11institutetext: Instituto de Astrofísica e Ciências do Espaço, Universidade
de Lisboa - OAL, Tapada da Ajuda, PT1349-018 Lisboa, Portugal 22institutetext:
Departamento de Física, Faculdade de Ciências da Universidade de Lisboa,
Edifício C8, Campo Grande, PT1749-016 Lisboa, Portugal 33institutetext:
Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto - CAUP,
Rua das Estrelas, PT4150-762 Porto, Portugal 44institutetext: Departamento de
Física e Astronomia, Faculdade de Ciências, Universidade do Porto, Rua do
Campo Alegre 687, PT4169-007 Porto, Portugal

# An investigation of the star-forming main sequence considering the nebular
continuum emission at low-$\mathrm{\it z}$

Henrique Miranda 1122 Ciro Pappalardo 1122 Polychronis Papaderos 112233 José
Afonso 1122 Israel Matute 1122 Catarina Lobo 3344 Ana Paulino-Afonso 33
Rodrigo Carvajal 1122 Silvio Lorenzoni 11 Duarte Santos 1122

###### Abstract

Context. Galaxy evolution has been studied by interpreting the spectral energy
distribution of galaxies using spectral synthesis codes. This method has been
crucial in discovering different pillars of modern galaxy evolution theories.
However, this analysis was mostly carried out using spectral synthesis codes
that are purely stellar, that is, they assume that the nebular contribution to
the total continuum is negligible. The code FADO is the first publicly
available population spectral synthesis tool that treats the contribution from
ionised gas to the observed emission self-consistently. This is expected to
have a particularly strong effect in star-forming (SF) galaxies.

Aims. We study the impact of the nebular contribution on the determination of
the star formation rate (SFR), stellar mass, and consequent effect on the
star-forming main sequence (SFMS) at low redshift.

Methods. We applied FADO to the spectral database of the SDSS to derive the
physical properties of galaxies. As a comparison, we used the data in the MPA-
JHU catalogue, which contains the properties of SDSS galaxies derived without
the nebular contribution. We selected a sample of SF galaxies with H$\alpha$
and H$\beta$ flux measurements, and we corrected the fluxes for the nebular
extinction through the Balmer decrement. We then calculated the H$\alpha$
luminosity to estimate the SFR. Then, by combining the stellar mass and SFR
estimates from FADO and MPA-JHU, the SFMS was obtained.

Results. The H$\alpha$ flux estimates are similar between FADO and MPA-JHU.
Because the H$\alpha$ flux was used as tracer of the SFR, FADO and MPA-JHU
agree in their SFR. The stellar mass estimates are slightly higher for FADO
than for MPA-JHU on average. However, considering the uncertainties, the
differences are negligible. With similar SFR and stellar mass estimates, the
derived SFMS is also similar between FADO and MPA-JHU.

Conclusions. Our results show that for SDSS normal SF galaxies, the additional
modelling of the nebular contribution does not affect the retrieved fluxes and
consequentially also does not influence SFR estimators based on the
extinction-corrected H$\alpha$ luminosity. For the stellar masses, the results
point to the same conclusion. These results are a consequence of the fact that
the vast majority of normal SF galaxies in the SDSS have a low nebular
contribution. However, the obtained agreement might only hold for local SF
galaxies, but higher-redshift galaxies might show different physical
properties when FADO is used. This would then be an effect of the expected
increased nebular contribution.

###### Key Words.:

galaxies: evolution - galaxies: fundamental parameters - galaxies: star
formation - galaxies: stellar content - techniques: spectroscopic - methods:
numerical

##  1 Introduction

Galaxies emit radiation that covers the whole range of the electromagnetic
spectrum. The analysis of this radiation enables us to study their physical
properties, formation, and evolution. In this context, a method that is widely
used to extract the physical properties of galaxies is spectral modelling. The
physical properties of extragalactic objects are retrieved and interpreted
based on their spectral energy distribution (SED) through spectral synthesis
methods. These methods were fundamental in the discovery of different pillars
of modern galaxy evolution theories (e.g. Kauffmann et al. 2003; Brinchmann et
al. 2004; Tremonti et al. 2004; Cid Fernandes et al. 2005; Gallazzi et al.
2005; Salim et al. 2007; Mannucci et al. 2010).

One of the challenges of these methods is distinguishing all the different
processes that contribute to the overall emission of the galaxy. This is
related to the intrinsic nature of the observed light, which is a combination
of emission from stars, gas, and dust. To fully model this emission, all these
mechanisms must be included in a consistent framework. In the past, inverse
population synthesis codes were built with the underlying assumption that the
contribution to the observed emission in the optical waveband due to ionised
gas was negligible. These codes are known as purely stellar codes. On the
other hand, most evolutionary synthesis codes include the nebular emission,
such as GALEV (Kotulla et al. 2009), Pégase (Fioc & Rocca-Volmerange 1997),
and STARBURST99 (Leitherer et al. 1999).

Most of the analysis of the spectroscopic data from one of the largest surveys
ever made, The Sloan Digital Sky Survey (SDSS; York et al. 2000), has been
carried out using inverse population synthesis codes. The SDSS provides one of
the most complete databases for studies of galaxy evolution and has been
fundamental in the investigation of this topic. Studies of the data obtained
by the SDSS demonstrated that the distribution of galaxy properties is
bimodal, which means that star-forming (SF) galaxies and passive galaxies have
distinct properties (Gavazzi et al. 2010). The SDSS enabled the extensive
measurement of fundamental galactic properties such as the star formation rate
(SFR) and stellar mass (Brinchmann et al. 2004; Kauffmann et al. 2003). These
two properties were found to be tightly correlated for SF galaxies, therefore
this relation was called star-forming main sequence (SFMS; Noeske et al.
2007). Since then, the SFMS has been extensively studied and was found to
exist up to a redshift of 4, maintaining approximately the same slope, but
increasing in normalisation factor (see Speagle et al. 2014, for a compilation
of results). This is generally considered to be the fundamental relation of
galaxy evolution theories today.

One particularly noteworthy spectroscopic analysis of the SDSS is provided by
the Max Planck Institute for Astrophysics-Johns Hopkins University (MPA-JHU)
catalogue, which analysed the spectroscopic data using a purely stellar
approach. The currently available MPA-JHU catalogue111https://wwwmpa.mpa-
garching.mpg.de/SDSS/DR7 provides a benchmark spectroscopic analysis of the
Data Release 7 of SDSS (SDSS-DR7; Abazajian et al. 2009). The MPA-JHU
catalogue has raw data with the object information (redshift, velocity
dispersion, sky position, etc.) and spectroscopic measurements (line fluxes,
equivalent widths (EWs), and continuum indices). The MPA-JHU catalogue has
also derived metadata estimating SFRs, based on Brinchmann et al. 2004
(hereafter B04), and stellar masses, based on fits to the photometry following
the philosophy of Kauffmann et al. (2003) and Salim et al. (2007).

All these analyses of the SDSS spectroscopic data have been performed so far
without considering the contribution of nebular gas. The modelling of the
nebular gas has been neglected because it is complex to implement this in
spectral synthesis codes and because the nebular component was expected to be
low for SDSS galaxies. However, the contribution from the nebular emission,
that is, the nebular gas surrounding SF regions, can rise up to 60% of the
total emission (Krüger et al. 1995; Izotov et al. 1997; Papaderos et al. 1998;
Leitherer et al. 1999; Anders & Fritze-v. Alvensleben 2003; Schaerer & de
Barros 2009). Hence, even if the assumption of negligible nebular emission
might be reasonable for galaxies with low SFRs, it requires some caution when
applied to galaxies with higher SFRs (Guseva et al. 2007; Cardoso et al. 2019;
Pappalardo et al. 2021).

The inclusion of the nebular component in the analysis affects the estimation
of the continuum. In the case of a purely stellar code, the continuum is fit
considering it is only due to stellar contribution, whereas in FADO, the
continuum is fit with a combination of the stellar and the nebular emissions.
FADO considers the spectral shape of the continuum in the analysis, and this
leads to a different parameter estimation. The essence of the problem is not
primarily the total luminosity fraction of the nebular continuum, but its
spectral shape: whereas the ionising stellar continuum is blue, the ionised
gas (specifically, the nebular continuum) is reddish/flat (in the optical,
excluding the Balmer and Paschen discontinuity). From the observational point
of view, this was discussed in Papaderos et al. (1998) (see their Fig. 8), who
presented a spectral decomposition into the stellar and nebular continuum.
Their figure and also Fig. 9 of Izotov et al. (2011) shows that the relative
contribution of the nebular continuum increases with wavelength. In other
words, the nebular continuum makes the total SED redder than the pure SED of
the stellar component. As pointed out in Izotov et al. (2011), this reddish
continuum forces a purely stellar code to invoke a much too high fraction of
red stellar populations (i.e. old and with a high mass-to-light ratio),
consequently to overestimate the stellar mass. On the other hand, to compute
SFRs based on emission line fluxes, the local continuum needs to be
subtracted, which is also handled differently in different codes.

