}_{\scaleto{\star}{5pt}}$ uncertainties
from 0.12-0.16 dex. The added variance in the unresolved/compact sources is
due to the fact that most are lower S/N and have significantly fewer complete
colors ($<$30%; see Fig. 1a and Table 1), but may also reflect the limitations
with estimating “total” fluxes using ALLWISE aperture-corrected measurements
of compact (and likely resolved to some extent) galaxies.

Finally, we consider the resulting $\Upsilon_{\bigstar}^{3.4\mu m}$ ,
comparing the derived M/L to the fiducial counterpart, shown in Fig. 5b. The
resulting M/L distribution (black line), has a peak or mode at
$\Upsilon_{\bigstar}^{3.4\mu m}$ = 0.34, a mean value of 0.343, and a FWHM
width of 0.21 (bearing in mind the distribution is not exactly Gaussian).
Nearly all sources have a M/L between 0.2 to 0.6. Compared to the fiducial
$\Upsilon_{\bigstar}^{3.4\mu m}$ values (i.e., the ‘truth’ set; see green line
histogram), which has a mode near 0.35, the fiducial distribution is slightly
wider in width (and hence, a larger range in M/L), notably the extreme or
wings of the distribution. The slightly narrower range in M/L for the WISE
values is a reminder that using only limited broad-band information to derive
the global stellar mass does have its limitations in tracing the full
diversity of M/L that galaxy populations exhibit. Nevertheless, with these new
scaling relations, it will be possible to estimate the Log
M${}_{\scaleto{\star}{5pt}}$ for any galaxy in the sky with 25-50% fidelity.

##  5 Practical Matters: Simplified k-corrections and Stellar Mass

The method for estimating stellar masses presented thus far uses the W1
luminosity and two colors of WISE. These are then weight-combined into a mean
value based on their excepted variance. Moreover, the measurements have been
rest-frame corrected. The latter is particularly daunting with the WISE bands
(see the Appendix). In this section, we offer some simple prescriptions for
estimating masses that help overcome the data complexities. Although the
resulting masses may not be as reliable as the robust determination presented
thus far, they nevertheless will be estimates that suffice for most needs.

Figure 6: WISE mid-IR stellar masses derived using only the WISE W1 luminosity
(Eq. 2). Compare with the complete method results, Fig. 5, the scatter has
increased (left) and the M/L is more restricted to the mean value $\sim$0.36.
It should be noted that although the random scatter is reasonable given the
restricted M/L range, they are in fact strongly correlated with M/L; i.e.
different errors for galaxy types (e.g., early vs late-types).

Figure 7: WISE mid-IR stellar masses derived using only the WISE $\rm W1{-}\rm
W2$ color (Eq. 3). Compare with the complete method results, Fig. 5, there are
fewer sources at low mass (left), while the M/L covers nearly all the range
observed in the fiducial sample.

Step one is to apply rest-frame correction to the measured fluxes. As we show
in the Appendix, the diversity of galaxy types and their associated variation
in energy distributions means that it is non-trivial to carry out these
corrections for the mid-IR, requiring careful SED modeling. We note, however,
that the W1 3.4 $\mu$m correction is approximated by an exponential law, that
works well for most galaxies and appears robust for redshifts $<$ 0.5 (see Eq.
A.1 and Table 2 for the correction to the W1 flux). Likewise, a simple
polynomial (cubic) works well for the $\rm W1{-}\rm W2$ color; see Eq. 2 and
Table 2. However, even in these instances, there will be some variation across
galaxy types; graphically, these k-corrections are shown in Fig. 9 of the
Appendix.

For the ISM-sensitive color, $\rm W1{-}\rm W3$ (and $\rm W3{-}\rm W4$), there
is no simple prescription, and we, therefore, do not recommend using a
simplified function. The k-corrections for these redward bands are graphically
shown in Fig. 10. On the other hand, if a source has a clear W3 detection, it
likely has dusty star-formation and you can be sure it will be a later type
(Sb, Sc, etc). You can use the curves in the Appendix with the appropriate
correction and attempt a rest-frame correction to the W3 flux (in which case
Eq. 4 may then be used).

