11institutetext:  National Centre for Nuclear Research, Pasteura 7, 02-093
Warsaw, Poland 11email<EMAIL_ADDRESS>22institutetext: Aix
Marseille Univ. CNRS, CNES, LAM, Marseille, France 33institutetext:
Astronomical Observatory of the Jagiellonian University, Orla 171, PL-30-244
Cracow, Poland 44institutetext: Instituto de Alta Investigación, Universidad
de Tarapacá, Casilla 7D, Arica, Chile 55institutetext: Department of
Astronomy, Indiana University, Bloomington, Indiana 47405, USA
66institutetext: School of Physics, University of New South Wales, NSW 2052,
Australia 77institutetext: Institute of Astrophysics, Facultad de Ciencias
Exactas, Universidad Andrés Bello, Sede Concepción, Talcahuano, Chile
88institutetext: Centre for Astrophysics and Supercomputing, Swinburne
University of Technology, Hawthorn, VIC 3122, Australia 99institutetext:
Department of Physics and Astronomy, The University of Sheffield, Sheffield,
S3 7RH, U.K 1010institutetext: INAF Osservatorio Astronomico di Padova, Vicolo
dell’Osservatorio 5, 35122, Padova, Italy 1111institutetext: Instituto de
Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059,
Valparaíso, Chile 1212institutetext: Instituto de Estudios Astrofísicos,
Facultad de Ingeniería y Ciencias, Universidad Diego Portales, Av. Ejército
441, Santiago, Chile 1313institutetext: University of California, Riverside,
900 University Ave, Riverside, CA 92521, USA 1414institutetext: INAF,
Osservatorio Astrofisico di Catania, Via S. Sofia 78, I-95123 Catania, Italy
1515institutetext: SISSA, Via Bonomea 265, 34136, Trieste, Italy
1616institutetext: Centro de Astronomía (CITEVA), Universidad de Antofagasta,
Avenida Angamos 601, Antofagasta, Chile 1717institutetext: Instituto de
Astronomía, Universidad Nacional Autónoma de Mexico, A.P. 70-264, 04510,
México, D.F., Mexico 1818institutetext: INAF Osservatorio Astronomico
d’Abruzzo, Via Maggini SNC 64100 Teramo, Italy 1919institutetext: INAF
Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, 80131 Napoli,
Italy 2020institutetext: Kapteyn Astronomical Institute, University of
Groningen, PO Box 800, 9700 AV Groningen, The Netherlands 2121institutetext:
Departamento de Astrofísica, Universidad de La Laguna, E-38206, La Laguna,
Tenerife, Spain

# Attenuation proxy hidden in surface brightness – colour diagrams.

A new strategy for the LSST era.

K. Małek 1122 Junais 11 A. Pollo 1133 M. Boquien 44 V. Buat 22 S. Salim 55 S.
Brough 66 R. Demarco 77 A. W. Graham M. Hamed 8811 J. R. Mullaney 99 M. Romano
111010 C. Sifón 1111 M. Aravena 1212 J. A. Benavides 1313 I. Busà 1414 D.
Donevski 111515 O. Dorey 1616 H. M. Hernandez-Toledo A. Nanni 1717111818 W. J.
Pearson 11 F. Pistis 11 R. Ragusa 1919 G. Riccio 11 J. Román 20202121

###### Abstract

Aims. Large future sky surveys, such as the Legacy Survey of Space and Time
(LSST), will provide optical photometry for billions of objects. Reliable
estimation of the physical properties of galaxies requires information about
dust attenuation, which is usually derived from ultraviolet (UV) and infrared
(IR) data. This paper aims to construct a proxy for the far-UV (FUV)
attenuation ($\rm A_{FUVp}$) from the optical data alone, enabling the rapid
estimation of the star formation rate (SFR) for galaxies that lack UV or IR
data. This will accelerate and improve the estimation of key physical
properties of billions of LSST–like observed galaxies (observed in the optical
bands only).

Methods. To mimic LSST observations, we used the deep panchromatic optical
coverage of the Sloan Digital Sky Survey (SDSS) Photometric Catalogue, Data
Release 12, complemented by the estimated physical properties for the SDSS
galaxies from the GALEX-SDSS-WISE Legacy Catalog (GSWLC) and inclination
information obtained from the SDSS Data Release 7. We restricted our sample to
the 0.025-0.1 spectroscopic redshift range and investigated relations among
surface brightness, colours, and dust attenuation in the FUV range for star-
forming galaxies obtained from the spectral energy distribution (SED).

Results. Dust attenuation is best correlated with colour measured between u
and r bands $\left(u-r\right)$ and the surface brightness in the u band
($\rm\mu_{u}$). We provide a dust attenuation proxy for galaxies on the star-
forming main sequence. This relation can be used for the LSST or any other
type of broadband optical survey. The mean ratio between the catalogue values
of SFRs and those estimated using optical-only SDSS data with the $\rm
A_{FUVp}$ prior calculated as $\Delta$SFR=log(SFR${}_{\tiny{\mbox{this
work}}}$/SFR${}_{\tiny{}\texttt{GSWLC}}$) is found to be less than 0.1 dex,
while runs without priors result in an SFR overestimation larger than 0.3 dex.
The presence or absence of the $\rm A_{FUVp}$ has a negligible influence on
the stellar mass ($\rm M_{star}$) estimation (with $\Delta$Mstar in the range
from 0 to $-0.15$ dex).

Conclusions. We note that $\rm A_{FUVp}$ is reliable for low-redshift main
sequence galaxies. Forthcoming deep optical observations of the LSST Deep
Drilling Fields, which also have multi-wavelength data, will enable one to
calibrate the obtained relation for higher redshift galaxies and, possibly,
extend the study towards other types of galaxies, such as early-type galaxies
off the main sequence.

###### Key Words.:

galaxies: evolution - galaxies: fundamental parameters - galaxies: star
formation - galaxies: statistics.

##  1 Introduction

The modelling of galaxies’ spectral energy distribution (SED) is a well-
established method for measuring key physical properties of galaxies, such as
their stellar mass ($\rm M_{star}$) and star formation rate (SFR). Both are
used as the primary building blocks to classify galaxies as quiescent, star-
forming, or starbursting and to reconstruct the evolutionary pathways of
galaxies (Brinchmann et al., 2004; Noeske et al., 2007; Elbaz et al., 2007;
Speagle et al., 2014; Pearson et al., 2018, 2023; Graham et al., 2024). The
complex nature of the baryonic components of galaxies, including stars, gas,
dust, and active galactic nuclei, and how they interact add considerable
complexity to modelling the SED.

To link, via the SED fitting process, $\rm M_{star}$ and SFR in a galaxy, the
star formation history (SFH) must be considered. Moreover, galaxy merger
events also have an influence on SFHs as it boost the SFR and increase $\rm
M_{star}$. A complex interplay between evolved and newborn stars and dust
inevitably accompanying star formation makes both measurements surprisingly
challenging (for example Walcher et al., 2011; Conroy, 2013), as dust strongly
affects the shape of the SED. Astrophysical dust originates from stellar
evolution and is one of the key components of the interstellar medium (ISM) of
galaxies. The presence of dust particles is wide-ranging: dust plays a
fundamental role in star and planet formations, molecule production, and
galaxy evolution (for example Galliano et al., 2018). Small dust particles,
called grains, typically ranging in size from 5 to 250 nm (Weingartner &
Draine, 2001), are highly influential. Dust grains impact the observations of
stars and gas by absorbing and scattering short-wavelength photons and then
re-emitting energy in much longer wavelengths. Moreover, the star-to-dust
geometry can change the effect of dust in a non-negligible way (e.g. Buat et
al., 2019; Hamed et al., 2023a).

Since star-forming regions are dust-enshrouded in the dense cores of molecular
clouds, the earliest stages of star formation can be observed at millimetre
wavelengths. When the clouds collapse and the proto-stars form, the dust near
them starts emitting in the near– and mid-infrared (IR) range. In the next
step in the formation of stars, the warmest regions of the cloud around the
newly formed stars are heated by stars’ ultraviolet (UV) emission, and this
energy is re-radiated in the IR domain. This process makes dust emission a
powerful indicator of star formation if IR-sub-millimetre detections are
accessible. With the advent of IR and sub-millimetre facilities such as
Spitzer, Hershel, WISE, ALMA, SCUBA2, SPT, and NOEMA, the galaxy’s dust
content (dust mass and dust emission) is measured routinely at low and high
redshift (e.g. Dunne et al., 2011; Cortese et al., 2012; Santini et al., 2014;
Shirley et al., 2019; Harikane et al., 2020; Hamed et al., 2023a; Zavala et
al., 2023).

Modified by dust grains, photons hold information about young and evolved
stellar populations, active galactic nuclei, or even interactions with other
galaxies, for example, merger events. Unfortunately, the primary information
from the UV–optical spectra is distorted by dust grains and diffused along
different wavelengths. This process can be described by the dust attenuation
curve, which refers to the total effect of dust absorption and scattering on a
galaxy SED. Though the issue is very complex, detailed studies of the
attenuation curves in galaxies are numerous, and various strategies are used.

Calzetti et al. (1994, 2000) used observational spectra of local UV-bright
star-forming galaxies to derive an empirical law for the dust attenuation.
Another method is to estimate the attenuation in a galaxy and to calculate the
SFR in the modelling of its SED. This method has also been used on larger
samples of galaxies both at low and high redshifts (Wild et al., 2011;
Battisti et al., 2016). In the literature, two prominent attenuation curves,
with some additional modifications, are those of Calzetti et al. (2000) and
Charlot & Fall (2000). The Calzetti et al. (2000) attenuation law is described
by a single curve for the continuum and a differential reddening with respect
to emission lines while Charlot & Fall (2000) assumed a different attenuation
to the ISM and to the birth cloud regions. Moreover, the Charlot & Fall (2000)
attenuation law is not a universal curve since it depends on the metallicity
of the ISM, as it was shown in Shivaei et al. (2020) and also on the relative
distribution of dust between star-forming regions and the ISM (e.g. Boquien et
al., 2022, and others), and thus, on the SFH. In addition to these two, Lo
Faro et al. (2017) introduced yet another attenuation recipe for $z\sim 2$
ultra luminous IR galaxies, although this recipe is similar in concept to the
two-component attenuation curve of Charlot & Fall (2000).

It has been shown, however, that attenuation laws are not universal, and a
single attenuation law cannot reproduce the physical properties of a large,
varied sample of galaxies (e.g. Buat et al., 2012, 2014; Małek et al., 2018;
Salim et al., 2018; Hamed et al., 2023a). Even galaxies with optical – far–IR
(FIR) observations are best modelled with different attenuation laws,
resulting in slightly different estimated SFRs (Buat et al., 2019; Hamed et
al., 2021). There are different ways to check which attenuation curve is the
closest to the physical one. Among these methods we can list the comparison of
the reduced $\chi^{2}$ of the SED modelled assuming different attenuation laws
(Małek et al., 2018; Buat et al., 2019; Hamed et al., 2021), calculation of
Bayesian information criterion (BIC) between different models (used for
example in works of Ciesla et al., 2018; Buat et al., 2019, 2021), or the
comparison with radiative transfer on a library of hydrodynamic simulations
for isolated disk and mergers (i.e. Chevallard et al., 2013; Roebuck et al.,
2019, and checked with SED models in Buat et al., 2018). Yet another method is
based on the IRX-$\beta$ diagram (Meurer et al., 1999; Takeuchi et al., 2012;
Salim & Boquien, 2019; Hamed et al., 2023b) which relates the slope of the UV
continuum ($\beta$) and the ratio between the IR and FIR luminosities (the IR
excess, IRX).