In face of these issues, and because it has been shown in particular that the
contribution of the nebular component can be significant for some galaxies, it
is fundamental to evaluate its effect on the relations between galactic
properties that have been established throughout the years. This required a
proper modelling of the stellar and nebular contribution to the observed
emission of a galaxy.

The code FADO222https://spectralsynthesis.org/fado (fitting analysis using
differential evolution optimization; Gomes & Papaderos 2017) is the first
population spectral synthesis code capable of fitting the optical SED due to
the stellar and nebular components self-consistently. The usage of FADO and
its novel approach to SED fitting is expected to play a decisive role in
shedding light into the impact of the nebular contribution when spectroscopic
data are analysed and how this might have influenced our current understanding
of the processes taking place in galaxies. The capabilities of FADO were
tested in a series of benchmark publications (Cardoso et al. 2019; Pappalardo
et al. 2021).

Cardoso et al. (2022) compared FADO with STARLIGHT (Cid Fernandes et al. 2005)
by applying these tools to a set of synthetic spectra of galaxies with
different star formation histories (SFHs). The consideration of different SFHs
allowed the comparison of the results between FADO and STARLIGHT as a function
of the level of the nebular contribution. The stellar masses, light- and mass-
weighted stellar ages, and metallicity estimates of the two tools were
compared, and relevant differences were found. For galaxies with a significant
nebular contribution, STARLIGHT can overestimate the stellar mass and mass-
weighted mean stellar age up to $\sim$2 dex and $\sim$4 dex, respectively. On
the other hand, STARLIGHT underestimates the mean metallicity and light-
weighted mean stellar age by up to $\sim$0.6 dex. Compared to these results,
FADO obtains significantly better estimates because these properties can be
estimated with a precision of $\sim$0.2 dex, even for evolutionary stages
where the nebular contribution is highly prominent. Pappalardo et al. (2021)
followed a similar method to study the relation between the signal-to-noise
ratio (S/N) and the accuracy of the estimated physical properties. Moreover,
this work related the overestimation of the mass-weighted quantities by pure
stellar codes to their difficulty in modelling the region of the spectrum
around the Balmer jump.

The next step after these initial tests was to apply FADO to real spectra, and
with this goal in mind, FADO was applied to the SDSS-DR7 (Cardoso et al.
2022). Cardoso et al. (2022) carried out a similar analysis to that of Cardoso
et al. (2019), but this time to real data. In Section 7, this work and the
results obtained are discussed in more detail.

In this paper, we intend to follow the line of investigation of re-evaluating
previously established relations between physical properties of galaxies in
the light of the novel self-consistent approach of FADO, focusing on the SFMS.
This is relevant because modelling the nebular continuum is expected to be
particularly influential for SF galaxies, in which the nebular contribution is
higher in general compared to other types of galaxies. Studying the SFMS with
FADO in a well-known data set such as SDSS-DR7 is therefore expected to
clarify the importance of modelling the nebular continuum for SF galaxies at
low z. Moreover, it will serve as a reference test, providing insights for
future studies of the SFMS with FADO at higher redshifts. This will help the
current effort to understand the best tool for modelling data from future
surveys at higher redshifts.

The aim of this paper is to quantify the impact of the nebular contribution on
the estimation of galactic properties in SF galaxies. To do this, results from
the application of FADO to the SDSS-DR7 are used to estimate the SFR and
stellar mass. Next, we compare our results with a previous analysis of the
SDSS-DR7, quantifying the differences due to the nebular contribution. In this
way, the results from FADO are compared to the data in the MPA-JHU catalogue,
which was obtained using a purely stellar approach. In summary, our objectives
are a) to quantify and understand the influence of the nebular component in
the determination of the SFR, stellar mass, and SFMS, abd b) to relate the
obtained results with the contribution of the nebular emission to the total
emission.

The paper is organised as follows: Section 2 provides an overall view of the
SDSS and the selection criteria of the studied sample. Section 3 details the
spectral fitting tool FADO. In Sects. 4 and 5 we explain how the SFR and
stellar mass were derived, respectively. Section 6 presents the results of the
comparison of the SFR, stellar mass, and SFMS between FADO and MPA-JHU. In
Section 7 we discuss the results. Finally, Section 8 provides a summary of the
main results of this work and their relevance. We use a cosmology with $H_{0}$
= 70 km s-1 Mpc-1, $\Omega_{M}$ = 0.3, and $\Omega_{\Lambda}$ = 0.7. We
consider the Chabrier (2003) initial mass function (IMF) to obtain the SFR and
stellar mass estimates. To convert these into a Kroupa (2001) IMF, the
estimates should be multiplied by 1.06 (Zahid et al. 2012; Speagle et al.
2014; Madau & Dickinson 2014).

##  2 Sample

The SDSS is a survey of photometric and spectroscopic data across $\pi$ sr,
conducted with the 2.5 m telescope at the Apache Point Observatory. The
galaxies analysed in this study were drawn from the SDSS-
DR7333http://classic.sdss.org/dr7/ (Abazajian et al. 2009), which has
spectroscopic data for a wavelength coverage of 3800 - 9200 Åwith a resolution
of 1800 - 2200 and an S/N $>$ 4 per pixel at $g$ = 20.2 for 926 246 objects.
The spectra were obtained with fibres that have a sky aperture of 3 arcsec.
The SDSS-DR7 also has photometric data for these objects in five bands ($u$,
$g$, $r$, $i$, $z$).

We considered the application of FADO to SDSS-DR7 carried out Cardoso et al.
(2022)444A repository with the data is being prepared and will be soon
publicly available.. We started by finding the galaxies in common in the FADO
data and the galaxies available in the MPA-JHU catalogue. There are 916 118 in
common galaxies in the two sets of data. To remove low flux values, the
following conditions were defined for H$\alpha$ and H$\beta$ for both FADO and
MPA-JHU: $-$20 $<$ $\log$(F[erg s-1cm-2]) and S/N $>$ 3 (680 432 galaxies were
removed with these criteria). We set a lower mass limit of 105 M⊙ for both
FADO and MPA-JHU to ensure that no low-mass peculiar objects or massive
globular clusters were included in the sample (19 517 galaxies were removed
with this criterion). We removed these objects because they might have a non-
linear relation between SFR and stellar mass. With the described constraints,
the sample was reduced to 216 169 galaxies, approximately 23% of the common
sample. From this sample, only the SF galaxies were selected, as described
next.

FADO uses a different galaxy classification scheme with respect to MPA-JHU
(for details, see Brinchmann et al. 2004). For both FADO and MPA-JHU, the
classification is based on the emission line properties of the galaxies
following the Baldwin, Phillips & Terlevich (1981, hereafter BPT) diagram.
FADO classifies the galaxies strictly using their position on the BPT diagram,
leading to the following categories: Pure SF, SF, Composite, LINER, and
Seyfert. On the other hand, MPA-JHU considers in addition to the position on
the BPT diagram also the S/N of the BPT lines, resulting in the following
categories: SF, low S/N SF, Composite, AGN non-LINER, and low S/N LINER.
Moreover, for both FADO and MPA-JHU, galaxies that lack a flux measurement for
one of the BPT lines are said to be Unclassifiable.

To select a common sample of SF galaxies, two subsamples were defined: SF
galaxies, and non-SF galaxies. The subsample of SF galaxies for FADO includes
Pure SF and SF galaxies, and for MPA-JHU, the sample includes SF and low S/N
SF galaxies. The subsample of non-SF galaxies for FADO and MPA-JHU includes
all the remaining categories except for the Unclassifiable galaxies. Finally,
the galaxies equally classified as SF or non-SF in both FADO and MPA-JHU were
selected, from which we obtained a sample of 182 933 SF galaxies and 25 220
non-SF galaxies. This information is summarised in Table 1. The SF sample of
182 933 galaxies is the final sample used in this work.

