# The Star Formation Rate of the Milky Way as seen by Herschel

D. Elia INAF - Istituto di Astrofisica e Planetologia Spaziali, Via Fosso del
Cavaliere 100, I-00133 Roma, Italy  S. Molinari INAF - Istituto di Astrofisica
e Planetologia Spaziali, Via Fosso del Cavaliere 100, I-00133 Roma, Italy  E.
Schisano INAF - Istituto di Astrofisica e Planetologia Spaziali, Via Fosso del
Cavaliere 100, I-00133 Roma, Italy  J. D. Soler INAF - Istituto di Astrofisica
e Planetologia Spaziali, Via Fosso del Cavaliere 100, I-00133 Roma, Italy  M.
Merello Departamento de Astronomía, Universidad de Chile, Casilla 36-D, Correo
Central, Santiago, Chile  D. Russeil Aix Marseille Univ., CNRS, CNES, LAM,
Marseille, France  M. Veneziani  Science & Technology Corporation, Olof
Palmestraat 14, 2616 LR Delft, The Netherlands  A. Zavagno Aix Marseille
Univ., CNRS, CNES, LAM, Marseille, France  Institut Universitaire de France,
Paris, France  A. Noriega-Crespo Space Telescope Science Institute, 3700 San
Martin Dr., Baltimore, MD 21218, USA  L. Olmi INAF - Osservatorio Astrofisico
di Arcetri, Largo E. Fermi 5, 50125, Firenze, Italy  M. Benedettini INAF -
Istituto di Astrofisica e Planetologia Spaziali, Via Fosso del Cavaliere 100,
I-00133 Roma, Italy  P. Hennebelle Université Paris-Saclay, Université Paris
Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France  R. S. Klessen Universität
Heidelberg, Zentrum für Astronomie, Institut für theoretische Astrophysik,
Albert-Ueberle-Str. 2, D-69120 Heidelberg, Germany  Universität Heidelberg,
Interdiszipliäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld
205, D-69120 Heidelberg, Germany  S. Leurini INAF - Osservatorio Astronomico
di Cagliari, Via della Scienza 5, I-09047 Selargius (CA), Italy  R. Paladini
Infrared Processing Analysis Center, California Institute of Technology,
Pasadena, CA 91125, USA  S. Pezzuto INAF - Istituto di Astrofisica e
Planetologia Spaziali, Via Fosso del Cavaliere 100, I-00133 Roma, Italy  A.
Traficante INAF - Istituto di Astrofisica e Planetologia Spaziali, Via Fosso
del Cavaliere 100, I-00133 Roma, Italy  D. J. Eden Astrophysics Research
Institute, Liverpool John Moores University, IC2, Liverpool Science Park, 146
Brownlow Hill, Liverpool, L3 5RF, UK  Armagh Observatory and Planetarium,
College Hill, Armagh BT61 9DB, UK  P. G. Martin Canadian Institute for
Theoretical Astrophysics, University of Toronto, 60 St. George Street,
Toronto, ON M5S 3H8, Canada  M. Sormani Universität Heidelberg, Zentrum für
Astronomie, Institut für theoretische Astrophysik, Albert-Ueberle-Str. 2,
D-69120 Heidelberg, Germany  A. Coletta INAF - Istituto di Astrofisica e
Planetologia Spaziali, Via Fosso del Cavaliere 100, I-00133 Roma, Italy
Dipartimento di Fisica, Sapienza Universitá di Roma, Piazzale Aldo Moro 2,
I-00185, Rome, Italy  T. Colman Université Paris-Saclay, Université Paris
Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France  R. Plume Department of
Physics & Astronomy, University of Calgary, 2500 University Dr. NW, Calgary,
AB, T2N1N4, Canada  Y. Maruccia INAF - Istituto di Astrofisica e Planetologia
Spaziali, Via Fosso del Cavaliere 100, I-00133 Roma, Italy  C. Mininni INAF -
Istituto di Astrofisica e Planetologia Spaziali, Via Fosso del Cavaliere 100,
I-00133 Roma, Italy  S. J. Liu INAF - Istituto di Astrofisica e Planetologia
Spaziali, Via Fosso del Cavaliere 100, I-00133 Roma, Italy

###### Abstract

We present a new derivation of the Milky Way’s current star formation rate
(SFR) based on the data of the Hi-GAL Galactic plane survey. We estimate the
distribution of the SFR across the Galactic plane from the star-forming clumps
identified in the Hi-GAL survey and calculate the total SFR from the sum of
their contributions. The estimate of the global SFR amounts to $2.0\pm 0.7$ M⊙
yr-1, of which $1.7\pm 0.6$ M⊙ yr-1 coming from clumps with reliable
heliocentric distance assignment. This value is in general agreement with
estimates found in the literature of last decades. The profile of SFR density
averaged in Galactocentric rings is found to be qualitatively similar to
others previously computed, with a peak corresponding to the Central Molecular
Zone and another one around Galactocentric radius $R_{\mathrm{gal}}\sim 5$
kpc, followed by an exponential decrease as
$\log(\Sigma_{\mathrm{SFR}}/[\mathrm{M}_{\odot}~{}\mathrm{yr}^{-1}~{}\mathrm{kpc}^{-2}])=a\,R_{\mathrm{gal}}/[\mathrm{kpc}]+b$,
with $a=-0.28\pm 0.01$. In this regard, the fraction of SFR produced within
and outside the Solar circle is 84% and 16%, respectively; the fraction
corresponding to the far outer Galaxy ($R_{\mathrm{gal}}>13.5$ kpc) is only
1%. We also find that, for $R_{\mathrm{gal}}>3$ kpc, our data follow a power
law as a function of density, similarly to the Kennicutt-Schmidt relation.
Finally, we compare the distribution of the SFR density across the face-on
Galactic plane and those of median parameters, such as temperature,
luminosity/mass ratio and bolometric temperature, describing the evolutionary
stage of Hi-GAL clumps. We found no clear correlation between the SFR and the
clump evolutionary stage.

stars: formation — stars: protostars — ISM:general

††journal: ApJ††facilities: Herschel

##  1 Introduction

The star formation rate (SFR) is a widely used parameter to globally
characterize galaxies (see Kennicutt & Evans, 2012, and references therein).
In the absence of small-scale description of the star forming activity in
external galaxies, their SFR expresses the fundamental relationship between
the stellar mass and the gas reservoir on which our understanding of galaxy
formation and evolution is based (see, e.g., McKee & Ostriker, 2007; Krumholz,
2014).”

Several estimates of the SFR have been computed for the Milky Way as well.
Considering, in principle, the total mass of available molecular gas and the
free-fall time in typical conditions of clouds in the Galaxy would lead to a
prediction of a global SFR of 300 M⊙ yr-1 (reducible to 46 M⊙ yr-1 if only
bound clouds are considered, Evans et al., 2021). However, the actual
estimates based on direct star formation indicators converge towards a value
of a few M⊙ yr-1 (see Licquia & Newman, 2015, and references therein). As
shown by Fraser-McKelvie et al. (2019) and Boardman et al. (2020), this is a
typical SFR for galaxies with characteristics similar to those of the Milky
Way, while in different galaxies values up to a few tens of M⊙ yr-1 can be
reached (see, e.g., Chomiuk & Povich, 2011; Mutch et al., 2011).

Table 1: Star formation rate (SFR) estimates for the Milky Way Method | SFR | Reference
---|---|---
| M⊙ yr-1 |
Ionization rate from radio free-free | 0.35aaThis value is not taken from the original article, but was re-scaled by Chomiuk & Povich (2011) to normalize all literature results to the same IMF (see text). | Smith et al. (1978)
Ionization rate from radio free-free |  $2.0\pm 0.6$aaThis value is not taken from the original article, but was re-scaled by Chomiuk & Povich (2011) to normalize all literature results to the same IMF (see text). | Guesten & Mezger (1982)
Ionization rate from radio free-free |  $1.6\pm 0.5$aaThis value is not taken from the original article, but was re-scaled by Chomiuk & Povich (2011) to normalize all literature results to the same IMF (see text). | Mezger (1987)
Ionization rate from [N ii] 205 $\mu$m (COBE) |  $2.6\pm 1.3$aaThis value is not taken from the original article, but was re-scaled by Chomiuk & Povich (2011) to normalize all literature results to the same IMF (see text). | Bennett et al. (1994)
Ionization rate from [N ii] 205 $\mu$m (COBE) |  $2.0\pm 1.0$aaThis value is not taken from the original article, but was re-scaled by Chomiuk & Povich (2011) to normalize all literature results to the same IMF (see text). | McKee & Williams (1997)
O/B Star Counts |  $1.8\pm 0.6$aaThis value is not taken from the original article, but was re-scaled by Chomiuk & Povich (2011) to normalize all literature results to the same IMF (see text). | Reed (2005)
Nucleosynthesis from 26Al (INTEGRAL) |  $2.0\pm 1.2$aaThis value is not taken from the original article, but was re-scaled by Chomiuk & Povich (2011) to normalize all literature results to the same IMF (see text). | Diehl et al. (2006)
Continuum emission at 100 $\mu$m (COBE) | $1.9\pm 0.8\tablenotemark{a}$ | Misiriotis et al. (2006)
Ionization rate from microwave free-free (WMAP) |  $2.4\pm 1.2$aaThis value is not taken from the original article, but was re-scaled by Chomiuk & Povich (2011) to normalize all literature results to the same IMF (see text). | Murray & Rahman (2010)
YSO counts (Spitzer) |  $1.1\pm 0.4$aaThis value is not taken from the original article, but was re-scaled by Chomiuk & Povich (2011) to normalize all literature results to the same IMF (see text). | Robitaille & Whitney (2010)
YSO counts (MSX) | $1.8\pm 0.3$ | Davies et al. (2011)
Combination of literature values | $1.9\pm 0.4~{}$ | Chomiuk & Povich (2011)
Continuum emission at 70 $\mu$m (Herschel) | $2.1\pm 0.4~{}$ | Noriega-Crespo (2013)
Combination of literature values | $1.65\pm 0.19~{}$ | Licquia & Newman (2015)
FIR clump counts (Herschel) | $2.0\pm 0.7$ | This work

