# Modelling the UV to radio SEDs of nearby star-forming galaxies: new PARSEC
SSP for GRASIL

I.A. Obi1, A. Bressan1, F. Perrotta1, L. Silva2 O. Vega3, Y. Chen4, A. Lapi1
E-mail<EMAIL_ADDRESS>C. Mancuso1, L. Girardi5, G.L. Granato2, P. Marigo4,
and A. Slemer4.
1SISSA, via Bonomea 265, I-34136 Trieste, Italy
2INAF - Osservatorio Astronomico di Trieste, via Tiepolo 11, 34131 Trieste,
Italy
3INAOE, Luis Enrique Erro 1, 72840 Tonantzintla, Puebla, Mexico
4Dipartimento di Fisica e Astronomia Galileo Galilei, Universitá di Padova,
Vicolo dell’Osservatorio 3, I-35122 Padova, Italy
5INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5,
I-35122 Padova, Italy

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

By means of the updated PARSEC database of evolutionary tracks of massive
stars, we compute the integrated stellar light, the ionizing photon budget and
the supernova rates of young simple stellar populations (SSPs), for different
metallicities and IMF upper mass limits. Using CLOUDY we compute and include
in the SSP spectra the nebular emission contribution. We also revisit the
thermal and non-thermal radio emission contribution from young stars. Using
GRASIL we can thus predict the panchromatic spectrum and the main
recombination lines of any type of star-forming galaxy, including the effects
of dust absorption and re-emission. We check the new models against the
spectral energy distributions (SEDs) of selected well-observed nearby
galaxies. From the best-fit models we obtain a consistent set of star
formation rate (SFR) calibrations at wavelengths ranging from ultraviolet (UV)
to radio. We also provide analytical calibrations that take into account the
dependence on metallcity and IMF upper mass limit of the SSPs. We show that
the latter limit can be well constrained by combining information from the
observed far infrared, 24 $\micron$, 33 GHz and H$\alpha$ luminosities.
Another interesting property derived from the fits is that, while in a normal
galaxy the attenuation in the lines is significantly higher than that in the
nearby continuum, in individual star bursting regions they are similar,
supporting the notion that this effect is due to an age selective extinction.
Since in these conditions the Balmer decrement method may not be accurate, we
provide relations to estimate the attenuation from the observed 24 $\micron$
or 33 GHz fluxes. These relations can be useful for the analysis of young high
redshift galaxies.

###### keywords:

radio continuum: galaxies–infrared: galaxies–ISM: dust,
extinction–stars:formation–galaxies: high-redshift

††pubyear: 2017

##  1 Introduction

Modelling the SEDs of galaxies has proven to be a very powerful tool in our
current understanding of the different physical processes that come into play
in the formation and evolution of galaxies. Various physical properties of
galaxies like stellar, metal and dust content, star formation rate, dust
obscuration, etc are estimated by fitting the theoretical SEDs to the observed
ones. In the UV to infrared (IR) spectral regions of a SED, dust plays an
important role. It absorbs (as well as scatters) the UV-near-infrared (NIR)
light and re-radiates it in the IR. This IR emission (3 - 1000 $\micron$) may
arise from (a) the emission from dust heated by young OB stars (neglecting
heating from AGN), (b) the emission from the circumstellar envelopes of
evolved stars and (c) the cirrus emission from dust distributed throughout in
the ISM and heated by the general interstellar radiation field. Point (b)
leads to wrong estimates of the physical properties related to star formation.
The radio band which is not sensitive to dust attenuation is usually used to
check and complement the interpretations arrived by using the optical/IR
bands. Radio emission from normal star-forming galaxies is usually dominated
by the non-thermal component (up to $\approx$ 90 per cent of the radio flux)
which is believed to be due to the synchrotron emission from relativistic
electrons accelerated into the shocked ISM by core-collapse supernovae (CCSN)
explosions from massive young stars (above $\approx 8~{}\rm{M}_{\sun}$) ending
their lifes (Condon, 1992). Recent advances in hydrodynamical simulations of
CCSN have indicated a range of stellar masses where the stars fail to explode
but rather end up directly as a black hole. This raises concern about the
widely accepted notion that all stars more massive than $\approx
8~{}\rm{M}_{\sun}$ end up as a CCSN.

Over the years, great progress has been made both in the development of tools
and models use to extract the information encoded in the SEDs and in multi-
wavelength surveys that sample the UV to radio SED of local and high redshift
galaxies. Furthermore, new observing facilities with unprecedented resolution
and sensivity (e.g. SKA, JWST, EVLT etc) will be put in place in the nearest
feature. At the same time, semi-analytic models of galaxy formation, in the
context of the current cosmological standard model, are appearing and
providing realistic predictions of the physical properties of galaxies with
which the results of the SED fitting can be compared.

Stellar population synthesis (SPS) still remains the basis of SED modelling.
The most common method used in computing the SEDs of SSPs is that of isochrone
synthesis (Chiosi, Bertelli & Bressan, 1988; Maeder & Meynet, 1988; Charlot &
Bruzual, 1991) which uses the locus of stars in an isochrone to integrate the
spectra of all stars along an isochrone to get the total flux. This involves
computing first the stellar evolutionary tracks for different masses and metal
contents and building the isochrones from the tracks. With the isochrones, the
resulting stellar spectrum is computed using stellar atmosphere libraries. In
recent years, significant efforts from different research groups have been put
into providing homogeneous sets of evolutionary tracks and improving the
stellar libraries. As a result of these improvements, SPS models can
remarkably reproduce well the UV-NIR SEDs and high-resolution spectra in the
optical wavelength band. However, despite these improvements, there are still
challenges in the field, especially in the treatment of the phases of stellar
evolution that are weakly understood. The most important being the short lived
phases: massive stars, thermally pulsing asymptotic giant branch (TP-AGB)
stars, blue stragglers and extreme horizontal branch stars. The TP-AGB stars
have however increasingly received attention leading to a rapid progress in
their modelling (Maraston, 2005; Marigo & Girardi, 2007; Marigo et al., 2017).

The stellar evolution code used in Padova to compute sets of stellar
evolutionary tracks that are of wide useage in the astronomical community has
recently been thoroughly revised and updated, going by the name,
PARSEC(PAdova-TRieste Stellar Evolution Code). More details of this code can
be found in Bressan et al. (2012, 2013); Chen et al. (2014) and will be
briefly discussed later in the text. In this paper, our main aim is to use the
PARSEC evolutionary tracks of massive stars to compute the quantities (the
integrated light, ionizing photon budget and the supernova rates) of young
SSPs, thereby updating the SSPs used by GRASIL in predicting the panchromatic
spectrum of star-forming galaxies. We also revisit the thermal and non-thermal
radio emissions. We finally check these new models with selected nearby well-
observed galaxies. We perform an analysis of the resulting best-fit SEDs, with
the aim of obtaining a new set of SFR calibrations and investigating the
dependence of the dust attenuation properties on galaxy types.

The structure of the rest of paper is as follows: In Section 2, we use the new
PARSEC code database of stellar evolutionary tracks of massive stars to
compute the ionizing photon budget, the integrated light and the supernova
rates predicted by young SSP models. Using the integrated spectra of the SPPs
in CLOUDY, we computed the nebular emissions. In Section 3, we revise (a) the
prescription used in computing the thermal radio emission. (b) the previous
non-thermal radio emission model originally described in Bressan, Silva &
Granato (2002, hereafter B02) while taking into account recent advances in
CCSN explosion models. In Section 4, we check our new radio emission models
and SSPs with GRASIL using selected well-observed nearby galaxies. Finally, we
discuss the resulting best fit SEDs of all galaxies studied in this paper in
Section 5. We draw our conclusions in Section 6.

Throughout this paper we adopt $\rm{12}+\rm{log}(\rm{O/H})=8.69$ (Asplund et
al., 2009) as the solar oxygen abundance. Our models adopt a Kennicutt (1983)
IMF (slope of -1.4 for $0.1-1~{}\rm{M}_{\sun}$ and -2.5 for $\geq
1~{}\rm{M}_{\sun}$). The cosmological parameters we adopt assume
$H_{0}=70~{}\rm{km}\;s^{-1}\;\rm{Mpc}^{-1},\Omega_{\Lambda}=0.7,\Omega_{M}=0.3$

##  2 Simple Stellar Populations with PARSEC

PARSEC is the latest version of the Padova-Trieste stellar evolution code with
thorough update of the major input physics including new and accurate
homogenised opacity and equation of state tables fully consistent with any
adopted chemical composition. More details may be found in Bressan et al.
(2012, 2013); Chen et al. (2014). The evolutionary tracks span a wide range in
metallicities, $0.0001\leq Z\leq 0.04$, and initial masses, from very low
$(M~{}=~{}0.1~{}\rm{M}_{\sun})$ to very massive
$(M~{}=~{}350~{}\rm{M}_{\sun})$ stars, starting from the pre-main sequence
phase and ending at central carbon ignition. Tang et al. (2014) and Chen et
al. (2015) computed new evolutionary tracks of massive stars and tables of
theoretical bolometric corrections that allow for the conversion from
theoretical HR to the observed colour-magnitude diagrams. A preliminary
comparison of the new models with color-magnitude diagrams of star-forming
regions in nearby low metallicity dwarf irregular galaxies was performed by
Tang et al. (2014). The full set of new evolutionary tracks and the
corresponding isochrones may be found in
http://people.sissa.it/~sbressan/parsec.html and http://stev.oapd.inaf.it/cgi-
bin/cmd, respectively.

###  2.1 Integrated SSP spectra

To build the integrated spectra of SSPs, we adopt the spectral library
compilation by Chen et al. (2015) who homogenized various sets of existing
stellar atmosphere libraries encompassing a wide range of parameters of both
cool and hot stars (i.e. masses, evolutionary stages and metallicities). The
atmosphere models adopted by Chen et al. (2015) are the ATLAS9 models (Kurucz,
1993; Castelli & Kurucz, 2004), suitable for intermediate and low mass stars,
and the Phoenix models (Allard et al., 1997) for the coolest stars. For hot
massive stars Chen et al. (2015) generated new wind models with the WM-BASIC
code Pauldrach, Puls & Kudritzki (1986) and adopted new models of WR stars
from the Potsdam group (see e.g. Sander et al., 2015). Since different sets of
libraries have different metallicities, they were merged to obtain homogeneous
sets of spectral libraries as described in Girardi et al. (2002) and Chen et
al. (2014).

Figure 1: SSP Integrated spectra per solar mass at ages of $t~{}=~{}$2, 5 and
40 Myr, for metallicities Z = 0.02, 0.004 and 0.0001 and $\rm{M}_{up}$ of 40,
120 and 350 $\rm{M}_{\sun}$ indicated by the solid blue, dotted red and dashed
green lines respectively. At 2 Myr, the spectra for the various upper mass
limits at any of the three metallicities is quite distinguishable. At 5 Myr
and Z = 0.02, it is distinguishable only at wavelengths below
$912~{}\text{\AA}$, the maximum for the lyman ionizing photons. At this age,
notice that the spectra for the $\rm{M}_{up}$ of 120 and 350$~{}\rm{M}_{\sun}$
below $912~{}\mbox{\AA}$ are superimposed, owing to the fact they must have
evolved off the main sequence at this age. At 5 Myr and Z = 0.0001, the
reverse is the case, the spectra is distinguishable only at wavelengths above
$912~{}\text{\AA}$ because at this metallicity, stars lifetimes are a bit
longer than at Z = 0.02. At 40 Myr, the spectra for all upper mass limits and
metalicities are all superimposed and the Lyman ionizing photons are no longer
produced.

The properties of SSP are obtained by integration along the corresponding
isochrones assuming a two-slopes power law Kennicutt initial mass function.
Results are presented for three different values of the upper mass limit,
$\rm{M}_{up}$ of 40,120, 350 $\rm{M}_{\sun}$. As an example, Figure 1 shows
the synthesized spectra of star clusters at ages of $t~{}=~{}$2, 5 and 40 Myr
and for three values of the initial metallicity, Z = 0.02, 0.004 and 0.0001.
The effect of assuming different $\rm{M}_{up}$ values, at fixed total initial
mass, is more pronounced at an age of 2 Myr. At 5 Myr, when stars with masses
of 120 $\rm{M}_{\sun}$ and 350 $\rm{M}_{\sun}$ already left the main sequence
or already died out, the spectra of SSP with upper mass limits of 120
$\rm{M}_{\sun}$ and 350 $\rm{M}_{\sun}$ are superimposed, while those of 40
$\rm{M}_{\sun}$ remains distinct. At 40 Myr the spectra are almost independent
of $\rm{M}_{up}$, though it is evident that the spectrum of the SSP with
larger $\rm{M}_{up}$ has a lower luminosity, because of the higher mass budget
stored in massive stars We already note from this figure that the effects of
$\rm{M}_{up}$ on the number of ionizing photons disappears at ages larger than
about 5 Myr, as discussed below.

###  2.2 Ionizing Photon Budget

The number of Lyman ionizing photons per sec ($\rm{Q(H)}$) and per unit mass
emitted by young stellar populations is controlled by hot massive stars, i.e.
O-B main sequence stars and Wolf Rayet stars. This number is thus critically
dependent on the shape of the initial mass function in the domain of massive
stars. In Figure 2, we show the time evolution of $\rm{Q(H)}$ for selected
SSPs with different upper mass limit ($\rm{M}_{up}$) and metallicities. As
already said, the adopted IMF is a two-slope power law Kennicutt (1983). The
lower limit is $\rm{M}_{low}$ = 0.1 $\rm{M}_{\sun}$ and the upper limits are
$\rm{M}_{up}$ = 40 $\rm{M}_{\sun}$, $\rm{M}_{up}$ = 120 $\rm{M}_{\sun}$ and
$\rm{M}_{up}$ = 350 $\rm{M}_{\sun}$. The slope of the IMF is X = 1.4 between
$\rm{M}_{low}$ and $\rm{M}$ = 1$\rm{M}_{\sun}$ and X = 2.5 between $\rm{M}$ =
1$\rm{M}_{\sun}$ and $\rm{M}_{up}$

Figure 2: Variation of the instantenous number of ionizing photons per second
per unit ionizing solar mass of the cluster with age (Myr). Different symbols
correspond to different metallicities as illustrated in the plot

.

From this figure we may appreciate the role of age, metallicity and IMF on the
ionizing photon rate $\rm{Q(H)}$. For a given Z and $\rm{M}_{up}$, $\rm{Q(H)}$
generally increases to a maximum value and then, once a threshold age is
reached, it rapidly decreases to negligible values. At fixed $\rm{M}_{up}$,
both the maximum value of $\rm{Q(H)}$ and the threshold age decrease, at
increasing metallicity. In general this is also true at varying age, i.e. at
given $\rm{M}_{up}$, $\rm{Q(H)}$ decreases at increasing metallicity. However
there are some cases where this is not true, in particular for the SSP of
solar metallicity. The effect of $\rm{M}_{up}$ is strong. A star cluster with
$\rm{M}_{up}$ of 350 $\rm{M}_{\sun}$ produces about seven times more ionizing
photons than a cluster with a $\rm{M}_{up}$ of 40 $\rm{M}_{\sun}$, at constant
total mass. Moreover, the age to attain the maximum $\rm{Q(H)}$ becomes lesser
at increasing upper mass limit, reflecting the larger relative weight of more
massive stars in the ionizing photon budget.