The mean redshift of our sample is $\langle z\rangle=0.07\pm 0.04$. The
redshift distribution of the galaxies is presented in Figure 1.

Figure 1: Redshift distribution of the galaxies in our sample. Total number of galaxies: 182 933. Table 1: Galaxy classification according to the categories defined in this work. In the FADO and MPA-JHU columns, the total number of galaxies is 216 169, and for the last column, it is 208 153. | FADO | MPA-JHU | FADO & MPA-JHU
---|---|---|---
Galaxy type | Number | Percentage | Number | Percentage | Number | Percentage
SF | 183 460 | 85 | 190 374 | 88 | 182 933 | 88
Non-SF | 31 184 | 14 | 25 787 | 12 | 25 220 | 12
Unclassifiable | 1 525 | 1 | 8 |  $>$ 1 | 0 | 0

##  3 Method

FADO (Gomes & Papaderos 2017) is the first population spectral synthesis code
that self-consistently fits the optical emission due to the stellar and
nebular components. This allows a spectroscopic analysis that takes into
account the contribution of the nebular gas emission to the observed SED,
which is a novelty compared to previous spectral synthesis codes.

FADO employs a differential genetic optimisation approach known as
differential evolution optimisation (DEO; Storn & Price 1996). The benefit of
this approach is a convergence to solutions in a stable way within a
reasonable computational time. Another innovation of FADO is the consistent
treatment of the main nebular characteristics, in particular, the luminosities
and EWs of H$\alpha$ and H$\beta$ and the continuum near the Balmer and
Paschen jumps when determining the physical and evolutionary properties of SF
galaxies. The fitting algorithm is responsible for this consistent treatment
of the nebular component, ensuring that during the minimisation process, the
various stellar components are consistent with the expected line and continuum
nebular emission characteristics. For instance, at the end of the convergence
procedure, the production of ionising radiation by the stellar component is
consistent with the nebular continuum observed and the Balmer line fluxes.
This is possible because the estimation and inclusion of the nebular continuum
in the best-fitting SED and the calculation of the electron density and
temperature of the ionised gas is made on the fly.

The model SED for FADO, $F_{\lambda}$, results from a linear combination of
$N_{*}$ spectral components dependent on the considered set of base spectra.
The model SED has a stellar component to which the emission from stars
contributes, plus a nebular component that is due to the nebular continuum and
the hydrogen recombination lines. This model SED is given by (Gomes &
Papaderos 2017)

$\displaystyle F_{\lambda}$
$\displaystyle=\overbrace{\sum_{j=1}^{N_{*}}M_{j,\lambda_{0}}L_{j,\lambda_{0}}\times
10^{-0.4A_{v}q_{\lambda}}\otimes
S\left(v_{*},\sigma_{*}\right)}^{Stellar\,Term}$ (1)
$\displaystyle+\overbrace{\Gamma_{\lambda}\left(n_{e},T_{e}\right)\times
10^{-0.4A_{v}^{neb}q_{\lambda}}\otimes
N(v_{\eta},\sigma_{\eta})}^{Nebular\,Term}$

On the stellar term, $M_{j,\lambda_{0}}$ is the stellar mass of the jth
spectral component at the normalisation wavelength $\lambda_{0}$,
$L_{j,\lambda}$ is the spectrum of the jth element, $A_{v}$ is the V band
extinction, and $S\left(v_{*},\sigma_{*}\right)$ represents the stellar
kinematics. On the nebular term, $\Gamma_{\lambda}\left(n_{e},T_{e}\right)$ is
the nebular continuum spectrum where $n_{e}$ and $T_{e}$ are the electron
density and temperature, respectively, $A_{v}^{neb}$ is the gas extinction,
and $N(v_{\eta},\sigma_{\eta})$ represents the nebular kinematics. In both
terms, $q_{\lambda}$ is the ratio of $A_{\lambda}$ over the V-band extinction.

The self-consistency of FADO means that both terms of Eq. 1 are calculated
simultaneously while ensuring their consistency. Pure stellar codes only
provide information regarding the first term, requiring the use of
photoionisation codes to model the nebular contribution in a second step of
the analysis. This separate treatment of the stellar and nebular components
leads to a loss of consistency between the two components. It should be noted
that there have been other attempts at a consistent fitting of the emission
from gas and stars using different approaches (e.g. Guseva et al. 2001;
Acquaviva et al. 2011; Amorín et al. 2012; Chevallard & Charlot 2016).
However, these approaches did not strictly follow the principles of population
spectral synthesis. For example, Guseva et al. (2001) and Amorín et al. (2012)
only considered old and young SSPs instead of a full library of SSPs. In
summary, the main aspects of FADO are i) that both the stellar and nebular
continuum are included in the fit, ii) the emission lines are automatically
masked, iii) the extinction law is set as an input parameter, and iv)
instrumental and physical broadening effects are included.

The parameter space is explored and the minimum $\chi^{2}$ between the
observed and fitted SED is searched for while ensuring that the various
stellar components are consistent with the expected line and continuum nebular
emission characteristics. This consistency in the fitting of the stellar and
nebular components ensures the previously mentioned self-consistency between
the stellar and nebular optical emission.

The application of FADO to the SDSS-DR7 spectroscopic data was fully described
in Cardoso et al. (2022), and the spectrum pipeline analysis used to derive
the data in the MPA-JHU catalogue was presented in Tremonti et al. (2004).
Here, we only summarise the procedures adopted in these two works.

Cardoso et al. (2022) corrected the raw SDSS-DR7 spectra for the foreground
extinction considering the dust maps from Schlegel et al. (1998) and were then
converted into the observer restframe. Next, the spectra were fitted with FADO
in the wavelength range between 3400 and 8900 Å with a base library from
Bruzual & Charlot (2003) composed of 150 simple stellar populations (SSPs),
containing 25 ages (between 1 Myr and 15 Gyr) and six metallicities (1/200,
1/50, 1/5, 2/5, 1, and 2.5 Z⊙). A Chabrier (2003) IMF was considered to match
the base library, Padova 1994 evolutionary tracks (Alongi et al. 1993; Bressan
et al. 1993; Fagotto et al. 1994a, b; Girardi et al. 1996), and the Calzetti
et al. (2000) extinction law.

To derive the data in the MPA-JHU catalogue, the continua of the spectra were
corrected for the foreground extinction following the Schlegel et al. (1998)
dust maps and were then fitted in the wavelength range between 3200 and 9300 Å
following the models by Bruzual & Charlot (2003) with ten ages (between 5 Myr
and 10 Gyr) and three metallicities (1/5, 1 and 2.5 Z⊙). A Kroupa (2001) IMF,
Padova 1994 evolutionary tracks (Alongi et al. 1993; Bressan et al. 1993;
Fagotto et al. 1994a, b; Girardi et al. 1996), and the Charlot & Fall (2000)
extinction law were used.

There are differences between the two approaches that affect the derived
results, in particular the IMF and extinction law. A different IMF impacts the
SFR and stellar mass estimates. However, this impact can be straightforwardly
solved by applying a conversion factor between the estimates obtained with the
two IMFs. In the case of the extinction law, Dickinson et al. (2003) and
Papovich et al. (2001) suggested that the impact of using different extinction
laws is negligible. Thus, the fact that different extinction laws are used in
the spectral modelling should not play a significant role in the derived
results. The differences in the age and metallicity are not expected to have a
significant impact on the final estimations because a different choice of SSP
ages will mostly affect the star formation history, while a different range of
metallicity does not change the stellar mass estimates for galaxies at $z$ ¡
1, as discussed in Speagle et al. (2014).

Moreover, the fluxes in the MPA-JHU catalogue were renormalised to match the
photometric fiber magnitude in the r band, which is a different
spectrophotometric calibration than the one considered in SDSS-DR7. This
renormalisation factor is provided in the MPA-JHU catalogue and was applied to
the flux and mass estimates used in this work so that they are comparable
between the two data sets.

##  4 SFR determination

We focused on SF galaxies. A common tracer of the SFR used in the literature
is the H$\alpha$ flux (e.g. Gallagher et al. 1984; Kennicutt 1998; Shim et al.
2011; Zahid et al. 2012; Sobral et al. 2013), and it was also the tracer we
used. The relation between the H$\alpha$ flux and the SFR is discussed in
Sect. 4.2.