State-of-the-art estimates of the SFR, summarized in Table 1, provide an
illustration of the variations found using different methods. The previous
compilation, presented in Chomiuk & Povich (2011, their Table 1), renormalized
some of the archival values from the literature, to conform all of them to the
same initial mass function (IMF, Kroupa & Weidner, 2003; Kennicutt et al.,
2009), as also recommended by Davies et al. (2011).

Misiriotis et al. (2006), using a direct relation between COBE/DIRBE
$100~{}\mu$m emission and SFR already adopted for external galaxies, provided
a value of 2.7 M⊙ yr-1 for the Milky Way’s SFR. Robitaille & Whitney (2010)
estimated a SFR from 0.68 to 1.45 M⊙ yr-1, starting from young stellar objects
(YSOs) revealed in the Spitzer/IRAC GLIMPSE Galactic plane survey, and using
model spectral energy distributions (SEDs) to predict the brightness and color
of the synthetic YSOs at different wavelengths.

More recent results do not use new data sets, but are derived by reconsidering
values existing in the literature: Chomiuk & Povich (2011) determine a SFR of
$1.9\pm 0.4~{}$M⊙ yr-1 by combining and re-normalizing literature results from
1978 to 2011. Licquia & Newman (2015) derived a SFR of $1.65\pm 0.19~{}$M⊙
yr-1 by statistically combining the prior measurements of this quantity in the
literature through a hierarchical Bayesian method.

An independent estimate of the SFR can be given through a technique which uses
far-infrared Herschel111Herschel is an ESA space observatory with science
instruments provided by European-led Principal Investigator consortia and with
important participation from NASA. satellite (Pilbratt et al., 2010) data, and
in particular photometric observations carried out by its PACS (at 70 and 160
$\mu$m, Poglitsch et al., 2010) and SPIRE (at 250, 350 and 500 $\mu$m, Griffin
et al., 2010) cameras. Veneziani et al. (2013) developed such method that
starts from the compact sources detected in Herschel maps. They generally
correspond to unresolved clumps that already host active star formation, or,
alternatively, are starless but show conditions for future activity. By
considering the former category of sources, Veneziani et al. (2013) evaluated
the contribution of each individual clump to the total SFR in two
$2^{\circ}\times 2^{\circ}$-tiles of Hi-GAL (Herschel InfraRed Galactic Plane
Survey, Molinari et al., 2010), namely the Open Time Key Project for
unbiasedly surveying the whole Galactic plane with PACS and SPIRE. For
comparison, they also derived the SFR of the same two regions through the
method of Li et al. (2010) based on assuming a direct relation between SFR and
70 $\mu$m emission. Subsequently, by assuming that these two fields are
representative of the entire Milky Way, Noriega-Crespo (2013) extrapolated for
the entire Galactic SFR an estimate of $2.1\pm 0.4~{}$M⊙ yr-1.

Other works also discussed the profile of the Milky Way SFR as a function of
the Galactocentric distance $R_{\mathrm{gal}}$, both to quantify the ability
of different zones of our Galaxy to form stars, and to make a comparison with
external galaxies, for which this relation is observed. A typical approach is
to study the SFR density, averaged in concentric Galactic rings, as a function
of the middle radius of the ring (e.g., Guesten & Mezger, 1982; Portinari &
Chiosi, 1999; Kennicutt & Evans, 2012; Lee et al., 2016). It is generally seen
that the profile of this quantity has an absolute maximum in correspondence of
the Central Molecular Zone (CMZ), another local maximum around at
$R_{\mathrm{gal}}\sim 4-5$ kpc (with a dip in the middle, centered at
$R_{\mathrm{gal}}\sim 2$ kpc), and an exponential decrease at larger radii.

In this paper, by extending the method of Veneziani et al. (2013), refined by
Veneziani et al. (2017), to the clump catalog of the entire Hi-GAL survey, we
present a direct and self-consistent estimate of the SFR for the whole Milky
Way, illustrating both the global SFR and its distribution across the Galactic
plane.

In Section 2 we show how Hi-GAL protostellar clump properties can be used to
obtain an estimate of the current Galactic SFR. In Section 3 we discuss how
the SFR is distributed across the Milky Way, adopting both the $[\ell,b]$ and
the pole-on perspective. We also make a test of the Kennicutt-Schmidt law with
our data, and investigate possible links between local SFR and clump average
evolutionary stage. Finally, in Section 4 we draw our conclusions.

##  2 The Herschel-based Milky Way’s SFR estimation

###  2.1 Source selection

We calculated the Galactic SFR using the physical properties of clumps
identified in the Hi-GAL survey (Elia et al., 2021). These objects were
identified and cataloged as follows.

The detection and 5-band photometry of Hi-GAL compact sources (i.e. unresolved
or poorly resolved) were performed through the CuTEx algorithm (Molinari et
al., 2011) in the five Hi-GAL bands, to obtain the single-band catalogs
presented by Molinari et al. (2016) and Molinari et al. (in prep.).

Starting from these lists, Elia et al. (2021) obtained a band-merged catalog,
from which only spectral energy distributions (SEDs) with a shape eligible for
a modified black body fit in the range from 160 to 500 $\mu$m were selected
(further details can be also found in Elia et al., 2013, 2017).

Heliocentric distances were estimated by Mège et al. (2021) in three steps:
$i$) by extracting the most reliable velocity component along the line of
sight from available spectroscopic surveys of the Galactic plane; $ii$) by
applying the rotation curve of Russeil et al. (2017) and solving the near/far
distance ambiguity towards the inner Galaxy by considering Hi absorption; and
$iii$) by complementing this information with independent estimates such as
Hii region distances or maser parallaxes. The medians of the absolute and
relative error on distances of Mège et al. (2021) are 0.54 kpc and $16\%$,
respectively.

As pointed out in Elia et al. (2017, 2021), the Hi-GAL compact sources are
distributed along a wide range of heliocentric distances and consequently
achieve a variety of physical sizes corresponding to different kinds of
structures: from single cores to larger overdensities hosting a more complex
but unresolved morphology. In particular, the majority of Hi-GAL sources
fulfill the definition of clump based on physical diameter ($0.3<D<3$ pc,
Bergin & Tafalla, 2007).

As a rough criterion to classify the clumps in star forming vs quiescent
(designated for brevity as protostellar and starless, respectively), Elia et
al. (2021) adopted the availability/lack of a detection at 70 $\mu$m,
respectively.

The modified black body fit provided temperatures for the clump SEDs and, for
cases with an available distance estimate, also the mass and the bolometric
luminosity. The mass was used to further classify the starless clumps as
gravitationally bound (called pre-stellar) vs unbound, by using a
gravitational stability criterion based on the so-called third Larson’s
relation (Larson, 1981). For pre-stellar sources the luminosity was estimated
from the whole integral of the best-fitting modified black body. For
protostellar sources, generally showing an excess of emission at
$\lambda<70\,\mu$m with respect to the modified black body (e.g., Dunham et
al., 2008; Giannini et al., 2012; Elia et al., 2017), the luminosity is
estimated by considering the sum of the integrals of the observed SED for
$\lambda<160\,\mu$m and of the best-fitting modified black body for
$\lambda\geq 160\,\mu$m, respectively.

The same considerations made above about the physical size can be extended to
the other distance-dependent observables mass and luminosity. Their ranges of
variability extend over several orders of magnitude not only due to intrinsic
differences among objects (that can be appreciated among sources located at
the same distance), but also - and above all - due to the wide range of
underlying distances, as figures 9 and 13 of Elia et al. (2021) clearly
illustrate. More quantitatively, 99% of Hi-GAL clumps of Elia et al. (2021)
provided with a distance estimate have masses ranging from 0.1 to $10^{4}$ M⊙,
and luminosities ranging from 0.1 to $10^{5}$ L⊙.

To compute the SFR through the method described in Section 2.2, first we
consider all 29880 protostellar clumps provided with a heliocentric distance.
Subsequently, in Section 2.2.1 we evaluate and suggest a reasonable additional
term which accounts for the contribution by further 5532 protostellar clumps
lacking a distance determination.

