The star formation histories, star formation efficiencies and ionizing sources of ATLASGAL clumps with HII regions
J. W. Zhou Sami Dib Pavel Kroupa





Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany  

jwzhou@mpifr-bonn.mpg.de
Max Planck Institute für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany 

dib@mpia.de
Helmholtz-Institut für Strahlen- und Kernphysik (HISKP), Universität Bonn, Nussallee 14–16, 53115 Bonn, Germany 

pkroupa@uni-bonn.de
Charles University in Prague, Faculty of Mathematics and Physics, Astronomical Institute, V Holešovičkách 2, CZ-180 00 Praha 8, Czech Republic 




1226 ATLASGAL clumps with HII regions (HII-clumps) were matched with radio sources in the CORNISH-North/South surveys, and 392 of them have corresponding radio sources. We determined the stellar luminosity L_*, T84 according to the Lyman continuum flux N_Ly.  When the bolometric luminosity of HII-clumps is less than log_10(L_ bol, obs/L_⊙) ≈ 3.7, corresponding to a clump mass log_10(M_ cl/M_⊙) ≈ 2.55, the values of L_*, T84 derived from N_Ly overestimate the actual stellar luminosities, because the accretion onto the protostars contributes significantly to the radio emission. After subtracting the  accretion luminosity from L_*, T84, we obtained reasonable estimates of the stellar luminosity. Using the 0.5 Myr isochrone, we calculated the stellar masses according to the stellar luminosities, and found that they roughly follow the m_ max-M_ ecl relation of embedded clusters, consistent with the ionizing sources representing the most massive stars in the embedded clusters of HII-clumps. We also studied the contribution of the possible flaring activity to the observed stellar luminosity and found that they can be neglected. We further studied the change of SFE with the clump mass. According to the derived mass of the most massive star in each HII-clump, using the theoretical m_ max-M_ ecl relation, we calculated the mass of the corresponding embedded cluster and then the SFE of the clump. The SFE decreases with  increasing clump mass, with a median value of ≈0.3. We also independently derived the SFE for each HII-clump based on the model developed in our previous work. The SFEs of HII-clumps derived from the observation and the model are in good agreement. 
Concerning the star formation histories of the ATLASGAL clumps, low-mass clumps may reach the peak of star formation earlier than high-mass clumps, consistent with the shorter free-fall time of low-mass clumps.




Star formation properties of ATLASGAL clumps
J. W. Zhou, Sami Dib, Pavel Kroupa

The star formation histories, star formation efficiencies and ionizing sources of ATLASGAL clumps with HII regions
    J. W. Zhou<ref> 
Sami Dib <ref>
Pavel Kroupa<ref>
<ref>

    Accepted XXX. Received YYY; in original form ZZZ
==================================================================================================================





§ INTRODUCTION


Understanding the contribution of stars in different mass ranges to the total energetics and dynamics of star clusters is crucial.
A majority, if not all, of the stars we observe were born within embedded clusters <cit.>, which adds significant intricacy to pinpointing and analyzing individual protostellar objects. 
Unlike their lower-mass counterparts, massive stars evolve towards the main sequence while still deeply embedded within their parental clumps. This renders their initial stages invisible, even when observed using mid-infrared wavelengths. Nowadays,
our comprehension of massive star and cluster formation has significantly advanced, owing to the combination of Galactic plane surveys and high-angular resolution images obtained through submillimeter facilities <cit.>. Surveys such as HiGAL <cit.> and ATLASGAL <cit.> have played pivotal roles by providing unbiased compilations of dense clumps that track the early phases of massive star and cluster formation. Explorations employing representative clump samples can provide valuable insights into the efficiency of converting molecular gas into stellar clusters <cit.>.

In <cit.>, 5007 ATLASGAL clumps have been categorized into four distinct evolutionary stages, with the most advanced stage being referred to as HII-clumps, i.e. ATLASGAL clumps with HII regions. 
In <cit.> (paper I), we synthesized the embedded clusters within HII-clumps under the assumption that the stellar initial mass function (IMF) follows a universal optimal distribution function rather than a probability density <cit.>.  
We utilized the 0.1 Myr isochrone to estimate the bolometric luminosity of individual stars within an embedded cluster, augmenting this with the accretion luminosity of each star in the cluster. The cumulative bolometric luminosity of the synthetic embedded clusters aligns closely with the observed bolometric luminosity of HII-clumps, validating the effectiveness of the method. As a follow up study based on paper I, in this work, we focus on the ionizing sources in HII-clumps. They may represent the most massive stars in the embedded clusters. We explore their physical properties by comparing the observations with numerical simulations. 