###  4.1 Emission line properties

Prior to computing the SFR, it should be insightful to characterise and
compare the properties of the emission lines retrieved by FADO and MPA-JHU. To
begin, it is important to have a sense of the relation between the observed
flux of an emission line characteristic of SF regions for FADO and MPA-JHU.
This information is a first indicator of the relation between the estimated
SFR for FADO and MPA-JHU.

Figure 2 compares the observed flux of H$\alpha$ between FADO and MPA-JHU. The
H$\alpha$ flux distributions are consistent, showing negligible differences.
This indicates a similar SFR estimate.

Figure 2: Comparison of the observed H$\alpha$ flux between FADO and MPA-JHU.
Left panel: Histogram of $\log$[F(H$\alpha$)] for FADO (blue) and for MPA-JHU
(orange). Right panel: Scatter plot of the logarithm of the ratio of the
H$\alpha$ flux of FADO and MPA-JHU as a function of $\log$[F(H$\alpha$)] for
FADO.

In addition to the flux, another important characteristic of a spectral line
is the EW. Figure 3 compares the EW of H$\alpha$ between FADO and MPA-JHU.

Figure 3: Comparison of the observed H$\alpha$ equivalent width for FADO and
MPA-JHU. Left panel: Histogram of $\log$[EW(H$\alpha$)] for FADO (blue) and
for MPA-JHU (orange). Right panel: Scatter plot of the logarithm of the ratio
of the H$\alpha$ EW of FADO and MPA-JHU as a function of $\log$[EW(H$\alpha$)]
for FADO.

The peak of the distribution for FADO corresponds to higher EW values than the
peak for MPA-JHU. The right panel of Figure 3 clearly shows that FADO obtains
higher EWs than MPA-JHU. Almost all the galaxies are distributed above the
blue reference line which marks equality between the estimates. This means
that for practically all SF galaxies, FADO retrieves higher EWs than MPA-JHU.
Their median difference shows that FADO estimates higher values than MPA-JHU
by 0.1 dex ($\approx$26%).

FADO measures higher EWs because it considers a different definition of the
level of the continuum. For FADO, the continuum is the monochromatic continuum
level at $\lambda_{0}$, which takes into account possible absorption features
and stellar velocity dispersion. On the other hand, for MPA-JHU, the continuum
is obtained by averaging the level of the continuum adjacent to the emission
or absorption line. The latter approach is most commonly used in the
literature. Because the stellar absorption is taken into account by FADO, the
EW values are higher than in MPA-JHU.

A consequence of the alternative (but more physically motivated) definition of
the continuum level in FADO is the observed measurement of larger H$\alpha$
EWs than in MPA-JHU. This difference in the definition of the continuum is
only relevant for recombination lines, such as H$\alpha$, because of the
absorption feature below the emission line. For forbidden lines, such as
[OIII]5007, both continuum definitions lead to compatible EWs.

As explained, the level of the continuum from which the emission line
properties are measured is different between FADO and MPA-JHU. From the
definition of EW, the level of the continuum is given by the ratio of the flux
and the EW. Thus, because FADO obtains in median H$\alpha$ EWs 0.1 dex higher
than MPA-JHU, then the continuum is in median 0.1 dex higher for MPA-JHU than
for FADO. Figure 4 shows the different definitions of the continuum for FADO
and MPA-JHU for the H$\alpha$ emission line.

Figure 4: Comparison of the definition of the continuum for FADO and MPA-JHU
for the H$\alpha$ line. The red line is the observed spectrum, and the green
line is the fitted model. The dashed orange line represents the continuum as
defined from averaging the level of the continuum adjacent to the emission
line (MPA-JHU definition). The dashed blue line represents the continuum
considering the bottom of the absorption feature below the emission line (FADO
definition).

Before using the H$\alpha$ flux to derive the SFR, we corrected the flux for
the intrinsic extinction caused by the gas and dust in the line of sight of
the observer. The flux was corrected for the nebular extinction through the
Balmer decrement (H$\alpha$/H$\beta$) and using the extinction law from
Charlot & Fall (2000) to be consistent with the procedure from MPA-JHU. The
procedure is fully described in Appendix A.

###  4.2 SFR from the H$\alpha$ flux

Young and massive stars emit a large amount of ionising radiation that ionises
the atoms in the surrounding gas, thus producing emission lines. In this way,
strong emission lines are characteristic of SF regions, in particular, the
lines from the Balmer series such as H$\alpha$ and H$\beta$. Because the most
significant contribution to the ionising flux comes from stars with masses
higher than 10 M⊙ and lifetimes short than 20 Myr, the emission lines are a
tracer of the current SFR (Kennicutt 1998). The H$\alpha$ line is a strong
line and is located in the optical wavelength range. It is therefore typically
used as a tracer of the current SFR.

The H$\alpha$ luminosity can be calculated from the H$\alpha$ flux, and these
two quantities are related by

$L(H\alpha)=F(H\alpha)4\pi d_{L}^{2},$ (2)

where L(H$\alpha$) and F(H$\alpha$) are the H$\alpha$ luminosity and flux,
respectively, and $d_{L}$ is the luminosity distance. The SFR and the
H$\alpha$ luminosity are related by

$\textrm{SFR}=\dfrac{L(H\alpha)}{\eta(H\alpha)},$ (3)

where L(H$\alpha$) is the H$\alpha$ luminosity and $\eta$(H$\alpha$) is the
conversion factor to the SFR. B04 used the Kennicutt (1998) conversion factor
$\eta$(H$\alpha$)=10 41.28 erg s-1M${}_{\odot}^{-1}$yr. For consistency with
B04, we assumed the same conversion factor, but calibrated to a Chabrier
(2003) IMF, $\eta$(H$\alpha$)=10 41.31 erg s-1M${}_{\odot}^{-1}$yr.

To derive a conversion factor between SFR and L(H$\alpha$), a set of
assumptions were considered, such as a continuous star formation over a period
of about 100 Myr and a metallicity equal to solar abundances. Weilbacher &
Fritze-v. Alvensleben (2001) showed that deviations from these assumptions can
lead to significant errors in the estimated SFR for particular types of
galaxies, such as star-bursting dwarf galaxies. However, when our sample
selection criteria are taken into account, galaxies with significant
deviations from the mentioned assumptions, such as starburst galaxies,
represent a very small fraction of the galaxies in our sample. Additionally,
because we derived the SFR through the same procedure for FADO and MPA-JHU
both, the concerns raised in Weilbacher & Fritze-v. Alvensleben (2001) do not
affect our goal of comparing SFR estimates between FADO and MPA-JHU.

In summary, with Equation 2, the H$\alpha$ flux can be converted into
H$\alpha$ luminosity. With Equation 3 , the SFR can then be estimated.

###  4.3 Conversion from fibre SFR into total SFR

The SDSS is a fibre-based survey. In this way, the SFR derived from the flux
corresponds only to the fibre SFR (hereafter, referred to as SFRfib), that is,
the SFR of the galactic region sampled by the fibre. The fibre only samples
about one-third of the total galaxy light at the median redshift of the SDSS
(Brinchmann et al. 2004). Hence, when the total SFR (SFRtot) is to be
obtained, the SFRfib needs to be corrected for aperture effects.

B04 corrected the SFR estimates for aperture effects. However, Salim et al.
(2007) showed that the method used in B04 overestimated the SFR for galaxies
with low SFRs. They also reported that it was possible to remove this bias. To
improve the aperture correction, the light outside the fibre was calculated
for each galaxy, and then stochastic models similar to those used by Salim et
al. (2007) were fitted to this photometry. These improved SFRtot estimates are
available on the MPA-JHU website.

There have been more recent attempts to empirically correct fiber-based SFRs
for aperture effects, taking advantage of integral field spectroscopy data
(e.g. Iglesias-Páramo et al. 2013; Duarte Puertas et al. 2017). However,
because we compare our data with the data from the MPA-JHU catalogue, we
decided to use its aperture corrections in order to be consistent with their
procedure. Because the aperture correction is a factor for each galaxy, the
SFRfib estimates with FADO and MPA-JHU are equally affected. This means that
our final goal of comparing the SFRtot between FADO and MPA-JHU is independent
of the applied aperture correction.