###  2.2 Computation of the SFR

To determine the SFR we followed the method described by Veneziani et al.
(2013) and refined by Veneziani et al. (2017).

In short, the contribution of each protostellar source to the total SFR is
computed through theoretical evolutionary tracks reported in the bolometric
luminosity vs mass diagram (Molinari et al., 2008; Baldeschi et al., 2017a;
Elia et al., 2021). At each time step, together with the clump mass-luminosity
pair, the mass of the internal protostar being formed is also computed. At the
end point of the theoretical track both the final mass of the star and the
corresponding total elapsed time are known. The ratio between these two
quantities constitutes the SFR contribution from the given clump.

The evolutionary tracks were initially obtained by Molinari et al. (2008) for
the case of an object forming a single central star but, especially at higher
masses, we actually deal with clumps possibly hosting the formation of entire
protoclusters (e.g. Baldeschi et al., 2017b). Veneziani et al. (2017) took
into consideration this aspect by applying a Monte Carlo procedure to an
evolutionary model of turbulent cores to account for the wide multiplicity of
sources produced during the clump collapse. We implemented this refinement in
this work as well (for details, see Veneziani et al., 2017).

Furthermore, given the wide range of clump heliocentric distances quoted in
the Hi-GAL catalog (Elia et al., 2017, 2021), a distance bias might be
expected to affect the estimate of physical parameters for these objects.
Baldeschi et al. (2017b) proposed a method to evaluate this possible bias by
simulating the appearance of nearby star forming regions observed by the
Herschel Gould Belt Survey (André et al., 2010) if they were moved to larger
distances (from 0.75 to 7 kpc), comparable to those typically found in the Hi-
GAL survey. Subsequently, in Baldeschi et al. (2017a) this approach was
applied, in particular, to the estimation of the SFR. For three nearby regions
they compared the SFR obtained from YSO counts (based on an updated version of
Equation 1 of Lada et al., 2010) to that estimated through the method of
Veneziani et al. (2017) from the clumps detected in the Herschel maps
“displaced” at different virtual distances. They found that the two estimates
remain consistent with each other within a factor 2 at various probed
distances.

Based on these premises we feel confident to adopt the method of Veneziani et
al. (2017). As mentioned above, the output of the algorithm we used is the
contribution of each clump $\Sigma_{\textrm{cl}}$ to the global SFR. This
quantity depends on the evolutionary track in the $L/M$ plot, which, in turn,
is defined by the initial clump mass $M_{\textrm{cl}}$. Although not
explicitly stated in the original articles, this algorithm associates the
final star mass to the input clump mass by interpolating the locus of final
masses of known evolutionary tracks with a power law. Here we explicitly
report the corresponding formula we used:

$\textrm{SFR}_{\textrm{cl}}=(5.6\pm 1.4)\times
10^{-7}~{}(M_{\textrm{cl}}/\textrm{M}_{\odot})^{0.74\pm
0.03}~{}\textrm{M}_{\odot}~{}\textrm{yr}^{-1}.$ (1)

We propose this relation hereafter as an operational prescription for an
immediate application of the method of Veneziani et al. (2017).

The global SFR, estimated by adding up all these contributions, amounts to
$1.7\pm 0.6$ M⊙ yr-1. The uncertainty is given by the error propagation,
combining in quadrature the uncertainties affecting the clump masses and the
parameters appearing in Equation 1.

Adopting a classification of inner vs outer Galaxy delimited by the Solar
circle with a radius of 8.34 kpc (as in Mège et al., 2021; Elia et al., 2021),
it is possible to give separately the contributions to the global SFR from
these two zones: 1.5 (i.e. the 84% of the total) and 0.3 M⊙ yr-1 (16%),
respectively.

####  2.2.1 Contribution from sources devoid of a distance estimate

Figure 1: Top: distributions of Hi-GAL protostellar clumps with and without a
heliocentric distance estimate (black and red histograms, respectively) in
$5^{\circ}$-wide bins of Galactitc longitude. Bottom: number ratio between
protostellar clumps without a distance estimate and the total number. in the
same longitude bins as in the top panel.

It is to notice that the SFR estimate reported above should be intended as a
lower limit, since the input data, namely the protostellar clump list, is
incomplete for the reasons described in the following. First, since the mass
completeness in the Hi-GAL catalog is a function of the heliocentric distance
(Elia et al., 2017), faint and/or far and/or cold protostellar clumps might
remain undetected at all. We address this issue in Appendix A. Second,
protostellar clumps undetected at 70 $\mu$m (for the same reasons written
above), but detected at the other ones, result misclassified as starless and
are not involved in the SFR calculation. In fact, in the literature examples
can be found of spectroscopic signatures of active star formation detected in
70 $\mu$m-dark clumps (e.g., Traficante et al., 2017). Third, 5532 sources
classified as protostellar in the Hi-GAL catalog are not provided with a
distance, and consequently not involved in our previous calculation (see
Figure 1).

Figure 2: Two examples (black and red histograms, respectively, in bins of 100
pc) of realizations of heliocentric distance distributions to be assigned, as
a test, to Hi-GAL protostellar clumps devoid of a distance estimate, compared
with the distance distribution for sources provided with a distance estimate
(gray histogram, Elia et al., 2021).

Regarding the last point, it is possible to give an estimate of the missing
contribution from this subsample by simulating a plausible realization of
distances for sources whose distance is unknown. In this way, masses can be
obtained also for them, to be inputted in the SFR calculation.

We simulated a set of 5532 distances whose distribution follows the one of the
protostellar sources provided with a distance estimate. Then we randomly
assigned each simulated distance to a clump devoid of distance, computed mass
and luminosity accordingly, and estimated the contribution to the SFR with the
usual procedure. This random assignation of distances to clumps has been
repeated 100 times (two of which are shown in Figure 2), to keep safe from
specific effects possibly induced by a single realization. The 100
corresponding estimates of additional SFR contribution are found to be
confined in the relatively narrow range 0.20 to 0.23 M⊙ yr-1, with an average
of 0.22 M⊙ yr-1 and a standard deviation of 0.01 M⊙ yr-1. Notice that this
standard deviation is expected to decrease by increasing the number of
simulation as desired. The real uncertainty, instead, is dominated by the
typical error bar associated to the SFR contribution evaluated for each
simulation, which is $0.08$ M⊙ yr-1.

In conclusion, a reasonable estimate of the total Galactic SFR taking into
account the correction calculated above is $2.0\pm 0.7$ M⊙ yr-1.

Figure 3: Probable contribution to the overall SFR from sources lacking a
distance estimate if they were all located at the same virtual distance. The
dotted and dashed lines represent the median and the average of the real
distance distribution, respectively.

The above test is based on the assumption that the real (unknown) distances of
sources lacking this information follow the same distribution of the known
ones. However, the lack of a distance estimate, i.e. the impossibility of
individuating a clear and/or reliable spectral component along the line of
sight of the Hi-GAL source (Mège et al., 2021), could be, in principle, due to
specific local conditions. This could then produce a departure from the
assumed behavior of distance distributions. In this respect, the most
unfavorable hypothetical case would be to have all sources located at the same
distance. Therefore, supposing that all sources without distance were located
at a minimum and at a maximum distance, respectively, helps us to identify a
range of maximum variability of their contribution to the total SFR. We
therefore analyzed the behavior of such contribution for a common distance
varying between 1 and 13 kpc (these limits are suggested by the distributions
shown in Figure 2). From Figure 3 we see that for a virtual common distance of
1 kpc the contribution to the SFR would be negligible, while for 13 kpc it
would amount to 0.57 M⊙ yr-1. Clearly, both scenarios (and, consequently, the
corresponding contributions to SFRs) are unrealistic, but this test proves
that, even in case of an extremely unfavorable distance distribution of clumps
without distance assignment, the order of magnitude of the SFR increment
estimated through our test would be confirmed. Finally, we notice that,
assuming that all sources without known distance were placed at the median and
the average of the known distance distribution, their contribution would
amount to 0.13 and 0.19 M⊙ yr-1, respectively (see Figure 3).

A final test we propose consists in a hybrid approach between the two followed
above. Indeed, the top panel of Figure 1 suggests that the distribution in
longitude of clumps lacking a distance assignment is far from being uniform
and, moreover, it does not strictly follow the behavior of sources with a
distance (as testified by the high degree of scatter of the ratio between the
former distribution and the overall one, shown in the bottom panel).
Statistically significant amounts of sources without a distance are found in
correspondence of the innermost Galactic longitudes
($0^{\circ}\lesssim\ell\lesssim 20^{\circ}$), and in the direction of the
Carina arm ($270^{\circ}\lesssim\ell\lesssim 310^{\circ}$). The first case can
be treated with relative ease, by arbitrarily placing all the sources without
distance and with $|\ell|<20^{\circ}$ at the Galactic center distance of 8.34
kpc, and proceeding for the remaining sources by simulating distances
distributed as the known ones. In this case, the additional term to the global
SFR would amount to $0.17\pm 0.06$ M⊙ yr-1.