###  2.3 Nebular Emission with PARSEC’s SSP

The integrated spectra of the SSPs are used to calculate the nebular emission
from the surrounding H ii regions which is then added to the original spectrum
to obtain the integrated spectra containing both the stellar continuum (with
absorption lines and eventually wind emission features) and the nebular
features (continuum and lines). For this purpose, star clusters are assumed to
be the central ionizing sources of the H ii regions that are modelled using
the photoionization code CLOUDY (Ferland, 1996). As a further input to the
CLOUDY code, we specify that the H ii region is modelled as a thin shell of
gas with constant density, $n_{H}=100~{}cm^{-3}$, placed at a distance $R_{H}$
= 15 pc from the central source. The evolution of the ionizing star clusters
is followed from 0.1 Myr to 20 Myr, for five different values of the
metallicity (Z = 0.0001, 0.0005, 0.004, 0.008, 0.008 and 0.02) and three
values of $M_{up}$ of 40, 120 and 350 M☉. We note that our goal is not that of
providing a detailed dependence of a large ensemble of emission lines on the
critical parameters of the H ii nebulae. Instead we aim at obtaining a
reasonable estimate of the intensities of the main recombination lines and of
free-free emission to increase the diagnostic capabilities of our population
synthesis codes. Line emission and the corresponding nebular continuum are
much less dependent on the characteristic properties of the H ii regions (e.g.
ionization parameter, individual abundance of heavy elements etc.) than e.g.
excitation lines, for which a more detailed set of initial parameters would be
more appropriate (see e.g. Panuzzo et al., 2003; Wofford et al., 2016). The
main output of this process is a library of emission line intensities, nebular
continuum properties and electron temperatures (Te) of the H ii regions. Then,
emission lines and nebular continuum are used to suitably complement the
integrated SED of SSPs from the far-UV to radio wavelengths.

##  3 RADIO CONTINUUM EMISSION

Radio emission associated with the presence of young massive stars comprises
essentially two components, the thermal radio emission (also referred to as
free-free emission) and the non-thermal radio emission (also referred to as
non-thermal radio emission). The former emission comes from the nebular free
electrons originating from the ionizing radiation of massive stars. The non-
thermal emission is instead believed to be synchrotron radiation that
originated from the interaction of relativistic electrons, produced in the
ejecta of core-collapsed supernovae (CSSN), with the ambient magnetic field.
The radio continuum is thus a tracer of the number of massive stars formed
(and exploded) and hence an optimal indicator of the very recent (if not the
current) star formation rate.

The fact that the radio emission is a good SFR tracer is supported by the
remarkably tight correlation between FIR and non-thermal radio emission. At
1.49 GHz, this correlation is quantified by the q-parameter (Helou, Soifer &
Rowan-Robinson, 1985):

$q=\rm{log}\frac{F_{\rm{FIR}}/(3.75\times
10^{12}Hz)}{F_{\nu}(1.49\rm{GHz})/(W\;m^{-2}\;Hz^{-1})}\approx 2.35\pm 0.2$
(1)

where $F_{\rm{FIR}}=1.26\times
10^{-14}(2.58S_{60\micron}+S_{100\micron})\rm{W}\rm{m}^{-2}$ (Young et al.,
1989), $S_{60\micron}$ and $S_{100\micron}$ are IRAS flux densities in
$\rm{Jy}$.

###  3.1 Thermal Radio Emission

At sufficiently high frequency such that free-free self-absorption is
negligible, the relation between the specific luminosity of free-free emission
$L_{\rm{ff}}(\nu)$ and the rate of ionizing photons $\rm{Q(H)}$ can be written
as (Rubin, 1968; Condon, 1992)

$L_{\rm{ff}}=\frac{\rm{Q(H)}}{C_{1}}\left(\frac{\rm{T}_{e}}{10^{4}\,\hbox{K}}\right)^{0.3}\,{\rm
G_{dra}}(\nu,\hbox{T}_{e})\hskip 3.0pt$ (2)

where $\rm{T}_{e}$ is the electron temperature, $C_{1}=\sqrt{3}/\pi\times
6.86\times 10^{26}$ and ${\rm G_{dra}}(\nu,\hbox{T}_{e})$ is the velocity
averaged gaunt factor (Draine, 2011):

$\displaystyle\hbox{G}_{\text{dra}}(\nu,\rm{T}_{e})=$
$\displaystyle\hbox{ln}\left\\{\rm{exp}\left[5.960-{\frac{\sqrt{3}}{\pi}}\rm{ln}\left(Z_{i}{\frac{\nu}{\rm{GHz}}}\left({\frac{\rm{T_{e}}}{10^{4}\rm{K}}}\right)^{-1.5}\right)\right]+e\right\\}$
(3)

where $\rm{Z_{i}}$ is the charge of the ions in the H ii region. An
approximate velocity averaged gaunt factor were obtained earlier by Oster
(1961):

$\displaystyle\hbox{G}_{\text{ost}}(\nu,\rm{T}_{e})=$
$\displaystyle\hbox{ln}\left[4.955\times
10^{-2}\left({\nu\over\hbox{GHz}}\right)^{-1}\right]+1.5\
\hbox{ln}\left({\hbox{T}_{e}\over\hbox{K}}\right)$ (4)

Taking into account that
$\hbox{G}_{\text{dra}}(\nu,\rm{T}_{e})\simeq\sqrt{3}/\pi\times\hbox{G}_{\text{ost}}(\nu,\rm{T}_{e})$
and adopting a common approximation used in literature, we obtain for the
free-free emission (Condon, 1992)

$\frac{L_{\rm
ff}}{\hbox{erg}\,\hbox{s}^{-1}\,\hbox{Hz}^{-1}}={\frac{Q(H)}{6.3\times~{}10^{25}}}\left(\frac{\rm{T}_{e}}{10^{4}\,\hbox{K}}\right)^{0.45}\,\left({\nu\over\hbox{GHz}}\right)^{-0.1}$
(5)

The latter equation is often used by assuming a fixed electronic temperature
(104 K) to provide useful analytical approximations at radio frequencies
especially in the lack of nebular emission (see discussion in B02). Here we
directly estimate the intensity of the thermal radio emission of our SSP from
Cloudy with the procedure already described in Section 2.3. In Figure 3, we
show the evolution of the 1.49 GHz and 33 GHz thermal radio emission for
different metallicities, Z = 0.0001, 0.0005, 0.004, 0.008 and 0.02, and
different $M_{up}$ of $40,120$ and $350~{}\rm{M}_{\sun}$. The effects of
metallicity and $M_{up}$ on the thermal emission are easily noted. For
example, at Z = 0.0001, the 1.4 GHz thermal emission for
$M_{up}=120~{}\rm{M}_{\sun}$ is about 3 - 7 times larger than that for
$M_{up}=40~{}\rm{M}_{\sun}$ In general by increasing the metallicity from Z =
0.0001 to Z = 0.02 the thermal radio emission decreases by a factor of about
3. Instead by increasing $M_{up}$ from 40 to 350 $\rm{M}_{\sun}$, thermal
emission increases by about one order of magnitude. These important factors,
that must be considered in the calibration of the relation between star
formation rate and thermal radio emission, are discussed below.

Figure 3: Variation of the 1.49GHz (red) and 33GHz (blue) thermal radio
emission with age at different metallicities for different values of $M_{up}$.
The five different line styles correspond to the five different values of
metallicity.

####  3.1.1 Metallicity effects

Equations 2, 3 and 5 contain an explicit dependence on the electron
temperature, that is generally neglected in analytical approximations which
assume a constant value of $\rm{T_{e}}$ = 104 K. Since $\rm{T_{e}}$ is known
to depend on the metallicity of the H ii regions, with a variation of more
than a factor of two in the range of the observed values, we provide in
Appendix A some useful analytical relations. We first list in Table 2 the
values of the electronic temperature derived using CLOUDY for our SSP models
at various metallicities and upper mass limits. In Figure 17 of Appendix A, we
show the relations between oxygen abundance and the electronic temperature. In
this figure, crosses, ’X’s and asterisks indicate $M_{up}=40$, $120$ and
$350~{}\rm{M}_{\sun}$ respectively. The average of the empirical fits derived
by López-Sánchez et al. (2012) for high-ionization O iii and for low-
ionization O iii zones is shown as the dashed black lines. We provide a
multiple regression fitting relation (Equation 34) between $\rm{T_{e}}$,
$\rm{M_{up}}$ and Z that could be easily included in analytical
approximations.

####  3.1.2  $\rm{SFR}-\rm{Q(H)}$ calibration

In a young stellar system the ionizing photon budget is dominated by massive
stars and thus there must be a tight relation between the current star
formation rate and the rate of ionizing photons, that ultimately produce the
thermal radio emission Condon & Yin (1990). In this section, we provide a
calibration of the $\rm{SFR}-\rm{Q(H)}$ relation. For this purpose we consider
the integrated spectrum of a galaxy up to a time $t$:

$f_{\nu}^{gal}(t,Z)=\int_{0}^{t}f_{\nu}^{SSP}(t-t^{\prime},Z)SFR(t^{\prime})dt^{\prime}$
(6)

where SFR(t’) is the star formation rate at time $t^{\prime}$ and
$f_{\nu}^{SSP}(t-t^{\prime},Z)$ is the stellar spectrum of a SSP of age
$t_{ssp}~{}=~{}t-t^{\prime}$ and given metallicity $Z$

$f_{\nu}^{SSP}(t_{ssp},Z)=\int_{M_{low}}^{M_{up}}\phi(m)f_{\nu}(m,t_{ssp},Z)dm$
(7)

In the latter equation, $f_{\nu}(m,t_{ssp},Z)$ is the spectrum of an
individual star in a SSP which depends on the fundamental parameters, its mass
$m$, age $t_{ssp}$, metal abundance $Z$ and the IMF, $\phi(m)$. Using the SSPs
already described, a constant SFR of 10$~{}\rm{M_{\sun}}\;\rm{yr^{-1}}$ and
adopting the Kennicutt (1983) IMF, we obtain the integrated number of ionizing
photons $IQ(H)$, shown in figure 4. Since $IQ(H)$ is dominated by the most
massive stars, the number of ionizing photons emitted by a galaxy with
constant SFR will initially grow and soon saturate to a constant maximum
value, when there is a balance between the newly formed ionizing massive stars
and the ones that die. Looking at Figure 2 we see that, almost independently
from the metallicity, the characteristic time of the saturation is set by the
rapid drop of $IQ(H)$ above about 6 Myr. The effect of $\rm{M}_{up}$ on
$IQ(H)$ is a direct consequence of the variation of the $Q(H)$ of SSPs shown
in Figure 2. The variation of $\rm{IQ(H)}$ with metallicity decreases with
increasing upper mass limit.

Figure 4:  Evolution of the integrated number of ionizing photons per second
per unit solar mass initially formed as a function of the age in yr. Different
line colors represents different metallciities. The variation of Q(H) with
metallicity decreasis at increasing IMF upper mass limit.

After denoting by $C_{2}$ the calibration coefficient between SFR and Q(H) in
the equation below

$\left({\frac{\rm{SFR}}{\rm{M}_{\sun}\;\rm{yr}^{-1}}}\right)=C_{2}\times\left(\frac{IQ(H)}{\rm{s^{-1}}}\right)$
(8)

we collect in Table 2 the values of $C_{2}$ obtained with our constant SFR
models, for different SSPs parameters. To illustrate the significant
variations of $C_{2}$ with metallicity at a given $M_{up}$, we show in Figure
18 the plot of $C_{2}$ against metallicity for $M_{up}$ = 40, 120 and 350
$M_{\sun}$ Using the values of $C_{2}$ given in Table 2 for different
$\rm{M}_{up}$ and $Z$ , we provide a multiple regression fitting relation
between $C_{2}$ , $\rm{M}_{up}$ and metallicity ($Z$), Equation 35. As an
example, using this fitting relation, for $M_{up}=120~{}\rm{M}_{\sun}$ and Z =
0.02 Equation 8 is:

$\left({\frac{\rm{SFR}}{\rm{M}_{\sun}\;\rm{yr}^{-1}}}\right)=\left({\frac{IQ(H)}{s^{-1}}}\right)$
(9)

We may compare equation 9 with the one obtained by Murphy et al. (2012) using
the Starburst99 stellar population model with a Kroupa (2001) IMF, a
metallicity of Z = 0.02 and a constant star formation over 100 Myr.

$\left({\frac{\rm{SFR}}{\rm{M}_{\sun}\;\rm{yr}^{-1}}}\right)=\left({\frac{IQ(H)}{s^{-1}}}\right)$
(10)

We see that the Murphy et al. (2012) calibration constant is fairly in
agreement with ours.

####  3.1.3 SFR - $L_{ff}$ Calibration

By combining equation 8 and equation 2, we derive the relation between the SFR
and thermal radio emission:

$\frac{\rm{SFR}}{\rm{M_{\sun}yr^{-1}}}=\frac{L_{\rm{ff}}}{C_{3}}\left({\frac{\rm{T}_{e}}{10^{4}\hbox{K}}}\right)^{-0.3}\,\left(\frac{1}{G_{dra}}\right)$
(11)

where $C_{3}=1/(C_{1}\times C_{2})$. The values of the coefficient $C_{3}$ for
different SSP parameters are provided in Table 2 and the variation of $C_{3}$
with metallicity and $M_{up}$ is shown in Figure 19. Using these values of
$C_{3}$ given in Table 2 for different $\rm{M}_{up}$ and $Z$, we also, as we
did for the case $C_{2}$, provide a multiple regression fitting relation
between $C_{3}$, $M_{up}$ and metallicity ($Z$), Equation 36. As an example,
using this fitting relation for $M_{up}=120~{}\rm{M}_{\sun}$ and Z = 0.02 and
assuming the already quoted approximation for the Gaunt factor, Equation 11
becomes

$\frac{\rm{SFR}}{\rm{M_{\sun}\;yr^{-1}}}=\left(\frac{L_{\rm{ff}}}{2.77\times
10^{27}}\right)\left({\frac{\rm{T}_{e}}{10^{4}\hbox{K}}}\right)^{-0.45}\,\left(\frac{\nu}{\rm{GHz}}\right)^{0.1}$
(12)

A similar equation is provided by Murphy et al. (2012):

$\frac{\rm{SFR}}{\rm{M_{\sun}\;yr^{-1}}}=\left(\frac{L_{\rm{ff}}}{2.17\times
10^{27}}\right)\left(\frac{T_{e}}{10^{4}\rm{K}}\right)^{-0.45}\left(\frac{\nu}{\rm{GHz}}\right)^{0.1}$
(13)

####  3.1.4 SFR versus H$\alpha$ and H$\beta$ calibrations

Using well-known relations between $IQ(H)$ and the intensity of recombination
lines (Osterbrock, 1989), we may also obtain the corresponding calibrations
for the SFR. For H$\alpha$ and H$\beta$ we have

$\left(\frac{\rm{IQ(H)}}{\rm{s}^{-1}}\right)=7.31\times
10^{11}\left(\frac{L(H\alpha)}{\rm{erg}\;\rm{s}^{-1}}\right)$ (14)
$\left(\frac{\rm{IQ(H)}}{\rm{s}^{-1}}\right)=0.21\times
10^{13}\left(\frac{L(H\beta)}{\rm{erg}\;\rm{s}^{-1}}\right)$ (15)

Using the above relations in equation 8, we write in Appendix A analytical
equations for the SFR-H$\alpha$ (equation 39) and SFR-H$\beta$(equation 40)
calibrations, as a function of Z and Mup. For the case with
$M_{up}~{}=~{}120~{}\rm{M}_{\sun}$ and Z = 0.02 we obtain with these
analytical equations

$\left({\frac{\rm{SFR}}{\rm{M}_{\sun}\;\rm{yr}^{-1}}}\right)=4.79\times
10^{-42}\left({\frac{L(H\alpha)}{\rm{erg}\;\rm{s}^{-1}}}\right)$ (16)
$\left({\frac{\rm{SFR}}{\rm{M}_{\sun}\;\rm{yr}^{-1}}}\right)=0.14\times
10^{-40}\left({\frac{L(H\beta)}{\rm{erg}\;\rm{s}^{-1}}}\right)$ (17)

Our calibration coefficient in equation 16 is in good agreement with the value
of $4.55\times{10^{-42}}$ obtained by Bicker & Fritze-v. Alvensleben (2005)
using GALEV synthesis code.