As shown in Cluver et al. (2014) and in this current study, the rest-frame
corrected WISE color $\rm W1{-}\rm W2$ is a reasonable and reliable tracer of
the mass-to-light, and it alone (with the W1 luminosity) will provide a very
effective stellar mass estimate. We thus recommend that the simple relation
given in Eq. 3 be used for this purpose, assuming a $\rm W1{-}\rm W2$ color
that has a S/N of at least 5. For those cases where W2 is faint or undetected,
resort to the relation given in Eq. 2, with an expected uncertainty of about
30%.

In summary, to derive a stellar mass using a minimal set of information, we
recommend the following:

  1. 1.

Convert the observed W1 mag (total mags are best) to flux density (e.g., the
zero point correction is 309.68 Jy for W1 Vega mags; see Jarrett et al.
(2011)).

  2. 2.

Correct the W1 flux to the rest frame, you can look up the scale factor
relevant to your source redshift from Fig. 9, or apply the exponential law
relation given in Eq. A1. and Table 2. The rest flux is found by scaling by
this factor; accordingly, fW1[rest] = $\frac{f_{\rm
W1}[rest]}{f_{o}}\times$fW1[obs].

  3. 3.

Compute the W1 in-band luminosity, LW1, by first converting your rest-frame
flux to absolute magnitude (use the distance modulus after converting your
redshift to luminosity distance), and applying Eq. 1.

  4. 4.

If you have a W2 detection and the resulting $\rm W1{-}\rm W2$ uncertainty is
better than 0.2 mag, you should attempt a k-correction using Fig. 9 or using
the polynomial given in Eq. A2 and Table 2. The rest-frame color is [$\rm
W1{-}\rm W2$]rest = [$\rm W1{-}\rm W2$]${}_{obs}-\delta$[$\rm W1{-}\rm W2$].
Note the two redshift ranges for the polynomial coefficients specified in
Table 2.

  5. 5.

The stellar mass may be estimated from simply the LW1 using Eq. 2. However if
you have a good $\rm W1{-}\rm W2$ color, we recommend instead using Eq. 3 (or
a weighted combination of the two), in which case a better range in mass-to-
light is covered. The stellar mass is then: LW1$\times$
$\Upsilon_{\bigstar}^{3.4\mu m}$.

This simplified stellar mass prescription is still very effective, producing
accurate stellar masses, with only a few% additional scatter compared to the
comprehensive treatment. This is shown in Figures 6 and 7, where we show the
stellar masses compared to the fiducial for the two simple cases: (1) using
only the W1 luminosity, and (2) using only the $\rm W1{-}\rm W2$ color.
Although the random errors look reasonable, we note that errors are strongly
correlated with M/L; i.e. different errors for galaxy types (e.g., early vs
late-types).

In the first case, lower luminosity sources have estimated masses, but they
also have a restricted range in M/L, between 0.3 to 0.42. The total scatter
has increased by 5-10%. In the second case, there are far fewer sources with
good quality $\rm W1{-}\rm W2$ color, but for those that do have a color, the
results are excellent, with only 2% increased scatter compared to the
comprehensive derivation, while the full range in $\Upsilon_{\bigstar}^{3.4\mu
m}$ is covered (almost identical to the fiducial sample used to calibrate the
masses). It is clearly advantageous to have color available to estimate
accurate stellar masses.