§ SAMPLE



The physical parameters of 1246 HII-clumps have been calculated and listed in <cit.>. 1226 HII-clumps with mass (M_ cl) and bolometric luminosity (L_ bol, obs) measurements were matched with radio sources in the catalogs of the 
CORNISH-North <cit.>
and the CORNISH-South <cit.> surveys. The separations between the central coordinates of a radio source and a HII-clump was required to be smaller than the radius of the clump. 

Generally, one HII-clump matches with one radio source. If one HII-clump includes more than one radio source, we only consider the radio source with the strongest radio emission, because we are only interested in the most massive star in a HII-clump. Finally, the numbers of radio sources matched with HII-clumps are 244 and 148 in the CORNISH-South and the CORNISH-North survey catalogs, respectively.

Fig. <ref>(a) displays the samples of HII clumps in the L_ bol, obs-M_ cl diagram. We can see that the samples are mainly concentrated at the upper end, similar to Fig. 5 in <cit.>, where the HII-clumps were divided into two populations, i.e. radio-loud and radio-quiet. Our samples are radio-loud, which are more luminous than the radio-quiet HII-clumps. This indicates that they are hosting high-mass stars.


In Fig. <ref>(b), the radii of radio sources are systematically smaller than the radii of the HII-clumps. These radio sources should be excited by the most massive stars in the embedded clusters of HII-clumps. From the radio emission, we can infer some physical parameters of the massive stars.





§ RESULTS AND DISCUSSION






 §.§ Lyman continuum flux



As discussed in <cit.>, assuming optically thin radio sources at 5 GHz (CORNISH-North) and 5.5 GHz (CORNISH-South) is justified for our samples. We calculated the physical parameters of the matched radio sources following the formalism of <cit.> and <cit.> using the equations collected in <cit.>.
The emission measure (EM) and the electron density (n_e) were calculated using:


    [EM/cm^-6 pc] = 
    1.7×10^7 [S_ν/Jy]
    [ν/GHz]^0.1[T_e/K]^0.35[θ_s/]^-2,

and

    [n_e/cm^-3] = 
    2.3×10^6 [S_ν/Jy]^0.5[ν/GHz]^0.05[T_e/K]^0.175[d/pc]^-0.5[θ_s/]^-1.5,


where S_ν is the integrated radio flux density, T_e is the electron temperature assumed to be 10^4 K, θ_s is the angular diameter of the source, d is the distance, and ν is the observing frequency.
S_ν and θ_s are assigned the values in the catalogs of <cit.> and <cit.>. 

The values of d are the distances of the corresponding HII-clumps listed in <cit.>.
The number of Lyman-continuum photons per second (N_Ly; hereafter Lyman continuum flux) is calculated from the flux density and distance as:



    [N_Ly/photons s^-1] = 
    8.9×10^40 [S_ν/Jy]
    [ν/GHz]^0.1[T_e/10^4K]^-0.45[d/pc]^2.





Table. 1 of <cit.> gives the relationship between the stellar parameters and Lyman continuum flux.
For stars with masses above 60 M_⊙, they made the assumption that Lyman continuum flux per unit area is constant, which is likely to slightly underestimate the Lyman flux of very massive stars.
In Fig. <ref>(a), we compared the data in Table. 1 of <cit.> with Table. 1 of <cit.>. 
It seems that Table. 1 of <cit.> preforms better at the end with high N_Ly and stellar luminosity, L_*. The relationship between
stellar mass and luminosity in <cit.> was taken from <cit.>, which is consistent with the isochrones of 0.5 Myr or 1 Myr from the MESA Isochrones and Stellar Tracks (MIST) project <cit.>, as shown in Fig. <ref>(b).