###  3.2 Non-Thermal Radio Emission

We know quite little about the source of non-thermal (NT) radio emission in
star-forming galaxies. The mechanism is believed to be synchrotron emission
from relativistic electrons that are accelerated by ISM shocks in the
outskirts of CCSN explosions (Condon, 1992). B02 derived the following
relation between NT radio emission $L^{NT}$ and the CCSN rate

$\displaystyle\frac{L_{NT}/\frac{\nu_{CCSN}}{yr^{-1}}}{10^{30}~{}\rm{erg}\;\rm{s}^{-1}\;\rm{Hz}^{-1}}$
$\displaystyle=$ $\displaystyle
E^{SNR}_{1.49}(\frac{\nu}{1.49})^{-0.5}+E^{NT}_{1.49}(\frac{\nu}{1.49})^{-\alpha}$
(18)

In the above equation $\nu_{CCSN}$ is the CCSN rate, $E^{SNR}_{1.49}$ is the
average non thermal radio luminosity due to a young Supernova Remnant (SNR)
and $E^{NT}_{1.49}$ is the average injected energy in relativistic electrons
per CCSN event. Moreover, B02 estimated that, at 1.49 GHz,

$E^{SNR}\approx 0.06E^{NT}.$ (19)

$E^{NT}_{1.49}$ can be estimated from observations. For our Galaxy B02 adopt
$L^{NT}=6.1\times 10^{21}\;\rm{W}\;\rm{Hz}^{-1}$ (Berkhuijsen, 1984) at
0.4$\rm{GHz}$ and $\nu_{CCSN}=0.015$. They obtain for a radio slope
$\alpha(\frac{-d\;\rm{log}S_{\nu}}{-d\;\rm{log}\nu})=0.8$, a value of
$E^{NT}_{1.49}$ = 1.44. We note that the final fate of massive stars is a
critical assumption for estimating their contribution to thermal radio
emission. The formation of NSs and BHs depends on the amount of mass lost by
the massive star through stellar winds and on the hydrodynamics of the
explosion. One of the the most troublesome facts is that the variations shown
by these pre-supernova stars in their structural properties, like for e.g. the
Fe-core and O-core masses, are non-monotonic and pronounced even within small
differences in the ZAMS masses (Sukhbold & Woosley, 2014). Some recent works
in the attempt to characterize the parameters of successful and failed
supernovae have yielded some structural parameters that can be utilized in
predicting the fate of SNe (eg. Ertl et al., 2015; Ugliano et al., 2012;
Janka, 2012; O’Connor & Ott, 2011). Using these parameters, Spera, Mapelli &
Bressan (2015) and Slemer et al. 2017 (to be submitted) were able to
characterize the final fate of PARSEC massive stars for the different criteria
adopted for successful CCSN explosion. Following Spera, Mapelli & Bressan
(2015) and Slemer et al. 2017 we assume that stars with initial mass above
about $M=30~{}\rm{M}_{\sun}$ do not contribute to non thermal radio emission,
contrary to B02 where all stars of masses above $M=8~{}\rm{M}_{\sun}$ were
thought to explode as CCSN. The region between about $24-30~{}\rm{M}_{\sun}$
is critical because the explosion criterion by O’Connor & Ott (2011) produces
a much higher number of NSs than that produced by Ertl et al. (2015)
criterion. Finally we stress that we assume that progenitors undergoing pair-
instability SNe either collapse to a BH or are completely incinerated by a
thermonuclear explosion without producing relativistic electrons, similarly to
what is assumed for SNIa. This assumption on the CCSN distribution with
initial mass modifies the expected non thermal luminosity of star-forming
galaxies and requires a re-calibration of the GRASIL model. After accounting
for the failed and successful SNe in the evaluation of non thermal emission
from Equation 18, we show in Figure 5 the variation of the 1.49 GHz non-
thermal radio emissions with age, for different upper mass limits and
metallicities. We note that non-thermal radio emission is not sensitive to the
upper mass limit of the IMF, as long as it is larger than the assumed
threshold for failed SN, M = 30$~{}\rm{M}_{\sun}$. In this case non thermal
emission begins at an age of $\sim$7 Myr and not earlier ($\sim$3 Myr) as is
the case in B02. By coincidence, this critical time corresponds to about the
epoch at which thermal emission abruptly fades down, as depicted in Figure 5.
Non thermal emission remains effective up to about 35 - 38 Myr, depending on
the metallicity. In subsequent sections we will show how thermal and non-
thermal radio emission are checked and eventually re-calibrated using the
GRASIL code.

Figure 5: Variation of the 1.49 GHz and 33 GHz non-thermal (free-free)
emission with age at different metallicities and different IMF upper mass
limits. The red lines indicate thermal radio emission and the blue lines non-
thermal radio emission. The plot for the thermal radio emission has already
been shown in Figure 3 but added here for comparison with the non-thermal
radio emission. As easily noticed in the figure, the non-thermal emission is
almost unchanged for each of the different IMF upper mass limits, owing to the
fact that masses above 30 $\rm{M}_{\sun}$ collapse directly to BH and do not
explode to produce non-thermal emission. For this same reason, the non thermal
emission for all plots starts at about an age of 7 Myr. For comparison
purposes, the plot of the thermal radio emission is included to show it’s
pronounced variation with upper mass limits. At 1.4 GHz (upper panel), the
non-thermal radio emission dominates that at 33 GHz (lower panel) whereas at
33 GHz, the thermal emission dominates.

##  4 Calibration of the new SSP suite with GRASIL

In the previous sections, we described a new suite of SSPs that will supersede
the ones used in the current version of GRASIL. The new suite differs in many
aspects from the previous one as we briefly list in the followings. (1) The
SSP are based on the most recent PARSEC stellar evolutionary tracks with an
updated physics, in particular new mass-loss recipes and finer and wider
coverage in initial mass and metallicity; (2) they include the most recent
advances in our understanding of the CCSN explosion mechanism and account for
the so called "failed SN"; (3) a more accurate gaunt factor and metallicity-
dependent electron temperature were incorporated in the relation between the
integrated ionization photon luminosity and thermal radio luminosity (4) in
the current suite, we may adopt several values of the IMF upper mass limit
Because of all these differences, we need to check and recalibrate some
parameters of the SED produced by GRASIL, in particular to check whether we
are able to reproduce the canonical value of $q_{1.4GHz}=2.35$, observed in a
prototype normal star-forming galaxiy, see Eq. 1. For this purpose, in the
next section we will use GRASIL with the new SSPs to reproduce the SEDs of a
some selected well studied galaxies.

###  4.1 SED Modelling with GRASIL

GRASIL is a spectro-phometric code able to predict the SED of galaxies from
the FUV to the radio, including state-of-the-art treatment of dust
reprocessing (Silva et al., 1998, 2011; Granato et al., 2000), production of
radio photons by thermal and non-thermal processes (BO2) and an updated
treatment of PAH emission (Vega et al. (2005)). The reader is referred to
these original papers for a fully detailed description of the code. It is also
worth noting that nebular emission was already included in GRASIL by Panuzzo
et al. (2003) but with a completely different procedure which also accounted
for different electron densities. In this respect our procedure is more simple
because nebular emission is added directly to the SSPs, but it allows a more
versatile use of GRASIL.

For sake of convenience, we briefly summarise here the main features of
GRASIL. The first step is to compute a chemical evolution model that provides
the star formation history (SFH) and the metallicity and mass of the gas.
Other quantities are computed by the code CHE$-$EVO, such as mass in stars, SN
rates, detailed elemental abundances, but they are not used for the spectro-
photometric synthesis. GRASIL’s main task is to compute the SED resulting from
the interaction between the stellar radiation from CHE$-$EVO and dust, using a
relatively flexible and realistic geometry where stars and dust are
distributed in a spheroidal and/or a disk profiles. A spherical symmetric
distribution with a King profile is adopted in the case of spheroidal systems
while, for disk-like systems, a double exponential of the distance from the
polar axis and from the equatorial plane is adopted. The dusty environments
are made up of dust (i) in interstellar HI clouds heated by the general
interstellar radiation of the galaxy (refered to as the cirrus component),
(ii) associated with star-forming molecular clouds (MC) and (iii) in the
circumstellar shells of Asymptotic Giant Branch (AGB) stars. GRASIL performs
the radiative transfer of starlight through these environments. We remind the
reader that the reprocessing of starlight by dust in envelopes of AGB stars is
already taken into account in our SSPs. For the intrinsic dust properties,
GRASIL adopts grains (graphite and silicate) in thermal equilibrium and a
mixture of smaller grains and PAHs fluctuating in temperature. The dust
parameters are tuned in order to match with the observed emissivity and
extinction properties of the local ISM (Vega et al., 2005).

####  4.1.1 M100: a close to solar metallicity galaxy

The galaxy M100 was selected from the Spitzer Nearby Galaxies Survey (SINGS)
(Kennicutt et al., 2003), a collection of 75 nearby galaxies with a rich data
coverage from the far-UV to the radio, because it is one of the best sampled
objects, including the intensities of main emission lines. This will provide
the opportunity to check, for the first time in GRASIL, the consistency of the
SED continuum and emission lines fitting. To obtain the chemical evolution
model we adopt the parameters of the chemical evolution code CHE-EVO given by
Silva et al. (1998) who was able to well-reproduce M100. They are infall time
scale $t_{inf}~{}=~{}4.0$ Gyr, efficiency of the Schmidt-law star formation
law $\nu_{sch}~{}=~{}0.75$ and the total infall mass $m_{inf}$ = $2\times
10^{11}~{}M_{\sun}$. Figure 7 shows the SFH resulting from the adopted
chemical evolution parameters. The SFH is indicated by the solid blue line and
the gas mass history by the dotted purple line. To perform the SED fit, we
consider several time steps along the chemical evolution of the galaxy for
which we built a library of SED models with different GRASIL parameters. We
note that since the observed SED refers to the whole galaxy, the derived
parameters correspond to a luminosity average of the physical properties, as
commonly obtained by this kind of fitting process. As a particular check, we
also investigate the effect of changing the upper mass limit of the IMF,
$\rm{M}_{up}~{}=~{}40$,120 and $350~{}M_{\sun}$, on some physical quantities
derived from our best-fit SED, in particular the SFR. The main GRASIL
parameters varied are the molecular gas fraction $f_{mol}$ , the escape time
$t_{esc}$ and the optical depth of molecular clouds (MC) at 1$\micron$
$\tau_{1}$. The best fits were obtained by minimizing the merit function
$\chi^{2}$ which is given by

$\chi^{2}=\frac{1}{N}\sum^{N}_{i=1}\left(\frac{F_{mod}(i)-F_{obs}(i)}{Err(i)}\right)^{2}$
(20)

where $F_{mod}(i)$, $F_{obs}(i)$ and $Err(i)$ are the model flux values, the
observed fluxes and observational errors respectively. $N$ is the number of
photometric bands used to obtain the best fit. Our GRASIL best-fit SEDs of
M100 for $M_{up}=40~{}\rm{M}_{\sun}$ and $M_{up}=120~{}\rm{M}_{\sun}$, are
shown in the left and right panels of Figure 6 respectively, while the
corresponding best fit parameters are summarized in Table 1. We do not show
the case with $M_{up}=350~{}\rm{M}_{\sun}$, because its thermal-radio emission
was significantly exceeding the observed one. This discrepancy could not be
cured by varying any other parameter in the fit. This result is, by itself,
quite interesting and shows that for a normal star-forming galaxy it is
difficult to have an average IMF extending up to such high initial masses. The
different dust components in diffuse medium (cirrus) and molecular clouds are
shown and indicated by the blue and red dashed lines, respectively. The
thermal and non-thermal radio emission components are indicated by the cyan
and purple dashed dotted lines, respectively. The thermal component can be
seen to be negligible for the case of $M_{up}=40~{}\rm{M}_{\sun}$. We include,
for the first time in GRASIL, selected emission lines in the SED. The labels
in the plots refer to some important quantities derived from our best fit: the
bolometric luminosity from UV to radio (BOL), the total IR luminosity from 3 -
1000$\micron$ (FIR), the predicted extinction uncorrected $H_{\alpha}$
luminosity (Hagra), the observed $H_{\alpha}$ luminosity (Hadat), $H_{\alpha}$
attenuation (A(H$\alpha$)), attenuation in the V($0.55\micron$) band (A(V)),
the SFR averaged over the last 100 Myr ($<\rm{SFR}>$), the q-parameter as
defined by equation 1 (q(1.4GHz)), the coefficients of the SFR(Hα) and SFR(IR)
calibrations ($<\rm{SFR}/Ha>$ and $<\rm{SFR}/FIR>$ respectively). A summary of
other important quantities derived from our best fit is provided in Tables 3
and 4. For both values of the adopted $M_{up}$, the best fits match very well
the overall observed UV-radio SEDs. However, for $M_{up}$ =
120$~{}\rm{M}_{\sun}$, the model over-predicts by factor of $\sim$3 the
observed (i.e. non corrected by extinction, hereafter ’transmitted’)
H${\alpha}$ luminosity, $1.23\times 10^{41}\text{ergs}\;\text{s}^{-1}$ (taken
from Kennicutt et al. (2009)). This upper mass limit also produces a higher UV
emission and a slight excess of thermal emission. On the other hand, the
observed H${\alpha}$ luminosity is well reproduced in the case of
$M_{up}=40~{}\rm{M}_{\sun}$. The over-predicted H${\alpha}$ luminosity
resulting from adopting $M_{up}$ = 120 $\rm{M}_{\sun}$ could be lowered by
increasing the escape time of young stars in their parent clouds, leading to a
larger absorption in the MCs. This will also lower the predicted far-UV
emission which, in the current best fit, is slightly larger than the observed
value. However, a larger absorption from MCs will increase the predicted
$24~{}\micron$ flux above the observed one. Note that at $24~{}\micron$ the
cirrus component has a pronounced minimum and its contribution to the overall
MIR emission is only a small fraction of the total. Thus, the $24~{}\micron$
flux, being dominated by the MC emission, is indeed a strong diagnostic for
the amount of light reprocessed by the MC component. From our best fit, we
obtain a CCSN rate to NT radio luminosity ratio (Equation 18) that is about a
factor of 1.35 larger than the value obtained by B02 using the previous radio
model. That is,

$E^{NT}_{1.49}(\rm{Thiswork})=1.944=E^{NT}_{1.49}(\rm{B02})\times 1.35$ (21)

We anticipate here that this value is confirmed by all subsequent SED fits. We
also note that this value is almost independent from the adoption of $M_{up}$
for the reasons already discussed previously. The predicted average SFR
resulting from the panchromatic fit is only slightly affected by the adoption
of a different value of $M_{up}$. By increasing $M_{up}$ from 40 to 120
$~{}\rm{M}_{\sun}$ the average SFR decreases by about 16 per cent.

Figure 6: M100 GRASIL best-fit SED for $M_{up}$ = 40 $\rm{M}_{\sun}$ (right
panel) and $M_{up}=120~{}\rm{M}_{\sun}$(left panel). The different dust
components, diffuse medium and molecular clouds are represented by the dashed
blue and red lines respectively. The thermal and non thermal radio components
are represented by the cyan and purple dashed dotted lines. The thermal
component can be seen to be negligible for the case of
$M_{up}=40~{}\rm{M}_{\sun}$. For the different $M_{up}$, note the differences
in the predicted quantities labelled in the plots (see text for more details),
in particular the Hα luminosities, attenuations (A(Hα)) and SFR(Hα)
calibration coefficient. We estimated $E^{NT}_{1.49}=1.944$, a factor of 1.35
larger than that obtained in B02. Our average value of q = 2.4 which is in
line with the observed value of 2.35 (Equation 1) is an indication of our good
calibration.  Figure 7:  Star formation history of M100 adopted in our CHE-EVO
model. The model parametrs are labelled: galaxy’s age (solid red line) ,
infall time (dashed green line) and schmidt-law efficiency. These parameters
were adopted by Silva et al. (1998) based on constraints to reproduce the
observed final mass of M100. The SFH is indicated by the solid blue lines
while the gas mass history by the dotted purple line.

