11institutetext: INAF — Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5,
I-50125, Florence, Italy 11email<EMAIL_ADDRESS>22institutetext:
Department of Astronomy, The Ohio State University, 140 West 18th Avenue,
Columbus, OH 43210, USA 33institutetext: Department of Physics and Astronomy,
McMaster University, 1280 Main St. West, Hamilton ON L8S 4M1 Canada
44institutetext: Canadian Institute for Theoretical Astrophysics (CITA),
University of Toronto, 60 St George Street, Toronto, ON M5S 3H8, Canada
55institutetext: Argelander-Institut für Astronomie, Universität Bonn, Auf dem
Hügel 71, D-53121 Bonn, Germany 66institutetext: Centro de Astronomía
(CITEVA), Universidad de Antofagasta, Avenida Angamos 601, Antofagasta, Chile
77institutetext: Max-Planck-Institut für extraterrestrische Physik (MPE),
Giessenbachstrasse 1, D-85748 Garching, Germany 88institutetext: Departamento
de Astronomía, Universidad de Chile, Camino del Observatorio 1515, Las Condes,
Santiago, Chile 99institutetext: Department of Physics and Astronomy,
University of Wyoming, Laramie, WY 82071, USA 1010institutetext:
Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität
Heidelberg, Mönchhofstraße 12-14, D-69120 Heidelberg, Germany
1111institutetext: Universität Heidelberg, Zentrum für Astronomie, Institut
für theoretische Astrophysik, Albert-Ueberle-Straße 2, D-69120, Heidelberg,
Germany 1212institutetext: Research School of Astronomy and Astrophysics,
Australian National University, Canberra, ACT 2611, Australia
1313institutetext: ARC Centre of Excellence for All Sky Astrophysics in 3
Dimensions (ASTRO 3D), Australia 1414institutetext: International Centre for
Radio Astronomy Research, University of Western Australia, 7 Fairway, Crawley,
6009, WA, Australia 1515institutetext: Universität Heidelberg,
Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld
205, D-69120 Heidelberg, Germany 1616institutetext: Observatorio Astronómico
Nacional (IGN), C/Alfonso XII, 3, E-28014 Madrid, Spain 1717institutetext:
Departamento de Física de la Tierra y Astrofísica, Universidad Complutense de
Madrid, E-28040 Madrid, Spain 1818institutetext: Instituto de Física de
Partículas y del Cosmos IPARCOS, Facultad de CC Físicas, UCM, E-28040, Madrid,
Spain 1919institutetext: Max-Planck-Institute for Astronomy, Königstuhl 17,
D-69117 Heidelberg, Germany

# Calibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ maps

Francesco Belfiore Calibration of hybrid resolved star formation rate recipes
based on PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved
star formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps
Adam K. Leroy Calibration of hybrid resolved star formation rate recipes based
on PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps Jiayi
Sun Calibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$
mapsCalibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps Ashley
T. Barnes Calibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps
Médéric Boquien Calibration of hybrid resolved star formation rate recipes
based on PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved
star formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps
Yixian Cao Calibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps Enrico
Congiu Calibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps Daniel
A. Dale Calibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps Oleg
V. Egorov Calibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps Cosima
Eibensteiner Calibration of hybrid resolved star formation rate recipes based
on PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps Simon
C. O. Glover Calibration of hybrid resolved star formation rate recipes based
on PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps
Kathryn Grasha Calibration of hybrid resolved star formation rate recipes
based on PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved
star formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$
mapsCalibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps Brent
Groves Calibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps Ralf
S. Klessen Calibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$
mapsCalibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps
Kathryn Kreckel Calibration of hybrid resolved star formation rate recipes
based on PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved
star formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps
Lukas Neumann Calibration of hybrid resolved star formation rate recipes based
on PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps Miguel
Querejeta Calibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps
Patricia Sanchez-Blazquez Calibration of hybrid resolved star formation rate
recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid
resolved star formation rate recipes based on PHANGS–MUSE H$\alpha$ and
H$\beta$ mapsCalibration of hybrid resolved star formation rate recipes based
on PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps Eva
Schinnerer Calibration of hybrid resolved star formation rate recipes based on
PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps Thomas
G. Williams Calibration of hybrid resolved star formation rate recipes based
on PHANGS–MUSE H$\alpha$ and H$\beta$ mapsCalibration of hybrid resolved star
formation rate recipes based on PHANGS–MUSE H$\alpha$ and H$\beta$ maps

(Received XXX; accepted XXX)

Mapping star-formation rates (SFR) within galaxies is key to unveiling their
assembly and evolution. Calibrations exist for computing SFR from a
combination of ultraviolet and infrared bands for galaxies as integrated
systems, but their applicability to sub-galactic (kpc) scales remains largely
untested. Here we use integral field spectroscopy of 19 nearby ($D<$ 20 Mpc)
galaxies obtained by PHANGS–MUSE to derive accurate Balmer decrements
(H$\alpha$/H$\beta$) and attenuation-corrected H$\alpha$ maps. We combine this
information with mid-infrared maps from WISE at 22 $\rm\mu m$, and ultraviolet
maps from GALEX in the far-UV band, to derive SFR surface densities in nearby
galaxies on resolved (kpc) scales. Using the H$\alpha$ attenuation-corrected
SFR as a reference, we find that hybrid recipes from the literature
overestimate the SFR in regions of low SFR surface density, low specific star-
formation rate (sSFR), low attenuation and old stellar ages. We attribute
these trends to heating of the dust by old stellar populations (‘IR cirrus’).
We calibrate this effect by proposing functional forms for the coefficients in
front of the IR term which depend on band ratios sensitive to the sSFR. Such
recipes return SFR estimates in agreement with those in the literature at high
sSFR ($\rm log(sSFR/yr^{-1})>-9.9$). Moreover, they lead to negligible bias
and $<$0.16 dex scatter when compared to our reference attenuation-corrected
SFR from H$\alpha$. These calibrations prove reliable as a function of
physical scale. In particular, they agree within 10% with the attenuation
corrections computed from the Balmer decrement on 100 pc scales. Despite small
quantitative differences, our calibrations are also applicable to integrated
galaxy scales probed by the MaNGA survey, albeit with a larger scatter (up to
0.22 dex). Observations with JWST open up the possibility to calibrate these
relations in nearby galaxies with cloud-scale ($\sim$100 pc) resolution mid-IR
imaging.

###### Key Words.:

Galaxies: ISM – Galaxies: star formation – ISM: general

##  1 Introduction

Estimates of the recent star-formation rate (SFR) play a central role in
studies of galaxy evolution, chemical enrichment, stellar feedback, and the
physics of stellar birth. Estimating the SFR in distant galaxies entails
measuring light from young massive stars, either directly (e.g. in the
ultraviolet, UV), or indirectly, via recombination line emission (e.g.
H$\alpha$) or the radiation re-emitted by dust in the infrared (IR).

From the cosmological perspective, for example, knowledge on how the dust
attenuation affects UV emission is fundamental to correctly reconstruct the
peak and decline of the cosmic star formation history at $z<4$ (Lilly et al.,
1996; Gruppioni et al., 2013; Madau & Dickinson, 2014; Koprowski et al.,
2017), and interpret the properties of the first galaxies, which may already
be subject to substantial dust obscuration (Casey et al., 2018; Gruppioni et
al., 2020; Algera et al., 2022). The physics of the star formation process, on
the other hand, can be probed in the extragalactic context by studying the so-
called ‘star formation laws’, relating gas content with SFR for entire
galaxies or sub-galactic regions (Kennicutt, 1998; Bigiel et al., 2008;
Bacchini et al., 2019). Accurate dust corrections have been instrumental in
reducing the systematic uncertainties in the measurement of slope of these
relations (Kennicutt & De Los Reyes, 2021), and therefore in discriminating
between competing physical models (Sun et al., submitted).

Unfortunately, both UV and IR emission can arise from sources other than young
stars, and different SFR tracers are dependent on the age of the stellar
population and other physical properties of the interstellar medium (ISM) in
complex ways. As a result, calibrating SFR estimates and linking between
different systems remains an important, open topic (for reviews see Kennicutt,
1998; Murphy et al., 2011; Wuyts et al., 2011; Kennicutt & Evans, 2012;
Calzetti, 2013; Boquien et al., 2016).

In this paper, we focus on measurements of the SFR per unit area
($\Sigma_{\mathrm{SFR}}$) on kiloparsec scales, and attempt to link three of
the most commonly used calibration methods: (1) H$\alpha$ emission corrected
for dust attenuation by using the Balmer decrement (e.g. Groves et al., 2012a,
and posited as early as Berman 1936); (2) H$\alpha$ emission corrected using
an empirically-calibrated combination with WISE 22 $\rm\mu m$ (or Spitzer
24$\mu$m, IRAS 25$\mu$m) emission (e.g. Kennicutt et al., 2007; Calzetti et
al., 2007); and (3) UV emission combined with WISE 22 $\rm\mu m$ emission
(e.g. Meurer et al., 1999; Leroy et al., 2008; Hao et al., 2011; Boquien et
al., 2016).

The first technique contrasts the expected ratio of H$\alpha$ to H$\beta$ line
emission for Case B recombination with the observed ratio to calculate the
ratio of attenuations. Combined with an adopted attenuation curve, the paired
measurement of H$\alpha$ and H$\beta$ line emission yields the attenuation
affecting H$\alpha$, $A_{\mathrm{H\alpha}}$, and the attenuation-corrected
H$\alpha$ flux, $F_{\rm H\alpha}^{\rm corr}$. Although this method has a long
pedigree, it has been used more widely in recent years thanks to the broad
availability of optical integral field spectroscopy (IFS) from surveys such as
CALIFA (Sánchez et al., 2012), SAMI (Croom et al., 2012), and MaNGA (Bundy et
al., 2015).

The second method consists of combining H$\alpha$ with mid-IR maps, generally
(but not always) via a linear combination. The ratio of mid-IR to H$\alpha$
emission is treated as an indicator of $A_{\mathrm{H\alpha}}$ or, put another
way, the mid-IR emission is used to trace the portion of H$\alpha$ emission
absorbed by dust. Although this approach can be theoretically motivated, its
calibration is fundamentally empirical. Namely, the mid-IR flux is scaled by a
(pre)factor set by benchmarking against some ‘gold standard’ reference SFR
tracer, most commonly attenuation-corrected hydrogen recombination lines as
described in the previous paragraph (e.g. Calzetti et al., 2007; Kennicutt et
al., 2009).

The third approach combines UV and IR emission. We consider linear
combinations of individual bands (e.g. Thilker et al., 2007; Leroy et al.,
2008; Hao et al., 2011; Leroy et al., 2012; Boquien et al., 2016; Leroy et
al., 2019), but previous versions of this approach also include multi-band or
higher-order calibrations, and summing full bolometric estimates of the IR and
UV emission (e.g. Meurer et al., 1999). In a broad sense, this approach
represents a simplified adaptation of the increasingly common practice of
fitting population synthesis models to the spectral energy distribution (SED)
of galaxies in the UV, optical, and IR bands (e.g. Da Cunha et al., 2008;
Conroy et al., 2013; Salim et al., 2016, 2018; Nersesian et al., 2019; Hunt et
al., 2019).

