The Star Formation Rate of the Milky Way as Seen by Herschel

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 five-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 S. Molinari et al. (2022, in preparation).

Starting from these lists, Elia et al. (2021) obtained a band-merged catalog, from which only SEDs with a shape eligible for a modified blackbody fit in the range from 160 to 500 μm 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 toward the inner Galaxy by considering H i 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 versus quiescent (designated for brevity as protostellar and starless, respectively), Elia et al. (2021) adopted the availability/lack of a detection at 70 μm, respectively.

The modified blackbody 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) versus 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 blackbody. For protostellar sources, generally showing an excess of emission at λ < 70 μm with respect to the modified blackbody (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 λ < 160 μm and of the best-fitting modified blackbody for λ ≥ 160 μm, 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 the Hi-GAL clumps of Elia et al. (2021) provided with a distance estimate have masses ranging from 0.1 to 10⁴ M_⊙ and luminosities ranging from 0.1 to 10⁵ L_⊙.

To compute the SFR through the method described in Section 2.2, first we consider all 29,880 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 a further 5532 protostellar clumps lacking a distance determination.

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 versus mass diagram (Molinari et al. 2008; Baldeschi et al. 2017b; 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. 2017a). 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. (2017a) 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. (2017b) 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, Σ_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_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:

$$\begin{eqnarray}&&{{\rm{SFR}}}_{\mathrm{cl}}={(5.6\pm 1.4)\times {10}^{-7}\,({M}_{\mathrm{cl}}/{M}_{\odot })}^{0.74\pm 0.03}\,{M}_{\odot }\,{{\rm{yr}}}^{-1}.\end{eqnarray} \tag{ 1 }$$

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 ± 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 versus outer Galaxy delimited by the solar circle with a radius of 8.34 kpc (as in Elia et al. 2021; Mège 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

It should be noted that the SFR estimate reported above is 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 μm (for the same reasons written above), but detected at the other ones, result in a misclassification 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 μm 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 are not involved in our previous calculation (see Figure 1).

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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 0.20–0.23 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.

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In conclusion, a reasonable estimate of the total Galactic SFR, taking into account the correction calculated above, is 2.0 ± 0.7 M_⊙ yr⁻¹.

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 the 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).

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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° ≲ ℓ ≲ 20°), and in the direction of the Carina Arm (270° ≲ ℓ ≲ 310°). The first case can be treated with relative ease, by arbitrarily placing all the sources without distance and with ∣ℓ∣ < 20° 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 ± 0.06 M_⊙ yr⁻¹.

Adopting the value of 2.0 ± 0.7 M_⊙ yr⁻¹ as our final estimate for the Milky Way SFR, and comparing with the 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).

A first view of SFR distribution in the Galactic plane can be given in the [ℓ, 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).

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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° < ℓ < 339° and −0.2° < b< 0.0° (containing the G337.342–0.119 region; Jackson et al. 2018), 27° < ℓ < 30° and 0.0° < b < 0.2°, and 18° < ℓ < 21° and −0.2° < b < 0.0°, respectively.

Through the [ℓ, 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_CMZ estimates, each of them derived in a different area around the Galactic center, and the output of our method for the same area.

Table 2. Star Formation Rate Estimates for the Central Molecular Zone

Method	Area Boundaries	SFRCMZ (lit.)	References	SFRCMZ (this Work)
		(M⊙ yr−1)		(M⊙ yr−1)
YSO counts (Spitzer)	$| {\ell }| \lt 1^\circ ,| b| \lt 10^{\prime}$	0.14	Yusef-Zadeh et al. (2009)	0.04 ± 0.02
Continuum emission at 60, 100 μm (IRAS)	∣ℓ∣ < 3°, ∣b∣ < 1° a	0.12	Crocker et al. (2011)	0.12 ± 0.05
Continuum emission at 60, 100 μm (IRAS)	∣ℓ∣ < 0.8°, ∣b∣ < 0.3°	0.08	Crocker et al. (2011)	0.04 ± 0.02
YSO counts (Spitzer)	∣ℓ∣ < 1.5°, ∣b∣ < 0.5°	0.08	Immer et al. (2012)	0.08 ± 0.03
Ionization rate from radio free–free	2.5° < ℓ < 3.5°, ∣b∣ < 0.5°	0.035	Longmore et al. (2013)	0.11 ± 0.04
Ionization rate from radio free–free	as above, but ∣b∣ < 1° for ∣ℓ∣ < 1°	0.06	Longmore et al. (2013)	0.12 ± 0.05
Continuum emission at 24 μm (Spitzer)	∣ℓ∣ < 1°, ∣b∣ < 0.5°	0.09 ± 0.02
Continuum emission at 70 μm (Spitzer)	∣ℓ∣ < 1°, ∣b∣ < 0.5°	0.10 ± 0.02	Barnes et al. (2017) b	0.06 ± 0.02 c
Cont. emission at 5.8–500 μm (Spitzer, Herschel)	∣ℓ∣ < 1°, ∣b∣ < 0.5°	0.09 ± 0.03

Notes.

^a Differently from other rectangular areas quoted in the table, this one has an elliptical shape with axes aligned along ℓ and b, and semiaxes of 3° and 1°, respectively. ^b In Barnes et al. (2011), their Table 2, a further list of SFR_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. ^c The three SFR_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.

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We found a good agreement with the SFR_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_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 ± 0.02 M_⊙ yr⁻¹.

Our results, however, appear to disagree with the SFRs estimated for the CMZ in Yusef-Zadeh et al. (2009) and Longmore et al. (2013). The SFR_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 ± 0.02 M_⊙ yr⁻¹) within the uncertainties.

