11institutetext: Dipartimento di Fisica, Università di Roma La Sapienza, P.le
A. Moro 2, I–00185 Roma, Italy 22institutetext: Istituto di Astrofisica e
Planetologia Spaziali (INAF–IAPS), Via Fosso del Cavaliere 100, I–00133, Roma,
Italy

# Calibration of mid- to far-infrared spectral lines in galaxies

Sabrina Mordini, E-mail<EMAIL_ADDRESS>Luigi Spinoglio,
E-mail<EMAIL_ADDRESS>Juan Antonio Fernández-Ontiveros, E-mail:
<EMAIL_ADDRESS>
###### Abstract

Context. Mid- to far-infrared (IR) lines are suited to study dust obscured
regions in galaxies, because dust extinction is strongly decreasing with
wavelength and therefore IR spectroscopy allows us to explore the most hidden
regions of galaxies, where heavily obscured star formation as well as
accretion onto supermassive black-holes at the nuclei of galaxies occur. This
is mostly important for the so-called “Cosmic Noon”, i.e. at redshifts of
1$<$z$<$3, when most of the baryonic mass in galaxies has been assembled.

Aims. Our goal is to provide reliable calibrations of the mid- to far-IR ionic
fine structure lines, the brightest H2 pure rotational lines and the
Polycyclic Aromatic Hydrocarbons (PAHs) features, that will be used to analyse
current and future observations in the mm/submm range from the ground, as well
as mid-IR spectroscopy from the upcoming James Webb Space Telescope.

Methods. We use three samples of galaxies observed in the local Universe: star
forming galaxies (196), active galactic nuclei (AGN; 90–150 for various
observables) and low-metallicity dwarf galaxies (40). For each population we
derive different calibrations of the observed line luminosities versus the
total IR luminosities.

Results. Through the resulting calibrations, we derive spectroscopic
measurements of the Star Formation Rate (SFR) and of the Black Hole Accretion
Rate (BHAR) in galaxies using mid- and far-IR fine structure lines, H2 pure
rotational lines and PAH features. In particular, we derive robust star-
formation tracers based on: the [CII]158$\,\rm{\rm{\upmu m}}$ line; the sum of
the two far-IR oxygen lines, the [OI]63$\mu$m line and the [OIII]88$\mu$m
line; a combination of the neon and sulfur mid-IR lines; the bright PAH
features at 6.2 and 11.3$\,\rm{\rm{\upmu m}}$, as well as – for the first time
– the H2 rotational lines at $9.7$, $12.3$ and $17\,\rm{\rm{\upmu m}}$. We
propose the [CII]158$\mu$m line, the combination of the two neon lines
([NeII]12.8$\mu$m and [NeIII]15.5$\mu$m) and, for solar-like metallicity
galaxies that may harbour an AGN, the PAH11.3$\mu$m feature as the best SFR
tracers. On the other hand, a reliable measure of the BHAR can be obtained
using the [OIV]25.9$\,\rm{\rm{\upmu m}}$ and the [NeV]14.3 and
24.3$\,\rm{\rm{\upmu m}}$ lines. For the most commonly observed fine-structure
lines in the far-IR we compare our calibration with the existing ALMA
observations of high redshift galaxies. We find an overall good agreement for
the [CII]158$\,\rm{\rm{\upmu m}}$ line for both AGN and star forming galaxies,
while the [OIII]88$\,\rm{\rm{\upmu m}}$ line in high-z galaxies is in better
agreement with the low metallicity local galaxies (dwarf galaxies sample) than
with the star forming galaxies, suggesting that high-z galaxies might have
strong radiation fields due to low metal abundances, as expected.

###### Key Words.:

galaxies: active – galaxies: evolution – galaxies: star formation – infrared:
galaxies – techniques: spectroscopic telescopes

##  1 Introduction

One of the major and still unsolved problems in astrophysics is related to the
lack of a clear understanding of how galaxies evolve, from the time of
structure formation till today, and what are the dominant processes that have
influenced their evolution (Somerville & Davé 2015; Naab & Ostriker 2017;
Bullock & Boylan-Kolchin 2017; Wechsler & Tinker 2018; Förster Schreiber &
Wuyts 2020). From the observational scenario that has been consolidated in the
last tens of years, we know that the two main energy production mechanisms
along galaxy evolution are, on one side, star formation and subsequent stellar
evolution and, on the other side, accretion onto the super-massive black holes
(SMBHs) that form at the centre of galaxies.

The bulk of star formation and BH accretion took place at the so-called Cosmic
Noon ($1<z<3$), with a steep decline toward the present epoch, in heavily
obscured environments, embedded in large amounts of gas and dust, where
optical and UV detected radiation corresponds to only $\sim 10\%$ of the total
emitted light (Madau & Dickinson 2014) and with most of the radiated energy
being absorbed by dust and re-emitted at longer wavelengths. Both star
formation and black hole accretion contribute to the dust re-emission, and
therefore measuring the integrated IR continuum, through photometric
observations, does not discriminate easily between both components. The
spatial resolution required to isolate the bulk of the nuclear IR emission
from the host galaxy ($\lesssim 100\,\rm{pc}$) cannot be attained in high-$z$
galaxies, while spectral decomposition techniques (e.g., Berta et al. 2013)
are ultimately dependent on templates of local active galactic nuclei (AGN),
whose inner workings and dust distribution are still far from being understood
(e.g. Lyu & Rieke 2020). Thus measuring star formation rate (SFR) and black
hole accretion rate (BHAR) in galaxies across cosmic time, which is one of the
major observational goals in galaxy evolution studies, has to be done through
spectroscopic observations at wavelengths long enough to overcome dust
absorption.

The mid to far-IR range is populated by a large number of atomic and molecular
lines and features. In particular, the atomic and ionic fine-structure lines
cover a wide range of physical parameters in terms of excitation, density and
ionisation, as can be seen in fig. 4 of Spinoglio et al. (2012), showing the
critical density for collisional de-excitation versus the ionisation potential
of the IR fine-structure lines. These lines can easily discriminate among the
different gas excitation conditions, from the Narrow Line Regions (NLR)
excited by the active nucleus, to the Hii and photo-dissociation regions
(PDRs), whose origin is due to stellar excitation (see, e.g., Spinoglio &
Malkan 1992; Tommasin et al. 2010; Spinoglio et al. 2015; Fernández-Ontiveros
et al. 2016), offering an ideal tool to probe the highly opaque and dust
obscured regions. Most of these lines can be assessed only from space IR
telescopes, while only a few transitions happen to lie in the atmospheric
windows, e.g. in the spectral region of $8$–$13\,\rm{\rm{\upmu m}}$.

The intermediate ionisation fine-structure lines are good tracers of the star
formation and in particular the sum of the fluxes of [NeII] and [NeIII] has
been shown to give a measure of the Star Formation Rate (Ho & Keto 2007;
Zhuang et al. 2019) in galaxies. The low ionisation [CII] line at
158$\,\rm{\rm{\upmu m}}$ is one of the brightest emission lines in star
forming galaxies, and can thus trace star formation activity both in the local
(De Looze et al. 2014; Herrera-Camus et al. 2015) and high redshift (Leung et
al. 2020) Universe.

The mid-IR spectra of star forming galaxies between $3$ and $19\,\rm{\rm{\upmu
m}}$ are often dominated by emission features attributed to Polycyclic
Aromatic Hydrocarbons (PAHs), due to infrared (IR) fluorescence by these large
molecules containing 50-100 C-atoms, pumped by single FUV photons (Allamandola
et al. 1989; Smith et al. 2007; Tielens 2008). Because they trace the FUV
stellar flux, they can be used to measure star formation. These features are
strong when compared to fine structure lines in star forming galaxies, and are
detected not only in local galaxies, but also at redshifts up to z$\sim$4
(Kirkpatrick et al. 2015; Riechers et al. 2014; Sajina et al. 2012). Their
emission accounts for $\sim 10-20\%$ of the total IR radiation by dust, and
originate from PDRs near star forming regions. PAHs can be used to trace SFR
not only in star forming galaxies (Shipley et al. 2016; Xie & Ho 2019), but
also in sources where an AGN contribution is present but not the dominant
source of integrated light (Shipley et al. 2013). In these AGN, the equivalent
width of the PAH features can be used to estimate the star formation
contribution to the total IR luminosity (Armus et al. 2007; Tommasin et al.
2010). However, in more extreme AGN, strong UV and X-ray radiation fields can
also suppress the PAH emission in the vicinity of the AGN by photo-
dissociation, while increasing the mid-IR continuum emission (Lacy et al.
2013). Analogously, also in Low Metallicity Galaxies (LMG, i.e. below 12 +
log(O/H)$\sim$8.2), because of the reduced formation efficiency and the
increased stellar radiation hardness, the strength of the PAH features is
reduced and only detected at metallicities above 1/8-1/10 Z⊙ (Engelbracht et
al. 2008; Cormier et al. 2015; Galliano et al. 2021).

Among the molecular lines in the mid-IR, the pure rotational transitions of H2
are particularly important as the typical physical conditions of the gas
associated with these lines can be found both in AGN and star forming
galaxies. Rigopoulou et al. (2002) found that, while in star forming
environments the H2 emission can originate from PDRs, with a small
contribution from shocks, in AGN dominated galaxies the X-ray emission from
the central AGN plays an important role in heating large amounts of gas
boosting the H2 emission.

High ionisation lines trace AGN activity: all lines whose ionisation potential
is higher than the one necessary to doubly ionise helium ($>54.4\,\rm{eV}$)
cannot be efficiently produced by stellar radiation in significant amounts.
Therefore, the detection of the [NeV]14.3$\,\rm{\rm{\upmu m}}$, [NeV]24.3
$\,\rm{\rm{\upmu m}}$ or [NeVI]7.65$\,\rm{\rm{\upmu m}}$ lines probes the
presence of an AGN, while the [OIV]25.9$\,\rm{\rm{\upmu m}}$ line, which can
also be found in energetic starbursts and LMG, is however much stronger in
AGN, having an equivalent width one order of magnitude larger compared to Star
Forming Galaxies (SFG) (Tommasin et al. 2010). Typical IR line ratios used to
measure the strength of the active nucleus with respect to the star formation
component in a galaxy are: the [NeV]14.3$\,\rm{\rm{\upmu m}}$ or
24.3$\,\rm{\rm{\upmu m}}$ to [NeII]12.8$\,\rm{\rm{\upmu m}}$ ratio, the
[OIV]25.9$\,\rm{\rm{\upmu m}}$/[NeII]12.8$\,\rm{\rm{\upmu m}}$ line ratio and
the [OIV]25.9$\,\rm{\rm{\upmu m}}$/[OIII]52$\,\rm{\rm{\upmu m}}$ or
88.35$\,\rm{\rm{\upmu m}}$ ratio (Sturm et al. 2002; Armus et al. 2007;
Tommasin et al. 2010; Spinoglio et al. 2015; Fernández-Ontiveros et al. 2016).

The main goals of this work are: (i) to revise the calibration of the mid- to
far-IR lines including ionic fine-structure lines, the brightest H2 pure
rotational lines and the PAH features and (ii) to provide a local calibration
of spectroscopic SFR and BHAR tracers, that can also be applied to
measurements at high-z. We will follow the study presented in Spinoglio et al.
(2012), update the IR spectroscopic observations presented there and include a
sample of LMG, to extend the calibration to these objects. These latter are
included to characterise the response of the lines to the conditions of low
metallicity (average $1/5\,\rm{Z_{\odot}}$) (Madden et al. 2013). Low-
metallicity AGN are not included because they are rare in the local Universe
and only a few examples are found (e.g. Circinus, Oliva et al. 1999), but the
broad- and narrow-line regions in AGN show essentially little or no chemical
evolution up to $z\sim 7$ (Nagao et al. 2006; Juarez et al. 2009; Onoue et al.
2020), suggesting that the quasar phase appears mostly when galaxies are
already chemically mature objects. Therefore including the metallicity
dependence in the calibrations for AGN is not as relevant as in the case of
star forming galaxies, where the chemical evolution with redshift is well
known (Sanders et al. 2020). This work extends and update a previous study
(Spinoglio et al. 2021) aimed at the preparation of the SPICA mission
(Roelfsema et al. 2018) where the authors give a calibration of the most
important features that were used to plan spectroscopic observations with that
mission.

A further motivation of this study resides in the need to exploit the great
potential of extragalactic IR spectroscopy for high redshift galaxies, which
is being already explored by the Atacama Large Millimeter/submillimeter Array
(ALMA, Wootten & Thompson 2009; Carpenter et al. 2020), and will have a
dramatic boost with the next space IR telescopes, such as the James Webb Space
Telescope (JWST, Gardner et al. 2006) and in a more far future, possibly by
the Origins Space Telescope111https://asd.gsfc.nasa.gov/firs.

The paper is organized as follows: section 2 describes the samples of
galaxies, observed in the local Universe, that we have used to derive the
correlations; section 3 reports our results, in particular: section 3.1
presents the new correlations between the line luminosities and the total IR
luminosities, while sections 3.2 and 3.3 give simple recipes to measure the
two main parameters of the SFR and BHAR. In section 4 we discuss our results:
section 4.3 compares our study to previous ones, in section 4.4 we discuss the
metallicity effect on the SFR tracers, section 4.5 presents how the
observations at high redshift compare with the correlations we have derived
and section 4.6 shows how our results can be used to interpret present and
future IR/(sub)-mm observations. Our conclusions are presented in section 5.

##  2 The selected lines and features and the samples of galaxies

We use the most representative samples of star forming galaxies (SFG), AGN,
and low metallicity galaxies (LMG) in the local Universe for which IR
spectroscopy is available, mainly from Spitzer-IRS (Houck et al. 2004) and
Herschel-PACS (Poglitsch et al. 2010), but also from the Infrared Space
Observatory (ISO) SWS and LWS spectrometers (de Graauw et al. 1996; Clegg et
al. 1996). For each of the three galaxy populations, we derive linear
relations in logarithmic space between the line luminosity and the total IR
luminosity. Then we derive the best tracers of the SFR and the BHAR, using the
discussed IR lines and features, and compare them with what has been reported
in the literature

In this analysis we considered, in order of decreasing ionisation/excitation:

  * •

Four high-ionisation fine structure lines, typical of AGN:
[NeVI]7.65$\,\rm{\rm{\upmu m}}$, [NeV]14.32$\,\rm{\rm{\upmu m}}$,
[NeV]24.32$\,\rm{\rm{\upmu m}}$ and [OIV]25.89$\,\rm{\rm{\upmu m}}$;

  * •

Ten intermediate ionisation fine structure lines, typical of stellar/Hii
regions: [SIV]10.51$\,\rm{\rm{\upmu m}}$, [NeII]12.81$\,\rm{\rm{\upmu m}}$,
[NeIII]15.55$\,\rm{\rm{\upmu m}}$, [SIII]18.71$\,\rm{\rm{\upmu m}}$,
[SIII]33.48$\,\rm{\rm{\upmu m}}$, [OIII]51.81$\,\rm{\rm{\upmu m}}$,
[NIII]57.32$\,\rm{\rm{\upmu m}}$, [OIII]88.36$\,\rm{\rm{\upmu m}}$,
[NII]121.9$\,\rm{\rm{\upmu m}}$ and [NII]205$\,\rm{\rm{\upmu m}}$;

  * •

Five low-ionisation/neutral fine structure lines, typical of PDR:
[FeII]25.99$\,\rm{\rm{\upmu m}}$, [SiII]34.81$\,\rm{\rm{\upmu m}}$,
[OI]63.18$\,\rm{\rm{\upmu m}}$, [OI]145.5$\,\rm{\rm{\upmu m}}$ and
[CII]157.7$\,\rm{\rm{\upmu m}}$;

  * •

Four H2 pure rotational lines at 9.67, 12.28, 17.03 and 28.22 $\,\rm{\rm{\upmu
m}}$;

  * •

Five PAH features at 6.2, 7.7, 8.6, 11.3 and 17$\,\rm{\rm{\upmu m}}$.