With the aim of applying the described correction to our SFRfib estimates, we
calculated the ratio of SFRtot and SFRfib from the MPA-JHU catalogue. This
ratio represents the relation between the SFRtot and the SFRfib for each
galaxy, so that by multiplying this ratio by the estimated SFRfib , we
obtained an estimate of the SFRtot. The distribution of the ratio of the
SFRtot and SFRfib from the MPA-JHU catalogue for our SF sample is presented in
Figure 5. We also show the effect of the correction on the SFRfib estimated
with FADO. The effect of this correction in the SFRfib estimates for MPA-JHU
is not shown since it is the same correction factor as for the FADO estimates.

Figure 5: Conversion from fibre into total SFR for the common SF sample. Left
panel: Histogram of the logarithm of the ratio of the fibre and total SFR for
MPA-JHU. Middle panel: Scatter plot of the logarithm of the ratio of the fibre
and total SFR as a function of the logarithm of the fibre SFR for MPA-JHU.
Right panel: Histogram of the fibre SFR (blue) and total SFR (green) for FADO.

The distribution of the SFRfib to SFRtot conversion factor (left panel of
Figure 5) gives an insight into the relation between the two estimates. The
most common conversion factor is $\log$(SFRtot/SFRfib) $\approx$ 0.3, which
means that the SFRtot is twice as high as the SFRfib. After this peak, the
distribution steadily decreases to $\log$(SFRtot/SFRfib) $\approx$ 2, after
which there are practically no galaxies. The comparison of the SFRfib and
SFRtot estimates for FADO (right panel of Figure 5) shows that the
distribution of the latter is narrower, covering one order of magnitude less
at low values, but it maintains approximately the same upper limit. Moreover,
the peak of the SFRtot distribution occurs at higher values. These facts lead
to the interpretation that the galaxies with lower SFRfib are most strongly
affected by the aperture correction. On the other hand, the galaxies with the
highest SFRfib are almost not affected by the correction, as confirmed by the
middle panel of Figure 5.

The final SFRtot estimates for FADO and MPA-JHU considered in this work were
obtained applying the above correction. Hereafter, only the SFRtot is
considered, and so it is simply referred to as the SFR.

##  5 Stellar mass determination

The stellar masses available in the MPA-JHU catalogue were obtained following
a similar philosophy as was used in Kauffmann et al. (2003). Kauffmann et al.
(2003) performed a fit to the SDSS spectra following various models from
Bruzual & Charlot (2003). In these fits, two spectral indices were considered:
the 4000Å break strength, and the Balmer absorption-line index H$\delta_{A}$.
From these indices, the stellar mass-to-light ratio was estimated, which was
then used to obtain the stellar mass. The stellar masses available in the MPA-
JHU catalogue differs from this method as the fits are based on the broadband
$u$, $g$, $r$, $i$ , and $z$ photometry. However, the differences between the
two methods are very small (median difference of 0.01 dex), and they are
discussed on the MPA-JHU website.

For FADO, the stellar mass is derived through a fit to the spectrum. The
stellar mass has to be corrected for aperture effects as well. We followed the
same procedure as for the SFR, calculating the ratio of the total and fibre
stellar mass for the MPA-JHU data and then applying this factor to the fibre
stellar estimate of FADO in order to obtain the total stellar mass. The
distribution of the ratio of total and fibre stellar mass from the MPA-JHU
catalogue is presented in Figure 6. We also show the effect of the correction
on the stellar mass estimated with FADO.

Figure 6: Conversion from fibre to total stellar mass for the common SF
sample. Left panel: Histogram of the logarithm of the ratio of the fibre and
total stellar mass for MPA-JHU. Middle panel: Scatter plot of the logarithm of
the ratio of the fibre and total stellar mass as a function of the logarithm
of the fibre stellar mass for MPA-JHU. Right panel: Histogram of the fibre
stellar mass (blue) and total stellar mass (green) for FADO.

The distribution of the conversion factor between fibre and total stellar mass
shows that the most common difference between these quantities is $\approx$0.5
dex, which corresponds to a stellar mass about three times higher than the
fibre stellar mass. After the peak, the distribution decreases to a difference
between total and fibre stellar mass of $\approx$1.5 dex. This correction
affects the fibre stellar mass distribution by shifting it to higher values,
maintaining approximately the overall shape. Hence, it can be said that this
conversion factor affects most of the galaxies in approximately the same way.
With the total stellar mass estimates for FADO obtained, the MPA-JHU estimates
were converted from a Kroupa (2001) in to a Chabrier (2003) IMF by dividing
the estimates by 1.06 (Zahid et al. 2012; Speagle et al. 2014; Madau &
Dickinson 2014).

##  6 Results

###  6.1 Star formation rate

The comparison between the SFR calculated from the H$\alpha$ flux from FADO
and from MPA-JHU is presented in Figure 7. The SFR distributions for FADO and
MPA-JHU are coincident, which was expected because the H$\alpha$ flux
distributions are also coincident. The 1:1 plots also show that FADO and MPA-
JHU obtain similar SFRs. In particular, the linear fit applied to the
comparison between the two estimates is practically parallel to the 1:1
reference line. This fact indicates that the relation between the two
estimates remains the same in the considered range, that is, there is no
change in the difference between the SFR estimate by FADO and MPA-JHU with
increasing SFR. The difference between the SFR estimate of FADO and MPA-JHU
differs by only 0.01 dex ($\approx$2%).

In order to quantify the SFR estimates of FADO in more detail, the sample was
divided into three subsamples: galaxies forming less than 1 M⊙yr-1, $\log$(SFR
[M⊙yr-1]) $<$ 0, galaxies forming between 1 and 10 M⊙yr-1, 0 $\leq$ log(SFR
[M⊙yr-1]) $<$ 1, and galaxies forming more than 10 M⊙yr-1, $\log$(SFR
[M⊙yr-1]) $\geq$ 1\. For each subsample, we computed the number of galaxies,
the median value, and the standard deviation of the $\log$(SFR). In Table 2 we
present the results of the analysis of these subsamples as well as the full
sample. This analysis provides insight into the star formation level of the
galaxies in our sample.

In the complete sample, FADO estimates a median $\log$(SFR) of 0.48
($\approx$3 M⊙yr-1). The 0 $\leq$ $\log$(SFR) $<$ 1 subsample contains
$\approx$62% of the complete SF sample, and the subsample with the highest
SFRs, $\log$(SFR) $\geq$ 1, has the fewest galaxies ($\approx$15%), with a
median value $\log$(SFR) of 1.21 ($\approx$16 M⊙yr-1). This SFR
characterisation shows that the galaxies in our sample do not have a very high
star-formation level in general. Consequently, the nebular emission of these
galaxies should also be low. Thus, our sample is likely dominated by galaxies
with a low nebular contribution.

Table 2: Analysis of the SFR, in $\log$(SFR [M⊙yr-1]), estimated with FADO. We present the number of galaxies, the median value, and the standard deviation of the three subsamples and the complete SF sample. Sample | No. Galaxies | Median $\pm$ $\sigma$
---|---|---
$\log$(SFR) $<$ 0 | 40 230 | -0.35$\pm$0.51
0 $\leq$ $\log$(SFR) $<$ 1 | 114 530 | 0.52$\pm$0.27
$\log$(SFR) $\geq$ 1 | 28 173 | 1.21$\pm$0.24
SF Sample | 182 933 | 0.48$\pm$0.64
Figure 7: Comparison of the SFR between FADO and MPA-JHU. Left panel:
Histogram of $\log$(SFR) for FADO (blue) and MPA-JHU (orange). Middle panel:
Scatter plot of the logarithm of the ratio of the SFR from FADO and MPA-JHU as
a function of $\log$(SFR) for FADO. Right panel: 1:1 contour plot showing 20,
40, 60, 80, and 100% of the sample and linear fit to the data (green line).

###  6.2 Stellar mass

Figure 8 compares the stellar mass for FADO and MPA-JHU. The two distributions
have a similar shape, but the peak occurs at higher stellar mass values for
FADO than for MPA-JHU. This fact means that FADO retrieves slightly higher
stellar masses than MPA-JHU on average, as is also shown in the middle and
right panel of Figure 8. However, as the stellar mass increases, the
difference between FADO and MPA-JHU decreases. This can also be seen for high
masses, where the distributions are consistent.

The complete sample was divided into two subsamples based on the stellar mass:
galaxies with a stellar mass lower and higher than 1010 M⊙. Table 3 presents
the results of the analysis.