####  4.1.2 NGC 6946: individual star-bursting regions

In the previous section we were able to reproduce fairly well the observed
SEDs of the prototype galaxy M100, in particular, we used the best fit model
to calibrate the constant $E_{NT}$ of the non-thermal radio emission model. In
this section we wish to check the thermal component of our radio emission
model. For this purpose, we compare our synthetic SEDs with those of selected
extranuclear regions of NGC 6946, a well studied starburst galaxy at a
distance of 6.8 Mpc. This galaxy is dominated by very young starburst regions
and shows relatively large thermal over non-thermal ratios (eg Israel &
Kennicutt, 1980; Heckman et al., 1983), and thus is particularly well suited
for a test of the free-free emission originating from star formation. Murphy
et al. (2010) observed these regions at 1.4, 1.5, 1.7, 4.9, 8.5 and 33 GHz and
complemented their full SEDs with existing data from the UV to the submm
range. For eight observed regions Murphy et al. (2010) estimated an average 33
GHz thermal component of $\approx$ 85 per cent of the total, while for one
region (named extra-nuclear region 4), this percentage was found to be
significantly lower, $\approx$ 42 per cent, likely for the presence of a so-
called anomalous dust emission component which suppresses the thermal
component. Particulary interesting for our purpose is that Murphy et al.
(2010) also provide the intensity of the H$\alpha$ emission for each region,
that can be directly compared to the predictions of our model.

To model the SEDs of the extra-nuclear star-forming regions of NGC 6946 we
adopt a spherical symmetric distribution with a King profile for the stars and
dust. Most important, we use the data not corrected for the local background
emission of the galaxy, though (Murphy et al., 2011) provide also data
corrected for this emission. We thus adopt, for each region, a chemical
evolution model composed by a star-burst superimposed to a quiescent star
formating component. Our choice is dictated by the fact that the subtraction
by (Murphy et al., 2011) may bias our results and we prefer to eventually
split the contribution of the starburst and the old disk components directly
from our fits.

Figure 8: SFH of one of the 8 extranuclear regions, NGC 6946_enuc_3 for
$M_{up}=40~{}M_{\sun}$ case. A starburst age of $t_{sb}^{1}$ = 14 Myr (green
solid line) produced the best fit but we added $t_{sb}^{2}$= 5 Myr (blue
dotted line) case only to illustrate that we do not expect any contribution to
the non-thermal emission from the starburst at this age according to our radio
model in Figure 5. At burst ages between 7 and 30 Myr, we expect a
contribution to the non-thermal radio emission also from the starburst. The
red dotted-dashed line indicates the age (11.000 Gyr) at which the SED of this
region was observed.

For purposes of clarity we show in Figure 8 a plot of the star formation
history of the extra-nuclear region $\sharp$3 for
$M_{up}=~{}40~{}\rm{M}_{\sun}$. The corresponding chemical evolution model
parameters are labelled in the left side of the figure. For these models, the
galaxy is observed at a given age $t_{gal}~{}=~{}11$ Gyr. The ongoing
starburst has an exponentially declining SFR that begins at
$t_{0}~{}=~{}t_{gal}-t_{sb}$, where tsb is the age of the starburst. Two SFR
examples are given for a starburst age of tsb = 5 Myr and tsb = 14 Myr
respectively. In the first case we are not expecting any contribution to the
non-thermal emission from the starburst while, in the second case since the
age is larger than 7 Myr (see Figure 5) we expect a contribution to the non-
thermal radio emission also from the starburst. Thus any age between tsb = 7
and 30 Myr can provide SEDs with different percentages of thermal vs non-
thermal radio emission. We note that this can be used to mimic possible
different contributions of non thermal emission from the underlying disk. To
obtain the GRASIL best fit models we mainly varied the following parameters:
the escape time of young stars from their birth clouds $t_{esc}$, the optical
depth of molecular clouds at 1$\micron$ $\tau_{1}$, molecular gas fraction
$f_{mol}$, the submm dust emissivity index $\beta$ and the start time of the
burst $t_{0}$ (represented by the starburst age $t_{sb}$). The parameters of
the best fit SEDs are shown in Table 1 while, the corresponding plots, are
shown in Figure 9. In this figure, left panels show the results obtained
adopting $M_{up}~{}=~{}40M_{\sun}$ and right panels show those for
$M_{up}~{}=~{}120~{}M_{\sun}$.

Figure 9:  GRASIL best fit SEDs for the extra-nuclear regions of NGC 6946 (red
diamonds) for upper mass limit of 40 (left panels) and 120
$\rm{M}_{\sun}$(right panels). To test our radio decomposition, we over-
plotted on top our thermal radio component (dotted-dashed cyan lines) the
local background subtracted radio data flux (cyan diamonds) by Murphy et al.
(2011).  Figure 10: continued

For all regions the observed IR SED can be fairly well reproduced by using
either of the IMF upper mass limits. In general, the value of $q_{1.4}$ is
found to lie between 2.5 and 2.6. This is about 0.2 dex larger than the value
observed in the normal star-forming galaxy M100 ($q_{1.4}$ = 2.35) implying
that, in these star-forming regions, the ratio between radio and FIR
luminosity is about a factor 1.6 lower than in the normal star-forming galaxy
M100. The estimated young starburst ages support the notion that this is due
to a lack of non-thermal radio emission as predicted by the original models by
B02. This is also evident from the radio slope which is found to be flatter
than the value observed in the normal star-forming galaxy ($\alpha\simeq$\-
0.8). It is worth reminding here that we adopted the calibration
($E^{NT}_{1.49}$) of non-thermal radio luminosity obtained from the fits to
M100. We note also that since the radio emission is dominated by the thermal
component, which has no free parameters, the SFR in these regions is very well
determined by the 33 GHz point, modulo the IMF. Another characteristics of the
fits is the difficulty of reproducing the UV data, in spite of a fairly well
fit in all the other bands from mid-IR to radio bands. In particular the runs
with $M_{up}=40~{}M\sun$ tend to show larger UV fluxes than the corresponding
cases made with $M_{up}=120~{}M\sun$, which is against what could be expected
from the dependence of the UV luminosity on the IMF. This can be noted in
Figure 9 where we see that adopting $M_{up}~{}=~{}120~{}M\sun$ we either
reproduce or underpredic the far-UV data while, adopting
$M_{up}~{}=~{}40~{}M\sun$, we either reproduce or overpredict them. At the
same time, the average SFR in the models with $M_{up}~{}=~{}40~{}M\sun$ is
about two to three times larger than that obtained with the models that adopt
$M_{up}~{}=~{}120~{}M\sun$. This is in evident contrast with what we have
found in the fits of normal galaxies where we see that the SFR, obtained with
different values of $M_{up}$, differ by 20 per cent at maximum and it is a
consequence of the young age of such star bursting regions, where the
bolometric luminosity is dominated by the most massive stars. Finally, the
attenuation in the models with $M_{up}~{}=~{}40~{}M\sun$ is always lower than
that obtained with the models adopting $M_{up}~{}=~{}120~{}M\sun$. As the non-
thermal radio emission starts about 7 Myr after the beginning of the burst of
star formation (see Figure 5), we have the opportunity to perform an accurate
decomposition of the radio flux into thermal and non thermal components. We
present in Table 4 the thermal fraction resulting from this decomposition and
we show the two radio components in Figure 9. To show the quality of our
decomposition, we also show in Figure 9 and only for the case with
$M_{up}~{}=~{}120~{}M\sun$, the background subtracted radio data fluxes (cyan
diamonds) by Murphy et al. (2011). In the next section, we will discuss the
SFR calibrations derived using our best fit model of the galaxies studied in
this work.

Table 1:  GRASIL best fit parameters (for upper mass limits of 40 and
120$~{}\rm{M}\sun$) for the modelled SEDs of M100 and NGC 6956 SF regions.

ID | $t_{gal}/t_{sb}$ | $t_{esc}$ | $\beta$ | $\tau_{1}$ | $f_{mol}$
---|---|---|---|---|---
| (Gyr/Myr) | (Myr) |  |  |
(1) | (2) | (3) | (4) | (5) | (6)
|  |  | $M_{up}$ | = 40 M☉ |
M100 | 12.0 | 2.5 | 2.0 | 12.0 | 0.10
NGC 6946_1 | 8.0 | 1.0 | 2.1 | 9.07 | 0.30
NGC 6946_2 | 9.0 | 0.5 | 1.7 | 12.00 | 0.05
NGC 6946_3 | 8.0 | 1.0 | 2.0 | 24.48 | 0.20
NGC 6946_5 | 8.0 | 0.3 | 2.0 | 4.15 | 0.25
NGC 6946_6 | 12.0 | 1.2 | 2.2 | 5.33 | 0.40
NGC 6946_7 | 7.0 | 1.0 | 2.1 | 14.81 | 0.30
NGC 6946_8 | 7.0 | 0.6 | 2.1 | 6.12 | 0.22
NGC 6946_9 | 8.0 | 1.0 | 2.1 | 5.33 | 0.30
|  |  | $M_{up}$ | = 120 M☉ |
M100 | 12.0 | 0.9 | 2.0 | 14.81 | 0.10
NGC 6946_1 | 11.0 | 0.6 | 2.2 | 5.33 | 0.12
NGC 6946_2 | 11.0 | 1.0 | 1.7 | 16.60 | 0.15
NGC 6946_3 | 8.0 | 0.9 | 2.2 | 12.00 | 0.14
NGC 6946_5 | 18.0 | 1.0 | 1.8 | 18.75 | 0.50
NGC 6946_6 | 11.0 | 0.7 | 2.2 | 5.33 | 0.12
NGC 6946_7 | 8.0 | 1.0 | 2.2 | 5.33 | 0.16
NGC 6946_8 | 8.0 | 0.9 | 2.2 | 5.33 | 0.12
NGC 6946_9 | 9.0 | 0.6 | 2.2 | 5.33 | 0.12

Column (2) gives the age of the galaxy ($t_{gal}$) in Gyr for the normal star-
forming galaxy M100 and the age of the burst ($t_{sb}$) in Myr for the NGC
6946 star-bursting regions. Parameters in other columns are as described in
text.

##  5 Discussion

In this section we discuss the results obtained from the best fits of the SEDs
of M100 and the extra-nuclear regions of NGC 6946. We begin with the SFR
calibrations obtained with the new SSPs and then discuss the impact of the
newly added emission line prediction on the resulting attenuation.

###  5.1 Star formation rate calibrations from GRASIL best fits

We first show in Table 3 the luminosities of the best fit models in the
selected photometric bands, from the UV to radio wavelengths. The upper panel
refers to the results obtained adopting $M_{up}~{}=~{}40~{}M_{\sun}$ while,
the lower panel refers to those obtained with $M_{up}~{}=~{}120~{}M_{\sun}$.
The FIR luminosity in column 11 was obtained by integrating the IR specific
luminosity from 3 to 1000 $\micron$. We also show the predicted intrinsic
(suffix int) and transmitted (suffix tra) intensities of the H$\beta$ and
H$\alpha$ emission lines, the value of $q_{1.4}$ (Equation 1), the relative
contribution to the 3 - 1000 $\micron$ FIR luminosity by the molecular clouds
component and by the cirrus component, and the ratio between the 3 - 1000
$\micron$ FIR and the bolometric luminosity. In the columns of radio
luminosities, we enclose in parenthesis the fractional contribution of the
thermal radio component to the total radio emission, derived from the models.

Effects of Mup
By construction, i.e. since we are discussing best fits SEDs, the predicted
luminosities obtained with the two different $M_{up}$ are very similar, but
for those in the emission lines and in the UV bands. These are the most
difficult to model because they are the most sensitive to the attenuation.
This is not the only reason however, because even the intrinsic H$\alpha$ and
H$\beta$ luminosity differs by a significant factor in the two $M_{up}$ cases.
Indeed, in the two cases the ionizing photon flux (and the far and near-UV)
differ much more than the corresponding bolometric luminosities. Said in
another way, one may be able to obtain the same bolometric flux with two
different IMF upper limits but the amount of ionizing photons may be
significantly different, and this mainly affect the intensity of the
recombination lines and the free-free radio emission. Note however that the 33
GHz luminosity does not show the same strong dependence on the adopted Mup as
that shown by the recombination lines, and this is likely due to the fact that
a non negligible contribution of the non-thermal luminosity is present even at
this high radio frequency, which should be larger in the case of lower
$M_{up}$. The best fit transmitted intensities of the H$\alpha$ emission lines
are compared with the observed ones in Figure 11, for both cases of $M_{up}$.
We do not include M100 in this plot to better show the case of the star-
forming regions. The solid line indicates the one-to-one correlation. We see
that, in the case of extra-nuclear regions, the predicted values of the case
with $M_{up}~{}=~{}40~{}\rm{M}_{\sun}$ are larger than those of the case with
$M_{up}~{}=~{}120~{}\rm{M}_{\sun}$.

Figure 11: Comparisons between our model’s transmitted intensities of the
H$\alpha$ emission lines with the observed ones, for $M_{up}~{}=~{}40M_{\sun}$
(red daimonds) and $M_{up}~{}=~{}120$$M_{\sun}$ (blue triangles) cases. The
solid line indicates the one-to-one correlation whereas the horizontal lines
indicate the errors in the observed H$\alpha$ line. The green daimond
indicates the case with $M_{up}~{}=~{}350~{}\rm{M}_{\sun}$ for the extra-
nuclear region 8.

In general our best fit models are able to reproduce the observed values with
an accuracy of about 50 per cent. Looking to individual regions, we see that
regions $\sharp$1 ,$\sharp$2, $\sharp$3, $\sharp$5, $\sharp$6 are compatible
with $M_{up}=40~{}\rm{M}_{\sun}$ while, for regions $\sharp$9 and $\sharp$7, a
larger value of $M_{up}$ seems suggested, up to
$M_{up}~{}=~{}120~{}\rm{M}_{\sun}$. From the figure it also appears that, in
order to fit region $\sharp$8, an even higher IMF upper mass limit (see the
green daimond in Figure 11 for the case of $M_{up}~{}=~{}350~{}\rm{M}_{\sun}$)
should be used. Furhermore, we note that the in all cases where we are able to
reproduce the observed H$\alpha$ emission we are also able to reproduce fairly
well the observed FUV luminosity, which is also strictly related to the
intrinsic ionizing photon flux. On the contrary the NUV flux is not well
reproduced by our models, likely indicating that its origin is not strictly
related to the star formation process. Since all model reproduce the observed
bolometric luminosity and, in particular, the 24 $\micron$ and the 33 GHz
data, we conclude that, by combining in a consistent way this relevant
information, it is possible to obtain a fairly robust estimate of the upper
mass limit of the IMF . Indeed we stress that the FIR luminosity and those at
H$\alpha$, 33 GHz and 24 $\micron$ bracket, on one side, the intrinsic
ionizing photon flux and, on the other, the attenuation and re-emission by
molecular clouds where the stars reside during their initial phase, which are
the most sensitive to the fractional contribution of the most massive stars.

As far as the prototype normal galaxy M100 is concerned, we remind that the
observed H$\alpha$ luminosity was matched by the model with
$M_{up}~{}=~{}40~{}M_{\sun}$. It is important, at this stage, to remind that
these conclusions should be taken with care because the current SSP adopted do
not account for binary evolution. Indeed it is known that binary evolution can
produce more ionizing photons at later stages than single star evolution,
because of the loss of the envelope by binary interaction (Wofford et al.,
2016). In particular we may say that regions $\sharp$1 ,$\sharp$2, $\sharp$3,
$\sharp$5, $\sharp$6 may be reproduced fairly well without invoking a high
$M_{up}$ and/or significant effects by binary evolution. The latter effect
could instead explain the location of region number $\sharp$8 that cannot be
reproduced even with $M_{up}~{}=~{}350~{}M_{\sun}$, as indicated by the green
diamond in Figure 12). This effect, which add a new dimension to the problem,
is under study in PARSEC and it will be included in the nearest future.

The q ratio.
The second thing that we note by inspecting Table 3 is that the value of the
q1.4 does not significantly depend on the upper limit of the IMF. This is
because a significant fraction of both the bolometric luminosity and the radio
luminosity are contributed by stars less massive than M = $40~{}M_{\sun}$.
Indeed after a few Myr, the upper mass of the SSP is already significantly
lower than M = $120~{}M_{\sun}$. Furthermore, the q1.4 ratio is generally
higher in the extranuclear regions of NGC 6946 than in the normal SF galaxy
M100, by a factor of $\simeq$1.4. We remind that in our model, CCSNs,
responsible for the non thermal emission, are produced only by the mass range
$10~{}M_{\sun}$ $\leq$ M$\leq$ $30~{}M_{\sun}$ and the individual star-
bursting regions have an age which is younger than the threshold for CCSN
production. Thus they did not yet reach a stationary equilibrium for non-
thermal radio emission and not only their radio slopes appear flatter but also
their q1.4 are higher than that of normal star-forming galaxies B02.