###  2.3 Comparison with the previous literature

Adopting the value of $2.0\pm 0.7$ M⊙ yr-1 as our final estimate for the Milky
Way SFR, and comparing with values in Table 1, it can be seen that it is
consistent, within the error bars, with all but one (namely Smith et al.,
1978) estimates collected and normalized by Chomiuk & Povich (2011), and with
the estimates of Chomiuk & Povich (2011) themselves and of Licquia & Newman
(2015). Remarkably, our estimate is also perfectly consistent, within the
errors, with that of Noriega-Crespo (2013), confirming that his choice to
extrapolate the SFR estimate from an area of 8 sq. deg. to the entire 720 sq.
deg was quite reasonable.

Our result, in essence, corroborates the common idea - supported by the fact
that a variety of approaches leads to similar values - that the current SFR of
the Milky Way lies in the range 1.5-2.5 M⊙ yr-1, which is consistent with the
values found for spiral galaxies (e.g., Kennicutt & Evans, 2012).

##  3 The SFR across the Galactic plane

We studied the SFR distribution across the Galactic Plane. From here on, we
will only consider the contribution to SFR given by clumps whose heliocentric
distance is known, for which the position in the Galaxy can be determined.

###  3.1 Mapping SFR in Galactic coordinates: the Central Molecular Zone

Figure 4: Top: Star formation rate (SFR) toward the Galactic plane in
$3^{\circ}\times 0.2^{\circ}$ bins in Galactic longitude and latitude,
respectively. It is to notice that in the outer Galaxy the Hi-GAL latitude
coverage follows the variations in the midplane position, or Galactic warp
(Westerhout, 1957). For a better rendering of the image dynamic range, the
color scale follows a power law with exponent 0.4. The dotted lines indicate
the location of the three most prominent SFR peaks. Bottom: SFR as a function
of longitude, estimated from the sum of the results reported in the top panel
across Galactic latitude.

A first view of SFR distribution in the Galactic plane can be given in the
$[\ell,b]$ plane, as shown in Figure 4, top. It can be seen that,
approximately, most of the total SFR (90%) comes from the inner 120∘ (Figure
4, bottom).

For the used binning of 3∘ in longitude and 0.2∘ in latitude, the three most
prominent local maxima are found in pixels corresponding to the ranges
$336^{\circ}<\ell<339^{\circ}$ and $-0.2^{\circ}<b<0.0^{\circ}$ (containing
the G337.342−0.119 region, Jackson et al., 2018), $27^{\circ}<\ell<30^{\circ}$
and $0.0^{\circ}<b<0.2^{\circ}$, and $18^{\circ}<\ell<21^{\circ}$ and
$-0.2^{\circ}<b<0.0^{\circ}$, respectively.

Table 2: Star formation rate estimates for the Central Molecular Zone  Method | Area boundaries | SFR${}_{\textrm{CMZ}}$ (lit.) | Reference | SFR${}_{\textrm{CMZ}}$ (this work)
---|---|---|---|---
|  | M⊙ yr-1 |  | M⊙ yr-1
YSO counts (Spitzer) | $|\ell|<1^{\circ},|b|<10\arcmin$ | 0.14 | Yusef-Zadeh et al. (2009) | $0.04\pm 0.02$
Continuum emission at 60, 100 $\mu$m (IRAS) |  $|\ell|<3^{\circ},|b|<1^{\circ}$ aaDifferently from other rectangular areas quoted in the table, this one has an elliptical shape with axes aligned along $\ell$ and $b$, and semi-axes of $3^{\circ}$ and $1^{\circ}$, respectively. | 0.12 | Crocker et al. (2011) | $0.12\pm 0.05$
Continuum emission at 60, 100 $\mu$m (IRAS) | $|\ell|<0.8^{\circ},|b|<0.3^{\circ}$ | 0.08 | Crocker et al. (2011) | $0.04\pm 0.02$
YSO counts (Spitzer) | $|\ell|<1.5^{\circ},|b|<0.5^{\circ}$ | 0.08 | Immer et al. (2012) | $0.08\pm 0.03$
Ionization rate from radio free-free | $2.5^{\circ}<\ell<3.5^{\circ},|b|<0.5^{\circ}$ | 0.035 | Longmore et al. (2013) | $0.11\pm 0.04$
Ionization rate from radio free-free | as above, but $|b|<1^{\circ}$ for $|\ell|<1^{\circ}$ | 0.06 | Longmore et al. (2013) | $0.12\pm 0.05$
Continuum emission at 24 $\mu$m (Spitzer) | $|\ell|<1^{\circ},|b|<0.5^{\circ}$ | $0.09\pm 0.02$ | $\left.\begin{array}[]{l}\\\ \\\ \\\ \end{array}\right\\}$ |
Continuum emission at 70 $\mu$m (Spitzer) | $|\ell|<1^{\circ},|b|<0.5^{\circ}$ | $0.10\pm 0.02$ |  Barnes et al. (2017)bbIn Barnes et al. (2011), their Table 2, a further list of SFR${}_{\textrm{CMZ}}$ estimates derived within the same area from monochromatic fluxes or bolometric luminosities available in the literature from 1998 to 2011 are quoted (see text). We omit them in this table for the sake of brevity. |  $0.06\pm 0.02$ccThe three SFR${}_{\textrm{CMZ}}$ estimates by Barnes et al. (2017) are derived within the same area in the sky through different methods; obviously, for the same area we can provide only one estimate.
Cont. emission at 5.8-500 $\mu$m (Spitzer, Herschel) | $|\ell|<1^{\circ},|b|<0.5^{\circ}$ | $0.09\pm 0.03$ |  |

Through the $[\ell,b]$ view of the Galactic SFR it is also possible to make a
direct comparison with SFRs estimated in the literature for the CMZ, generally
taken as a rectangular coordinate box in such plane. In Table 2 we report a
list of recent SFR${}_{\textrm{CMZ}}$ estimates, each of them derived in a
different area around the Galactic center, and the output of our method for
the same area.

We found a good agreement with the SFR${}_{\textrm{CMZ}}$ estimates reported
in Crocker et al. (2011) and Immer et al. (2012). We also found a fairly good
agreement with Barnes et al. (2017) on SFR${}_{\textrm{CMZ}}$, which those
authors estimated from monochromatic fluxes or bolometric luminosities
available in the literature from 1998 to 2011, and ranging from 0.07 to 0.12
M⊙ yr-1, to be compared with our estimate of $0.06\pm 0.02$ M⊙ yr-1.

Our results, however, appear to disagree with the SFR estimated for the CMZ in
Yusef-Zadeh et al. (2009) and Longmore et al. (2013). The
SFR${}_{\textrm{CMZ}}$ in Yusef-Zadeh et al. (2009) is above our result, which
is most likely due to an excessive number of early YSOs involved in the
calculation, as discussed in Koepferl et al. (2015), who quantify this
overestimate in a factor of 3. Correcting by this factor, the estimate of
Yusef-Zadeh et al. (2009) would turn out to be consistent with our value
($0.04\pm 0.02$ M⊙ yr-1) within the uncertainties.

Longmore et al. (2013) estimated SFR${}_{\textrm{CMZ}}=0.035$ M⊙ yr-1 in the
range $2.5^{\circ}<\ell<3.5^{\circ}$ and $|b|<0.5^{\circ}$, and 0.06 M⊙ yr-1
if the wider latitude range $|b|<1^{\circ}$ is considered in the central
$|\ell|<1^{\circ}$. Correspondingly, we calculate SFR${}_{\textrm{CMZ}}=0.11$
and 0.12 M⊙ yr-1 for these two zones, respectively. The former one is about 3
times larger than that of Longmore et al. (2013), and not compatible with it
within the associated error bar. Of course, the estimate of Longmore et al.
(2013) is also the lowest among the estimates from the literature collected in
Table 2. The authors indeed suggest that the choice of the assumed IMF, which
did not account for possible peculiar conditions of the CMZ, could affect this
estimate down to a factor 1/3.

Figure 5: Map of the decimal logarithm of SFR density (divided by M⊙ yr-1
kpc-2), across the Galactic plane. The spatial bins are of $0.5\times 0.5$
kpc2, so that the absolute SFR inside each pixel can be obtained by
exponentiating the displayed logarithm, and then multiplying by a factor 4
kpc2. The Galactic center is at the [0,0] position, while the gray X symbol
indicates the position of the Sun. Gray dashed lines mark Galactic octants.
Four spiral arms according to the prescription of Hou et al. (2009) are
overplotted with four different tones of green; the correspondence between
arms and colors is explained in the legend at the bottom left corner.

###  3.2 SFR Galactocentric profile

Using the information about the heliocentric distance, a pole-on view of SFR
density in the Milky Way, mapped on a grid of $0.5\times 0.5$-kpc2 pixels, can
be built. This is a simplified 2-D representation of the Galactic disk which
has a structure along the third dimension, an idea of which can be taken from
Figure 4. The design of the Hi-GAL survey took into account the warp of far-
infared emission already observed by IRAS towards the outer Galaxy, which
however is less prominent than the Galactic warp observed in atomic gas
(Westerhout, 1957; Soler et al., 2022), corresponding to Galactocentric
distances not achieved by objects in our sample ($R_{\mathrm{gal}}\gtrsim 16$
kpc, Voskes & Butler Burton, 2006).