We focus here on the linear combination of UV and the $22\leavevmode\nobreak\
\mu$m mid-IR band. The mid-IR offers several advantages with respect to the
far-IR. Physically, the mid-IR continuum is expected to be directly sensitive
to the incident radiation field.Empirically, the warm mid-IR dust appears to
be physically associated with regions of massive star formation (e.g. Helou et
al., 2004; Relaño et al., 2010) with a smaller contribution from cool dust
heated by older stellar populations than the far-IR. Crucially, Spitzer had
excellent performance at $24\leavevmode\nobreak\ \mu$m, and more recently WISE
(Wright et al., 2010) provided mid-IR data at sufficient resolution
($\theta_{\rm FWHM}\approx 12\arcsec$ at $22\leavevmode\nobreak\ \mu$m) to map
nearby star-forming galaxies (Jarrett et al., 2013). Further emphasizing the
importance of the mid-IR, the James Webb Space Telescope (JWST) is providing
maps of the mid-IR emission at even higher resolution and sensitivity for both
local and distant galaxies.

Despite the utility of the mid-IR based methods, their calibration remains
exclusively empirical. Fortunately, studies of nearby galaxies have
established self-consistent calibrations for the mid-IR in combination with UV
or H$\alpha$ to estimate the SFR for whole galaxies (Kennicutt et al., 2009;
Hao et al., 2011; Catalán-Torrecilla et al., 2015; Leroy et al., 2019).
Another line of studies have established calibrations that can be applied to
individual star-forming regions inside of galaxies (including Kennicutt et
al., 2007; Calzetti et al., 2007; Murphy et al., 2011; Boquien et al., 2016).
These calibrations are generally found to be accurate to $\sim$ 0.1–0.15 dex
(Kennicutt et al., 2009; Leroy et al., 2012), but systematically differ from
each other at the level of $\sim$ 0.2 dex.

Physically, mid-IR emission reflects the combined effect of dust abundance,
relative geometry of dust and stars, and dust heating by the surrounding
stellar populations. While young stars may provide the bulk of the heating in
individual star-forming regions, in more quiescent regions older stars also
play a substantial role in heating the dust and powering the emission in the
IR (e.g. Groves et al., 2012b; Li et al., 2013). High-resolution studies
($\lesssim$ 100 pc) can isolate individual regions and effectively avoid or
filter out this diffuse background or ‘infrared cirrus’ (e.g. Calzetti et al.,
2007). However, on the larger scales of kiloparsecs or whole galaxies, this IR
cirrus will be an unavoidable part of the measurement (Boquien et al., 2016).
This fact leads to a non-linear relation between total infrared luminosity and
the SFR that is dependent on the specific SFR ($\rm sSFR=SFR/M_{\star}$, e.g.
Cortese et al. 2008; Boquien & Salim 2021). The contribution of the IR cirrus
may be physically modeled and subtracted (e.g. Leroy et al., 2012). Or the
calibration of the IR term may take into account the contribution of this
diffuse dust component (Boquien et al., 2016).

So far, the lack of quality reference maps has been a major obstacle to
calibrating the IR term in SFR prescriptions to apply at sub-galactic scales.
In this work, we revisit these questions prompted by a unique combination of
new data. The PHANGS–MUSE survey (Emsellem et al., 2022) has recently obtained
comprehensive integral field spectroscopy (IFS) mapping of the star-forming
discs of $19$ nearby galaxies (distance $<20$ Mpc), using the MUSE (Multi-unit
Spectroscopic Explorer) instrument (Bacon et al., 2010) on the ESO Very Large
Telescope. These observations give us access to high signal-to-noise maps of
H$\alpha$ and H$\beta$, free from the continuum subtraction systematics
associated with narrow-band images. Because PHANGS–MUSE targets nearby
galaxies, the relatively coarse $\sim 5{-}15\arcsec$ resolution of mid-IR maps
from WISE or Spitzer translates to $\lesssim 1$ kpc. This scale is
fortuitously set, since it allows us to define regions with significant
numbers of the relatively rare massive stars, yet not so large as to merge
multiple adjacent star-forming regions.

Extensive, homogenised multi-wavelength imaging is available for the
PHANGS–MUSE targets from GALEX far-UV (FUV, $\lambda\sim 154$ nm) and near-UV
(NUV, $\lambda\sim 231$ nm), as well as WISE Band 1, 2, 3, and 4 imaging
($\lambda=3.4$, $4.5$, $12$, and $22\leavevmode\nobreak\ \mu$m) with
15$\arcsec$ or $\sim 1$ kpc resolution (Leroy et al., 2019). Such a data set
allows us to check for consistency among the three methods of estimating SFR
discussed above, quantify systematic uncertainties, and derive new empirical
coefficients appropriate for this scale and sample, using the attenuation-
corrected H$\alpha$ maps from MUSE as a reference.

A key motivation for this new calibration effort is the desire to derive SFR
maps that are anchored in the Balmer-corrected MUSE data and combine WISE IR
maps with new narrow-band H$\alpha$ observations of nearby galaxies (Razza et
al., in preparation). Moreover, we wish to integrate the UV+IR-based SFR maps
widely available for the nearest galaxies ($D\lesssim 50$ Mpc, e.g. Jarrett et
al. 2013; Leroy et al. 2019) into a common framework with the attenuation-
corrected H$\alpha$ derived from optical IFS mapping that has been obtained
for large samples of more distant ($D\sim 20{-}150$ Mpc) galaxies (Belfiore et
al., 2018; Medling et al., 2018; Sánchez, 2020). Finally, observations in the
mid-IR with JWST now provide us with a high-resolution view of embedded star
formation in the nearby Universe, therefore making it urgent to verify the
validity of SFR calibrations across a range of physical scales.

With these goals in mind, Section 2 presents our compilation of $15\arcsec$
resolution MUSE, UV, and IR measurements, and describes our methods and
assumptions. Section 3 compares results for $\Sigma_{\mathrm{SFR}}$ and
$A_{\mathrm{H\alpha}}$ calculated using previous calibrations and presents
updated recipes for kpc-scale SFR calibrations using H$\alpha$+IR and UV+IR.
In Sect. 4, we discuss the impact on our new calibrations together with their
limitations. We summarise our results in Sect. 5.

##  2 Data and methods

The starting point of this work is the sample of 19 galaxies targeted by the
PHANGS-MUSE survey. These objects were selected to be nearby ($D<20$ Mpc),
close to the star formation main sequence, and moderately inclined
($i<57.3^{\circ}$). The main properties of the sample, which spans the stellar
mass range $\log(M_{\star}/M_{\odot})=9.4{-}11.0$, are presented in Emsellem
et al. (2022) and summarised in Table 1.

Table 1: Key properties of the PHANGS–MUSE sample of galaxies used in this work. Distances are taken from the compilation of Anand et al. (2021). Stellar masses, SFR, and offset from the star-formation main sequence ($\Delta_{\rm SFMS}$) are taken from the PHANGS sample paper (Leroy et al., 2021), inclination ($i$) are taken from Lang et al. (2020). $\rm\theta_{15}$ is the size in kpc of the 15′′ resolution element used in this analysis. Name | Distance | ${\rm Log}(M_{\star})$ | $\rm Log(SFR)$ | $i$ | $\rm\theta_{15}$
---|---|---|---|---|---
| Mpc | [$M_{\odot}$] | [$M_{\odot}\leavevmode\nobreak\ \mathrm{yr}^{-1}$] | deg | kpc
NGC 0628 | 9.8 | 10.34 | 0.18 | 8.9 | 0.7
NGC 1087 | 15.9 | 9.93 | 0.33 | 42.9 | 1.2
NGC 1300 | 19.0 | 10.62 |  $-$0.18 | 31.8 | 1.4
NGC 1365 | 19.6 | 10.99 | 0.72 | 55.4 | 1.4
NGC 1385 | 17.2 | 9.98 | 0.50 | 44.0 | 1.3
NGC 1433 | 18.6 | 10.87 |  $-$0.36 | 28.6 | 1.4
NGC 1512 | 18.8 | 10.71 |  $-$0.21 | 42.5 | 1.4
NGC 1566 | 17.7 | 10.78 | 0.29 | 29.5 | 1.3
NGC 1672 | 19.4 | 10.73 | 0.56 | 42.6 | 1.4
NGC 2835 | 12.2 | 10.00 | 0.26 | 41.3 | 0.9
NGC 3351 | 10.0 | 10.36 | 0.05 | 45.1 | 0.7
NGC 3627 | 11.3 | 10.83 | 0.19 | 57.3 | 0.8
NGC 4254 | 13.1 | 10.42 | 0.37 | 34.4 | 1.0
NGC 4303 | 17.0 | 10.52 | 0.54 | 23.5 | 1.2
NGC 4321 | 15.2 | 10.75 | 0.21 | 38.5 | 1.1
NGC 4535 | 15.8 | 10.53 | 0.14 | 44.7 | 1.1
NGC 5068 | 5.2 | 9.40 | 0.02 | 35.7 | 0.4
NGC 7496 | 18.7 | 10.00 | 0.53 | 35.9 | 1.4
IC 5332 | 9.0 | 9.67 | 0.01 | 26.9 | 0.7

We compiled and analysed a data set that includes intensities and associated
uncertainties at a $15\arcsec$ common resolution from MUSE, GALEX, and WISE.
The limiting resolution is set by WISE W4 band centered at
$22\leavevmode\nobreak\ \mu$m. For the distances of our galaxy sample
($D=5.2-19.6$ Mpc), this angular resolution corresponds to a median physical
resolution of 1.1 kpc, varying across our sample from 400 pc to 1.4 kpc (see
Table 1).

###  2.1 PHANGS–MUSE, optical IFS

Each galaxy in the PHANGS–MUSE sample was observed with several (three to
fifteen) MUSE pointings, each $1\arcmin\times 1\arcmin$. Observing strategy,
data reduction and analysis of this data set are described in detail in
Emsellem et al. (2022). The data reduction was carried out via pymusepipe, a
python wrapper to the ESOREX MUSE reduction recipes (Weilbacher et al., 2020).
The point spread function (PSF) and sky background of each MUSE pointing were
determined by comparison with wide-field $R$-band imaging (Razza et al., in
preparation). Reconstructed photometry from MUSE is consistent with Sloan
Digital Sky Survey (SDSS) imaging with a typical scatter of 0.04 mag. In this
work we mostly use datacubes where the native-resolution mosaicked cube was
convolved with a suitable kernel in order to deliver a common Gaussian
resolution of 15$\arcsec$ full width at half maximum (FWHM) and spaxels
2$\aas@@fstack{\prime\prime}$4 in size. In Sect. 3.3.1 we study the effect of
spatial resolution on our results and we therefore use the ‘optimally
convolved’ (copt) datacubes, described in Emsellem et al. (2022). In these
data products the PSF was homogenised across each individual mosaic and as a
function of wavelength using the pypher tool (Boucaud et al., 2016) to obtain,
for each galaxy, a best possible common Gaussian PSF. The median PSF FWHM of
the copt data is 0$\aas@@fstack{\prime\prime}$95.

