11institutetext:  Astronomical Institute, Academy of Sciences, Boční II 1401,
CZ-14131 Prague, Czech Republic
11email<EMAIL_ADDRESS>22institutetext: Institute of Astrophysics, FORTH, GR-71110 Heraklion, Greece 33institutetext: Department of Physics, Univ. of Crete, GR-70013 Heraklion, Greece 44institutetext: Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA 55institutetext: Department of Astronomy, Indiana University, Bloomington, IN 47404, USA

# Star-formation rate and stellar mass calibrations based on infrared
photometry and their dependence on stellar population age and extinction

K. Kouroumpatzakis<EMAIL_ADDRESS>0000-0002-1444-2016 A. Zezas 334422 0000-0001-8952-676X E. Kyritsis 3322
0000-0003-1497-1134 S. Salim 55 J. Svoboda 11 0000-0003-2931-0742

(Received February 01, 2023; accepted March 16, 2023)

###### Abstract

Context. The stellar mass ($M_{\star}$) and the star-formation rate (SFR) are
among the most important features that characterize galaxies. Measuring these
fundamental properties accurately is critical for understanding the present
state of galaxies, their history, and future evolution. Infrared (IR)
photometry is widely used to measure the $M_{\star}$ and SFR of galaxies
because the near-IR traces the continuum emission of the bulk of their stellar
populations (SPs), and the mid/far-IR traces the dust emission powered by
star-forming activity.

Aims. This work explores the dependence of the IR emission of galaxies on
their extinction, and the age of their SPs. It aims at providing accurate and
precise IR-photometry SFR and $M_{\star}$ calibrations that account for SP age
and extinction while providing quantification of their scatter.

Methods. We use the CIGALE spectral energy distribution (SED) fitting code to
create model SEDs of galaxies with a wide range of star-formation histories,
dust content, and interstellar medium properties. We fit the relations between
$M_{\star}$ and SFR with IR and optical photometry of the model-galaxy SEDs
with the Markov-chain Monte-Carlo (MCMC) method. As an independent
confirmation of the MCMC fitting method, we perform a machine-learning random
forest (RF) analysis on the same data set. The RF model yields similar results
to the MCMC fits validating the latter.

Results. This work provides calibrations for the SFR using a combination of
the WISE bands 1 and 3, or the JWST NIR-F200W and MIRI-F2100W. It also
provides mass-to-light ratio calibrations based on the WISE band-1, the JWST
NIR-F200W, and the optical $u{-}r$ or $g{-}r$ colors. These calibrations
account for the biases attributed to the SP age, while they are given in the
form of extinction-dependent and extinction-independent relations.

Conclusions. The proposed calibrations show robust estimations while
minimizing the scatter and biases throughout a wide range of SFRs and stellar
masses. The SFR calibration offers better results, especially in dust-free or
passive galaxies where the contributions of old SPs or biases from the lack of
dust are significant. Similarly, the $M_{\star}$ calibration yields
significantly better results for dusty/high-SFR galaxies where dust emission
can otherwise bias the estimations.

###### Key Words.:

galaxies:general – galaxies:star formation – galaxies:stellar content –
galaxies:ISM – infrared:galaxies – (ISM:) dust, extinction

##  1 Introduction

Galaxies are among the most important structures of the Universe hosting the
bulk of its cold baryonic mass. The fundamental process of star formation (SF)
which takes place within galaxies, transforms the gas into stars and
therefore, plays a significant role in all aspects of galaxy evolution, and
their current state. For example, the current star-formation rate (SFR) of
galaxies is strongly correlated with their stellar mass ($M_{\star}$; e.g.
Elbaz et al. 2007, a.k.a. main sequence of galaxies), with their galactic
winds (e.g. Heckman et al. 1990; Lehnert & Heckman 1996), the number of
supernovae (e.g. Mac Low et al. 2005), and their X-ray luminosity originating
from high-mass X-ray binaries (e.g. Mineo et al. 2012; Kouroumpatzakis et al.
2020).

Similarly, the $M_{\star}$ of galaxies is strongly correlated with their
kinematic and morphological parameters (e.g. Bundy et al. 2005; Genzel et al.
2006) while it is also related to the nature of their stellar content and the
properties of their interstellar medium (ISM), for example through the
mass–metallicity relation (e.g. Tremonti et al. 2004). Moreover, there is a
strong correlation between the $M_{\star}$ of the bulges of galaxies with the
mass of their central supermassive black hole (e.g. Kormendy & Ho 2013).
Therefore, being able to accurately estimate the SFR and $M_{\star}$ is
crucial for most studies involving galaxies, and studies of the evolution of
the Universe.

Galaxies come in many forms and shapes, and in different states with respect
to their current star-forming activity or ISM properties. This is the main
reason for discrepancies in measuring the SFR because the various SFR tracers
are based on different emission mechanisms related to SF. These tracers
include: the ultraviolet (UV) light coming directly from the photospheres of
massive stars, emission lines from nebulae that have been ionized by star-
forming activity (e.g. H$\alpha$, [O III]), infrared (IR) emission from dust
that was heated by the young stars’ UV and optical emission, radio emission
from relativistic electrons related to supernovae activity, and several others
(for a review see Kennicutt & Evans 2012).

Depending on the availability of observations different methods to estimate
the SFR are used. However, it has been shown that there can be significant
discrepancies between these methods, sometimes up to an order of magnitude
(e.g. Kennicutt & Evans 2012). In particular, although monochromatic IR-based
SFR indicators are good for dusty/high-SFR galaxies, they can lead to
significant underestimation of the true SFR in low-mass or low-metallicity
dust-deficient galaxies (e.g. Calzetti et al. 2007; Kouroumpatzakis et al.
2021). Moreover, IR SFR tracers can be also biased by the age of the stellar
population (SP; e.g. Cortese et al. 2008; Leroy et al. 2012; Boquien et al.
2014; Cerviño et al. 2016; Nersesian et al. 2019; Kouroumpatzakis et al. 2020)
or stochastic heating of the dust (e.g. Camps et al. 2015; Lianou et al.
2019). On the other hand, the use of UV or H$\alpha$ emission as SFR tracers
require precise measurements of extinction in order to provide reliable
estimations.

A possible solution is to use one of the hybrid SFR indicators (e.g. 24$\mu$m
+ H$\alpha$, FIR + UV; e.g. Kennicutt et al. 2009; Hao et al. 2011). They
combine emission that traces SF in a more direct way but can be affected by
absorption, along with IR emission which is not. Thus, these hybrid tracers
account for the energy lost from the UV/optical light that has suffered
absorption by measuring the corresponding IR light. However, it is relatively
harder to have observations in both wavelength regimes due to the lack of deep
UV or spectral observations. Spectral energy distribution (SED) fitting can
provide robust estimations for both the $M_{\star}$, and SFR, but require
extended photometry in multiple bands, while the lack of UV or IR photometry
can still lead to large biases (e.g. Lanz et al. 2013).

As a result of these limitations, most catalogs providing SFRs, and stellar
masses for local Universe galaxies were limited in the SDSS footprint where
the sources were mainly characterized through SED fitting combining the
optical with WISE (Wright et al. 2010) and/or GALEX (Martin et al. 2005)
photometry (e.g. Chang et al. 2015; Salim et al. 2016, 2018). Other surveys
not limited to the SDSS footprint (e.g. Sheth et al. 2010; Ashby et al. 2011;
Leroy et al. 2019; Bianchi et al. 2018; Nersesian et al. 2019) combined IR and
UV GALEX observations, however, all of them were limited to relatively nearby
galaxies and small samples. Catalogs targeting the whole sky were based solely
on IR photometry (e.g. Jarrett et al. 2013; Kovlakas et al. 2021).

A significant problem in measuring SFRs through monochromatic IR emission is
that they do not account for the different degrees of extinction between
galaxies, especially in low-metallicity, dust-poor galaxies. As a result, an
IR SFR indicator would underestimate the SFR for a galaxy with less dust
because in this case a larger amount of UV photons, produced by the star-
forming activity, would have escaped with respect to a higher-extinction
galaxy. Moreover, the fact that the dust can be excited but other means than
reprocessing of UV radiation of young stars (e.g. stochastic heating, heating
by older SPs, post-AGB stars, e.t.c. Sauvage & Thuan 1994; Draine & Li 2001;
Nersesian et al. 2019; Zhang & Ho 2022) adds another complication for the IR
SFR indicators. In fact, several studies have shown that in quiescent galaxies
the mid/far-IR emission is higher with respect to their SFR (e.g. Davis et al.
2014; Simonian & Martini 2017; Leroy et al. 2019). Therefore, for these cases,
an IR SFR indicator would overestimate their SFR.

Similarly, the $M_{\star}$ measurements can be obtained by means of SED
fitting, which requires photometry in an extended wavelength range. However,
most commonly they are based on near-IR (NIR) photometry (e.g. 2MASS $K_{\rm
S}$, WISE band-1 or band-2) which traces the thermal/continuum emission of
low-mass SPs. Such observations provided $M_{\star}$ measurements for a large
amount of relatively nearby galaxies through the all-sky IR surveys (e.g.
IRAS, AKARI, 2MASS, WISE). However, it has been shown that the relation
between NIR emission and the $M_{\star}$ depends strongly on the age of the
SPs. In the past, optical (e.g. Bell et al. 2003; Zhu et al. 2010; Wen et al.
2013) or IR colors (e.g. Jarrett et al. 2013, 2023) were used as tracers of
the SP age and thus, as a way to correct for their contribution in the NIR
emission. However, these corrections depend on the color of choice and its
ability to trace the SP age or the biases induced by interstellar reddening.
The existing calibrations are based on the intrinsic optical colors whereas in
their application the observed colors are used. Therefore, results from
various calibrations show discrepancies and large scatter in the estimations
of the $M_{\star}$ (e.g. Bonfini et al. 2021). Of course, not accounting for
the effect of extinction further exacerbates these discrepancies.

This work investigates the biases introduced by the SP age and extinction in
the widely used calibrations of photometric IR luminosity ($L_{\rm IR}$)-to-
SFR, and mass-to-light ratio, and proposes new calibrations that try to
mitigate these effects. The analysis is based on creating a large grid of
galaxy models with different ISM (e.g. metallicity, ionization parameter U)
and dust properties, for a large range of star-forming conditions. The
obtained SEDs and monochromatic luminosities of these galaxy models are then
compared with the model SFRs and stellar masses in order to derive the
corresponding calibrations.

In Section 2 we discuss the motivation of this work and give an example that
reveals the physical reasons behind the discussed biases. The used model suite
is presented in Section 3. The details about the analysis and the SFR and
$M_{\star}$ calibrations are provided separately in Section 4. In Section 5
the proposed calibrations are compared with previous works and observations,
and in Section 6 we present the conclusions. This work considers only rest-
frame luminosities. Throughout this work, unless stated otherwise, given
values correspond to the modes of the distributions and the uncertainties to
68% percentiles.

##  2 Biases in star-formation rate and stellar mass estimations from infrared
emission: an example

In order to explore the possible biases in measuring the SFR, and $M_{\star}$
from IR photometry we plot in Figure 1 the young stellar, old stellar, and
dust components from the SEDs of two model galaxies with low ($\rm
sSFR=10^{-11.75}\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\
yr^{-1}/M_{\odot}$) and high ($\rm sSFR=10^{-8.5}\leavevmode\nobreak\
M_{\odot}\leavevmode\nobreak\ yr^{-1}/M_{\odot}$) specific SFR (${\rm
sSFR}\equiv{\rm SFR}/M_{\star}$) using the code CIGALE (Burgarella et al.
2005; Noll et al. 2009; Boquien et al. 2019). Plotted in Figure 1 are also the
transmissions of the WISE bands 1 and 3, and JWST NIR-F200W, MIRI-F2100W
(Gardner et al. 2006; Gordon et al. 2022) bandpasses showing the parts of the
spectrum used to extract information about the $M_{\star}$ and the star-
forming activity.