In Figure 5 one can appreciate that, with respect to the Galactic center (to
which we assign the [0,0] position), the SFR shows a certain degree of
circular symmetry.

Figure 6: Top: Star formation rate (SFR) density profile estimated from the
Hi-GAL observations in 0.5-kpc-wide concentric rings around the Galactic
center, shown in black filled circles. Fitting the relation in Eq. 2 to the
portion at $R_{\mathrm{gal}}\geq 5$ kpc gives $a=-0.28\pm 0.01$ and
$b=-0.30\pm 0.16$. The fit is plotted as a black dashed line. The green line
shows the results of the visible-band observations considered in Guesten &
Mezger (1982), re-scaled by Kennicutt & Evans (2012) to the total SFR of 1.9
M⊙ yr-1 reported in Chomiuk & Povich (2011). The blue line corresponds to the
estimates obtained in Lee et al. (2016) by combining YSO NIR photometry and
cloud properties from millimeter line observations. The red line shows the SFR
profile of the galaxy NGC 6946, presented in Schruba et al. (2011) and adapted
by Kennicutt & Evans (2012). The pink-shaded area is the region occupied by
the models of Evans et al. (2022). Bottom: Cumulative SFR profile as a
function of the Galactocentric radius, also expressed as a percentage with
respect to the total SFR ($y$-axis on the right-hand side). The vertical
dotted lines indicate, from left to right, the Galactocentric radii
corresponding to 50%, 90%, and 99% of the total SFR, respectively.

It makes sense, therefore, to analyze the behavior of the SFR density
$\Sigma_{\mathrm{SFR}}$, averaged in concentric 500-pc wide rings, vs the
Galactocentric radius $R_{\mathrm{gal}}$. In Figure 6, top panel, a peak
corresponding to the inner circle222To correctly compare this plot with Figure
5, it must be considered that in this case the SFR density is obtained by
dividing by the ring area, proportional to $R_{\mathrm{gal}}^{2}$ , while in
Figure 5 the denominator consists of a local area extending over 0.25 kpc2.
Also notice that, as discussed in Section 2.2.1, the distribution in longitude
of sources without a distance assignment shows a remarkable concentration at
$0^{\circ}\lesssim\ell\lesssim 20^{\circ}$. Assigning the distance of the
Galactic center to all of them would turn out in an even higher value for this
peak. is followed by a wide dip between 1 and 4 kpc, mostly due to shortage of
sources at these Galactocentric distances (Elia et al., 2021). This is seen
also in other works (see below), and is due in part to the lack of SFR data in
two lobes extending south-west and south-east of the Galactic center position
over an extent of about 4 kpc (recognizable as inner void areas in Figure 5),
which can not be populated with heliocentric distances since the observed
clump velocities are forbidden by the adopted rotation curve (Russeil et al.,
2017). In particular, in these areas absolute values of radial velocity are
expected to be larger than 100 km s-1 (Ellsworth-Bowers et al., 2015, their
Fig. 1), while most of sources observed in the corresponding longitude ranges
show lower values. The kinematics of the Milky Way along these lines of sight
is indeed generally dominated by the Galactic bar (Binney et al., 1991;
Ellsworth-Bowers et al., 2015; Li et al., 2016). As explained by Mège et al.
(2021) and Elia et al. (2021), only in cases of small deviations, the distance
of the corresponding tangent point is assigned to sources, which can therefore
be involved in the calculation of the SFR. Additionally, in the longitude
range corresponding to the void lobe in the fourth quadrant, Mège et al.
(2021) found a relatively high occurrence of noise in the original spectral
survey data which in many cases prevented an estimation of the radial velocity
and hence of the kinematic distance. In any case, since the two void areas are
oriented approximately along Galactocentric radial directions, their influence
is distributed over a certain number of rings (and therefore attenuated).

The dip in the SFR density profile is followed by another local peak at around
5 kpc. A peak in this position was predicted by Krumholz & McKee (2005) as a
consequence of the distribution of the molecular gas surface density, and was
found also by Portinari & Chiosi (1999) and Chiappini et al. (2001), but at
$\sim 4$ kpc. It should correspond to the so-called Molecular Ring (which is,
more realistically, an effect of spiral arm arrangement, see Dobbs & Burkert,
2012; Roman-Duval et al., 2016; Miville-Deschênes et al., 2017), but it does
not mirror exactly the peak of source number distribution found by Elia et al.
(2017) at $R_{\mathrm{gal}}\simeq 6$ kpc. Finally, at larger distances,
$\Sigma_{\mathrm{SFR}}$ definitely shows a systematic decrease.

This global behavior is qualitatively similar to those shown by Kennicutt &
Evans (2012) (starting from the data of Guesten & Mezger, 1982) and Lee et al.
(2016): an absolute maximum corresponding to the CMZ, followed by a local
minimum and by another local peak at $R_{\mathrm{gal}}=$1 and 5 kpc,
respectively, with a decreasing behavior at larger Galactocentric distances.
This general behavior was successfully modeled by Evans et al. (2022) by
taking into account $i$) a dependence of the conversion factor from CO
luminosity to cloud mass on metallicity (which in turn depends on
$R_{\mathrm{gal}}$), and $ii$) a dependence of the star formation efficiency
on the virial parameter. The envelope of the six models produced by Evans et
al. (2022, their Fig. 3) in the $\Sigma_{\mathrm{SFR}}$ vs $R_{\mathrm{gal}}$
diagram delimits a belt (with a vertical width of 1-2 dex) in which the curve
observed by Lee et al. (2016) is contained. Being the latter quite similar to
ours in many points (Figure 6, top), a similar consistency is expected also
for our data. In fact, our $\Sigma_{\mathrm{SFR}}$ curve is found, in turn, to
be enclosed within the region occupied by the models of Evans et al. (2022),
running closer to the top of it around the $R_{\mathrm{gal}}\simeq 5$ kpc
peak, and to the bottom of it at larger Galactocentric distances, in the range
corresponding to the final decrease.

It is also important to compare the SFR profile found for Milky Way with that
of external galaxies, to examine if some macroscopic bias is introduced when
observing the Galaxy from the inside, rather than as a whole from the outside.
The SFR Galactocentric profile of NGC 6946 galaxy obtained by Schruba et al.
(2011) and adapted by Kennicutt & Evans (2012) shows, in turn, a qualitatively
similar, but flatter behavior. More recently, Logroño-García et al. (2019),
using H$\alpha$ emission, estimated for NGC 3994 and NGC 3995 SFR profiles
which, although not quantitatively similar to that of the Milky Way, again
broadly follow the same sequence of peaks and dips. Interestingly, an similar
inner dip was also found by Lian et al. (2018) to be typical of local massive
star-forming galaxies ($10.5<\log(M/M_{\odot})<11$).

Returning to SFR density profile of the Milky Way and to its decrease for
$R_{\mathrm{gal}}\gtrsim 5$ kpc, Misiriotis et al. (2006) found a good
agreement between the radial profile of SFR density collected from the
literature and their model of spatial distribution of dust, stars, and gas in
the Milky Way, constrained through COBE/DIRBE data. Such model has a
functional form

$\log(\Sigma_{\mathrm{SFR}}/[\mathrm{M}_{\odot}~{}\mathrm{yr}^{-1}~{}\mathrm{kpc}^{-2}])=a\,R_{\mathrm{gal}}/[\mathrm{kpc}]+b\>,$
(2)

with $a\sim-0.14$. Fitting the same relation to our data for
$R_{\mathrm{gal}}\geq 5$ kpc, we obtain $a=-0.28\pm 0.01$ (Figure 6, top).

The linear fit in the same range of Galactocentric distance for the Guesten &
Mezger (1982) has a much shallower slope ($a=-0.11\pm 0.02$), while the more
recent curve of Lee et al. (2016), which shows more scattering, has a slope
consistent with ours: $a=-0.25\pm 0.05$. Finally, in the NGC 6946 case the
slope shows an intermediate behavior ($a=-0.20\pm 0.01$).

The slope of -0.28 corresponds to an exponential scale length of 1.55 kpc,
which is considerably shorter than those typically found for the stellar count
profile in the Galactic thin disk, e.g., 2.15 kpc by Licquia & Newman (2015),
2.6 kpc by Bland-Hawthorn & Gerhard (2016, see also further references
therein), and 2.2 kpc by Xiang et al. (2018). However more recently, by using
LAMOST (Cui et al., 2012; Deng et al., 2012) Data Release 4, and Gaia Data
Release 2 (Gaia Collaboration et al., 2018; Lindegren et al., 2018), Yu et al.
(2021) derived shorter scale lengths. In particular, they considered
separately stellar populations characterized by different chemical abundances;
the scale length that most closely approaches ours is that found for stars
with Solar-type abundances, 1.28 kpc (but for $R_{\mathrm{gal}}>8$ kpc).