Emission-line fluxes and other stellar population properties were obtained by
using the PHANGS data analysis pipeline (dap), as described in Emsellem et al.
(2022). The dap was used to perform three full spectral fitting steps: the
first optimised for the recovery of the stellar kinematics, the second aimed
at determining the stellar population properties, and the final one optimised
for extraction of fluxes and kinematics of gas emission lines. The core of all
three spectral fitting modules is the pPXF python module (Cappellari &
Emsellem, 2004; Cappellari, 2017). The stellar continuum was fitted using a
set of simple stellar population models from the E-MILES library (Vazdekis et
al., 2012). From the dap output we use maps of H$\alpha$ and H$\beta$ line
emission (corrected for Galactic extinction), and also consider the mean
light-weighted age of the stellar population in Sect. 3.1.1. We applied a set
of custom-designed masks to the output MUSE maps. In particular, for the
15$\arcsec$-resolution data we masked foreground stars brighter than 16.5 mag
in Gaia G-band from Gaia Data Release 2 (Gaia Collaboration et al., 2018) and
applied a few additional masks to stars which were missed and the central
regions of galaxies hosting active galactic nuclei (AGN).

###  2.2 Mid-IR and UV maps

IR and UV imaging was obtained from z0MGS ($z=0$ multi-wavelength galaxy
synthesis, Leroy et al. 2019), an atlas of WISE and GALEX (Martin, 2005)
images of nearby ($D<$ 50 Mpc) galaxies. z0MGS is based on the unWISE
reprocessing by Lang (2014) and draws heavily on HyperLEDA (Makarov et al.,
2014) for the atlas construction.

Images used in this work were taken from the publicly available atlas, and
were convolved to a common 15$\arcsec$ FWHM resolution, with a pixel size of
5$\aas@@fstack{\prime\prime}$5\. Masking based on GAIA DR2, 2MASS (Skrutskie
et al., 2006), and HyperLEDA was carried out to remove foreground stars or
other galaxies in the field, and a local background subtraction was performed
for each object. GALEX imaging was corrected for Galactic extinction following
Peek & Schiminovich (2013) and the Schlegel et al. (1998) dust maps. Stellar
mass surface density maps were derived according to the prescription of Leroy
et al. (2019), who used the WISE $\rm 3.4\mu m$ W1 band in combination with a
mass-to-light ratio that scales with SFR to obtain masses in agreement with
Salim et al. (2018). As already noted in Leroy et al. (2021), this mass
estimate agrees well with the masses derived from full spectral fitting of the
MUSE data, up to a roughly constant multiplicative offset, which makes the
MUSE masses larger by 0.09 dex.

Since much of the work on SFR calibration in the literature is based on the
Spitzer MIPS 24 $\rm\mu m$ band, we have compared the WISE W4 maps from z0MGS
with Spitzer MIPS 24 $\rm\mu m$ mosaics for a subset of 8 galaxies with
available Spitzer images from either the SINGS (Kennicutt et al., 2003) or LVL
(Dale et al., 2009) programs (NGC 0628, NGC 1512, NGC 1566, NGC 3351, NGC
3627, NGC 4254, NGC 4321, NGC 5068). The Spitzer MIPS 24 $\rm\mu m$ images
were convolved to the same common 15$\arcsec$ FWHM resolution. While the
Spitzer maps are substantially deeper than the WISE ones (by a factor of 6 on
average), the fluxes in regions which are detected in both maps ($S/N>$ 5 in
the WISE maps) are in excellent agreement. In particular, we find a small
multiplicative offset, which goes in the direction of making the MIPS flux
densities smaller by 11% (0.05 dex), in line with the previous comparisons
(Jarrett et al., 2011, 2013; Brown et al., 2014).111The WISE data used here
are calibrated to $\rm MJy\,sr^{-1}$ units following the WISE documentation
Sect. 4.4h, and Wright et al. (2010) assuming a $\propto\nu^{-2}$ reference
spectrum with no colour-corrections. Brown et al. (2014) show that the W4
colour correction, which depends on the spectral index of the source, can be
significant. However, we do not perform such a correction because because we
wish to directly compare with Leroy et al. (2019) and Salim et al. (2018), who
do not perform such a correction to their W4 fluxes, and because we we wish to
derive prescriptions that can be applied to the WISE data without such
preprocessing. In any case, the SED fitting necessary to determine it is
outside the scope of this work. In energy units, which are used in the next
section to define the SFR calibration,
$\log{(\nu_{\mathrm{MIPS}}F_{\nu,\mathrm{MIPS}})}=\log{(\nu_{\mathrm{W4}}F_{\nu,\mathrm{W4}})}-0.08$,
with an uncertainty due to galaxy-to-galaxy scatter of 0.03 dex.

Figure 1: Comparison between $\rm\Sigma_{SFR}/(M_{\odot}\ kpc^{-2}\ yr^{-1})$
estimates at 15$\arcsec$ resolution from extinction-corrected H$\alpha$
emission and UV+IR (FUV + W4) using the Leroy et al. (2019) prescription for
three example galaxies. The maps are shown on the original pixel size of
5$\aas@@fstack{\prime\prime}$5, even though the analysis in this paper is
based on sampling the data at $\sim$ 15$\arcsec$, and are scaled to the same
level for ease of comparison. The white contours in the UV+IR-based map
represent the boundary of the MUSE mosaics. The right column shows the
difference between the two SFR estimates in dex (the average uncertainty in
the measured ratio due to the error in the fluxes is 0.03 dex). The hatched
regions represent masked areas, namely regions contaminated by foreground
stars and the AGN in NGC 1566.

###  2.3 Data processing

In order to generate a catalog of distinct regions at our working resolution,
the H$\alpha$ and H$\beta$ 15$\arcsec$-resolution maps are resampled with
pixels of 14$\aas@@fstack{\prime\prime}$4 (comparable to the 15$\arcsec$
FWHM). The z0MGS data are reprojected (using the reproject python module) onto
the same world coordinate system and their pixel size is matched to that of
the MUSE data. A signal-to-noise cut of 5 is applied to the H$\alpha$ and
H$\beta$ lines, but this only affects a negligible fraction of our data set
($<3$% of the regions). This procedure leads to a sample of 1759 regions,
covering a total area of $\rm\sim 100\ arcmin^{2}$ or $\rm
10^{4}\leavevmode\nobreak\ kpc^{2}$ across our $19$ targets.

Stellar and SFR surface density measurements are corrected for the effect of
inclination by multiplication by cosine of the inclination angles compiled in
Leroy et al. 2021.

Figure 2: Dependence of the ratio between SFR measured from FUV+W4 (and the
Leroy et al. 2019 coefficient) and that measured from attenuation-corrected
H$\alpha$ on various physical properties for a sample of 1759 $\sim$ kpc-sized
regions taken from the 19 galaxies of the PHANGS-MUSE sample: a) sSFR, b) SFR
surface density (both measured from attenuation-corrected H$\alpha$), c)
$E(B-V)$ of the ionised gas measured from the Balmer decrement, d) stellar
mass surface density, e) deprojected galactocentric radius normalised to $\rm
R_{eff}$, f) light-weighted mean age of the stellar population, derived from
fits to the MUSE stellar continuum,. The grey points represent individual
$\sim$ 15$\arcsec$ regions, while the red hexagons and the dashed red lines
represent the median and scatter ($\rm 16^{th}-84^{th}$ percentiles) of the
distribution. In Panel c) we also show the result of calculating the SFR from
the FUV flux, without considering the IR term (blue lines). This demonstrates
that the inclusion of the IR term correctly reproduces the H$\alpha$-based SFR
for highly attenuated regions, but it overestimates the attenuation level for
those regions with low Balmer decrement. These trends are consistent with an
increasing contamination from old stellar population to the dust heating when
moving to lower SFR levels. Figure 3: Same as Fig. 2 but using H$\alpha$+W4
instead of FUV+W4. The coefficient $C_{\rm W4}^{\rm H\alpha}$ is taken from
Calzetti et al. (2007).

###  2.4 Formalism for computing SFR

The attenuation-corrected H$\alpha$ flux is obtained as

$L_{\rm H\alpha,corr}=L_{\rm H\alpha}10^{0.4\leavevmode\nobreak\ A_{\rm
H\alpha}}=L_{\rm H\alpha}10^{0.4\leavevmode\nobreak\ k_{\rm
H\alpha}\leavevmode\nobreak\ E(B-V)},$ (1)

where $k_{\rm H\alpha}$ is the value of the reddening curve at the H$\alpha$
wavelength. We quote all luminosities in energy units ($\rm erg\,s^{-1}$),
i.e. $L\equiv\nu L_{\nu}$, where $\nu$ is the (effective) frequency of the
band considered.

We measure $E(B-V)$ from the Balmer decrement (BD) assuming Case B
recombination, temperature $T=10^{4}$ K and density $n_{\rm
e}=10^{2}\,\mathrm{cm}^{-2}$, leading to $L_{\rm H\alpha,corr}/L_{\rm
H\beta,corr}=2.86$. This ratio is almost independent of density for the range
of densities relevant to H ii regions, and has only a small temperature
dependence (Osterbrock & Ferland, 2006). Under these assumptions

$E(B-V)_{\rm BD}=\frac{2.5}{k_{\rm H\beta}-k_{\rm
H\alpha}}\log_{10}\left[\frac{L_{\rm H\alpha}/L_{\rm H\beta}}{2.86}\right].$
(2)

On the scales probed by our observations dust will be mixed with the gas,
leading to the need to consider an effective attenuation law, and not just the
effect of dust extinction along the line of sight. We nonetheless use a
foreground screen attenuation model with total-to-selective extinction
$R_{V}=3.1$ equivalent to the Milky Way extinction law of O’Donnell (1994),
which represents a small modification to the one proposed by Cardelli et al.
(1989). There is some observational support for the use of such a simplified
foreground screen model as applied to the nebular lines in galaxies (Calzetti
et al., 1996; Wild et al., 2011). Moreover, the use of a Milky-Way-like
extinction law follows the conventions of the field, making our results
directly comparable with the literature (Calzetti et al., 2000; Kennicutt et
al., 2009; Boquien et al., 2019). We discuss the effect of making different
assumptions regarding the attenuation curve for the nebular component in Sect.
4.

The conversion factors $C$ for different monochromatic SFR estimators (e.g.
H$\alpha_{\mathrm{corr}}$) are defined by

$\mathrm{SFR}\leavevmode\nobreak\
[\mathrm{M_{\sun}yr^{-1}}]=C_{\mathrm{H\alpha}}L_{\mathrm{H\alpha,corr}}\leavevmode\nobreak\
[\rm erg\leavevmode\nobreak\ s^{-1}].$ (3)

For H$\alpha$ and FUV values of the parameter $C$ can be derived from stellar
population synthesis models by making some suitable simplifying assumption for
the star-formation history, the metallicity, and the initial mass function
(IMF). This is particularly important for the FUV, which traces longer
timescales ($\sim 100$ Myr) than H$\alpha$ ($\sim$ 5 Myr, Kennicutt & Evans
2012). We adopt the $C_{\rm H\alpha}$ value presented by Calzetti et al.
(2007), computed using Starburst99 Leitherer et al. (1999), a constant star-
formation history, age of 100 Myr, solar metallicity and a Kroupa (2001) IMF.
Using a Chabrier (2003) IMF changes this factor by less than 5%, while a
Salpeter (1955) IMF requires dividing by a factor of $\sim$ 0.66 (Madau &
Dickinson, 2014). For reference, this value is within $\approx 2\%$ of the
value of $C_{\rm H\alpha}$ suggested by Murphy et al. (2011) and endorsed by
Kennicutt & Evans (2012). For FUV emission, we adopt the value of $C_{\rm
FUV}$ recommended by Leroy et al. (2019), based on the SED fitting results of
Salim et al. (2018), using a Chabrier (2003) IMF. The $C_{\rm FUV}$ factor has
a larger dependence on the star formation history, and consequently the
position of galaxies in the SFR-$\rm M_{\star}$ plane (see Leroy et al. 2019,
their Fig. 25), than $C_{\rm H\alpha}$. The value adopted here is 0.07 dex
higher than that recommended by Kennicutt & Evans (2012).