Figure 1: Intensity as a function of wavelength for the total attenuated
intensity (gray), and the components of the young stellar (blue), old stellar
(orange), and dust emission (red) for two model galaxies: a) in the top panel
is a galaxy with low star-forming activity ($\rm SFR=0.01\leavevmode\nobreak\
M_{\odot}\leavevmode\nobreak\ yr^{-1}$, $\rm
sSFR=10^{-11.75}\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\
yr^{-1}/M_{\odot}$) and low-extinction ($E(B-V)<0.001$), and b) in the bottom
panel is a dusty ($E(B-V)=0.55$) galaxy with high star-forming activity ($\rm
SFR=68\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\ yr^{-1}$, $\rm
sSFR=10^{-8.5}\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\
yr^{-1}/M_{\odot}$). The dashed-dotted lines with brown and purple colors
represent the WISE bands 1 and 3 transmission bandpasses respectively. The
dotted fuchsia and red lines represent the JWST NIR-F200W, and MIRI-F2100W
bandpasses respectively.

These two extreme SEDs demonstrate that the emission in these IR bands
corresponds to different physical processes when galaxies are in different
star-formation states. In the NIR (e.g. WISE band-1, JWST NIR-F200W), for the
low SF activity galaxy, the stellar emission is orders of magnitude higher
compared to the dust emission. Calibrations of mass-to-light ratio are based
on emission in this wavelength range by tracing the thermal emission of the
SPs. However, we see that in a highly star-forming galaxy, the dust component
dominates the emission even in the NIR wavelengths, and therefore if it is not
taken into account it will significantly bias the stellar-mass estimations.

Similarly for the high SF galaxy, in the mid-IR (MIR; e.g. WISE band-3, JWST
MIRI-F2100W) wavelengths, emission from the dust component dominates the SED
and it is orders of magnitude higher compared to the stellar continuum
emission. However, we see that for a low-dust/low-sSFR galaxy, the stellar
thermal emission can dominate the emission in that bandwidth, enough to
significantly bias the SFR estimation. This example shows that both the SP age
and extinction have to be taken into account to properly estimate the SFR and
$M_{\star}$ through the IR emission across the full range of galactic ISM
environments.

It should be noted that Figure 1 shows that regardless of the level of the
star-forming activity the shape of the spectrum attributed to the dust
component is not changing. This is because CIGALE is not performing radiative
transfer that would allow a more detailed description of the SED based on the
amount and chemical composition of the dust, the geometric effects, and the
local thermal equilibrium of the dust accounting for the SPs in the region of
the dust clouds. However, radiative-transfer analysis is significantly more
computationally intensive and could not be performed for this analysis which
requires a large number of galaxy models needed to cover the variety of star-
formation histories (SFHs) and ISM conditions. Therefore, in the following
analysis dust temperature variations are not being considered, but we expect
them to play a minor role in comparison to the total luminosity due to the
fact they affect mainly longer wavelengths than 24 $\mu$m (e.g. Nersesian et
al. 2021), and the relative scatter in the $L_{\rm IR}$–SFR relation being
considerably less compared to that caused by extinction.

##  3 Modeling the SED of galaxies

###  3.1 Initial library of model-galaxy SEDs

In order to examine the correlation between the IR emission as a tracer of the
SFR or the $M_{\star}$, we created a large library of mock-galaxy SEDs that
cover a wide range of SFHs, stellar masses, and ISM conditions. Modeling
galaxies with a SED-fitting code allows us to know a-priori their SFR and
$M_{\star}$, which are an outcome of the given SFHs, while the corresponding
band-luminosities are given by the convolution of the filter transmissions and
the resulting SEDs. For the generation of the model-galaxy SEDs, we adopted
the commonly used SED fitting code CIGALE.

CIGALE is based on the principle of energy balance, where the energy
corresponding to the UV/optical light that has been absorbed by the dust is
re-emitted in the mid and far-IR parts of the spectrum. This makes CIGALE the
ideal tool to examine the dependence of IR SFR and $M_{\star}$ tracers on the
amount of dust within a galaxy and the corresponding extinction. The generated
grid of galaxies covered the extinction range for $E(B-V)$ between 0.001 and
1, with steps of $E(B-V)=0.05$ magnitudes. This $E(B-V)$ refers to the
nebular, not the stellar-continuum extinction.

Moreover, a wide range of different SFHs, covering continuously the
distribution of star-forming activity between passive and highly star-forming
galaxies, is required in order to investigate the dependence of the IR tracers
of SFR and $M_{\star}$ on the SP age. We adopted the sfhdelayed module, which
models the coexistence of an old SP, along with a delayed and recent star-
formation burst that generates the young SPs. By giving a large range to the
modulation parameters (Table A1) the initial library covers a wider range of
SFHs and star-forming conditions. Throughout this analysis, the SFR coming
from modeling with the CIGALE code refers to the sfh.sfr10Myrs that
corresponds to the average SFR over 10 Myrs. In addition, the generated grid
of galaxies covered a wide range of metallicities from extremely low
($Z=0.0001$) to high ($Z=0.05$). The configuration of the SED creation modules
is given in Table A1. This initial setup generated 1,854,720 SEDs of galaxies.
This library was extended by renormalizing the stellar masses, SFRs, and
corresponding luminosities creating a grid of model galaxies with a large
range of SFRs, $M_{\star}$, and ISM conditions.

Figure 2 shows the relation between the luminosity of WISE band-3 ($L_{\rm
W3}$) per unit $M_{\star}$ and the sSFR. In this work, in addition to the
generally used $E(B-V)$ extinction metric that can be obtained through various
ways, we also utilize the ratio of WISE band-4 to the optical $u$, or $g$ band
luminosities as infrared excess tracers where ${\rm IRXu}={\rm
log}\leavevmode\nobreak\ L_{\rm W4}/L_{u}$ and ${\rm IRXg}={\rm
log}\leavevmode\nobreak\ L_{\rm W4}/L_{g}$. The latter are chosen because of
the large coverage for galaxies in these bands through wide-area surveys (e.g.
PANSTARSS; Chambers et al. 2016). It should be noted that this IRX is not the
same as the one given by CIGALE which refers to the ratio between the dust,
and GALEX far-UV (FUV) luminosities.

Figure 2: WISE band-3 luminosity ($L_{\rm W3}$) per $M_{\star}$ unit, as a
function of sSFR for the generated library of galaxy SEDs. All panels show the
same sources but in the top left they are color-coded based on their $E(B-V)$,
in the top right based on their metallicity ($Z$), in the bottom left on their
infrared excess (${\rm IRXu}={\rm log}\leavevmode\nobreak\ L_{\rm W4}/L_{u}$),
and in the bottom right based on their infrared excess using the g band
instead of u (${\rm IRXg}={\rm log}\leavevmode\nobreak\ L_{\rm W4}/L_{g}$).

The generated galaxies show a continuous coverage of sSFRs ranging from
extremely low to highly star-forming galaxies. Moreover, Figure 2 shows that
the relation between $L_{\rm W3}/M_{\star}$ and sSFR: a) is not continuously
linear: the linearity breaks in low sSFRs where beyond a point the $L_{\rm
W3}$ remains constant while the sSFR drops. Previous works based on
observational data (e.g. Salim et al. 2016) showed that this break occurs near
sSFR $\simeq 10^{-11}\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\
yr^{-1}/M_{\odot}$ which is similar to what this analysis shows; b) there is a
strong dependence on extinction which can result in offsets of the
sSFR–$L_{\rm W3}/M_{\star}$ correlation of up to 2 orders of magnitude; c)
metallicity is an additional reason for scatter. The fact that in low-sSFR
galaxies the $L_{\rm W3}$ remains almost constant although the sSFR continues
to drop is a representation of the contribution of the old SPs in the MIR
emission (see also Section 2). Omitting the common denominator $M_{\star}$ in
the sSFR–$L_{\rm W3}/M_{\star}$ comparison shows that the above conclusions
also stand for the relation between SFR and $L_{\rm W3}$.

By binning the set of simulated photometries in different metallicity bins and
calculating the scatter with respect to the best SFR calibration in each bin
we can disentangle the role of metallicity and extinction in the observed
scatter. We note that this metallicity refers to the metallicity of the stars
and because in our simulation the SP and the ISM components are independent
there is no intrinsic correlation between the gas-phase metallicity (and
therefore the extinction) and the stellar metallicity used as a parameter in
our analysis. The scatter induced by metallicity in the relation between log
SFR and log $L_{\rm W3}$ is maximized for extremely low SFRs ($\rm SFR\simeq
10^{-6}\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\ yr^{-1}$) and low-
extinction ($E(B-V)\simeq 0.1$) galaxies at about 0.3 dex. Therefore, the
metallicity-induced scatter is by far smaller compared to that related to
extinction. In addition, because measuring metallicity requires spectroscopic
observations which are more costly, we do not account for the metallicity
differences in the following analysis. However, we include galaxies that fully
cover sub-solar to super-solar metallicities as given by CIGALE ($0.0001\leq
Z\leq 0.05$). Thus, the following relations and the evaluation of their
scatter encompass the dispersion caused by metallicity differences.

###  3.2 Fitting sample

The final master sample of model-galaxy SEDs used in the following analysis is
part of the initial library, and it was selected to fully and evenly cover the
plethora of star-forming conditions between passive and starburst, or dwarf
and massive galaxies. It is a result of sampling randomly the initial library
in order to produce uniform $M_{\star}$ and SFR distributions whose ratio, the
sSFR, is a distribution that can be described as a normal distribution with
mean $\rm\left<log\leavevmode\nobreak\ sSFR/(M_{\odot}\leavevmode\nobreak\
yr^{-1}/M_{\odot})\right>=-10.02$ and $\sigma=1.45$. This sample of model-
galaxy SEDs covers the range $10^{6.5}<M_{\star}/(M_{\odot})<10^{12}$, $\rm
10^{-6}<SFR/(M_{\odot}\leavevmode\nobreak\ yr^{-1})<10^{3}$, and $\rm
10^{-13}<sSFR/(M_{\odot}\leavevmode\nobreak\ yr^{-1}/M_{\odot})<10^{-7.5}$.
Thus, this sample would not bias the analysis towards any particular group of
galaxies. The SFR-$M_{\star}$ plane for this master sample is shown in Figure
3. The best linear fit is:

$\displaystyle\rm log\leavevmode\nobreak\
\frac{SFR}{(M_{\odot}\leavevmode\nobreak\ yr^{-1})}=-8.93\leavevmode\nobreak\
(\pm 0.02)+0.89\leavevmode\nobreak\ (\pm 0.01)\leavevmode\nobreak\
log\leavevmode\nobreak\ \frac{M_{\star}}{(M_{\odot})}\quad.$ (1)

Figure 3: The SFR, $M_{\star}$ plane for the final sample of CIGALE model
galaxies used for the rest of the analysis and fitting. The color code
indicates their specific SFR [${\rm sSFR}\equiv{\rm
SFR}/M_{\star}\leavevmode\nobreak\ (M_{\odot}\leavevmode\nobreak\ {\rm
yr^{-1}}/M_{\odot})$]. The best linear fit is represented with a red line. The
green dashed, yellow dashed-dotted, magenta dotted, and blue dashed-dotted
lines represent the main sequence for diverse samples of local Universe
galaxies from Elbaz et al. (2007), Speagle et al. (2014), Popesso et al.
(2019), and Kouroumpatzakis et al. (2021) respectively.