The fundamental parameters for the fine-structure lines considered in this
analysis are reported in Table 2.

In this analysis we have included those lines observed with the Spitzer-IRS
high resolution (HR) channel in the 10-35$\mu$m range, for which good spectra
are available in the literature. The only exception is the line of
[NeVI]7.65$\mu$m, an exclusive AGN line, for which we included ISO-SWS
observations. Other well known fine-structure lines, such as [ArII]6.98$\mu$m,
[ArIII]8.99$\mu$m and [NeIII]36.0$\mu$m, which could indeed have a relevant
role for future observations, especially in view of the JWST launch, at the
present time do not have enough high quality spectra in the literature to be
included in the analysis. We do not consider upper limits to derive our
correlations. This is because, in general, our statistics are quantitatively
appropriate, thus making the inclusion of upper limits not necessary.
Moreover, where the statistics are less precise, upper limits are usually not
available for the considered lines.

The definition of the various samples of galaxies chosen to compute the
correlations are described in the following sections and summarised in Table
1, with the instruments used to observe the spectral lines and features, the
total number of objects selected and the references for each sample.

Table 1: Characteristics of the different samples used in this analysis. For
each class of objects we report, for the different lines and features, the
original sample (heterogeneous indicates that the sources do not belong to a
specific well-defined sample of galaxies), the facility and instrument that
observed the sample of galaxies, the number of objects used in the analysis
and the references where the original data can be found.

Type | Lines/features | Sample | Observed by | N. of objects | References
---|---|---|---|---|---
AGN |  [NeVI]7.7$\mu$m | heterogeneous | ISO-SWS | 8 | Sturm et al. (2002)
MIR lines in the 10-35$\mu$m range | 12MGS | Spitzer-IRS HR | 88 | Tommasin et al. (2008); Tommasin et al. (2010)
including H2 rotational lines
PAH features | 12MGS | Spitzer-IRS LR | 103 | Wu et al. (2009)
FIR lines in the 50-205$\mu$m range | heterogeneous | Herschel-PACS and SPIRE | 149 | Fernández-Ontiveros et al. (2016)
SFG |  MIR lines in the 10-35$\mu$m range |  | GOALS sample
---
\+ heterogeneous
\+ heterogeneous
Spitzer-IRS HR |  | 153
---
10
28
| Inami et al. (2013)
---
Bernard-Salas et al. (2009)
Goulding & Alexander (2009)
H2 rotational lines |  | GOALS sample
---
\+ heterogeneous
\+ heterogeneous
Spitzer-IRS HR |  | 153
---
10
28
| Stierwalt et al. (2014)
---
Bernard-Salas et al. (2009)
Goulding & Alexander (2009)
PAH features |  | GOALS sample
---
\+ heterogeneous
Spitzer-IRS LR |  | 179
---
12
| Stierwalt et al. (2014)
---
Brandl et al. (2006)
FIR lines in the 50-205$\mu$m range |  | GOALS sample
---
\+ heterogeneous
\+ heterogeneous
| Herschel-PACS
---
Herschel-PACS and SPIRE
ISO-LWS
| 153
---
20
23
| Díaz-Santos et al. (2017)
---
Fernández-Ontiveros et al. (2016)
Negishi et al. (2001)
LMG |  MIR lines in the 10-35$\mu$m range | DGS | Spitzer-IRS HR and LR | 40 | Cormier et al. (2015)
FIR lines in the 50-158$\mu$m range | DGS | Herschel-PACS | 40 | Cormier et al. (2015)

###  2.1 The AGN sample

The AGN sample has been drawn from the $12\,\rm{\rm{\upmu m}}$ selected active
galaxies sample (12MGS, Rush et al. 1993), which is the brightest complete and
unbissed sample of Seyfert galaxies in the local Universe. For the mid-IR fine
structure lines and the H2 rotational lines, we have used the sub-sample of
the 12MGS observed by Spitzer IRS at high spectral resolution (R$=600$) which
contains 88 AGN (Tommasin et al. 2008; Tommasin et al. 2010). For the PAH
features at 6.2 $\mu$m and 11.2 $\mu$m we have used the Spitzer-IRS data at
low spectral resolution (R$\sim 60$–$120$; Buchanan et al. 2006; Wu et al.
2009) of the 12MGS (103 objects), because this setting matches better the
intrinsic width of these features. We note here that Wu et al. measure the PAH
features using a spline function to determine the continuum level. In order to
make these measurements comparable to those obtained using automated fitting
procedures (e.g. PAHFIT, Smith et al. 2007), we apply a correction factor of
$1.7$ and $1.9$ to increase the fluxes of the $6.2\,\rm{\mu m}$ and
$11.3\,\rm{\mu m}$ PAH in Wu et al., respectively, following the differences
found by Smith et al. (2007).

The adoption of the complete 12MGS as the main catalog, from which we cover
about 75% of the total sample, allows us to derive statistically robust
calibrations.

For the [NeVI]7.65$\,\rm{\rm{\upmu m}}$ line, we could not use the 12MGS,
because this line was not detected by Spitzer at low resolution and was
outside its spectral range at high-resolution. Therefore we had to use the
data from Sturm et al. (2002), which contain 8 detections of the
[NeVI]7.65$\,\rm{\rm{\upmu m}}$ line with ISO-SWS at medium resolution (R
$\sim$ 1500), and thus not contaminated by the PAH emission at 7.7
$\,\rm{\rm{\upmu m}}$.

For the far-IR spectral range, namely the $50$–$205\,\rm{\rm{\upmu m}}$, we
used the catalogue of AGN assembled by Fernández-Ontiveros et al. (2016),
which includes all the Seyfert galaxies and the quasars of the Véron-Cetty &
Véron (2010) catalogue which have far-IR spectra observed by Herschel-PACS.
The sample of the AGN observed in the far-IR lines counts 170 galaxies and
contains about 50% of objects from the 12MGS, while the others do not come
from a complete sample.

###  2.2 The SFG sample

The SFG sample was constructed using the The Great Observatories All-Sky LIRG
Survey (GOALS sample, Armus et al. 2009), from which we extracted 158
galaxies, with data from Inami et al. (2013), who report the fine structure
lines at high resolution in the $10$–$36\,\rm{\rm{\upmu m}}$ interval, and
Stierwalt et al. (2014), who include the detections of the H2 molecular lines
and the PAH features at low spectral resolution. For those galaxies in the
GOALS sample that have a single IRAS counterpart, but more than one source
detected in the emission lines, we have added together the line/feature fluxes
of all components, to consistently associate the correct line/feature emission
to the total IR luminosity computed from the IRAS fluxes. To cover also lower
luminosity galaxies, as the GOALS sample only includes Luminous IR Galaxies
(LIRGs) and Ultra-Luminous IR Galaxies (ULIRGs), we have included 38 galaxies
from Bernard-Salas et al. (2009) and Goulding & Alexander (2009), to reach the
total sample of 196 galaxies with IR line fluxes in the
$5.5$–$35\,\rm{\rm{\upmu m}}$ interval in which an AGN component is not
detected. For the Bernard-Salas et al. (2009), Goulding & Alexander (2009) and
the GOALS samples we excluded all the composite starburst-AGN objects
identified as those with a detection of [NeV] either at $14.3$ or
$24.3\,\rm{\rm{\upmu m}}$. It is worth to note that the original samples from
Goulding & Alexander (2009) and Bernard-Salas et al. (2009) have spectra
covering solely the central region of the galaxies. To estimate the global
SFR, we corrected the published line fluxes of the Spitzer spectra by
multiplying them by the ratio of the continuum reported in the IRAS Point
Source Catalogue to the continuum measured on the Spitzer spectra extracted
from the CASSIS database (Lebouteiller et al. 2015). We assume here that the
line emission scales, at first order, with the IR brightness distribution. In
particular, we considered the continuum at $12\,\rm{\rm{\upmu m}}$ for the
[NeII]$12.8\,\rm{\rm{\upmu m}}$ and [NeIII]$15.6\,\rm{\rm{\upmu m}}$ lines,
and the continuum at 25 $\,\rm{\rm{\upmu m}}$ for the
[OIV]$25.9\,\rm{\rm{\upmu m}}$, [FeII]$26\,\rm{\rm{\upmu m}}$,
[SIII]$33.5\,\rm{\rm{\upmu m}}$ and [SiII]$34.8\,\rm{\rm{\upmu m}}$ lines.
This correction was not needed for the AGN sample and the GOALS sample because
of the greater average redshift of the galaxies in the 12MGS and GOALS
samples. In particular, the 12MGS active galaxies sample has a mean redshift
of 0.028 (Rush et al. 1993), while the GOALS sample has a mean redshift of
0.026. The galaxies presented by Bernard-Salas et al. (2009) have instead an
average redshift of 0.0074, while the sample by Goulding & Alexander (2009)
has an average redshift of 0.0044. For the other lines in the
$10$–$36\,\rm{\rm{\upmu m}}$ interval, Goulding & Alexander (2009) do not
report a detection, and we used the data presented in Bernard-Salas et al.
(2009) for a total of 15 objects. Both Bernard-Salas et al. (2009) and
Goulding & Alexander (2009) report data from the high-resolution Spitzer-IRS
spectra. Data in the $50$–$205\,\rm{\rm{\upmu m}}$ interval were taken from
Díaz-Santos et al. (2017) for the GOALS sample, 20 starburst galaxies from
Fernández-Ontiveros et al. (2016), and 23 objects from the ISO-LWS
observations of Negishi et al. (2001), resulting in a total sample of 193
objects. Lastly, the PAH features fluxes were measured from the low resolution
Spitzer-IRS spectra by Brandl et al. (2006), including 12 objects from the
sample of Bernard-Salas et al. (2009) and 179 objects from Stierwalt et al.
(2014).

###  2.3 The LMG sample

The LMG sample was selected from Cormier et al. (2015), where the
$10$–$36\,\rm{\rm{\upmu m}}$ interval was observed by Spitzer-IRS in both high
and low resolution, and the $50$–$158\,\rm{\rm{\upmu m}}$ interval was
observed by Herschel-PACS. For the Spitzer-IRS data, we only considered the
high resolution results, for a total sample of 40 objects.

##  3 Results

We derive, for each line or feature in each galaxy sample, the correlation
between the logarithms of the total IR luminosity in the
$8$–$1000\,\rm{\rm{\upmu m}}$ range – computed from the IRAS fluxes following
Sanders & Mirabel (1996) – and the line luminosity, according to the equation:

$\log L_{\rm Line}=(a\pm\delta a)\log L_{IR}+(b\pm\delta b)$ (1)

with all luminosities expressed in units of $10^{41}\,\rm{erg\,s^{-1}}$. We
report in Table 3 of the Appendix D the best-fit parameters obtained for each
line/feature using the orthogonal distance regression fit (Boggs & Rogers
1990), the number of objects $N$, and the Pearson correlation coefficient $r$.
We use the orthogonal distance regression because the two variables are
independent each other, instead of the ordinary least-square minimisation,
where one variable is dependent from the other one. This is also particularly
useful to derive the inverse relation between the two variables from the best-
fit coefficients.

Using these correlations, we derive the tracers for the SFR (Section 3.2) and
the BHAR (Section 3.3). For the SFR, we used the [CII]158$\,\rm{\rm{\upmu m}}$
luminosity, various combinations of the luminosities of the
[NeII]12.8$\,\rm{\rm{\upmu m}}$, [NeIII]15.6$\,\rm{\rm{\upmu m}}$,
[SIII]18.7$\,\rm{\rm{\upmu m}}$ and [SIV]10.5$\,\rm{\rm{\upmu m}}$ lines, the
luminosity of the PAH features at 6.2$\,\rm{\rm{\upmu m}}$ and
11.3$\,\rm{\rm{\upmu m}}$ and the luminosity of the H2 rotational lines at
9.7$\,\rm{\rm{\upmu m}}$, 12.3$\,\rm{\rm{\upmu m}}$ and 17.3$\,\rm{\rm{\upmu
m}}$. For the BHAR, we have used the luminosities of the
[OIV]25.9$\,\rm{\rm{\upmu m}}$, [NeV]14.3$\,\rm{\rm{\upmu m}}$ and
24.3$\,\rm{\rm{\upmu m}}$ lines.

###  3.1 Spectral lines and features vs. total IR luminosity

Figure 1: (a: left) The [CII]158$\,\rm{\rm{\upmu m}}$ line luminosity as a
function of the total IR luminosity. Blue squares represent detections in AGN,
red stars indicate SFG and green triangles LMG. The solid red line represents
the linear relation calculated for SFG, the blue dotted line shows the
relation for AGN and the green dashed line the one for LMG. (b: centre) The
[NeII]12.8$\,\rm{\rm{\upmu m}}$ line luminosity as a function of the total IR
luminosity. (c: right) The [NeIII]15.6$\,\rm{\rm{\upmu m}}$ line luminosity as
a function of the total IR luminosity. In (b) and (c) we use the same
notations as in (a).

Figure 2: (a: left) The [OIII]88$\,\rm{\rm{\upmu m}}$ line luminosity as a
function of the total IR luminosity. (b: centre) The [OIV]25.9$\,\rm{\rm{\upmu
m}}$ line luminosity as a function of the total IR luminosity. (c: right) The
luminosity of the PAH feature at 11.3 $\,\rm{\rm{\upmu m}}$ as a function of
the total IR luminosity. The same legend as in Fig. 1 was used.

Among the different correlations derived in this work and presented in Table 3
of the Appendix D, we highlight in this section the main results, while we
refer the reader to Appendix B for the whole set of figures illustrating the
correlations with the total IR luminosity.

In Fig. 1a, we present the correlation between the [CII]158$\,\rm{\rm{\upmu
m}}$ line luminosity and the total IR luminosity for the three galaxy
populations considered. We find that the SFG and the LMG samples follow a
tight relation over six orders of magnitude in LIR with a consistent
regression slope and offset values within the uncertainties, from low-
luminosity dwarf galaxies to LIRGs and ULIRGs, confirming the results of De
Looze et al. (2014). This will be further discussed in Section 3.2.2. On the
other hand, AGN have higher IR luminosities for a given
[CII]158$\,\rm{\rm{\upmu m}}$ line luminosity with respect to the SFG and LMG
samples, likely due to the contribution from the AGN continuum emission to the
total IR luminosity.This effect is also seen in low ionisation transitions
where the AGN contribution to the line emission is expected to be small, such
as the [NII]122$\mu$m line (see Fig. 1c). However, the same effect is not
observed for [OI]63$\mu$m (see Fig. 20b). This might be related to the higher
critical density of the [OI]63$\mu$m line, which becomes an efficient coolant
in X-ray dissociation regions in the presence of an AGN (Maloney et al. 1996;
Dale et al. 2004). High-excitation lines where the AGN completely dominates
the line emission show the opposite behaviour, that is, the sources are
shifted towards higher line intensities at a given IR luminosity (e.g.
[OIV]25.9$\mu$m in Fig. 2b). This suggests that the lack of a noticeable shift
between AGN and SFGs in intermediate excitation lines such as [NeII]12.8$\mu$m
or [SIII]18.7,33.5$\mu$m (Figs. 1b, 17b and 19a) might be caused by a
comparable AGN contribution to both the IR luminosity and the line intensity.

We note that various authors (e.g. Herrera-Camus et al. 2015; Croxall et al.
2017) have reported an observed deficit in [CII] luminosity with the increase
of total LIR, in particular in ULIRGs. Different mechanisms have been proposed
to explain the lower [CII] emission (Sutter et al. 2019, 2021, and references
therein). In particular, Sutter et al. (2019) find that the [CII] deficit is
particularly evident when the emission arises from ionised gas, while the
effect is negligible when the emission comes predominantly from PDR regions.

For the [NeII]12.8 $\,\rm{\rm{\upmu m}}$ line (Fig. 1b) there is a slight
difference between SFGs and the other samples, but the differences are not
statistically relevant, being consistent within 3$\sigma$ of each other.