The SFR calibration.
Using the luminosity in the different bands, as presented in Table 3, and the
average SFR obtained from the GRASIL fits, we show in Table 4, the
corresponding SFR calibrations C(b) = SFR/L(b), where b indicates the band.
The average SFR is derived by considering the last 100 Myr time interval for
the normal star-forming galaxy M100, and the age of the burst in the star-
forming regions of NGC 6946, which is shown in column 16 of Table 4 for the
NGC 6946 extranuclear regions. This choice is suggested by the fact that
within an entire galaxy, the star-forming regions distribute continuously with
time while, this is obviously not the case for individual regions. The average
SFR ($\rm{M}_{\sun}\;\rm{yr}^{-1}$) so computed is shown in the second column
of Table 4. These calibrations are directly obtained by GRASIL. For sake of
comparison, we also add, in Table 4, some calibrations that can be obtained by
using the un-attenuated fluxes derived from GRASIL best fit models, either
directly or by using the analytical relations obtained by means of the SSPs.
These are useful to compare the differences between the SFR derived from a
panchromatic fit that realistically account for ISM processes to analytical
fits made with SSP fluxes. For example, the FUV, H$\beta$ and H$\alpha$ values
shown in column 3, 4 and 6, refer to the calibrations obtained by dividing the
average SFR by the corresponding un-attenuated intrinsic fluxes of the models,
listed in Table 3. For H$\beta$ and H$\alpha$ we also show, in columns 5 and
7, the values obtained by inserting their corresponding intrinsic fluxes in
the analytical calibrations obtained from simple SSPs models, Equations 40 and
39 respectively. For the 33 GHz radio emission, we add in column 15, the value
of the calibration C(33)SSP = SFR/L(33 GHz), directly derived from the SSP
models (Equation 38). The last rows in the first and second sections,
indicated by $<NGC~{}6946>$, show the median values of all SFR calibrations
derived from the star-bursting regions of NGC 6946. In the third section we
collect some common SFR calibrations from the literature, with the
corresponding bibliographic sources given in the caption.

Figure 12:  Left panel: SFR calibration constants C($\nu$), in units of
$\rm{M}_{\sun}\rm{yr}^{-1}10^{28}\rm{erg}^{-1}\;\rm{s}\;\rm{Hz}$, at $\nu$ =
1.4, 4.9, 8.5 and 33.0$\rm{GHz}$ (Table 4), for the star-bursting regions and
using the total radio emission. Triangles and daimonds indicate cases of
$M_{up}~{}=~{}40~{}\rm{M}_{\sun}$ and $M_{up}~{}=~{}120~{}\rm{M}_{\sun}$
respectively. Right panel: The same as in the left panel but for M 100 and
using both the total radio (crosses) and non-thermal radio emission (xs). In
both panels, all red symbols and lines indicate cases of
$M_{up}~{}=~{}40~{}\rm{M}_{\sun}$ and blue symbols and lines cases of
$M_{up}~{}=~{}120~{}\rm{M}_{\sun}$. Also all lines represent least-square
fits.

In Figure 12 we plot the calibration constants C($\nu$) at $\nu$ = 1.4, 4.9,
8.5 and 33.0 $\rm{GHz}$, in units of
$\rm{M}_{\sun}\rm{Yr}^{-1}10^{28}\rm{erg}^{-1}\;\rm{s}\rm{Hz}$, against the
corresponding radio frequencies, for both the star-bursting regions (left
panel) and M100 (right panel) and for both cases of $M_{up}$. The relations
(in logarithmic units) are almost linear above 1.4 GHz, and we show the best
fit regressions as solid lines. Concerning the dependence on Mup, there is a
clear difference between the starburst and the normal star formation regime in
the sense that, in the former case the constant is about a factor of three
larger independently from the frequency. Again this effect is due to the young
age of the starburst regions coupled with the dependence of the thermal radio
emission on Mup. Indeed for the normal SFR regime, the dependence on Mup
becomes significant only at high frequencies, where the thermal emission start
dominating. From the fits we derive the following general relations between
SFR (in $\rm{M}_{\sun}\rm{yr}^{-1}$) and radio luminosity (in
$\rm{erg}\;\rm{s}^{-1}\rm{Hz}^{-1}$). For Star-bursting regions, with average
metallicity $\sim$ 0.006, we have at any frequency between 1.4 $\rm{GHz}$ and
33 $\rm{GHz}$, to better than 10 per cent,

$\displaystyle\rm{log}(\frac{SFR}{L\nu})^{tot}~{}=~{}0.43\ \rm{log}(\nu)-0.96\
\rm{log}(\frac{M_{up}}{120})-27.91$ (22)

while, for the normal star-forming galaxy, M100, with slightly more than solar
metallicity, we have for the total radio emission

$\displaystyle\rm{log}(\frac{SFR}{L\nu})^{tot}~{}=~{}0.73\ \rm{log}(\nu)-0.16\
\rm{log}(\frac{M_{up}}{120})-28.31$ (23)

and for the non-thermal radio emission, we have

$\displaystyle\rm{log}(\frac{SFR}{L\nu})^{nth}~{}=~{}0.58\ \rm{log}(\nu)-0.43\
\rm{log}(\frac{M_{up}}{120})-28.33$ (24)

Relations similar to Equations 22, 23 and 24 above were provided by Murphy et
al. (2011), assuming Kroupa IMF (with $M_{up}=100~{}\rm{M}_{\sun}$ ), solar
metallicity and a constant SFR over a timescale of 100 Myr. For total radio
emission

$\left(\frac{SFR}{L\nu}\right)^{tot}_{Mur}=10^{-27}\left(2.18\nu^{-0.1}(\frac{T_{e}}{10^{4}})^{0.45}+15.1\nu^{-0.85}\right)^{-1}$
(25)

and for non-thermal radio emission

$\left(\frac{SFR}{L\nu}\right)^{nth}_{Mur}=10^{-27}(6.64\times
10^{-29}\nu^{0.85})$ (26)

These relations are in between our relations in 22 and 23, running only
$\lesssim$ 20 per cent lower than our one for normal star-forming galaxies,
below 33 GHz. However at frequencies higher than 33 GHz the difference grows
rapidly and at 100 GHz our values are $\sim$40 per cent and $\sim$200 per cent
higher than that of Murphy et al. (2011) for starburst and normal galaxies,
respectively. Interestingly, the relation derived by Schmitt et al. (2006)
with Starburst99, assuming a Salpeter IMF (with
$M_{up}~{}=~{}100~{}\rm{M}_{\sun}$), solar metallicity and a continuous SFR of
1 $\rm{M}_{\sun}\;yr^{-1}$

$\left(\frac{SFR}{L\nu}\right)^{tot}_{Sch}~{}=~{}10^{-27}(8.55\nu^{-0.8}+1.6\nu^{-0.1})$
(27)

agrees well with our relations in 22 and 23 at frequencies from about 33 GHz
to 100 GHz. Above 100 GHz the dust emission contribution becomes dominant.

It is interesting to compare the GRASIL calibration at 33 GHz with that
derived from the SSP models (Equation 38), i.e. columns 14 and column 15 in
Table 4. The values show some discrepancies, especially for normal galaxies.
However we remind that the values in column 15 refer to free-free emission
alone while the 33 GHz flux in column 14 may include a non negligible
contribution by synchrotron emission. If we take this contamination into
account, using the percentage contribution of the thermal radio component
enclosed in parenthesis, the corrected values of column 14 are in fair
agreement with those predicted by Equation 38. This is expected because, at
these wavelengths, attenuation, including also the free-free one (Vega et al.,
2008), should not be important. The above overall agreement indicate a
surprising robustness of the radio calibration, irrespective of the
differences in the underlying models. This will be particularly important for
the high redshift galaxies especially in their early evolutionary phases.

As a further comparison, we provide at the bottom of Table 4 other SFR
calibrations taken from the literature. We remind the reader that, in this
work, we adopt a Kennicutt (1983) IMF and a typical time-scale for averaging
the SFR rate of 100 Myr for normal galaxies and equal to the age of the
starburst in the extra-nuclear star bursting regions. The details on the
parameters adopted by different authors may be found in the quoted papers.

###  5.2 Dust attenuation properties

It has been already shown in (Silva et al., 1998; Granato et al., 2000) that
the attenuation is the result of the interplay between the extinction
properties of dust grains and the distribution between dust and stars in space
and in time. Further discussion can also be found in (Charlot & Fall, 2000;
Panuzzo et al., 2003). We now investigate the properties of the attenuation
curves of the galaxies studied here, with particular interest to understand
its dependence on the galaxy type, i.e. starburst vs. normal regime, and also
the effects of using different parameters in the analysis, such as Mup in the
IMF.

We first show in Figure 13, the attenuation curves, Aλ/AV, of M100 (green
solid line) and of the NGC 6946 extra-nuclear regions studied in this work
(black solid lines). They are simply derived by comparing the transmitted
galaxy flux to the intrinsic one, as a function of the wavelength. The top
panel refers to the case of $M_{up}~{}=~{}40~{}\rm{M}_{\sun}$ while the case
with $M_{up}~{}=~{}120~{}\rm{M}_{\sun}$ is shown in the bottom panel.

Figure 13: Attenuation curves, Aλ/AV, for M100 and NGC 6946 extra-nuclear
regions, for the cases of $M_{up}~{}=~{}40$ (top panel) and
$M_{up}~{}=~{}120~{}\rm{M}_{\sun}$ (bottom panel). The black solid lines are
the attenuations curve of the eight regions of NGC 6946 while the dotted blue
line represents the median of these curves. The median values of $R_{V}$ and
${E(B-V)}$ for the star-forming regions are also given in the labels. The
green line is the attenuations curve of M100. For comparison purposes, we
added the attenuation curves of Calzetti et al. (2000) (red dashed line) and
Cardelli, Clayton & Mathis (1989) (purple dotted-dashed line) The later
attenuation seems to agree very well with that of M100 at wavelengths above
$\sim$ 4000 Å.

The dotted blue lines represent the median curve of the NGC 6946 extra-nuclear
regions obtained by sampling the curves of individual regions at selected
wavelengths. The median values of the quantities $E(B-V)$ and RV = AV/E(B-V)
for the star-bursting regions are also given in the labels. We have also
labelled some prominent emission lines ($H\alpha$ line is in brown). For
comparison purposes, we added the attenuation curves of Calzetti et al. (2000)
(red dashed lines) and Cardelli, Clayton & Mathis (1989) (purple dotted-dashed
lines). We also collect in Columns (2 - 9) of Table 5 the attenuation in
H$\alpha$, H$\beta$, in the FUV (0.16 $\micron$) and NUV (0.20 $\micron$), in
the B-band(0.45 $\micron$), and in the interpolated continuum at H$\beta$
(named A48), in the V-band(0.55 $\micron$) and in the interpolated continuum
at H$\alpha$ (named A65). Columns 10 and 11 give the optical depth of the
molecular cloud at 1 $\micron$ derived from GRASIL and the $R_{V}$ ratio,
respectively. In Column 12 we show the difference A48-A65 which corresponds to
$E(48-65)$. Using the Balmer decrement method for the $H\beta$ and $H\alpha$
lines, we also obtained the same quantity, $E(H\beta-H\alpha)$:

$E(H\beta-H\alpha)=1.086\left({\rm
ln}(\frac{H\beta^{int}}{H\alpha^{int}})-{\rm
ln}(\frac{H\beta^{tra}}{H\alpha^{tra}})\right)$ (28)

and show its value in Column 13. We assume an intrinsic line intensity ratio,
$(H\alpha^{int}/H\beta^{int})$ = 2.86, which is valid for $T_{e}$ =10,000 K
(Hummer & Storey, 1987). Column 14 gives the escape time of young stars from
their birth cloud in Myr. Column 15 gives the FIR flux in
$\rm{ergs}\,\rm{s}^{-1}\,\rm{cm}^{-2}$ in the 40 - 120 $\micron$ interval
derived using the 60 and 120 $\micron$ fluxes (Helou et al., 1988).

Several points can be noted by inspection of Figure 13 and Table 5. First of
all, we see that, while in normal galaxies the attenuation in the lines is
significantly higher than that in the surrounding continuum, (a factor of 4 on
average for M100), this is not true for the star bursting regions where this
factor is not more than 1.3. The fact that the attenuation in the lines is
significantly higher than that in the surrounding continuum is well known and
it is a manifestation that young stars are more dust enshrouded than older
stars. However, in the case of the star-bursting regions the underlying
continuum is essentially produced by the same stellar populations, with a
small contribution from the older populations, so that the attenuation in the
lines is not much different from that in the continuum. To get more insights
from the models we compare the characteristic R values in the lines and in the
nearby continuum. We see that for star-forming galaxies, RHα =
A${}_{H\alpha}/($AHβ-AHα) is much larger than the value obtained in the
corresponding interpolated underlying continuum regions
(R${}_{65}=A_{65}/(A_{48}-A_{65}$). For example, for M100, RHα = 16(10) while
R65 = 4(4), for Mup = 40(120) M☉. In the starburst regions, RHα and R65 have
similar values between 6 and 10. The high values so determined for RHα would
suggest a high neutral absorption in these objects but this is not the case
because we have not modified the grain size distribution which, by default in
GRASIL is the one of the Galactic ISM. We also see that in M100, R65, in the
continuum, has a value of $\sim$3, compatible with the common extinction laws
provided in literature (Calzetti et al., 2000; Cardelli, Clayton & Mathis,
1989). Instead this effect is due to the fact that the attenuation within
molecular clouds is so high (see the values of $\tau_{1\micron}$ in Table 5)
that the emission of young stars is almost completely absorbed, until they
escape from the clouds. This effect mimics the neutral absorption. This effect
is enhanced for the emission lines because they are generated only by stars
younger than about 6 Myr, so that the fraction of line flux absorbed by
molecular clouds with respect to the total one, is even higher. Adopting a
$t_{esc}$ larger than 7 Myr will erase line emission in our model,
irrespective of the galaxy types considered here. A major consequence of the
age selective attenuation described above is that it leads to wrong estimates
of the attenuation in the line emission, if one uses methods involving
attenuation-affected observables, like the Balmer decrement method. A more
accurate method to estimate the intrinsic H$\alpha$ luminosity (hence the
H$\alpha$ attenuation) is by using the 33 GHz and 24 $\micron$ luminosities,
since they are optimal tracers of the radiation emitted by the most massive.
Adopting a median value of 78.5 per cent as the percentage contribution of the
thermal radio component in young star-bursts, we first obtain the thermal
radio emission component of our model’s 33 GHz total radio emission
luminosity. Using the analytical relation given in Equation 2 (Te and gaunt
factors used here are as given in Equations 34 and 3 respectively), we derive
the number of ionizing photons corresponding to this luminosity. With Equation
35, we then estimate the corresponding intrinsic H$\alpha$ luminosity. A plot
of the estimated intrinsic H$\alpha$ luminosity (in ${\rm erg}\;{\rm s}^{-1}$)
against our model’s 33 GHz thermal radio luminosity (L33 in ${\rm erg}\;{\rm
s}^{-1}{\rm Hz}^{-1}$) is presented in Figure 14. We added in this figure also
the normal SF galaxy M100, with an average thermal fraction of 16.0 per
cent(42.0 per cent) for Mup = 40 $\rm{M}_{\sun}$(Mup = 120 $\rm{M}_{\sun}$),
respectively. The fitting relation (solid line) considering only the star
bursting regions for both cases of Mup, is given by

${\rm log}({\rm L_{H\alpha}^{int}})={\rm log}(0.785{\rm L(33)})+14.10$ (29)

Figure 14: Plot of our model’s 33 GHz thermal radio luminosity against the
intrinsic H$\alpha$ luminosity. Solid line represents linear fit for the star-
bursting regions. The H$\alpha$ luminosity here is derived from the thermal
radio luminosity using Equations 2 and 35. We adopted a median thermal
fraction of 78.5 per cent in estimating the thermal radio component of the 33
GHz total radio luminosity. We also added in this plot the M100 which has an
average thermal fraction of 16.0 per cent(42.0 per cent) for Mup = 40
$\rm{M}_{\sun}$(Mup = 120 $\rm{M}_{\sun}$).

luminosity (L33 in erg s?1Hz?1) is presented in Figure 12.