For hybridised SFR estimators (e.g. W4 22 $\rm\mu m$ \+
H$\rm\alpha_{\mathrm{obs}}$) we define the conversion factors $C$ in a similar
fashion as

$\mathrm{SFR_{H\alpha+W4}}=C_{\mathrm{H\alpha}}L_{\mathrm{H\alpha}}+C_{\mathrm{W4}}^{\mathrm{H\alpha}}L_{\mathrm{W4}},$
(4)

where, by definition, $C_{\mathrm{H\alpha}}$ is taken to be the same value as
the monochromatic estimator (e.q. 3), so that the sum in Eq. 4 may be
interpreted as the combination of an ‘unobscured’ and an ‘obscured’ SFR term.
The conversion factor of the W4 $\rm 22\leavevmode\nobreak\ \mu m$ emission is
written as $C_{\mathrm{W4}}^{\mathrm{H\alpha}}$ to emphasize that it
multiplies the W4 $\rm 22\leavevmode\nobreak\ \mu m$ luminosity when
hybridised with H$\alpha$. In all cases the conversion factors $C$ have units
of $\rm M_{\sun}yr^{-1}/(erg\,s^{-1})$, which we omit for clarity from now on.
We adopt the same approach as Leroy et al. (2019) to construct the UV+IR
hybrid SFR calibration, given by

$\mathrm{SFR_{FUV+W4}}=C_{\mathrm{FUV}}L_{\mathrm{FUV}}+C_{\mathrm{W4}}^{\mathrm{FUV}}L_{\mathrm{W4}}.$
(5)

We express $C_{\mathrm{W4}}^{\mathrm{H\alpha}}$ in this form because our main
goal is to infer SFRs, but the drawback of this formalism is that it makes the
IR-coefficient dependent upon the assumed SFR conversion factor for H$\alpha$
($C_{\mathrm{H\alpha}}$). In fact, since by equating Equations 3 and 4,
$C_{\mathrm{W4}}^{\mathrm{H\alpha}}\propto C_{\mathrm{H\alpha}}$, our inferred
value of $C_{\mathrm{W4}}^{\mathrm{H\alpha}}$ can be directly rescaled to a
different assumed $C_{\mathrm{H\alpha}}$.

We summarise values for coefficients of monochromatic and hybridized SFR
estimators from the literature in Table 2. Where relevant, coefficients
derived using Spitzer MIPS 24 $\rm\mu m$ were revised downward by 0.08 dex to
account for the bandpass difference with WISE W4 (Sec. 2.2).

Coefficient $C$ | Band(s) | $\log_{10}(C)$ | reference
---|---|---|---
$\rm[M_{\sun}yr^{-1}/(erg\,s^{-1})]$ |  |  |
$C_{\rm FUV}$ | FUV | $-43.42$ | (1)
$C_{\rm W4}^{\rm FUV}$ | FUV+W4 |  $-$42.73 | (1)
$C_{\rm W4}^{\rm FUV}$ | FUV+W4 |  $-$42.91 | (2)⋆, K12
$C_{\rm H\alpha}$ | H$\alpha$ |  $-$41.26 | (3), K12
$C_{\rm W4}^{\rm H\alpha}$ | H$\alpha$+W4 |  $-$42.86 | (3)⋆
$C_{\rm W4}^{\rm H\alpha}$ | H$\alpha$+W4 |  $-$43.05 | (4)⋆, K12
Table 2: Summary of coefficients from luminosity to SFR, as defined in Eq. 3,
4 and 5, taken from the literature and used in this work. ⋆ implies that the
original calibration was derived with Spitzer MIPS and has been revised
downward by 0.08 dex for application with 22 $\rm\mu m$ WISE W4. References 1:
Leroy et al. (2019), 2: Hao et al. (2011), 3: Calzetti et al. (2007), 4:
Kennicutt et al. (2009), K12: Kennicutt & Evans (2012).

##  3 Results

###  3.1 Comparisons among results for existing prescriptions

In this section we focus on comparing the SFR obtained from the Balmer
decrement-corrected H$\alpha$ against the FUV+W4 prescription of Leroy et al.
(2019). In Fig. 1, we show SFR maps for three example galaxies from the
PHANGS–MUSE sample. NGC 5068 is a low-mass galaxy and the closest object in
our sample (15$\arcsec$ = 400 pc). NGC 628 is an archetypal grand-design
spiral galaxy, and finally NGC 1566 is a massive barred Seyfert 1 galaxy.

The comparison between H$\alpha$ and FUV+W4 SFR shows that deviations from
equality are spatially correlated with the intensity of star formation
activity. In particular, the FUV+W4 SFR is biased high in regions of low
$\Sigma_{\mathrm{SFR}}$, while the trend is inverted for regions of high
$\Sigma_{\mathrm{SFR}}$. Such a trend is evident in all three galaxies shown
in Fig. 1, despite clear differences in the $\Sigma_{\mathrm{SFR}}$
distributions, and is also seen in the other PHANGS–MUSE galaxies in our
sample. The AGN in NGC 1566 shows excess IR emission, but the region is masked
in further analysis as already discussed in Sect. 2.1.

####  3.1.1 Secondary dependencies

In this section we study how well the SFR estimated from FUV+W4 agrees with
the SFR estimated from attenuation-corrected H$\alpha$ as a function of
several physical properties (Fig. 2).

The ratio $\rm\log(SFR_{FUV+W4}/SFR_{H\alpha,corr})$ decreases as a function
of both $\Sigma_{\mathrm{SFR}}$ and $\rm sSFR=\Sigma_{SFR}/\Sigma_{\star}$. On
average, our data demonstrate that most regions have a larger $\rm
SFR_{FUV+W4}$ than $\rm SFR_{H\alpha},corr$, but the two estimates are in good
agreement at the highest sSFR and $\Sigma_{\mathrm{SFR}}$ values present in
our sample, $\rm log(sSFR/yr^{-1})\gtrsim-10$ and $\rm
log(\Sigma_{SFR}/M_{\odot}\leavevmode\nobreak\ yr^{-1}\leavevmode\nobreak\
kpc^{-2})>-2$.

In Panel c, we show the dependence of
$\rm\log(SFR_{FUV+W4}/SFR_{H\alpha},corr)$ on $E(B-V)$ measured from the
Balmer decrement. The vertical scale in this panel has been extended to show
in blue the median relation obtained by calculating the SFR using the FUV
alone, without any IR correction. Using FUV+IR, we obtain an overestimate of
the SFR at low $E(B-V)$, which is driven by an overestimate of the attenuation
from the IR term. In fact, at larger values of $E(B-V)$ the IR term provides a
correction to the FUV flux which brings the inferred SFR in agreement with the
H$\alpha$ SFR estimate.

No trend is evident between $\rm\log(SFR_{FUV+W4}/SFR_{H\alpha,corr})$ and
stellar mass surface density or deprojected galactocentric radius (Fig. 2,
panels d & e) . Considering the light-weighted age of the stellar population
(panel f), the $\rm\log(SFR_{FUV+W4}/SFR_{H\alpha,corr})$ ratio increases for
older ages, in agreement with the sSFR trend noted above.

We do not find any dependence on distance or on the physical scale
corresponding to our 15$\arcsec$ resolution ($\approx 400{-}1400$ pc). In
particular, the median trends are consistent within the scatter when
considering two sub-samples of galaxies divided according to the median
distance of the sample (15.85 Mpc). We have checked that the scatter within
individual galaxies is comparable to the sample-wide scatter, indicating small
galaxy-to-galaxy variations.

Trends associated with the H$\alpha$+W4 calibration using the Calzetti et al.
(2007) coefficient are similar and are presented in Fig. 3. The main
difference is the substantially smaller scatter, as expected since in this
case both SFR estimates rely on H$\alpha$ for the un-attenuated portion.

Figure 4: Comparison between $E(B-V)$ estimated via the Balmer Decrement and
the same quantity calculated via FUV+W4 (left) and H$\alpha$+W4 (right),
colour-coded by sSFR. In the left panel we show as a dashed line the ratio
between the differential reddening of gas and stars inferred by Calzetti et
al. (2000), $\rm E(B-V)_{stars}=0.44\leavevmode\nobreak\ E(B-V)_{gas}$. Solid
black lines in both panels represent the one-to-one line. In both panels
attenuation is overestimated in regions of low sSFR because of IR cirrus.

####  3.1.2 Comparison of the implied E(B-V)

By equating Equations 4 (or equivalently Equation 5 for FUV) and 1 , one can
write $E(B-V)$ as a function of the W4/H$\alpha$ or the W4/FUV ratio. In
calculating $E(B-V)$ one needs to assume both the hybridisation coefficients
(we use Table 2) and an attenuation law to determine $k_{\lambda}$. In
particular,

$E(B-V)=\frac{2.5}{k_{\lambda}}\log{\left(1+\frac{C_{W4}^{\lambda}L_{W4}}{C_{\lambda}L_{\lambda}}\right)},$
(6)

where $\lambda$ refers to either H$\alpha$ or the FUV band. As discussed in
Sect. 2.4, we assume a Milky Way-like foreground screen as parameterised by
O’Donnell (1994) for the H$\alpha$ line, and the Calzetti (2001) law for the
FUV.

In Fig. 4, we plot the resulting $E(B-V)$, inferred from FUV + W4 and
H$\alpha$ \+ W4, versus the $E(B-V)_{\mathrm{BD}}$ inferred from the Balmer
decrement for the $\sim$ 15$\arcsec$ regions, colour-coded by sSFR. On
kiloparsec scales, the attenuation of the stars in the FUV is expected to be
lower than that of the ionised gas because nebular emission is more closely
linked to the most recent star formation, and therefore to dust. Calzetti et
al. (2000) report the differential reddening between the stars and the gas to
be a factor of 0.44, which we show as a dashed line in the left panel of Fig.
4 (equivalent to the relation $A_{\rm FUV}=1.78\leavevmode\nobreak\ A_{\rm
H\alpha}$). More recent work on nearby galaxies based on MaNGA (Greener et
al., 2020) supports a somewhat lower ratio, between 0.25 and 0.44. The subset
of our regions with the largest sSFR lies close to the latter value, with an
average ratio of $E(B-V)_{\mathrm{FUV+W4}}/E(B-V)_{\mathrm{BD}}=0.52$ for $\rm
log(sSFR/yr^{-1})>-10$. On the other hand, regions of low sSFR, with $\rm
log(sSFR/yr^{-1})<-10.8$, show
$E(B-V)_{\mathrm{FUV+W4}}>E(B-V)_{\mathrm{BD}}$.