###  5.1 Example

As an example of using these prescriptions, we consider WXSC-2dF source 96128;
Ra, Dec(J2000) $=$ (358.70312, -33.04600 ). The SED is shown in Fig. 8, which
includes 2MASS and WXSC measurements, as well as the template used to
restframe-correct the observed fluxes. The observed W1 mag is 13.200$\pm$0.015
(1.6222$\pm$0.0225 mJy), and colors(mag) $\rm W1{-}\rm W2$$=$ 0.178$\pm$0.027,
$\rm W1{-}\rm W3$$=$ 3.794$\pm$0.038, $\rm W3{-}\rm W4$$=$1.653$\pm$0.190. The
CMB-frame redshift is 0.06057 (helio is 0.06150), and the corresponding
luminosity distance is 271.6 Mpc (co-moving, 256 Mpc), so the distance modulus
is 37.17 mag. The Log LW1 and M${}_{\scaleto{\star}{5pt}}$ from the WXSC
(which uses SED-fitting for k-corrections and the full-up
M${}_{\scaleto{\star}{5pt}}$ method) are, respectively, 10.811$\pm$0.028 L⊙
and 10.272$\pm$0.079 M⊙. The M/L is thus 0.29.

K-correction: For the redshift and using exponential function Eq A1. with
coefficients (1.0 and -2.615; see Table 2), the scaling factor is 0.854. That
is to say, the W1 flux is brightened by 20% due to the R-J tail shifting into
the 3.4 $\mu$m band. So we apply the correction to give a rest flux of
1.6222$\times$0.854 = 1.387 mJy. Converting this flux back to magnitudes
(13.37 mag), and applying the distance modulus, the resulting absolution
magnitude, MW1, is -23.80 mag. And finally using Eq. 1, we arrive at the W1
luminosity, Log LW1 = 10.816 L⊙. (note: if you do not apply a k-correction to
the flux, the Log luminosity would be $\sim$0.08 dex too large). From Eq. 2,
the estimated Log M${}_{\scaleto{\star}{5pt}}$ is 10.32 M⊙, and the resulting
$\Upsilon_{\bigstar}^{3.4\mu m}$ is 0.32.

Since the $\rm W1{-}\rm W2$ color has a good quality S/N, we proceed with a
rest-frame correction. At the redshift of the source, the change in color is
(Eq. A2, or look up with Fig. 9) is 0.036 mag, but may also range from 0.02 to
0.06 mag depending on the galaxy type (see the figure). Using the average
value of 0.036 mag, the rest frame color is then $0.178-0.036=0.142$ mag. It
then follows from Eq. 3, that $\Upsilon_{\bigstar}^{3.4\mu m}$ is 0.30. Hence,
the resulting Log M${}_{\scaleto{\star}{5pt}}$ is 10.29 M⊙. This mass is 7%
smaller than the estimate using only the luminosity because the mass-to-light
is slightly smaller, appropriate to a star-forming galaxy (which it clearly
is, see Fig. 8. It is also nearly equivalent to the full-up value, showing
that even the simplest of prescriptions works very effectively. If no
k-correction was applied to the color, the M/L is lower (0.27) and the stellar
mass is correspondingly lower by 0.04 dex (10%).

##  6 Discussion

Galaxies, enclosed within their dark-matter-dominated halos, are comprised of
stars in all stages of their growth and evolution. The aggregate stellar mass
will depend on the stellar content, which we observe and estimate through its
light. Given the observed band or set of bands, and the initial mass function,
we can infer the mass distribution and hence, total mass from stars. It is
clearly a complex problem for extragalactic sources given that only aggregate
(population-combined) measurements are typically available, thus washing out
any differences galaxies may have from one to another. Typically a single M/L
is adopted for a given band for convenience (and standardization advantages),
notably with large surveys where limited information is available. However, it
has been shown that adding some simple additional information – notably the
colors – provides some stellar population information that informs a more
diverse range in $\Upsilon_{\bigstar}$ and increases fidelity to the resulting
M${}_{\scaleto{\star}{5pt}}$. Adding additional information, such as star-
formation history and morphological structure may further improve the M/L
coverage.