The corresponding attenuation at H$\alpha$ can be obtained by using the
intrinsic value provided by equation 29

${\rm A_{H\alpha}}=-2.5{\rm log}\left(\frac{H\alpha^{obs}}{0.785{\rm
L_{33}}}\right)+35.25$ (30)

As mentioned earlier, this attenuation is larger than the one that can be
obtained with the Balmer decrement method because the latter might not
accounts for the strong obscuration in the early life of massive stars (see
Table 5). Thus, in presence of an estimate of the radio flux at 33 GHz,
Equation 30 should be preferred for young starbursts. A similar relation,
though with somewhat larger scatter, can be obtained for the 24 ${\micron}$
flux, which is shown in Figure 15. The relation between the model’s intrinsic
H$\alpha$ (in ${\rm erg}\;{\rm s}^{-1}$) and 24 $\micron$ luminositiy, L24 =
$\lambda$Lλ in ${\rm erg}\;{\rm s}^{-1}$, is

${\rm log}({\rm L_{H\alpha}^{int}})=0.59{\rm log}({\rm L_{24}})+15.73$ (31)

almost independently from Mup. Correspondingly, the derived attenuation in
H$\alpha$ is

${\rm A_{H\alpha}}=-2.5{\rm log}\left(\frac{H\alpha^{obs}}{1.48\rm
L_{24}}\right)+39.33$ (32)

Figure 15: Plot of the 24 $\micron$ luminosity against the intrinsic H$\alpha$
luminosity, both extracted directly from our best-fit models. Solid line
represents linear fit. We excluded M100 in this plot so as to better represent
the star-bursting regions.

It is also interesting to provide a relation between the attenuation at 1600 Å
(Column 4 of Table 5) and the ratio of FIR (Column 11 of Table 3) to
F${}_{1600}^{tra}$ (Column 3 of Table 3). The plot is shown in Figure 16,
along with the linear fitting relation obtained for the star-bursting regions
(solid line), for both Mup cases.

$A_{1600}~{}=~{}1.209{\rm log}\left(\frac{FIR}{F_{1600}^{obs}}\ \right)+0.254$
(33)

We note that the normal galaxy M100 is out from the relation. This may likely
be due to the non negligible contribution to the FIR intermediate age stellar
populations that are not contributing to the far UV.

Figure 16:  Plot of attenuation at 1600 Å against the ratio, $FIR/F_{1600}$,
both extracted from our best-fit models. The solid black line represents our
linear fit as given in Equation 33. The red crosses and blue ’X’s indicate
M${}_{up}~{}=~{}40$ and 120 $\rm{M}_{\sun}$ cases respectively.

##  6 Conclusions

In this paper we have used the recent PARSEC tracks to compute the integrated
stellar light, the ionizing photon budget and the supernova rates predicted by
young SSP models for different IMF upper mass limits. The SSPs spans a wide
range in metallicities, from 0.0001 to 0.04 and initial masses ranging from
0.1 to 350M$\sun$. In building the integrated spectra, we have adopted the
spectral library compiled by Chen et al. (2015). This library is a result of
homogenising different sets of libraries through a process fully described in
Girardi et al. (2002) and Chen et al. (2014). Using this integrated spectra in
the photoionization code CLOUDY, we have also calculated and included in the
integrated spectra the nebular continum and the intensities of some selected
emission lines. With the new SSPs we can then be able to predict the
panchromatic spectrum and the main recombination lines of star-forming
galaxies of any type, with GRASIL.

We made also two major revisions in the radio emission model. First, we
revisited the relation between free-free radio emission, number of ionizing
photons and radio frequency and came up with a relation that (a) takes into
account the full dependence of the electron temperature on the metallicity and
the effect of considering different IMF upper mass limits and (b) incorporated
a more accurate gaunt factor term. The free-free radio emission shows a
significant variation with metallicity and IMF upper mass limit, decreasing by
a factor of about 3 when Z increases from 0.0001 to 0.02 and inceasing by more
than an order of of magnitude when the mass limit is increased from 40 to 350
$M\sun$, at constant total mass. These differences cannot be negelected,
especially when calibrating SFRs using tracers that depend strongly on the
ionizing photon budget. Hence, the corresponding SFR calibrations (H$\alpha$,
H$\beta$ and thermal radio emission) take into account this dependence on
metallicity and IMF upper limits.

Second, we revised the non-thermal radio emission model originally described
by B02, taking into account recent advances in the CCSN explosion mechanisms
which indicates a range of stellar masses where the stars fail to explode, the
so called Failed SNe. We adopt a threshold value of 30 M$\sun$, above which
stars do not produce non-thermal radio emission. This has two immediate
effects: (a) the beginning of the non-thermal radio emission is delayed by
about 7 Myr, $\sim$ the lifetime of a 30 M$\sun$ star; (b) the non-thermal
radio emission is independent from the IMF upper mass limit as long as it is
above 30 M$\sun$. It is also assumed that pair-instability SNe, discussed by
Slemer et al. (2017) do not produce synchrotron radiation.

With these revisions in the new SSPs and in the radio emission model, there is
the need to check some parameters of the GRASIL code. We first re-calibrate
the proportionality constant between the supernova rate and the corresponding
non-thermal radio luminosity, $E_{1.49}^{NT}$), using one of the best-sampled
nearby normal star-forming galaxy, M100. We are able to reproduce very well
the far-UV to radio SED of this galaxy, for IMF upper mass limits of 40
$M_{\sun}$ and 120 $M_{\sun}$, with an average $q_{1.4}$ of 2.42. We obtain a
value of $E_{1.49}^{NT}$ = 1.94 which is larger than that obtained previously
by B02 by a factor of 1.35, but it is similar to that obtained by Vega et al.
(2005). We find that using M${}_{up}~{}=~{}350~{}\rm{M}_{\sun}$ produces a
thermal radio emission that significantly exceeds the observed one. This
excess thermal radio emission cannot be cured by varying any other parameter
in the fit. This result suggests that, for a normal star-forming galaxy, it is
difficult to have an average IMF extending up to such high initial masses.
What depends strongly on the upper mass limit of the IMF is the number of
ionizing photons, which beqars on the intensity of the recombination lines and
on the thermal radio emission. Indeed, in spite of being able to reproduce the
FIR and 1.4 GHz radio emission, the model that adopts Mup = 120 $M_{\sun}$
over predicts the H$\alpha$ emission by a factor of about three while, in the
case of Mup = 40 $M_{\sun}$, the discrepancy is only of a few percent.

We then check the new thermal radio emission model against the well studied
thermal radio dominated star-forming regions in NGC 6946. Even in this case we
are able to reproduce very well the observed SEDs from NIR to radio
wavelengths for both cases of the IMF upper mass limits. In fitting the SEDs
of these thermal radio-dominanted star-bursting regions, we adopt the value of
$E_{1.49}^{NT}$ resulting from the non-thermal radio calibration with the
normal star-forming galaxy M100. The estimated values of the $q_{1.4}$ ratio
in these regions lie between 2.5 and 2.6, implying a relatively lower non-
thermal emission than in normal star-forming galaxies. The additional evidence
of flatter radio slopes supports the notion that there is lack of the non-
thermal emission as predicted by the original non-thermal radio emission
models by B02. The resulting ages of the bursts, range from 7 to 12 Myr,
confirming that these observations can be used to determine the star-burst
ages. The fit obtained with the two IMF upper mass limits exhibit interesting
differences, in particular in the predicted H$\alpha$ and UV luminosities
which are the most sensitive to the IMF and to dust attenuations. We show
that, by combining information from the FIR, 24 $\micron$, 33 GHz and
H$\alpha$, we can determine the preferred value for Mup. This is the first
time, to the best of our knowledge, that the IMF upper mass limit can be
consistently determined. However, for region $\sharp$8 we cannot reproduce the
observed H$\alpha$ flux even with M${}_{up}~{}=~{}350~{}\rm{M}_{\sun}$. This
could be a case where binary evolution, which is known to produce an excess of
ionizing photons due to the loss of the envelope of massive stars by binary
interaction, may be required.

With luminosities and averaged SFR extracted from our best fit models (Table
3), we derive SFR calibrations in different bands for both normal galaxies and
star-burst regions (Table 4). Since thermal radio emission and that in the
recombination lines depend strongly on the upper mass limit of the IMF, we
provide multiple regression fitting relations with Mup and metallicity (e.g.
Equations 38, 39 40).

Finally, exploiting the realistic treatment of dust performed by GRASIL, we
investigate the properties of the attenuation curves of the galaxies studied
in this work with the aim of understanding their dependence on the galaxy
type. We find that, while in the normal SF galaxy M100 the attenuation in the
lines is significantly higher than that in the surrounding continuum, for the
star bursting regions of NGC 6946 the two attenuations are similar. A common
property is that the predicted R values obtained for the ${H\alpha}$ line is
large, mimicking a significant neutral absorption. For star bursting regions,
large values are found also for the continuum while, in the case of M100, the
R value of the continuum is normal. Large R values have been found in high
redshift star-forming galaxies by Fan et al. (2014), using a completely
different population synthesis tool. A major disturbing consequence of the age
selective attenuation, which could not be revealed in a foreground screen
model, is that it leads to wrong estimates of attenuation when using methods
involving observed line fluxes, like the Balmer decrement. Of course, more
accurate methods are those that combine UV or line fluxes with fluxes that are
not affected by attenuation, such as the well known relation between the FUV
(at 1600 Å) attenuation and the flux ratio ratio, FIR/F1600. For the star-
forming regions we revisit the former relation (Equation 33) and we provide
new relations between the H$\alpha$ attenuation and the observed H$\alpha$and
24 $\micron$ (Equation 30) or 33 GHz (Equation 32) fluxes. The above mentioned
relations, which we show to be almost independent from Mup, can be extremely
useful in estimating the attenuations in young high redshift galaxies. For
this purpose we are working on a larger galaxy sample to increase the
statistical significance of our results.

## Acknowledgments

We thank ?? for helpful discussions. A. Bressan and L. Girardi acknowledge
financial support from INAF through grants PRIN-INAF-2014-14. P. Marigo and Y.
Chen acknowledge support from ERC Consolidator Grant "STARKEY", G.A. n.
615604. F. Perrotta was supported by the RADIOFOREGROUNDS grant of the
European Union’s Horizon 2020 research and innovation programme
COMPET-05-2015, grant agreement number 687312.

## References

  * Allard et al. (1997) Allard F., Hauschildt P. H., Alexander D. R., Starrfield S., 1997, ARA&A, 35, 137
  * Asplund et al. (2009) Asplund M., Grevesse N., Sauval A. J., Scott P., 2009, ARA&A, 47, 481
  * Berkhuijsen (1984) Berkhuijsen E. M., 1984, A&A, 140, 431
  * Bicker & Fritze-v. Alvensleben (2005) Bicker J., Fritze-v. Alvensleben U., 2005, A&A, 443, L19
  * Bressan et al. (2013) Bressan A., Marigo P., Girardi L., Nanni A., Rubele S., 2013, in European Physical Journal Web of Conferences, Vol. 43, European Physical Journal Web of Conferences, p. 3001
  * Bressan et al. (2012) Bressan A., Marigo P., Girardi L., Salasnich B., Dal Cero C., Rubele S., Nanni A., 2012, MNRAS, 427, 127
  * Bressan, Silva & Granato (2002) Bressan A., Silva L., Granato G. L., 2002, A&A, 392, 377
  * Calzetti et al. (2000) Calzetti D., Armus L., Bohlin R. C., Kinney A. L., Koornneef J., Storchi-Bergmann T., 2000, ApJ, 533, 682
  * Calzetti et al. (2007) Calzetti D. et al., 2007, ApJ, 666, 870
  * Cardelli, Clayton & Mathis (1989) Cardelli J. A., Clayton G. C., Mathis J. S., 1989, ApJ, 345, 245
  * Castelli & Kurucz (2004) Castelli F., Kurucz R. L., 2004, ArXiv Astrophysics e-prints
  * Charlot & Bruzual (1991) Charlot S., Bruzual A. G., 1991, ApJ, 367, 126
  * Charlot & Fall (2000) Charlot S., Fall S. M., 2000, ApJ, 539, 718
  * Chen et al. (2015) Chen Y., Bressan A., Girardi L., Marigo P., Kong X., Lanza A., 2015, MNRAS, 452, 1068
  * Chen et al. (2014) Chen Y., Girardi L., Bressan A., Marigo P., Barbieri M., Kong X., 2014, MNRAS, 444, 2525
  * Chiosi, Bertelli & Bressan (1988) Chiosi C., Bertelli G., Bressan A., 1988, A&A, 196, 84
  * Condon (1992) Condon J. J., 1992, ARA&A, 30, 575
  * Condon & Yin (1990) Condon J. J., Yin Q. F., 1990, ApJ, 357, 97
  * Draine (2011) Draine B. T., 2011, Physics of the Interstellar and Intergalactic Medium
  * Ertl et al. (2015) Ertl T., Janka H.-T., Woosley S. E., Sukhbold T., Ugliano M., 2015, ArXiv e-prints
  * Fan et al. (2014) Fan L.-L., Lapi A., Bressan A., Nonino M., De Zotti G., Danese L., 2014, Research in Astronomy and Astrophysics, 14, 15
  * Ferland (1996) Ferland G., 1996, Hazy: A Brief Introduction to CLOUDY. Univ. of Kentucky Physics Department Internal Report
  * Girardi et al. (2002) Girardi L., Bertelli G., Bressan A., Chiosi C., Groenewegen M. A. T., Marigo P., Salasnich B., Weiss A., 2002, A&A, 391, 195
  * Granato et al. (2000) Granato G. L., Lacey C. G., Silva L., Bressan A., Baugh C. M., Cole S., Frenk C. S., 2000, ApJ, 542, 710
  * Heckman et al. (1983) Heckman T. M., van Breugel W., Miley G. K., Butcher H. R., 1983, AJ, 88, 1077
  * Helou et al. (1988) Helou G., Khan I. R., Malek L., Boehmer L., 1988, ApJS, 68, 151
  * Helou, Soifer & Rowan-Robinson (1985) Helou G., Soifer B. T., Rowan-Robinson M., 1985, ApJ, 298, L7
  * Hummer & Storey (1987) Hummer D. G., Storey P. J., 1987, MNRAS, 224, 801
  * Israel & Kennicutt (1980) Israel F. P., Kennicutt R. C., 1980, Astrophys. Lett., 21, 1
  * Janka (2012) Janka H.-T., 2012, Annual Review of Nuclear and Particle Science, 62, 407
  * Kennicutt (1983) Kennicutt, Jr. R. C., 1983, ApJ, 272, 54
  * Kennicutt (1998) Kennicutt, Jr. R. C., 1998, ARA&A, 36, 189
  * Kennicutt et al. (2003) Kennicutt, Jr. R. C. et al., 2003, PASP, 115, 928
  * Kennicutt et al. (2009) Kennicutt, Jr. R. C. et al., 2009, ApJ, 703, 1672
  * Kroupa (2001) Kroupa P., 2001, MNRAS, 322, 231
  * Kurucz (1993) Kurucz R. L., 1993, in Astronomical Society of the Pacific Conference Series, Vol. 44, IAU Colloq. 138: Peculiar versus Normal Phenomena in A-type and Related Stars, Dworetsky M. M., Castelli F., Faraggiana R., eds., p. 87
  * Lawton et al. (2010) Lawton B. et al., 2010, ApJ, 716, 453
  * Li et al. (2010) Li Y., Calzetti D., Kennicutt R. C., Hong S., Engelbracht C. W., Dale D. A., Moustakas J., 2010, ApJ, 725, 677
  * López-Sánchez et al. (2012) López-Sánchez Á. R., Dopita M. A., Kewley L. J., Zahid H. J., Nicholls D. C., Scharwächter J., 2012, MNRAS, 426, 2630
  * Maeder & Meynet (1988) Maeder A., Meynet G., 1988, A&AS, 76, 411
  * Maraston (2005) Maraston C., 2005, MNRAS, 362, 799
  * Marigo & Girardi (2007) Marigo P., Girardi L., 2007, A&A, 469, 239
  * Marigo et al. (2017) Marigo P. et al., 2017, ApJ, 835, 77
  * Murphy et al. (2012) Murphy E. J. et al., 2012, ApJ, 761, 97
  * Murphy et al. (2011) Murphy E. J. et al., 2011, ApJ, 737, 67
  * Murphy et al. (2010) Murphy E. J. et al., 2010, ApJ, 709, L108
  * O’Connor & Ott (2011) O’Connor E., Ott C. D., 2011, ApJ, 730, 70
  * Oster (1961) Oster L., 1961, Reviews of Modern Physics, 33, 525
  * Osterbrock (1989) Osterbrock D. E., 1989, Astrophysics of gaseous nebulae and active galactic nuclei. University Science Books, Mill Valley, CA
  * Panuzzo et al. (2003) Panuzzo P., Bressan A., Granato G. L., Silva L., Danese L., 2003, A&A, 409, 99
  * Pauldrach, Puls & Kudritzki (1986) Pauldrach A., Puls J., Kudritzki R. P., 1986, A&A, 164, 86
  * Rubin (1968) Rubin R. H., 1968, ApJ, 154, 391
  * Sander et al. (2015) Sander A., Shenar T., Hainich R., Gímenez-García A., Todt H., Hamann W.-R., 2015, A&A, 577, A13
  * Schmitt et al. (2006) Schmitt H. R., Calzetti D., Armus L., Giavalisco M., Heckman T. M., Kennicutt, Jr. R. C., Leitherer C., Meurer G. R., 2006, ApJ, 643, 173
  * Silva et al. (1998) Silva L., Granato G. L., Bressan A., Danese L., 1998, ApJ, 509, 103
  * Silva et al. (2011) Silva L. et al., 2011, MNRAS, 410, 2043
  * Spera, Mapelli & Bressan (2015) Spera M., Mapelli M., Bressan A., 2015, MNRAS, 451, 4086
  * Sukhbold & Woosley (2014) Sukhbold T., Woosley S. E., 2014, ApJ, 783, 10
  * Sutherland & Dopita (1993) Sutherland R. S., Dopita M. A., 1993, ApJS, 88, 253
  * Tang et al. (2014) Tang J., Bressan A., Rosenfield P., Slemer A., Marigo P., Girardi L., Bianchi L., 2014, MNRAS, 445, 4287
  * Ugliano et al. (2012) Ugliano M., Janka H.-T., Marek A., Arcones A., 2012, ApJ, 757, 69
  * Vega et al. (2008) Vega O., Clemens M. S., Bressan A., Granato G. L., Silva L., Panuzzo P., 2008, A&A, 484, 631
  * Vega et al. (2005) Vega O., Silva L., Panuzzo P., Bressan A., Granato G. L., Chavez M., 2005, MNRAS, 364, 1286
  * Wofford et al. (2016) Wofford A. et al., 2016, MNRAS, 457, 4296
  * Young et al. (1989) Young J. S., Xie S., Kenney J. D. P., Rice W. L., 1989, ApJS, 70, 699
  * Zhu et al. (2008) Zhu Y.-N., Wu H., Cao C., Li H.-N., 2008, ApJ, 686, 155