In Figure 6, bottom, the behavior of the SFR profile as a function of
$R_{\mathrm{gal}}$ is further analyzed by means of its cumulative. Although
this curve is reminiscent of the SFR density profile shown in the top panel,
it additionally accounts for the increasing area of the ring at increasing
$R_{\mathrm{gal}}$. For our data we notice three most evident changes of slope
in the cumulative at around 4 (marked steepening), 7 (getting shallower), and
10 kpc (flattening), respectively. We find that the 50%, 90%, and 99% of total
SFR are achieved at $R_{\mathrm{gal}}=$ 5.8, 9.2, and 13.4 kpc, respectively.
This means that, according to our results, half of the Galactic SFR comes from
within the Molecular Ring, and just 1% from the far outer Galaxy
($R_{\mathrm{gal}}>13.5$ kpc, according to the definition by Heyer et al.,
1998).

###  3.3 Testing the Kennicutt-Schmidt relation

The SFR density profile computed in rings of $R_{\mathrm{gal}}$ and plotted in
the top panel of Figure 6 can be also used for checking if the SFR in the
Milky Way follows the Kennicutt-Schmidt (KS) law.

This is an empirical scaling relation (Schmidt, 1959; Kennicutt, 1998)
generally found between star formation rate density $\Sigma_{\mathrm{SFR}}$
and mean gas (atomic + molecular) surface density $\Sigma_{\mathrm{gas}}$ in
disk galaxies (the atomic and molecular gas density,
$\Sigma_{\mathrm{H\textsc{i}}}$ and $\Sigma_{\mathrm{H}_{2}}$, respectively,
can be also considered separately, see Bigiel et al., 2008; Kennicutt & Evans,
2012). It is expressed by a power law such as

$\Sigma_{\mathrm{SFR}}\propto\Sigma_{\mathrm{gas}}^{n}\,,$ (3)

with $n$ generally falling in the range 1 to 2, depending on the tracer used
and the linear scales considered. In particular, the “classic” slope of
Kennicutt (1998) is $n=1.40\pm 0.15$, with $\Sigma_{\mathrm{gas}}$ expressed
in units of M⊙ pc-2. Finally, in the literature descriptions of
$\Sigma_{\mathrm{SFR}}$ as a power law of the single atomic
($\Sigma_{\mathrm{H\textsc{i}}}$), or molecular component
($\Sigma_{\mathrm{H_{2}}}$), respectively, can be also commonly found.

Whereas a vast literature is available about external galaxies following the
KS relation (see, e.g., Kennicutt, 1998; Miettinen et al., 2017; Orr et al.,
2018, and references therein), this has been poorly tested for the Milky Way,
also due to well understood intrinsic difficulties related to observing the
Galaxy “from within”.

The KS relation for a set of regions in the Southern Milky Way was explored by
Luna et al. (2006), who considered only $\Sigma_{\mathrm{H_{2}}}$ and found an
exponent $n=1.2\pm 0.2$. Similarly, Sofue & Nakanishi (2017) found $n=1.12\pm
0.05$ for a set of $(\Sigma_{\mathrm{SFR}},\Sigma_{\mathrm{H_{2}}})$ pairs
obtained through binning the Galactic plane in $0.2\times 0.2$ kpc2 boxes. No
clear correlation was found, instead, between $\Sigma_{\mathrm{SFR}}$ and both
$\Sigma_{\mathrm{H\textsc{i}}}$ and $\Sigma_{\mathrm{gas}}$. A steeper slope
($3.7\pm 1.6$), again considering only $\Sigma_{\mathrm{H_{2}}}$, was found by
Gutermuth et al. (2011). However, this value was based only on the analysis of
a small sample of nearby star forming regions, which are not necessarily
representative of the entire Galaxy.

Since in our case the available data set includes the entire Galactic plane,
the approach we follow here is to plot the
$(\Sigma_{\mathrm{SFR}},\Sigma_{\mathrm{H}_{2}})$ pairs azimuthally averaged
in bins of Galactocentric radius (i.e. rings). This was done, for example, in
Boissier et al. (2003), who found that the KS relation is satisfied in the
range $4\leq R_{\mathrm{gal}}\leq 15$ kpc with a slope of 2.06, and in
Misiriotis et al. (2006), who obtained a slope of 2.18 across the entire range
of $R_{\mathrm{gal}}$.

To do that, for the $x$ axis we use the molecular $\Sigma_{\mathrm{H}_{2}}$
averaged over Galactocentric bins of 1 kpc by Miville-Deschênes et al. (2017),
starting from the CO survey assembled by Dame et al. (2001), and the atomic
$\Sigma_{\mathrm{H\textsc{i}}}$ from Nakanishi & Sofue (2016). Therefore, for
consistency, we recalculated the SFR density in the same bins (which are twice
wider than those used for Figure 6)333See Onodera et al. (2010) and Kruijssen
& Longmore (2014) for detailed discussions of minimum scales to be preserved
while binning, to avoid incomplete sampling and consequent break down of star
formation relations..

Figure 7: Star formation rate (SFR) estimated from the Hi-GAL observations
versus the average molecular gas surface density (from Miville-Deschênes et
al., 2017) in 1-kpc-wide concentric rings around the Galactic center whose
central radius is indicated by the colors. The bins corresponding to
$R_{\mathrm{gal}}\leq 3$ kpc, and $R_{\mathrm{gal}}>3$ kpc are marked with
square and circle symbols, respectively. The dotted and the dashed lines
indicate the linear fit to the SFR estimates at $R_{\mathrm{gal}}\geq 3$ kpc
(with slope $n=1.14\pm 0.07$), and the KS relation
($\Sigma_{\mathrm{SFR}}=2.5\times 10^{-4}\,\Sigma_{\mathrm{gas}}^{1.4}$),
respectively.

In Figure 7 the log-log plot of $\Sigma_{\mathrm{SFR}}$ vs
$\Sigma_{\mathrm{H_{2}}}$ is shown. Differently from the plots built with the
atomic and total gas density (see Appendix B), a linear trend can be
recognized here, except for points corresponding to the innermost radii. For
this reason, in the estimation of the linear fit, following the indication of
Boissier et al. (2003), we do not consider the points corresponding to three
first Galactocentric rings ($R_{\mathrm{gal}}\leq 3$ kpc). Notice that, as it
can be seen in the bottom panel of Figure 6, the fraction of SFR contained
within this radius correspond to only the 7% of the entire Milky Way SFR. From
the fit to the remaining points, we obtain $n=1.14\pm 0.07$.

First of all, from the qualitative point of view, it is important to confirm
that the largest part of the Milky Way disk follows a power-law behavior,
testifying a direct and clear connection between the availability of molecular
gas and the rate of its conversion into stars.

The slope $n$ we found appears in good agreement with those derived by Luna et
al. (2006) and Sofue & Nakanishi (2017), while it is shallower than those
found by Boissier et al. (2003) and Misiriotis et al. (2006). Furthermore, we
note the agreement of our result with that obtained by Onodera et al. (2010)
for the M33 galaxy ($n=1.18\pm 0.11$). The comparison with the classic slope
of Kennicutt (1998), instead, makes little sense, since it refers to the total
$\Sigma_{\mathrm{gas}}$.

Actually, it is known that a variety of slopes is generally found for the KS
relation (see, e.g., Kennicutt & Evans, 2012, and references therein).
Furthermore, it is not difficult to figure out how varying some assumptions
made in calculating the plotted variables can produce a change of the slope
(even though preserving the global power-law behavior). For example, surface
densities were computed by Miville-Deschênes et al. (2017) by converting CO to
H2 by using the constant factor $X_{\mathrm{CO}}=10^{20}$ cm${}^{-2}~{}($K km
s${}^{-1})^{-1}$. However, if an increasing trend of $X_{\mathrm{CO}}$ at
increasing $R_{\mathrm{gal}}$ was adopted (e.g., Nakanishi & Sofue, 2006;
Pineda et al., 2013), the plot in Figure 7 would get steeper (see also
Boissier et al., 2003). Moreover, the masses used here as input for the SFR
calculation were derived by Elia et al. (2021) by assuming a constant gas-to-
dust ratio of 100. However, a possible increase of this ratio at increasing
$R_{\mathrm{gal}}$ is suggested by Giannetti et al. (2017), and invoked by
Elia et al. (2021) themselves to better explain the Galactocentric behavior of
the surface density of Hi-GAL clumps. Taking into account such dependence
would produce a flattening of our $\Sigma_{\mathrm{SFR}}$ vs
$\Sigma_{\mathrm{H}_{2}}$ plot.

###  3.4 Face-on view of the Milky Way’s SFR

Mapping the SFR in the Galactic plane as in Figure 5 offers the chance of
comparing the local values of this observable with those of other quantities
that can be mapped starting from the Hi-GAL clump distribution in the plane.

Figure 8: Face-on view of the Milky Way representing the following quantities
derived (with the same spatial binning used for Figure 5) from the Hi-GAL
observations…: $a$) logarithm of number of pre-stellar Hi-GAL clumps per bin,
$N_{\rm pre}$; $b$) logarithm of number of protostellar Hi-GAL clumps per bin,
$N_{\rm pro}$; $c$) star formation fraction, SFF; $d$) median temperature
(from modified black body fit) of protostellar clumps, $\left<T\right>$; $e$)
median logarithm of luminosity-to-mass ratio normalized to the solar values,
$\left<(L/M)/(\mathrm{L_{\odot}}/\mathrm{M_{\odot}})\right>$; $f$) median
bolometric temperature of protostellar clumps,
$\left<T_{\mathrm{bol}}\right>$. In each panel, the [0,0] position corresponds
to the Galactic center position, and the gray X symbol indicates the position
of the Sun.