Our new scaling relations, derived simply from the rest wavelength WISE W1
(3.4 $\mu$m) total fluxes, and ($\rm W1{-}\rm W2$) and ($\rm W1{-}\rm W3$)
colors, provide M${}_{\scaleto{\star}{5pt}}$ estimates for galaxies that range
from dwarf to giant galaxies (107 \- 1011.5 M⊙) with $\sim$25-40% accuracy
(relative to the fiducial masses used to create the relations) while covering
a M/L range between 0.2 to 0.6 that reflects the dominant stellar populations
in the individual galaxies. The mean $\Upsilon_{\bigstar}^{3.4\mu m}$ we
compute is 0.34 $\pm$ 0.11, consistent with recent derivations using the mid-
infrared (e.g. Leroy et al., 2019), but somewhat lower than earlier work with
Spitzer and WISE that had values close to 0.5 to 0.8 (e.g. Meidt et al., 2014,
yet for perspective, these larger $\Upsilon_{\bigstar}$ values are seemingly
appropriate to early type, spheroidal galaxies).

Our newly derived masses should be slightly lower than previous estimates
since we are now using the new GAMA/KIDS/VIKING-G23 stellar masses derived
from sophisticated SED fitting with opt-IR multi-band photometry, as well as
our new, updated WISE resolved galaxy photometry. These masses should have a
typical (normal galaxy) uncertainty of 25-30% or 0.10-0.12 dex in the Log, but
most importantly, be more appropriate to those with significant population
differences; e.g., early vs late-types; passive vs. active types, which can be
significantly (2$\times$) different in M/L value ($cf.$ Schombert et al.,
2022). Even the simpler prescriptions we provide to estimate stellar masses
from only the W1 flux or the $\rm W1{-}\rm W2$ color (see Section 5) have
relatively good (30-50%) mass estimates for all luminosity/mass ranges, and at
least with the single color method, cover a wide range in
$\Upsilon_{\bigstar}^{3.4\mu m}$. Since galaxy evolution work is focused on
how galaxies grow and evolve in their home environment over cosmic time, thus
covering the full range of masses that populate the field, groups, and
clusters, it is vital that an accurate M/L and aggregate stellar mass be
estimated for the entire range, especially the most difficult to observe and
study, the dwarf galaxy regime.

These new relations should improve ongoing and future studies, provide a
stronger ‘control’ to cross-compare physical parameters and construct more
representative SFR-M${}_{\scaleto{\star}{5pt}}$ diagrams, the so-called ‘star
formation main sequence’ that is a powerful diagnostic tool of past-to-present
SF history. It is particularly important now to have this capability since
next-generation large-area surveys are happening in this decade, notably from
radio, optical and X-ray surveys, which all depend on having a stellar mass
estimate for individual galaxies. WISE covered the entire sky, reaching depths
of 20 to 30 $\mu$Jy, and hence may provide the necessary information to
estimate M${}_{\scaleto{\star}{5pt}}$. If the galaxy is local, $z$ $<$ 0.1 to
0.2, then it is likely resolved in the 3.4 $\mu$m band, requiring a more
careful extraction, typically not conducive to automated pipelines because of
blending, contamination, and shredding issues that plague galaxy photometry.

The WISE resolved catalog, WXSC, was created to fill this need, to carefully
measure local universe galaxies, and to extract basic properties that include
colors and total fluxes. It was effectively used to derive scaling relations
with M${}_{\scaleto{\star}{5pt}}$ and SFR (e.g., Cluver et al., 2017; Jarrett
et al., 2019), and is being used to study local galaxy samples (e.g. Ogle et
al., 2019; Yao et al., 2020; Naluminsa et al., 2021; Bok et al., 2020, 2022;
Healy et al., 2021). Since the WXSC is not fully complete for the local
universe (as of this writing, it contains about 200,000 sources or an order of
magnitude less than is required for the total volume), it remains an ongoing
task to build and hence estimate the stellar masses for galaxies in the local
universe.

##  7 Summary

Updating from the original work in Cluver et al. (2014), we derive new mid-IR
based scaling relations that are used to estimate global stellar masses for
galaxies. The primary photometric data are drawn from the WISE Extended Source
Catalogue (WXSC), developed to extract and characterize galaxies that are
resolved in the W1 3.4 $\mu$m band. Compact and unresolved galaxies are drawn
from the archival ALLWISE catalog, with further corrections to these
measurements to capture the total flux of compact-resolved sources. The
stellar masses, used as the fiducial calibration set, come from the new DR4
Catalogue of the the GAMA-KiDS-VIKING survey of the southern hemisphere G23
field.