In the case of limited IR measurements, this topic becomes even more complex,
as galaxies with different dust properties can appear similar in the optical
wavelength range (e.g. both young dusty galaxies and old dust-free galaxies
look red in the optical part of the spectrum, more detailed description of
classification problems related to the limited wavelength spectrum; a more
detailed description can be found, for example in Siudek et al., 2018).
Galaxies with full SED coverage, from UV to FIR, are rarely available,
creating obstacles to studying them at a significant statistical level. This
problem will become even more urgent and important in the upcoming era of the
Legacy Survey of Space and Time (LSST, Ivezić et al., 2019) from the Vera C.
Rubin Observatory, where types of galaxies still poorly understood and
difficult to observe, such as faint low-surface brightness (LSB) galaxies, or
even ultra-diffuse galaxies (e.g. Sandage & Binggeli, 1984; van Dokkum et al.,
2015) are expected to be routinely discovered. While LSB galaxies were usually
assumed to be dust-free, Junais et al. (2023) found that a non-negligible
fraction of them (4% of their sample, namely 23 LSB galaxies from their
sample) can actually contain enough dust to affect the shapes of their SEDs,
with attenuation in the V band, $\rm A_{V}$$\sim$0.8 mag.

The 10 year LSST observations will provide high-quality optical data in the
ugrizy bands for $\sim$20 billion galaxies (Ivezić et al., 2019; Robertson et
al., 2019). However, most of these galaxies will have no counterparts in
existing (or forthcoming) IR catalogues. Another issue is that with a large
number of galaxies observed by LSST, the traditional SED fitting method will
be very computationally expensive. Planned joint observations of LSST and
near-IR satellites, including Euclid (Laureijs et al., 2011) for Deep Drilling
Fields and the Nancy Grace Roman Space Telescope (formerly the Wide Field
Infrared Survey Telescope, WFIRST, Spergel et al., 2015) for follow-up
observations, will shed light on the near-IR properties of the observed LSST
galaxies but will not be sufficient to analyse the entire LSST sample.
Moreover, planned FIR missions like The Far-IR Spectroscopy Space Telescope
(FIRSST), The SPace Interferometric Cosmology Explorer (SPICE), The Single
Aperture Large Telescope for Universe Studies (SALTUS) or The PRobe far-
Infrared Mission for Astrophysics (PRIMA) can help to obtain dust measurements
for LSST galaxies in the future, although non of these future projects will
match the area-depth combination of the LSST. Furthermore, with such deep
data, the existing IR maps may suffer from source blending (e.g. Hurley et
al., 2017; Pearson et al., 2017), resulting in flux inaccuracy, further
complicating the SED fitting processes (Pearson et al., 2018). As a result, an
extremely valuable data set from the LSST observations will suffer from a poor
understanding of the dust attenuation and, consequently, mis-estimated SFR. As
shown by Riccio et al. (2021), the estimation of the LSST SFR for normal star-
forming galaxies up to $z\sim 1$ can be greatly overestimated, with a strong
redshift-dependent bias. The issue can be even more problematic for hitherto
poorly known populations of faint galaxies, including LSB galaxies. Graham &
de Blok (2001) and Graham (2001) reported on dust and opacity in LSB galaxies
and provided simple dust corrections for the surface brightness. Those faint
LSB galaxies are not that different from known and well-studied brighter
galaxies — they are also a mixture of stars, gas, and dust (even though only
recently we have found IR counterparts for those unfamiliar objects; see
Junais et al., 2023, for the first statistical analysis of the dust properties
in LSB galaxies).

LSB galaxies undergo similar processes, such as dust attenuation and emission,
essential to explain their physical properties. Considering the depth of the
forthcoming LSST observations ($\sim$27.5 mag in the r in the 10-years
observations, and $\sim$28.5 mag band for Deep Drilling Fields, equivalent to
$\rm\mu_{r}\sim 30-33$ mag/arcsec2, Robertson et al., 2019; Brough et al.,
2020), it is expected to detect a significant number of LSB galaxies and other
types of faint galaxies that have remained undetected in current surveys.
However, this vast dataset presents a significant challenge: how to account
for attenuation when calculating, for instance, SFR. The LSST catalogue will
require additional IR and spectroscopic observations to address this issue.
The study of the Deep Drilling Fields holds the promise of providing valuable
knowledge that can be harnessed by, for example, machine learning techniques
to calculate the physical properties (e.g. $\rm M_{star}$, SFR, bolometric and
IR luminosities) of these faint sources. On the other hand, LSST will deliver
unprecedented high-quality flux and morphology data for observed galaxies. In
this study, we aim to investigate if optical LSST data can suffice — at least
to some extent — to construct a prior for dust attenuation of young stellar
populations. Such a prior can then be used as a preliminary input for SED
modelling (Bogdanoska & Burgarella, 2020; Riccio et al., 2021). After
obtaining a first estimate of the main physical parameters, it can be replaced
with more refined priors derived from other LSST pipelines and ancillary data.

More precisely, in this paper, we study the possibility of using LSST–like
observables to estimate the prior of the dust attenuation in the far UV
regime. We do not aim to estimate the actual $\rm A_{FUV}$ but only the prior
value for each optically detected galaxy that can be further used in SED
modelling. Additionally, we check to what extent we can reduce the number of
parameters in the fit without a significant decrease in the estimate of the
main physical properties of the LSST-like sources in order to reduce the
computing time.

This paper is structured as follows: in Sect. 2 we describe the data used for
our study. Section 3 presents the sample selection and all additional
calculations of parameters needed for the next steps of the analysis. The main
analysis of LSST-like observables and the resultant attenuation proxy is
presented in Sect. 4. The reliability of the obtained dust attenuation prior
is checked in Sect. 5. The results are discussed in Sect. 6, and the summary
and future perspectives conclude this paper in Sect. 7. Throughout this paper,
we adopt the stellar IMF of Chabrier (2003) and $\Lambda$CDM cosmology
parameters Komatsu et al., 2011, WMAP7,): H0 = 70.4 km s-1 Mpc-1, $\Omega_{M}$
= 0.272, and $\Omega_{\Lambda}$ = 0.728, the default from the CIGALE SED
fitting tool.

##  2 Data

To construct a prior for the dust attenuation in the FUV from the
observational data, we used three catalogues: (1) The GALEX-SDSS-WISE Legacy
Catalog (GSWLC-X2, Salim et al., 2016,
2018)111https://salims.pages.iu.edu/gswlc/, (2) the Sloan Digital Sky Survey
Photometric Catalogue, Data Release 12 (SDSS, Alam et al., 2015), and the SDSS
Data Release 7 spectroscopic main galaxy sample with morphological parameters
(Meert et al., 2015).

###  2.1 Key physical properties: $\rm M_{star}$, SFR, and $\rm A_{FUV}$

GSWLC is a catalog of local galaxies based on the 10th SDSS Data Release (Ahn
et al., 2014), which covers $\sim$8 000 deg2. Three different catalogues were
produced depending on the GALEX exposure time (GSWLC-A, -M and -D for all-sky
shallow, medium and deep surveys , respectively), providing a total of 659 229
objects ($\sim$90% of SDSS DR10 objects) at $0.01<z<0.3$, with additional
selection on the brightness of SDSS objects: $r_{petro}<$18 $[\rm mag]$, which
is the magnitude limit for SDSS galaxies in the r band. All three primary
catalogues listed above, GSWLC-A, M, and D, yield reliable SFRs for main-
sequence galaxies, as the SFR were obtained through a simple conversion factor
between IR and SFR and then calibrated using mid-IR luminosity and H$\alpha$
line. Galaxies on which the calibration was performed were selected via BPT
diagram (Baldwin et al., 1981). For quiescent or nearly quiescent galaxies,
the simple conversions of IR luminosity do SFR produce overestimation of SFR
(specific SFR reaches the overestimation up to 2 dex, Salim et al., 2016).
GSWLC-M and D are recommended for galaxies off the main sequence. For GSWLC,
the photometry was taken from (i) the GALEX GR6/7 final release (Bianchi et
al. 2014; (ii) the 2MASS Extended Source Catalog (XSC, Jarrett et al. 2000);
(iii) the SDSS DR10; (iv) and WISE from the AllWISE Source Catalog and uWISE
(Lang et al., 2016). The SDSS and GALEX photometry were corrected for galactic
extinction based on Peek & Schiminovich (2013) and Yuan et al. (2013)
coefficients. Moreover, additional corrections are used for GALEX data , when
the most significant one is correction due to blending. This correction is a
function of the difference in SDSS g magnitude and the range of separations
between sources in the SDSS catalogue. This correction is the same for GALEX
FUV and NUV bands. The other two corrections deal with (1) edge-of-detector
correction required for NUV band when the distance from the centre of the tile
to the location of the galaxy is larger than 0.47 degrees; (2) and the
centroid shift between optical and UV positions due to lower accuracy of the
GALEX astrometry, applied when the shift between SDSS and GALEX position is
larger than 0.7 arcsecond. All those corrections are described in detail in
Salim et al. 2016. In our analysis, we used the second version of the
catalogue, namely GSWLC–X2, which is the master catalogue taking the deepest
of GSWLC-A, M and D (659,229 galaxies). All the details concerning the
construction of the catalogue can be found in Salim et al. (2016) and Salim et
al. (2018).

We selected GSWLC in order to have a homogeneous associated catalogue of
physical parameters, which was obtained with the Code Investigating GALaxy
Emission (CIGALE, Burgarella et al., 2005; Noll et al., 2009; Boquien et al.,
2019). To model the stellar population of galaxies, Salim et al. (2016) used
Bruzual & Charlot (2003) models with four different metallicities from 0.2 to
2.5 $Z_{\odot}$ (according to Gallazzi et al., 2005, these values are in the
proper range for a majority of SDSS galaxies). A two-component exponential
model of the star formation history was used, and the modified, using a
variable slope $\delta$, Calzetti et al. (2000) dust attenuation curve with an
additional burst was used to model physical parameters for GSWLC. The total
dust luminosity $\rm L_{TIR}$, (8–1000 $u$m, Sanders & Mirabel, 1996) was
estimated by interpolating the Chary & Elbaz (2001) IR templates. The final
GSWLC catalogue contains a list of estimated parameters ($\rm M_{star}$, SFR,
$\rm A_{FUV}$) and flags which we used in our analysis as described in Sect.
3.1.