The right panel of Fig. 4 compares $E(B-V)_{\mathrm{H\alpha+W4}}$ with
$E(B-V)_{\mathrm{BD}}$ and confirms our findings. For regions with $\rm
log(sSFR/yr^{-1})>-10$, the two estimates of the $E(B-V)$ of the nebular
emission agree to within 10%. At low sSFR, however,
$E(B-V)_{\mathrm{H\alpha+W4}}$ is significantly larger, indicating a likely
contribution from IR cirrus (see Sect. 4).

###  3.2 A multi-wavelength calibration scheme for SFR at kiloparsec scales

Figure 5: The hybridisation coefficients of W4 with either FUV ($\rm
C^{FUV}_{W4}$, Panels a, b, c) or H$\alpha$ ($\rm C^{H\alpha}_{W4}$, Panels d,
e) as a function of sSFR (left), $L_{\rm H\alpha}/L_{\rm W1}$ (middle) and
$L_{\rm FUV}/L_{\rm W1}$ (right). These coefficients are calibrated to match
the SFR obtained from Balmer-decrement-corrected H$\alpha$ for individual
15$\arcsec$ ($\sim$ kpc-scale) regions (grey dots). Red hexagons and shaded
areas show the median trends and scatter. Median relations are fitted with a
broken power law (equation 7) and the best-fit model is shown as a dashed blue
line. The scatter $\sigma$ of the data with respect to the best-fit is
presented in the bottom-right. Values of $C_{\rm W4}^{\rm FUV}$ and $C_{\rm
W4}^{\rm H\alpha}$ from the literature are shown as black dashed horizontal
lines. Our data for kpc-scale regions agree well with the literature values at
high sSFR, but show a systematic deviation for lower sSFR which is captured
well by the power-law model.

In order to bring the FUV+W4 and H$\alpha$+W4 SFR into agreement with the
star-formation rates obtained from attenuation-corrected H$\alpha$, we
calibrate the hybridisation coefficients $C_{\rm W4}^{\rm band}$ as a function
of physical quantities that trace the ratio of young to old stars (e.g.
Boquien et al., 2016). From Sect. 3.1 we determined that sSFR,
$\Sigma_{\mathrm{SFR}}$, and stellar population light-weighted age show well-
defined trends, all potentially attributable to IR cirrus. We focus on sSFR,
as well as H$\alpha$/W1 and FUV/W1 as empirical proxies for sSFR, because they
are intensive quantities (unlike $\Sigma_{\mathrm{SFR}}$), less model-
dependent than the light-weighted age derived from full spectral fitting, and
are widely available for nearby galaxies thanks to the combination of GALEX
and WISE.

Assuming the attenuation correction from the Balmer decrement, we use Equation
4 and 3 to determine $C_{\rm W4}^{\rm FUV}$ and $C_{\rm W4}^{\rm H\alpha}$,
and plot these coefficients as a function of sSFR (Fig. 5, a & d). Both
coefficients show a strong positive correlation with sSFR (Spearman $r$ =
0.71–0.72). We fit the median relation with a broken power law of the form

$\log{C_{W4}^{\rm band}}=\cases{a}_{0}+a_{1}\,\log{Q}&Q<Q_{\rm max},\\\
\log{C_{\rm max}}Q>Q_{\rm max},\\\ {}$ (7)

where $Q$ refers to the quantity of interest (e.g. sSFR) and
$a_{0}=\log{C_{\rm max}}-a_{1}\log{Q_{\rm max}}$ for the function to be
continuous at $\log{Q_{\rm max}}$. The best-fit broken power law relations are
shown in Fig. 5 as blue dashed lines and the values of the best-fit parameters
($C_{\rm max},Q_{\rm max}$ and $a_{1}$) are given in Table 3. These relations
should not be extrapolated past the range populated by our data, we therefore
also show in Table 3 the 5th percentile of the distribution of the quantity of
interest, log($Q_{\rm min}$). We do not recommend using the best fits for
values lower than log($Q_{\rm min}$).

A broken power law of this form is chosen in order to allow for a constant
coefficient at high sSFR. The choice reflects the expectation that the
contribution to emission at 22$\rm\leavevmode\nobreak\ \mu m$ from dust heated
by old stellar populations becomes negligible in regions dominated by young
stars, and therefore the hybridisation coefficients should approach a constant
value. In fact, such a flattening is evident in Fig. 5 (Panels a & d) for both
$C_{\rm W4}^{\rm FUV}$ and $C_{\rm W4}^{\rm H\alpha}$ and $\log{\rm
sSFR/yr^{-1}}\gtrsim-9.9$. The best-fit value of the coefficients for large
sSFR are $C_{\rm W4}^{\rm FUV}=-42.68\pm 0.05$ and $C_{\rm W4}^{\rm
H\alpha}=-42.88\pm 0.04$. We consider these values to be our best estimates
for the hybridisation coefficients in star-forming regions. The value of
$\log{C_{\rm max}}$ for $C_{\rm W4}^{\rm FUV}$ agrees within the error with
the estimate by Leroy et al. (2019), based on galaxy-integrated fluxes
($C_{\rm W4}^{\rm FUV}=-42.73$). Regarding the $C_{\rm W4}^{\rm H\alpha}$
coefficient, our inferred value of $\log{C_{\rm max}}$ is virtually identical
to the value preferred by Calzetti et al. (2007) ($C_{\rm W4}^{\rm
H\alpha}=-42.87$, based on star-forming regions) and higher then that of
Kennicutt et al. (2009) ($C_{\rm W4}^{\rm H\alpha}=-43.06$, based on galaxy-
integrated photometry).

The parametrisation of $\log{C_{\rm W4}^{\rm band}}$ in terms of sSFR is
instructive, but cannot be used directly to infer the SFR. If a stellar mass
measurement is available, however, an iterative approach can be used to
determine the SFR. Nonetheless, we also investigate alternative
parametrisations in terms of band ratios that can be used as proxies for sSFR.
In particular, we consider the luminosity ratios of $\rm H\alpha/W1$ and $\rm
FUV/W1$ (Fig. 5 middle and right columns). We find that a broken power law of
the form of Equation 7 is a good description for the relation between $C_{\rm
W4}^{\rm FUV}$ and $C_{\rm W4}^{\rm H\alpha}$ and the $\rm H\alpha/W1$
luminosity ratio. The scatter between the data and the best-fit models (0.25
dex in the case of $C_{\rm W4}^{\rm FUV}$ and 0.20 dex for $C_{\rm W4}^{\rm
H\alpha}$) is comparable to that obtained for the sSFR parametrisation
discussed above. Therefore we consider these fits our recommended approach for
calculating SFR when H$\alpha$ fluxes are available.

The use of $\rm FUV/W1$ instead of $\rm H\alpha/W1$ leads to larger scatter,
and the resulting median relation between the hybridisation coefficients and
$\rm FUV/W1$ is less well described by a broken power law. We therefore only
recommend the use of $\rm FUV/W1$ when H$\alpha$ data are not available (i.e.
only for $C_{\rm W4}^{\rm FUV}$ coefficient). The scatter in the
parametrisation in this case goes up to 0.3 dex (Fig. 5, Panel c).

Band | combined with | as a function of | $a_{1}$ | $\log{Q_{\rm max}}$ | $\log{C_{\rm max}}$ | $\log{Q_{\rm min}}$ | $\sigma({\rm logC})$
---|---|---|---|---|---|---|---
FUV | W4 | sSFR | $0.46\pm 0.14$ | $-9.87\pm 0.19$ | $-42.68\pm 0.04$ | $-11.00$ | 0.24
FUV | W4 | $L_{\rm H\alpha}/L_{\rm W1}$ | $0.52\pm 0.15$ | $-1.48\pm 0.16$ | $-42.66\pm 0.06$ | $-2.45$ | 0.25
FUV | W4 | $L_{\rm FUV}/L_{\rm W1}$ | $0.23\pm 0.14$ | $0.60\pm 0.68$ | $-42.73\pm 0.12$ | $-0.64$ | 0.30
H$\alpha$ | W4 | sSFR | $0.40\pm 0.12$ | $-9.91\pm 0.18$ | $-42.88\pm 0.04$ | $-11.00$ | 0.18
H$\alpha$ | W4 | $L_{\rm H\alpha}/L_{\rm W1}$ | $0.45\pm 0.16$ | $-1.52\pm 0.19$ | $-42.88\pm 0.04$ | $-2.45$ | 0.20
Table 3: Coefficients for predicting the hybridization coefficients $C_{\rm
W4}^{\rm FUV}$ and $C_{\rm W4}^{\rm H\alpha}$ according to Equation 7 as a
function of quantity $Q$. $a_{1}$ corresponds to the slope at low $Q$,
$\log{C_{\rm max}}$ to the value of the coefficient for $Q>Q_{\rm max}$.
$\log{Q_{\rm min}}$ is the 5th percentile of the distribution of $\log{Q}$ in
the PHANGS data. We do not recommend extrapolation below this value.
$\sigma({\rm logC})$ refers to the scatter between the data and the best-fit
broken power law model.

Finally, we check the effect of our suggested calibrations on the inferred
$E(B-V)$. In particular we focus on the case where W1, W4 and H$\alpha$ data
are available, since it is relevant to the galaxies observed by the
PHANGS-H$\alpha$ narrow band survey. We calculate $E(B-V)$ via Equation 6,
using the calibration of $C_{\rm W4}^{\rm H\alpha}$ as a function of
H$\alpha$/W1 given by Equation 7. Comparing the $E(B-V)$ computed in this
fashion with those obtained from the Balmer decrement, we find that they agree
very well on average (median offset of ¡ 0.01 mag) with a scatter of 0.08 mag.
This represents a substantial improvement over the situation shown in Fig. 4,
where the $E(B-V)$ obtained using a constant H$\alpha$+W4 was on average 0.14
mag larger than the one obtained from the Balmer decrement. An evaluation of
the performance of our calibrations is delayed to Sect. 4.1.

###  3.3 Validity of calibrations as a function of physical scale

In this section we study the validity of our proposed calibrations as a
function of physical scale. In particular, we address the question of whether
the $\Sigma_{\mathrm{SFR}}$ estimates obtained at kpc scales are consistent
with those measured at 100 pc scales accessible in MUSE surveys of nearby
galaxies such as PHANGS-MUSE (Emsellem et al., 2022; Pessa et al., 2021) or
the MUSE Atlas of Discs (Erroz-Ferrer et al., 2019). We then explore whether
the calibrations can be applied to spatially-integrated (galaxy-wide) scales,
in order to connect to the SFR measurements obtained by large spectroscopic
surveys in the nearby Universe.

####  3.3.1 Dust extinction from kpc to 100 pc scales

The mixing of regions with different physical conditions within the same
resolution element leads to an underestimate of the overall attenuation
correction, since the more attenuated regions contribute less to the
integrated light in the larger aperture. Vale Asari et al. (2020) showed
analytically that, assuming the attenuation law does not vary across a galaxy,
a low-resolution measurement of the attenuation-corrected H$\alpha$ luminosity
always underestimates the result obtained at high spatial resolution. The
magnitude of this underestimation depends in a complex fashion on the spatial
distribution of the dust emission and the strength of the correlation between
dust attenuation and H$\alpha$ surface brightness. Vale Asari et al. (2020)
used MaNGA data to demonstrate that this effect is negligible in practice (3%
on average) when comparing measurements taken on kpc-scale resolution elements
to entire galaxies, but they cautioned against extending this result to
smaller scales.