Restricting to redshifts $<$ 0.15 and focusing on normal galaxies that span 6
orders of magnitude in W1 luminosity, we construct three scaling relations:
the LW1 versus the M${}_{\scaleto{\star}{5pt}}$, and the WISE $\rm W1{-}\rm
W2$, $\rm W1{-}\rm W3$ colors versus the 3.4 $\mu$m mass-to-light
($\Upsilon_{\bigstar}^{3.4\mu m}$). The color relations are sensitive to a
variety of galaxy types from early-type or passive to late-type or star-
forming. Combining the scaling relations using inverse-variance weighting, we
produce a mean stellar mass estimate that has a residual scatter (relative to
the fiducial GAMA stellar masses) better than 0.10-0.12 dex accuracy for most
galaxies (with W1 luminosities $>10^{9}$M⊙) and 0.15-0.25 dex for lower mass
dwarf galaxies. These new scaling relations will enable stellar mass estimates
for any galaxy in the sky detected by WISE with $\sim$25-50% fidelity.

Detailed highlights include the following:

  * •

The DR4 G23 catalog provides 16,000 high-quality stellar masses that are used
as the fiducial calibration set. Cross-matching with the WISE photometry
catalog returns 3374 and 3297 resolved and compact sources, respectively, with
S/N $>$ 5\. A slighter smaller fraction of resolved sources have a full set of
WISE colors.

  * •

Essentially the Log LW1 scales linearly with Log M${}_{\scaleto{\star}{5pt}}$,
although we fit a stiff (cubic) polynomial that accounts for a small upturn at
the high-mass end (see Eq. 2). The roughly linear relation means that a single
or tight-range in mass-to-light adequately ($\sim$30-50%) describes the
stellar mass in the W1 band. We find a mean $\Upsilon_{\bigstar}^{3.4\mu m}$
value of 0.35 $\pm$ 0.11. Relative to the fiducial mass, the derived mass has
a statistical scatter of about 0.12 dex, but several % higher for the low mass
regime. It should be noted, however, embedded are strong differential
systematics for early and late-types – which can amount to as much as 0.6 dex
(peak to trough) in log mass. These systematics cannot be meaningfully
encapsulated with simple Gaussian statistics.

  * •

Unlike the single luminosity scaling relation, colors trace a more
representative range in M/L. The WISE $\rm W1{-}\rm W2$ color scales linearly
with the Log M${}_{\scaleto{\star}{5pt}}$, see Eq. 3. It covers the expected
range that normal galaxies, from passive to dusty star-forming, exhibit, with
a resulting $\Upsilon_{\bigstar}^{3.4\mu m}$ range of 0.7 to 0.16,
respectively. It has a 25-30% residual scatter.

  * •

Color $\rm W1{-}\rm W3$ has a more complex trending with Log
$\Upsilon_{\bigstar}^{3.4\mu m}$, steepening from linearity at colors redder
than 3.1 mag. The flat $\Upsilon_{\bigstar}^{3.4\mu m}$ regime and roll-off to
much lower values conveniently fit with a Schechter Function, arriving at a
scaling relation (Eq. 4) that has a residual scatter from 23-38%, the higher
variance at the red color end where the mass-to-light is changing more rapidly
and less constrained.

  * •

Using inverse-variance weighting, we create a mean-combined stellar mass from
the three scaling relations. In order to cover a wider range in mass-to-light
and create an optimal, maximally informative combination, we de-weight the
first scaling relation to emphasize the power of the mid-IR colors to better
sample the variety of real galaxies. The resulting scatter in the Log
M${}_{\scaleto{\star}{5pt}}$ is $\sim$0.11 (in Log dex) for most galaxies, 109
to 1012 M⊙ in W1 luminosity, rising to 0.16-0.17 dex for those with faint flux
levels and lacking full-color information.