In our estimates, the SFR is the sum of the contributions from each
protostellar clump, which is a function of its mass. In this respect, the
spatial distribution of SFR in bins of $[x,y]$ coordinates mirrors that of
total mass of protostellar clumps in those bins. This total mass, in turn,
does not necessarily depend on the number of protostellar clumps
$N_{\mathrm{pro}}$ found in each bin (shown in Figure 8, $b$)444Spatial
distributions in panels $a$ and $b$ clearly appear asymmetric because at
shorter heliocentric distances a larger number of sources (having relatively
smaller physical sizes and masses) is detected. Conversely, at higher
distances sources tend to appear blended in physically larger structures
(Section 2.1). Since the SFR depends on the available mass (Section 2.2), the
final appearance of the SFR density map in Figure 5 appears much more
symmetric than that of source counts. Moreover, the selection bias with
distance has to be taken into account; this is done in Appendix A., and does
not depend at all on the number of pre-stellar clumps $N_{\mathrm{pre}}$
(shown in Figure 8, $a$). These two quantities were combined together by Ragan
et al. (2016) to obtain the star formation fraction (hereafter SFF), defined
as the ratio
$N_{\mathrm{pro}}/\left(N_{\mathrm{pre}}+N_{\mathrm{pro}}\right)$. The
relative populations of these two classes depend on their corresponding
lifetimes, which in turn depend on the mass (Motte et al., 2010; Urquhart et
al., 2014; Elia et al., 2021). In this respect, the local SFF is related not
only to the overall evolutionary state of clumps in a given region, but also,
indirectly, to their mass spectrum. Using Hi-GAL data, Ragan et al. (2016)
highlighted a decreasing behavior of SFF as a function of the Galactocentric
radius over the 3.1 kpc $<R_{\mathrm{gal}}<8.6$ kpc range. This was confirmed
by Elia et al. (2021), who however found a more scattered behavior just
outside this range. This 1-D description of the SFF is further developed here
in the 2-D mapping shown in the panel $c$ of Figure 8, in which we can
recognize the aforementioned decrease of this quantity at intermediate
Galactocentric radii as a blueish ring-like area.

Figure 9: Star formation rate (SFR), evaluated in bins of 0.5 kpc in the
$[x,y]$ plane, as a function of star formation fraction ($i$), and medians of
temperature ($ii$), $L/M$ ratio ($iii$) and bolometric temperature ($iv$) of
protostellar clumps, respectively, all calculated in the same spatial bins. In
this respect, they represent a pixel-to-pixel plot Figure 5 vs panels $c$,
$d$, $e$ and $f$ of Figure 8, respectively. Gray symbols correspond to bins
with $N_{\mathrm{pro}}+N_{\mathrm{pre}}<20$, considered statistically poor. In
the leftmost panel, several of them give rise to vertical features
corresponding to recurring values like
$N_{\mathrm{pro}}/(N_{\mathrm{pro}}+N_{\mathrm{pre}})=\nicefrac{{1}}{{1}},\nicefrac{{1}}{{2}},\nicefrac{{1}}{{3}},\nicefrac{{1}}{{4}}$,
etc.).

It makes sense to compare here the maps of SFR and SFF, since on the one hand
they have no direct link a priori, and on the other hand this allows us to
check whether locally the star formation rate is somehow correlated to the
average evolutionary stage of the region. A bin-to-bin comparison between SFR
and SFF is shown in panel $i$ of Figure 9. Apparently, no correlation emerges
from the plot, except for the fact that a small number of bins with very low
SFF ($\ll 1$) also exhibit relatively low SFR values.

As a further check of the absence of a trend with the average evolutionary
stage of clumps in different parts of the disk, we also investigate the trend
of the SFR versus the medians of meaningful evolutionary indicators for
protostellar clumps, such as modified black body temperature $T$, bolometric
luminosity over mass ratio $L/M$, and bolometric temperature
$T_{\mathrm{bol}}$ (Cesaroni et al., 2015; Elia et al., 2017, 2021). Median
values were evaluated for them in the same spatial bins of Figure 5, and
displayed in Figure 8, panels $d$, $e$, and $f$, respectively.

For the median temperature and $L/M$ ratio of protostellar clumps, Elia et al.
(2021) did not find relevant variations in the 1-D profile as a function of
$R_{\mathrm{gal}}$, apart from an increase in the far outer Galaxy, but
supported by poor statistics. On the contrary, the trend found for median
$T_{\mathrm{bol}}$ is to slightly raise at increasing $R_{\mathrm{gal}}$. This
is what can be seen also in the 2-D view provided in Figure 8. Therefore, the
bin-to-bin comparison of quantities nearly constant with $R_{\mathrm{gal}}$
such as $T$ and $L_{\mathrm{bol}}/M$ and an observable with a more complex
spatial pattern (SFR) is expected to show no correlation between the two, as
confirmed by Figure 9, panels $ii$ and $iii$. A mild indication seems to be
provided by bins with poor statistics (i.e. containing less than 20 clumps, so
an even smaller number of protostellar ones), generally corresponding to large
$R_{\mathrm{gal}}$, for which a higher $L/M$ but a small SFR (due to deficit
of clumps) can be found. A similar, but clearer, behavior can be seen also
considering the median $T_{\mathrm{bol}}$ (panel $iv$), essentially due to the
increase of this observable with $R_{\mathrm{gal}}$.

The large degree of scatter between the local SFR and median evolutionary
indicators suggests that while the SFR, as it is calculated here, essentially
depends on the local availability of mass, it is quite insensitive to the
average evolutionary stage of clumps. In fact, it increases with the number of
protostellar clumps and with their masses. Therefore, massive star forming
regions surely provide a relevant contribution to the SFR of the Milky Way,
however this does not seem to depend - once their mass is given - on an
earlier or later mean evolutionary stage. This confirms the result found by
Komugi et al. (2018) for M 33, namely the fact that the SFR is clearly
correlated with the cloud density, in a somehow extended meaning of the KS
relation, but not with the evolutionary stage.

###  3.5 SFR and spiral arm locations

In several portions of the spatial distribution of SFR
density$\Sigma_{\mathrm{SFR}}$ in the Galactic plane shown in Figure 5 it is
possible to recognize stretches of spiral arm features, similarly to what has
been observed in external galaxies (Rebolledo et al., 2015; Elagali et al.,
2019; Yu et al., 2021).

The SFR arm-like features are not much different from the pattern visible also
in the distribution of protostellar clumps (Figure 8, panel $b$), i.e. the
elements from which the SFR is computed. However, as already discussed in Elia
et al. (2017, 2021), a relevant amount of clumps with “inter-arm” distances is
also found, due to a large spread in velocities detected along many lines of
sight of spectral surveys (Mège et al., 2021). Additionally, each different
analytic spiral arm prescription available in the literature is able to
successfully reproduce the observations over wide ranges of longitude, but can
fail to fit observed arm-like features along other directions.

Figure 10: Top: histograms of logarithm of SFR density for pixels of the map
in Figure 5, classified as on-arm (black) and off-arm (gray), based on the
fact that their minimum distance from at least one of the spiral arms
displayed in Figure 5 is shorter or longer than 0.5 kpc, respectively. Middle:
the same as in the top panel, but for pixels of the map of SFF in Figure 8,
panel $c$. Bottom: the same as in the previous panels, but for pixels of the
map of $\left<L_{\mathrm{bol}}/M\right>$ in Figure 8, panel $e$.

In Figure 5 we show, as an example, the four-arm prescription by Hou et al.
(2009). It appears in good agreement with enhancements of our SFR density map
along a significant portion of the Perseus arm; a good agreement is also found
for the Norma and Carina-Sagittarius arms in the third quadrant, and for the
Carina-Sagittarius and Crux-Scutum arms in the first octant of the fourth
quadrant, respectively. Nevertheless, there are other regions with high SFR
density, but not directly intersected by one of the displayed arms. All these
aspects are mirrored in the distribution of the $\Sigma_{\mathrm{SFR}}$ values
in Figure 5, if pixels closer than 0.5 kpc from a whatever spiral arm
candidate plotted in the figure are considered separately from the remaining
ones (Figure 10, top). The distribution of the “on-arm” pixels is visibly more
left-skewed than the one of the “off-arm” pixels. In short, we observe a
sufficiently clear spiral arm structure in the SFR spatial distribution, which
elects spiral arms as preferential (although not unique) places for star
formation activity in the Galaxy.

A similar spiral arm-like structure, on the contrary, is not seen in Figure 8
in the distributions of both SFF (panel $c$) and median evolutionary
indicators as $T$ ($d$), $L_{\mathrm{bol}}/M$ ($e$), and $T_{\textrm{bol}}$
($f$). Figure 10, middle, contains a comparison between the on-arm and off-arm
distributions of SFF. These two distributions appear to be much less
distinguishable than in the case of $\Sigma_{\mathrm{SFR}}$ (top panel). A
similar behavior can be also seen for $\left<L_{\mathrm{bol}}/M\right>$
(Figure 10, middle), taken as an example for the three considered evolutionary
indicators.