###  2.2 Photometric measurements and radii

We cross-matched GSWLC with the SDSS DR 12 catalogue (Alam et al., 2015),
which contains not only spectroscopic redshifts but also Petrosian magnitudes
and Petrosian radii in u, g, r, and i bands222There are no Petrosian radii and
magnitudes for the z band in this catalogue.. Petrosian magnitudes and radii
are a good first-order proxy for more precise magnitude and radii from the
LSST pipeline. This makes the SDSS DR 12 catalogue a perfect sample to study
possible changes in the magnitudes and sizes calculated based on the Petrosian
measurements as a function of dust attenuation. The SDSS DR 12 contains 469
053 874 primary sources plus 324 960 094 secondary sources. More than 3 500
000 objects have spectroscopic data. It is the final release of the SDSS III,
and, at the same time, a perfect LSST-like sample to study. The main
difference between DR 10 used by Salim et al. (2016) and DR 12 used in our
analysis are additional dedicated surveys included in the catalogue (BOSS,
APOGEE, and MARVELS) as well as the publication of Petrosian data for all four
bands.

###  2.3 Inclination

The SDSS DR 12 catalogue does not include information about the angular sizes
of galaxies. The minor-to-major axis ratio is crucial for our analysis, as it
indicates the galaxy’s inclination, which can strongly influence attenuation
due to non-spherically symmetric dust distribution. To obtain morphological
information for our sample of galaxies, we use the catalogue of two-
dimensional photometric decomposition from the SDSS DR7 spectroscopic main
galaxy sample (Meert et al., 2015). This catalogue provides a robust set of
morphological parameters obtained for the SDSS r band, using the GALFIT (Peng
et al., 2002) and PYMORPH (Vikram et al., 2010) software. The Meert et al.
(2015) catalogue includes measurements for 607 722 galaxies. We cross-matched
GSWLC catalogue with axis ratio measurements for r-band detections from Meert
et al. (2015).

##  3 Sample selection

###  3.1 Cleaning of the GSWLC catalogue

As recommended by Salim et al. (2018), for statistical studies of the main
sequence galaxies, we used galaxies from the SDSS Main Galaxy Survey (flag_mgs
= 1) catalogue, known as GSWLC-X2. We focus on the main sequence galaxies, as
the SFR estimated for GSWLC are shown to be reliable (Salim et al., 2016,
2018). For galaxies in this catalogue, the accuracy of estimated SFRs is
similar in three versions of GSWLC (A: shallow all-sky catalogue containing
640 659 galaxies, corresponding to 88% of DR10 targets; M: medium-deep
catalogue with 361 328 galaxies, 49% of the SDSS DR10; D: deep catalogue,
which contains 48 401 galaxies, 7% of the SDSS DR10). Selection based on the
flag_mgs = 1 results in 610 518 galaxies. Additionally, we only keep galaxies
with a good fit to their SED (FLAG_SED=0, also recommended by Salim et al.,
2016 and Salim et al., 2018). This selection gives us an initial catalogue of
603 615 main sequence galaxies in the redshift range $0.01-0.30$. In the next
step, we perform further cleaning.

The analysis presented in the following sections aims to construct and test a
possible prior for attenuation of the young stellar population ($\rm
A_{FUV}$). As the accuracy of the $\rm A_{FUV}$ and the SFR depends on the
depth of the GALEX observation, we remove all shallow UV detections (all-sky,
GSWLCA). Therefore, we perform the analysis based on the medium-deep and deep
GALEX observations (we use UV_SURVEY flags 2 and 3). After this selection, we
are left with a sample of 404 830 galaxies. We also remove all objects that
belong to the shallow, all-sky, GALEX catalogue. Thus cutting down the
selection by a 152 385 galaxies.

The GSWLC-X2 catalogue contains galaxies whose total IR luminosity ($\rm
L_{TIR}$) was calculated based on the 12 $\mu$m or 22 $\mu$m detection and
corrected for mid-IR AGN emission. To homogenise the data used for the
analysis, we decided to use galaxies with $\rm L_{TIR}$ estimated based on the
22 $\mu$m WISE detection. This cut removes all AGN-corrected galaxies, which
means that the sample should not contain any AGNs. As the IR counterpart is
necessary in a standard SED fitting process to estimate reliable attenuation
and SFR, we have to limit our analysis to galaxies bright enough to be visible
in the WISE bands. It creates a bias by removing a large fraction of LSB
galaxies but not all of them. Moreover, in the future, we are planning the
next calibration based on more sensitive Euclid measurements. After that
selection, the catalogue contains 82 116 galaxies. We also reject all objects
with REDCHISQ flag, which stands for the reduced goodness-of-fit value
($\chi_{red}^{2}$) for the SED fitting, larger than five (following Salim et
al., 2016 and Salim et al., 2018). After all these steps, the final subsample
of the GSWLC-X2 catalogue used in this analysis contains 78 725 galaxies.

###  3.2 Cleaning of the SDSS catalogue

Based on the flags used for the photometric measurements of the Alam et al.
(2015) catalogue, we remove all objects with class flag equal to six (stars
from the SDSS catalogue), and objects that do not have magnitude and
Petriosian radii measurements in all four bands (u, g, r, and i). This
criterion allows us to check all possible relations between $\rm A_{FUV}$ and
future LSST data. This initial cleaning resulted in 78 723 galaxies (i.e. only
two galaxies from the above sample were removed).

We also use flags describing the quality of the estimation of the radii. We
remove all galaxies where no valid Petrosian radius was found (NONPETRO) or
with multiple Petrosian radii (MANYPETRO). We also remove measurements with
Petrosian radius larger than the radial profile (NOPETRO_BIG) or with more
than one radius including 50% or 90% of the light (MANYR50) and MANYR90). We
do not use radii that were measured at the edge of the frame
(INCOMPLETE_PROFILE), those rejected because of low surface brightness level
(PETROFAINT) or objects larger than 4 arcmin (TOO_LARGE). Additionally, we
remove all measurements with possible saturation deception as the centre of
the radii is close to the saturated pixel (SATUR_CENTER) or interpolated pixel
(INTERP_CENTER). On top of those flags, we also remove galaxies detected with
a very low sky level, which results in the centre pixel of the galaxy being
negative (BADSKY) or at the edge of the frame (EDGE). Yet another flag which
indicates a possible problem with the image is CANONICAL_CENTER. This flag is
set for objects for which it is impossible to measure the centre in the r
band. We also remove all possibly moving objects (MOVED) or galaxies detected
at a level larger than 200$\sigma$ in the r band (BRIGHT)333The detailed
description of all SDSS flags can be found on the
https://www.sdss.org/dr12/algorithms/flags_detail/ webpage.

Figure 1: Main physical properties used in our analysis. Panels above show
spectroscopic redshift, stellar mass ($\rm log(M_{star}/M_{\odot}$)),
attenuation in the FUV band ($\rm A_{FUV}$), SFR and specific SFR (sSFR), both
in logarithmic scale. The last bottom right panel shows the axis ratio (semi-
minor/semi-major) from Meert et al. (2015). Black histograms represent
distributions obtained from the whole sample of 29 487 galaxies
($0.01<z<0.30$), while maroon hatched histograms show distributions for the
final sample used for the analysis (7 986 galaxies). Legends show median
values for all parameters calculated for the initial (black histograms) and
final (maroon hatched histograms) samples.

We use the same quality condition for the u,g,r, and i bands. In total, we
remove 34 676 galaxies from the initial sample created based on the Salim et
al. (2018). This selection allows us to create a catalogue of 44 047 galaxies
with good photometric measurements in all four SDSS bands, with UV and mid-
infrared detections, and reliably estimated key physical parameters from Salim
et al. (2018), namely SFR, $\rm M_{star}$, $\rm A_{FUV}$, and AV.

###  3.3 Cross match with the SDSS DR7 2D decomposition catalogue

Next, we cross-match the catalogue with the Meert et al. (2015) SDSS DR7
catalogue to obtain information about angular sizes and the inclination of
galaxies. This cross-matching reduces the sample significantly to 29 487
galaxies, that is, removing one third of the sample. From this sample, we
remove all galaxies with an axis ratio (semi-minor/semi-major) lower than
zero. This cut removes an additional 106 galaxies.

###  3.4 Final sample

The selection described above yields our final sample of 29 487 normal star-
forming galaxies in the redshift range 0.01$<z<$0.3. In Table 1, we list all
steps performed to obtain the final sample. This final sample provides by
reliable measurements of magnitudes and radii in all four SDSS bands, sets of
morphological parameters for the SDSS ugri bands, and proper estimation of the
main physical parameters ($\rm M_{star}$, SFR, attenuation in the FUV band,
etc) from the GSWLC-X2 catalogue.

Table 1: Sample selection discussed in Section 3.  Selection criteria | Number of selected sources | % of the initial sample
---|---|---
catalogue of physical properties (Salim et al., 2016, 2018)
GSWLC-X2 | 659 229 | 100,0%
Objects with $\rm A_{FUV}$ estimation | 650 597 | 98.69%
Main Galaxy Survey flag_msg=1 | 610 518 | 92.61%
SED fitting flag=0 (all SDSS photometry, no broad-line spectrum) | 603 615 | 91.56%
At least one GALEX detection (FUV or NUV) | 404 830 | 61.41%
Medium and deep UV exposure time (GSWLC-A and D) | 252 445 | 38.29%
$\rm L_{TIR}$ estimated based on the WISE 22 $\mu$m | 82 116 | 12.45%
REDCHISQ$<$5 (goodness of the fit, following Salim et al., 2016, 2018) | 78 725 | 11.94%
photometric catalogue (Alam et al., 2015)
Cross-matching with SDSS Alam et al. (2015) catalogue | 78 725 | 11.94%
Cleaning based on the SDSS flags (Sec. 3.2) | 44 047 | 6.68%
2D photometric decompositions catalogue (Meert et al., 2015)
Cross-matching with Meert et al. (2015) catalogue | 29 593 | 4.49%
The axis ratio (b/a) of the total fit $>0$ | 29 487 | 4.47%
Selection based on Sect. 3.4:
Cut for the $\rm A_{FUVs\mbox{ }err}$$<$0.25 [mag] | 15 004 | 2.28%
Redshift range 0.025–0.1 | 9 837 | 1.49%
Cut between 1st and the 99th percentile of the $\rm A_{FUV}$ | 9 641 | 1.46%
Main sequence galaxies | 7 986 | 1.21%

Black histograms presented in Fig. 1 show distributions of the main physical
parameters for the whole sample of 29 487 galaxies ($0.001<z<0.3$). Six panels
of this figure show the distribution of the spectroscopic redshift, as well as
the main physical properties: stellar mass ($\rm log(M_{star}/M_{\odot}$)),
attenuation in the FUV band ($\rm A_{FUV}$), SFR, and specific SFR (sSFR),
both in logarithmic scale. The bottom right panel shows the axis ratio (semi-
minor/semi-major) from Meert et al. (2015).

Figure 2: Attenuation in the FUV band ($\rm A_{FUV}$) as a function of
redshift. Two horizontal lines, dashed and dotted, represent the 1st and the
99th percentile of the $\rm A_{FUV}$ distribution, respectively. The solid
vertical line indicates redshift equal to 0.1, while the dashed double-dotted
line represents a redshift cut at 0.025. Above that redshift line, the sample
cannot probe the most extreme $\rm A_{FUV}$ values.

From the sample, we remove all galaxies with $\rm A_{FUV\mbox{ }err}$ larger
than 0.25 [mag], as in the next step of our analysis, we want to bin galaxies
regarding the $\rm A_{FUV}$ value, and too large errorbars can influence our
binning. The final sample, after this cut, contains 15 004 galaxies. We stress
here that this cut has negligible influence on the main physical properties of
the final sample, as shown in Fig. 1.