  * •

In terms of mass-to-light, the derived (weight-combined) WISE
$\Upsilon_{\bigstar}^{3.4\mu m}$ has a mean value of 0.34 and a FWHM width of
0.21 (insofar as a Gaussian can be modeled), while the extreme wings of the
distribution reach values as low as 0.2 and as high as 0.6.

  * •

We also provide simpler prescriptions for estimating the mass, using (1) only
the rest-frame W1 luminosity, or (2) only the rest-frame $\rm W1{-}\rm W2$
color. In conjunction, we provide a (far) simpler prescription for estimating
the rest-frame correction to the W1 flux and the colors (details in the
Appendix). These simple prescriptions produce satisfying results, with with a
$\sim$few% more scatter compared to the full-up method, while also (using
color) covering the full range in $\Upsilon_{\bigstar}^{3.4\mu m}$.

T.H.J. acknowledges support from the National Research Foundation (South
Africa). M.C. is a recipient of an Australian Research Council Future
Fellowship (project No. FT170100273) funded by the Australian Government. This
publication makes use of data products from the Wide-field Infrared Survey
Explorer, which is a joint project of the University of California, Los
Angeles, and the Jet Propulsion Laboratory/California Institute of Technology,
funded by the National Aeronautics and Space Administration. GAMA is a joint
European-Australasian project based around a spectroscopic campaign using the
Anglo-Australian Telescope. The GAMA input catalog is based on data taken from
the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey.
Complementary imaging of the GAMA regions is being obtained by a number of
independent survey programs including GALEX MIS, VST KIDS, VISTA VIKING, WISE,
Herschel-ATLAS, GMRT, and ASKAP providing UV to radio coverage. GAMA is funded
by the STFC (UK), the ARC (Australia), the AAO, and the participating
institutions. The GAMA website is located at http://www.gama- survey.org/.

##  Appendix A WISE Rest-frame Corrections

Unlike the near-infrared, rest-frames corrections are non-trivial for the WISE
mid-IR bands due to strong spectral features, including the [Si] absorption at
10 $\mu$m, broad and prominent PAH emission bands at 6, 8, 11.3 $\mu$m and
[NeII] emission at 12.8 $\mu$m, in addition to the underlying stellar
continuum and warm dust continuum. In general, simple power laws are limited
to describing the shift in flux with (1+z) photometric band-shifting. From one
galaxy to the next, there are varying contributions from the stellar continuum
(current and past history), emission lines, and ISM components (e.g. dust)
that arise from current star formation activity, rendering a composite
spectrum that reflects the diversity of galaxies observed in the local
universe.

The approach adopted by the WXSC team is to model the spectrum using composite
(pre-constructed) templates that range from the blue (optical) to the mid-
infrared ($<$40 $\mu$m) and measure the synthetic fluxes (integrated across
the photometric bandpasses), comparing them to the observed photometry. We
identify the best fit template, scaling the “observed” measurements to the
equivalent “rest” measurements. The list of composite templates we deploy are
from Brown et al. (2014) and Spitzer-SWIRE/GRASIL Polletta et al. (2006,
2007); Silva et al. (1998), totaling over 135 composite templates from a wide
range of morphological and activity types. Further details of the SED template
fitting are given in Jarrett et al. (2013, 2017).

Figure 8: Spectral energy distribution (SED) of the WXSC-2dF source 96128,
whose redshift is 0.0606. The near-infrared data (green points) are total
fluxes from the 2MASS Extended Source Catalog, and the mid-IR (blue points) is
from the WISE XSC. The best fit (black line) was found to be that of the NGC
4321 template (from Brown et al., 2014), which has a morphological type of
SABbc. LEFT panel shows the observed (i.e., redshifted) fluxes, and RIGHT
panel the k-corrected rest-frame fluxes (note the pink triangles are the
expected fluxes based on synthetic photometry). Also shown are the WISE filter
traces (scaled to unity; blue, green, orange, and red lines), as given in
Jarrett et al. (2011).