A goal of this work is to provide an extinction-dependent calibration that can
correct for the variance introduced by extinction in the relation between
$L_{\rm IR}$ and SFR. However, because estimations of extinction are not
always available, this work also provides calibrations that do not necessarily
require the input of extinction. From the master sample, two subsets were
created in order to account for the differences between the extinction-
independent and extinction-dependent calibrations. For the extinction-
dependent relations the sample followed a uniform $E(B{-}V)$ distribution in
order for the MCMC fits to be evenly affected by galaxies with low and high
extinction. Thus, the extinction-dependent calibration can compensate and
correct for the dependence of the $L_{\rm IR}$–SFR relation on extinction
(Figure 2) using as an input the $E(B-V)$ or the IRX. This $E(B-V)$-uniform
sample used for the extinction-dependent calibration included 397,256 galaxy
models.

The extinction-independent calibration does not have as an input an extinction
indicator to correct for the variance it introduces in the $L_{\rm IR}$–SFR
relation. But still, this calibration is being applied to estimate the SFR of
galaxies whatever their extinction may be. Thus, the distribution of
extinction of the sample used for the fitting of the calibration plays a
significant role in determining where the normalization between the $L_{\rm
IR}$ and the SFR will be. The uniform E(B-V) sample has on average more
galaxies with higher extinction compared to galaxies found in nature (Fig. 4).
Therefore, if it was used for the fitting, it would have shifted the
calibration towards dustier galaxies for whom the SFR corresponds to higher
$L_{\rm IR}$ compared to average-extinction galaxies. This would have
mistakenly led to lower normalization, and thus, lower SFRs for the same
$L_{\rm IR}$ that would be inappropriate for the bulk of the galaxies.

Therefore, for the extinction-independent relations, the distribution of the
nebular $E(B{-}V)$ of the model galaxies was matched to follow the
distribution of $E(B-V)$ of the SDSS spectroscopic MPA-JHU catalog (Kauffmann
et al. 2003; Brinchmann et al. 2004; Tremonti et al. 2004). The $E(B-V)$ for
the SDSS spectroscopic MPA-JHU catalog was calculated using the flux ratio of
the $\rm H\alpha$ and $\rm H\beta$ emission lines based on the Balmer
decrement adopting the conversion of Domínguez et al. (2013):

$\displaystyle E(B{-}V)=1.97\,{\rm log}\left[\frac{(f_{\rm H\alpha}/f_{\rm
H\beta})}{2.86}\right]\quad,$ (2)

using the reddening law of Calzetti et al. (2000). Sources with
$E(B-V)/E(B-V)_{\rm err}<3$ were omitted in order to keep only galaxies with
reliable extinction estimations. The distribution of the $E(B-V)$ color excess
for the models and the SDSS galaxies (used as a reference) are shown in Figure
4. This SDSS-matched sample included 123,908 galaxy models.

Figure 4:  The distribution of $E(B-V)$ for the uniform (green histogram) and
the SDSS-matched samples (red histogram) of model galaxies. With blue color is
the nebular $E(B-V)$ distribution based on the Balmer decrement for sources in
the SDSS spectroscopic MPA-JHU catalog considering only star-forming galaxies
with $E(B-V)/E(B-V)_{\rm err}>3$. The ordinate corresponds to density which is
equal to the ratio of the count of each bin to the total number of counts
multiplied by the bin width with the integral of the area under the histogram
being equal to one.

Finally, for each of these parent samples, we created by sampling randomly two
separate subsets for the fitting and testing of the best-fit models. The
fitting and testing samples included 70% and 30% of the parent samples
respectively ensuring that there is no overlap between them.

##  4 Analysis and results

###  4.1 Fitting methods

In all cases in the following analyses, the calibration models were fitted
with the Markov-chain Monte-Carlo (MCMC) technique using the Python emcee
package (Foreman-Mackey et al. 2013). The MCMC fitting used 64 walkers over
10000 iterations, while the first 500 were for the burn-in phase. The initial
model parameters were selected by running a maximum likelihood fit on the
parameter space based on the scipy minimization algorithm (Virtanen et al.
2020). A uniform prior was adopted covering a wide range for each parameter.
The models and best-fit results are presented separately for the $L_{\rm
IR}$-to-SFR and the mass-to-light ratios in the following sections.

Additionally, in order to independently test the calibrations given by the
MCMC fitting method, we performed machine-learning analysis on the same
datasets. We adopted the widely-used supervised machine learning algorithm
Random Forest (RF) in a regression mode (for a detailed review see Louppe
2014). The building blocks of RF are the decision trees. Each decision tree is
a non-parametric model which can be trained to learn the relation between the
target variable (e.g SFR, M⋆) and the feature values (e.g observables as W1,
W3, $E(B-V)$, etc.) by using a set of continuous nodes in a tree-like
structure. During the training, the algorithm searches for the feature and its
corresponding value that leads to the best condition of separation at each
tree node. This process is repeated recursively until a decision tree reaches
its final nodes, namely very well-separated and homogenized subsets of the
initial training data set. We adopted the Gini impurity (Baron 2019) for the
calculation of the best separation, which is also the default criterion of the
sklearn RF routine.

One of the advantages of the RF regression algorithm is that it can handle
problems where the parameter space is highly complex and non-linear, as in our
case. This makes the RF method the best independent test of the MCMC
calibration results which is the standard fitting method followed in this
work.

We trained exactly the same calibration datasets that were fitted by MCMC, for
consistency between the comparisons of the two methods. As with the MCMC
method, in all RF models, we considered as training dataset the 70$\%$ of the
initial sample and as the test dataset the 30$\%$. We also tuned the basic
hyper-parameters of the RF algorithm (such as the number of trees in the
forest, the maximum depth of the tree, etc.) in order to avoid overfitting or
underfitting. We used the implementation of the RF classifier
sklearn.ensemble.RandomForestClassifier() provided by the scikit-
learn111https://scikit-learn.org/stable/ version 1.1.2 (Pedregosa et al. 2011)
package for Python 3.

###  4.2 Star-formation rate calibrations

In order to account for the contribution of old SPs in the MIR, we fit the
relation between the MIR luminosity ($L_{\rm MIR}$) and the SFR including a
tracer of the $M_{\star}$. The NIR (e.g. $K_{\rm S}$, WISE 1 bands) traces
part of the stellar continuum emission of low-mass stars and it has been
traditionally used to measure the $M_{\star}$ in galaxies (e.g. Cole et al.
2001; Bell et al. 2003). In this work, the $L_{\rm MIR}$ and SFR are
calibrated including an observable component that traces the $M_{\star}$. This
allows to partially disentangle the contribution of old SPs in the MIR
emission which can be significant, especially in low-SFR galaxies (see also
Section 2, Figure 1). The form of this relation is described in the following
Eq. 3:

${\rm log}L_{\rm MIR}={\rm log}(\alpha M_{\star}+\beta{\rm SFR})$ (3)

where we assume that the observed luminosity arises from two components: a
young component that scales linearly with SFR and an older component that
scales linearly with stellar mass. The older component could be either the
tail of the continuum emission of the stars or associated with stochastically
heated dust.

As discussed in Section 3 our goal is to define a SFR indicator based on NIR
and MIR photometry which (a) is applicable to dust rich and dust poor
galaxies, and (b) galaxies with intense and low-level star-forming activity
(where the contribution of an older stellar component in the MIR emission
might be not negligible). Based on the photometric bands we adopt, we have
that $M_{\star}=k\leavevmode\nobreak\ L_{\rm NIR}$. We include the effect of
dust by parametrizing the parameter $\beta$ in Eq. 3 as a logistic function:
$\beta=\delta^{\prime}/(1+\epsilon^{\prime}\leavevmode\nobreak\
10^{-\zeta^{\prime}e})$ where $\delta^{\prime}$, $\epsilon^{\prime}$, and
$\zeta^{\prime}$ are fitted parameters and $e$ is an extinction metric (Eq.
4). We adopt as an extinction metric the color excess $E(B-V)$ or the IRX
index (as defined in Section 3). However, because stochastically heated dust
may also have a contribution in the NIR bands we also include an extinction
term in the parameter $\alpha$. We find that this is best parametrized with a
second-order polynomial of the extinction metric:
$\alpha=\alpha^{\prime}+\beta^{\prime}e+\gamma^{\prime}e^{2}$.

In order to account for cases where there are no available extinction
measurements we also fit the model data with an extinction-independent
parametrization, after dropping all the extinction-sensitive terms. This
extinction-independent analysis is performed on the SED models that include
the effect of extinction but follow the $E(B-V)$ distribution of SDSS galaxies
(Section 3.2).

For the WISE photometric system, as a SFR tracer, we adopted the WISE band-3
(12.1 $\mu$m), and as a $M_{\star}$ tracer the WISE band-1 (3.4 $\mu$m). The
12.1 $\mu$m band covers a polycyclic aromatic hydrocarbons (PAHs) band which
has been used as a SFR indicator (e.g. Jarrett et al. 2013), while the 3.4
$\mu$m band is mostly dominated by stellar continuum emission with a small
contribution by a PAH component (e.g. Figure 1). For the JWST photometric
system, as a SFR tracer, we adopted the MIRI-F2100W (21 $\mu$m) which probes
thermal emission from dust, and as a $M_{\star}$ tracer the NIR-F200W (2
$\mu$m) which is very similar to the $K_{\rm S}$ band. Both JWST filters have
the best effective responses in the wavelength area of interest, and were
designed for general purposes thus, are not significantly dominated by
emission from PAHs which could lead to additional biases.

Solving Eq. 3 for the SFR term, the calibration of SFR as a function of the
measured photometric quantities is:

$\displaystyle\frac{{\rm SFR}}{\rm(M_{\odot}yr^{-1})}=\frac{L_{\rm MIR}-\alpha
L_{\rm NIR}}{10^{21}\beta}$ (4) $\displaystyle L_{\rm MIR}=\frac{L_{\rm
W3}}{{\rm(Watt\leavevmode\nobreak\ Hz^{-1})}},\leavevmode\nobreak\ {\rm
or}\leavevmode\nobreak\ L_{\rm MIR}=\frac{L_{\rm F2100W}}{({\rm
Watt\leavevmode\nobreak\ Hz^{-1}})}$ $\displaystyle L_{\rm NIR}=\frac{L_{\rm
W1}}{({\rm Watt\leavevmode\nobreak\ Hz^{-1}})},\leavevmode\nobreak\ {\rm
or}\leavevmode\nobreak\ L_{\rm NIR}=\frac{L_{\rm
F200W}}{{\rm(Watt\leavevmode\nobreak\ Hz^{-1})}}$
$\displaystyle\alpha=\alpha^{\prime}+\beta^{\prime}e+\gamma^{\prime}e^{2}$
$\displaystyle\beta=\frac{\delta^{\prime}}{1+\epsilon^{\prime}10^{-\zeta^{\prime}e}}$
$\displaystyle e=E(B-V),\leavevmode\nobreak\ e={\rm IRXu},\leavevmode\nobreak\
e={\rm IRXg}\quad.$

The factor $10^{21}$ was introduced in order to homogenize the values of SFR
($\rm M_{\odot}\leavevmode\nobreak\ yr^{-1}$) and the IR luminosities ($\rm
Watt\leavevmode\nobreak\ Hz^{-1}$). Following the method outlined in Section
4.1 we fit the above relation using the emcee implementation of the MCMC
fitting scheme. Because the photometry is based on SED models given the
different SFH scenarios, they come without uncertainties. Therefore the fitted
likelihood functions do not include terms accounting for uncertainties in
photometry or in the SFR. The likelihood function for the extinction-
independent model is:

$\displaystyle
p(y|x,z,\alpha,\beta)=-\frac{1}{2}\sum_{n}\left[y_{n}-\frac{x_{n}-\alpha
z_{n}}{10^{21}\beta}\right]^{2}\quad,$ (5) $\displaystyle{\rm
and\leavevmode\nobreak\ for\leavevmode\nobreak\ the\leavevmode\nobreak\
extinction-dependent\leavevmode\nobreak\ model\leavevmode\nobreak\ is}:$
$\displaystyle
p(y|x,z,e,\alpha^{\prime},\beta^{\prime},\gamma^{\prime},\delta^{\prime},\epsilon^{\prime},\zeta^{\prime})=$
$\displaystyle-\frac{1}{2}\sum_{n}\left[y_{n}-\frac{x_{n}-(\alpha^{\prime}+\beta^{\prime}e+\gamma^{\prime}e^{2})z_{n}}{10^{21}(\frac{\delta^{\prime}}{1+\epsilon^{\prime}10^{-\zeta^{\prime}e}})}\right]^{2}\quad,$
$\displaystyle{\rm where:}\leavevmode\nobreak\ y={\rm SFR}/({\rm
M_{\odot}\leavevmode\nobreak\ yr^{-1}})$ $\displaystyle x=\frac{L_{\rm
W3}}{({\rm Watt\leavevmode\nobreak\ Hz^{-1}})}\leavevmode\nobreak\
{\rm,\leavevmode\nobreak\ or}\leavevmode\nobreak\ x=\frac{L_{\rm
F2100W}}{({\rm Watt\leavevmode\nobreak\ Hz^{-1}})}$ $\displaystyle
z=\frac{L_{\rm W1}}{({\rm Watt\leavevmode\nobreak\
Hz^{-1}})}\leavevmode\nobreak\ {\rm,\leavevmode\nobreak\
or}\leavevmode\nobreak\ z=\frac{L_{\rm F200W}}{({\rm Watt\leavevmode\nobreak\
Hz^{-1}})}$ $\displaystyle e=E(B-V),\leavevmode\nobreak\ {\rm
or}\leavevmode\nobreak\ e={\rm IRX}\quad.$

The best-fit results for Eq. 4 are given in Table 1.

Table 1: Best-fit results for the Eq. 4 SFR calibrations.

WISE band-1, and band-3
---
| $\alpha$ | $\beta$
$[$W1, W3$]$ | $0.348\pm 0.001$ | $16.437\pm 0.002$
| $\alpha^{\prime}$ | $\beta^{\prime}$ | $\gamma^{\prime}$ | $\delta^{\prime}$ | $\epsilon^{\prime}$ | $\zeta^{\prime}$
$[$W1, W3, $E(B{-}V)]$ | $-0.834\pm 0.000$ | $2.801\pm 0.003$ | $2.921\pm 0.003$ | $16.395\pm 0.001$ | $6.889\pm 0.002$ | $9.771\pm 0.001$
$[$W1, W3, ${\rm IRXu^{\star}}]$ | $0.629\pm 0.004$ | $-0.625\pm 0.004$ | $0.509\pm 0.001$ | $24.112\pm 0.001$ | $20.695\pm 0.000$ | $1.068\pm 0.000$
$[$W1, W3, ${\rm IRXg^{{\dagger}}}]$ | $15.630\pm 0.004$ | $18.765\pm 0.004$ | $1.282\pm 0.001$ | $0.386\pm 0.001$ | $-0.998\pm 0.000$ | $0.310\pm 0.001$
JWST NIR-F200W, and MIRI-F2100W
| $\alpha$ | $\beta$
$[$F200W, F2100W$]$ | $0.215\pm 0.001$ | $27.887\pm 0.002$
| $\alpha^{\prime}$ | $\beta^{\prime}$ | $\gamma^{\prime}$ | $\delta^{\prime}$ | $\epsilon^{\prime}$ | $\zeta^{\prime}$
$[$F200W, F2100W, $E(B{-}V)]$ | $0.001\pm 0.000$ | $-1.944\pm 0.002$ | $6.135\pm 0.003$ | $30.511\pm 0.001$ | $23.391\pm 0.003$ | $12.798\pm 0.001$

  * •

${\rm{}^{\star}IRXu}={\rm log}(L_{\rm W4}/L_{u})$,
${\rm{}^{{\dagger}}IRXg}={\rm log}(L_{\rm W4}/L_{g})$.

####  4.2.1 Comparisons with the star-formation rates of the model galaxies

In order to quantify the scatter in the derived SFR calibrations and any
biases with respect to the input values or SFRs measured with other methods,
we calculate the difference between the results of Eq. 4 applied to the model
SED photometry, and their true SFRs which are an outcome of the assumed SFHs.
Table 2 summarizes the comparisons with the true SFRs for low and high SFRs,
and for the uniform, and SDSS-matched extinction samples. Comparisons with
SFRs calculated using the calibrations of Jarrett et al. (2013), Chang et al.
(2015), and Cluver et al. (2017) are also given as reference. It should be
noted that the 68% percentiles provided in Table 2 are dominated by the
intrinsic scatter of the data due to the various dependencies that affect the
relation between the SFR and the IR luminosities (e.g. extinction distribution
of the applied sample, metallicity, ionization parameter, dust content
e.t.c.). On the other hand, 68% uncertainties given in Table 1 refer only to
the statistical uncertainties of the MCMC fitting procedure and the
corresponding parametrization.

Table 2:  The modes and 68% percentiles for the logarithm of the ratio between the estimated SFR for the model SEDs using Eq. 4, based on WISE band-1 and band-3 or NIR-F200W and MIRI-F2100W, over the true SFR as given by the SFHs of the SED modeling. Similarly, estimations based on WISE band-3 for the relations of Jarrett et al. (2013), Chang et al. (2015), and Cluver et al. (2017) are given. The two columns are for comparisons using the uniform-$E(B-V)$ samples, and the SDSS-matched $E(B-V)$ samples in the SFR range $\rm-4<logSFR/M_{\odot}\leavevmode\nobreak\ yr^{-1})<-1$ (left), and $\rm-1<log(SFR/M_{\odot}\leavevmode\nobreak\ yr^{-1})$ (right).  | $\rm<log\frac{SFR_{model}}{SFR_{true}}>$
---|---
SFR range | $\rm-4<log\leavevmode\nobreak\ \frac{SFR}{(M_{\odot}\leavevmode\nobreak\ yr^{-1})}<-1$ | $\rm log\leavevmode\nobreak\ \frac{SFR}{(M_{\odot}\leavevmode\nobreak\ yr^{-1})}>-1$
Model $E(B-V)$ distribution | uniform | SDSS | uniform | SDSS
$[$W1, W3] | $0.15^{+1.04}_{-0.44}$ | $0.17^{+1.11}_{-0.2}$ | $0.04^{+0.36}_{-0.38}$ | $-0.01^{+0.32}_{-0.16}$
$[$W1, W3, $E(B-V)$] | $0.07^{+1.06}_{-0.2}$ | $0.05^{+0.91}_{-0.16}$ | $-0.07^{+0.32}_{-0.1}$ | $-0.01^{+0.26}_{-0.15}$
$[$W1, W3, ${\rm IRXu}$] | $0.03^{+0.87}_{-0.2}$ | $0.04^{+0.93}_{-0.18}$ | $-0.08^{+0.3}_{-0.17}$ | $-0.08^{+0.32}_{-0.16}$
$[$W1, W3, ${\rm IRXg}$] | $0.16^{+1.21}_{-0.31}$ | $-0.01^{+1.05}_{-0.22}$ | $-0.17^{+0.42}_{-0.2}$ | $-0.32^{+0.49}_{-0.09}$
$[$F200W, F2100W] | $0.19^{+1.24}_{-0.28}$ | $0.04^{+1.14}_{-0.17}$ | $-0.05^{+0.39}_{-0.25}$ | $-0.1^{+0.35}_{-0.12}$
$[$F200W, F2100W, $E(B-V)$] | $0.0^{+1.17}_{-0.18}$ | $0.05^{+1.23}_{-0.19}$ | $-0.07^{+0.29}_{-0.15}$ | $-0.08^{+0.33}_{-0.17}$
Jarrett et al. (2013) [W3] | $0.04^{+1.3}_{-0.33}$ | $-0.06^{+1.3}_{-0.2}$ | $-0.3^{+0.44}_{-0.32}$ | $-0.23^{+0.38}_{-0.16}$
Chang et al. (2015) [W3] | $0.17^{+1.3}_{-0.33}$ | $0.07^{+1.3}_{-0.2}$ | $-0.17^{+0.44}_{-0.32}$ | $-0.1^{+0.38}_{-0.16}$
Cluver et al. (2017) [W3] | $0.76^{+1.2}_{-0.33}$ | $0.63^{+1.2}_{-0.19}$ | $0.3^{+0.45}_{-0.32}$ | $0.26^{+0.42}_{-0.24}$

Figure 5 shows comparisons between the true SFR based on the assumed SFHs
($\rm SFR_{true}$; referring to the average SFR over 10 Myrs from CIGALE), and
the calculated SFR based on the model photometry and the application of Eq. 4
(with the best-fit results; Table 1), and the RF models for different
extinction values. As we see in Figure 5 and Table 2 the extinction-dependent
relation of Eq. 4 results in excellent agreement with the true SFR of the
galaxies regardless their extinction over the full extinction range. Our
analysis using the extinction-dependent parametrization provides a remarkable
improvement in the reliability of the SFR measurements in low extinction
galaxies in comparison to the extinction-independent parametrization or other
metrics that do not account for extinction. In the case of higher extinction
galaxies, the agreement is also excellent (Figure 5, lower-right panel). In
very low SFRs ($\rm\lesssim 10^{-1}\leavevmode\nobreak\
M_{\odot}\leavevmode\nobreak\ yr^{-1}$) we see worse performance with a larger
scatter.

The extinction-independent relation of Eq. 4 results in an overall good
agreement, that is excellent for galaxies with average, and high extinction.
In very low SFRs the calibrations tend to overestimate the SFRs which,
however, cannot be reliably measured also due to SP sampling effects. This
behavior affects all SFR calibrations due to stochasticity driven by
relatively small numbers of massive stars in the probed SPs (e.g. Kennicutt &
Evans 2012). The relations based on the JWST photometric system are not shown
in Figure 5 in order to avoid congestion but they yield similar results as
those based on WISE (Table 2).

Figure 5:  Comparison between the true SFR as given by the CIGALE output, and
the best-fit models of Eq. 4 (MCMC; Table 1), and the random forest machine
learning method (RF) using the WISE bands 1 and 3. The $y$-axis represents the
logarithm of the ratio between the SFR of each model and the true SFR, thus
the black dashed line at zero represents equality. Blue, red, pink, and purple
colors show the extinction-independent, and extinction-dependent ($E(B-V)$,
IRXu, IRXg) relations of Eq. 4 respectively. With light-blue and orange colors
are the extinction-independent, and extinction-dependent models based on the
RF method respectively. The linear log $L_{\rm W3}$–log ${\rm SFR}$
calibration by Cluver et al. (2017) is also displayed in green color. The
lines represent the distribution modes and the shaded areas show the 68%
percentiles of the $\rm log\leavevmode\nobreak\ (SFR_{model}/SFR_{true})$ as a
function of the log $\rm SFR_{true}$. Each panel shows the comparison of
galaxy models with one of the discrete $E(B-V)$ values, shown in the bottom
left corner of each panel. This comparison is performed with a
uniform-$E(B-V)$ sample, therefore, there is an equal number of sources for
each extinction value. At the bottom of each panel are the modes and 68%
percentiles of the $\rm log\leavevmode\nobreak\ (SFR_{model}/SFR_{True})$
distribution for all galaxy models with the specific $E(B-V)$, shown with
corresponding colors to the relations used to derive the SFR. These
comparisons are also summarized in Table 2.