The number of objects in the sample drops above redshift $z\sim 0.1$, which
can be seen in the upper left panel in Fig. 1. A similar drop (but much
steeper) can be seen below redshift 0.025. In Fig. 2, we check the $\rm
A_{FUV}$ distribution as a function of redshift. This figure shows that below
redshift 0.025 and above redshift 0.1, the values of $\rm A_{FUV}$ are not
spread across the full range of this parameter. To obtain a representative
sample of galaxies across the attenuation and redshift space range, we remove
all galaxies with $\rm A_{FUV}$ below the 1st (0.67 mag) and above the 99th
(3.71 mag) percentile of the distribution. Moreover, we introduce additional
cuts in redshift, removing galaxies outside the 0.025 and 0.1 redshift bin.
Those two cuts allow us to keep a statistically significant galaxy sample
characterised by an almost complete distribution of $\rm A_{FUV}$. Therefore,
we decide to use in the following analysis only galaxies within the redshift
range $0.025<z<0.1$ (which reduces the sample to 9 837 galaxies), and with
$0.67<\rm A_{FUV}<3.71$ mag, further reducing the sample to 9 641 galaxies.

As the last step of the sample selection, we remove all galaxies in the tail
of the sSFR distribution. From the SDSS distribution, we remove the tail of
the main sequence distribution by selecting only galaxies located within $\rm
4\sigma$ of the sSFR distribution (i.e.
$-10.34<\mathrm{log(sSFR/yr^{-1})}<-9.49$, as illustrated in Fig. 3). This cut
removes 1 655 objects.

Distributions of the main parameters used in the analysis are shown in Fig. 1.
The full sample of 29 487 galaxies, without our internal cuts, is shown in
black histograms, while the final sample of 7 986 galaxies is presented as
maroon-hatched histograms. The $\rm A_{FUV}$ distribution, together with the
$\rm M_{star}$ distribution, show that cuts based on the $\rm A_{FUV\mbox{
}err}$, redshift, $\rm A_{FUV}$ and sSFR, do not change the main properties of
the key physical properties, but only remove the most massive and in the same
time the most active in the star formation processes, and the most attenuated
galaxies.

Figure 3: Distribution of log(sSFR/yr$-1$) for the selected 7 986 galaxies in
the redshift range 0.025$<z<$0.1. A vertical solid line and two vertical
dotted lines represent the mean value of sSFR of the sample, and the mean
value of the sSFR decreased/increased by $\rm 4\sigma$ of the distribution,
respectively.

###  3.5 Surface brightness calculation

For the final sample of 7 986 galaxies, we calculate the surface brightness in
each of the SDSS bands using the equation:

$\rm\mu_{x}=\texttt{mag}_{x}+2.5\cdot log_{10}(2\pi\texttt{r}_{x}^{2}),$ (1)

where x stands for u, g, r and i band, mag for Petrosian magnitude and r for
circular Petrosian radius. We note that we use the Petrosian radius for the
surface brightness measurements, unlike the half-light radius, which is
generally used in the literature (e.g. Paudel et al., 2017; Pérez-Montaño et
al., 2022). As we do not have a half-light radius measurement for all the
photometric bands used in this work, we chose to adopt the Petrosian radius in
all the bands for consistency (Meert et al. 2015, 2016 provides half-light
radius measurements only for the g, r and i bands, without the u-band). Graham
et al. (2005) shows that the Petrosian radius for a galaxy with Sérsic index
$n=1$ (which is a reasonable assumption for the main-sequence galaxies studied
in this work) is about twice larger than its Sérsic half-light radius. This
will result in our surface brightness measurements $\sim$1.5 mag/arcsec2
fainter than those estimated using the half-light radii. However, such a
systematic offset does not affect any of the trends studied in this work.
Therefore, from here upon, we adopt the surface brightness measurements
obtained using the Petrosian quantities.

We apply the correction for inclination (following Zhong et al., 2008; Pahwa &
Saha, 2018):

$\rm\mu_{x,corrected}=\mu_{x}+2.5\cdot log_{10}\cdot(b/a)-10\cdot
log_{10}(1+z),$ (2)

where  b/a  represents the ratio of a galaxy’s minor and major axis. From now
on, we always use only ’corrected’ surface brightness in the analysis, and
thus we drop the subscript corrected from the definition of $\mu_{\rm
x,corr}$. Figure 10 shows the distribution of the magnitudes and calculated
surface brightness based on the Eqs. 1 and 2 for the final sample used in the
analysis.

###  3.6 $\rm A_{FUV}$– LSST-like observables relations

We look for possible relations between observed LSST-like data (fluxes,
magnitudes, colours, surface brightness in different bands, as well as the
ratio between different surface brightness) in four SDSS bands (u, g, r, and
i) and the attenuation in the FUV band estimated via fitting the UV to IR SED.
The $\rm A_{FUV}$ relation with colours and surface brightness calculated in
different bands are presented in App. C. We also tried to use colours, but the
relations are too narrow to separate different attenuation levels. GSWLC-X2
$\rm A_{FUV}$ values were obtained using GALEX, SDSS, and WISE photometry
calibrated on the Hershel ATLAS (Salim et al., 2018), ensuring proper dust
attenuation estimation. The method used by Salim et al. (2018) combined SED
constrained with CIGALE (Burgarella et al., 2005; Noll et al., 2009; Boquien
et al., 2019) fitting code with infrared luminosity (SED+$\rm L_{TIR}$
fitting; more details can be found in Salim et al., 2018). Our main goal is to
find a simple proxy for $\rm A_{FUV}$ based on the observational LSST-like
data.

We checked all possible relations between colours, magnitudes, surface
brightnesses, and their ratios. As a result, we find a promising relation
between $\left(u-r\right)$ colour, the surface brightness calculated in the u
band, and the $\rm A_{FUV}$, which is characterised by a monotonic rise of the
ratio of $\left(u-r\right)$ colour and the u surface brightness with the $\rm
A_{FUV}$ values, and an extensive parameter locus (more than 0.6 magnitude in
colour; for example, the (g$-$i) colour gives only $\sim$0.3 magnitude width,
which makes the $\rm A_{FUV}$ analysis more complicated taking into account
the uncertainties of photometric measurements, and so forth). We also find a
very similar relation using $\left(u-i\right)$ instead of $\left(u-r\right)$.
The two main changes between both relations (the chosen one
$\left(u-r\right)$–$\rm\mu_{u}$–$\rm A_{FUV}$ and the second best one
$\left(u-i\right)$–$\rm\mu_{u}$–$\rm A_{FUV}$) are the larger global slope
uncertainty for $\left(u-i\right)$ shown in Fig. 13, and larger uncertainties
for the final $\rm A_{FUVp}$ equation based on larger errors for the intercept
equation (Eq. 3). The $\left(u-r\right)$ or $\left(u-i\right)$ colour is a
natural indicator of dust attenuation since dust affects the slope of the
galaxy SED. Both colours also cover the Balmer break, so they are very good
indicators of the age of the stellar population. We are aware that we have a
degeneracy between age and dust attenuation; since we have no information on
the ages of the GSWLC and a very narrow redshift range, we subsequently
analyse this degeneracy in the forthcoming analysis using much smaller but
more informative, reference catalogues. The surface brightness in the band
closest to UV SDSS is an indicator of the SFR as it traces light from the
young stellar populations. The combination of these two parameters can be thus
expected to be sensitive to dust attenuation for young stellar populations.
However, this is the first time, to our knowledge, that these parameters have
been combined to derive the proxy for $\rm A_{FUV}$. In Sections 4 and 5, we
present and analyse this relation in detail.

##  4 $\rm A_{FUV}$ prior: $\left(u-r\right)$ colour versus surface brightness
in the u band

Figure 4: Distribution of $\rm A_{FUV}$ for the final sample of 7 814 main
sequence galaxies in the redshift range $0.025-0.1$. The palette of colours
represents the $\rm A_{FUV}$ bins used in our analysis. The mean $\rm A_{FUV}$
value for each bin ($\rm\overline{A}_{FUV}$) is denoted in the legend.

To study the relation between the colour and the surface brightness, we divide
our sample into 14 $\rm A_{FUV}$ bins to check if different attenuation values
follow different relations in the colour $-$ surface brightness space and if
they can be separated. These bins are presented in Tab. 2 and graphically
shown in Fig. 4. Each bin contains at least one percent of the final sample
(at least 70 galaxies). Due to small numbers of galaxies in the GSWLC-X2
having $\rm A_{FUV}$ lower than 1 mag, we use a variable bin width to probe as
densely as possible the lowest attenuation range, which is underrepresented in
the catalogue. Thus, the first bin has a width of 0.05 mag, the second is 0.10
mag wide, and the third is 0.15 mag wide. Starting from the fourth, the width
is greater at 0.20 mag. Additionally, to increase the number of galaxies in
each bin, as well as to include possible uncertainties arising from the
redshift estimation or physical properties, we add overlaps between bins.
These overlaps increase with increasing $\rm A_{FUV}$ according to the
relation: bin${}_{n}\cdot$0.005, where binn refers to the bin number. This
helps us to gather enough galaxies in the less populated bins of high $\rm
A_{FUV}$ (larger than 2 mag), but also to take into account the uncertainties
of the $\rm A_{FUV}$ estimated by Salim et al. (2018), as for larger $\rm
A_{FUV}$ we also have larger $\rm A_{FUV\mbox{ }err}$ (it can be seen later in
the right bottom panel of Fig. 7, blue circles).

Table 2: $\rm A_{FUV}$ bins discussed in Section 4 and used in our analysis.

$\rm A_{FUV}$ bin | bin width | $\rm\overline{A}_{FUV}$ | # gal. | % sample
---|---|---|---|---
0.72 – 0.77 | 0.06 | 0.74 | 91 | 1.14
0.76 – 0.87 | 0.11 | 0.81 | 199 | 2.49
0.86 – 1.02 | 0.16 | 0.95 | 481 | 6.02
1.00 – 1.22 | 0.22 | 1.12 | 1098 | 13.75
1.20 – 1.42 | 0.23 | 1.31 | 1303 | 16.32
1.39 – 1.62 | 0.23 | 1.51 | 1290 | 16.15
1.59 – 1.82 | 0.24 | 1.70 | 1267 | 15.87
1.78 – 2.02 | 0.24 | 1.89 | 1077 | 13.49
1.98 – 2.22 | 0.25 | 2.09 | 812 | 10.17
2.17 – 2.42 | 0.25 | 2.28 | 654 | 8.19
2.37 – 2.62 | 0.25 | 2.47 | 472 | 5.91
2.56 – 2.82 | 0.26 | 2.67 | 267 | 3.34
2.76 – 3.02 | 0.27 | 2.86 | 161 | 2.02
2.95 – 3.22 | 0.27 | 3.06 | 117 | 1.47

Notes: The first column represents the minimal and the maximal value of $\rm
A_{FUV}$ in each bin, the second – the bin width, and the third is the mean
value of $\rm A_{FUV}$ in each bin. The fourth column presents the number of
galaxies in each bin, while the fifth column is the percentage of the full
sample of 7 934 galaxies.