In this work, we determined the stellar luminosity L_*, T84 corresponding to the calculated N_Ly by interpolating the
stellar luminosity given as a function of the Lyman continuum flux in Table. 1 of <cit.>, shown in Fig. <ref>(a).




 §.§ The turning points



The stellar luminosity L_*, T84 of the brightest star should be smaller than the bolometric luminosity L_ bol, obs of the corresponding HII-clump.
However, in Fig. <ref>(a), 
for low-luminosity HII-clumps, L_*, T84 is significantly larger than L_ bol, obs, which is abnormal.
In Eq. <ref>, N_Ly∝ S_ν, and in Fig. <ref>, N_Ly∝ L_*. If the bolometric luminosity L_ bol, obs of an HII-clump is mainly from the stars, L_ bol, obs∝ L_*.
Finally, L_ bol, obs∝ S_ν. However, in Fig. <ref>(b), there is a turning point around log_10(L_ bol, obs/L_⊙) ≈ 3.7. For HII-clumps with luminosity lower than this value, there is no clear correlation between L_ bol, obs and radio flux density S_ν.


The turning point log_10(L_ bol, obs/L_⊙) ≈ 3.7 corresponds to the HII-clump mass log_10(M_ cl/M_⊙) ≈ 2.55 in Fig. <ref>(a).







Using the 0.5 Myr isochrone, we estimated the stellar mass m_*, iso according to the stellar luminosity L_*, T84.
The ionizing sources represent the most massive stars in HII-clumps, i.e. m_*, iso≈ m_ max. 
Considering the m_ max-M_ ecl
relation <cit.>, a correlation between the embedded star cluster mass M_ ecl and the mass of the most massive star m_ max,
there should be a correlation between 
m_*, iso and 
the mass of the corresponding embedded cluster. The clump mass M_ cl and its embedded cluster mass M_ ecl satisfy M_ ecl = SFE × M_ cl, where SFE is the final star formation efficiency of the clump, and which we assumed to be SFE=0.33 <cit.>, see Sec.<ref> for more discussion.
For 100 embedded clusters with age < 5 Myr, the table in <cit.> gives the values of their m_ max and M_ ecl. This sample fits the m_ max-M_ ecl relation well.
In Fig. <ref>(c), we compared our sample with the sample from <cit.>. When log_10(M_ ecl/M_⊙) > 2.1, the m_*, iso-M_ ecl relation follows the m_ max-M_ ecl relation.
Assuming SFE=0.33, from the turning point
log_10(M_ ecl/M_⊙) ≈ 2.1, we have
log_10(M_ cl/M_⊙) ≈ 2.55, consistent with the findings above.



 §.§ Synthetic embedded clusters



We adopted the optimal sampling algorithm <cit.> to produce a population of stars for an embedded cluster with the mass M_ ecl using the publicly available GalIMF code. Detailed descriptions of the optimal sampling and the publicly available GalIMF code that optimally populates clusters and galaxies with stars are available in <cit.> and <cit.>. The code has three input parameters: 
the mass of the embedded cluster, M_ ecl, its age, and a metallicity associated with this adopted age. In this work, we assume that the metallicity of the clumps and their associated embedded clusters is the solar metallicity ([Fe/H] = 0). For the age, following <cit.>, it is taken to be 1 Myr.


In Fig. <ref>, we picked out the most massive star with mass m_ max in each synthetic embedded cluster and compared it with the corresponding m_*, iso. When 
log_10(m_ max/M_⊙) > 1, m_ max and m_*, iso are roughly comparable. If the mass of the most massive star in a synthetic embedded cluster is log_10(m_ max/M_⊙) ≈ 1, which corresponds to log_10(M_ cl/M_⊙) ≈ 2.55 or log_10(M_ ecl/M_⊙) ≈ 2.1, consistent with the results in Sec. <ref>. Moreover,
using the fitted m_ max-M_ ecl relation, i.e. equation.1 in <cit.>, when log_10(M_ ecl/M_⊙) ≈ 2.1, we can also obtain log_10(m_ max/M_⊙) ≈ 1.