An example SED is shown in Fig. 8, a typical spiral/disk galaxy from the
GAMA/KiDS-s sample in the WXSC. It shows the observed fluxes, as measured by
2MASS and WISE (left panel), with the best fit spectral template indicated by
the solid line. It features the well-known stellar ”bump” that peaks in the
H-band (1.6 $\mu$m), characteristic of the luminous evolved population. The
R-J tail of the continuum extends into the mid-IR where the WISE W1 3.4 $\mu$m
and W2 $\mu$m bands pick up the emission. It then abruptly transitions to ISM
emission, warm dust continuum which now rises with longer wavelength, PAH and
[NeII] emission in the broad W3 12 $\mu$m band, and finally steep rising dust
continuum in the W4 23 $\mu$m band. Correcting to the rest frame using the
synthetic photometry of the spectral template, the resulting rest-frame SED is
shown in Fig. 8, right panel.

When the rest fluxes land on the expected synthetic fluxes (pink triangles),
the fit quality should be good, as indicated by the $\chi^{2}$ metric, as is
the case in the example given here. There is a finite set of templates, and
the photometric quality determines fit quality, and hence not all fits are
high quality, notably for those with only a few points (i.e., faint) or with
extreme properties that are not well characterized by our suite of templates.
For the most part, the k-correction imparts less than 5-10% uncertainty for
most sources that have redshifts $<$ 0.3 (see also the Appendix in Yao et al.
(2022)).

A cautionary note about rest-corrections: as pointed out by Hogg et al.
(2002), there is an additional (1+z) scaling that must be applied when using
spectral luminosities ($\nu$Lν). For example, star formation rates are
computed using spectral luminosities (e.g., Cluver et al., 2017) and hence
would have the luminosity normalized by (1+z). Whereas for bolometric
luminosities, this factor does not apply, which is the case for the stellar
masses in which the W1 in-band luminosity (see Eq. 1) is used for the
$\Upsilon_{\bigstar}^{3.4\mu m}$ scaling relations.

To show the complexity of the rest frame corrections, as a function of
redshift, we plot the expected correction of the W1 flux (mJy units) and the
WISE colors (in Vega mag units) for a set of templates that range from early
to late types (bulgy to disk). These are normal galaxies and their expected
optical-infrared emission spectra, from which we compute synthetic fluxes,
comparing the observed with the rest flux and colors. (note, we are not
showing the exotica, which may look completely different; e.g. QSOs).

Similar to the near-infrared K-band band (at 2 $\mu$m), the WISE W1 3.4 $\mu$m
band gets brighter with (1+z) because of the ‘peak’ stellar bump at 1.6 $\mu$m
shifts longward toward the K and W1 bands, increasing their flux (relative to
rest, ignoring the 4$\pi$r2 flux dilution). As shown in Fig. 9, upper left
panel, the expected change in flux with redshift is well-behaved, an
exponential function (or power-law), similar for all galaxy types. This
demonstrates how most normal galaxies have the same near-infrared (1-4 $\mu$m)
spectrum (i.e., the stellar bump from evolved, luminous K/M stars), nearly a
”standard candle” which may be exploited in photometric-redshift studies. The
k-correction is the scale factor shown on the Y-axis, it is less than unity to
reduce the observed flux to the rest value. Do note that the correction is
significant, 25% at z=0.1, and nearly 75% at z = 0.5. By doing no correction
and just assuming the fluxes and colors are ’rest-like’, the “rest” flux will
be highly over-estimated, badly skewing results. It is important to apply a
k-correction to the flux (and colors, see below). Below we provide simpler
descriptions which, even if limited, are better than doing nothing.