The RF method agrees well with the MCMC fitting method with respect to its
behavior regarding extinction. However, for the extinction-dependent
calibrations, this method shows slightly improved behavior for model galaxies
with extremely low extinction and slightly worse results for model galaxies
with extremely high extinction. Conversely, for extinction-independent
calibrations, the RF performs better in high-extinction galaxies and worse in
extremely low-extinction galaxies. This is probably because of the large
flexibility of the RF method with respect to MCMC because it does not fit an
explicit functional form to the data. Due to the behavior of the RF models, in
the following part of this work we refer to and provide only analytical
calibrations which are a result of the MCMC fitting. However, the fact that
the RF method yields similar results to the MCMC indicates that the SFRs from
the proposed calibrations of Eq. 4 do not strongly depend on the fitting
method.

The Eq. 4 relations offer excellent agreement while minimizing the scatter. It
is worth noting that the results between the true and inferred SFRs for the
extinction dependent and independent relations are almost the same for the
SDSS-matched-$E(B-V)$ sample. This is mainly because the SDSS-matched-$E(B-V)$
distribution mainly consists of average extinction galaxies ($E(B-V)\simeq
0.3$), while it completely lacks galaxies with $E(B-V)<0.05$ (Figure 4). Thus,
the use of the extinction-independent relation will not significantly bias the
SFR estimations for medium extinction galaxies, or samples of galaxies similar
to that of SDSS (e.g. local Universe). However, for a sample including
galaxies with extreme (low or high) values of extinction, the use of the
extinction-dependent relations yields better results with less scatter. As
also shown in Figure 5 this can be particularly important for dwarf, and dust-
free galaxies.

Table 2 shows the modes and 68% C.I.s of the distributions of the $\rm
log(SFR_{model}/SFR_{true})$ for the calibrations of Eq. 4, and that of
Jarrett et al. (2013), Chang et al. (2015), and Cluver et al. (2017) for both
the SDSS-matched-$E(B-V)$, and uniform-$E(B-V)$ samples considered here,
separated to low, and high-SFR galaxies. The relation of Cluver et al. (2017)
shows good agreement for average/high-extinction and high-SFR galaxies but the
discrepancy increases as the extinction decreases. These comparisons show that
the Cluver et al. (2017) calibration can lead to an underestimation of the SFR
for dust-free galaxies by more than an order of magnitude. A probable
explanation for these discrepancies is the selection of the sample. The
calibration of Cluver et al. (2017) used the SINGS/KINGFISH sample of galaxies
supplemented with some dwarf and some ultra-luminous IR galaxies (ULIRGs). On
average the agreement is better and the scatter is reduced using the
calibrations of Eq. 4 as shown from the modes and 68% percentiles of the
distributions of $\rm SFR_{model}/SFR_{true}$ (in the bottom of the individual
panels; Table 2).

Moreover, regardless of the extinction, the commonly used linear $L_{\rm
W3}$–SFR calibrations fail to recover the true SFR for passive galaxies.
Figure 5 shows that the Cluver et al. (2017) calibration overestimates the
SFRs in the low-SFR regime regardless of their extinction. This is because in
low-SFR galaxies (e.g. $\rm\lesssim 0.01\leavevmode\nobreak\
M_{\odot}\leavevmode\nobreak\ yr^{-1}$) the increasing contribution of the
emission from old SPs becomes dominant in the MIR photometric bands like WISE
band-3 (e.g. Figures 1, 2). On the other hand, the relation of Cluver et al.
(2017) shows better agreement for the high-SFR sample that follows the SDSS
extinction distribution although it still has a tendency to overestimate the
SFRs as shown from the upper 68% percentile of the distribution. Overall, the
SFRs based on the linear $L_{\rm W3}$–SFR calibrations show a non-linear trend
between the estimated SFR and $\rm SFR_{true}$.

As shown in Table 2 the calibrations of Jarrett et al. (2013), and Chang et
al. (2015) tend to underestimate the SFR on average for high SFR galaxies. The
68% percentiles of the ratio between these indicators and the true SFR when
they are applied to the model SEDs shows that they tend to overpredict the
SFR, especially in the case of low-SFR galaxies (Table 1). This bias is
significantly reduced (but not absent) in higher SFR galaxies. This behavior
is interpreted as the result of older SPs contributing to the emission of the
MIR bands we consider. The calibrations of Jarrett et al. (2013), and Chang et
al. (2015) are not shown in Figure 5 in order to avoid congestion.

####  4.2.2 Mid-IR emission dominated by old stellar populations

Although the data used to derive the parameters of Eq. 4 are based on SED
fitting tools built on the principle of energy balance, the large scatter in
the W3 luminosity at low SFR (SFR $\rm<0.01\leavevmode\nobreak\
M_{\odot}\leavevmode\nobreak\ yr^{-1}$) combined with the negative term in Eq.
4 (introduced to account for the contribution of old SPs) may lead to negative
SFR for galaxies with low SFR or low sSFR. This is because Eq. 4 gives an
average relation between SFR, MIR, and NIR luminosity while the significant
scatter resulting from the wide variety of SFHs and ISM parameters may result
in stronger NIR emission with respect to the MIR emission. This effect may be
exacerbated by the presence of stochastically heated dust (which is more
likely in low sSFR galaxies). The 2–4 $\mu$m bands include a strong dust
emission feature that can be significantly excited by the general interstellar
radiation field (e.g. Camps et al. 2015). For these cases, the SFR can not be
calculated through Eq. 4.

The comparisons with the test dataset (Section 4.1) show that sources showing
negative SFRs using Eq. 4 have on average $L_{\rm old}\simeq
65\leavevmode\nobreak\ L_{\rm young}$. The results are similar for both the
extinction-dependent and extinction-independent relations. For the extinction-
independent calibration the cutoff value for the W3–W1 color, below which
galaxies give negative SFRs, is ${\rm log}(L_{\rm W3}/L_{\rm W1})=-0.371$,
while this is not constant for the extinction-dependent relations. The
galaxies whose SFR can not be measured are relatively few (${\sim}6\%$, and
${\sim}12\%$ for the extinction-independent and extinction-dependent relations
respectively) considering that this analysis included galaxies with extremely
low sSFR down to sSFR $\rm\simeq 10^{-13}\leavevmode\nobreak\
M_{\odot}\leavevmode\nobreak\ yr^{-1}/M_{\odot}$. Therefore, galaxies that
yield negative SFRs based on Eq. 4 have extremely small star-forming activity
and their IR emission is dominated by old SPs. Overall, the Eq. 4 calibrations
offer reliable estimations even in low SFRs.

Additionally, we examine the capability of the IR–SFR calibrations to recover
the SFR throughout the range of SF activity. Figure 6 shows the ratio of the
estimated SFR, from the proposed and existing IR–SFR calibrations, and the
true SFR as a function of the model-galaxies sSFR. Previous works already
showed that in passive galaxies with $\rm sSFR\lesssim
10^{-11}\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\ yr^{-1}/M_{\odot}$
the dust is not heated by young star and thus, SFR estimations based on the
MIR emission for these galaxies are not reliable (e.g. Kennicutt 1998; Salim
et al. 2009, 2016). Figure 6 shows that all the IR–SFR calibrations compared
here fail to recover and overestimate the SFR in galaxies with $\rm
sSFR\lesssim 10^{-10.7}\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\
yr^{-1}/M_{\odot}$. However, the proposed calibrations from this work, and
especially the extinction dependent, tend to reduce the biases in the SFR
estimation. This is demonstrated in Figure 6 by the slightly better agreement
between the modes of the proposed calibrations (red and blue lines) with the
equality (black dashed) line in relatively low sSFRs ($\rm sSFR\simeq
10^{-10.5}\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\
yr^{-1}/M_{\odot}$) and in high sSFRs ($\rm sSFR\simeq
10^{-8}\leavevmode\nobreak\ M_{\odot}\leavevmode\nobreak\ yr^{-1}/M_{\odot}$).

Figure 6:  The logarithm of the ratio of the estimated SFR based on the IR
emission over the true SFR as given by the CIGALE models, as a function of
their true specific SFR (sSFR). The comparisons show the extinction-dependent
and extinction-independent calibrations of Eq. 4, and the $L_{\rm IR}$–SFR
calibrations of Jarrett et al. (2013), Chang et al. (2015), and Cluver et al.
(2017) with blue, red, purple, yellow, and green lines respectively for model
galaxies separated in groups of low (left), moderate (center), and high
(right) extinction.

###  4.3 Stellar mass calibrations

Similarly to the SFR-$L_{\rm IR}$ relation, we explore the dependence of the
mass-to-light ratio on the SP age and dust content/extinction. For this
analysis, we use the same data as with the SFR analysis, as described in
Section 3.2. Previous studies have identified the SP age as the main cause of
uncertainty in the calculation of the $M_{\star}$ (e.g. Rix & Rieke 1993).
Here, as an observational tracer of the SP age, we adopt the $u{-}r$, and
$g{-}r$ colors, which have also been used in other studies (e.g. Bell et al.
2003; Wen et al. 2013). However, this work also explores the possibility of
increased scatter in the inferred mass-to-light ratio induced by the use of
extinction-corrected colors as is usually the case. Dust emission, heated by
the star-forming activity is mainly in the mid, and far-IR parts of the
spectrum.

However, emission from PAH molecules and stochastically heated dust grains may
have a significant contribution to the NIR emission. The relation between the
mass-to-light ratio with the optical colors shows a large scatter (e.g. Figure
7). This may be the result of extinction affecting the $u{-}r$, and $g{-}r$
colors of the SPs creating multiple mass-to-light ratio tracks for different
values of extinction. In fact, if we account for the extinction (i.e. correct
the $u{-}r$ color for the extinction based on the $E(B-V)$) the scatter is
slightly reduced. The advantage of our analysis is that it accounts for the
effect of dust in two ways: (a) by its effect on the colors of the SPs and (b)
by its effect on the NIR emission through stochastically heated dust. There is
a tertiary effect originating from the contribution of the H$\alpha$ emission
in the $r$ band photometry. This is also indirectly accounted for through the
(loose) correlation between extinction in the ISM and star-forming activity.

Figure 7 shows the mass-to-light ratio of the CIGALE model galaxies, as a
function of the $u{-}r$ color. Each point represents a single model SED and it
is color-coded based on its extinction. Figure 7 reveals the scatter induced
by the differences in extinction, where especially for blue galaxies it can be
more than an order of magnitude. The scatter increases for decreasing $u{-}r$
color. However, in highly star-forming galaxies ($u-r\lesssim 1.5$ or
$g-r\lesssim 0.75$) the scatter is by far larger. In these color regimes,
galaxies have higher SFRs, and therefore, IR emission from the dust is
brighter, and its respective dependence on extinction is stronger. Moreover,
the $g{-}r$ color has an overall narrower range and shows a larger scatter as
a function of the mass-to-light ratio compared to $u{-}r$, and therefore is
not as good as a SP age indicator for the purposes of this work.