 §.§ Accretion




The turning point log_10(M_ cl/M_⊙) ≈ 2.55 is used to judge the relative importance of accretion luminosity in <cit.>, also presented in Fig. <ref>(a). 
Therefore, the abnormal tail in Fig. <ref> and Fig. <ref> on the left of the vertical dashed lines means that if the accretion luminosity is dominant, the formalism in Sec. <ref> cannot be used, because the main Lyman continuum flux or radio emission are not predominantly from the stars.
The high radio flux density of low-luminosity and low-mass HII-clumps in Fig. <ref>(b) indicate that accretion is an effective way to produce radio emission. Therefore, the Lyman continuum flux N_Ly derived from radio flux density S_ν includes a significant contribution from the accretion process. The previous studies of HII regions in <cit.> have noted that there is an excess of Lyman photons over what would be expected from the ionizing stars compared to the bolometric luminosities. The inclusion of the luminosity contribution from accretion may provide an explanation.

In Sec. <ref>, we obtained the stellar luminosity L_*, T84 from the Lyman continuum flux N_Ly. Actually, L_*, T84 is not only the stellar luminosity, which includes the real stellar luminosity L_*, but the sum of L_* and the accretion luminosity L_acc, i.e. L_*, T84=L_*+L_acc. 


Using the method described in <cit.>, we calculated the accretion luminosity L_acc of the most massive stars of the synthetic embedded clusters, and subtracted L_acc from the corresponding L_*, T84. 
Then we repeated the analysis in Sec. <ref> using L_* rather than L_*, T84.
As can be seen in Fig. <ref>(a), the abnormal tail disappears.









 §.§ Flares




The value of L_* still remains somewhat higher in Fig. <ref>(a). The abnormal tail in Fig. <ref>(b) is not completely eliminated, although it shows a substantial improvement over the case without considering accretion, presented in Fig. <ref>. 



Therefore,
the stellar luminosity L_* may still be overestimated. There are likely other physical processes contributing to the luminosity that have not been considered, similar to the accretion luminosity, which also needs to be subtracted.


Flaring represents a common occurrence of magnetic activity in low-mass stars, including our Sun. While flares, originating from both the Sun and other stars, are predominantly observed in the soft X-ray band, the majority of the emitted energy is released at optical/UV wavelengths <cit.>.

For a  sample of pre-main-sequence (PMS) stars in the NGC 2264 star-forming region, <cit.> detected seventy-eight X-ray flares with optical and/or mid-infrared counterparts. 

The optical emission of flares, encompassing both emitted energy and peak flux, demonstrates a strong correlation with, and notably surpasses, the X-ray emission.
The luminosities in X-ray (L_ x) and optical (L_ opt) bands satisfy L_ opt= a × L_ x^b with a ranging from 5.2 to 14.7 and b between 0.6 and 1.0. Here we take the average values, i.e. L_ opt≈ 10 × L_ x^0.8. 

<cit.> identified a sample of 1086 X-ray superflares and megaflares.

These events are generated by young stars across all masses throughout evolutionary stages spanning from protostars to stars without disks.
Only the sample of <cit.> has mass estimates of the PMS stars. We converted the X-ray luminosity of the flare measured in <cit.> into optical luminosity using the relation fitted in <cit.>. In Fig. <ref>(a), the distribution of L_ opt is highly scattered, and it has almost no correlation with stellar mass. The ratio of the optical and bolometric luminosities (L_ opt/L_ bol) of the PMS stars decreases rapidly as mass increases. Even for low-mass stars, the ratio is always less than 1. Thus, the luminosity from the flares can be neglected, and does not make a significant contribution to the abnormal tail in Sec. <ref>.











 §.§ Star formation efficiency











  §.§.§ Observations


In Fig. <ref>(c), for low-mass clusters, there is a mismatch between our sample and the sample in <cit.>. As discussed in Sec. <ref>, we may be overestimating the stellar mass. 
Another possibility is that the low-mass clumps have higher SFEs > 0.33. Assuming all measured maximum stellar masses follow the m_ max-M_ ecl relation, we can calculate the mass of the embedded cluster and then the star formation efficiency for each clump.
The third-order polynomial fit to the observed m_ max-M_ ecl relation presented in <cit.> has large errors.
As an alternative, we derived a theoretical m_ max-M_ ecl relation directly from the initial mass function (IMF). The details are described in Sec.<ref> and the embedded cluster mass, M_ ecl, is given by


    M_ecl≈5.37/0.77/m_max^1.3 - 0.001 - 3.33/m_max^0.3(0.77/m_max^1.3 - 0.001),


where we use the m_ max values displayed in Fig .<ref>(c) to derive the corresponding value of M_ ecl for each embedded cluster.