We perform a linear fit in each $\rm A_{FUV}$ bin in the
$\left(u-r\right)$–$\rm\mu_{u}$ parameter space, as this relation shows the
most prominent slope of the general relation (0.0796 $\pm$ 0.0024)444The
second most prominent relation is $\left(u-i\right)$–$\rm\mu_{u}$ with slope
0.0799 $\pm$ 0.0047, however, the slope uncertainty is almost twice larger
than for the slope uncertainty of the $\left(u-r\right)$–$\rm\mu_{u}$ plane.,
Fig. C. Moreover, the $\left(u-r\right)$ colour space is wide enough to
separate different attenuation levels, taking into account measurement errors
for future LSST observations. Since the $\rm\mu_{u}$ range is much wider than
the range of $\left(u-r\right)$, and more often is contaminated by outliers
caused by uncertainties in calculating Petrosian radii and Pertosian
magnitudes, we perform the fit only between the 10th and the 90th percentile
of the $\rm\mu_{u}$ distribution in each bin. Figure 14 shows fits of the
relation $\left(u-r\right)$ versus $\rm\mu_{u}$ fits for all $\rm A_{FUV}$
bins, while Fig. 5 presents combined linear fits for all 14 bins. In Figure 5,
it is evident that for all $\rm A_{FUV}$ bins, the
$\left(u-r\right)$-$\rm\mu_{u}$ relation maintains a consistent slope, with
the intercept gradually increasing as $\rm A_{FUV}$ rises. We have interpreted
the flattening observed in the lowest and highest $\rm A_{FUV}$ values as
being related to the much less densely populated part of our sample. This can
be seen in Table 2 and Figure 4, where $\rm A_{FUV}$ values lower than
$\sim$0.8 mag and greater than 2.7 mag constitute only about 8% of the total
sample analysed in our manuscript. Figure 11 shows the same relation between
$\left(u-r\right)$$-$$\rm\mu_{u}$ and $\rm A_{FUV}$, but with an additional
background of galaxies used in our analysis colour $-$ coded according to the
value of $\rm A_{FUV}$, and with the interpolated linear fit with $\rm\mu_{u}$
in a range of $22.5-27.5$ [mag/arcsec2]. For the simplicity in the main
manuscript, we show only fits with the $\rm\pm 1\sigma$ uncertainty around
estimated lines.

Figure 5: Relations fitted between observed $\left(u-r\right)$ colours and
$\rm\mu_{u}$ for 14 $\rm A_{FUV}$ bins. The sequence of colours represents the
one used for $\rm A_{FUV}$ bin in Fig. 4. Filled areas display the $\pm
1\sigma$ uncertainty around estimated lines.

###  4.1 Global $\left(u-r\right)$ – $\rm\mu_{u}$ – $\rm A_{FUV}$ relation

Figures 5 and 11 indicate that there is a possibility to create a global
relation between observed $\left(u-r\right)$ colour, surface brightness in the
u band and the attenuation in the FUV band. To find a relation, we examine the
slopes and the intercepts of the relation between $\left(u-r\right)$ and
$\rm\mu_{u}$ for each $\rm A_{FUV}$ bin. We show in Fig. 6, a relation between
both slopes and intercepts as a function of $\rm A_{FUV}$, together with two
linear fits: one for the slopes and one for the intercepts, both as a function
of $\rm A_{FUV}$:

$\displaystyle\rm slope=(-0.02\pm 0.00)\cdot A_{FUV}+0.12\pm 0.01{,}$ (3)
$\displaystyle\rm intercept=(0.65\pm 0.08)\cdot{}A_{FUV}-1.68\pm 0.16{.}$

The increase in scatter for slope ad intercept with decreasing $\rm A_{FUV}$
can be explained by smaller bin sizes and less representative samples in the
global distribution of $\rm A_{FUV}$.

We derived a solution for this set of two equations based on the fitted
relations (slopes and intercepts with respect to $\rm A_{FUV}$ shown in Eq.
3), resulting in a linear expression that characterises $\rm A_{FUV}$ through
the combination of $\left(u-r\right)$ and $\rm\mu_{u}$. This final relation
(Eq. 4) incorporates all three values: two entirely observational
($\left(u-r\right)$ and $\rm\mu_{u}$) and one physical property ($\rm
A_{FUV}$) obtained from the SED fitting from GSWLC:

$\rm A_{FUVp}=\frac{(u-r)-(0.12\cdot\mu_{u})+1.68}{(-0.02\cdot\mu_{u})+0.65}.$
(4)

This equation provides a proxy for the $\rm A_{FUV}$ when only optical
measurements (fluxes and radii) are available, as will be the case for the
majority of the LSST survey555As discussed above, we obtained a very similar
result for the $\left(u-i\right)$–$\rm\mu_{u}$ plane:  $\rm
A_{FUVp}=\frac{(u-i)-(0.13\cdot\mu_{u})+1.68}{(-0.02\cdot\mu_{u})+0.65}.$ (5)
The main difference between equations are the larger uncertainties for slopes
and intercepts in Eq. 3. This proxy can significantly shorten the time needed
to estimate all physical parameters through the SED fitting, as the grid for
the dust attenuation properties can be much narrower and more specific.

We have also checked that using half-light instead of Petrosian radii will not
change our relation significantly. Using approximately $\sim$1.5 mag/arcs2
brighter values of surface brightness (see Sec. 3.5) results in changes of two
values from Eq. 4: from 1.68 to 2.50 and from 0.65 to 0.74. This change
results in the mean difference between$\rm A_{FUVp}$ obtained with Petrosian
end effective values equal to 0.05 [mag], with $\sigma$=0.45 [mag].

Figure 6: Slopes and intercept from $\left(u-r\right)$-$\rm\mu_{u}$ fits.
Interpolated slopes (left axis) and intercepts (right axis) obtained from
linear fits for all 14 bins of $\rm A_{FUV}$ (Fig. 14), as a function of $\rm
A_{FUV}$. The left axis represents slopes (purple full diamonds), and the
right axis represents intercepts (red full circles). Linear fits and
associated fitted parameters are presented using dashed-dotted lines for
slopes and dashed lines for intercepts.

Future LSST observations will provide more precise magnitude and morphology
measurements than the data employed in this manuscript, where we adopt
Petrosian radii and magnitudes from SDSS DR12. We plan to perform a similar
test for the data acquired from LSST Deep Drilling Fields to better calibrate
Eq. 4 as soon as the observations, both optical from LSST and from other
ground-based and satellite observatories are collected. For those fields,
near-IR data will also be available, for example, VISTA-NIR, which will
further help us to constrain reliable dust properties, including attenuation
in the FUV band in the broader $\rm A_{FUV}$ range. LSST will enable the
investigation of the lower $\rm A_{FUV}$ range, since a substantial percentage
of galaxies observed by LSST will be LSB galaxies. While low in comparison to
many other types of galaxy, FUV attenuation levels among LSB galaxies are
still non-negligible (i.e. $\rm A_{FUV}$¡0.4 mag), as shown recently by Junais
et al. (2023). While waiting for observations and estimates of the main
physical properties of galaxies from the Deep Drilling Field, we can use Eq. 4
to calculate a proxy representing dust attenuation for star-forming galaxies.
This relation will be used to prepare a software pipeline to estimate physical
properties from future LSST data by members of the LSST Galaxy Science
Collaboration (Robertson et al., 2017).

###  4.2 Final $\rm A_{FUV}$ prior

The final distribution of the obtained $\rm A_{FUVp}$, as well as the
comparison with $\rm A_{FUV}$ from the original work of Salim et al. (2018),
is shown in Fig. 7. Panel (a) in this Figure shows the $\rm A_{FUV}$ and $\rm
A_{FUVp}$ distributions. It is clearly seen that the distributions are
different and that the prior obtained from $\left(u-r\right)$ and $\rm\mu_{u}$
can reach both much lower (to the $\rm A_{FUVp}$=0 mag) and larger (up to $\rm
A_{FUVp}$$>4$ mag) values than the original $\rm A_{FUV}$ from Salim et al.
(2018). The Kolmogorov–Smirnov test, which is a commonly used non-parametric
test based on the distance between two cumulative distributions, confirms that
$\rm A_{FUV}$ and $\rm A_{FUVp}$ do not come from the same distribution
($p_{KS}=1.11\cdot 10^{-128}$). For this plot, we have removed 426 galaxies
for which the value of $\rm A_{FUVp}$ from Eq. 4 is less than 0. All galaxies
from the removed sample occupy $\left(u-r\right)$-$\rm\mu_{u}$ loci not
included in our analysis (just below the bin with the lowest value of mean
$\rm A_{FUV}$ ($\overline{A}_{FUV}=0.74$ [mag]). Fig. 12 shows the position of
all 426 galaxies in the $\left(u-r\right)$-$\rm\mu_{u}$ plane.

We notice here the sharp cut in the low-end of the $\rm A_{FUV}$ distribution
(visible also in Fig. 1, right upper panel). It can be related to the specific
sets of parameters used in Salim et al. (2018) or the data set for which the
SED fitting was performed. As shown in Osborne et al. (2023), the GALEX data,
which give direct insight into the young stellar population and were used to
create GSWLC, are partially affected by blending. The new catalogue built with
a new software pipeline EMphot, which uses forced photometry from the SDSS
catalogue, presented by Osborne et al. (2023), revealed that magnitudes used
in GSWLC were systematically fainter (up to 0.5 mag) due to insufficient
background subtraction for faint sources. The new, deblended, GALEX catalogue
of Osborne et al. (2023) shows that $\sim$15% of galaxies in the GSWLC
catalogue were moderately affected by blending (contamination $>$0.2 mag), and
2.4% of galaxies were contaminated at the level of more than 1 magnitude. To
summarise, the NUV and FUV GALEX magnitudes originally used to estimate $\rm
A_{FUV}$ by Salim et al. (2018) were fainter than the corrected deblended
magnitudes, but only a small percentage of galaxies in our sample can be
affected by this effect. It means that GALEX data used by Salim et al. (2016)
has negligible influence on the lack of low $\rm A_{FUV}$ values in the
original GSWLC-X2 catalogue for the main sequence galaxies666Based on private
communication with S. Salim, we have found that the $\rm A_{FUV}$
distributions based on the previous GALEX data used for the GSWLC catalogue
and the $\rm A_{FUV}$ obtained with the new, deblended GALEX measurements from
Osborne et al. (2023) have a statistically negligible change..

The mean difference between $\rm A_{FUV}$ and $\rm A_{FUVp}$ equals $0.01$
mag, with a $\sigma=0.74$ magnitude (see middle panel of Fig. 7). From
hereupon, we use this $\sigma$, which is the scatter in the difference between
fiducial values of $\rm A_{FUV}$ and $\rm A_{FUVp}$, as a constant uncertainty
for our estimated $\rm A_{FUVp}$ (hereafter: $\rm A_{FUVp\mbox{ }err}$). We
want to stress that, in the future, with the LSST-like observations, it will
also be possible to calculate the $\rm A_{FUVp\mbox{ }err}$ directly for
individual sources based on the uncertainties in the observed colour, surface
brightness, and the fit coefficients shown in Eq. 4. However, we are currently
restricted to a limited number of galaxies, which affects the uncertainties of
our fits from Eq 3. Additionally, we have significant photometric errors (for
both radii and magnitudes). Furthermore, the fiducial $\rm A_{FUV}$ is not
free of uncertainties (limited to $\rm A_{FUV\mbox{ }err}$ $<0.25$ magnitude,
based on our selection in Table 3). Therefore, for the simplicity of this work
and to avoid significant overestimations of errors, we decided to use a
constant uncertainty of $\rm A_{FUVp\mbox{ }err}$=0.74 magnitude.