---

---

Figure 7: Mass-to-light ratio as a function of $u{-}r$ (left column) and
$g{-}r$ (right column) color for the SDSS-matched sample of the modeled
galaxies. In the bottom row, the $u{-}r$ and $g{-}r$ colors are corrected for
extinction while in the top row, they are not. The galaxies are color-coded
based on their $E(B-V)$. The dashed magenta line represents the extinction-
independent relation of Eq. 6, and the dashed-dotted lines the extinction-
dependent, drawn for some discrete $E(B-V)$ values as shown in the legend
using the parameters presented in Table 3. The dotted black line shows the
relations of Wen et al. (2013).

We describe the correlation between the mass-to-light ratio and the optical
color as a power law with negative power values, thus as
$M/L=\alpha(u-r)^{\beta}$. Moreover, in this analysis, we fit the mass-to-
light ratio relation with the optical color taking also into account the
extinction. We include the effect of dust by parametrizing the parameter
$\beta$ as a linear function of extinction with the form
$\beta=\delta^{\prime}+\epsilon^{\prime}\leavevmode\nobreak\ e$ where
$\delta^{\prime}$ and $\epsilon^{\prime}$ are fitted parameters and $e$ is an
extinction metric. We adopt as an extinction metric ($e$) the color excess
$E(B-V)$ or the IRX index (as defined in Section 3). However, because
extinction also affects the $u{-}r$ color we also include an extinction term
in the parameter $\alpha$ which is best parametrized with a second-order
polynomial of the extinction metric:
$\alpha=\alpha^{\prime}+\beta^{\prime}e+\gamma^{\prime}e^{2}$. In order to
account for cases where there are no available extinction measurements we also
fit the data with an extinction-independent parametrization. These fits
involve the SED models that include the effect of extinction but follow the
$E(B-V)$ distribution of SDSS galaxies (Section 3.2).

The final mass-to-light calibrations are presented in Eq. 6:

$\displaystyle{\rm log}\leavevmode\nobreak\ \frac{M_{\star}/{(\rm
M_{\odot})}}{L_{\rm NIR}}=\alpha(x+1)^{\beta}+0.5$ (6) $\displaystyle
x=u-r,\leavevmode\nobreak\ {\rm or}\leavevmode\nobreak\ x=g-r$ $\displaystyle
L_{\rm NIR}=\frac{L_{\rm W1}}{(L_{\rm\odot,W1})},\leavevmode\nobreak\ {\rm
or}\leavevmode\nobreak\ L_{\rm NIR}=\frac{L_{\rm F200W}}{(L_{\rm\odot})}$
$\displaystyle\alpha=\alpha^{\prime}+\beta^{\prime}e+\gamma^{\prime}e^{2}$
$\displaystyle\beta=\delta^{\prime}+\epsilon^{\prime}e$ $\displaystyle
e=E(B-V),\leavevmode\nobreak\ {\rm or}\leavevmode\nobreak\ e={\rm
IRXu},\leavevmode\nobreak\ {\rm or}\leavevmode\nobreak\ e={\rm IRXg}$

The addition of 1 to the $u{-}r$ or $g{-}r$ color, and the term $0.5$ were
introduced in order to avoid singularities in the model. $L_{\rm W1}$ is
measured in the in-band equivalent solar luminosity (see e.g. Jarrett et al.
2013). It is equal to scaling $\nu L_{\nu}$ by 22.883. The $L_{\rm F200W}$ is
measured in solar luminosities. The best-fit results based on the MCMC fitting
method for the parameters $\alpha,\leavevmode\nobreak\ \beta$ and
$\alpha^{\prime},\leavevmode\nobreak\ \beta^{\prime},\leavevmode\nobreak\
\gamma^{\prime},\leavevmode\nobreak\ \delta^{\prime},\leavevmode\nobreak\
\epsilon^{\prime}$ are given in Table 3. Moreover, we performed a RF model for
the same datasets and the same sets of features with the MCMC in order to
independently compare the result of the MCMC method.

Table 3: Best-fit results for the Eq. 6 mass-to-light ratio calibrations.

Model | WISE band-1
---|---
| $\alpha$ | $\beta$
$[$W1, $u{-}r$] | $-2.307\pm 0.013$ | $-0.847\pm 0.010$
| $\alpha^{\prime}$ | $\beta^{\prime}$ | $\gamma^{\prime}$ | $\delta^{\prime}$ | $\epsilon^{\prime}$
$[$W1, $u{-}r$, $E(B-V)]$ | $-1.689^{+0.010}_{-0.011}$ | $-1.493^{+0.048}_{-0.047}$ | $-0.201^{+0.061}_{-0.058}$ | $-0.637^{+0.009}_{-0.009}$ | $-0.394^{+0.016}_{-0.017}$
$[$W1, $u{-}r$, $\rm IRXu^{\star}]$ | $-1.520\pm 0.013$ | $-0.201\pm 0.010$ | $-0.168\pm 0.05$ | $-0.532\pm 0.010$ | $-0.133\pm 0.006$
| $\alpha$ | $\beta$
$[$W1, $g{-}r$] | $-1.555^{+0.005}_{-0.006}$ | $-0.924^{+0.013}_{-0.010}$
| $\alpha^{\prime}$ | $\beta^{\prime}$ | $\gamma^{\prime}$ | $\delta^{\prime}$ | $\epsilon^{\prime}$
$[$W1, $g{-}r$, $E(B-V)]$ | $-1.266^{+0.006}_{-0.007}$ | $-0.765^{+0.034}_{-0.034}$ | $-0.115^{+0.037}_{-0.036}$ | $-0.608^{+0.012}_{-0.012}$ | $-0.896^{+0.024}_{-0.023}$
$[$W1, $g{-}r$, $\rm IRXg^{{\dagger}}]$ | $-1.135^{+0.005}_{-0.005}$ | $-0.135^{+0.006}_{-0.005}$ | $-0.156^{+0.003}_{-0.003}$ | $-0.661^{+0.013}_{-0.013}$ | $-0.106^{+0.009}_{-0.009}$
| JWST NIR-F200W
| $\alpha$ | $\beta$
$[$F200W, $u-r]$ | $-1.193\pm 0.015$ | $-1.691\pm 0.028$
| $\alpha^{\prime}$ | $\beta^{\prime}$ | $\gamma^{\prime}$ | $\delta^{\prime}$ | $\epsilon^{\prime}$
$[$F200W, $u{-}r$, $E(B-V)]$ | $-0.838^{+0.012}_{-0.012}$ | $-0.800^{+0.079}_{-0.078}$ | $-0.815^{+0.114}_{-0.114}$ | $-1.509^{+0.029}_{-0.030}$ | $-0.719^{+0.059}_{-0.061}$
| $\alpha$ | $\beta$
$[$F200W, $g-r]$ | $-0.547\pm 0.005$ | $-1.843\pm 0.031$
| $\alpha^{\prime}$ | $\beta^{\prime}$ | $\gamma^{\prime}$ | $\delta^{\prime}$ | $\epsilon^{\prime}$
$[$F200W, $g{-}r$, $E(B-V)]$ | $-0.455^{+0.007}_{-0.007}$ | $-0.148^{+0.038}_{-0.038}$ | $-0.376^{+0.042}_{-0.042}$ | $-1.123^{+0.030}_{-0.029}$ | $-2.542^{+0.071}_{-0.069}$

  * •

${\rm{}^{\star}IRXu}={\rm log}(L_{\rm W4}/L_{u})$,
${\rm{}^{{\dagger}}IRXg}={\rm log}(L_{\rm W4}/L_{g})$.

As mentioned in Section 4.2, because the SEDs are an output of modeling, their
photometries, and their stellar masses come without uncertainties, and thus
they are not included in the likelihoods of the MCMC fitting. The likelihood
function for the extinction-independent model is:

$\displaystyle{\rm log}\leavevmode\nobreak\
p(y|x,\alpha,\beta)=-\frac{1}{2}\sum_{n}\left(y_{n}-\alpha
x_{n}^{\beta}\right)^{2}$ (7) $\displaystyle{\rm and\leavevmode\nobreak\
for\leavevmode\nobreak\ the\leavevmode\nobreak\
extinction{-}dependent\leavevmode\nobreak\ model\leavevmode\nobreak\ is:}$
$\displaystyle{\rm log}\leavevmode\nobreak\
p(y|x,e,\alpha^{\prime},\beta^{\prime},\gamma^{\prime},\delta^{\prime},\epsilon^{\prime})=$
$\displaystyle-\frac{1}{2}\sum_{n}\left[y_{n}-(\alpha^{\prime}+\beta^{\prime}e+\gamma^{\prime}e^{2})\leavevmode\nobreak\
x_{n}^{(\delta^{\prime}+\epsilon^{\prime}e)}\right]^{2}$ $\displaystyle{\rm
where:}$ $\displaystyle y={\rm log}\leavevmode\nobreak\ \frac{M_{\star}/L_{\rm
W1}}{(M_{\odot}/L_{\rm\odot,W1})}-0.5,\leavevmode\nobreak\ {\rm
or}\leavevmode\nobreak\ y={\rm log}\leavevmode\nobreak\ \frac{M_{\star}/L_{\rm
F200W}}{(M_{\odot}/L_{\rm\odot})}-0.5$ $\displaystyle
x=u-r+1\leavevmode\nobreak\ ({\rm mag}),\leavevmode\nobreak\ {\rm
or}\leavevmode\nobreak\ x=g-r+1\leavevmode\nobreak\ ({\rm mag})\quad.$

####  4.3.1 Comparisons with the stellar masses of the model galaxies

Figure 7 shows the extinction-independent relation of Eq. 6 with a dashed
magenta line. The dashed-doted lines show the extinction-dependent relation
for some representative extinction values. The extinction-independent relation
follows the median position of the points. The lines drawn based on the
extinction-dependent relation follow well the data points of similar
extinction values. As a reference, we also compare with the calibration of Wen
et al. (2013) which is empirically calibrated based on the W1 emission and
stellar masses estimated from the MPA–JHU analysis. The mass-to-light ratio in
this relation is parametrized as a polynomial function of the $u{-}r$ or
$g{-}r$ colors. The relation of Wen et al. (2013) shows good agreement for
average $u{-}r$ colors ($u-r\simeq 1.5$), as well as, for moderate extinction
[$E(B-V)\simeq 0.5$]. However, it fails to recover the $M_{\star}$ for
galaxies with low or high extinction, while it shows larger discrepancies near
the lower and upper limits of $u{-}r$ color.

Figure 8 shows the logarithm of the ratio between the calculated stellar
masses for the MCMC, and RF models (using the WISE band-1) over the true
$M_{\star}$, as a function of the true $M_{\star}$. The true $M_{\star}$ is an
output of CIGALE based on the given SFHs of the modeled galaxies. We also plot
the calculated stellar masses based on the relation of Wen et al. (2013) as a
comparison reference. Figure 8 does not show the results based on the JWST-NIR
F200W band for clarity because these are very similar to those using the WISE
band-1. Table 4 gives the comparisons for the proposed relations for all the
models considered in this analysis, separated in low $u{-}r$ (younger SPs or
lower extinction), and high $u{-}r$ (older SPs or higher extinction) color.
These comparisons are also given separately for the sample having a uniform
distribution in extinction, and the SDSS-matched sample (Section 3.2) in order
to reveal possible biases based on the selected sample of galaxies.

The MCMC relations show robust estimations throughout the $M_{\star}$ range.
Figure 8 demonstrates that the use of the extinction feature in the
$M_{\star}$ calculations is important in low and high-extinction galaxies. The
extinction-independent relation of Eq. 6 is in good agreement for mid-range
and high-extinction sources. However, in all cases, the scatter is increased
when the extinction is not taken into account. Overall, the RF yields similar
results to the MCMC indicating the results of the proposed calibrations are
independent of the method.