The uncertainty presented for the FADO estimates (0.2 dex) is a conservative
upper limit error for the stellar mass obtained in previous studies of FADO
(Gomes & Papaderos 2017; Cardoso et al. 2019; Pappalardo et al. 2021). The
uncertainties given directly by FADO for the stellar mass estimates are the
formal errors from the minimisation algorithm, related to the error of the
$\chi^{2}$ distribution, and not the physical error affecting the stellar mass
estimate. Since these formal errors were unrealistically low, we decided to
fix the value of 0.2 dex, based on the aforementioned previous assessments.
For MPA-JHU, the uncertainties presented are the 16th and 84th percentile of
their stellar mass estimate. The uncertainty of the median difference between
FADO and MPA-JHU is the result of adding in quadrature the FADO and MPA-JHU
uncertainties.

For the complete sample, the median difference between FADO and MPA-JHU is
0.11 dex. This means that in median, FADO estimates stellar masses that are
about 29% higher than MPA-JHU. For both subsamples, the median values of FADO
are higher than those of MPA-JHU. The median difference between FADO and MPA-
JHU for the subsample of stellar masses lower and higher than 1010 M⊙ is 0.14
dex and 0.10 dex, respectively. These results are consistent with the trend
observed in the linear fit to the 1:1 contour plot (right panel of Figure 8)
of a decrease in the difference between FADO and MPA-JHU with increasing
stellar mass.

To test the nature of the origin of the observed differences, we applied the
statistical Kolmogorov-Smirnov test and obtained a p-value of $\sim$0\. This
value is lower than any of the commonly used confidence levels. The hypothesis
that the distributions are similar in statistical terms can therefore be
rejected. However, the measured difference between the stellar masses derived
by FADO and MPA-JHU are in any case within 1$\sigma$ of the uncertainties for
each method. This implies that even if the distributions are statistically
different, the uncertainties associated with the measurements dominate and the
compatibility of the two estimates cannot be ruled out.

This is also true for the observed trend of a decrease in the difference
between FADO and MPA-JHU with increasing stellar mass, and an interpretation
of this trend is not statistically significant. It is worth mentioning that
although the stellar masses are derived through different approaches in FADO
and MPA-JHU, as stated in Section 5, the estimates still agree.

Figure 8: Comparison of the total stellar mass between FADO and MPA-JHU. Left
panel: Histogram of the logarithm of the stellar mass for FADO (blue) and MPA-
JHU (orange). Middle panel: Scatter plot of the logarithm of the ratio of the
stellar mass from FADO and MPA-JHU as a function of the logarithm of the
stellar mass. Right panel: 1:1 contour plot showing the 20, 40, 60, 80, and
100% of the sample and linear fit to the data (green line). Table 3: Analysis
of the stellar mass, in $\log$(M∗ [M⊙]), for FADO and MPA-JHU. We present the
number of galaxies, the median value for FADO and MPA-JHU, and the median
difference between FADO and MPA-JHU for two subsamples and the complete
sample. The uncertainty associated with FADO is a conservative upper limit,
and for MPA-JHU, it is the 16th and 84th percentile of the estimate. The
uncertainty associated with the median difference is the result of adding the
FADO and MPA-JHU uncertainties in quadrature.

|  $\log$(M∗) $<$ 10 |  $\log$(M∗) $\geq$ 10 | Complete Sample
---|---|---|---
| Nº Gal | Median | No. Gal | Median | No. Gal | Median
FADO | 63 412 | 9.6$\pm$0.2 | 119 521 | 10.4$\pm$0.2 | 182 933 | 10.2$\pm$0.2
MPA-JHU | 9.45${}^{+0.09}_{-0.07}$ | 10.36${}^{+0.10}_{-0.09}$ | 10.12${}^{+0.10}_{-0.08}$
FADO$-$MPA-JHU | 0.14${}^{+0.22}_{-0.21}$ | 0.10${}^{+0.22}_{-0.22}$ | 0.11${}^{+0.22}_{-0.22}$

###  6.3 Main sequence

We showed and quantified that FADO obtains similar SFRs and stellar masses as
MPA-JHU. Since the SFMS is the combination of these two galactic properties,
small differences are expected between the SFMSs obtained using the results
from FADO and MPA-JHU.

The SFMS obtained with FADO and MPA-JHU data are plotted in Figure 9. A linear
fit is applied to the SFMS, and the resulting equation presented in the plot.
The contours and linear fit of the SFMS for FADO are overplotted on the SFMS
for MPA-JHU in order to easily compare both. We also present a cross with the
median error affecting the SFR and stellar mass estimates.

Figure 9: Comparison of the star-forming main sequence with FADO and MPA-JHU
data. Left panel: Contour plot for FADO and linear fit to the data (green
line). Right panel: Contour plot for MPA-JHU (colours) and linear fit to the
data (blue line) with the contours (black lines) and linear fit (green line)
for FADO overplotted. The median error of the data is presented in the bottom
right corner, and the linear fit equation is presented in the upper left
corner. The contour plots show 20, 40, 60, 80, and 100% of the sample.

The obtained equations for the linear fits, as well as the formal errors from
the fitting, are

$\begin{aligned} \log(\textrm{SFR})&=(0.774\pm 0.001)\log(M_{*})-(7.43\pm
0.02)\qquad\mbox{FADO}\\\ \log(\textrm{SFR})&=(0.757\pm
0.001)\log(M_{*})-(7.18\pm 0.01)\ \mbox{MPA-JHU}\end{aligned}.$ (4)

As expected, the SFMSs are consistent with each other, and the linear fits
applied to the data present the same relation between SFR and stellar mass.
Taking into account the error cross presented in the plot, we conclude that
the linear fits are not statistically different from each other. Moreover, the
contour plot shows that the distribution of the galaxies within the SFMS is
similar for FADO and MPA-JHU.

##  7 Discussion

As discussed in Section 3, the main novelty of FADO from the astrophysical
point of view is the inclusion of the nebular component in the fitting process
of the SED. The overall continuum of the FADO model SED is therefore the
result of the contribution from the stellar and nebular continuum. On the
other hand, previous spectral synthesis methods, such as the one used to
estimate the physical parameters in the MPA-JHU catalogue, only consider the
stellar contribution. Therefore, the overall continuum is assumed to be purely
stellar.

The impact of considering the nebular contribution in the fitting process of
the SED was recently studied in Cardoso et al. (2022) and Breda et al. (2022).
In both these works, FADO, with its self-consistent treatment of the stellar
and nebular continuum, was compared to the purely stellar synthesis code
STARLIGHT. Because these works addressed a similar topic as ours, it is worth
analysing our results in the context of their findings.

Cardoso et al. (2022) reevaluated the evolutionary trend of light- and mass-
weighted quantities related to the stellar age, metallicity, and mass of SF
galaxies in light of the new approach followed by FADO. This comparison was
carried out considering different subsamples, defined in terms of the star
formation level of the galaxies using the EW as indicator. In this way, the
stellar properties of the galaxies in the different subsamples were related to
the level of the nebular continuum contribution. Of the different results
presented in Cardoso et al. (2022), the most relevant result for this work is
that the estimated stellar mass is minimally affected by taking into
consideration the nebular continuum because FADO obtains stellar masses just
slightly higher than STARLIGHT (see Table 1 of Cardoso et al. 2022).

Breda et al. (2022) studied a sample of 414 extreme emission line galaxies
(EELGs) and 697 normal SF galaxies from the SDSS, providing insight into how
stellar properties, such as stellar mass and mean stellar age, are affected by
the additional modelling of the nebular continuum. The specific SFR (sSFR) and
the sum of the flux of the most prominent emission lines were used as
indicators of the degree of star formation activity, thus enabling the
interpretation of the results considering the level of the nebular
contribution. In EELGs, STARLIGHT estimates stellar masses higher than FADO as
a consequence of the increased level of nebular emission. Normal SF galaxies
are only a slightly overestimated by STARLIGHT, however.

As stated in Section 3, we considered a specific subsample of the application
of FADO to SDSS-DR7 from Cardoso et al. (2022). We obtained that the H$\alpha$
flux estimates of FADO and MPA-JHU are consistent with each other. The
consistency between the fluxes estimated by FADO and those in the MPA-JHU
catalogue is a test of the reliability of the flux estimates from FADO.
Furthermore, our results show that FADO estimates H$\alpha$ EWs that are
$\approx 26\%$ higher than the values in the MPA-JHU catalogue. This is
expected because FADO uses the monochromatic continuum level of the SED at the
central wavelength of emission lines to determine the EW, whereas the approach
followed in MPA-JHU considers the continuum as the average level of the
continuum adjacent to the emission or absorption line (see Section 4.1 for
further details).