We do not observe any redshift dependence (panel (c), Fig. 7); however, the
redshift range used in this analysis is very narrow (0.025–0.100). We can
reasonably expect that the much deeper LSST data will require adding a
redshift-dependent calibration.

The obtained priors do not follow a 1:1 relation with the $\rm A_{FUV}$ values
calculated directly from fits to the UV-IR range SED. This is due to many
reasons, where the most important ones are (1) uncertainties of the original
$\rm A_{FUV}$, $\left(u-r\right)$ and $\rm\mu_{u}$, which were not taken into
account when constructing the $\left(u-r\right)$ \- $\rm\mu_{u}$ \- $\rm
A_{FUV}$ relation; (2) the quality of the SED fits obtained by Salim et al.
(2018) (our only selection is based on REDCHISQ$<$5); (3) the quality of the
data used for full fitting — there is no information about the signal-to-noise
ratio for specific bands or the goodness of the measurement; (4) but even more
importantly, the $\left(u-r\right)$-$\rm\mu_{u}$ relation is not tight as the
$\left(u-r\right)$ colour depends both on the age of the stellar population
and on the dust. Nevertheless, the median difference between $\rm A_{FUV}$
obtained based on the careful fitting of broadband photometry from UV to IR
and $\rm A_{FUVp}$ determined from $\left(u-r\right)$ and $\rm\mu_{u}$
observed quantities is only 0.10 magnitude larger than the median $\rm
A_{FUVp\mbox{ }err}$ (0.12 magnitude).

Figure 7: Main properties of obtained $\rm A_{FUV}$ priors. Panel (a) shows
the distributions of original $\rm A_{FUV}$ from the Salim et al. (2018) work
obtained from careful SED fitting based on measurements from UV to mid-IR
($\rm A_{FUV}$, blue left hatched histogram), and $\rm A_{FUVp}$ calculated
based on Eq. 4 (orange right hatched histogram). Panel (b) presents the
distribution of the difference between $\rm A_{FUV}$ and $\rm A_{FUVp}$ and
denotes its median value. In panel (c), the difference between $\rm A_{FUV}$
and $\rm A_{FUVp}$ is shown as a function of redshift. We removed for clarity
from panel (a) 426 galaxies, for which calculated $\rm A_{FUVp}$ was lower
than 0.

##  5 Reliability of $\rm A_{FUVp}$ derived from LSST-like observations

Even if, as mentioned above, the agreement between $\rm A_{FUVp}$ $\pm$ $\rm
A_{FUVp\mbox{ }err}$ and $\rm A_{FUV}$ estimated via SED fitting by Salim et
al. (2016, 2018) is not perfect, the correlation is evident, and the
advantages of such an approach are numerous. An $\rm A_{FUV}$ prior obtained
from an optical-only dataset (without any information about dust emission or
proxy from UV observations) can help to reduce the number of parameters needed
to estimate the main physical properties of studied galaxies. It can also
reduce the risk of overfitting by decreasing the number of free parameters
used for SED fitting based on five optical broadbands only.

To check whether the obtained $\rm A_{FUVp}$ values can lead to reliable
estimates of the main physical properties of galaxies, we perform a set of
tests using only optical broadband data with and without two priors: the
original $\rm A_{FUV}$ from GSWLC obtained by Salim et al. (2018), which we
call now $\rm A_{FUVs}$ (and $\rm A_{FUVs\mbox{ }err}$) to distinguish between
both $\rm A_{FUV}$ values used in the test, and the $\rm A_{FUVp}$ (and $\rm
A_{FUVp\mbox{ }err}$) from Eq. 4.

To test using $\rm A_{FUVp}$ obtained via Eq. 4 as a prior, we perform six
CIGALE runs to fit the SED of our sample. We use SDSS DR 12 ugriz measurements
from Alam et al. (2015) (in case of Salim et al., 2016, 2018, they used SDSS
DR10). Salim et al. (2016) and Salim et al. (2018) made use of additional data
from GALEX and WISE which are not taken into account in our analysis. A simple
run based on five optical bands only is intended to reproduce the future LSST-
like observations (without the LSST y band). We employ the same SED fitting
code as used for the GSWLC data set. Parameters and modules used are described
in Table 3. As our SED coverage consists of only five optical SDSS data
points, we do not include any dust emission module.

We categorise the SED fitting parameters into two main groups: FULL--run
(based on the description given by Salim et al., 2016, 2018) and LIGHT--run,
with a significantly reduced number of parameters describing dust attenuation
and the age of the late burst for the star formation history module (details
are listed in Table 3). Runs based on FULL--run parameters produce 332 640
templates per redshift bin. In contrast, utilising LIGHT--run parameters
reduces the number of templates to only 5 540 per redshift bin. Thus, the
number of generated templates decreases by 98%. We performed our runs for 7
934 galaxies using Intel Core i9–9900K CPU @ 3.60 GHz processor with 64 GB
memory and 8 cores (16 threads). FULL--run required 188 seconds to compute
models while the LIGHT--run did the same in 11 seconds. The Bayesian estimates
of the physical properties for these templates for the FULL--run CIGALE took
203 seconds, while for the LIGHT--run, only one second. Thus, both runs used
the same time to estimate best-fit properties for all galaxies (85 seconds).
Thus, the LIGHT--run required only 3% of the time spent on the FULL--run. This
reduction in the number of templates is of particular significance for ‘big
data’ galaxy samples like the LSST. For LSST-like surveys, with billions of
observed galaxies, running full, detailed SED fitting will be impossible due
to CPU and memory limitations.

We used the LSST-like data set, and we performed the SED fitting using FULL--
run and LIGHT--run parameter sets. For FULL--run and LIGHT--run parameters we
further divide runs into three groups: ( 1) without any priors (tagged as NO
prior), without any indication of a preferable $\rm A_{FUV}$ value, ( 2) with
the $\rm A_{FUVs\mbox{ }err}$ and $\rm A_{FUVs\mbox{ }err}$ used as priors for
the CIGALE run, and (3) with $\rm A_{FUVp}$ and $\rm A_{FUVp\mbox{ }err}$. To
add priors, we used the properties option from the original CIGALE tool777 To
run CIGALE with the properties option, one has to add to the initial input
file additional columns filled with prior values and corresponding errors. In
the case of this work, we added two columns: attenuation.AFUV and
attenuation.AFUV_err, and we filled them with $\rm A_{FUVp}$ from Eq. 4 and
$\rm A_{FUVp\mbox{ }err}$ equal to 0.74 magnitude, respectively. , while for
runs without priors, we left the properties option empty. Runs with and
without priors still need to be supported by the input parameters for the dust
attenuation module. In all six cases, we used input parameters as listed in
Table 3.

In Fig. 8, we present the difference between estimated $\rm M_{star}$ and SFRs
obtained via our six runs and the original (fiducial) values from the GSWLC
catalogue. We stress that for all six runs, only u, g, r, i, and z SDSS
broadband photometry was used. The resultant loci of the obtained six main
sequences are shown in Fig. 9.

It is important to note that processing LIGHT--run data with CIGALE without
$\rm A_{FUV}$ prior leads to large overestimates of the SFR (see also Riccio
et al., 2021). Priors, even if based only on optical measurements, can
reproduce the physical parameters of the fiducials well enough. Moreover,
priors can also help to reduce the number of parameters, and hence the CPU
time, required to analyse large numbers of galaxies.

Table 3: Input parameters for the code CIGALE. Parameters |  Values
---|---
Star formation history:
double exponential (delayed with additional burst)
e-folding time of the main stellar population model (Myr) |  500, 1000, 3000, 5000, 8000, 10000, 15000, 20000
e-folding time of the late starburst population model (Myr) |  20000
Mass fraction of the late burst population |  0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35,0.4, 0.45, 0.5
Age of the main stellar population (Myr) |  6500
Age of the late burst (Myr) |  10, 30∗, 100∗, 300, 1000, 3000, 5000
Single stellar population Bruzual & Charlot (2003)
Initial mass function |  Chabrier (2003)
Metallicities (solar metallicity) |  $0.02$
Age of the separation between the young and the old star population (Myr) |  $10$
Nebular
Ionisation parameter |  -3.0
FULL--run: Dust attenuation law Calzetti et al. (2000)
$E(B-V)$: the colour excess of the stellar continuum light for the young population |  0, 0.01, 0.02, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.5, 0.6, 0.7, 0.8
Amplitude of the UV bump at 217.5 nm |  0, 2.0, 4.0, 6.0
Slope of the power law modifying the attenuation curve |  0.4, 0.2, 0, -0.2, -0.4, -0.6, -0.8, -1., -1.2
LIGHT--run: Dust attenuation law Calzetti et al. (2000)
$E(B-V)$: the colour excess of the stellar continuum light for the young population |  0, 0.05, 0.15, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8
Amplitude of the UV bump at 217.5 nm |  0
Slope of the power law modifying the attenuation curve |  0
∗ Notes: values 30 and 100 was removed from the LIGHT--run.

##  6 Discussion

Figure 8: Difference between SFR and Mstar from the original GSWLC catalogue
(Salim et al., 2018), and those obtained in this work. The difference $\Delta
M_{star}$ was calculated as log(M${}_{star\mbox{ \tiny{this work}}}$/Mstar);
and $\Delta SFR$ was obtained in an analogous way. Panels (a) and (b) show the
difference in the stellar masses and SFRs, correspondingly, as a function of
redshift, with shaded areas represent standard deviation of the scatter;
panels (c) and (d) present the relation between the fiducial values and the
estimates obtained in this work , with contours showing the distribution of
the data. Runs based on GSWLC set of parameters, FULL--run, are shown in blue-
coloured lines, while those obtained from LIGHT--run – in an orange-coloured
set of lines. Black solid lines visible in panels (c) and (d) represent 1:1
relations.

Figures 8 and 9 show the differences between the original (fiducial) physical
parameters from the GSWLC catalogue and the physical parameters obtained with
two sets of parameters: FULL--run, and highly reduced LIGHT--run.
Additionally, different $\rm A_{FUV}$ priors (and also no prior for even more
detailed comparison) were applied in these runs. Table 4 shows the mean SFR
and the mean $\rm M_{star}$ accompanied by their uncertainties for fiducial
parameters from the GSWLC and those obtained from six additional runs
described in Sec. 5.

data set | log(SFR/M⊙yr-1) | log($\rm M_{star}$/M⊙)
---|---|---
fiducial values | 0.19 $\pm$ 0.07 | 10.09 $\pm$ 0.05
FULL--run/ NO prior | 0.54 $\pm$ 0.41 | 9.96 $\pm$ 0.17
FULL--run/ $\rm A_{FUVs}$ prior | 0.20 $\pm$ 0.14 | 10.06 $\pm$ 0.09
FULL--run/ $\rm A_{FUVp}$ prior | 0.30 $\pm$ 0.21 | 10.04 $\pm$ 0.11
LIGHT--run/ NO prior | 0.68 $\pm$ 0.36 | 10.01 $\pm$ 0.11
LIGHT--run / $\rm A_{FUVs}$ prior | 0.23 $\pm$ 0.07 | 10.06 $\pm$ 0.08
LIGHT--run/ $\rm A_{FUVp}$ prior | 0.28 $\pm$ 0.10 | 10.05 $\pm$ 0.08
Table 4: Mean values of log(SFR) and log($\rm M_{star}$) for the fiducial
catalogue of 7 986 galaxies and those estimated from all six runs described in
Sec. 6, and presented in Figs. 8 and 9.