In Fig. 1c, we report the correlation obtained for the
[NeIII]15.6$\,\rm{\rm{\upmu m}}$ line with the total IR luminosity, showing
that the AGN and the SFG have a comparable correlation, while LMG have the
[NeIII]15.6$\,\rm{\rm{\upmu m}}$ line more than one order of magnitude
brighter, at a given IR luminosity (see also Cormier et al. 2012).

In Fig. 2a, b and c we report the correlations obtained for the
[OIII]88$\,\rm{\rm{\upmu m}}$, the [OIV]25.9$\,\rm{\rm{\upmu m}}$ line
luminosities and the luminosity of the PAH feature at 11.3 $\,\rm{\rm{\upmu
m}}$ with the total IR luminosity, respectively. For the
[OIII]88$\,\rm{\rm{\upmu m}}$ line, LMG are on average two orders of magnitude
brighter, at a given IR luminosity (see also Cormier et al. 2012), compared to
the other two classes of galaxies, that are almost overlapping. This could be
due to the differences in the ionising spectra and conditions in the ISM of
LMG with respect to the SFG and the AGN. In LMG, an increased number of
photons can ionise gas at greater distances from the star forming regions,
thus facilitating cooling through ionised gas emission. Additionally, the
lower dust-to-gas ratios in LMG is also expected to cause a decrease of the IR
luminosities, at a fixed stellar mass, in these galaxies.

AGN have about one order of magnitude brighter [OIV]25.9$\,\rm{\rm{\upmu m}}$
emission compared to SFG (see also Tommasin et al. 2010), LMG have a shallower
relation with IR luminosity, with a decreasing slope at higher luminosities
(Fig. 2b). The flatter slope could be related to the high ionisation potential
of the [OIV]25.9$\mu$m line, that is, the few ionising photons beyond 54.4 eV
produced by LMGs might not scale linearly with the overall luminosity.
However, no firm conclusion can be drawn because of the relatively poor
statistics of LMG, for which only 16 detections of the [OIV] line are
available. We note here that in order to obtain a reliable correlation for the
AGN sample, we considered only objects with an AGN component at the
19$\,\rm{\rm{\upmu m}}$ continuum, as defined in Tommasin et al. (2010),
greater than 85$\%$. We apply this limit only in this case, to minimise a
possible contamination in this line from emission due to strong starburst
activity (see, e.g. Lutz et al. 1998). This approach is motivated by the use
of the [OIV]25.9$\mu$m line as a BHAR tracer in Sect. 3.3, where the same
reduced sample is adopted.

Fig. 2c shows that the 11.3$\,\rm{\rm{\upmu m}}$ PAH feature is present both
in AGN and SFG and correlates well with the total IR luminosity, while in this
analysis we do not consider the PAH detections in LMG, because detections are
available only in a few cases, and thus are not enough to obtain a
statistically significant result. As shown by different authors (Madden 2000;
Engelbracht et al. 2005; Wu et al. 2006; Smith et al. 2007; Calzetti et al.
2007), in LMG there is evidence for a deficit of PAH emission and the
available measurements present significantly weaker features than SFG.
Moreover, the higher ionising continuum present in LMG contribute to destroy
these features (Engelbracht et al. 2008; Cormier et al. 2015).

As will be shown in Section 3.2.5, while there is a difference of $\sim$0.3
dex between the emission of SFG and AGN, the slopes of the two correlations
are comparable within the errors, with the difference linked to a higher LIR
for equal PAH emission in AGN.

We note here that, while we report in Tab. 3 the correlation derived for the
[NeVI]7.6$\,\rm{\rm{\upmu m}}$ line, this correlation was obtained for a
sample of only 8 AGN, while for the majority of AGN this line is not detected.
We therefore conclude that this calibration has to be taken with caution,
because it may be bissed toward AGN with a high ionisation parameter
(U$\sim$-1), while for a lower ionisation we expect the [NeVI] line to be
considerably less prominent, as discussed in Satyapal et al. (2021, in
particular in their Fig. 5).

###  3.2 Star formation rate tracers

####  3.2.1 Determination of SFR

In this section, we propose different SFR tracers. In order calibrate the
proposed SFR tracers, we use two different methods. For SFG, we measure the
SFR directly from the total IR luminosity LIR, following Kennicutt (1998):

$SFR=k_{IR}*L_{IR}$ (2)

where $k_{IR}=4.5\times 10^{-44}$M⊙yr-1erg-1s. We apply the same conversion
also to AGN sources (see Sect. 3.2.5), but limiting our sample to the sources
with L${}_{IR}\leq 10^{45}$erg s-1. We apply this limit to minimise the AGN
effect on the LIR and avoid overestimating the SFR because of AGN activity.

For the LMG, we adopt the SFR derived from Hα and corrected by the total IR
luminosity, as reported in Rémy-Ruyer et al. (2015). The total IR luminosity
alone does not accurately represent the total SFR in LMG due to the lower
dust-to-gas ratio in their ISM. The inclusion of the optical component for
computing the SFR is necessary to properly account for the emission not
reprocessed by dust.

####  3.2.2 L[CII]158μm–SFR relation

The [CII]158 $\,\rm{\rm{\upmu m}}$ line can be used as a tracer of the SFR, as
proposed by different authors (see, e.g., De Looze et al. 2014). We compare
the line intensity of SFG and LMG to the SFR. Applying the orthogonal distance
regression fit, we find a strong correlation between the two quantities:

$\log\left(\frac{SFR}{\rm M_{\odot}\,yr^{-1}}\right)=(0.62\pm 0.02)\\\
+(0.89\pm 0.02)\left(\log\frac{L_{[CII]}}{\rm 10^{41}\,erg\,s^{-1}}\right)$
(3)

Figure 3: (a: left) Correlation between the [CII]158 $\,\rm{\rm{\upmu m}}$
line luminosity and the SFR derived from the total LIR. Red stars represent
SFG, green triangles the LMG. Purple squares show the ULIRG population of SFG,
and the shaded green area indicates the 95$\%$ confidence interval. (b:
centre) Comparison of the logL[CII]-log(SFR) relation obtained in this work
(black solid line) with the results obtained by De Looze et al. (2014): the
green dashed line represents a sample of HII/star forming galaxies, the blue
diamond line shows the results for the low metallicity dwarf sample, and the
pink dash-dotted line considers the whole sample. The red dotted line shows
the results obtained by Sargsyan et al. (2012). (c: right) Correlation between
the [NeII]12.8$\,\rm{\rm{\upmu m}}$ and [NeIII]15.6$\,\rm{\rm{\upmu m}}$
summed emission lines luminosity, in units of $10^{41}\,\rm{erg\,s^{-1}}$ and
the SFR derived from the total IR luminosity (black dashed line) for SFG (red
star) and from the Hα luminosity (corrected for the IR luminosity) for LMG
(green triangles). The shaded green area indicates the 95$\%$ confidence
interval. The purple squares highlight the ULIRG population in the SFG sample.
The blue dashed line shows the results obtained by Zhuang et al. (2019) for
the same relation .

In Fig. 3a we show the correlation obtained for 227 galaxies, of which 37 are
LMG and 190 are SFG. The relation is nearly linear, with a Pearson r-value of
$r$ = 0.92, and covers six orders of magnitude in SFR, indicating that the
[CII] line emission is an overall good tracer of SFR for local star forming
galaxies independently of their metallicity. We excluded the AGN from this
correlation, since we derive the SFR from the LIR and AGN have an excess IR
continuum emission not due to star formation.

In Fig. 3b, we compare our results with those obtained in De Looze et al.
(2014) and Sargsyan et al. (2012). We note that, while in this work and in the
work by Sargsyan et al. the SFR was determined following Kennicutt (1998), De
Looze et al. trace the SFR using the GALEX FUV emission (Cortese et al. 2012)
and the Spitzer-MIPS 24$\,\rm{\rm{\upmu m}}$ emission (Rieke et al. 2004).
When compared to the results obtained by De Looze et al. (2014) for their
total sample, we find good agreement at luminosities of L${}_{[CII]}>10^{41}$
erg s-1, while there is a difference of $\sim$0.3 dex for lower luminosities.
In De Looze et al. the dwarf sample, when considered alone, shows a flatter
slope than the total sample, equal to $0.80\pm 0.5$ (see Fig. 3b, the blue
diamond line). The authors link the flatter slope to an underestimation of the
SFR based on the far-UV emission. We observe a similar flattening of the slope
when considering the LMG sample alone, equal to $0.69\pm 0.05$, consistent
with the result by De Looze et al. within 3$\sigma$ of each other.

De Looze et al. (2014) find that the ULIRG population presents a scatter of
almost one order of magnitude in the [CII]-SFR relation, when compared to the
total sample. We find the same result in our correlation, with the ULIRG
population (composed of 16 objects) lying between 0.2 and 1.0 dex above the
correlation derived for the total sample. Given the small number of sources,
we do not derive a specific [CII]-SFR correlation for the ULIRG sample.

####  3.2.3 Oxygen based SFR tracer

Figure 4: Correlation between the [OI]63$\mu$m and [OIII]88$\mu$m summed
emission line luminosities, in units of $10^{41}\,\rm{erg\,s^{-1}}$, and the
SFR derived from the total IR luminosity (black dashed line) for a composite
sample of SFG (red stars) and from the Hα luminosity (corrected for the IR
luminosity) for LMG (green triangles). Purple squares indicate the sample of
ULIRGs included in the SFG sample.

Besides C+, also O and O2+ are two important coolants of the ISM. The
[OI]63$\mu$m line traces the warm and/or dense PDRs, while the [OIII]88$\mu$m
emission line originates from diffuse, highly ionised regions near young, hot
stars.

The [OIII] line can be an important SFR tracer in LMG (De Looze et al. 2014),
where PDRs are weak or absent in the ISM and the ionisation field is stronger.
In order to trace the SFR in SFG, however, a tracer that can account for PDRs
needs to be included. For this reason, combining these two lines can provide
an accurate estimate of the SFR, both in SFG and LMG, probing both neutral and
ionised medium.

Fig. 4 shows the correlation obtained for a sample of 151 objects, 24 LMG and
124 SFG, of which 22 ULIRGs. The correlation, with a Pearson correlation
coefficient of $r=0.94$, can be expressed by:

$\log\left(\frac{SFR}{\rm M_{\odot}\,yr^{-1}}\right)=(0.30\pm 0.04)\\\
+(1.25\pm 0.04)\log\left(\frac{L_{\rm[OI]+[OIII]}}{\rm
10^{41}\,erg\,s^{-1}}\right)$ (4)

####  3.2.4 Neon- and sulfur-based SFR tracers

Figure 5: (a: left) Correlation between the [SIV]10.5$\,\rm{\rm{\upmu m}}$ and
[SIII]18.7$\,\rm{\rm{\upmu m}}$ summed emission lines luminosity, expressed in
units of $10^{41}\,\rm{erg\,s^{-1}}$ and the SFR derived from the total IR
luminosity (black dashed line) for a composite catalogue of SFG (red star) and
from the Hα luminosity (corrected for the IR luminosity) for LMG (green
triangles). The shaded green area in the three plots indicates the 95$\%$
confidence interval. (b: centre) Correlation between the
[NeII]12.8$\,\rm{\rm{\upmu m}}$ and [SIV]10.5$\,\rm{\rm{\upmu m}}$ summed
luminosities and the SFR derived from the total IR luminosity (black dashed
line) for SFG (red star) and from the Hα luminosity (corrected for the IR
luminosity) for LMG (green triangles). (c: right) Correlation between the
[NeIII]15.6$\,\rm{\rm{\upmu m}}$ and [SIII]18.7$\,\rm{\rm{\upmu m}}$ summed
emission lines and the SFR derived from the total IR luminosity (black dashed
line) for SFG (red star) and from the Hα luminosity (corrected for the IR
luminosity) for LMG (green triangles). In all three figures, the purple
squares highlight the ULIRG population in the SFG sample.

A promising SFR tracer, proposed by Ho & Keto (2007); Zhuang et al. (2019), is
the sum of the [NeII]12.8$\,\rm{\rm{\upmu m}}$ and the
[NeIII]15.6$\,\rm{\rm{\upmu m}}$ emission line fluxes. By adding the two
lines, this tracer is fairly independent from the effects related to the
hardness of the radiation field, which are stronger at lower metallicities
affecting single line diagnostics. For instance, while the
[NeII]12.8$\,\rm{\rm{\upmu m}}$ line intensity scales consistently in LMG and
SFG, the [NeIII]15.6$\,\rm{\rm{\upmu m}}$ line becomes remarkably brighter in
LMG (Fig. 1). Although some dependency on density is still expected, the high
critical density of the sulphur ($>7\times 10^{3}\,\rm{cm^{-3}}$; see Table 2)
and the neon lines ($>5\times 10^{4}\,\rm{cm^{-3}}$) used in this Section
guarantees a minor effect on these tracers for the vast majority of the galaxy
population.

In Fig. 3c we show the correlation found between the summed luminosity of the
[NeII] and [NeIII] emission lines and the SFR. This relation, obtained from
data of 203 local star forming galaxies, of which 182 SFG and 21 LMG, can be
expressed as:

$\log\left(\frac{SFR}{\rm M_{\odot}\,yr^{-1}}\right)=(0.88\pm 0.03)\\\
+(0.96\pm 0.03)\log\left(\frac{L_{\rm[NeII]+[NeIII]}}{\rm
10^{41}\,erg\,s^{-1}}\right)$ (5)

where $\log$(L[NeII]+[NeIII]) is the luminosity corresponding to the sum of
the fluxes of the [NeII]12.8$\,\rm{\rm{\upmu m}}$ and the
[NeIII]15.6$\,\rm{\rm{\upmu m}}$ lines. This relation has a Pearson
coefficient of $r$=0.91. We report in Fig. 3c the comparison between the
theoretical relation obtained by Zhuang et al. (2019), and our empirical one.
The relation by Zhuang et al. is derived following the assumption that the
neon lines trace all the ionising photons in a star forming region. Our
relation, derived from observational data, shows a lower slope, and thus a
larger offset, thus suggesting a lower efficiency in reprocessing the ionising
photons by the neon transitions.

Analogously to the case of the two neon lines, we also take into consideration
the sum of the two sulfur lines, i.e. [SIII]18.7$\,\rm{\rm{\upmu m}}$ and
[SIV]10.5$\,\rm{\rm{\upmu m}}$, as a tracer of the SFR. We plot in Fig. 5a the
correlation of the luminosity derived by summing [SIII]18.7$\,\rm{\rm{\upmu
m}}$ and [SIV]10.5$\,\rm{\rm{\upmu m}}$ with the SFR derived from the total IR
luminosity:

$\log\left(\frac{SFR}{\rm M_{\odot}\,yr^{-1}}\right)=(1.07\pm 0.06)\\\
+(1.04\pm 0.05)\log\left(\frac{L_{\rm[SIII]+[SIV]}}{\rm
10^{41}\,erg\,s^{-1}}\right)$ (6)

For this relation were used 52 SFG and 25 LMG, obtaining a relation with a
Pearson $r$ coefficient of $r$=0.90.

As shown in Table 2, Ne+ has a ionisation potential (IP) of 21.56 eV, while
Ne2+ has a IP of $40.96\,\rm{eV}$ and a potential of the next stage at
$63.45\,\rm{eV}$, thus covering the $\sim$ 20 – 60 eV interval in energy. This
is roughly the same ionisation interval covered by the [NeII]+[SIV], with
[SIV] having a IP of 34.79 eV and the next stage IP at 47.22 eV, or by
[NeIII]+[SIII], with [SIII]18.7$\,\rm{\rm{\upmu m}}$ that has a IP of 23.34
eV. Analyzing the former pair, we find the linear correlation between
L[NeII]+[SIV] and SFR shown in Fig. 5b and expressed as:

$\log\left(\frac{SFR}{\rm M_{\odot}\,yr^{-1}}\right)=(0.85\pm 0.05)\\\
+(0.94\pm 0.04)\log\left(\frac{L_{\rm[NeII]+[SIV]}}{\rm
10^{41}\,erg\,s^{-1}}\right)$ (7)

This relation has been calculated using 77 galaxies, of which 56 SFG and 21
LMG, and has a Pearson correlation coefficient $r$ = 0.93. The small number of
SFG used for determining this relation is due to the lack of [SIV]
measurements available in literature, due to the intrinsic line faintness in
star forming galaxies (see Table 3) and the position of the line in the
silicate absorption band in the 9-11$\,\rm{\rm{\upmu m}}$ range.