The abnormal tail in Fig. <ref>(b) is visible in Fig. <ref>(a) again, presented as the SFEs close to 1. We excluded this tail in the subsequent analysis by requiring the clump mass larger than 10^1.5 M_⊙.
Fig. <ref>(b) displays a strong correlation between the SFE and the clump mass. The SFE decreases with increasing clump mass, with an upper limit of ≈0.65, This is also visible in Fig.5 of <cit.>. We fit the relation between M_ cl and the SFE with a power law of the form SFE∝ M_ cl^k and find a value of k=-0.44±0.02. 
This anti-correlation between the mass of the clumps and their SFE has been predicted theoretically by <cit.>. For the simple case of uniform density clumps, <cit.> predicted the clump mass and the SFE to follow SFE∝ M_ cl^-0.6 and to be the result of an increasingly stronger effect of stellar feedback in clumps with higher masses which leads to a faster gas expulsion and limits the value of the SFE. Interestingly, the median value of the SFEs is ≈0.3, consistent with our previous work <cit.>. An SFE≈ 0.33 is also consistent with the value obtained from hydrodynamic calculations including self-regulation <cit.> and also with observations of embedded systems in the solar neighborhood <cit.>.









  §.§.§ Models


In order to assess the uncertainty in the observations, now we independently study the clump's SFE using theoretical models. In Fig. <ref>(b), the upper limit of the SFE is ≈0.65. In Fig. 4 of <cit.>, a SFE larger than 0.5 is necessary to fit the low-mass HII-clumps.
Therefore, we consider the SFEs that fall in the range [0.05, 0.65] with a step of 0.05. Then each HII-clump has 13 alternative SFEs. 
We consider both the stellar luminosity and accretion luminosity of each star in the embedded cluster of a HII-clump, and also apply the pre-main sequence evolutionary track for each star. 
There is certainly an age spread of the stellar populations in embedded clusters within HII-clumps. Therefore, selecting the appropriate isochrones according to the stars' ages is essential. Additionally, not all stars in HII-clumps are in the accretion phase, as this too is age-dependent. We should identify the fraction of stellar objects that remain protostars to properly add their accretion luminosities. In order to address these issues, we apply the method described in <cit.> to generate an age distribution for the stellar population in each synthetic cluster. The age distribution of stars within a HII-clump depends on the star formation history (SFH) of the clump. As discussed in <cit.>, compared to a constant
SFH, burst-like and time-dependent SFHs can better fit the observational data. Thus, we only consider the time-dependent SFHs in this work. As described in <cit.>, there are 12 different SFHs with different age peaks (t_ p) and age standard deviations (σ_ t), i.e. t_ p= 0.1, 0.25, 0.5, 1 Myr and σ_ t= 0.25, 0.5, 1 Myr. Finally, each clump has 156 different combinations of SFEs and SFHs, resulting in 156 distinct bolometric luminosities. The combination of a SFE and a SFH that produces the bolometric luminosity of the synthetic embedded cluster closest to the observed bolometric luminosity of the corresponding HII-clump is considered as the optimal one. The optimal SFE and SFH distributions of all HII-clumps are shown in Fig. <ref>. The optimal combination has the smallest difference from the observed bolometric luminosity.
To test the robustness of the model, we also selected the second optimal combination, which has the second smallest difference from the observed bolometric luminosity. As shown in Fig. <ref>, the optimal and the second optimal combinations give almost the same results.




Compared with Fig. <ref>, overall, the SFEs of HII-clumps derived from the observation and the model are comparable. 
From Fig. <ref>(b) and (c), we have


    log_10 (SFE) = (-0.37 ± 0.01) ×log_10 (M_cl) + (0.42 ± 0.04),

and

    log_10 (M_cl) = (1.02 ± 0.02) ×log_10 (M_ecl) + (0.52 ± 0.05).