Figure 6: Mean ratio of the attenuation-corrected H$\alpha$ luminosity
computed in low-resolution apertures of varying size to the luminosity derived
at the highest spatial scales accessible with the PHANGS-MUSE data ($\sim
1\arcsec$), as a function of the low-resolution aperture size. Trends are
shown for each of the 19 galaxies in the PHANGS-MUSE sample, colour-coded by
total stellar mass. The data for each galaxy is resampled on apertures of 5,
10, 15, 20, and 30$\arcsec$. The alternative x-axis on the top of the plot
shows the equivalent physical size in kpc at the median distance of the
sample. At 15$\arcsec$ resolution, the resolution of the WISE W4 data, the
median difference is is 10% ($\rm<L_{low\ res}/L_{high\ res}>=0.92$).

Here we reconsider the effect of spatial sampling on the attenuation
corrections based on the Balmer decrement, focusing on the spatial scales of
interest in this work (1–15$\arcsec$, or from $\sim$ 70 to 1100 pc at the
average distance of our targets). To do so, we resample the MUSE H$\alpha$ and
H$\beta$ copt maps with pixels of 5$\arcsec$, 10$\arcsec$, 15$\arcsec$ (the
sampling adopted in the previous sections to match WISE W4), 20$\arcsec$, and
30$\arcsec$. For simplicity we do not convolve the data before resampling, but
the effect of the convolution process does not significantly change the
outcome of our results. Using these resampled maps we compute the Balmer
decrement and the attenuation-corrected H$\alpha$ luminosity ($L_{\rm low\
res}$) in each coarser pixel. We then compare this estimate with that obtained
by correcting individual pixels in the original high-resolution maps and
summing the attenuation-corrected H$\alpha$ flux ($L_{\rm high\ res}$). We
follow this procedure for each galaxy in the sample and compute the average
ratio of $L_{\rm low\ res}/L_{\rm high\ res}$ as a function of aperture size.

Fig. 6 demonstrates that the ratio $L_{\rm low\ res}/L_{\rm high\ res}$ varies
across the galaxies in our sample, presumably due to differences in the
overall dust morphology. At 15$\arcsec$ resolution the underestimation in the
attenuation correction with respect to the estimate obtained at the copt
(arcsec-resolution) data is on average 10%, with a 0.1 dex scatter within
different regions of individual galaxies. This is a small source of
uncertainty in the overall uncertainty budget of the attenuation-corrected
H$\alpha$.

We checked for a correlation between the slope of the $L_{\rm low\ res}/L_{\rm
high\ res}$ versus aperture size and other quantities of galaxies. In
particular, we considered the effect of inclination, distance, $\rm
M_{\star}$, SFR and sSFR. The strongest correlation manifests itself with
stellar mass (Spearman $r=-0.6$), indicating that low-mass galaxies tend to
have flatter trends in this space (as already evident in the colour-coding in
Fig. 6). A mild inverse correlation is also seen with inclination (Spearman
$r=-0.32$), where more inclined galaxies have flatter slopes in this space. We
have checked the effect of distance on these correlations by rerunning the
analysis using apertures of fixed physical size in kpc. We find that the trend
with mass persists (Spearman $r=-0.51$), but the trend with inclination is
considerably weaker (Spearman $r=-0.09$).

The result of this exercise is dependent on our approach to the low-S/N area
in the arcsec-resolution copt maps. In particular, while H$\alpha$ is detected
in nearly 100% of the pixels of our maps, H$\beta$ is only detected with S/N
$>$ 3 in 85% of the pixels in each map on average (with some galaxies having
up to 50% of their pixels undetected in H$\beta$ at the 3 $\sigma$ level).
These spaxels contain on average 3% of the total H$\alpha$ flux, but in some
galaxies, this fraction can go up to 10%. It is important, therefore, to
exclude the possibility that the undetected H$\beta$ hides large amounts of
attenuated star formation.

In order to estimate the importance of dust attenuation in regions where
H$\beta$ is undetected, we make use of the adaptive binning procedure
described in Belfiore et al. (2022) to obtain detection of H$\beta$ in the low
surface brightness regions via binning. This binning procedure was optimised
for recovery of emission line fluxes in the diffuse ionised gas, and therefore
generates larger bins in regions of faint H$\alpha$ emission (e.g. inter-arm
regions). We find that low-surface-brightness pixels lie on the extrapolation
of the power law relation between H$\alpha$ and $\rm E(B-V)$ observed at high
surface brightness. At H$\beta$ surface brightness corresponding to a 3
$\sigma$ detection limit in the individual spaxels of our copt maps
($\rm\log{L_{H\beta}/erg\ s^{-1}kpc^{-2}}=37.5$), the median $\rm E(B-V)$ is
$\sim$ 0.1 mag ($-$0.1/+0.15). Given this modest level of average attenuation,
the relative contribution of H$\beta$ undetected regions to the attenuation-
corrected H$\alpha$ would remain small, and we therefore treat spaxels with
S/N in H$\beta$ less than 3 as having zero attenuation for the purposes of our
test. We test the effect of assigning to all the H$\beta$ undetected spaxel an
$\rm E(B-V)=0.1$ mag and the results do not change substantially.

We conclude that attenuation corrections calculated via H$\alpha$ \+ W4 (Sec.
3.2) on kpc scales are in agreement with those derived from arcsec-resolution
($\sim$100 pc physical resolution) Balmer decrement maps.

####  3.3.2 SFR calibration for whole galaxies from integrated UV, IR and
H$\alpha$ fluxes

We test the validity of our approach for whole galaxies using integrated
fluxes for a sample of galaxies from the MaNGA IFS survey. MaNGA is the
largest optical IFS survey of the local Universe ($0.01<z<0.15$), offering
measurements of the H$\alpha$ and H$\beta$ fluxes out to at least 1.5 $r$-band
effective radii. Aperture corrections in calculating the SFR are negligible,
as demonstrated by Belfiore et al. (2018). In order to test our hybrid SFR
estimator on MaNGA data we cross-match the MaNGA catalog from the SDSS Data
Release 17 (Abdurro’uf et al., 2022), and in particular the summary DAPALL
catalog file generated by the MaNGA Data Analysis Pipeline (Westfall et al.,
2019; Belfiore et al., 2019), with the integrated GALEX and WISE photometry
derived by Salim et al. (2018) as part of
GSWLC2222https://salims.pages.iu.edu/gswlc/ (GALEX-SDSS-WISE legacy catalog).
We consider the version of the GSWLC2 catalog which contains the deepest
available GALEX data, denoted as GSWLC2-X2. The MaNGA DR17 catalog contains
10248 galaxies, 86% of which have a match in GSLWC-X2. We further restrict the
sample to galaxies that are star forming according to the Kewley et al. (2001)
criteria in the $\rm[SII]\lambda 6717,31/H\alpha$ versus $\rm[OIII]\lambda
5007/H\beta$ Baldwin-Phillips-Terlevich (Baldwin et al., 1981; Phillips et
al., 1986) diagnostic diagram, have S/N$>3$ in GALEX FUV, WISE W1 and W4 bands
and have a semi-axis ratio $b/a>0.4$ (inclination $\sim 66^{\circ}$), to
exclude highly inclined systems. These cuts lead to a final sample of 1955
galaxies. We also tested the criterion of Stern et al. (2012) for selecting
AGN using the WISE bands, and find that this leads to the exclusion of only 13
additional galaxies. The results do not depend on whether this additional cut
is performed or not.

We first compare the SFR derived by Salim et al. (2018) with those obtained
using the attenuation-corrected H$\alpha$ data from MaNGA. Salim et al. (2018)
compute SFRs using the CIGALE SED-fitting code applied on UV-optical (GALEX-
SDSS) data, and constraining the total IR luminosity via the $\rm
22\leavevmode\nobreak\ \mu m$ WISE 4 flux (or 12 $\rm\mu m$ WISE 3, if the
object is not detected in WISE 4). Such an approach allows Salim et al. (2018)
to also fit for the slope of the attenuation curve without having access to
data in the far-IR. The SFR estimates from Salim et al. (2018) agree well with
those we obtain from attenuation-corrected H$\alpha$, with a relative scatter
of 0.16 dex and an offset of 0.09 dex. The CIGALE-derived SFR estimates are
larger than those obtained from attenuation-corrected H$\alpha$ and the offset
is independent of the sSFR. The reason for the discrepancy is unclear, but may
include differences in the treatment of the attenuation law.

To test the applicability of our derived hybrid calibrations, we repeat the
analysis in Sect. 3.2 using the MaNGA/GSWLC data. In Fig. 7 we show the
dependencies of $C_{\rm W4}^{\rm FUV}$ and $C_{\rm W4}^{\rm H\alpha}$ on sSFR,
$\rm L_{H\alpha}/L_{W1}$ and $\rm L_{FUV}/L_{W1}$ for the MaNGA data (grey
points). All the physical quantities involved are computed in the same way as
for the PHANGS/z0MGS data to allow for a direct comparison. We find that the
trends present in the MaNGA data go in the same direction as those observed in
PHANGS: galaxies with lower sSFRs show lower values of the $C$ coefficients.
At high sSFR, however, the MaNGA data tend to lie below the best-fit model
from PHANGS (shown in blue in Fig. 7). This discrepancy is more evident for
the H$\alpha$+IR calibration than for the FUV+IR one.

The results from MaNGA are consistent with the value given by Kennicutt et al.
(2009), who also used integrated measurements of galaxies for their
calibration, but lower than the best estimate from Calzetti et al. (2007), who
used data on smaller spatial scales. Depending on a proxy for the sSFR,
Boquien et al. (2016) find an amplitude of the variation of $\rm
C_{FUV}^{MIPS24}$ of a factor 2–3 on kpc-sized regions in nearby galaxies, a
value that is very similar to what is found for integrated MaNGA galaxies
here. Leroy et al. (2019) find a similar sSFR-dependent trend for $C_{\rm
W4}^{\rm FUV}$ using the SFR from Salim et al. (2018). In Fig. 7 we show the
median sSFR-dependent trend obtained by Leroy et al. (2019) (from their Table
5), scaled by the 0.09 dex offset discussed above to bring the Salim et al.
(2018) and H$\alpha$ attenuation-corrected SFR in agreement. After performing
this correction we are in excellent agreement with Leroy et al. (2019).

The trends in Fig. 7 show that the contribution of old stellar populations to
the heating of the dust is significant even at high sSFR because of the
unavoidable mixing of emission from a variety of regions (e.g. arm and
interarm) when considering galaxies as a whole. This mixing implies a lower
$C$ coefficient than that measured for kpc-scale regions of high sSFR. The
slope of the trend at lower sSFR is also shallower for MaNGA than for PHANGS,
which can also be qualitatively explained by an averaging effect. Finally, we
show as blue diamonds the results obtained on integrated scales for the 19
galaxies in the PHANGS-MUSE sample. In particular, for each galaxy we
integrate the relevant fluxes within the MUSE mosaic coverage, taking the
masks into account. We find that the PHANGS sample agrees within the scatter
with the general population of local galaxies from MaNGA.

The MaNGA sample leads to relations that show a larger scatter than for PHANGS
regions, highlighting the wider range of conditions found in the overall
galaxy population. The trends do not change if we make a more aggressive
inclination cut, e.g. $b/a>0.6$ ($i<53^{\circ}$), although a stricter cut
decreases the number of systems with low FUV/W1 ratios. In fact, we find that
the larger sampling of low FUV/W1 values in MaNGA with respect to PHANGS is
primarily due to inclination effects. Because of the larger scatter, we do not
provide new fits to the median relations based on the MaNGA data. We evaluate
the effect on the estimated SFR of using the PHANGS-based corrections on the
MaNGA data in the next section.