The relation between SFR and the [NeIII]+[SIII] emission sum is shown in Fig.
5c and can be expressed as:

$\log\left(\frac{SFR}{\rm M_{\odot}\,yr^{-1}}\right)=(1.16\pm 0.04)\\\
+(1.03\pm 0.04)\log\left(\frac{L_{\rm[NeIII]+[SIII]}}{\rm
10^{41}\,erg\,s^{-1}}\right)$ (8)

This relation has been calculated using 165 galaxies, of which 140 SFG and 25
LMG, and has a Pearson correlation coefficient $r$ = 0.86.

####  3.2.5 PAH SFR tracer

Figure 6: (a: left) Correlation between the PAH emission feature at
6.2$\,\rm{\rm{\upmu m}}$, expressed in units of $10^{41}\,\rm{erg\,s^{-1}}$
and the SFR derived from the total IR luminosity (black line) for a composite
catalogue of SFG (red star) and AGN dominated galaxies (blue square). The
shaded area in the three plots indicates the 95$\%$ confidence interval. (b:
centre) Correlation between the PAH emission feature at 11.3$\,\rm{\rm{\upmu
m}}$, expressed in units of $10^{41}\,\rm{erg\,s^{-1}}$ and the SFR derived
from the total IR luminosity (black line) for a composite catalogue of SFG
(red star) and AGN dominated galaxies(blue square). In panels (a) and (b) we
exclude ULIRGs from both AGN and SFG samples due to the known PAH deficit in
these sources, plus AGN with luminosities above $10^{45}\,\rm{erg\,s^{-1}}$
that could dominate the IR continuum used to estimate the SFR. (c: right)
Comparison between the relation between PAH total luminosity and SFR derived
by Shipley et al. (2016) (black solid line line) and using our sample (red
dashed line). Red stars indicate SFG, purple squares indicate the ULIRG sub-
sample included in the GOALS sample, and grey circles represent a sample of
local galaxies used in Xie & Ho (2019).

In the mid-IR range, the emission features due to the PAH molecules arise from
PDR around Hii regions embedding young stars (e.g., Draine & Li 2007), and can
be used as SFR tracers. We analyse the PAH features at 6.2 and
11.3$\,\rm{\rm{\upmu m}}$, considering the same sample of 155 SFG and the
sample of 103 AGN galaxies described in section 2 for the PAH analysis. As
discussed in Section 3.1, we have not computed the correlation between the PAH
and the IR luminosity in LMG, therefore we exclude the LMG for the SFR
determination with the PAH.

The use of the PAH as a measure of the SFR was originally proposed by Wu et
al. (2005), who used the Spitzer-IRAC camera (Fazio et al. 2004)
8$\,\rm{\rm{\upmu m}}$ band, whose flux density is dominated by the strongest
PAH feature, i.e that at 7.7$\,\rm{\rm{\upmu m}}$. They derive a calibration
of the 8$\,\rm{\rm{\upmu m}}$ SFR using the radio VLA emission at 1.4GHz and
the H$\alpha$ luminosity from the SFR-radio luminosity relation given by Yun
et al. (2001) and the SFR-H$\alpha$ luminosity from Kennicutt (1998),
respectively.

More recently, the relatively bright PAH features at 6.2, 7.7 and
11.3$\,\rm{\rm{\upmu m}}$ have been used to derive a total PAH luminosity and
has been correlated to the extinction corrected H$\alpha$ luminosity of a
sample of 227 galaxies (Shipley et al. 2016). For star-forming galaxies (105
galaxies), the total PAH luminosity correlates linearly with the extinction-
corrected H$\alpha$ luminosity.

In Fig. 6 we show the correlation of the single PAH features and the SFR:
panel (a) shows the relation calculated for the 6.2$\,\rm{\rm{\upmu m}}$
feature using 142 SFG and 56 AGN, while panel (b) shows the relation for the
11.3 $\,\rm{\rm{\upmu m}}$ feature, for which were used 142 SFG and 77 AGN.
When computing these correlations, we excluded all ULIRGs from the original
sample of AGN and SFG, following the approach by Pope et al. (2013) to avoid
saturation effects in the PAH-to-continuum luminosities. Moreover, when
considering the AGN sample, we discarded all sources with a total IR
luminosity greater than $L_{IR}=10^{45}\,\rm{erg\,s^{-1}}$, where the total IR
luminosity, and therefore the derived SFR, could be severely contaminated by
the AGN. Nevertheless, these are a few sources that do not affect the derived
fit parameter when they are included in the fit. The two relations are
described respectively by the following equations, which also report the
Pearson $r$ coefficient relative to each equation:

$\log\left(\frac{SFR}{\rm M_{\odot}\,yr^{-1}}\right)=(0.37\pm 0.04)\\\
+(0.76\pm 0.03)\log\left(\frac{L_{PAH6.2}}{\rm
10^{41}\,erg\,s^{-1}}\right),r=0.87$ (9)

$\log\left(\frac{SFR}{\rm M_{\odot}\,yr^{-1}}\right)=(0.41\pm 0.03)\\\
+(0.73\pm 0.02)\log\left(\frac{L_{PAH11.3}}{\rm
10^{41}\,erg\,s^{-1}}\right),r=0.89$ (10)

Both results show a shallow slope, significantly lower than unity. This is a
consequence of including the AGN sample. Considering only the SFG sample, the
slopes would increase, becoming 0.87 for the PAH feature at 6.2$\mu$m, and of
0.90 for the 11.3$\mu$m feature. On the other hand, while we excluded all
ULIRGs, including these objects does not result in a significant change in the
slope, that remains comparable within the errors. In particular, including the
entire AGN and SFG samples for the PAH feature at 6.2$\mu$m results in a slope
of 0.81$\pm$0.03, while for the PAH feature at 11.3$\mu$m the slope would be
of 0.78$\pm$0.03.

We compared our results with the one proposed by Shipley et al. (2016): in
order to properly determine the SFR using the Hα and 24$\,\rm{\rm{\upmu m}}$
luminosities, as done by these authors, we selected only the SFG, excluding
the AGN, leaving our sample composed of 100 sources with the 6.2, 7.7, 11.3
PAH features, the 24$\,\rm{\rm{\upmu m}}$ and the Hα data.We find that our
estimate of the SFR is higher by $\sim$0.25 dex than that derived by Shipley
et al. (2016) and almost linear in slope, (see Fig. 6c). We obtain:

$\log\left(\frac{SFR}{\rm M_{\odot}\,yr^{-1}}\right)=(-1.28\pm 0.23)+\\\
(1.08\pm 0.10)\log\left(\frac{L_{\Sigma PAH}}{\rm
10^{41}\,erg\,s^{-1}}\right)$ (11)

with a Pearson r coefficient of $r$=0.66. While the slope of our result is
comparable, within the error, to that obtained by Shipley et al., we attribute
the difference in the intercept to the significant difference in sample
characteristics. In particular, the disagreement arises when we extrapolate
the results obtained by Shipley et al. for local galaxies. The sample used by
Shipley et al. is in fact composed of galaxies with redshift in the $z\sim
0.2-0.6$ range, while our sample has a mean redshift of $z\sim 0.027$.
Moreover, while our sample is primarily composed of LIRGs and ULIRGs, the
sample used by these authors is mainly constituted of galaxies in the
$L_{IR}\sim 10^{9}-10^{12}L_{\odot}$ interval. We additionally compared our
derived relation to a sample of local SFG described by Xie & Ho (2019), for
which was available in the literature the SFR derived from H$\alpha$,
corrected by the flux at 24$\mu$m. This sample is composed of local galaxies
and, while the bulk of these galaxies present a lower total PAH luminosity, it
follows the same relation derived from the GOALS sample. This suggests that
the difference observed between our result and the result by Shipley et al. is
indeed due to an intrinsic difference of the sample used.

####  3.2.6 H2 SFR tracers

Figure 7: (a: left) Correlation between the H2 molecular line at
9.67$\,\rm{\rm{\upmu m}}$, expressed in units of $10^{41}\,\rm{erg\,s^{-1}}$
and the SFR derived from the total IR luminosity (black line) for a catalogue
of SFG (red star). (b: centre) Correlation between the H2 molecular line at
12.28$\,\rm{\rm{\upmu m}}$, expressed in units of $10^{41}\,\rm{erg\,s^{-1}}$
and the SFR derived from the total IR luminosity (black line) for a catalogue
of SFG (red star). The green triangle show LMG detection, not used to derive
the correlation. (c: right) Correlation between the H2 molecular line at
17.03$\,\rm{\rm{\upmu m}}$, expressed in units of $10^{41}\,\rm{erg\,s^{-1}}$
and the SFR derived from the total IR luminosity (black line) for a catalogue
of SFG (red star). In all three figures, the shaded ares shows the 95$\%$
confidence interval of the relations, and the purple squares highlight the
ULIRG population in the SFG sample.

Molecular hydrogen is the most abundant molecule in the Universe and can be
found in various environments. It can be excited by UV fluorescence (Black &
van Dishoeck 1987), shocks (Hollenbach & McKee 1989) or X-ray illumination
(Maloney 1997), thus probing various astrophysical environments. H2 forms on
the surface of dust grains, affecting the ISM chemistry, and acts as a
coolant. It is particularly important in all processes that regulate star
formation and galaxy evolution, where the principal mechanism of these lines
is associated with the UV radiation from massive stars.

Comparing theoretical models to observations, Rigopoulou et al. (2002) found
evidence that an important fraction of H2 emission in SFG can originate in
PDRs. In this section we test the use of different H2 molecular lines as SFR
tracers. In Fig. 7a we show the correlation between the H2 (S(3)) molecular
line at 9.67$\,\rm{\rm{\upmu m}}$ and the SFR determined from the total LIR
for the SFG in the GOALS sample. We find a relation expressed by:

$\log\left(\frac{SFR}{\rm M_{\odot}\,yr^{-1}}\right)=(1.57\pm 0.02)\\\
+(0.71\pm 0.03)\log\left(\frac{L_{S(3)}}{\rm 10^{41}\,erg\,s^{-1}}\right)$
(12)

determined using 168 SFG, with a Pearson $r$ coefficient of r=0.87. Fig. 7b
shows the correlation between the H2 (S(2)) line at 12.28$\,\rm{\rm{\upmu m}}$
and the SFR, determined with a sample of 126 SFG, with a Pearson coefficient
of $r$=0.86 and expressed by:

$\log\left(\frac{SFR}{\rm M_{\odot}\,yr^{-1}}\right)=(1.71\pm 0.02)\\\
+(0.87\pm 0.04)\log\left(\frac{L_{S(2)}}{\rm 10^{41}\,erg\,s^{-1}}\right)$
(13)

Finally, Fig. 7c shows the correlation between the H2 (S(1)) line at
17.03$\,\rm{\rm{\upmu m}}$ and the SFR, derived from a sample of 154 SFG, with
r=0.85 and expressed by:

$\log\left(\frac{SFR}{\rm M_{\odot}\,yr^{-1}}\right)=(1.39\pm 0.02)\\\
+(0.92\pm 0.04)\log\left(\frac{L_{S(1)}}{\rm 10^{41}\,erg\,s^{-1}}\right)$
(14)

In Fig. 7b and c we also report, in green, seven detections in LMG of S(2) and
S(1), respectively. We did not include the LMG sample in the correlation due
to the small number of sources, limiting ourselves to a comparison of the
results. While a good correlation is lacking for the S(1) line, for the S(2)
line we have a good agreement between the LMG population and the SFG one. This
can be of particular interest in those LMG in which the CO emission is
considerably lower than what it would correspond to the estimated SFR,
implying the presence of CO-dark molecular gas that is not traced by the CO
emission in LMG (Togi & Smith 2016). We speculate that the sub-linear slopes
obtained for the S(2) and S(3) lines might be linked to the different gas
excitation temperature associated with each line. In particular, by moving to
higher excitation lines, and thus to shorter wavelengths, the H2 transitions
are originated by an increasingly warmer gas in a thinner layer of the
molecular gas clouds. Tracing the warmest material might cause the flattening
of the slope, since colder star-forming clouds are not detected by the higher
transitions. Additionally, S(2) and S(3) lines could have a larger
contribution from other excitation mechanisms, as suggested by the different
excitation temperatures measured for these transitions in the Boltzmann
diagrams for nearby galaxies (e.g. Tommasin et al. 2010).

We have not included the AGN sample in this Section due to the different
excitation mechanisms that can contribute to the H2 rotational lines in these
sources, such as shocks or X-ray illumination. For the same reason H2 lines
are neither used as BHAR tracers in Section 3.3, since they respond to the
excitation temperature rather than the hardness of the ionising continuum.
This makes quite difficult to establish a clear connexion between the line
intensities and the SFR or the accreted mass onto the black hole when both
contributions are present.

###  3.3 BHAR tracers

The [OIV]25.9$\,\rm{\rm{\upmu m}}$ and the [NeV]24.3$\,\rm{\rm{\upmu m}}$
lines can be used to trace AGN activity. From our catalogue of AGN, derived
from the 12$\,\rm{\rm{\upmu m}}$ sample, we compiled from the literature the
2-10 keV X-rays fluxes corrected for absorption. We then selected all objects
with hydrogen column density of NH $\leq$ 5$\times$10${}^{23}\,\rm{cm^{-2}}$,
because we want to exclude Compton-thick objects for which the 2-10 keV X-rays
can be substantially absorbed, thus obtaining a sub-catalogue of 42 objects.
For these objects, we investigated the correlation between the
[OIV]25.9$\,\rm{\rm{\upmu m}}$ line luminosity and the 2-10 keV X-ray
luminosity (see Fig. 8a).

Figure 8: (a: left) Linear correlation between the [OIV]25.9$\,\rm{\rm{\upmu
m}}$ and the 2-10 keV X-ray luminosity. (b: centre) Linear correlation between
the [NeV]24.3$\,\rm{\rm{\upmu m}}$ line luminosity and the 2-10 keV X-ray
luminosity. (c: right) Linear correlation between the
[OIV]25.9$\,\rm{\rm{\upmu m}}$ line luminosity and the 19$\,\rm{\rm{\upmu m}}$
luminosity. All luminosities are expressed in units of
$10^{41}\,\rm{erg\,s^{-1}}$.