Our comparison of the stellar mass between FADO and MPA-JHU showed that the
median values are similar considering the uncertainty range, but the
distributions are slightly different. The estimate of comparable stellar
masses by FADO relative to a purely stellar approach to SED fitting was also
observed in Cardoso et al. (2022). However, Breda et al. (2022) observed that
not considering the nebular contribution leads to an overestimation of the
stellar mass.

Since FADO considers the contribution of the stellar and nebular component to
the total continuum, but a purely stellar approach only the stellar component,
then the stellar continuum for FADO is at a lower level. Consequently, the
stellar mass estimates of FADO are expected to be lower than those of a pure
stellar code. However, this difference will only be significant for galaxies
for which the contribution of the nebular continuum to the SED becomes
significant.

In this way, the apparent disagreement of the results from Breda et al. (2022)
with the results reported here and in Cardoso et al. (2022) is probably
related to the level of the nebular continuum contribution of the galaxies
that is included in the sample used in each of these works. In our sample, the
median value of the H$\alpha$ EW is 34 Å and only 6% of the galaxies in the
sample have an EW(H$\alpha$)¿100 Å (in Figure 3 is presented the EW(H$\alpha$)
distribution). In Cardoso et al. (2022), only $\approx 1.3\%$ of the galaxies
have an EW(H$\alpha$)¿75 Å (see Fig.2 of Cardoso et al. 2022, for the
EW(H$\alpha$) distribution). This means that these samples are strongly
dominated by galaxies with a negligible nebular continuum, which is not
expected to appreciably influence purely stellar spectral fits. Thus, an
agreement between the physical properties retrieved by FADO and by a pure
stellar code is observed, as is the case in what regards the MPA-JHU
catalogue. On the other hand, in Breda et al. (2022) the sample of EELGs has
an average H$\alpha$ EW of $\approx$400 Å (see Fig.1 of Breda et al. 2022, for
the EW(H$\alpha$) distribution). Therefore, the nebular contribution to the
galaxies is so high that it is not negligible, and in order to retrieve
accurate physical properties, such as the stellar mass, it is fundamental to
model the nebular continuum.

Our comparison of FADO and MPA-JHU is worth extending to EELGs. We selected a
subsample of EELGs from our SF sample by adopting the same cuts in EW as were
followed in Breda et al. (2022). Thus, we selected the galaxies with
EW(H$\beta$) $>$ 30 Å and EW([OIII]) $>$ 100 Å for both FADO and MPA-JHU. With
these criteria, we obtained a subsample of 2759 EELGs, which corresponds to
$\approx$ 1.5% of our SF sample. The EELGs sample has a median H$\alpha$ EW of
316 Å , while for the SF sample the median H$\alpha$ EW is 34 Å.

In order to compare the EELGs sample with the complete SF sample, the sSFR of
these two samples for FADO and MPA-JHU are presented in Figure 10. There is a
clear distinction between the EELGs and normal SF galaxies. The peak in the
distributions of the EELGs is substantially shifted towards higher values,
which means that the EELGs generally have higher sSFR than normal SF galaxies.

The sSFR for MPA-JHU is higher than for FADO in both samples. This is a
consequence of the difference in the stellar mass estimates between FADO and
MPA-JHU because the SFR estimates are similar between the two. However, the
difference between FADO and MPA-JHU is higher for the EELGs than for the
normal SF galaxies. This trend is opposite to the results in Breda et al.
(2022), where FADO obtains even higher sSFRs for the EELGs than for the normal
SF galaxies compared to the pure stellar approach. However, the uncertainties
are again larger than the differences, even though the distributions are
clearly different, and the compatibility of both estimates cannot be ruled
out.

This difference in trend could be related to the differences in the spectral
basis used in these works, both in terms of the age, in Breda et al. (2022)
the age upper limit of the basis depends on the redshift of the galaxy, and of
the metallicity, in Breda et al. (2022) the basis spans a smaller range,
particularly at sub-solar metallicities. These differences in the basis lead
to different population vectors, thus changing the derived physical
properties, including the stellar mass. Finally, even if the same spectral
basis is considered, different approaches could lead to different results due
to the effect of the fitting algorithm on the derived population vector
Cardoso et al. (2022). Thus, the fact that in this work FADO is compared with
MPA-JHU, but in Breda et al. (2022) it is compared with STARLIGHT, and that
the three approaches use different fitting algorithms could lead to relevant
differences.

These points underline the difficulty of comparing different approaches to SED
fitting because the chosen spectral basis and the employed fitting algorithms
play a non-negligible role. Another relevant point to take into consideration
in future works is to refine the error estimates for these physical parameters
so that the effect of the nebular contribution can be better quantified.

Figure 10: Histograms of the logarithmic sSFR of the SF galaxies and EELG
samples for FADO (in blue and green, respectively) and MPA-JHU (in orange and
red, respectively).

In summary, our results show that the stellar mass is consistent between FADO
and MPA-JHU because the vast majority of SF galaxies in our sample have a low
nebular contribution. Thus, the modelling or lack thereof of the nebular
contribution does not affect the retrieved physical properties of these
galaxies. However, there is a subsample of SDSS galaxies with a significant
high nebular contribution, such as the EELGs studied in Breda et al. (2022) or
luminous compact galaxies (e.g. Izotov et al. 2011; Amorín et al. 2012;
Papaderos & Östlin 2012), for which the modelling of the nebular continuum is
fundamental to retrieve more reliable physical parameters.

In addition to particular types of SDSS galaxies, FADO will probably be proven
valuable at higher redshifts, where the galaxies have higher SFRs and nebular
contribution (Schaerer & de Barros 2010; Atek et al. 2011). Consequently,
applying FADO to galaxies at redshifts higher than the redshift coverage of
SDSS will be crucial to understand the impact of the nebular contribution at
these redshifts, where we expect to probe a population that is generally less
evolved and more dust free. SDSS galaxies with a high nebular contribution are
considered to be prototypes of higher-redshifts galaxies. Hence, results such
as those presented in Breda et al. (2022) indicate that a self-consistent
treatment of the stellar and nebular continuum is required at higher redshifts
in order to obtain accurate estimates of the physical properties of galaxies.
The application of FADO to galaxies at higher redshifts than SDSS should also
take into account that the high nebular contribution might not be the only
difference relative to local galaxies because the stellar populations that
compose these galaxy might also be different (Olivier et al. 2021; Umeda et
al. 2022).

##  8 Summary and conclusions

FADO is an innovative spectral synthesis code that considers both the stellar
and nebular components in a self-consistent way while fitting the SED of a
galaxy. We considered the application of FADO to the SDSS-DR7 carried out in
Cardoso et al. (2022). Our goal was to study the impact of the nebular
contribution on the estimation of the physical properties of SF main-sequence
galaxies. We selected a sample of 182 933 local SF galaxies and compared the
results to the well-established MPA-JHU catalogue, whose high quality and
reliable results were used in previous benchmark publications.

The SFR was estimated from the flux of the H$\alpha$ emission line. Firstly,
the flux and EW of H$\alpha$ were studied. FADO retrieves in median H$\alpha$
fluxes similar to MPA-JHU and EWs 0.1 dex ($\approx$26%) higher. This
difference is a consequence of a different definition of the level of the
continuum between FADO and MPA-JHU. FADO uses the monochromatic continuum
level of the SED at the central wavelength of emission lines to determine the
EW, whereas the usual approach, which is followed in MPA-JHU, is to consider
the continuum as the average level of the continuum adjacent to the
emission/absorption line.

In order to correct the fiber-determined SFRs and stellar masses for aperture
effects, the same aperture corrections as in MPA-JHU (Brinchmann et al. 2004;
Salim et al. 2007) were adopted. These corrections have a typical value of 0.3
and 0.5 dex for the SFR and stellar mass, respectively, and can reach values
as high as 1 dex for both quantities. Considering the magnitude of these
corrections, they are crucial to the slope and scatter of the SFMS.

Proceeding to determine the SFR, we observed an agreement between FADO and
MPA-JHU, as for the H$\alpha$ flux, with a median difference between the two
data sets of 0.01 dex ($\approx$3%). In order to analyse the estimates in more
detail, the sample was divided into three bins, $\log$(SFR) $<$ 0, 0 $\leq$
$\log$(SFR) $<$ 1, and $\log$(SFR) $\geq$ 1\. This analysis of the SFR shows
that our sample is likely dominated by galaxies with a low nebular
contribution (see Figure 7 and Table 2 for details).