Longmore et al. (2013) estimated SFR_CMZ = 0.035 M_⊙ yr⁻¹ in the range 2.5° < ℓ < 3.5° and ∣b∣ < 0.5°, and 0.06 M_⊙ yr⁻¹ if the wider latitude range ∣b∣ < 1° is considered in the central ∣ℓ∣ < 1°. Correspondingly, we calculate SFR_CMZ = 0.11 and 0.12 M_⊙ yr⁻¹ for these two zones, respectively. The former one is about three 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 of 1/3.

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 × 0.5 kpc² pixels, can be built. This is a simplified 2D 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 FIR emission already observed by IRAS toward 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_gal ≳ 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.

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It makes sense, therefore, to analyze the behavior of the SFR density Σ_SFR, averaged in concentric 500 pc wide rings, versus the Galactocentric radius R_gal. In Figure 6, top panel, a peak corresponding to the inner circle ¹⁹ is followed by a wide dip between 1 and 4 kpc, mostly due to a 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 southwest and southeast 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 Figure 1), while most of the 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).

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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 ∼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_gal ≃ 6 kpc. Finally, at larger distances, Σ_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_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_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 Figure 3) in the Σ_SFR versus R_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. The latter being quite similar to ours in many points (Figure 6, top), a similar consistency is expected also for our data. In fact, our Σ_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_gal ≃ 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 the Milky Way with those 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 the 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 Hα 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, a similar inner dip was also found by Lian et al. (2018) to be typical of local massive star-forming galaxies ($10.5\lt \mathrm{log}(M/{M}_{\odot })\lt 11$).

Returning to the SFR density profile of the Milky Way and to its decrease for R_gal ≳ 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:

$$\begin{eqnarray}&&\mathrm{log}({{\rm{\Sigma }}}_{\mathrm{SFR}}/[{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}])=a\,{R}_{\mathrm{gal}}/[\mathrm{kpc}]+b,\end{eqnarray} \tag{ 2 }$$

with a ∼ −0.14. Fitting the same relation to our data for R_gal ≥ 5 kpc, we obtain a = −0.28 ± 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 ± 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 ± 0.05. Finally, in the NGC 6946 case the slope shows an intermediate behavior (a = −0.20 ± 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 Data Release 4 (Cui et al. 2012; Deng et al. 2012) 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_gal > 8 kpc).

In Figure 6, bottom, the behavior of the SFR profile as a function of R_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_gal. For our data we notice three very 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_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_gal > 13.5 kpc, according to the definition by Heyer et al.1998).

The SFR density profile computed in rings of R_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 KS law.

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

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{SFR}}\propto {{\rm{\Sigma }}}_{\mathrm{gas}}^{n},\end{eqnarray} \tag{ 3 }$$

with n generally falling in the range 1–2, depending on the tracer used and the linear scales considered. In particular, the "classic" slope of Kennicutt (1998) is n = 1.40 ± 0.15, with Σ_gas expressed in units of M_⊙ pc⁻². Finally, in the literature descriptions of Σ_SFR as a power law of the single atomic (Σ_{H I}), or molecular component (${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$), 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 ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ and found an exponent n = 1.2 ± 0.2. Similarly, Sofue & Nakanishi (2017) found n = 1.12 ± 0.05 for a set of $({{\rm{\Sigma }}}_{\mathrm{SFR}},{{\rm{\Sigma }}}_{{{\rm{H}}}_{2}})$ pairs obtained through binning the Galactic plane in 0.2 × 0.2 kpc² boxes. No clear correlation was found, instead, between Σ_SFR and both Σ_{H I} and Σ_gas. A steeper slope (3.7 ± 1.6), again considering only ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$, 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 $({{\rm{\Sigma }}}_{\mathrm{SFR}},{{\rm{\Sigma }}}_{{{\rm{H}}}_{2}})$ 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 ≤ R_gal ≤ 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_gal.

To do that, for the x-axis we use the molecular ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ 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 Σ_{H 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). ²⁰

In Figure 7 the log–log plot of Σ_SFR versus ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ 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_gal ≤ 3 kpc). Notice that, as can be seen in the bottom panel of Figure 6, the fraction of SFR contained within this radius corresponds to only the 7% of the entire Milky Way SFR. From the fit to the remaining points, we obtain n = 1.14 ± 0.07.

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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 value 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 ± 0.11). The comparison with the classic slope of Kennicutt (1998), instead, makes little sense, since it refers to the total Σ_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_CO = 10²⁰ cm⁻² (K km s⁻¹)⁻¹. However, if an increasing trend of X_CO at increasing R_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_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 Σ_SFR versus ${{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ plot.

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.

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 the total mass of protostellar clumps in those bins. This total mass, in turn, does not necessarily depend on the number of protostellar clumps, N_pro, found in each bin (shown in Figure 8(b)), ²¹ and does not depend at all on the number of pre-stellar clumps, N_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_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 1D description of the SFF is further developed here in the 2D 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.

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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 SFR 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 (≪1) also exhibit relatively low SFR values.

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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 blackbody temperature T, bolometric luminosity over mass ratio L/M, and bolometric temperature T_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 1D profile as a function of R_gal, apart from an increase in the far outer Galaxy, but supported by poor statistics. On the contrary, the trend found for median T_bol is to slightly raise at increasing R_gal. This is what can be seen also in the 2D view provided in Figure 8. Therefore, the bin-to-bin comparison of quantities nearly constant with R_gal such as T and L_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_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_bol (panel (iv)), essentially due to the increase of this observable with R_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 M33, 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.

In several portions of the spatial distribution of SFR density Σ_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.

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 Σ_SFR values in Figure 5, if pixels closer than 0.5 kpc from 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.

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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_bol/M (e), and T_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 Σ_SFR (top panel). A similar behavior can be also seen for $\left\langle {L}_{\mathrm{bol}}/M\right\rangle$ (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).