We find a correlation expressed by the equation:

$\log\left(\frac{L_{X}}{\rm 10^{41}\,erg\,s^{-1}}\right)=(1.85\pm 0.07)\\\
+(1.21\pm 0.10)\log\left(\frac{L_{[OIV]25.9}}{\rm
10^{41}\,erg\,s^{-1}}\right)$ (15)

with a Pearson $r$ coefficient of $r$=0.87. In order to obtain a measure of
the BHAR, it is necessary to convert the luminosity in the 2-10 keV band to
the bolometric luminosity of the object, and from there to the BHAR
$(L_{AGN}=\eta\dot{M}_{BH}c^{2})$. Different studies have been carried out to
determine the best bolometric correction to apply when considering the 2-10
keV luminosity, with those by Marconi et al. (2004) and Lusso et al. (2012)
being the most used in the literature. The resulting bolometric luminosities
obtained applying these corrections are different, giving us a range of
possible values. We applied both correction to our data, and then from the
resulting bolometric luminosities we calculated two linear relations linking
the [OIV]25.9$\,\rm{\rm{\upmu m}}$ line luminosity to the BHAR. Assuming a
radiative efficiency of $\eta$=0.1, we obtained in equation 16 the linear
correlation applying the bolometric correction from Lusso et al. (2012), and
in equation 17 the one applying the correction from Marconi et al. (2004). The
equations are reported each followed by its Pearson $r$ coefficient:

$\log\left(\frac{\dot{M}_{BH}}{\rm M_{\odot}\,yr^{-1}}\right)=(-1.65\pm
0.07)\\\ +(1.04\pm 0.09)\log\left(\frac{L_{[OIV]25.9}}{\rm
10^{41}\,erg\,s^{-1}}\right),\,\,r=0.86$ (16)

$\log\left(\frac{\dot{M}_{BH}}{\rm M_{\odot}\,yr^{-1}}\right)=(-1.66\pm
0.09)\\\ +(1.49\pm 0.12)\log\left(\frac{L_{[OIV]25.9}}{\rm
10^{41}\,erg\,s^{-1}}\right),\,\,r=0.87$ (17)

In a similar way, we first calculated the linear correlation between the
[NeV]24.3$\,\rm{\rm{\upmu m}}$ line luminosity and the 2-10 keV luminosity
(see right panel in Fig. 8). In this case, we have a total of 34 objects, due
to a smaller number of [NeV] available data, and we obtained a linear relation
described by the equation:

$\log\left(\frac{L_{X}}{\rm 10^{41}\,erg\,s^{-1}}\right)=(2.40\pm 0.08)\\\
+(0.95\pm 0.11)\log\left(\frac{L_{\rm[NeV]24.3}}{\rm
10^{41}\,erg\,s^{-1}}\right)$ (18)

with a Pearson coefficient of $r$=0.84. From this, we then applied the same
bolometric corrections, obtaining linear correlations between the line
luminosity and the BHAR, expressed by equations 19 and 20 for the Lusso et al.
(2012) and Marconi et al. (2004) correction respectively:

$\log\left(\frac{\dot{M}_{BH}}{\rm M_{\odot}\,yr^{-1}}\right)=(-1.11\pm
0.09)\\\ +(1.06\pm 0.12)\log\left(\frac{L_{\rm[NeV]24.3}}{\rm
10^{41}\,erg\,s^{-1}}\right),\,\,r=0.83$ (19)

$\log\left(\frac{\dot{M}_{BH}}{\rm M_{\odot}\,yr^{-1}}\right)=(-0.89\pm
0.11)\\\ +(1.47\pm 0.15)\log\left(\frac{L_{\rm[NeV]24.3}}{\rm
10^{41}\,erg\,s^{-1}}\right),\,\,r=0.84$ (20)

As a general trend, we find that both the [NeV]24.3$\,\rm{\rm{\upmu m}}$ and
[OIV]25.9$\,\rm{\rm{\upmu m}}$ lines correlate linearly with the 2-10 keV
X-ray luminosity, thus providing a good proxy to measure the AGN activity, as
shown in Fig. 8 a and b.

Following the work by Tommasin et al. (2010), we also analysed the correlation
between the [NeV]24.3 and [OIV]25.9$\,\rm{\rm{\upmu m}}$ lines with the
luminosity at 19$\,\rm{\rm{\upmu m}}$. The luminosity at 19$\mu$m (L19μm) was
used for two main reasons. On one hand, accurate Spitzer-IRS observations are
available for the 19$\mu$m flux density, from two different apertures, for the
considered sample of galaxies. On the other hand, the 19$\mu$m photometry data
are the only available accurate photometric data closest to the emission at
12$\mu$m. The 12$\mu$m emission is, in turn, the best proxy for the bolometric
flux of an active galaxy (Spinoglio et al. 1995). In particular, Tommasin et
al. (2010) found that the 19$\mu$m luminosity correlates with the
[NeV]14.3$\mu$m line luminosity.

The L19μm has been used by Tommasin et al. to compute the percentage of AGN
and starburst components in 51 sources of their sample. Following these
results, we selected all sources with an AGN component at 19$\,\rm{\rm{\upmu
m}}$ equal or above 85$\%$, and for this sub-sample of 35 objects we
determined the correlation between the [OIV]25.9$\,\rm{\rm{\upmu m}}$ and the
[NeV]24.3$\,\rm{\rm{\upmu m}}$ line luminosities with the 19$\mu$m luminosity.
These correlations are shown in Fig. 8c and Fig. 9a, respectively, and
expressed by Eq. 21 and 22 respectively, where $r$ indicated the Pearson
correlation coefficient, and $n$ the number of objects used to derive the
correlation.

$\log\left(\frac{L_{19{\mu}m}}{\rm 10^{41}\,erg\,s^{-1}}\right)=(2.72\pm
0.06)\\\ +(0.77\pm 0.10)\log\left(\frac{L_{\rm[OIV]25.9}}{\rm
10^{41}\,erg\,s^{-1}}\right),\,\,r=0.81,\,\,n=32$ (21)

$\log\left(\frac{L_{19{\mu}m}}{\rm 10^{41}\,erg\,s^{-1}}\right)=(3.12\pm
0.06)\\\ +(0.81\pm 0.10)\log\left(\frac{L_{\rm[NeV]24.3}}{\rm
10^{41}\,erg\,s^{-1}}\right),\,\,r=0.85,\,\,n=30$ (22)

Figure 9: (a: left) Linear correlation between the [NeV]24.3$\,\rm{\rm{\upmu
m}}$ line luminosity and the 19$\,\rm{\rm{\upmu m}}$ luminosity, expressed in
units of $10^{41}\,\rm{erg\,s^{-1}}$. (b: centre) Comparison of the three
different relations between [OIV]25.9$\,\rm{\rm{\upmu m}}$ line luminosity and
the BHAR: the blue dashed line reports the relation obtained from the Lusso et
al. bolometric correction, the red dotted line the relation from the Marconi
et al. bolometric correction, and the green solid line the results from the
Spinoglio et al. correction. (c: right) Same as panel (a), for the
[NeV]24.3$\,\rm{\rm{\upmu m}}$ line.

We then used these relations, (equations 21 and 22,) to determine the
bolometric luminosity of our sources, and then the accretion rate. In order to
calculate the bolometric luminosity, we used the relation, calculated by
Spinoglio et al. (1995), that links the bolometric luminosity to the
12$\,\rm{\rm{\upmu m}}$ luminosity. The 12$\,\rm{\rm{\upmu m}}$ luminosity for
our sample was determined using data from Deo et al. (2009): the authors
report continuum measurements at 5.5$\,\rm{\rm{\upmu m}}$,
14.7$\,\rm{\rm{\upmu m}}$ and 20$\,\rm{\rm{\upmu m}}$ taken from Spitzer low-
resolution spectroscopic observations. Where possible, we interpolated the
continuum slope and the 12$\,\rm{\rm{\upmu m}}$ continuum flux using the
5.5$\,\rm{\rm{\upmu m}}$ and 14.7$\,\rm{\rm{\upmu m}}$ measurements, otherwise
(but only in three cases) using the extrapolation from the
14.7$\,\rm{\rm{\upmu m}}$ and 20$\,\rm{\rm{\upmu m}}$ fluxes. We then matched
the Deo et al. sample with the Tommasin et al. sample, calculated the
bolometric luminosity starting from the monochromatic 12$\,\rm{\rm{\upmu m}}$
luminosity for those sources with a 85$\%$ AGN component. Similarly to what
has been done for the 2-10 keV luminosity, we determined the BHAR starting
from the bolometric luminosity derived from the 12$\,\rm{\rm{\upmu m}}$
luminosity, and its relation to the [NeV]24.3$\,\rm{\rm{\upmu m}}$ and
[OIV]25.9$\,\rm{\rm{\upmu m}}$ line luminosities, obtaining the following
relations:

$\log\left(\frac{\dot{M}_{BH}}{\rm M_{\odot}\,yr^{-1}}\right)=(-1.14\pm
0.07)\\\ +(0.67\pm 0.15)\log\left(\frac{L_{\rm[OIV]25.9}}{\rm
10^{41}\,erg\,s^{-1}}\right),\,\,r=0.68,\,\,n=26$ (23)

$\log\left(\frac{\dot{M}_{BH}}{\rm M_{\odot}\,yr^{-1}}\right)=(-0.77\pm
0.10)\\\ +(0.77\pm 0.16)\log\left(\frac{L_{\rm[NeV]24.3}}{\rm
10^{41}\,erg\,s^{-1}}\right),\,\,r=0.71,\,\,n=24$ (24)

The smaller number of objects used to derive these relations is due to the
lack of data for the determination of the continuum at 12$\,\rm{\rm{\upmu
m}}$.

Measuring the BHAR requires important approximations in terms of bolometric
correction, which can yield significantly different results. Between the three
proposed bolometric corrections, we note that the correlations obtained when
applying the Marconi correction are steeper than those obtained from the Lusso
and Spinoglio correction. In particular, we compare the results in Fig. 9: for
both the [OIV]25.9 $\mu$m (panel b, at the centre) and
[NeV]24.3$\,\rm{\rm{\upmu m}}$ (panel c, on the right) lines, we note that the
results derived from the corrections of Spinoglio et al. (1995) and Lusso et
al. (2012) show a similar, flatter slope, indicating, within the errors, the
expected linear relation between the line tracers and the BHAR. It is
important to note that the Marconi et al. and Lusso et al. bolometric
corrections are based on a third-degree polynomial transformation of the X-ray
luminosity. This necessarily leads to deviations from the slopes obtained in
Eq. 15 and 18.

We refer to the Appendix C for a discussion on the use of the mid-ionisation
lines of [NeIII]15.5 $\mu$m and [SIV]10.5 $\mu$m as alternative BHAR tracers.

##  4 Discussion

###  4.1 Applicability of our results to composite objects

Figure 10: Comparison of the ratio between the SFR obtained using the
[NeII]+[NeIII] tracer and the SFR obtained using the PAH feature at
11.3$\mu$m, against the EW of the PAH feature at 6.2$\mu$m for the GOALS
sample. The colour gradient indicates the percentage of the AGN component in
each object as determined by mid-IR tracers (Díaz-Santos et al. 2017). The
black dashed line shows the mean value of the SFR ratio for the population,
with the shaded area indicating where the 50$\%$ of the entire sample is
located. The red dotted lines represent the 1$\sigma$ interval around the mean
value.

The coexistence of AGN and star formation in galaxies is well known (see, e.g.
Pérez-Torres et al. 2021, for a review) and therefore we expect that many
galaxies are characterised by both components at work together. In general,
when considering composite objects, with both a SF and an AGN component
detectable, our results in Section 3.2 and Section 3.3 can still be applied.

We use the GOALS sample to determine whether the results on the SFR tracers
can be applied to composite objects (Section 3.2). The GOALS sample is
composed of SFG, but for part of the sample an AGN component is also present,
if not always detected. For our calibrations, we have excluded all sources for
which a [NeV] emission line is detected. Here we instead include all objects
of the sample. In Fig. 10 we plot the ratio of the SFR obtained using the
[NeII]+[NeIII] tracer to the SFR obtained using the 11.3$\mu$m PAH feature, as
a function of the Equivalent Width (EW) of the PAH feature at 6.2$\mu$m, as
determined by Stierwalt et al. (2014). The mean ratio for the entire
population is -0.049, with the median value equal to -0.023 and 50$\%$ of the
sample included in the [-0.18, 0.06] interval. While there is some dispersion
around zero, for the majority of the sample we obtain a similar SFR whether we
use the [NeII]+[NeIII] tracer or the PAH tracer. This suggests that for mixed
objects these two tracers are equivalent, and independent of the presence of
an AGN. Significant differences are observable for the extreme objects in the
sample: the sources with the larger AGN content are located in the top left
corner, for which the SFR obtained using the [NeII]+[NeIII] tracer is
significantly higher than the SFR obtained using the PAH. This is plausibly
due to an increase of the [NeIII] emission related to the presence of the AGN.

###  4.2 Comparison among the various tracers

Summarising the results presented in Section 3.2, among the various SFR
tracers presented, for galaxies dominated by SF processes we suggest that the
best tracers are the [CII]158$\mu$m line (slope $\alpha=0.89\pm 0.02$ and
correlation coefficient $r$ = 0.92) or the combination of the [OI] and [OIII]
lines (slope $\alpha=1.25\pm 0.04$ and $r$ = 0.94) for high-redshift galaxies
observed from the ground with submillimeter telescopes. For galaxies observed
from space or airborne facilities (but also from the new generation of the
very large optical ground based telescopes in the 8-13$\mu$m atmospheric
window, for local galaxies), the best SFR tracers are the combination of the
[NeII] and [NeIII] lines (slope $\alpha=0.96\pm 0.03$ and $r$ = 0.91) or the
combination of the other mid-IR fine-structure lines. For galaxies containing
an AGN, the PAH features can reliably be used and the PAH 11.3$\mu$m feature
is probably the best SFR tracer (slope $\alpha=0.75\pm 0.04$ and $r$ = 0.79).
As an alternative, the [NeII] and [NeIII] lines can still be used also in AGN,
with the correction to the total neon flux that can be computed using one of
the [NeV] lines, as suggested by Zhuang et al. (2019).

When measuring the BHAR, following the results presented in Section 3.3, the
[NeV] lines at 14.3 and 24.3$\mu$m are exclusive probes of AGN activity,
because their emission is a direct signature of the hard ionising spectrum due
to the accretion process. However, these lines are fainter than the
[OIV]25.9$\mu$m line, which can be detected more easily in faint objects. When
using the [OIV] line as a BHAR tracer, it is important to keep into account
the possible contamination due to SF processes, because its emission can be
attributed also, to some extent, to starburst activity (Lutz et al. 1998).

###  4.3 Comparison with previous line calibrations

We briefly compare here the results of the line calibrations obtained in this
work with those of Spinoglio et al. (2012, 2014) and Gruppioni et al. (2016),
while leaving all the details of the comparisons to Appendix E. We note that
in this work, to derive the correlations, we used the orthogonal distance
regression method, instead of the least-squares minimisation method, which was
used by the other authors, because we consider the total IR luminosity and the
line luminosity two independent variables.

The AGN sample used in this work and in Spinoglio et al. is the same for the
lines in the 10-35 $\,\rm{\rm{\upmu m}}$ interval, and the differences in the
correlation are only due to the different methods of analysis. We note that in
this work, in order to obtain a correlation between the total LIR and the
[OIV]25.9$\,\rm{\rm{\upmu m}}$ line luminosity that better represents the AGN
population, we did not use the entire sample of AGNs by Tommasin et al.
(2008); Tommasin et al. (2010), but a sub-sample of objects with an AGN
component in the 19$\,\rm{\rm{\upmu m}}$ luminosity of at least 85$\%$ (see
Section 3.1).

For the SFG sample, while in Spinoglio et al. the sample of galaxies described
by Bernard-Salas et al. (2009) was used, in this work we have expanded the
same sample by including the LIRGs and ULIRGs sample of GOALS, as described in
Section 2. Nonetheless, we obtain comparable results, except in the cases of
the PAH feature at 11.3$\,\rm{\rm{\upmu m}}$ and of the H2 line at
17.03$\,\rm{\rm{\upmu m}}$, for which our results show line luminosities an
order of magnitude higher. This is due to the presence, in our sample, of
LIRGs and ULIRGs, which shift the relation toward a steeper slope.

For the lines at wavelengths in the 50-158 $\mu$m range, we refer to the
Appendix E, where we discuss the differences and plot the results of the
different calibrations of Spinoglio et al. with respect to our results.

Because of the different analytical method used to determine the correlations
between total and line luminosities, for the comparison with Spinoglio et al.,
we have recomputed our calibrations using the least square fit and give the
results in the Appendix E. For the comparison with Gruppioni et al., instead,
we recomputed the correlations of the Gruppioni et al. sample, applying the
orthogonal distance regression and then compare these with our results. As a
general trend, we find good agreement between our results and those by
Gruppioni et al., with comparable slopes within 2$\sigma$ of each other. For
all the details and the plots of the results of the different calibrations, we
refer to the Appendix E.

We do not compare our results with those obtained by Bonato et al. (2019),
since the methods of analysis are widely different. In particular, while in
this work we calibrate the line luminosities leaving the slope of the
correlations as a free parameter, in Bonato et al. (2019) the slope of the
relation was fixed to unity, thus giving raise to substantial differences in
the results.