Figure 7: The hybridization coefficients of W4 with either FUV (Panels a, b, c) or H$\alpha$ (Panels d, e) as a function of sSFR (left), $L_{\rm H\alpha}/L_{\rm W1}$ (middle) and $L_{\rm FUV}/L_{\rm W1}$ (right) for the MaNGA sample (using galaxy-integrated fluxes). Best-fit relations obtained for the PHANGS 15$\arcsec$ data (Fig. 5) are shown in blue. Blue diamonds show the results obtained considering the integrated fluxes from the 19 PHANGS galaxies. Values of $C_{\rm W4}^{\rm FUV}$ and $C_{\rm W4}^{\rm H\alpha}$ from the literature are shown as black dashed horizontal lines and labelled in the figure (see text for additional detail). The scatter $\sigma$ of the data with respect to the best-fit is presented in the bottom-right. The MaNGA data show similar trends as the resolved regions in PHANGS galaxies, even though in general it displays flatter slopes and does not reach the same values of $C$ at the highest sSFR.  band | combined with |  $C_{W4}$ dependent on | recommended | reference | PHANGS | MaNGA
---|---|---|---|---|---|---
|  |  |  |  | median | scatter | median | scatter
|  |  |  |  | offset [dex] | [dex] | offset [dex] | [dex]
FUV | W4 | constant | no | (1) | 0.08 | 0.17 |  0.12 | 0.25
FUV | W4 | $\log(\rm FUV/W1)$ | yes | this work | 0.003 | 0.16 |  $-$0.007 | 0.22
FUV | W4 | $\log(\rm H\alpha/W1)$ | no | this work | 0.003 | 0.13 |  $-$0.02 | 0.20
H$\alpha$ | W4 | constant | no | (2) or (3) | 0.07 | 0.12 |  0.003 | 0.18
H$\alpha$ | W4 | $\log(\rm H\alpha/W1)$ | yes | this work |  $-$0.003 | 0.08 |  $-$0.002 | 0.14
Table 4: Quality assessment of the SFR recipes discussed in this work in terms
of their median offset and scatter with respect to H$\alpha$ attenuation-
corrected SFR. Literature recipes with constant coefficients are also shown
for comparison. In particular, for the case of H$\alpha$+W4, we use the value
from (2) for PHANGS and the value from (3) for MaNGA. Notes: (1) Leroy et al.
(2019), (2) Calzetti et al. (2007), (3) Kennicutt et al. (2009)

##  4 Discussion

###  4.1 Recommendations on estimating SFR from hybrid indicators

We evaluate the success of our kpc-scale calibrations based on PHANGS data
(Eq. 7) by comparing the median offset and scatter between the hybrid
estimators and the SFR inferred from attenuation-corrected H$\alpha$. In Fig.
8 we show the median log ratio and the scatter of the hybrid SFR estimators
considered in Sect. 3.2 as a function of sSFR for both kpc-scale regions in
PHANGS and for galaxy-integrated fluxes from MaNGA. We also consider the usage
of constant $C$ coefficients taken from the literature. For the case of the
H$\alpha$+W4 calibration, we use the Calzetti et al. (2007) value for PHANGS
and the the Kennicutt et al. (2009) value for MaNGA, in order to match each
coefficient to the range of spatial scales it was computed for. In fact, our
results confirm both coefficients are accurate and their difference is due to
the different level of cirrus contamination affecting typical galaxies as a
function of spatial scale. The median value of the offset and scatter for each
data set and calibration strategy are provided in Table 4.

For both calibrations and both data sets, the use of constant coefficients
(red points in Fig. 8) leads to an overestimation of the SFR using the hybrid
calibrations at low sSFR levels. We therefore recommend against the use of
constant coefficients, especially when investigators are interested in
comparing regions or galaxies of different sSFRs. We find, however, that the
coefficient are roughly constant for $\rm log(sSFR/yr^{-1})>-9.9$.

The H$\alpha$ \+ W4 calibration using a $C$ coefficient scaling with
H$\alpha$/W1 (green points in bottom panel, Fig. 8) leads to the least scatter
in both PHANGS and MaNGA (0.08 and 0.14 dex, respectively). This calibration
performs extremely well for PHANGS regions because of its negligible residual
dependence on sSFR. In case of the integrated galaxies from MaNGA, it shows a
slight sSFR-dependent tilt, underestimating the SFR at low sSFR and
overestimating at high sSFR. This tilt is the result of a flatter slope in the
$C_{\rm W4}^{H\alpha}$ versus H$\alpha$/W1 plane shown by the MaNGA data with
respect to PHANGS shown in Panel e of Fig. 7.

When H$\alpha$ data are not available, we recommend to use the FUV+W4
calibration with a $C$ coefficient depending on FUV/W1 (blue points in Fig.
8). This calibration also successfully removes the mean offset between the
hybrid SFR and the H$\alpha$-based SFR in both MaNGA and PHANGS. Despite being
calibrated on the PHANGS data, it performs better on MaNGA, where it more
successfully removes the residual sSFR dependence.

In summary, our best-effort SFR indicators require the use of WISE W1 band in
addition to the canonical bands used for hybridization: FUV (or H$\alpha$) and
W4. When compared to Balmer-decrement corrected H$\alpha$ SFR, our
calibrations using WISE W4 lead to a scatter smaller than 0.2 dex. Since WISE
is an all-sky survey, our prescriptions can be readily applied to a large
sample of local galaxies (e.g. z0MGS, Leroy et al. 2019).

The use of closely related bands, including IRAC1, J, H, K band instead of W1
and MIPS24 or JWST MIRI F2100W (21 $\rm\mu m$) band instead of W4 is expected
to lead to comparable results if appropriate conversion factors are applied.
In anticipation of a comparison with upcoming results from JWST we note the
mean ratio $\log{(L_{\rm F2100W}/L_{\rm W4})}=-0.10$ obtained by Leroy et al.
(submitted) for four galaxies in the PHANGS-MUSE sample (NGC 0628, NGC 1365,
NGC 7496, IC 5332). The use of the WISE W3 band (centered at $\rm
11.6\leavevmode\nobreak\ \mu m$) would lead to additional systematic
uncertainty in the SFR estimation because this band includes strong features
from polycyclic aromatic hydrocarbon (PAH). The flux ratio between PAH-tracing
bands and 24 $\rm\mu m$ emission is not constant, and may depend
systematically on metallicity and the intensity of the radiation field
(Calzetti et al., 2007; Lee et al., 2013). We nonetheless calculate the ratio
between W3 and W4 fluxes in our data to allow a first-order correction and
obtain a mean ratio of $\log{(L_{\rm W3}/L_{\rm W4})}=0.08$ for our PHANGS
kpc-scale regions. As already shown in Leroy et al. (2019), however, the
$C_{\rm W3}^{\rm FUV}$ factor shows significant changes across the
$M_{\star}$-SFR plane (their Fig. 21), therefore pointing to the importance of
additional physics in addition to the cirrus contamination we aim to correct
for in this work.

Figure 8: Comparison of the SFR estimated from hybrid tracers (FUV+W4 and
H$\alpha$+W4) and from attenuation-corrected H$\alpha$ as a function of sSFR.
We consider different calibrations, including the use of constant $C$
coefficients (red points, Leroy et al. 2019, L19 for the FUV+W4 and Calzetti
et al. 2007, C07, and Kennicutt et al. 2009, K09, for H$\alpha$+W4, see Table
2), and PHANGS-based calibrations with $C$(FUV/W1) (blue points) and
$C$(H$\alpha$/W1) (green points) derived in this work. Left panels show the
results for PHANGS (kpc-scale) 15$\arcsec$ regions, while the right panel show
the result for MaNGA galaxies. Dashed lines show the 1 $\sigma$ scatter while
hexagons the median relations. The mean offset and scatter for each
calibration, for both PHANGS and MaNGA, is presented in Table 4.

###  4.2 Caveats and future prospects

Our calibrations derived from the small PHANGS galaxy sample are found to
generalise well to the larger population of main sequence galaxies probed on
integrated scales by the MaNGA survey. However, our sample does not contain
dwarf galaxies or ultra-luminous infrared galaxies. PHANGS-MUSE data is also
limited to the inner star-forming disc and has limited coverage of the Hi-
dominated outer parts ($r_{\rm gal}\gtrsim 0.5\leavevmode\nobreak\ r_{25}$) of
disk galaxies. In particular, PHANGS-MUSE does not cover extended UV discs
(Thilker et al., 2007) outside the optical radius at all. Tomicic et al.
(2019), for example, study regions in the Hi-dominated disc of M31 with IFS
and find conversion factors for the IR term (both $\rm C_{\rm W4}^{\rm
H\alpha}$ and $C_{\rm W4}^{\rm FUV}$) a factor of 5-8 times larger than the
ones we obtain here. They attribute this difference to the high inclination of
M31 ($\sim 77^{\circ}$, Walterbos & Kennicutt 1988) and the flaring of the Hi
disc. Within the limits of our data, we do not find any hint of such a radial
trend, but it is reasonable to expect an increased mid-IR cirrus contribution
in the presence of a large diffuse ISM component. We highlight future
extension of this analysis to the Hi-dominated outer parts of galaxies as an
important future direction.

The impact of the old stellar populations on the calibration of hybrid SFR
calibrators on resolved scale has previously been discussed by Boquien et al.
(2016). They studied a sample of eight nearby galaxies and used SED fitting
based on UV-to-FIR imaging to derive the attenuation in the FUV, and therefore
attenuation-corrected SFR. In agreement with our work, they find a trend
between $\rm C_{W4}^{FUV}$ and sSFR and other band ratios used as sSFR proxies
(including FUV - 3.6 $\mu m$). In particular, they find a flattening in the
relation between $\rm C_{W4}^{FUV}$ and sSFR at $\rm sSFR\sim-9.75$,
comparable to what is found here. However, their conversion factors are
systematically higher than the ones derived here by $\sim$0.3 dex, perhaps due
to differing assumptions on the attenuation curve as they allowed for curves
steeper than a bona fide starburst curve. This shows the dependence of hybrid
estimators on indirect but necessary assumptions on the attenuation curve as
we discuss below.

The calibrations presented in this paper are consistent with the large body of
literature presenting SFR calibrations based, directly or indirectly, on
attenuation-corrected recombination line fluxes (Calzetti et al., 2007;
Kennicutt et al., 2009; Hao et al., 2011; Catalán-Torrecilla et al., 2015).
They represent an update, extending the validity of the calibrations to
smaller physical scales with respect to the recipes presented in the review of
Kennicutt & Evans (2012). Our calibrations are also in agreement with the SFRs
measured based on radio continuum by Murphy et al. (2012) for a sample of
nuclear and extra-nuclear star-forming regions.