Then we use Eq. <ref> to calculate the SFE or the mass of the embedded cluster for each HII-clump. As shown in Fig. <ref>,
after the modification of the SFE, the mismatch in Fig. <ref>(c) indeed disappears.





 §.§ Star formation history




Fig. <ref> displays the distributions of the optimal t_ p and σ_ t for HII-clumps with different masses. For each t_ p or σ_ t, we calculated the median and mean masses of the corresponding HII-clump groups. The median and mean clump masses are comparable for different σ_ t. However, there is a weak trend that clumps with larger masses have larger t_ p, indicating that low-mass clumps reach the peak of star formation earlier than high-mass clumps, possibly because star formation occurs more rapidly in low-mass clumps. The clump free-fall time (τ_ ff) is useful in providing a lower limit to the star formation time-scales. In Fig.<ref>, we compared the free-fall time of ATLASGAL clumps in the earliest ("quiescent") and the latest ("HII region") stages <cit.>. 
Low-mass clumps indeed have a shorter free-fall time.






§ CONCLUSION


In this work, 1226 HII-clumps from the ATLASGAL survey with mass M_ cl and bolometric luminosity L_ bol, obs estimates were matched with radio sources in the catalogs of the CORNISH-North and the CORNISH-South
surveys. 392 HII-clumps have corresponding radio sources in the CORNISH-North/South surveys. These HII-clumps are mainly concentrated at the
upper end in the L_ bol, obs-M_ cl diagram, which are more luminous than the remaining HII-clumps, indicating they host high-mass stars. Radio sources are excited by the most massive stars in the embedded clusters of HII-clumps. We calculated the emission measure (EM), the electron density (n_e) and the Lyman continuum flux (N_Ly) for each radio source. Using  the relationship between the stellar parameters and the Lyman continuum flux in Table. 1 of <cit.>, 
we determined the stellar luminosity L_*, T84 according to the calculated N_Ly. 
We found a turning point around log_10(L_ bol, obs/L_⊙) ≈ 3.7, which corresponds to a HII-clump mass log_10(M_ cl/M_⊙) ≈ 2.55. When the bolometric luminosity of HII-clumps is less than this value, L_*, T84 is significantly larger than L_ bol, obs. Thus, there is an abnormal tail in the L_ bol, obs-L_*, T84 diagram. Using the 0.5 Myr isochrone, we estimated the stellar mass m_*, iso according to the stellar luminosity L_*, T84.

To investigate the physical origin of this turning point, we adopted the optimal sampling algorithm to produce a population of stars for an embedded cluster using the publicly available GalIMF code.
Then, we picked out the most massive star with mass m_ max
in each synthetic embedded cluster and compared it with the corresponding m_*, iso. When log_10(m_ max/M_⊙) > 1, m_ max and m_*, iso are roughly comparable.
Actually, the turning point at log_10(M_ cl/M_⊙) ≈ 2.55 exactly corresponds to log_10(m_ max/M_⊙) ≈ 1, which is also used to judge the relative importance of accretion luminosity in <cit.>. 
The abnormal tail means that if the accretion luminosity is dominant, we can not directly estimate the stellar parameters from the radio emission, and have to subtract the contribution of accretion from the radio flux density. To address this issue, we calculated the accretion luminosity L_acc of the most massive star in each synthetic embedded cluster, and subtracted L_acc from the corresponding L_*, T84.
However, the cleaned stellar luminosity L_* (=L_*, T84-L_acc) may be still overestimated. There are likely other physical processes contributing to the luminosity apart from the accretion. For example, the flaring activity represents a common occurrence of magnetic activity in low-mass stars. We found that the ratio of the optical and bolometric luminosities (L_ opt/L_ bol) of pre-main sequence stars decreases rapidly as mass increases. Even for low-mass stars, the ratio is always less than
1. Thus, the luminosity from the flaring activity can be neglected. Other physical processes need to be further investigated.