###  4.4 Metallicity and SFR tracers

In Section 3.2.2 we revise the [CII]-SFR relation for a wide galaxy sample,
from LMG to extreme ULIRGs. In Section 3.2.4 we derive a measure of the SFR
through the neon and sulfur mid-IR lines and propose new SFR tracers using
different combinations of these lines. In this section we discuss the possible
effects that metallicity, and the associated changes in the ISM of these
galaxies, may have on these tracers.

First of all, for the sample of dwarf galaxies, we adopted SFR values derived
from the observed Hα luminosity and corrected from the total IR luminosity
(Rémy-Ruyer et al. 2015). This is motivated by the underestimation of the SFR
by the IR luminosity at very low metallicities ($12+\log(O/H)\lesssim 8.5$;
Lee et al. 2013), due to the lower metal abundance in these galaxies compared
to SFG. In principle, the lower dust to gas ratio of LMG should also have an
impact on the observed intensities of the fine-structure lines. This is,
however, balanced by the higher cooling rates in these transitions, as
discussed by De Looze et al. (2014).

Fig. 3a shows that the [CII]$158\,\rm{\rm{\upmu m}}$ emission in LMG follows
the trend found in SFG with no need to perform any additional correction for
metallicity in these galaxies. Similarly, the different combinations of
[NeII]$12.8\,\rm{\rm{\upmu m}}$, [NeIII]$15.6\,\rm{\rm{\upmu m}}$,
[SIII]$18.7\,\rm{\rm{\upmu m}}$, and [SIV]$10.5\,\rm{\rm{\upmu m}}$, shown in
Figs. 3 and 5, follow the correlation of solar-like metallicity galaxies. The
higher cooling rates in LMG are particularly evident when the neon and sulfur
transitions are considered. While the [NeII]$12.8\,\rm{\rm{\upmu m}}$ emission
in LMG scales with $L_{\rm IR}$ similarly as for SFG (Fig. 1b), the
[NeIII]$15.6\,\rm{\rm{\upmu m}}$ line becomes comparatively much brighter for
a given IR luminosity, more than one order of magnitude above the correlation
found for SFG, as can also be seen by the value of the constant $b$ in the
best fit equation for the [NeIII]15.6$\,\rm{\rm{\upmu m}}$ line in Table 2.
When considering the sulfur lines, this effect is even more pronounced (see
Fig. 17a and Table 2). This means that mid- to high-ionisation species such as
Ne2+ or S3+ trace a contribution to the star-formation that is not revealed by
either the Ne+ and $S^{2+}$ low-ionisation gas or the IR emission. Thus, the
combination of low and high-ionisation lines allows us to trace the total star
formation in both low- and solar-metallicity galaxies (Ho & Keto 2007; Zhuang
et al. 2019).

In the case of the [CII]158$\,\rm{\rm{\upmu m}}$ line, the Fig. 3a suggests
that this transition still remains a dominant coolant of the ISM at low
metallicities. Given its low ionisation potential (11.3 eV, see Table 2), this
line can originate from both neutral and ionised gas, and one could expect a
decreasing contribution from the neutral component as the ionisation field
becomes harder at low metallicities. However, Cormier et al. (2019)
demonstrated that the PDR contribution to the global [CII]$158\,\rm{\rm{\upmu
m}}$ emission is still dominant for the same LMG sample used in this work.
This is also in line with the results of Croxall et al. (2017), suggesting
that the [CII]$158\,\rm{\rm{\upmu m}}$ emission linked to ionised gas is of
the order of $\sim 10\%$ in LMG, and up to maximum $<40\%$ where a high-U is
required. Moreover, analytical models developed in these studies show a
decrease of [CII] emission from ionised gas with decreasing metallicity with
$\sim 55-75\%$ of the [CII] emission arising from PDRs in LMG, reaching almost
100$\%$ when the metallicity decreases below 1/4 Z⊙. Additionally, the
thickness of the [CII] layer increases for molecular clouds exposed to the
harder radiation fields typical of LMG. This is shown by the detection of
higher [CII]/CO(1-0) ratios in local LMG when compared to SFG with solar or
super-solar metallicity (e.g. Madden et al. 1997; Madden 2000; Hunter et al.
2001), and it is supported by PDR models (Bolatto et al. 1999; Röllig et al.
2006).

While a detailed study of the ionised gas and PDR structure is out of the
scope of the present work, the results discussed above suggest that both
[CII]$158\,\rm{\rm{\upmu m}}$ and the different combinations of neon and
sulphur lines are robust star formation tracers, virtually independent of dust
extinction, that can be applied to a wide diversity of environments with
different physical conditions and metallicities. Specifically, the variations
expected from the changes in the chemical abundances are mostly balanced by
the increase in the cooling rates of these transitions.

###  4.5 Comparison with high-z data

Figure 11: (a: left) Comparison between the LIR-L[CII]158 relation (black
dashes line) for local AGN dominated galaxies (gray squares) and high redshift
detections for QSOs (blue symbols). The shaded area shows the 95$\%$
prediction interval. (b: centre) Same for high redshift starburst galaxies
(pink symbols). (c: right) Comparison of the local log(L[CII]-log(SFR)
relation (black dashes line) and L[CII]-SFR values of high redshift sources
(pink symbols). Grey dots represent local star forming galaxies. The shaded
area shows the 95$\%$ prediction interval for the local relation.

Figure 12: (a: left) Comparison between the LIR-L[OIII]88 relation (black
dashes line) for local AGN dominated galaxies (gray squares) and high redshift
detections for QSOs (blue symbols). (b: centre) Comparison between the
LIR-L[OIII]88 relation (red dashes line) for local SFG (gray stars) and high
redshift detections for starburst galaxies (pink symbols). The red shaded area
shows the 95$\%$ prediction interval of this relation. For comparison, the
green dashed line shows the LIR-L[OIII]88 relation for local LMG, with gray
squares indicating local LMG and the green shaded area giving the 95$\%$
prediction interval for this relation. (c: right) Comparison of the local
LIR-L[NII]205 relation (black dashes line) for high redshift detections of SFG
(pink symbols). Grey stars represent local star forming galaxies. The shaded
area in all the figures shows the 95$\%$ prediction interval for the local
relation.

We compare the calibrations described in Section 3.1 obtained for local
galaxies to high-redshift detections of sources obtained with ALMA. In
particular, we consider detections of the [OIII]88$\,\rm{\rm{\upmu m}}$,
[NII]122$\,\rm{\rm{\upmu m}}$, [CII]158$\,\rm{\rm{\upmu m}}$ and
[NII]205$\,\rm{\rm{\upmu m}}$ lines at $z\geq 3$. Sources identified as QSOs
are compared to local AGN results, while sources for which a classification is
not given in the literature, or are classified as starburst galaxies, are
compared to local SFG.

Fig. 11a shows the comparison between local and high redshift detections of
the [CII]158$\,\rm{\rm{\upmu m}}$ line for QSOs galaxies, and in particular 27
detections at z$\sim$6 reported in Venemans et al. (2020), plus the detections
by Walter et al. (2018) for one source at z$\sim$6.08 and by Hashimoto et al.
(2018) for two sources at z$\sim$7.1, for a total of 30 sources. Fig. 11b
displays the comparison of local and high redshift SFG: 84 detections from the
ALPINE catalogue (Faisst et al. 2020) plus other 9 detections (Inoue et al.
2016; Vishwas et al. 2018; Walter et al. 2018; De Breuck et al. 2019;
Hashimoto 2019; Harikane et al. 2019; Rybak et al. 2019) for a total of 93
objects with redshift in the 4.2$\lesssim$z$\lesssim$7.2 range. While some
outliers are present, the bulk of the detections in both cases lies within the
prediction interval, that we show in the figures at the 95% level, suggesting
that the relations derived for local galaxies hold for high redshift sources.

We compare in Fig. 11c the L[CII]-SFR relation extrapolated for local galaxies
(see Section 3.2.2) with ALPINE detections of the [CII]158$\,\rm{\rm{\upmu
m}}$ emission line in starburst galaxies. We find that high redshift objects
follow the same L[CII]-SFR relation as local SFG and LMG, thus suggesting
little or no evolution in the L[CII]-SFR relation. In particular, no deficit
of [CII] emission is seen for the highest SFR values of $\sim
100\,\rm{M_{\odot}\,yr^{-1}}$, suggesting that starbursts at high-z might not
behave like local ULIRGs such as those shown in Fig. 3 (purple squares). An
analogues result was found by Schaerer et al. (2020), who analysed a large
sample of galaxies at high redshift (z$\sim$ 4-6) observed by ALMA. Leung et
al. (2020) explored the possibility of tracing the SFR at the Epoch of
Recombination with [CII]158$\,\rm{\rm{\upmu m}}$ using simulated data.
Although they find a good correlation, when compared to observed data, the
simulated sample shows an average [CII] luminosity lower than the one obtained
from ALMA pointed observations. A similar result was obtained for simulations
of high redshift systems (Ferrara et al. 2019) showing the presence of under-
luminous [CII] emission.

In Fig. 12a we show the comparison between the local LIR-L[OIII]88 relation
for AGN and high redshift detections in QSOs (Hashimoto et al. 2018; Walter et
al. 2018), while panel (b) shows the relation between local and high redshift
SFG (Inoue et al. 2016; Vishwas et al. 2018; Walter et al. 2018; De Breuck et
al. 2019; Harikane et al. 2019; Hashimoto 2019; Tamura et al. 2019). We note
that, while the high-redshift detections of [OIII]88$\,\rm{\rm{\upmu m}}$ in
QSOs are comparable with the relation derived from local AGN dominated
galaxies, in SFG the majority of high redshift detections appears to be one to
two orders of magnitude brighter at comparable LIR. These detections are
better represented by the LIR-L[OIII]88 relation derived for local LMG (see
Fig. 12b). This suggests that the [OIII]88$\,\rm{\rm{\upmu m}}$ line is
produced in an environment with a higher ionising spectrum, similar to the ISM
of local LMG (see Section 3.1).

Finally, the high redshift detections in panel (c) show the local
LIR-L[NII]205 relation for SFG and 17 high-redshift detections of
[NII]205$\,\rm{\rm{\upmu m}}$ (Cunningham et al. 2020) with the local
correlation that well represents also the high redshift detections. We report
in Fig. 21 the comparison between the local LIR-L[NII]122 relation and high
redshift SFG: in this case only one object (De Breuck et al. 2019) has been
detected, while three (Harikane et al. 2019) objects have upper limits. Thus,
only one object is located almost an order of magnitude higher than the local
relation for SFG, and therefore there is no statistical evidence for a
difference in the production mechanism of [NII] in high redshift objects,
especially considering the results for the [NII]205$\,\rm{\rm{\upmu m}}$ line.

###  4.6 Application to present and future facilities

The lines and features presented in this work can be observed in galaxies by
present and future IR/(sub)-mm facilities. The JWST will be the next NASA
observatory to explore the Universe in the near- and mid-IR spectral range. In
particular, the JWST Mid-InfraRed Instrument (MIRI, Rieke et al. 2015; Wright
et al. 2015) will be able to obtain imaging and spectra with unprecedented
sensitivity in the $4.9$–$28.9\,\rm{\rm{\upmu m}}$ spectral range. In Fig. 13a
we compare the MIRI wavelength interval to the observability of key mid-IR
lines at various redshifts. In particular, from the analysis presented in
Section 3.2, it will be possible to study the SFR up to redshift z$\sim$3.5
with the PAH feature at 6.2$\,\rm{\rm{\upmu m}}$, up to redshift z$\sim$1.5
with the PAH feature at 11.3$\,\rm{\rm{\upmu m}}$, or up to z$\sim$0.8 using
combinations of the neon and sulfur lines.

On the other hand, while we do not report in Fig. 13a the [NeV]24.3 and
[OIV]25.9$\,\rm{\rm{\upmu m}}$ lines due to the limited wavelength range
available for JWST, we show the [NeV]14.3$\,\rm{\rm{\upmu m}}$ line, for which
the same analysis carried out in Section 3.3 to derive the BHAR is applied and
reported in Appendix C. The [NeV]14.3$\,\rm{\rm{\upmu m}}$ line will be
observed by JWST-MIRI up to redshift z$\lesssim$1, while the
[NeV]24.3$\,\rm{\rm{\upmu m}}$ line will be observed up to redshift
z$\lesssim$0.15, and the [OIV]25.9 $\,\rm{\rm{\upmu m}}$ line will be observed
up to redshift z$\lesssim$0.1.

While JWST-MIRI will be able to observe molecular and atomic lines up to
redshift z$\sim$2, and the PAH up to higher redshift (z$\sim$3.5 for the
6.2$\mu$m feature), ALMA is able to trace the far-IR lines, like
[CII]158$\,\rm{\rm{\upmu m}}$, up to redshift z$\sim$8, as shown in Fig. 13b,
providing important information on the evolution of galaxies with cosmic time.
An analogues study can be found in Fig. 1 of Carilli & Walter (2013), where
the CO transitions and other key tracers of the ISM are shown as a function of
redshift versus frequency in terms of the observability by ALMA and JVLA (Karl
J. Jansky Very Large Array, Perley et al. 2011). We also report in Fig. 13(b)
the redshift of currently available data of ALMA detections: while lines like
[CII]158$\,\rm{\rm{\upmu m}}$ or [NII]205$\,\rm{\rm{\upmu m}}$ are detected
over a significant redshift interval, shorter wavelength lines, such as
[OI]63$\,\rm{\rm{\upmu m}}$, present very few results. This is due to the
difficulty of observing in the bands at higher frequency, for which the
atmospheric absorption allows good visibility only for 10$\%$ of the total
observational time.

In a near future the extremely large ground-based telescopes under
construction, with dedicated instruments for N-band spectroscopy will be able
to obtain observations of the mid-IR spectral range 8–13 $\mu$m and thus will
test the [SIV]10.5μm \+ [NeII]12.8μm relation, for local galaxies and AGN. The
ESO Extremely Large telescope (E-ELT; Gilmozzi & Spyromilio 2007) is expected
to obtain first light at the end of 2025, with the Mid-infrared ELT Imager and
Spectrograph (METIS, Brandl et al. 2021) instrument covering the N band with
imaging and low resolution spectroscopy, the Thirty Meter Telescope (TMT;
Schöck et al. 2009) will have completed first light and will be ready for
science at the end of 2027, and the Giant Magellan Telescope (GMT; Johns et
al. 2012) will be operational in 2029.

Figure 13: (Left) Observability of key mid-IR (dashed lines) lines compared to
the JWST-MIRI instrument (shaded area) in the 0-1.6 redshift interval. (Right)
Observability of key far-IR (dashed lines) lines compared to the ALMA bands
(shaded area) in the 1-8 redshift interval. Dots represent current detection
for each line.

##  5 Summary and conclusions

In this work we systematically revise the calibration of lines and features in
the 6-205$\mu$m spectral range. We report new line to LIR calibrations for
three galaxy types: active galactic nuclei (AGN), low-metallicity galaxies
(LMG) and star-forming galaxies (SFG), building well defined samples with
available IR spectroscopy mainly obtained from Herschel and Spitzer. The main
results of this work are:

  * •

Statistically robust calibrations to the total IR luminosity have been
obtained for the bright IR lines and features, including the PAH features, the
H2 pure rotational lines and the fine-structure lines, mostly for all the
three galaxy populations, excluding the PAH features for the LMG and the high-
ionisation fine structure lines for the SFG and LMG.

  * •

The [CII]158$\,\rm{\rm{\upmu m}}$ line can be used as SFR tracer for both LMG
and SFG, covering 6 order of magnitudes in luminosity, independently of the
source metallicity. Moreover, when compared to high redshift detections, the
correlation obtained for local galaxies can still be applied.

  * •

The sum of the two far-IR oxygen lines, the [OI]63$\mu$m line and the
[OIII]88$\mu$m line, can be used as a tracer for the SFR.

  * •

A combination of low- and intermediate-ionisation lines from neon and/or
sulfur is also a robust proxy for the SFR, independent of the metallicity. In
particular, the [SIV]$10.5\,\rm{\rm{\upmu m}}$+[NeII]$12.8\,\rm{\rm{\upmu m}}$
tracer will be accessible to ELTs facilities in the near future for galaxies
in the local Universe.