The calibrations presented in this work are subject to the systematic
uncertainties associated with attenuation corrections via the Balmer decrement
and with using H$\alpha$ as a tracer of star formation. Regarding the dust
correction, uncertainties are related to the temperature (and density)
dependence of the intrinsic H$\alpha$/H$\beta$ ratio and to the use of a
Milky-Way-like extinction law to model attenuation of the nebular lines. The
intrinsic Balmer decrement changes only weakly with temperature, increasing
from 2.76 for 5000 K to 3.05 at 20000 K, therefore causing changes of less
than 0.02 mag in the derived $E(B-V)$. The attenuation law (and in particular
the assumed value of $R_{V}$) for the nebular component represents a more
significant systematic uncertainty. Assuming a Calzetti et al. (2000)
attenuation law for the nebular continuum does not substantially change our
conclusions. However, radiative transfer models predict a change in the slope
of the attenuation law with dust optical depth (Chevallard et al., 2013; Salim
& Narayanan, 2020; Tacchella et al., 2022) and dust attenuation curves have
been shown to vary within and between galaxies at all redshifts (e.g. Salmon
et al., 2016; Buat et al., 2018; Salim et al., 2018; Decleir et al., 2019;
Reddy et al., 2020; Boquien et al., 2022, and many others). For example,
implementing the attenuation and slope of the curve presented by Tacchella et
al. (2022) implies substantially higher values of $A_{\rm H\alpha}$ than using
a Milky Way law, especially for the more attenuated regions, and would
therefore lead a substantial recalibration of the IR coefficients presented
here. More detailed observational work is required in order to confirm changes
in the slope of the nebular attenuation law as a function of optical depth.

The conversion between attenuation-corrected H$\alpha$ flux and SFR is
affected by additional systematic uncertainties, including the escape of
ionising radiation from galaxies, the absorption of ionising photons from
dust, the stochastic sampling of the IMF in low-mass clusters (Da Silva et
al., 2014), and to some extent the variability of star formation at low SFR.
The fraction of Lyman continuum photons absorbed by dust is found to be
significant in nearby galaxies and clusters, between 30-50% (Inoue et al.,
2001; Iglesias-Páramo et al., 2004), in agreement with some theoretical models
of radiatative transfer (Kado-Fong et al., 2020; Tacchella et al., 2022).
However, substantial uncertainty affect these estimates, and in detailed
models of dusty H ii regions dust competes with hydrogen only for high values
of the ionisation parameter (the ratio of ionising photons to hydrogen atoms),
a regime only relevant to young, compact H iiregions. Potential dust
absorption of ionising photons is not taken into account by our calibration,
and would imply a change in the $C_{\rm H\alpha}$ factor. For example,
assuming the fraction of Lyman continuum photons directly absorbed by dust to
be 30%, $C_{\rm H\alpha}$ would need to be increased by 0.23 dex. Stochastic
sampling of the IMF in low-mass clusters will, on the other hand, cause a bias
(and increased scatter) at the lowest SFR levels, but should only have a
limited effect on our kpc-scale data of the inner regions of discs.

At kpc resolution it is not possible to distinguish individual H ii regions
from the diffuse emission surrounding them. The nature of the diffuse emission
at H$\alpha$ and 22 $\rm\mu m$, however, represents one of the main systematic
uncertainties to the determination of SFR on even smaller spatial scales. The
diffuse H$\alpha$ emission (also known as diffuse ionised gas) is directly
powered by ionising radiation leaking from H ii regions (Ferguson et al.,
1996; Zurita et al., 2000; Haffner et al., 2009; Belfiore et al., 2022). In
particular, Belfiore et al. (2022) studied the same sample of galaxies used in
this work and showed that the DIG is consistent with being powered by leaking
radiation, with a mean free path of $\sim 2$ kpc, therefore having negligible
impact on the SFR estimates on kpc scales.

Diffuse emission at 22 $\rm\mu m$ results from the convolution of the dust
distribution with the local radiation field. Therefore it is potentially
sensitive to both heating due to the radiation field from old stars (Leroy et
al., 2008; Verley et al., 2009; Kennicutt et al., 2009; Leroy et al., 2012)
and the presence of an extended dust-bearing gas reservoir, especially atomic
gas, mixed with the old stellar population (e.g., Leroy et al., 2012).

The calibrations we present in this work effectively model this spatially
unresolved background emission by making the IR calibration coefficient
dependent on an sSFR-like quantity. Higher-resolution IR maps, capable of
resolving dust heated by H ii regions from the diffuse component, are needed
to directly test this interpretation. While Spitzer 24 $\rm\mu m$ offers
cloud-scale $\sim$ 100 pc resolution in the Local Group and in galaxies
$\lesssim 4$ Mpc (e.g. Helou et al. 2004; Verley et al. 2009; Faesi et al.
2014), these targets probe only a limited set of environments. The PHANGS-JWST
program will address this shortcoming by providing sub-arcsec-resolution 21
$\rm\mu m$ mapping of the 19 PHANGS-MUSE galaxies using the MIRI instrument on
board JWST. In combination with arcsec-resolution mapping of H$\alpha$ and
H$\beta$ from MUSE, and FUV maps at $\sim 2\arcsec$ resolution from AstroSat
(Hassani et al. in preparation), this combined data set will shed new light on
the nature of diffuse IR emission, small-scale behaviour of dust attenuation
in star-forming discs, and the statistics of fully-embedded regions (Prescott
et al., 2007).

As discussed of above, a complementary future direction will be to extend this
analysis to the outer, atomic gas-dominated parts of galaxies. In these
regions, we might expect much of the dust (i.e., the dust mixed with diffuse
atomic gas) to reside in an extended distribution rather than being
concentrated towards the peaks of star formation. Such a situation could
produce increased infrared cirrus emission even when the average sSFR remains
high and so will represent an important test of the generality of our
prescriptions.

##  5 Summary and conclusions

In this work we use H$\alpha$ maps corrected for attenuation using the Balmer
decrement to calibrate multi-wavelength hybrid SFR recipes for use in nearby
galaxies at kpc-scale resolution. We make use of a sample of 19 galaxies
observed with optical IFS as part of the PHANGS-MUSE program, and combine this
data with FUV imaging from GALEX and mid-IR imaging at 22 $\rm\mu m$ from WISE
W4. We focus on calibrating two commonly used hybrid recipes: FUV+W4 and
H$\alpha$+W4. We further assess the reliability of the resulting calibrations
as a function of physical scale. We summarise our main results below.

  * •

Calibrations of FUV+W4 and H$\alpha$+W4 from the literature overestimate the
SFR obtained from attenuation-corrected H$\alpha$ in regions of low sSFR, or
equivalently low $\Sigma_{\mathrm{SFR}}$, old light-weighed stellar age or low
E(B$-$V). We attribute this trend to the presence of ‘IR cirrus’: diffuse IR
emission originating from dust heated by the interstellar radiation field and
not associated with star formation.

  * •

In order to correct for IR cirrus we propose to modify the calibration term in
front of the IR term in the hybrid SFR recipes and make it a function of band
ratios which correlate with sSFR. We present broken power law fits (equation
7) for the dependence of $\rm C_{FUV}^{W4}$ on H$\alpha$/W1 and FUV/W1 and of
$\rm C_{H\alpha}^{W4}$ on H$\alpha$/W1 in Table 3. These calibrations lead to
SFR which agree with those obtained from attenuation-corrected H$\alpha$ with
negligible bias and scatter of less than 0.16 dex.

  * •

The SFR obtained on kpc scales with these recipes agrees well with the
estimates from attenuation-corrected H$\alpha$ calculated on scales of 100 pc
using the MUSE IFS data at its native resolution. In particular, we find that
based on our sample of 19 star-forming galaxies the lower-resolution SFR is
underestimated by less 10% on average.

  * •

We test the validity of our calibration on the scales of entire galaxies by
using attenuation-corrected H$\alpha$ data for star-forming galaxies from the
MaNGA IFS survey. We observe comparable trends in terms of the sSFR dependence
of the IR coefficient, albeit with quantitative differences compatible with
the inevitable mixing of regions with various levels of star formation
activity when considering galaxies as whole. Nonetheless, we find that the
calibrations derived from PHANGS kpc-scale data perform well with MaNGA,
leading to a scatter smaller than 0.2 dex when comparing hybrid and H$\alpha$
attenuation-corrected SFRs.

  * •

At high sSFR our calibrations are in good agreement with the coefficients
recommended by the Kennicutt & Evans (2012) review. In particular, for the
H$\alpha$+W4 calibration the kpc-scale analysis of the PHANGS-MUSE data is
comparable with the results of Calzetti et al. (2007) at high sSFR, and the
MaNGA analysis of flux-integrated galaxies is compatible with the coefficient
derived by Kennicutt et al. (2009). However, our calibrations will lead to
more accurate SFR measurements than obtained using constant coefficients from
the literature when considering samples of regions or galaxies sampling a wide
range of sSFR.

###### Acknowledgements.

This work has been carried out as part of the PHANGS collaboration. Based on
observations collected at the European Southern Observatory under ESO
programmes 094.C-0623 (PI: Kreckel), 095.C-0473, 098.C-0484 (PI: Blanc),
1100.B-0651 (PHANGS-MUSE; PI: Schinnerer), as well as 094.B-0321 (MAGNUM; PI:
Marconi), 099.B-0242, 0100.B-0116, 098.B-0551 (MAD; PI: Carollo) and
097.B-0640 (TIMER; PI: Gadotti). Science-level MUSE mosaicked datacubes and
high-level analysis products are provided via the ESO archive phase 3
interface333https://archive.eso.org/scienceportal/home?data_collection=PHANGS.
A full description of the the first PHANGS data release is presented in
Emsellem et al. (2022). The work of AKL was partially supported by the
National Science Foundation (NSF) under Grants No. 1615105, 1615109, and
1653300. The work of JS is partially supported by the Natural Sciences and
Engineering Research Council of Canada (NSERC) through a Canadian Institute
for Theoretical Astrophysics (CITA) National Fellowship. ATB would like to
acknowledge funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No.726384/Empire, PI Bigiel). MB gratefully acknowledges support by
the ANID BASAL project FB210003 and from the FONDECYT regular grant 1211000.
EC acknowledge support from ANID Basal projects ACE210002 and FB210003. OE and
KK gratefully acknowledge funding from the German Research Foundation (DFG) in
the form of an Emmy Noether Research Group (grant No. KR4598/2-1, PI Kreckel).
CE gratefully acknowledges funding from the Deutsche Forschungsgemeinschaft
(DFG) Sachbeihilfe, grant number BI1546/3-1. SCOG and RSK thank for funding
from the Heidelberg Cluster of Excellence EXC 2181 (Project-ID 390900948)
‘STRUCTURES’, supported by the German Excellence Strategy, from the ERC in the
Synergy Drant ‘ECOGAL’ (project ID 855130), from DFG via the Collaborative
Research Center (SFB 881, Project-ID 138713538) ‘The Milky Way System’
(subprojects A1, B1, B2, B8), and from the German Ministry for Economic
Affairs and Climate Action for funding in project ‘MAINN’ (funding ID
50OO2206). KG is supported by the Australian Research Council through the
Discovery Early Career Researcher Award (DECRA) Fellowship DE220100766 funded
by the Australian Government. MQ acknowledges support from the Spanish grant
PID2019-106027GA-C44, funded by MCIN/AEI/10.13039/501100011033. PSB
acknowledges financial support from the Spanish Ministry of Science,
Innovation and Universities under grant number PID2019-107427GB-C31. ES and
TGW acknowledge funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No. 694343, PI Schinnerer).

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