For most galaxies, the stellar masses retrieved by FADO are higher than in
MPA-JHU, with a median difference of 0.11 dex ($\approx$29%). The median
difference between FADO and MPA-JHU was evaluated for two different subsample:
galaxies with a stellar mass lower and higher than 1010 M⊙. The results show
that the differences decrease as the stellar mass increases. However, the
observed differences are contained within the uncertainty range, and
consequently, the stellar mass estimates are consistent with each other (see
Figure 8 and Table 3 for details).

The consistent estimates of the SFR and stellar mass between FADO and MPA-JHU
lead to a similar SFMS (see Figure 9). From these results, we conclude that
for local and normal SF galaxies, the nebular contribution does not affect the
determination of the SFMS significantly because the nebular continuum is weak
on average. Nonetheless, for specific SDSS galaxies, such as starburst
galaxies, the nebular contribution is sufficiently high to influence the
determination of the physical properties (Izotov et al. 2011; Amorín et al.
2012; Papaderos & Östlin 2012; Breda et al. 2022).

This work shows that taking into account the contribution of the nebular
component to the galaxy SED does not affect SFR tracers based on emission line
fluxes. Moreover, for local and normal SF galaxies, the inclusion of the
nebular component does not lead to significant differences in the stellar mass
estimates in the specific context of aperture-corrected estimates. In the end,
the linear relation between SFR ans stellar mass known as SFMS is not
significantly affected by modelling the nebular continuum for local and normal
SF galaxies.

From studies at higher redshifts, it seems clear that the nebular emission is
significantly higher than in local SF galaxies. Thus, it is fundamental to
study its influence on the physical properties of galaxies at higher redshifts
with FADO and other SED fitting codes. This line of research is particularly
relevant in order to understand the most reliable tool for modelling and
analysing the data from future high-redshift surveys, such as MOONS (Cirasuolo
& MOONS Consortium 2016; Cirasuolo et al. 2020; Maiolino et al. 2020), PFS
(Takada et al. 2014), WEAVE (Dalton 2016), and 4MOST (de Jong et al. 2019).

###### Acknowledgements.

This work was supported by Fundação para a Ciência e a Tecnologia (FCT)
through the research grants PTDC/FIS-AST/29245/2017, UID/FIS/04434/2019,
UIDB/04434/2020 and UIDP/04434/2020. C. P. acknowledges support from DL
57/2016 (P2460) from the ‘Departamento de Física, Faculdade de Ciências da
Universidade de Lisboa’. I. M. acknowledges support from DL 57/2016 (P2461)
from the ‘Departamento de Física, Faculdade de Ciências da Universidade de
Lisboa’. APA acknowledges support from the Fundação para a Ciência e a
Tecnologia (FCT) through the work Contract No. 2020.03946.CEECIND, and through
the FCT project EXPL/FIS-AST/1085/2021. R.C. acknowledges support from the
Fundação para a Ciência e a Tecnologia (FCT) through the Fellowship
PD/BD/150455/2019 (PhD:SPACE Doctoral Network PD/00040/2012). Funding for the
SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the
Participating Institutions, the National Science Foundation, the U.S.
Department of Energy, the National Aeronautics and Space Administration, the
Japanese Monbukagakusho, the Max Planck Society, and the Higher Education
Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The
SDSS is managed by the Astrophysical Research Consortium for the Participating
Institutions. The Participating Institutions are the American Museum of
Natural History, Astrophysical Institute Potsdam, University of Basel,
University of Cambridge, Case Western Reserve University, University of
Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the
Japan Participation Group, Johns Hopkins University, the Joint Institute for
Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and
Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences
(LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for
Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico
State University, Ohio State University, University of Pittsburgh, University
of Portsmouth, Princeton University, the United States Naval Observatory, and
the University of Washington.

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##  Appendix A Extinction correction

The effect of extinction $A(\lambda)$ can be quantified as the difference
between the observed magnitude $m(\lambda)$ and the magnitude in the absence
of dust and gas $m_{0}(\lambda)$,

$m(\lambda)-m_{0}(\lambda)=A(\lambda).$ (5)

Following the definition of magnitude, the difference between $m(\lambda)$ and
$m_{0}(\lambda)$ is related to the observed flux $F(\lambda)$ and the flux in
the absence of dust $F_{0}(\lambda)$ by

$m(\lambda)-m_{0}(\lambda)=-2.5\log\left(\dfrac{F(\lambda)}{F_{0}(\lambda)}\right).$
(6)

By joining Equation 5 and Equation 6, a relation between $F_{0}(\lambda)$ and
$F(\lambda)$ as a function of $A(\lambda)$ can be derived,

$A(\lambda)=-2.5\log\left(\dfrac{F(\lambda)}{F_{0}(\lambda)}\right)\Leftrightarrow
F_{0}(\lambda)=F(\lambda)10^{0.4A(\lambda)}.$ (7)

Equation 7 is only valid for an uniform layer of dust between the source and
the observer, that is, assuming that the extinction does not depend on the
geometry.

Dust does not affect the different wavelengths of the passing radiation
equally. The flux is more diminished for smaller wavelengths than for higher
wavelengths. In this way, to characterise the dust extinction, we compared the
flux from two different wavelengths since they are differently affected by the
presence of dust. It is reasonable to choose lines characteristic of regions
where the nebular lines are produced, thus we used the hydrogen emission lines
H$\alpha$ and $H\beta$. The ratio of the flux of H$\alpha$ and $H\beta$ is
known as the Balmer decrement, and it was used to correct the observed fluxes.
The difference between the extinction at $H\beta$ and H$\alpha$ is

$A(H\beta)-A(H\alpha)=2.5\log\left(\dfrac{(H\alpha/H\beta)_{obs}}{(H\alpha/H\beta)_{int}}\right),$
(8)

where (H$\alpha$/H$\beta$)obs and (H$\alpha$/H$\beta$)int are the observed and
intrinsic Balmer decrement, respectively. Assuming an electron density n = 100
cm-3 and electron temperature T = 104 K for case-B recombination, then
(H$\alpha$/H$\beta$)int = 2.86 (Osterbrock 1989).

With Equation 8, the difference between the extinction at $H\beta$ and
H$\alpha$ can be calculated, which is called colour excess. The broadband
colour excess $E(B-V)$ and an extinction law $k(\lambda)$ can be used to
calculate the extinction at any wavelength, which is the final objective, via

$A(\lambda)=k(\lambda)E(B-V).$ (9)

The previous equation can be used to write the difference between the
extinction at H$\beta$ and H$\alpha$,

$A(H\beta)-A(H\alpha)=[k(H\beta)-k(H\alpha)]E(B-V),$ (10)

where $k($H$\beta)$ and $k($H$\alpha)$ are the values of the extinction law at
H$\beta$ and H$\alpha$, respectively. Both Equation 8 and Equation 10 provide
a way in which the difference between the extinction at H$\beta$ and H$\alpha$
can be calculated. These two equations together read

$E(B-V)=\dfrac{2.5}{k(H\beta)-k(H\alpha)}\log\left(\dfrac{(H\alpha/H\beta)_{obs}}{(H\alpha/H\beta)_{int}}\right).$
(11)

In this way, the broadband colour excess can be derived from the observed
Balmer decrement.

Lastly, it is necessary to define an extinction law. We used the extinction
law from Charlot & Fall (2000), which is given by

$k_{Char}(\lambda)=\left(\dfrac{\lambda}{5500}\right)^{-0.7},$ (12)

where $\lambda$ is in Ångströms. Using this extinction law, we obtain
$k_{Char}(H\alpha)\simeq 0.88$ and $k_{Char}(H\beta)\simeq 1.09$.

Summarising, to correct the observed flux for the nebular extinction, the
procedure we followed was

  1. 1.

Calculate the ratio of the extinction at H$\beta$ and H$\alpha$ with Equation
8.

  2. 2.

Calculate the broadband colour excess $E(B-V)$ with Equation 11.

  3. 3.

Calculate the extinction $A(\lambda)$ at the desired wavelength with Equation
9.

  4. 4.

Calculate the flux in the absence of dust $F_{0}(\lambda)$ with Equation 7.