  * •

The brightest PAH features at 6.2 and 11.3 $\mu$m can be reliably used for SFR
tracers in both SFG and AGN, while the lack of these features prevents their
use in LMG.

  * •

We present for the first time a correlation between the SFR and the H2
molecular line fluxes at 9.7, 12.3 and 17.0$\mu$m.

  * •

The [CII]158$\mu$m line, the combination of the two neon lines
([NeII]12.8$\mu$m and [NeIII]15.5$\mu$m) and, for solar-like metallicity
galaxies that may harbour an AGN, the PAH11.3$\mu$m feature are the best SFR
tracers.

  * •

The [NeV]14.3$\mu$m, [NeV]24.3$\mu$m and [OIV]25.9$\mu$m lines are good
tracers of the BHAR, derived either from the 2-10 keV luminosity or from the
12 $\mu$m luminosity.

These results will assume particular relevance with the upcoming launch of the
JWST, which will observe with the MIRI instrument many of the mid-IR features
in galaxies of the Local Universe and beyond, up to redshift of $\sim$1 for
the brightest lines, thus allowing to measure with spectroscopy both the SFR
and the BHAR.

###### Acknowledgements.

We thank the anonymous referee for his/her comments which helped to improve
the present article. LS and JAFO acknowledge financial support by the Agenzia
Spaziale Italiana (ASI) under the research contract 2018-31-HH.0.

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##  Appendix A Characteristics of IR fine-structure lines

Table 2: Fine-structure lines in the mid- to far-IR range. The columns
correspond to the central wavelength, frequency, ionisation potential,
excitation temperature, critical density, spectral and spatial resolution of
the data presented in this work, and the number of AGN, starburst, and dwarfs
galaxies with line detections above $3\times$ rms. Critical densities and
excitation temperatures from: Launay & Roueff (1977); Tielens & Hollenbach
(1985); Greenhouse et al. (1993); Sturm et al. (2002); Cormier et al. (2012);
Goldsmith et al. (2012); Farrah et al. (2013).

[b]  Line $\lambda$ $\nu$ I.P. E $n_{\rm cr}$ Spec. Res. Ang. Res.
($\,\rm{\rm{\upmu m}}$) (GHz) (eV) (K) ($\rm{cm^{-3}}$) ($\rm{km\,s^{-1}}$)
(arcsec) [Ne vi]2P3/2–2P1/2 7.65 39188.56 126.21 1888 $2.5\times 10^{5}$ $\sim
1500$ $14\times 20$ H2 (0,0) S(3) 9.66 31034.41 4.48 2504 $9\times 10^{5}$a
$\sim 500$ $4.7\times 11.3$ [S iv]2P3/2–2P1/2 10.51 28524.50 34.79 1369
$5.39\times 10^{4}$ $\sim 500$ $4.7\times 11.3$ H2 (0,0) S(2) 12.28 24413.07
4.48 1682 $2\times 10^{5}$a $\sim 500$ $4.7\times 11.3$ [Ne ii]2P1/2–2P3/2
12.81 23403.00 21.56 1123 $7.00\times 10^{5}$ $\sim 500$ $4.7\times 11.3$ [Ne
v]3P2–3P1 14.32 20935.23 97.12 1892 $3\times 10^{4}$ $\sim 500$ $4.7\times
11.3$ [Ne iii]3P1–3P2 15.56 19266.87 40.96 925 $2.68\times 10^{5}$ $\sim 500$
$4.7\times 11.3$ H2 (0,0) S(1) 17.03 17603.78 4.48 1015 $2\times 10^{4}$a
$\sim 500$ $4.7\times 11.3$ [S iii]3P2–3P1 18.71 16023.11 23.34 769
$2.22\times 10^{4}$ $\sim 500$ $4.7\times 11.3$ [Ne v]3P1–3P0 24.32 12326.99
97.12 596 $5.0\times 10^{5}$ $\sim 500$ $11.1\times 22.3$ [O iv]2P3/2–2P1/2
25.89 11579.47 54.94 555 $10^{4}$ $\sim 500$ $11.1\times 22.3$ [Fe
ii]a6D7/2–a6D9/2 25.98 11539.35 7.9 553 $2.2\times 10^{6}$ $\sim 500$
$11.1\times 22.3$ H2 (0,0) S(1) 17.03 17603.78 4.48 1015 $2\times 10^{4}$a
$\sim 500$ $4.7\times 11.3$ [S iii]3P1–3P0 33.48 8954.37 23.34 430 $7.04\times
10^{3}$ $\sim 500$ $11.1\times 22.3$ [Si ii]2P3/2–2P1/2 34.81 8612.25 8.15 413
$3.4\times 10^{5}$a,$10^{3}$ $\sim 500$ $11.1\times 22.3$ [O iii]3P2–3P1 51.81
5787.57 35.12 441 $3.6\times 10^{3}$ $\sim 105$ 9.4c [N iii]2P3/2–2P1/2 57.32
5230.43 29.60 251 $3.0\times 10^{3}$ $\sim 105$ 9.4 [O i]3P2–3P1 63.18 4744.77
– 228 $4.7\times 10^{5}$a $\sim 86$ 9.4 [O iii]3P1–3P0 88.36 3393.01 35.12 163
$510$ $\sim 124$ 9.4 [N ii]3P2–3P1 121.90 2459.38 14.53 118 $310$ $\sim 290$
9.4 [O i]3P1–3P0 145.52 2060.07 – 98 $9.5\times 10^{4}$a $\sim 256$ 10.3 [C
ii]2P3/2–2P1/2 157.74 1900.54 11.26 91
$20$,$[2.2{\textsuperscript{a}},4.4{\textsuperscript{b}}\ ]\times 10^{3}$
$\sim 238$ 11.2 [N ii]3P1–3P0 205.3 1460.27 14.53 70 $48$ $\sim 297$ $\sim
16.8$

  * a

Critical density for collisions with hydrogen atoms.

  * b

Critical density for collisions with H2 molecules.

  * c

The beam size for Herschel/PACS is dominated by the spaxel size
($9\hbox{$.\\!\\!^{\prime\prime}$}4$) below $\sim 120\,\rm{\rm{\upmu m}}$.

##  Appendix B Line correlations

Figure 14: (a: left) The luminosity of the PAH feature at 6.2 $\,\rm{\rm{\upmu
m}}$ as a function of the total IR luminosity. (b: centre) The
[NeVI]7.65$\,\rm{\rm{\upmu m}}$ line luminosity as a function of the total IR
luminosity. (c: right) The luminosity of the PAH feature at 7.7
$\,\rm{\rm{\upmu m}}$ as a function of the total IR luminosity. The same
legend as in Fig. 1 was used.

Figure 15: (a: left) The luminosity of the PAH feature at 8.6 $\,\rm{\rm{\upmu
m}}$ as a function of the total IR luminosity. (b: centre) The luminosity of
the H2 molecular line at 9.7 $\,\rm{\rm{\upmu m}}$ as a function of the total
IR luminosity. (c: right) The [SIV]10.5$\,\rm{\rm{\upmu m}}$ line luminosity
as a function of the total IR luminosity. The same legend as in Fig. 1 was
used.

Figure 16: (a: left) The luminosity of the H2 molecular line at 12.28
$\,\rm{\rm{\upmu m}}$ as a function of the total IR luminosity. (b: centre)
The [NeV]14.3$\,\rm{\rm{\upmu m}}$ line luminosity as a function of the total
IR luminosity. (c: right) The luminosity of the PAH feature at 17
$\,\rm{\rm{\upmu m}}$ as a function of the total IR luminosity. The same
legend as in Fig. 1 was used.

Figure 17: (a: left) The luminosity of the H2 molecular line at 17.03
$\,\rm{\rm{\upmu m}}$ as a function of the total IR luminosity. (b: centre)
The [SIII]18.7$\,\rm{\rm{\upmu m}}$ line luminosity as a function of the total
IR luminosity. (c: right) The [FeIII]22.93$\,\rm{\rm{\upmu m}}$ line
luminosity as a function of the total IR luminosity. The same legend as in
Fig. 1 was used.

Figure 18: (a: left) The [NeV]24.3$\,\rm{\rm{\upmu m}}$ line luminosity as a
function of the total IR luminosity. (b: centre) The
[FeII]25.99$\,\rm{\rm{\upmu m}}$ line luminosity as a function of the total IR
luminosity. (c: right) The luminosity of the H2 molecular line at 28.22
$\,\rm{\rm{\upmu m}}$ as a function of the total IR luminosity. The same
legend as in Fig. 1 was used.

Figure 19: (a: left) The [SIII]33.5$\,\rm{\rm{\upmu m}}$ line luminosity as a
function of the total IR luminosity. (b: centre) The
[SiII]34.8$\,\rm{\rm{\upmu m}}$ line luminosity as a function of the total IR
luminosity. (c: right) The [OIII]52$\,\rm{\rm{\upmu m}}$ line luminosity as a
function of the total IR luminosity. The same legend as in Fig. 1 was used.

Figure 20: (a: left) The [NIII]57$\,\rm{\rm{\upmu m}}$ line luminosity as a
function of the total IR luminosity. (b: centre) The [OI]63.18$\,\rm{\rm{\upmu
m}}$ line luminosity as a function of the total IR luminosity. (c: right) The
[NII]122$\,\rm{\rm{\upmu m}}$ line luminosity as a function of the total IR
luminosity. The same legend as in Fig. 1 was used.

Figure 21: (a: left) The [OI]145$\,\rm{\rm{\upmu m}}$ line luminosity as a
function of the total IR luminosity. (b: right) The [NII]122$\,\rm{\rm{\upmu
m}}$ line luminosity as a function of the total IR luminosity. The same legend
as in Fig. 1 was used.

Figure 22: (a: left) Comparison between the local LIR-L[NII]122 relation for
local SFG (black dashed line) and high redshift detections of
[NII]122$\,\rm{\rm{\upmu m}}$ line. The shaded area shows the 95$\%$
prediction interval for the local relation. The pink $+$ sign indicates a
detection for the [NII]122$\,\rm{\rm{\upmu m}}$ line (De Breuck et al. 2019),
while symbols with arrows indicate upper limits for high redshift sources
(Harikane et al. 2019). (b: centre) Comparison between the local LIR-L[OI]63
relation for local SFG (black dashed line) and high redshift detections of
[OI]63$\,\rm{\rm{\upmu m}}$ line (pink symbol Rybak et al. 2019). (c: right)
Comparison between the local LIR-L[OI]145 relation for local SFG (black dashed
line) and high redshift detections of [OI]145$\,\rm{\rm{\upmu m}}$ line (pink
symbol De Breuck et al. 2019). The shaded area shows the 95$\%$ prediction
interval for the local relation. Grey stars show local SFG, while grey
triangles show local LMG.

##  Appendix C [NeV]14.3 $\,\rm{\rm{\upmu m}}$ as BHAR tracer and discussion
on the use of the mid-ionisation lines

Figure 23: (a: left) Linear correlation between the [NeV]14.3$\,\rm{\rm{\upmu
m}}$ and the 2-10 keV X-ray luminosity. Blue squares show the AGN sample, and
the shaded area shows the 95$\%$ confidence interval for the relation.(b:
centre) Linear correlation between the [NeV]14.3$\,\rm{\rm{\upmu m}}$ line
luminosity and the 19$\,\rm{\rm{\upmu m}}$ luminosity. (c: right) Comparison
between the [NeV]14.3$\,\rm{\rm{\upmu m}}$ line luminosity and the BHAR
derived from the bolometric correction by Spinoglio et al. (green solid line),
Marconi et al. (red dotted line) and Lusso et al. (blue dashed line).

In this section, following what has been done in Section 3.3, we report the
results for the [NeV]14.3$\,\rm{\rm{\upmu m}}$ line as BHAR tracer. In
particular, starting from the total sample of Tommasin et al. (2008); Tommasin
et al. (2010), excluding Compton-thick objects we obtain a sample of 38
objects. The correlation between the [NeV]14.3$\,\rm{\rm{\upmu m}}$ line
luminosity and the 2-10 keV X-ray luminosity is shown in Fig 23a and is
expressed by the equation:

$\log\left(\frac{L_{X}}{\rm 10^{41}\,erg\,s^{-1}}\right)=(2.58\pm
0.10)+(1.11\pm 0.13)\log\left(\frac{L_{\rm[NeV]14.3}}{\rm
10^{41}\,erg\,s^{-1}}\right)$ (25)

with a Pearson $r$ coefficient of $r=$0.79. From this result, we then apply
the bolometric corrections by Marconi et al. (2004) and Lusso et al. (2012) to
the 2-10 keV X-ray luminosity, in order to obtain the bolometric luminosity of
the AGN. From the bolometric luminosities we then calculate the linear
correlation linking the [NeV]14.3$\,\rm{\rm{\upmu m}}$ line luminosity to the
BHAR, with $L_{AGN}=\eta\dot{M}_{BH}c^{2}$. Assuming a radiative efficiency of
$\eta=0.1$, we report in equation 26 the linear correlation applying the
bolometric correction from Lusso et al. (2012), and in equation 27 the one
applying the Marconi et al. (2004) bolometric correction, and the relative
Pearson r coefficient:

$\log\left(\frac{\dot{M}_{BH}}{\rm M_{\odot}\,yr^{-1}}\right)=(-1.0\pm
0.10)+(1.04\pm 0.13)\log\left(\frac{L_{\rm[NeV]14.3}}{\rm
10^{41}\,erg\,s^{-1}}\right),\,\,r=0.76$ (26)

$\log\left(\frac{\dot{M}_{BH}}{\rm M_{\odot}\,yr^{-1}}\right)=(-0.73\pm
0.17)+(1.49\pm 0.13)\log\left(\frac{L_{\rm[NeV]14.3}}{\rm
10^{41}\,erg\,s^{-1}}\right),\,\,r=0.78$ (27)

We have then analysed the correlation between the [NeV]$14.3\,\rm{\rm{\upmu
m}}$ line with the luminosity at 19$\,\rm{\rm{\upmu m}}$ following the work by
Tommasin et al. (2010). As in Section 3.3, we select only those sources with
an AGN component at 19$\,\rm{\rm{\upmu m}}$ equal or above 85$\%$, obtaining a
subsample of 32 objects. The correlation between the [NeV]14.3$\,\rm{\rm{\upmu
m}}$ line luminosity and the 19$\,\rm{\rm{\upmu m}}$ luminosity is shown in
Fig. 23b, and expressed by:

$\log\left(\frac{L_{19}}{\rm 10^{41}\,erg\,s^{-1}}\right)=(3.09\pm
0.06)+(1.01\pm 0.10)\log\left(\frac{L_{\rm[NeV]14.3}}{\rm
10^{41}\,erg\,s^{-1}}\right)$ (28)

with a Pearson $r$ coefficient of r=0.85. We then calculate the bolometric
luminosity from the monochromatic 12$\,\rm{\rm{\upmu m}}$ luminosity extracted
from Deo et al. (2009) using the relation by Spinoglio et al. (1995). From the
bolometric luminosity we determine the BHAR and its relation to the
[NeV]14.3$\,\rm{\rm{\upmu m}}$ line luminosity, obtaining:

$\log\left(\frac{\dot{M}_{BH}}{\rm M_{\odot}\,yr^{-1}}\right)=(-0.66\pm
0.11)+(1.05\pm 0.19)\log\left(\frac{L_{\rm[NeV]14.3}}{\rm
10^{41}\,erg\,s^{-1}}\right),\,\,r=0.68$ (29)

for a sample of 26 objects, where $r$ is the Pearson coefficient. The small
number of objects used to derive this relation is due to the lack of data for
the determination of the continuum at 12$\,\rm{\rm{\upmu m}}$.

In Fig. 23c we compare the three different relation derived to determine the
BHAR from the [NeV]14.3$\,\rm{\rm{\upmu m}}$ line luminosity. as in Section
3.3, also for this line we find good agreement between the results obtained
with the Lusso et al. and Spinoglio et al. bolometric corrections, while the
Marconi et al. bolometric correction produces a significantly steeper
relationship.