##  Appendix A Variation of electron temperature and the constants, $C_{2}$
and $C_{3}$, with IMF Upper mass limits and metallicity

Table 2: Calibration constants, $C_{2}$ and $C_{3}$, for different
metallicities and upper mass limits.

$\rm{Z}$ | $IQ(H)$ | $C_{2}$ | $T_{e}$ | $C_{3}$
---|---|---|---|---
| $(10^{54}$ | $(10^{-54})$ | $(10^{4})$ | $(10^{26})$
(1) | (2) | (3) | (4) | (5)
$M_{up}~{}=~{}40~{}\rm{M}_{\sun}$ |  |  |  |
0.0200 | 0.554 | 18.03 | 0.64 | 1.47
0.0080 | 0.738 | 13.56 | 0.89 | 1.96
0.0040 | 0.943 | 10.61 | 0.11 | 2.51
0.0005 | 1.376 | 7.27 | 0.16 | 3.66
0.0001 | 1.823 | 5.48 | 0.17 | 4.85
$M_{up}~{}=~{}120~{}\rm{M}_{\sun}$ |  |  |  |
0.0200 | 1.730 | 5.78 | 0.65 | 4.60
0.0080 | 2.153 | 4.64 | 0.94 | 5.73
0.0040 | 2.520 | 3.97 | 0.12 | 6.70
0.0005 | 3.065 | 3.26 | 0.17 | 8.15
0.0001 | 3.761 | 2.66 | 0.19 | 10.00
$M_{up}~{}=~{}350~{}\rm{M}_{\sun}$ |  |  |  |
0.0200 | 2.819 | 3.54 | 0.65 | 7.50
0.0080 | 3.469 | 2.88 | 0.95 | 9.22
0.0040 | 3.739 | 2.67 | 0.12 | 9.94
0.0005 | 4.269 | 2.34 | 0.17 | 11.35
0.0001 | 5.144 | 1.94 | 0.20 | 13.68

Column(1): metallicty. Column(2): production rate of ionizing photons per unit
solar mass formed. Column(3): calibration constant ($C_{2}$) of the SFR-IQ(H)
relation as given in equation 8. Column(4): electron temperature in K computed
using CLOUDY. Column(5): calibration constant ($C_{3}$) of the $SFR$-$L_{ff}$
relation as given in equation 11.

Figure 17: Plot of electron temperature Te (obtained from CLOUDY) against
metallicity Z for $M_{up}$ = 40, 120 and 350$~{}\rm{M}_{\sun}$ indicated by
the red ’X’s, blue crosses and green xterisks respectively. The lower axis is
the metallicity while the upper one is the corresponding oxygen abundance,
$\rm{x}~{}=~{}12+\rm{log(O/H)}$. The blue dashed-dotted line indicates the
metallicity at Z = 0.0134 which we adopted in this work as the solar
metallicity. The average of the empirical fits derived by López-Sánchez et al.
(2012) for high-ionization O iii and for low-ionization O ii zones is shown by
the black dashed line.  Figure 18: Variation of the constant ${C_{2}}$ with
age at different metallicities (Z = 0.0001, 0.0005, 0.004, 0.008 and 0.02) and
different upper mass limits. The symbols used to indicate the different
$M_{up}$ are as in Figure 17. These variations of ${C_{2}}$, depending on the
set of upper limits used, can severely underestimate or overestimate the
estimated SFR using the thermal radio luminosity as a tracer. As clearly
noticed, there is a large difference between $C_{2}$ obtained with
$M_{up}=40~{}\rm{M}_{\sun}$ and $M_{up}=120~{}\rm{M}_{\sun}$ and that obtained
with $M_{up}~{}=~{}120~{}\rm{M}_{\sun}$ and $M_{up}=350~{}\rm{M}_{\sun}$.
$C_{2}$ is as defined in equation 8.  Figure 19: Same as in Figure 18 but for
the constant $C_{3}$. $C_{3}$ is as defined in equation 11.

In Figure 17, we show the plot of electron temperature Te (obtained from
CLOUDY) against metallicity Z for $M_{up}$ = 40, 120 and 350. The oxygen
abundance corresponding to the metallicity Z, written as $\rm{x}=12+\
log(\rm{O/H})$ where $\rm{O/H}~{}=~{}\rm{O/H}_{\sun}+\log(Z/Z_{\sun}$, is
shown on the upper axis. To convert from Z to (O/H), we adopt the solar oxygen
abundance given by Asplund et al. (2009),
$\rm{x}_{\sun}=12+\log(\rm{O/H})_{\sun}=8.69\pm 0.05$ and $Z_{\sun}=0.0134$
and thus x = 8.69 + log(Z/0.0134). The blue vertical dotted line indicates
this solar metallicity value of 0.0134. The average of the empirical fits
derived by López-Sánchez et al. (2012) for high-ionization O iii and for low-
ionization O ii zones, using the MAPPINGS III code (Sutherland & Dopita,
1993), is shown as the dashed black lines. In this figure, our $T_{e}$ values
(’X’s, crosses and xterisks) computed with CLOUDY for different metallicities
are compared with the above-mentioned empirical fit obatined by López-Sánchez
et al. (2012). It can be seen that our $T_{e}$ values start deviating from
those of López-Sánchez et al. (2012) at low metallicities. A multiple
regression fitting relation between $T_{e}$, $M_{up}$ and Z that can be easily
included in analytical approximations, is provided by equation 34.

$\rm{log}\left(\frac{\rm{T}_{e}(\rm{Z},\rm{M}_{up})}{10^{4}~{}K}\right)=-0.13-0.19\rm{log}(Z/0.02)+0.03\rm{log}(\rm{M}_{up}/120)$
(34)

Table 2 lists the values of the constants $C_{2}$ and $C_{3}$ in equations 8
and 11, obtained with our constant SFR models for different SSPs parameters.
These values are shown in Figure 18. The multiple regression fitting relation
is given by Equation 35.

$\rm{log}(C_{2})=0.16\rm{log}(Z/0.02)-0.61\rm{log}(M_{up}/120)-53.18$ (35)

Table 2 also lists the values of the constants $C_{3}$ in equation 11. These
values are shown in Figure 19. The corresponding multiple regression fitting
relation is given by Equation 36.

$\rm{log}(C_{3})=0.61\rm{log}(M_{up}/120)-0.16\rm{log}(Z/0.02)+26.61$ (36)

Using the relation 35 in Equation 8 we obtain the integrated ionizing photon
flux of a star-forming region of arbitrary constant metallicity and IMF upper
mass limit

$\displaystyle\rm{log}(IQ(H))=$
$\displaystyle-0.16\rm{log}(Z/0.02)+0.61\rm{log}(M_{up}/120)+53.18]+\rm{log}(SFR)$
(37)

while, using the relation 36 in Equation 11 ($\rm{T}_{e}$ in this Equation is
in $10^{4}~{}K$), we may get the corresponding SFR-thermal radio calibration:

$\displaystyle\rm{log}(SFR/L(\nu)_{\rm{ff}})=0.60\rm{log}(M_{up}/120)-0.10\rm{log}(Z/0.02)$
$\displaystyle-\rm{log}(G_{dra})-26.57$ (38)

where SFR and $\nu$ are in units of $\rm{M}_{\sun}\rm{yr}^{-1}$ and GHz
respectively. Similarly for SFR vs. H$\alpha$ calibration we may write:

$\rm{log}(SFR/H\alpha)=0.16\rm{log}(Z/0.02)-0.61\rm{log}(M_{up}/120)-41.32$
(39)

and for SFR vs. H$\beta$ calibration:

$\rm{log}(SFR/H\beta)=0.16\rm{log}(Z/0.02)-0.61\rm{log}(M_{up}/120)-40.87$
(40)

Comparisons between SFR calibrations derived using Equations 38 , 39 and 40
above and those obtained directly from our model are given in Table 4 and
discussed in Section 5.1. The former case is indicated by the superscript ssp
in this table.

##  Appendix B Best-fits derived quantities for M100 and NGC 6946 Star-forming
regions

Table 3:  GRASIL Best-fit model luminosities.

ID | ${FUV_{i}}$ | ${FUV_{t}}$ | $H\beta_{i}$ | $H\beta_{t}$ | $H\alpha_{i}$ | $H\alpha^{\rm{data}}$ | $H\alpha_{t}$ | $24$ | $70$ | $FIR$ | $1.4$ | $4.9$ | $8.5$ | $33$ | $q_{1.4}$ | $MC$ | $CIR$ | $FIR/BOL$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14) | (15) | (16) | (17) | (18) | (19)
| $10^{41}$ | $10^{41}$ | $10^{39}$ | $10^{39}$ | $10^{39}$ | $10^{39}$ | $10^{39}$ | $10^{41}$ | $10^{41}$ | $10^{42}$ | $10^{26}$ | $10^{26}$ | $10^{26}$ | $10^{26}$ |  |  |  |
$M_{up}~{}=~{}40~{}\rm{M}_{\sun}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
M100 | 653.10 | 319.0 | 102.15 | 33.70 | 298.61 | 123.0 | 105.0 | 130.0 | 616.0 | 167.0 | 793.0 (2) | 285.0 (5) | 186.0 (8) | 71.6 (18) | 2.38 | 19 | 81 | 0.39
NGC 6946_1 | 4.52 | 2.03 | 1.92 | 0.95 | 5.48 | 3.96 | 2.88 | 0.77 | 2.81 | 0.66 | 2.28 (24) | 1.1 (44) | 0.87 (54) | 0.55 (75) | 2.53 | 25 | 75 | 0.62
NGC 6946_2 | 10.65 | 6.73 | 5.01 | 3.47 | 14.28 | 12.10 | 10.30 | 1.34 | 4.50 | 1.02 | 4.11 (37) | 2.3 (59) | 1.89 (68) | 1.33 (85) | 2.46 | 35 | 65 | 0.44
NGC 6946_3 | 4.52 | 2.20 | 1.92 | 1.02 | 5.48 | 4.86 | 3.07 | 0.47 | 2.85 | 0.60 | 2.28 (24) | 1.1 (44) | 0.87 (54) | 0.55 (75) | 2.53 | 28 | 72 | 0.58
NGC 6946_5 | 2.71 | 1.54 | 1.15 | 0.76 | 3.29 | 2.90 | 2.28 | 0.20 | 1.28 | 0.30 | 1.37 (24) | 0.7 (44) | 0.52 (54) | 0.33 (75) | 2.42 | 12 | 88 | 0.50
NGC 6946_6 | 12.47 | 5.22 | 4.86 | 2.17 | 13.87 | 7.87 | 6.59 | 2.92 | 8.41 | 2.01 | 7.99 (16) | 3.6 (32) | 2.68 (42) | 1.55 (65) | 2.47 | 27 | 73 | 0.66
NGC 6946_7 | 13.93 | 5.79 | 6.47 | 2.87 | 18.40 | 7.81 | 8.67 | 2.96 | 9.42 | 2.26 | 5.36 (36) | 3.0 (58) | 2.45 (67) | 1.71 (84) | 2.69 | 37 | 63 | 0.66
NGC 6946_8 | 9.90 | 4.99 | 4.59 | 2.66 | 13.08 | 2.73 | 8.02 | 1.42 | 5.75 | 1.29 | 3.81 (36) | 2.1 (58) | 1.74 (67) | 1.21 (84) | 2.61 | 21 | 80 | 0.57
NGC 6946_9 | 10.80 | 4.86 | 4.61 | 2.29 | 13.10 | 5.08 | 6.92 | 2.07 | 6.58 | 1.58 | 5.47 (24) | 2.7 (44) | 2.09 (54) | 1.33 (75) | 2.52 | 24 | 76 | 0.62
$M_{up}~{}=~{}120~{}\rm{M}_{\sun}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
M100 | 729.94 | 379.0 | 260.66 | 131.0 | 759.97 | 123.0 | 406.0 | 120.0 | 687.0 | 175.0 | 734.0 (7) | 291.0 (15) | 204.0 (21) | 98.4 (42) | 2.45 | 18 | 82 | 0.42
NGC 6946_1 | 3.77 | 1.29 | 1.82 | 0.63 | 5.17 | 3.96 | 1.94 | 0.83 | 3.33 | 0.70 | 2.26 (24) | 1.1 (44) | 0.87 (54) | 0.55 (76) | 2.60 | 212 | 78 | 0.69
NGC 6946_2 | 6.76 | 3.29 | 4.53 | 2.07 | 12.95 | 12.10 | 6.23 | 1.29 | 4.29 | 0.95 | 4.23 (35) | 2.3 (57) | 1.91 (66) | 1.33 (84) | 2.44 | 42 | 58 | 0.60
NGC 6946_3 | 3.01 | 0.99 | 1.85 | 0.71 | 5.28 | 4.86 | 2.19 | 0.57 | 3.02 | 0.63 | 1.96 (30) | 1.0 (52) | 0.82 (61) | 0.55 (81) | 2.62 | 21 | 79 | 0.70
NGC 6946_5 | 1.74 | 0.68 | 1.02 | 0.38 | 2.92 | 2.90 | 1.16 | 0.27 | 1.25 | 0.30 | 1.48 (21) | 0.7 (40) | 0.54 (49) | 0.33 (72) | 2.38 | 29 | 71 | 0.70
NGC 6946_6 | 10.69 | 3.49 | 5.12 | 1.55 | 14.66 | 7.87 | 4.79 | 2.85 | 9.27 | 2.02 | 6.32 (24) | 3.1 (44) | 2.43 (54) | 1.55 (76) | 2.60 | 26 | 74 | 0.70
NGC 6946_7 | 9.24 | 2.80 | 5.73 | 1.87 | 16.24 | 7.81 | 5.79 | 2.57 | 8.99 | 2.01 | 6.06 (30) | 3.2 (52) | 2.55 (61) | 1.71 (81) | 2.61 | 24 | 76 | 0.72
NGC 6946_8 | 6.57 | 2.28 | 4.07 | 1.61 | 11.55 | 2.73 | 4.97 | 1.54 | 6.26 | 1.32 | 4.30 (30) | 2.2 (52) | 1.81 (61) | 1.21 (81) | 2.60 | 21 | 79 | 0.68
NGC 6946_9 | 9.15 | 3.03 | 4.51 | 1.43 | 12.82 | 5.08 | 4.43 | 2.41 | 8.06 | 1.75 | 5.00 (28) | 2.5 (49) | 2.02 (58) | 1.33 (79) | 2.64 | 26 | 74 | 0.69