Figure 8: Comparison between the true $M_{\star}$ as given by the CIGALE output, and the best-fit models of Eq. 6 (MC; Table 3), and the random forest machine learning method (RF) using the WISE band-1 and the $u{-}r$ color. The $y$-axis represents the logarithm of the ratio between the $M_{\star}$ of each model and the true $M_{\star}$, thus the black dashed line at zero represents equality. Blue and red colors show the extinction-independent, and extinction-dependent relation of Eq. 6 respectively. With light-blue and orange colors are the extinction-independent, and extinction-dependent models based on the RF method respectively. The $M_{\star}$ calculated based on the mass-to-light conversion from Wen et al. (2013) is also shown with green color. The lines represent the distribution modes and the shaded areas the 68% percentiles as a function of the log $M_{\rm\star\leavevmode\nobreak\ true}$. Each panel shows the comparison of galaxy models with one of the discrete $E(B-V)$ values, shown in the bottom left corner of each panel. This comparison is performed with the uniform-$E(B-V)$ sample, therefore, there is an equal number of sources for each extinction value. At the bottom of each panel are the modes and 68% percentiles of the $\rm log\leavevmode\nobreak\ M_{\star model}/M_{\star,True}$ distribution for all galaxy models with the specific $E(B-V)$, shown with corresponding colors to the relations used to derive the $M_{\star}$. These comparisons are also summarized in Table 4.  Table 4:  The modes and 68% percentiles for the logarithm of the ratio between the estimated $M_{\star}$ of the models, based on Equation 6 using the WISE band-1 or NIR-F200W photometries and the $u{-}r$ or $g{-}r$ optical colors, over the true $M_{\star}$ as given by CIGALE. Results using the Wen et al. (2013) calibrations are also given. The two columns are for comparisons using the uniform-$E(B-V)$ samples, and the SDSS-matched $E(B-V)$ samples in the color range $u-r\leq 1$ (left), and $u-r>1$ (right), and $g-r\leq 0.75$ (left), and $g-r>0.75$ (right).  | $\rm<log\frac{M_{\star,model}}{M_{\star,true}}>$
---|---
Model $E(B-V)$ distribution | uniform | SDSS | uniform | SDSS
| $u-r\leq 1$ | $u-r>1$
$[$W1, $u{-}r$] | $-0.02^{+0.4}_{-0.37}$ | $-0.04^{+0.28}_{-0.23}$ | $0.14^{+0.16}_{-0.24}$ | $0.1^{+0.13}_{-0.25}$
$[$W1, $u{-}r$, $E(B-V)$] | $-0.05^{+0.25}_{-0.19}$ | $-0.03^{+0.25}_{-0.18}$ | $0.09^{+0.09}_{-0.29}$ | $0.08^{+0.09}_{-0.29}$
$[$W1, $u{-}r$, IRXu] | $-0.09^{+0.29}_{-0.18}$ | $-0.08^{+0.25}_{-0.19}$ | $0.04^{+0.14}_{-0.21}$ | $0.06^{+0.11}_{-0.22}$
Wen et al. (2013) $[$W1, $u{-}r$] | $0.05^{+0.43}_{-0.28}$ | $0.17^{+0.32}_{-0.24}$ | $0.26^{+0.14}_{-0.26}$ | $0.26^{+0.1}_{-0.29}$
$[$F200W, $u{-}r$] | $-0.08^{+0.34}_{-0.23}$ | $0.15^{+0.18}_{-0.31}$ | $0.04^{+0.15}_{-0.2}$ | $0.05^{+0.14}_{-0.21}$
$[$F200W, $u{-}r$, $E(B-V)$] | $-0.06^{+0.28}_{-0.2}$ | $0.03^{+0.22}_{-0.23}$ | $0.05^{+0.14}_{-0.21}$ | $0.06^{+0.14}_{-0.21}$
| $g-r\leq 0.75$ | $g-r>0.75$
$[$W1, $g{-}r$] | $0.1^{+0.5}_{-0.22}$ | $0.18^{+0.46}_{-0.22}$ | $0.0^{+0.27}_{-0.18}$ | $-0.04^{+0.23}_{-0.18}$
$[$W1, $g{-}r$, $E(B-V)$] | $0.11^{+0.45}_{-0.23}$ | $0.17^{+0.45}_{-0.22}$ | $-0.03^{+0.27}_{-0.16}$ | $-0.03^{+0.23}_{-0.18}$
$[$W1, $g{-}r$, IRXg] | $0.22^{+0.32}_{-0.23}$ | $0.21^{+0.33}_{-0.22}$ | $0.0^{+0.19}_{-0.18}$ | $0.05^{+0.18}_{-0.2}$
Wen et al. (2013) $[$W1, $g{-}r$] | $0.14^{+0.49}_{-0.22}$ | $0.26^{+0.44}_{-0.22}$ | $0.2^{+0.24}_{-0.21}$ | $0.22^{+0.21}_{-0.24}$
$[$F200W, $g{-}r$] | $0.21^{+0.33}_{-0.25}$ | $0.19^{+0.33}_{-0.25}$ | $0.05^{+0.24}_{-0.18}$ | $-0.08^{+0.25}_{-0.11}$
$[$F200W, $g{-}r$, $E(B-V)$] | $0.21^{+0.32}_{-0.25}$ | $0.18^{+0.34}_{-0.26}$ | $-0.01^{+0.22}_{-0.14}$ | $0.07^{+0.23}_{-0.18}$

##  5 Comparison with observations

The comparisons of the SFR and mass-to-light calibrations of Equations 4 and 6
show excellent results in retrieving the SFR and $M_{\star}$ of the mock
galaxies used in our analysis. However, in order to test these calibrations in
a more realistic setting we compare them with observations of galaxies and
their SFRs and stellar masses inferred from different methods.

###  5.1 Comparison of the star-formation rate calibrations

Figure 9 shows comparisons between SFRs based on our method of extinction-
dependent and extinction-independent calibrations using Eq. 4 and the best-fit
relations in Table 1 with various methods including: a) SED fitting from Chang
et al. (2015), Salim et al. (2018), or MPA-JHU (Kauffmann et al. 2003;
Brinchmann et al. 2004; Tremonti et al. 2004), and b) those based on
monochromatic WISE band-3 emission (Jarrett et al. 2013; Chang et al. 2015;
Cluver et al. 2017). These comparisons involve star-forming and unclassified
but exclude LINER and AGN galaxies as they were classified by MPA-JHU
(BPTCLASS $\neq$ 4 and BPTCLASS $\neq$ 5; Brinchmann et al. 2004). The adopted
IR fluxes for the WISE bands are from the AllWISE source
catalog222http://wise2.ipac.caltech.edu/docs/release/allwise/ as given by
Chang et al. (2015) before the photometric corrections they applied. Distances
used to calculate the luminosities were based on the redshifts provided by the
MPA-JHU catalog. Their $E(B-V)$ was estimated through the Balmer extinction
based on the H$\alpha$ and H$\beta$ fluxes from MPA-JHU and the conversion of
Domínguez et al. (2013). Galaxies with $\rm SNR<5$ in the WISE bands 1 and 3
fluxes and the $E(B-V)$ were omitted ensuring good quality data.

Figure 9: Comparisons between SFRs estimated through a) the extinction-
dependent and extinction-independent relations of Eq. 4 (Table 1) which
utilize WISE bands 1 and 3, b) calibrations based only on WISE band-3 (Jarrett
et al. 2013; Chang et al. 2015; Cluver et al. 2017), and c) based on SED
fitting from Chang et al. (2015), Salim et al. (2018) and MPA-JHU (Kauffmann
et al. 2003; Brinchmann et al. 2004; Tremonti et al. 2004). In each subplot,
the ordinate indicates the logarithm of the ratio ($\rm
log\leavevmode\nobreak\ (SFR_{y}/SFR_{x})$) between the compared SFR
estimations, and the abscissa the logarithm of the SFR as indicated in the
axes labels. Red error bars represent the modes and 68% percentiles of the
distribution of galaxies within bins of log $\rm
SFR/M_{\odot}\leavevmode\nobreak\ yr^{-1}\simeq 0.4$ (12 bins within the range
$\rm-2.5<log\leavevmode\nobreak\ SFR/M_{\odot}\leavevmode\nobreak\
yr^{-1}<2.5$). The $\rm log\leavevmode\nobreak\ SFR_{y}/SFR_{x}$ modes and 68%
percentiles for all the galaxies compared in each panel are shown at the top
of each subplot. The points are color-coded based on the logarithm of the
absolute value of the equivalent width (EW) of the H$\alpha$ line of the
galaxies. The black dashed line represents equality.

The format of Figure 9 is similar to that of Figure 5 where the ordinate
indicates the logarithm of the ratio of the compared SFRs ($\rm
log\leavevmode\nobreak\ SFR_{y}/SFR_{x}$), and the abscissa the logarithm of
the SFRs as indicated in the axes labels. Therefore, the equality lies at
value zero of the y-axis. The points in Figure 9 are color-coded based on the
absolute value of the equivalent width of the H$\alpha$ line ($\rm
EW_{H\alpha}$) of the galaxies. The H$\alpha$ EW is strongly correlated with
the sSFR of the galaxies (e.g. Belfiore et al. 2018) and it is used here as an
independent method for tracing their current star-forming activity with
respect to their stellar component.

The SED fitting of Chang et al. (2015) included optical (SDSS), and IR (WISE),
but not UV photometry. The SED fitting of Salim et al. (2018) was based on a
SED + $L_{\rm IR}$ fitting method utilizing except the SDSS optical and GALEX
UV in the SED fitting, corrections based on the 22 $\mu$m or 12 $\mu$m
photometry from unWISE. The SFR estimations from MPA-JHU were based on the
galaxies’ emission lines and the method of Kauffmann et al. (2003) for
galaxies within the SDSS spectroscopic fiber. For galaxies that did not fit
within the fiber, SFRs were estimated using the ugriz photometry based on SED
fitting following Salim et al. (2007).

The proposed SFR calibrations of Eq. 4 show the best on-average agreement
compared to the SFRs from SED fitting. The extinction-dependent relation shows
an overall slightly better agreement with the other indicators compared to the
extinction-independent calibration. Moreover, the Eq. 4 calibrations show good
agreement throughout the SFR range compared to the SFRs from Salim et al.
(2018) with on-average no large deviations in the low or high SFRs. As
expected, the extinction-dependent relation shows better behavior in the lower
and higher SFR regimes. The scatter between the Eq. 4 calibrations compared to
results from SED fitting is reduced with respect to other IR-based SFRs.

The comparisons between the monochromatic IR–SFR calibrations (Jarrett et al.
2013; Chang et al. 2015; Cluver et al. 2017) reveals some differences between
them. While the best agreement is between the calibrations of Jarrett et al.
(2013), and Chang et al. (2015), the largest difference is between Jarrett et
al. (2013), and Cluver et al. (2017) (about 0.5 dex on average). As also shown
from the comparisons with the mock photometry used in our analysis (Figure 5),
comparing Eq. 4 with these calibrations show differences ranging from -0.25
dex (compared to Cluver et al. 2017) to +0.25 dex (compared to Jarrett et al.
2013) on average and non-linear dependence on the value of the SFR.

As revealed by the color-coding of Figure 9, the galaxies where the Eq. 4
calibrations infer lower SFRs compared to other IR–SFR calibrations have very
low EW ($\rm EW_{H\alpha}\lesssim 20$ Å) indicating that these are passive
galaxies which are mainly dominated by older SPs. Moreover, only the
extinction-dependent calibration of Eq. 4 does not show a monotonic $\rm
EW_{H\alpha}$ gradient in the comparisons with the SED-fitting SFRs. This
indicates the ability of the proposed calibration to account for the
contribution of older SPs and extinction in the dust heating while all other
IR–SFR calibrations tend to overestimate the SFR of passive galaxies and
underestimate the SFR of highly star-forming galaxies.