As presented in Section 3.3 and here, high ionisation lines, with an
ionisation potential above the threshold of doubly ionised helium (54.4 eV)
can only be efficiently produced by AGN activity, and thus are optimal BHAR
tracers. Mid-ionisation lines, however, are partially affected by AGN
activity, and could in theory be used as tracers for the BHAR. For this
reason, we have tested two mid-ionisation lines, namely the [SIV] 10.5$\mu$m
line, and the [NeIII]15.6$\mu$m line, to evaluate their use as tracers for the
X-ray luminosity in the 2-10 keV interval. Starting from the sample of
Tommasin et al. (2008); Tommasin et al. (2010), as in Sect. 3.3, we exclude
all Compton-thick objects and obtain a sub-sample of 44 objects for the
[NeIII] line, and of 39 objects for the [SIV] line. Fig. 24a shows the
correlation between the [NeIII] line luminosity and the 2-10 keV X-ray
luminosity, expressed as:

$\log\left(\frac{L_{X}}{\rm 10^{41}\,erg\,s^{-1}}\right)=(2.20\pm
0.10)+(1.33\pm 0.13)\log\left(\frac{L_{\rm[NeIII]15.6}}{\rm
10^{41}\,erg\,s^{-1}}\right)$ (30)

The correlation of the [SIV]10.5 $\mu$m line with the 2-10 keV X-ray
luminosity, shown in Fig. 24b, can be expressed as:

$\log\left(\frac{L_{X}}{\rm 10^{41}\,erg\,s^{-1}}\right)=(2.51\pm
0.09)+(0.98\pm 0.10)\log\left(\frac{L_{\rm[SIV]10.5}}{\rm
10^{41}\,erg\,s^{-1}}\right)$ (31)

For both correlations, we find a low Pearson correlation coefficient, equal to
$r=0.63$, lower than the values found for [OIV] ($r=0.87$) and the [NeV] lines
($r=0.79$ and $r=0.84$, for the 14.3$\mu$m and 24.3$\mu$m lines,
respectively). This suggests that, while mid-ionisation lines can correlate
with the X-ray luminosity, they may suffer from contributions due to star
formation activity, and thus are far less reliable as BHAR proxies. In
particular, AGN sources in low-metallicity star-forming galaxies would have a
significant contamination of the [SIV] and [NeIII] lines from the stellar
population, as shown by the LMG in our sample (Figs. 1c and 15c), in contrast
with the [OIV] and [NeV] lines.

Figure 24: (a: left) Linear correlation between the
[NeIII]15.6$\,\rm{\rm{\upmu m}}$ and the 2-10 keV X-ray luminosity. Blue
squares show the AGN sample, and the shaded area shows the 95$\%$ confidence
interval for the relation. (b: right) Linear correlation between the
[SIV]10.5$\,\rm{\rm{\upmu m}}$ and the 2-10 keV X-ray luminosity

##  Appendix D Line and PAH calibrations

Table 3: New calibration obtained for fine-structure lines and PAH bands. For
each class of objects, are reported the slope (a) and intercept (b) of the
linear correlation with relative errors, the number of data from which the
linear correlation was calculated (N) and the best-fit coefficients of
determination (r).

Line/feature | AGN | SFG | LMG
---|---|---|---
|  $a$ $\pm$ $\delta a$ |  $b$ $\pm$ $\delta b$ | N | $r$ |  $a$ $\pm$ $\delta a$ |  $b$ $\pm$ $\delta b$ | N | $r$ |  $a$ $\pm$ $\delta a$ |  $b$ $\pm$ $\delta b$ | N | $r$
PAH 6.2 |  1.04$\pm$0.07 |  -2.73$\pm$0.24 | 65 | 0.87 |  1.15$\pm$0.06 |  -2.99$\pm$0.24 | 150 | 0.81 | – | – | – | –
[NeVI]7.65 |  1.22$\pm$0.24 |  -4.12$\pm$0.82 | 8 | 0.89 | – | – | – | – | – | – | – | –
PAH 7.7 | – | – | – | – |  1.29$\pm$0.07 |  -2.98$\pm$0.27 | 150 | 0.81 | – | – | – | –
PAH 8.6 | – | – | – | – |  1.43$\pm$0.23 |  -4.25$\pm$0.67 | 16 | 0.84 | – | – | – | –
H29.67 |  1.20$\pm$0.28 |  -4.97$\pm$1.17 | 9 | 0.82 |  1.39$\pm$0.06 |  -5.43$\pm$0.26 | 137 | 0.87 | – | – | – | –
[SIV]10.51 |  1.37$\pm$0.11 |  -5.19$\pm$0.38 | 75 | 0.80 |  1.58$\pm$0.24 |  -7.00$\pm$0.94 | 51 | 0.59 |  1.03$\pm$0.08 |  -3.08$\pm$0.16 | 29 | 0.91
PAH 11.3 |  1.04$\pm$0.06 |  -2.82$\pm$0.20 | 90 | 0.88 |  1.11$\pm$0.06 |  -2.79$\pm$0.23 | 150 | 0.81 | – | – | – | –
H212.28 |  1.25 $\pm$ 0.10 |  -5.41 $\pm$ 0.37 | 31 | 0.91 |  1.18 $\pm$ 0.05 |  -4.79$\pm$ 0.19 | 114 | 0.91 | – | – | – | –
[NeII]12.81 |  1.19$\pm$0.07 |  -4.23$\pm$0.26 | 86 | 0.85 |  1.04$\pm$0.43 |  -3.48$\pm$0.17 | 186 | 0.84 |  1.37$\pm$0.10 |  -4.36$\pm$0.20 | 21 | 0.95
[NeV]14.32 |  1.32$\pm$0.11 |  -5.01$\pm$0.39 | 74 | 0.78 | – | – | – | – | – | – | – | –
[NeIII]15.56 |  1.44 $\pm$0.10 |  -5.16$\pm$0.36 | 88 | 0.80 |  1.13$\pm$0.06 |  -4.60$\pm$0.21 | 182 | 0.80 |  1.13$\pm$0.06 |  -3.11$\pm$0.12 | 28 | 0.96
PAH 17 | – | – | – | – |  1.14$\pm$0.06 |  -3.08$\pm$0.23 | 131 | 0.84 | – | – | – | –
H217.03 |  1.14$\pm$0.06 |  -4.64$\pm$0.19 | 76 | 0.91 |  1.22$\pm$0.05 |  -4.58$\pm$0.20 | 135 | 0.89 | – | – | – | –
[SIII]18.71 |  1.22$\pm$0.09 |  -4.62$\pm$0.30 | 70 | 0.83 |  1.26$\pm$0.11 |  -4.79$\pm$0.42 | 140 | 0.60 |  1.14$\pm$0.06 |  -3.42$\pm$0.12 | 25 | 0.96
[FeIII]22.93 | – | – | – | – |  1.33$\pm$0.13 |  -5.79$\pm$0.44 | 14 | 0.64 | – | – | – | –
[NeV]24.32 |  1.27$\pm$0.11 |  -4.73$\pm$0.37 | 65 | 0.80 | – | – | – | – | – | – | – | –
[OIV]25.89 |  1.27$\pm$0.14 |  -4.13$\pm$0.48 | 34 | 0.82 |  1.15$\pm$0.05 |  -5.00$\pm$0.20 | 144 | 0.86 |  0.88$\pm$0.13 |  -3.71$\pm$0.27 | 16 | 0.86
[FeII]25.98 | – | – | – | – |  0.80$\pm$0.18 |  -3.99$\pm$0.51 | 35 | 0.44 | – | – | – | –
H228.22 |  1.04$\pm$0.14 |  -4.44 $\pm$ 0.45 | 25 | 0.81 |  1.65$\pm$0.24 |  -6.51$\pm$0.82 | 11 | 0.91 | – | – | – | –
[SIII]33.48 |  1.10$\pm$0.07 |  -3.85$\pm$0.23 | 75 | 0.87 |  1.13$\pm$0.05 |  -3.94$\pm$0.20 | 170 | 0.83 |  1.11$\pm$0.08 |  -3.24$\pm$0.17 | 20 | 0.96
[SiII]34.81 |  1.21$\pm$0.08 |  -3.99$\pm$0.28 | 73 | 0.84 |  1.15$\pm$0.04 |  -3.72 $\pm$0.15 | 167 | 0.91 |  1.05$\pm$0.08 |  -3.14$\pm$0.18 | 17 | 0.95
[OIII]51.81 |  0.92$\pm$0.13 |  -3.10$\pm$0.54 | 19 | 0.85 | – | – | – | – | – | – | – | –
[NIII]57.32 |  0.98$\pm$0.06 |  -3.96$\pm$0.24 | 31 | 0.94 | - | - | - | - | - | - | - | -
[OI]63.18 |  1.07$\pm$0.04 |  -3.56$\pm$0.17 | 106 | 0.91 |  1.18$\pm$0.05 |  -3.86$\pm$0.21 | 183 | 0.83 |  1.07$\pm$0.06 |  -3.00$\pm$0.11 | 31 | 0.96
[OIII]88.36 |  1.18$\pm$0.06 |  -4.34$\pm$0.22 | 81 | 0.91 |  1.20$\pm$0.06 |  -4.27$\pm$0.25 | 117 | 0.85 |  1.05$\pm$0.06 |  -2.53$\pm$0.10 | 37 | 0.95
[NII]121.9 |  1.08$\pm$0.06 |  -4.43$\pm$0.22 | 77 | 0.89 |  0.90$\pm$0.06 |  -3.39$\pm$0.22 | 75 | 0.86 |  0.96$\pm$0.09 |  -4.17$\pm$0.23 | 8 | 0.97
[OI]145.5 |  1.06$\pm$0.07 |  -4.53$\pm$ 0.26 | 64 | 0.88 |  1.03$\pm$0.05 |  -4.14$\pm$0.16 | 13 | 0.99 |  1.05$\pm$0.07 |  -4.14$\pm$ 0.18 | 12 | 0.97
[CII]157.7 |  1.00$\pm$0.04 |  -3.18$\pm$ 0.14 | 149 | 0.90 |  0.97$\pm$0.05 |  -2.76$\pm$0.20 | 183 | 0.78 |  1.09$\pm$0.06 |  -2.88$\pm$0.10 | 40 | 0.95
[NII]205 |  0.88$\pm$0.05 |  -3.88$\pm$ 0.18 | 60 | 0.92 |  1.02$\pm$0.07 |  -4.36$\pm$0.23 | 13 | 0.97 | – | – | – | –

##  Appendix E Differences with previous results

We analyse here the differences of the line calibrations obtained in this work
with respect to those of Spinoglio et al. (2012, 2014) and Gruppioni et al.
(2016).

When considering the results by Spinoglio et al., we only analyse the fine
structure lines in the 50-160$\mu$m spectral interval, for which these authors
consider an heterogeneous sample of both AGN and SFG, while in this work we
consider three classes of galaxies. When comparing the results, we consider
the calibrations obtained by Spinoglio et al. and compare them to our results
for AGN and SFG. For this comparison, we apply to our samples the ordinary
least square method, which was used in Spinoglio et al. to derive their
correlations. Fig 25 shows the comparison for the [OIII]52$\mu$m,
[NIII]57$\mu$m and [OI]63$\mu$m lines, Fig 26 shows the comparison for the
[OIII]88$\mu$m, [NII]122$\mu$m and [OI]145$\mu$m lines, and Fig. 27 shows the
comparison for the [CII]158$\mu$m line.

When compared to our AGN sample, the results are consistent within the errors,
except for the [NII]57$\mu$m and [OIII]88$\mu$m lines, where the slopes are
comparable within 2$\sigma$ of each other. If compared to the SFG results, we
obtain results comparable within the errors except for [OIII]88$\mu$m and
[NII]122$\mu$m lines, for which the slopes are comparable within 2$\sigma$ of
each other.

Figure 25: (a: left) The [OIII]52$\mu$m line luminosity versus the total IR
luminosity. (b: centre) The [NIII]57$\mu$m line luminosity versus the total IR
luminosity. (c: right) The [OI]63$\mu$m line luminosity versus the total IR
luminosity. In the figures, Blue squares represent AGN while red stars show
the SFG sample. The dashed blue line represents the linear he dot-dashed red
line the relation for SFG, and the black solid line the relation shows the
relation obtained by Spinoglio et al. (2012, 2014).

Figure 26: (a: left) The [OIII]88$\mu$m line luminosity versus the total IR
luminosity. (b: centre) The [NII]122$\mu$ m line luminosity versus the total
IR luminosity. (c: right) The [OI]145$\mu$m line luminosity versus the total
IR luminosity. The same legend as in Fig. 25 was used. Figure 27: (a:) The
[CII]158$\mu$m line luminosity versus the total IR luminosity. The same legend
as in Fig. 25 was used

When considering the results presented by Gruppioni et al. (2016), we take
advantage of the published catalogue and apply the orthogonal distance
regression to the sample presented by the authors. Gruppioni et al. present a
sample of 76 AGN, divided in two sub-samples depending on the fraction of
5-40$\mu$m luminosity produced by the active nucleus [$f_{AGN}(5-40\mu m)$].
In particular, there are 30 objects with $f_{AGN}(5-40\mu m)\leq 0.4$ and 43
objects with $f_{AGN}(5-40\mu m)>0.4$ .

Figs. 28-31 show the comparison for the relations calculated for the
$f_{AGN}$(5-40$\,\rm{\rm{\upmu m}}$)$>0.4$ sub-sample, and Figs. 32-35 show
the comparison for the relations calculated for the
$f_{AGN}$(5-40$\,\rm{\rm{\upmu m}}$)$\leq 0.4$ sub-sample.

As a general trend, our sample of AGN and the sample by Gruppioni et al.
occupy the same region in the LIR-Lline space. We note here that, while the
data to calculate the correlations were available in the literature, Gruppioni
et al. only present the correlations for [NeV]14.3, 24.3$\mu$m,
[NeIII]15.6$\mu$m and [OIV]25.9$\mu$m for the $f_{AGN}$(5-40$\,\rm{\rm{\upmu
m}}$)$>0.4$ sub-sample. For these relations, we find that our results show
steeper slopes, but consistent within 3$\sigma$ of each other.

For the $f_{AGN}$(5-40$\,\rm{\rm{\upmu m}}$)$\leq 0.4$ sub-sample, there is a
better agreement between the relations, which are comparable within the
errors. A significant difference in slope is present for the [NeV]24.3$\mu$m
line, for which the relations are comparable within 3$\sigma$, and for the
[SIV]10.5$\mu$m line, comparable within 2$\sigma$.

If we compare the results obtained using the Gruppioni et al. sample with the
results we obtain using the SFG sample, we find comparable results only for
the PAH features for the $f_{AGN}$(5-40$\,\rm{\rm{\upmu m}}$)$>0.4$, and for
all lines in the $f_{AGN}$(5-40$\,\rm{\rm{\upmu m}}$)$\leq 0.4$ sub-sample,
excluding the [SIV]10.5$\mu$m and [OIV]25.9$\mu$m lines.

Figure 28: (a: left) The PAH 6.2$\mu$m luminosity versus the total IR
luminosity. Blue squares represent AGN, red stars SFG and black circles are
the sample by Gruppioni et al. (2016). The dashed blue line represents the
linear relation for AGN, the dot-dashed red line the relation for SFG, and the
black solid line the relation for the Gruppioni et al. sample. (b: centre) The
[SIV]10.5$\mu$m luminosity versus the total IR luminosity. (c: right) The PAH
11.3$\mu$m luminosity versus the total IR luminosity. In all three figures,
the Gruppioni et al. sample was limited to AGN fractions $>$ 0.4 in the
5-40$\mu$m spectral interval (see Sect.4.3).

Figure 29: (a: left) The [NeII]12.8$\mu$m line luminosity versus the total IR
luminosity. (b: centre) The [NeV]14.3$\mu$m line versus the total IR