Col.(1): ID. Col.(2-7): luminosities at FUV (0.16 $\micron$), $H\beta$ and
$H\alpha$. i and t indicates intrinsic and attenuated transmitted luminosities
respectively. Col.(8): observed attenuation uncorrected $H\alpha$ luminosity.
Cols.(9 and 10): luminosities at 24 $\micron$ and 70 $\micron$. Col.(11):
total (3-1000 $\micron$)IR luminosity. Cols.(12-15): radio luminosities at
1.4, 4.9, 8.5 and 33 GHz. Enclosed in parenthesis is the fraction in per cent
of the thermal radio component to the total radio emission. Col.(16):
q-parameter as defined by equation 1. Cols.(17 and 18) are the MC and cirrus
contribution (in per cent) to the total IR luminosity respectively. Col.(19):
ratio of the total IR and the bolometric luminosities All UV, optical and
infrared luminosities are in $\rm{erg}\;\rm{s}^{-1}$ while all radio
luminosities are in $\rm{erg}\;\rm{s}^{-1}\;\rm{Hz}^{-1}$. The corresponding
SFR calibrations at these luminosities are given in Table 4.

Table 4:  GRASIL Best-fit derived SFRs and their calibrations at various
bands.

$ID$ | $<SFR>$ | $C(FUV_{i})$ |  $C(H\beta_{i})$ |  $C(H\beta_{i})^{ssp}$ | $C(H\alpha_{i})$ | $C(H\alpha_{i})^{ssp}$ | $C(24)$ | $C(70)$ | $C(FIR)$ | $C(1.4)$ | $C(4.9)$ | $C(8.5)$ | $C(33)$ | $C(33)^{ssp}$ | $t$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14) | (15) | (16)
|  | $10^{-43}$ | $10^{-42}$ | $10^{-42}$ | $10^{-42}$ | $10^{-42}$ | $10^{-43}$ | $10^{-43}$ | $10^{-44}$ | $10^{-28}$ | $10^{-28}$ | $10^{-28}$ | $10^{-28}$ | $10^{-28}$ |
$\rm{M}_{up}~{}=~{}40~{}\rm{M}_{\sun}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
M100 | 5.58 | 0.85 | 54.67 | 45.22 | 18.70 | 16.04 | 4.30 | 0.91 | 3.35 | 0.70 (2) | 1.96 (5) | 3.00 (8) | 7.79 (18) | 25.71 | 12.0
NGC 6946_1 | 0.09 | 1.88 | 44.33 | 31.96 | 15.5 | 11.34 | 11.01 | 3.02 | 12.80 | 3.72 (24) | 7.62 (44) | 9.72 (54) | 15.4 (76) | 14.77 | 8.0
NGC 6946_2 | 0.20 | 1.85 | 39.31 | 31.95 | 13.8 | 11.33 | 14.70 | 4.38 | 19.30 | 4.79 (37) | 8.64 (59) | 10.40 (68) | 14.9 (85) | 14.75 | 9.0
NGC 6946_3 | 0.09 | 1.88 | 44.22 | 31.96 | 15.5 | 11.34 | 18.01 | 2.98 | 14.20 | 3.72 (24) | 7.62 (44) | 9.72 (54) | 15.4 (76) | 14.77 | 8.0
NGC 6946_5 | 0.05 | 1.88 | 44.28 | 31.96 | 15.5 | 11.34 | 25.63 | 3.98 | 17.00 | 3.72 (50) | 7.62 (71) | 9.72 (78) | 15.3 (90) | 14.77 | 8.0
NGC 6946_6 | 0.20 | 1.58 | 40.51 | 31.99 | 14.2 | 11.35 | 6.75 | 2.34 | 9.79 | 2.47 (24) | 5.53 (44) | 7.36 (54) | 12.8 (76) | 14.79 | 12.0
NGC 6946_7 | 0.29 | 2.06 | 44.38 | 31.95 | 15.6 | 11.34 | 9.70 | 3.05 | 12.70 | 5.36 (17) | 9.70 (33) | 11.70 (42) | 16.8 (66) | 14.76 | 7.0
NGC 6946_8 | 0.20 | 2.06 | 44.46 | 31.95 | 15.6 | 11.34 | 14.37 | 3.55 | 15.80 | 5.36 (37) | 9.70 (59) | 11.70 (68) | 16.8 (85) | 14.76 | 7.0
NGC 6946_9 | 0.20 | 1.88 | 44.06 | 31.96 | 15.5 | 11.34 | 9.81 | 3.09 | 12.90 | 3.72 (37) | 7.62 (59) | 9.72 (68) | 15.4 (85) | 14.77 | 8.0
$<NGC~{}6946>$ | 0.20 | 1.88 | 44.25 | 31.96 | 15.5 | 11.34 | 12.69 | 3.07 | 13.55 | 3.72 (30) | 7.62 (52) | 9.72 (61) | 15.4 (80) | 14.77 | 8.0
$\rm{M}_{up}~{}=~{}120~{}\rm{M}_{\sun}$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
M100 | 4.80 | 0.66 | 18.43 | 15.92 | 6.32 | 5.65 | 4.00 | 0.70 | 2.74 | 0.65 (24) | 1.65 (44) | 2.36 (54) | 4.88 (76) | 8.90 | 12.0
NGC 6946_1 | 0.03 | 0.80 | 16.44 | 11.46 | 5.80 | 4.07 | 3.61 | 0.90 | 4.30 | 1.34 (9) | 2.73 (19) | 3.48 (26) | 5.48 (48) | 5.27 | 11.0
NGC 6946_2 | 0.07 | 1.02 | 15.22 | 11.27 | 5.33 | 4.00 | 5.35 | 1.61 | 7.25 | 1.63 (26) | 2.98 (46) | 3.62 (56) | 5.20 (77) | 5.13 | 11.0
NGC 6946_3 | 0.03 | 0.90 | 14.62 | 11.46 | 5.11 | 4.07 | 4.78 | 0.89 | 4.27 | 1.38 (37) | 2.64 (59) | 3.27 (68) | 4.88 (85) | 5.26 | 8.0
NGC 6946_5 | 0.02 | 0.92 | 15.75 | 11.32 | 5.48 | 4.02 | 5.84 | 1.28 | 5.23 | 1.07 (32) | 2.28 (53) | 2.95 (63) | 4.79 (82) | 5.16 | 18.0
NGC 6946_6 | 0.09 | 0.80 | 16.59 | 11.46 | 5.80 | 4.07 | 2.98 | 0.92 | 4.19 | 1.34 (37) | 2.73 (59) | 3.48 (68) | 5.48 (85) | 5.27 | 11.0
NGC 6946_7 | 0.08 | 0.90 | 14.48 | 11.46 | 5.11 | 4.07 | 3.23 | 0.92 | 4.15 | 1.38 (23) | 2.64 (42) | 3.27 (52) | 4.88 (74) | 5.26 | 8.0
NGC 6946_8 | 0.06 | 0.90 | 14.51 | 11.46 | 5.11 | 4.07 | 3.83 | 0.94 | 4.48 | 1.38 (26) | 2.64 (46) | 3.27 (56) | 4.88 (77) | 5.26 | 8.0
NGC 6946_9 | 0.07 | 0.82 | 16.63 | 11.46 | 5.85 | 4.07 | 3.11 | 0.93 | 4.29 | 1.51 (32) | 2.97 (53) | 3.72 (63) | 5.68 (82) | 5.26 | 9.0
$<NGC~{}6946>$ | 0.06 | 0.90 | 15.49 | 11.46 | 5.41 | 4.07 | 3.72 | 0.93 | 4.29 | 1.38 (29) | 2.69 (50) | 3.38 (60) | 5.04 (80) | 5.26 | 10.0
Literature |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
|  |  0.44a |  75.8h |  |  7.94b |  |  2.46d |  0.97g |  4.55b |  0.64a |  1.4e |  2.0e |  6.53a |  |
|  |  |  |  |  5.30c |  |  4.80h |  0.92h |  3.88a |  0.62e |  |  |  |  |
|  |  |  |  |  6.62h |  |  |  0.58f |  3.99h |  |  |  |  |  |

Col.(1): ID. Col.(2): average SFR (in $M_{\sun}\;yr^{-1}$) derived by
considering the last 100 Myr time interval for the normal star-forming galaxy
M100 and the age of the burst in the star-forming regions of NGC 6946
Cols.(3-15): SFR calibrations obtained using the above averaged SFR and the
luminosities (intrinsic values for FUV, H$\alpha$ and H$\beta$ ) given in
Table 3. In Cols.5, 7 and 15, we rather present the calibrations obtained from
simple SSPs models, Equations 40 , 39 and 38 respectively for H$\beta$,
H$\alpha$ and 33GHz. The later refers to free-free emission. For the radio
-based SFR calibration, we enclosed in parenthesis the percentage contribution
of the thermal radio component to the total radio emission. Col.(16): age of
the galaxy in Gyr for M100 or the age of the burst in Myr, for NGC 6946
extranuclear regions. The last row in the first and second panels (with ID,
$<NGC~{}6946>$) gives the median values of all quantities given for NGC 6946
star-bursting regions. The values of the SFR calibrations given in the 3rd
panel were taken from the literature. The superscript on these values
indicates the reference as described below: aMurphy et al. (2011), bKennicutt
(1998), cCalzetti et al. (2007), dZhu et al. (2008), eSchmitt et al. (2006),
fLi et al. (2010), gLawton et al. (2010), hPanuzzo et al. (2003). These
authors adopted different IMF and evolutionary synthesis models.

Table 5:  GRASIL Best-fit derived quantities related to dust attenuation.

ID | $A_{H\alpha}$ | $A_{H\beta}$ | $A_{FUV}$ | $A_{NUV}$ | $A_{B}$ | $A_{48}$ | $A_{V}$ | $A_{65}$ | $\tau_{1}$ | $R_{V}$ | $E(H\beta-H\alpha)^{c}$ | $E(H\beta-H\alpha)^{l}$ | $t_{esc}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12) | (13) | (14)
| $M_{up}~{}=~{}40~{}\rm{M}_{\sun}$ |  |  |  |  |  |  |  |  |  |  |  |
M100 | 1.13 | 1.20 | 0.78 | 0.83 | 0.30 | 0.28 | 0.24 | 0.23 | 17.79 | 4.17 | 0.055 | 0.093 | 2.5
NGC 6946_1 | 0.70 | 0.76 | 0.87 | 0.91 | 0.60 | 0.63 | 0.54 | 0.57 | 12.66 | 8.08 | 0.060 | 0.059 | 1.0
NGC 6946_2 | 0.36 | 0.40 | 0.50 | 0.53 | 0.31 | 0.33 | 0.27 | 0.29 | 15.38 | 6.03 | 0.041 | 0.037 | 0.5
NGC 6946_3 | 0.63 | 0.69 | 0.78 | 0.82 | 0.53 | 0.56 | 0.47 | 0.51 | 26.84 | 7.34 | 0.056 | 0.055 | 1.0
NGC 6946_5 | 0.40 | 0.45 | 0.61 | 0.65 | 0.43 | 0.42 | 0.39 | 0.37 | 8.43 | 9.14 | 0.053 | 0.054 | 0.3
NGC 6946_6 | 0.81 | 0.88 | 0.95 | 0.99 | 0.67 | 0.71 | 0.60 | 0.66 | 8.51 | 8.62 | 0.056 | 0.067 | 1.2
NGC 6946_7 | 0.82 | 0.88 | 0.96 | 0.99 | 0.67 | 0.73 | 0.59 | 0.66 | 17.67 | 7.71 | 0.062 | 0.059 | 1.0
NGC 6946_8 | 0.53 | 0.59 | 0.75 | 0.79 | 0.53 | 0.54 | 0.48 | 0.48 | 9.63 | 9.48 | 0.059 | 0.053 | 0.6
NGC 6946_9 | 0.70 | 0.76 | 0.87 | 0.91 | 0.60 | 0.63 | 0.54 | 0.57 | 8.79 | 8.08 | 0.060 | 0.060 | 1.0
| $M_{up}~{}=~{}120~{}\rm{M}_{\sun}$ |  |  |  |  |  |  |  |  |  |  |  |
M100 | 0.68 | 0.75 | 0.71 | 0.77 | 0.30 | 0.30 | 0.24 | 0.23 | 20.16 | 4.18 | 0.070 | 0.087 | 0.9
NGC 6946_1 | 1.07 | 1.16 | 1.18 | 1.24 | 0.82 | 0.95 | 0.75 | 0.78 | 9.28 | 9.48 | 0.173 | 0.088 | 0.6
NGC 6946_2 | 0.79 | 0.85 | 0.78 | 0.84 | 0.53 | 0.77 | 0.45 | 0.58 | 19.10 | 5.28 | 0.188 | 0.055 | 1.0
NGC 6946_3 | 0.96 | 1.04 | 1.20 | 1.26 | 0.86 | 0.93 | 0.78 | 0.79 | 16.11 | 10.38 | 0.136 | 0.087 | 0.9
NGC 6946_5 | 0.99 | 1.06 | 1.01 | 1.08 | 0.73 | 0.96 | 0.65 | 0.77 | 21.31 | 7.51 | 0.190 | 0.062 | 1.0
NGC 6946_6 | 1.21 | 1.30 | 1.21 | 1.28 | 0.84 | 1.01 | 0.75 | 0.81 | 9.05 | 8.92 | 0.201 | 0.084 | 0.7
NGC 6946_7 | 1.13 | 1.22 | 1.30 | 1.36 | 0.91 | 1.02 | 0.82 | 0.87 | 9.08 | 9.71 | 0.152 | 0.086 | 1.0
NGC 6946_8 | 0.92 | 1.01 | 1.15 | 1.21 | 0.82 | 0.89 | 0.74 | 0.76 | 9.26 | 10.03 | 0.135 | 0.085 | 0.9
NGC 6946_9 | 1.16 | 1.25 | 1.20 | 1.27 | 0.83 | 1.00 | 0.75 | 0.80 | 9.11 | 9.10 | 0.194 | 0.087 | 0.6

Col.(1) :ID. Cols.(2 - 9) give the attenuations in H$\alpha$, H$\beta$, FUV
(0.16$\micron$), NUV (0.20 $\micron$), B-band(0.45 $\micron$), the
interpolated continum at H$\beta$, V-band(0.55 $\micron$) and the interpolated
continum at H$\alpha$. They were derived using the ratio of the intrinsic
unattenuated and attenuated (observed) fluxes. Col.(10): optical depth of the
molecular cloud at 1 $\micron$. Col.(11): $R_{V}$. Col.(12):The reddening
given simply by A48 \- A65. Col.(13):The reddening derived using the balmer
decrement method as given by Equation 28. Col.(14): escape time of young stars
from their birth cloud in Myr.