###  6.1 Influence of the $\rm A_{FUVp}$ or lack of it on the $\rm M_{star}$
estimation for the LSST-like observations

Results presented in Fig. 8, panels (a) and (c), show a negligible, $\leq 0.1$
dex, difference in the estimated $\rm M_{star}$ between all six runs and the
original $\rm M_{star}$ from the GSWLC catalogue. The consistency in the
estimations is actually expected given the complete set of the SDSS DR 12
optical broadband data used by Salim et al. (2018) and in our work. Detailed
coverage of the optical spectrum, from u to z bands, allows reconstruction of
the old stellar population in the galaxy (especially at low redshift), which
is the main ingredient of the total stellar mass of both quiescent and normal
star-forming galaxies.

However, it is worth mentioning that only for runs without prior $\rm\Delta
M_{star}$, defined as log(M${}_{\rm star\mbox{ \tiny{this work}}}$/$\rm
M_{star}$), is lower than $-0.07$, and monotonically decreases with redshift.
It implies that for optical data only, without any proxy for dust emission or
$\rm A_{FUV}$, the Bayesian method of estimating physical parameters prefers
to choose templates corresponding to somewhat less massive galaxies. It is
clearly visible when comparing runs without priors: for the run with a larger
number of templates (marked as FULL--run) $\rm\Delta$$\rm M_{star}$ is
systematically shifted towards lower values of $\rm M_{star}$ when compared to
the LIGHT--run. We want to stress here that this effect can be caused by
overfitting (it can be seen also Fig. 7 in Riccio et al., 2021, where the $\rm
M_{star}$ is very slightly, but still overestimated due to the number of used
templates). The number of parameters used in Salim et al. (2018) was allowed
thanks to a larger number of measurements available, which is not the case in
the LSST-like dataset. We stress that possible overfitting should be avoided
for the LSST-like data analysis, as it may cause the choice of templates of
systematically less massive galaxies, which is another reason why introducing
a prior to reduce the size of the parameter grid is needed.

###  6.2 Influence of the $\rm A_{FUVp}$ and lack of the prior on the SFR
estimation for the LSST-like observations

In this section, we check how the use of priors influences the estimations of
the SFR. Again, we calculate the ratio between the fiducial physical value
from GSWLC catalogue and those obtained in our runs. We define this ratio,
$\Delta$SFR, as log(SFR${}_{\tiny{\mbox{this work}}}$/SFR). The results are
shown in Fig. 8, panels (b) and (d).

####  6.2.1 No $\rm A_{FUV}$ prior involved

The first conclusion from this test, presented in the right panels of Fig. 8,
is that using optical data only without any prior (that is, without giving any
values for the properties option, see note 7) for the dust attenuation results
in a significant overestimation of the SFR, which decreases with redshift. The
same overestimation and its redshift dependence were found by Riccio et al.
(2021) for a sample of $\sim$50 000 main sequence galaxies. They used a sample
of observed galaxies to estimate the expected LSST fluxes in the ugrizy bands
and then performed SED fitting. They found that the $\rm M_{star}$ remains
well estimated (similarly to our result shown in Sect. 6.1) by the LSST–like
data set. However, at the same time, the SFR and the dust luminosity are
overestimated when the LSST–like sample alone is used. The SFR overestimation
found by Riccio et al. (2021) is redshift dependent and clearly decreases with
redshift, disappearing at about redshift $\sim$1\. This effect can be
explained by the wavelength range of the LSST observations. At redshift
$\sim$0 LSST probes mainly old stellar population, without any band probing
young stellar population or dust properties. As redshift increases, the LSST
$ugriz$ filters start to cover the UV rest frame, and the estimates of the SFR
significantly improve.

The lack of information about the UV and MIR rest-frame wavelengths for the
LSST low redshift sample causes a large overestimation of the attenuation
during the fitting (Riccio et al., 2021), which then translates into an
overestimation of the dust luminosity and, finally, the SFR. As shown in Fig.
8, both FULL--run and LIGHT--run sets of parameters, applied without any $\rm
A_{FUV}$ prior (blue and orange solid lines, respectively), result in
significant SFR overestimation: mean $\Delta$SFR for FULL--run and LIGHT--run
runs without priors equal to 0.34 and 0.48, respectively. Based on Eq. 1 from
Riccio et al. (2021), that is, $\Delta\mathrm{SFR}=\rm
SFR_{LSST}/SFR_{UV\mbox{-}FIR}$, the expected $\Delta$SFR at redshift 0.062
(which is the mean redshift of our galaxy sample) is equal to $\sim$0.5. As
seen from panels (b) and (d) of Fig. 8, our results, although limited to a
much narrower redshift range, are in agreement with predictions of Riccio et
al. (2021).

We stress that enlarging the parameter space in the SED fitting process, with
only a stellar population as a proxy, cannot solve the problem of the
overestimation of SFR. This conclusion is illustrated in Fig. 9. In this
figure, the black contours are based on the original catalogue of Salim et al.
(2018), while the orange contours show the estimates used in this work. It is
clearly seen that runs based on the optical data only without any priors
result in a significant systematic overestimation of SFR (panels (a) and (b)).
The overestimation occurs both for the FULL--run and the LIGHT--run parameter
grid. The overestimation of the SFR results in a shift of the main sequence
locus and can affect not only directly the physical analysis but also the
classification of normal star-forming and starbursting galaxies. Such
classification can propagate through the statistical analysis of evolutionary
paths and many other science cases.

####  6.2.2 Overcoming SFR overestimation with $\rm A_{FUV}$ and $\rm
A_{FUVp}$

In Riccio et al. (2021), it was also found that the $\rm M_{star}$–$\rm
A_{FUV}$ prior can solve the SFR overestimation for objects without UV or IR
observation. As seen in the panel (b) of Fig. 8, SFR estimated with $\rm
A_{FUV}$ priors (both FULL--run and LIGHT--run) are much closer to the
fiducial values than the SFR obtained without any priors. The $\Delta$SFR for
runs with priors are lower than 0.1.

The quality of SFR reconstruction can be seen in the central and the right
columns in Fig. 9. Adding $\rm A_{FUV}$ prior (original $\rm A_{FUVs}$ from
GSWLC catalogue or $\rm A_{FUVp}$ from Eq. 4) clearly reduces the
overestimation. The calculated SFR ratio between mean fiducial values from
Salim et al. (2018) and those obtained in this work with $\rm A_{FUVs}$ prior
is less than 0.05, and with $\rm A_{FUVp}$ – less than 0.1.

##### Original $\rm A_{FUV}$ prior.

Results presented as blue and orange dashed lines (FULL--run and LIGHT--run,
respectively) in Fig. 8 are based on the $\rm A_{FUVs}$ prior. Although the
difference with respect to the fiducial SFR values is almost negligible
($\Delta\mathrm{SFR}=0.1$), the relation is not 1:1 (which would correspond to
$\Delta\mathrm{SFR}=0$). In both cases, FULL--run and LIGHT--run sets of
parameters, the difference with respect to the fiducial value of SFR,
$\Delta$SFR, decreases with redshift.

This small difference results from a narrower wavelength coverage than used by
Salim et al. (2018). At the same time, these two runs demonstrate that $\rm
A_{FUVs}$ prior works almost equally well for the dense (FULL--run) and
significantly reduced (LIGHT--run) grids of parameters.

The central panels in Fig. 9 (c and d) show the main sequence built using the
original $\rm A_{FUVs}$ prior for FULL--run (panel c) and LIGHT--run (panel d)
runs. The original main sequence from the GSWLC catalogue is below the orange
contours. These two panels show that the use of the original $\rm A_{FUVs}$
prior reproduces the fiducial values of the SFR equally well with fewer
parameters (we remind that the number of templates in the LIGHT--run case is
98% lower than in the FULL--run case).

Of course, without ancillary data from other facilities, the actual value of
$\rm A_{FUV}$ will be unknown for most of LSST galaxies. The purpose of this
test was to demonstrate that (1) providing an $\rm A_{FUV}$ prior is
instrumental in the recovery of SFR and (2) that an $\rm A_{FUV}$ prior can be
used along with a small number of free parameters in the fitting process,
which will reduce the risks of overfitting in the case of a low number of data
points. However, galaxies at redshifts higher than $\sim 2$ will benefit from
the restframe FUV observations with the $u$ band of LSST. This will give
unprecedented power for probing the attenuation in this band specifically.
Therefore, attenuation in FUV can be used as a prior for high redshift
galaxies, correcting their SFR and $\rm M_{star}$ estimations when IR data are
missing.

##### Results based on the $\rm A_{FUVp}$ from $\left(u-r\right)$–$\rm\mu_{u}$
relation.

To test the effect of the prior obtained from Eq. 4, we run FULL--run and
LIGHT--run setup of parameters with $\rm A_{FUVp}$. SFRs obtained from this
test are shown with blue and orange dotted lines (FULL--run, and LIGHT--run,
respectively) in panels (b) and (d) of Fig. 8. Both results are very close to
the fiducial values ($\Delta\mathrm{SFR}<0.1$, very similar to the results
obtained with an original prior estimated from full UV–IR SED fitting, Salim
et al., 2018). The FULL--run parameter setup with an $\rm A_{FUVp}$ provides
values of SFR which have a small bias almost uniform along all the probed
ranges of the SFR, with both mean and median $\Delta$SFR equal to 0.11). In
the run with a smaller grid of parameters (LIGHT--run+$\rm A_{FUVp}$), a
similar small bias is observed (the median and the mean
$\Delta\mathrm{SFR}=0.09$. This small overestimation rises somewhat with
redshift.

The right column of Fig. 9 (panels e and f) shows the comparison of the main
sequences of 7 934 galaxies based on physical properties obtained from the
original GSWLC catalogue and those calculated using $\rm A_{FUVp}$ from Eq. 4.
It can be seen that for the case of a large number of parameters (FULL--run,
panel e) and a much smaller number of templates (LIGHT--run, panel f), the
obtained main sequence is narrower than the original one, and the
overestimation can be seen for more massive galaxies. The obtained main
sequence is, however, much closer to the fiducial one than in the case of
fitting with no priors. We emphasize that this may be related to a lower
fraction of massive galaxies in our sample (see Fig. 1) but also to the
representative original $\rm A_{FUVs}$ estimates, which have not been taken
into account in our fit of the slopes–intercepts.