In the above analysis, we assumed a constant SFE=0.33. We further studied the change of SFE with the clump mass. According to the derived mass of the most massive star in each HII-clump, using the theoretical m_ max-M_ ecl relation, we calculated the mass of the corresponding embedded cluster and then the SFE of the clump. We find a strong anti-correlation between the SFE and the clump mass. The SFE decreases with increasing clump mass in agreement with theoretical expectations <cit.>, with a median value of ≈0.3.   
We also independently derived the SFE for each HII-clump based on the model developed in <cit.>. We consider the SFEs that fall in the range [0.05, 0.65] with a step of 0.05. 
We employ 12 different star formation histories for each HII-clump to distribute the ages to the stars in the clump. According to the stars' age,
we consider both the stellar luminosity and accretion luminosity of each star in the embedded cluster of a HII-clump, and apply the pre-main sequence evolutionary track for each star. Finally,
each clump has 156 different combinations of SFEs and SFHs, resulting in 156 distinct bolometric luminosities. The optimal one has the smallest difference from the observed bolometric luminosity of the corresponding HII-clump. Overall, the SFEs of HII-clumps derived from the observations and the models are comparable. 
Using the quantitative relation between the SFE and the clump mass, we calculated the SFE and the mass of the embedded cluster for each HII-clump. As shown in Fig. <ref>,
after the modification of the SFE, the mismatch in Fig. <ref>(c) disappears.

For the optimal t_ p or σ_ t, we estimated the median and mean masses of the corresponding HII-clump groups. There is a weak trend that clumps with larger masses have larger t_ p, indicating that low-mass clumps reach the peak of star formation earlier than high-mass clumps, possibly because star formation occurs more rapidly in low-mass clumps, consistent with the shorter free-fall time of low-mass clumps.


We would like to thank the referee for the comments and suggestions that helped improve and clarify this work.


aa 









§ THE THEORETICAL M_ MAX-M_ ECL RELATION



The stellar IMF, ξ_⋆(m), is canonical <cit.>:

    ξ_⋆(m,M) =
     
    0,     m<0.08 M_⊙, 
    
    2k_⋆ m^-1.3,     0.08 M_⊙⩽ m<0.5 M_⊙, 
     
    k_⋆ m^-2.3,     0.5 M_⊙⩽ m<1M_⊙, 
    
    k_⋆ m^-α_3,     1 M_⊙⩽ m<m_max(M), 
    
    0,     m_max(M) ⩽ m,

where α_3=2.3 is the constant Salpeter-Massey index for the invariant canonical IMF but will change for larger ρ_cl (the density of star-forming clump) to account for IMF variation under star-burst conditions <cit.>.
0.08 M_⊙ in Eq. <ref> is about the lower mass limit of stars <cit.>. 
The mass conservation of the embedded cluster gives

    M_ecl=∫_0.08 M_⊙^m_maxm ξ_⋆(m) dm.

The optimal sampling normalization condition is

    1=∫_m_max^150 M_⊙ξ_⋆(m) dm,

where 150 M_⊙ is the adopted stellar upper mass limit <cit.>. When m_max > 1 M_⊙,
the optimal sampling normalization condition becomes

    1=∫_m_max^150 M_⊙ k_⋆ m^-α_3 dm.



For larger ρ_cl, ξ_⋆(m) becomes top-heavy where a α_3(ρ_cl) relation is adopted from <cit.>:

    α_3=
     
    2.3,    ρ_cl<9.5× 10^4, 
    
    1.86-0.43log_10(ρ_cl/10^6),    ρ_cl≥ 9.5× 10^4.

Here 

    ρ_cl=3M_cl/4π r_ h^3

in the unit of [M_⊙/pc^3] is the clump density when the embedded cluster is forming.

For M_cl and r_ h in the equation, we take the clump's mass and radius calculated in <cit.>.
As shown in Fig.<ref>, for ATLASGAL clumps with HII regions (HII-clumps), generally, they have ρ_cl<9.5× 10^4. Thus, we take α_3=2.3 in this work. 

Setting x ≡ m_max and solving the above equations, we obtain

    M_ecl=∫_0.08 M_⊙^xm ξ_⋆(m) dm
       
    =∫_0.08 M_⊙^0.5 M_⊙m 2k_⋆ m^-1.3 dm+∫_0.5 M_⊙^xm k_⋆ m^-2.3 dm
    
       ≈5.37/0.77/x^1.3 - 0.001 - 3.33/x^0.3(0.77/x^1.3 - 0.001)




§ ROBUST OF THE MODEL