These considerations reinforce the conclusions of Ragan et al. (2016) and Elia
et al. (2021) about the role of spiral arms, i.e. that no significant
differences in the mean evolutionary stage of star formation between spiral
arms and inter-arm regions are found. The main difference between these two
regions seems rather to consist in the larger amount of material available for
star formation in the former than in the latter (see also Moore et al., 2012).

##  4 Summary

We used the information contained in the catalog of Hi-GAL clump physical
properties to obtain a self-consistent estimate the Milky Way SFR. The main
points to be summarized are:

  * •

We extrapolated from the algorithm proposed by Veneziani et al. (2013) and
Veneziani et al. (2017) an operative analytic prescription to obtain the
contribution to the SFR from the mass of each far-infrared clump:
$\textrm{SFR}_{\textrm{clump}}=5.6\times
10^{-7}~{}(M_{\textrm{clump}}/\textrm{M}_{\odot})^{0.74}~{}\textrm{M}_{\odot}~{}\textrm{yr}^{-1}$.

  * •

Considering all clumps classified as protostellar (i.e. star forming) in the
catalog and provided with a heliocentric distance, we obtain SFR=$1.7\pm 0.6$
M⊙ yr-1.

  * •

If we also take into account the contribution from the protostellar clumps
without distance assignment, we estimate an additional contribution of 0.22 M⊙
yr-1 to the global SFR, assuming that their distances follow the same
distribution as the sources with known locations. This would lead to a final
SFR=$2.0\pm 0.7$ M⊙ yr-1. Also simulating the extreme case consisting in
placing all those sources at $d=13$ kpc, the corrective term would not surpass
0.57 M⊙ yr-1.

  * •

The profile of SFR surface density as a function of the Galactocentric radius
$R_{\mathrm{gal}}$ is qualitatively similar to other results in the literature
for the Milky Way and NGC 6946, and is consistent with most recent models
(Evans et al., 2022). The absolute maximum is found in correspondence of the
CMZ, and another local peak is found around $R_{\mathrm{gal}}=5$ kpc, after
which the logarithm of SFR density linearly decreases at increasing
$R_{\mathrm{gal}}$, with slope -0.28.

  * •

Studying the cumulative of the SFR as a function of the Galactocentric radius,
we find that 50% of the entire Milky Way SFR comes from within the so-called
Molecular Ring, and 84% from within the Solar circle (inner Galaxy,
$R_{\mathrm{gal}}<8.34$ kpc). The outer Galaxy is therefore responsible of 16%
of total SFR and, in particular, the far outer Galaxy ($R_{\mathrm{gal}}>13.5$
kpc) of 1%.

  * •

The relation between SFR density and the molecular gas surface density
Galactocentric profiles follows, for $R_{\mathrm{gal}}>3$ kpc (containing the
93% of the entire Galactic SFR), a Kennicutt-Schmidt behavior, with slope
$n=1.14\pm 0.07$.

  * •

We find no significant trend relating the maps of SFR and those of the
observables considered indicative of the mean local evolutionary stage of
clumps. We conclude that the local SFR is determined by the amount of
available mass rather than by the clump evolutionary stage.

  * •

In the distribution of SFR across the Galactic plane, arm-like enhancements
emerge over wide ranges of longitude. These enhancements are not seen in maps
of the clump evolutionary indicators, suggesting that they are produced by
source crowding within the arms.

This research has received funding from the INAF Mainstream Grant “The
ultimate exploitation of the Hi-GAL archive and ancillary infrared/mm data”
(1.05.01.86.09), and from the European Research Council synergy grant ECOGAL
(855130). SL acknowledges support by the INAF PRIN 2019 grant ONSET. AZ thanks
the support of the Institut Universitaire de France.

##  Appendix A Evaluating the completeness of the total SFR estimate for the
Milky Way

Here we provide an estimate of the validity of the total SFR derived for the
Milky Way in Section 2.2 in the light of possible incompleteness of the Hi-GAL
catalog for large heliocentric distances.

Elia et al. (2017) showed how the mass completeness limit for Hi-GAL, based on
the 350-$\mu$m band flux, varies with source temperature and distance. For
$T=16$ K, similar to the median temperature for Hi-GAL star forming clumps
(Elia et al., 2021), a distance $d=10$ kpc, and a typical flux completeness
limit of 5 Jy for the inner Galaxy (Molinari et al., 2016), the mass
completeness limit amounts to $\sim 90~{}\textrm{M}_{\odot}$. One can
therefore expect a shortage of detected clumps with mass lower than that,
ignored in the SFR calculation. Here we give a quantitative discussion of this
possible effect.

Figure 11: Cumulative SFR profile as a function of the heliocentric distance.
Since the distance $d$ appears as upper limit of integration, here the
integration variable is called $s$.

First, we compute the cumulative SFR as a function of the distance. In Figure
11 a sharp change from a steeply positive to an almost horizontal slope is
seen around $d=12$ kpc. The same can be affirmed, but in terms of source
number, noting that around the same distance a decreasing trend is seen to
start also in Figure 2. Such a change of slope can, in principle, be a
combination of the true distribution of the SFR density in the plane and the
selection effect due to distance we want to quantify. In the Hi-GAL catalog
distances $d>12$ kpc are achieved by sources in the first and fourth
quadrants555Notice that, since the relation between Galactocentric radius and
heliocentric distance is
$R_{\mathrm{GC}}=\sqrt{R_{0}^{2}+d^{2}+2R_{0}d\cos\ell}$, sources placed at a
same distance span a variety of Galactic locations, with the minimum
$R_{\mathrm{GC}}$ achieved for $\ell=0^{\circ}$. and the total SFR due to such
sources is 0.45 M⊙ yr-1.

Second, we evaluated the SFR from the symmetrical region on the opposite side
of the Milky Way, i.e. we considered clumps (essentially located in the second
and third quadrants) lying outside the circle centered on the symmetric point
of the Sun with respect to the Galactic center ($[x,y]=[0,+8.34]$ kpc) and
with a radius of 12 kpc. The SFR from this area amounts to 0.60 M⊙ yr-1, so
that, in the rough assumption of a circular symmetry for the “true” SFR
distribution in the $[x,y]$ plane, the “missing SFR” due to incompleteness is
comparable to the difference in SFRs measured from the two zones, namely 0.15
M⊙ yr-1. The extent of this correction shows that the effect of the catalog
incompleteness is not much strong, although not negligible: it should be added
as a further term to the total estimate of 1.7 M⊙ yr-1 based on clumps
provided with a distance estimate (Section 2.2), then representing a 8%
correction to it.

To evaluate an analogous additional term for the SFR of 2.0 M⊙ yr-1, estimated
by involving also clumps with no distances (Section 2.2.1) is more
complicated, because the distribution itself of known distances is involved in
the calculation. However, assuming a 8% correction also in this case, the
total Milky Way SFR would amount to 2.13 M⊙ yr-1.

##  Appendix B Testing the Kennicutt-Schmidt relation with atomic gas

To complete the discussion about the SFR vs gas density relation contained in
Section 3.3, here we show the plots of $\Sigma_{\mathrm{SFR}}$ vs
$\Sigma_{\mathrm{H\textsc{i}}}$ (Figure 12, left) and
$\Sigma_{\mathrm{gas}}=\Sigma_{\mathrm{H\textsc{i}}}+\Sigma_{\mathrm{H}_{2}}$
(right), respectively, obtained using the Galactocentric profile of Hi surface
density of Nakanishi & Sofue (2016, their Figure 4).

Comparing such profile with that of $\Sigma_{\mathrm{SFR}}$, peaks for these
two quantities are seen at very different values of $R_{\mathrm{GC}}$ (at
$\sim 10$ and $\sim 5$ kpc, respectively), preventing a power-law relation
between the two. Furthermore, for $R_{\mathrm{GC}}\lesssim 20$ kpc,
$\Sigma_{\mathrm{H\textsc{i}}}$ varies only within one order of magnitude
(namely between $\sim 1$ and 10 M⊙ pc-2), so that the plot assumes an almost
vertical appearance, as already highlighted by Sofue & Nakanishi (2017) and
Bacchini et al. (2019). In this last work, in particular, it was shown how a
power law behavior is recovered if the atomic+molecular gas volume density
instead of the surface density is considered, which implies to take into
account the increasing Galactic disk scale height at increasing
$R_{\mathrm{GC}}$.

Similar considerations can be formulated if one considers also the total gas
surface density, which, as it can be seen in Miville-Deschênes et al. (2017),
is dominated by the atomic component at $R_{\mathrm{GC}}\gtrsim 7$ kpc.

Figure 12: Left: the same as in Figure 7, but with atomic gas surface density
on the $x$ axis (from Nakanishi & Sofue, 2016), instead of the molecular one.
The KS relation is also plotted as a dashed line. Right: the same as in Figure
7 and in the left panel of this figure, but with total gas surface density on
the $x$ axis. In both panels the $x$-axis range has been kept identical to
that of Figure 7, to facilitate comparison.

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