For convenience to the WISE user, we fit an exponential (a power-law also
works well) to the ”Sb” template, it is a good middle-ground, straddling all
galaxy types; it has the form:

$\frac{f_{\rm W1}[rest]}{f_{o}}=a_{0}e^{\alpha z}$ (A1)

where we fit for the scaling $a_{0}$ and index $\alpha$. The rest flux follows
by scaling by this ratio; accordingly, fW1[rest] = $\frac{f_{\rm
W1}[rest]}{f_{o}}\times$fW1[obs]. We find that the redshift range between 0 to
0.5 is well fit. Beyond that redshift limit, it is best to do proper SED
modeling to derive the k-corrections. We emphasize an exponential or power-law
is just an approximation of the rest-frame correction required for the W1
flux. In Table 2, we provide the best fit coefficients for this limited
redshift range.

Figure 9: Rest-frame corrections for WISE W1 3.4 $\mu$m (left) and $\rm
W1{-}\rm W2$ (right) based on SED composite templates that cover the galaxy
types from early-to-late (bulge-to-disk dominated). The W1 scaling correction
is approximated by an exponential function, see Eq. A1, and the color by a
cubic polynomial (Eq. A2); both based on the Sb template.

Now consider the other bands or colors. The R-J stellar tail is still well-
detected in the W2 at 4.5 $\mu$m. Hence we expect the W2 correction to be
simple as well. However, the W2 band is subject to warm dust from the ISM and
from AGN, it can be enhanced in SF galaxies and active galaxies; e.g., see
Fig. 1, right panel. The resulting W1-W2 color k-correction has more variation
in galaxy type; see Fig. 9, upper right panel. Notably, the early types, E’s n
the diagram, are somewhat different from the disk galaxies, S types. At higher
redshifts, z $>$ 0.2, there is some mixing of types. At this point, it is
clear that more sophisticated modeling is necessary to estimate the rest-frame
correction for a given galaxy. However, in the spirit of deriving a simple and
convenient relation that may be used for a large number of cases (i.e., normal
disks in the local universe), we fit a 3rd order polynomial to the ”Sb”
template (again, a good middle ground, at least for z $<$ 0.2); it has the
form:

$\delta[W1-W2]=a_{0}+a_{1}z+a_{2}z^{2}+a_{3}z^{3}$ (A2)

where we fit for the a-coefficients. The rest-frame color is simply: [$\rm
W1{-}\rm W2$]rest = [$\rm W1{-}\rm W2$]${}_{obs}-\delta$[$\rm W1{-}\rm W2$].
Here we fit two redshift ranges, 0-0.25 and 0.25 - 0.5 to accommodate the non-
linear curve. The coefficients are specified in Table 2. Once again, this is
only an approximation, notably for disk galaxies.

Figure 10: Rest-frame corrections for WISE $\rm W1{-}\rm W3$ color (left) and
$\rm W3{-}\rm W4$color (right) based on SED composite templates that cover the
galaxy types from early to late.

Finally, we consider the longward bands of WISE. Fig. 10 shows the expected
k-corrections for the $\rm W1{-}\rm W3$ color (left panel), and the $\rm
W3{-}\rm W4$ color (right panel). As demonstrated in Fig. 8, these bands are
subject to a number of absorption, emission, and continuum components that are
unique to the galaxy type. At low redshifts, z $<$ 0.1, there are two paths (E
types, and S types) that may be approximated with an exponential function, for
higher redshifts, there is no common ground to form a simple k-correction
relation. It is best to use SED modeling to derive the needed corrections or
use these figures as a lookup table.

Table 2: k-correction Coefficients for Eq. A1 and A2 $a_{0}e^{\alpha z}$ |
---|---
W1 flux |  a0 | $\alpha$ |  |
z $<$ 0.5 | 1.0 | -2.614 |  |
$a_{0}+a_{1}z+a_{2}z^{2}+a_{3}z^{3}$
$\rm W1{-}\rm W2$ |  a0 |  a1 |  a2 |  a3
z $<$ 0.25 | 0.000 | 0.142 | 9.193 | -26.980
0.25 $<$ z $<$ 0.5 | -0.040 | 2.258 | -6.306 | 3.875

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