The linear relations found between $\rm\mu_{u}$ and the $\left(u-r\right)$
colours for all the$\rm A_{FUV}$ bins show that the morphological aspect of
galaxies might be an indicator of attenuation. Surface brightness encodes in
itself the spatial extent of galaxies, and indications of a correlation
between the amount of attenuation on the one hand and the compactness of
galaxies on the other hand were found in Buat et al. (2019); Hamed et al.
(2023a, b). The $\rm\mu_{u}$ can also be correlated with the age of stellar
populations, and so it helps to break the degeneracy of the $\left(u-r\right)$
colour (which depends both on the dust attenuation and the stellar population
age).

We conclude that the LIGHT--run+$\rm A_{FUVp}$ run results in only the small
effect of SFR overestimation, visible mainly for massive galaxies. For the
case of the optical SDSS data only, the reduced number of templates and $\rm
A_{FUVp}$ created based on $\left(u-r\right)$ colour and $\rm\mu_{u}$ are thus
optimal, although not perfect, solutions to reproduce the main physical
properties of galaxies even without IR or UV data.

Figure 9: Comparison of physical properties obtained in FULL--run, and LIGHT--
run runs. Upper panels present results obtained based on FULL--run parameter
sets (332 640 templates per redshift bin), while the results shown in bottom
panels were estimated based on the LIGHT--run sets of parameters (5 544
templates per redshift bin). Black contours shown in each panel correspond to
fiducial values from the GSWLC-X2 catalogue (SFR, $\rm M_{star}$), while
orange contours illustrate physical parameters estimated in this work. The
left column presents results obtained without priors, the middle column – with
$\rm A_{FUVs}$ from the GSWLC catalogue, and the right column – with $\rm
A_{FUVp}$ calculated from Eq. 4.

##  7 Summary

In this work, we aimed at constructing the proxy of dust attenuation in the
FUV wavelength range from the optical photometric LSST-like data only. To
imitate the LSST optical detections of low redshift main-sequence galaxies, we
used SDSS observations. Furthermore, for the analysis of the dust attenuation,
we selected only galaxies with IR auxiliary observations, with homogeneous
data analysis and estimated main physical parameters, such as $\rm M_{star}$,
SFR and $\rm A_{FUV}$. We selected a sample of 7 934 local ($0.025<z<0.1$)
main-sequence star-forming SDSS galaxies with properties previously measured
based on multiwavelength IR-to-UV photometric data. We used the GSWLC
catalogue of physical properties citepSalim2018 together with their SDSS DR 12
ugriz measurements (Alam et al., 2015) and inclination from
citeMeert2015MNRAS.446.3943M. We calculated their surface brightness in the
ugri bands and corrected for inclination and cosmological surface brightness
dimming.

We tested different setups of the CIGALE SED fitting code in order to find an
optimal method to estimate the main physical properties of galaxies when only
the $ugriz$ photometric data are available. Our results can be summarised as
follows:

  1. 1.

We find that the proxy for dust attenuation, $\rm A_{FUVp}$, can be
constructed based on the combination of the $\left(u-r\right)$ colour and the
galaxy surface brightness in the $u$ band ($\rm\mu_{u}$). The formula is given
by Eq. 4. The formula remains almost the same if the colour $\left(u-i\right)$
is used instead of $\left(u-r\right)$, Eq. 5 in the footnote.

  2. 2.

Surface brightness measured in bands other than $u$ do not provide an equally
good estimate for $\rm A_{FUV}$, as it can be seen in Fig. 13.

  3. 3.

The $\rm M_{star}$ can be well recovered by the SED fitting in the optical
range only, with no necessity to use additional priors.

  4. 4.

The SFR cannot be well measured based on the SED fitting method in the optical
range only due to missing information about dust attenuation. In the case of
CIGALE, it results in systematic and redshift-dependent overestimation of SFR
at the level of $\Delta\mathrm{SFR}>0.3$ dex, which is in agreement with the
findings of Riccio et al. (2021).

  5. 5.

The SFR can be recovered almost unbiased, with $\Delta\mathrm{SFR}<0.05$ dex,
if the fitting is performed with an $\rm A_{FUV}$ proxy. However, such a proxy
will not be available for all future LSST data without auxiliary data.

  6. 6.

The $\rm A_{FUVp}$ given by Eq. 4, used as a proxy for the SED fitting, allows
for very good recovery of SFR with only a small bias $\Delta\mathrm{SFR}\sim
0.1$ dex.

  7. 7.

This bias is both $\rm M_{star}$ and redshift dependent, which implies that
the $\rm A_{FUVp}$ proposed in Eq. 4 in the future will have to be refined and
generalised based on the calibration with deeper data and larger galaxy
samples.

Additionally, we have found that when a prior for dust attenuation is used,
the parameter grid used by the fitting code can be significantly reduced in
order to avoid overfitting, which, in particular, tends to lead to
overestimation of SFR. Also, thanks to the reduction of the number of
generated templates by $\sim$98%, we decrease the computing time needed for
fitting by a comparable factor, which will be of great importance for the ’big
data’ analysis of the LSST data.

Thus, this work proposes a strategy based only on optical photometric
measurements to reliably and efficiently measure galaxy physical properties
through SED fitting in future large surveys. The next steps will include the
analyses based on galaxy samples deeper than the SDSS, complemented by the
state-of-the-art simulations, which will allow for extensions of the proposed
strategy towards higher redshift and lower brightness data sets, as well as
cover different types of galaxies, which will likely require a diversified
approach.

###### Acknowledgements.

We acknowledge and thank the referee for a thorough and constructive report,
which helped improve this work.
This research was done as a part of the NCBJ’s In-kind Contributions to the
Vera C. Rubin Observatory Legacy Survey of Space and Time POL-NCB-S3. This
research was supported by the Polish National Science Centre grant
UMO-2018/30/M/ST9/00757 and the Polish Ministry of Science and Higher
Education grant DIR/WK/2018/12. K. M and J. have been supported by the
National Science Centre (UMO-2018/30/E/ST9/00082). This research was also
partially supported by the ’PHC POLONIUM’ programme (project number: 49136QB),
funded by the French Ministry for Europe and Foreign Affairs, the French
Ministry for Higher Education and Research and the Polish NAWA. The project is
co-financed by the Polish National Agency for Academic Exchange
(BPN/BFR/2022/1/00005). This research was supported by from COST Action
CA21136 – ’Addressing observational tensions in cosmology with systematics and
fundamental physics (CosmoVerse)’, supported by COST (European Cooperation in
Science and Technology). R. D. gratefully acknowledges support by the ANID
BASAL project FB210003. M. B., C. S., and M. A. acknowledge support from the
ANID through FONDECYT grants (no. 1211000, 11191125 and 1211951) and Agencia
Nacional de Investigación y Desarrollo (ANID) BASAL project FB210003. D. D.
acknowledges support from the National Science Center (NCN) grant SONATA
(UMO-2020/39/D/ST9/00720). J. R. M. is supported by STFC funding for UK
participation in LSST, through grant ST/Y00292X/1. A. N. and M. R. acknowledge
support from the Narodowe Centrum Nauki (UMO- 2020/38/E/ST9/00077), and M. R.
also acknowledge support from the Foundation for Polish Science (FNP) under
the program START 063.2023. O. D. acknowledges support from the Programa de
doctorado en Astrofísica y Astroinformática of Universidad de Antofagasta. W.
J. P. has been supported by the Polish National Science Center project
UMO-2020/37/B/ST9/00466. JR acknowledges funding from the University of La
Laguna through the Margarita Salas Program from the Spanish Ministry of
Universities ref. UNI/551/2021-May 26, and under the EU Next Generation. This
work made use of Astropy:888http://www.astropy.org a community-developed core
Python package and an ecosystem of tools and resources for astronomy (Astropy
Collaboration et al., 2013, 2018, 2022)

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##  Appendix A Distribution of used SDSS magnitudes and their corresponding
surface brightness

Figure 10 shows distributions of SDSS ugri magnitudes and calculated based on
Eqs. 1 and 2 their corresponding surface brightness.

Figure 10: Distributions of SDSS magnitudes and their corresponding surface
brightness. The left panel (grey histograms) show the distribution of ugri
magnitudes of SDSS 7 934 galaxies from the final sample used in our analysis,
while the right panel (dark red hatched histograms) their corresponding
surface brightness corrected for the inclination. Median values of all
distributions are added in each panel.

Figure 11 shows the same plot as Fig. 5 (fitted $\left(u-r\right)$ –
$\rm\mu_{u}$ relation for all 14 bins of $\rm A_{FUV}$) but with an additional
background of the whole sample, and also interpolated relations between 22.5
and 27.5 $\rm\mu_{u}$.

Figure 11: Relations fitted between observed $\left(u-r\right)$ colours and
$\rm\mu_{u}$ for 14 $\rm A_{FUV}$ bins with additional background showing all
galaxies used in our analysis. The grey scale colour bar axis shows the value
of the $\rm A_{FUV}$.

##  Appendix B Galaxies with $\rm A_{FUVp}$¡0

We present here the location in the $\left(u-r\right)$-$\rm\mu_{u}$ plane
galaxies for which the calculated based on Eq. 4 $\rm A_{FUVp}$ is lower than
zero. Fig. 12 shows the location of all 426 galaxies with $\rm A_{FUVp}$$<0$.

Figure 12: Fiducial $\rm A_{FUV}$ values from the GSWLC catalogue in the
$\left(u-r\right)$-$\rm\mu_{u}$ plane. Similarly, as in Fig. 11, grey scale
colour bar axis shows the value of the $\rm A_{FUV}$. Additionally, galaxies
with calculated $\rm A_{FUVp}$$<0$ are marked as open orange squares.

##  Appendix C Other colour$-$surface brightness relatons

Relations fitted between observed colours and surface brightness in different
configurations for 14 $\rm A_{FUV}$ bins are presented in Fig. 13. The
sequence of colours represents the one used for $\rm A_{FUV}$ bin in Fig. 4.
Filled areas mimic the $\pm 1\sigma$ uncertainty around estimated lines. The
black line on each panel represents the linear fit to all bins. The slope of
this mean relation is shown on each panel.

Figure 13: Exemplary colour – surface brightness-fits for all 14 $\rm A_{FUV}$
bins. Colors are the same as in Fig. 5 and Fig. 11. The black lines shown in
each panel represent the linear slope between colour and surface brightness
calculated for the whole sample.

##  Appendix D $\rm A_{FUV}$ bins fitting

Here, we present separate fits performed for all 14 $\rm A_{FUV}$ bins. Each
panel represent one bin. At the top of each panel, we give the $\rm A_{FUV}$
range and the number of galaxies from our sample. The fitted linear relation,
as well as the deviation of the $\left(u-r\right)$ colour from the linear fit
($\Delta_{ur}$) and the variability of $\rm A_{FUV}$ within the range of $\pm
1\sigma$, are given in each panel.

Figure 14: Relation between observed $\left(u-r\right)$ and $\rm\mu_{u}$ for
the sample of 7 934 galaxies divided into 14 $\rm A_{FUV}$ bins. Each panel
represents consecutive $\rm A_{FUV}$ bins (Fig. 4 and Table 2). The resulting
slope and intercept of the fit are provided within each corresponding panel.
The deviation of the $\left(u-r\right)$ colour from the linear fit
($\Delta_{ur}$) and the variability of $\rm A_{FUV}$ within the range of $\pm
1\sigma$ from the linear fit ($\sigma_{AFUV}$) are both indicated in each
panel.