The SED fitting and the use of emission lines have been proven powerful
methods for deriving the characteristics of galaxies. However, not using
simultaneously UV and IR photometry can lead to biases with respect to the
estimation of extinction due to not covering the energy balance between the
absorbed UV light, and that which is re-emitted in the IR (e.g. Lanz et al.
2013). This can be more important in the extremes of the SFR, and $M_{\star}$
range, for instance in dust-free galaxies when the UV is not taken into
account, or in dusty/high-SFR galaxies when the IR emission is not taken into
account. The fact that the calibrations of Eq. 4 show excellent agreement with
the SFRs from SED fitting demonstrates the advantage of this method with
respect to monochromatic IR SFR indicators and shows that their results can be
considered robust over a wide SFR range.

###  5.2 Comparison of the mass-to-light ratio calibrations

Figure 10 shows comparisons between stellar masses calculated using various
methods including: a) the extinction-dependent, and extinction-independent
relations of Eq. 6 which combine the WISE band 1 and the $u{-}r$, or $g{-}r$
color, b) SED fitting from Chang et al. (2015), Salim et al. (2018), and MPA-
JHU (Kauffmann et al. 2003; Brinchmann et al. 2004; Tremonti et al. 2004), and
c) the mass-to-light calibration of Wen et al. (2013) which is using the WISE
band-1 and the $u{-}r$ color. Similarly to Figure 9, this comparison involves
star-forming or unclassified galaxies but excludes LINER and AGN galaxies as
they have been classified by MPA-JHU (BPTCLASS $\neq$ 4 and BPTCLASS $\neq$
5). It involves galaxies with $\rm SNR>5$ in the $L_{\rm W1}$, SDSS $u$, $g$,
$r$ magnitudes, and $E(B-V)$ extinction metric. WISE band fluxes were based on
the AllWISE source catalog and were adopted from Chang et al. (2015). The
$E(B-V)$ was estimated through the Balmer extinction based on the H$\alpha$
and H$\beta$ fluxes from MPA-JHU and the conversion of Domínguez et al.
(2013). Distances used to calculate the luminosities were based on the
redshifts provided by the MPA-JHU catalog.

Figure 10: Comparisons between stellar masses estimated through a) the
extinction-dependent, and extinction-independent relations of Eq. 6 based on
the MCMC fitting (Table 3) which utilize the WISE band-1 photometry and the
$u{-}r$ or the $g{-}r$ color, b) the calibration of Wen et al. (2013) which
uses the WISE band 1 and the $u{-}r$ color and c) based on SED fitting from
Chang et al. (2015), Salim et al. (2018), and MPA-JHU (Kauffmann et al. 2003;
Brinchmann et al. 2004; Tremonti et al. 2004). In each subplot, the ordinate
indicates the logarithm of the ratio ($\rm log\leavevmode\nobreak\
(M_{\star,y}/M_{\star,x})$) between the compared $M_{\star}$ estimations, and
the abscissa the logarithm of the $M_{\star}$ as indicated in the axes labels.
Red error bars represent the modes and 68% percentiles of the distribution of
galaxies within bins of log $M_{\star}/M_{\odot}\simeq 0.6$ (8 bins within the
range $7<{\rm log}\leavevmode\nobreak\ M_{\star}/M_{\odot}<12$). The ${\rm
log}\leavevmode\nobreak\ M_{\star_{\rm y}}/M_{\star_{\rm x}}$ modes and 68%
percentiles for all the galaxies compared in each panel are shown at the top
right of each subplot. The points are color-coded based on the galaxies’
nebular $E(B-V)$. The black dashed line represents equality.

Overall, these comparisons show that calibrations using monochromatic NIR
photometry and an optical color agree on average with the stellar masses based
on SED fitting with a small standard deviation of about 0.1 dex. The
extinction-independent relation is in excellent agreement with the stellar
masses derived using the Wen et al. (2013) calibration showing $\sim-0.1$ dex
lower $M_{\star}$ on average without strong deviations in low or high stellar
masses. However, it shows a slightly better agreement in low and high
$M_{\star}$ galaxies compared to results based on SED fitting, while the
$u{-}r$ relation of Wen et al. (2013) shows a small overestimation for
galaxies with large $M_{\star}$ and high extinction.

The extinction-dependent relation of Eq. 6 is in good agreement with stellar
masses estimated through SED fitting. These comparisons show slightly better
agreement throughout the $M_{\star}$ range, even for very high and low-SFR
galaxies where the effect of reddening and SP age becomes gradually more
important. Moreover, as indicated by the color code of Figure 10 the
extinction-dependent relation of Eq. 6 tends to correct the overestimation of
high extinction galaxies, and vice-versa the underestimation of low extinction
galaxies, compared to extinction independent relations. The comparison with
the model SEDs, where we know a-priori the true $M_{\star}$ of the galaxies
(Section 4.3), indicates that the inferred stellar masses based on the
extinction-dependent relation of Eq. 6 and on SED fitting are closer to the
correct $M_{\star}$. These comparisons also demonstrate the importance of
taking extinction into account in the estimation of the stellar mass.

##  6 Conclusions

We used the SED fitting code CIGALE to create a large sample of mock galaxy
SEDs that covered a wide range of SFRs, stellar masses, and ISM conditions. We
compared the SFR and $M_{\star}$ of the model galaxies with their IR
luminosity for photometric bands of WISE and the JWST MIRI and NIRCam
instruments. This analysis showed that there is a strong dependence of the IR
tracers of galaxies’ SFR and $M_{\star}$ on the age of their SPs, and their
extinction. However, the combined use of IR bands that trace SPs of different
ages can provide more robust calibrations and minimize the biases and the
scatter they introduce in the SFR, and $M_{\star}$ calibrations. This is
particularly important for low-SFR galaxies where the contribution of old SPs
can be significant when measuring their SFR, and for high-SFR galaxies where
the contribution of dust emission can bias the $M_{\star}$ measurements. The
addition of extinction-dependent calibrations also offers more reliable SFR,
and $M_{\star}$ estimations with less scatter which are particularly better
for low-extinction/dust-free (e.g. dwarf) galaxies.

In summary, this work provides extinction-dependent and extinction-independent
calibrations and quantification of their scatter for measuring the:

  1. 1.

SFR based on the WISE band-3 and band-1 utilizing as extinction indicators the
$E(B{-}V)$, the $L_{\rm W4}/L_{u}$, and the $L_{\rm W4}/L_{g}$ ratio (Eq. 4,
Table 1)

  2. 2.

SFR based on the JWST NIR-F200W and MIRI-F2100W bands utilizing as extinction
indicator the $E(B{-}V)$ (Eq. 4, Table 1)

  3. 3.

mass-to-light ratio based on the WISE band-1 and the $u{-}r$ or the $g{-}r$
color utilizing as extinction indicators the $E(B{-}V)$, the $L_{\rm
W4}/L_{u}$, and the $L_{\rm W4}/L_{g}$ ratio (Eq. 6, Table 3)

  4. 4.

mass-to-light ratio based on the JWST NIR-F200W and the $u{-}r$ or $g{-}r$
color utilizing as extinction indicator the $E(B{-}V)$ (Eq. 6, Table 3).

The comparisons with both modeled and observed samples of galaxies show that
the proposed calibrations offer robust SFR, and $M_{\star}$ estimations for a
wide range of these values, minimizing the scatter. The random forest analysis
yields similar results to the MCMC method showing that the provided
calibrations are method independent.

###### Acknowledgements.

The authors thank the anonymous referee for providing comments and suggestions
that improved the clarity of this work. KK is supported by the project
”Support of the international collaboration in astronomy (Asu mobility)” with
the number: CZ.02.2.69/0.0/0.0/18_053/0016972. Asu mobility is co-financed by
the European Union. KK and AZ acknowledge funding from the European Research
Council under the European Union’s Seventh Framework Programme
(FP/2007-2013)/ERC Grant Agreement n. 617001 (A-BINGOS). This project has
received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie RISE action, grant
agreements No 691164 (ASTROSTAT), and No 873089 (ASTROSTAT-II). EK
acknowledges support from the Public Investments Program through a Matching
Funds grant to the IA-FORTH. SS is supported through NASA awards NNX12AE06G
and 80NSSC20K0440. JS acknowledges financial support from the Czech Science
Foundation under Project No. 22-22643S. This research has made use of: (a)
data products from the Wide-field Infrared Survey Explorer (WISE), which is a
joint project of the University of California, Los Angeles, and JPL,
California Institute of Technology, funded by NASA; (b) observations made with
the Spitzer Space Telescope, which was operated by JPL, California Institute
of Technology under a contract with NASA; (c) the NASA/IPAC Extragalactic
Database (NED), which is operated by the Jet Propulsion Laboratory (JPL),
California Institute of Technology, under contract with NASA; (d) the
NASA/IPAC Infrared Science Archive (IRSA), which is funded by NASA and
operated by the California Institute of Technology;

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## Appendix A

Table A1 shows the configuration of the CIGALE SED creation modules. See text
of Boquien et al. (2019) for a detailed description of the parameters.

Table A1: Configuration of the CIGALE SED creation modules. Module | sfhdelayed |
---|---|---
tau_main | 500, 1000, 2000, 4000, 6000, 8000 | e-folding time of the main SP model in Myr
age_main | 8000, 13000 | Age of the main SP in the galaxy in Myr
tau_burst | 10, 25, 50, 100, 250, 500, 1000, 2000 | e-folding time of the late starburst population
age_burst | 3000, 2000, 1000, 500, 250, 100, 50, 25, 10 | Age of the late burst in Myr
f_burst | 0.25, 0.1, 0.05, 0.01, 0.001, 0.0001, 0.0 | Mass fraction of the late burst population
sfr_A | 10,000 | Multiplicative factor controlling the SFR
normalise | False | Normalise SFH to produce one solar mass
Module | bc03 |
imf | 0 | Salpeter initial mass function
metallicity | 0.0001, 0.0004, 0.004, 0.008, 0.02, 0.05 | Z
separation_age | 10 | Young and old SPs separation age in Myr
Module | nebular |
logU | -1.0, -4.0 | Ionisation parameter
f_esc | 0.1 | Fraction of escaping $Ly$ continuum photons
f_dust | 0.1 | Fraction of absorbed $Ly$ continuum photons
lines_width | 300.0 | Line width (km/s)
emission | True | Include nebular emission
Module | dustatt_modified_starburst |
E_BV_lines | 0.001, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3 | color excess of the nebular lines
| 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65 |
| 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1.0 |
E_BV_factor | 0.3 | Reduction factor applied on E_BV_lines.
uv_bump_wavelength | $217.5$ | Central wavelength of the UV bump (nm)
uv_bump_width | $35.0$ | Width (FWHM) of the UV bump (nm)
uv_bump_amplitude | $0.0$ | Amplitude of the UV bump
powerlaw_slope |  $-1.5$, $-0.5$, $0.0$, $0.5$ | Modifying slope $\delta$
Ext_law_emission_lines | 1 | 1 corresponds to MW extinction law
$R_{\rm V}$ | 3.1 | $A_{\rm V}/E(B-V)$
Module | dale2014 |
fracAGN | 0.0 | AGN fraction
alpha | 2.0 | Alpha slope
Module | restframe_parameters |
beta_calz94 | True | UV slope as in Calzetti et al. (1994)
D4000 | True | As in Balogh et al. (1999)
IRX | True | based on GALEX FUV and dust luminosity
Module | redshifting |
redshift | 0 |