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Issue
A&A
Volume 611, March 2018
Article Number A6
Number of page(s) 17
Section Interstellar and circumstellar matter
DOI https://doi.org/10.1051/0004-6361/201732168
Published online 15 March 2018

© ESO 2018

1 Introduction

In the Galactic disk, star formation appears to occur only in dense regions (spatial density n(H2) ≳ 104 cm−3) composed ofmolecular gas (Lada et al. 2010; Ginsburg et al. 2015). High-mass stars form in massive clumps with typical size of order ~1 pc (e.g. Dunham et al. 2010, 2011; Rosolowsky et al. 2010; Urquhart et al. 2014; He et al. 2015; Wienen et al. 2015; König et al. 2017; Yuan et al. 2017). High-mass stars influence the surrounding environment and subsequent star formation through their feedback such as outflows, winds, and UV radiation. However, the details of high-mass star formation process and how their feedback may affect the initial conditions of high-mass stars in their formation process are still far from clear and require, as a basis, the precise determination of kinetic temperature and density.

The Atacama Pathfinder EXperiment (APEX) Telescope Large Area Surveyof the GALaxy (ATLASGAL; Schuller et al. 2009), presenting observations in a Galactic longitude and latitude range of ±60∘ and ±1.5∘, respectively,introduces a global view on star formation at 870 μm and identifies ~10 000 massive clumps in various stages of evolution undergoing high-mass star formation in the inner Galaxy (Contreras et al. 2013; Urquhart et al. 2014, 2018; Csengeri et al. 2014). The most fundamental physical parameters, kinetic temperature and spatial density of the clumps, affect chemistry, star formation, and could also impact the stellar initial mass function. Accurate measurements of these physical parameters are indispensable for a general understanding of the physical processes involved in these massive star-forming clumps.

Formaldehyde (H2CO) is a ubiquitous molecule in interstellar clouds (Downes et al. 1980; Bieging et al. 1982; Henkel et al. 1991; Zylka et al. 1992; Mangum et al. 2008, 2013a; Ao et al. 2013; Tang et al. 2013; Ginsburg et al. 2015, 2016, 2017; Guo et al. 2016). As a slightly asymmetric rotor molecule, H2 CO exhibits a large number of millimetre and submillimetre transitions. This molecule is a reliable tracer of physical conditions such as temperature and density (Henkel et al. 1980, 1983; Mangum & Wootten 1993; Mühle et al. 2007; Ginsburg et al. 2011, 2015, 2016; Ao et al. 2013). Since the relative populations of the Ka ladders of H2 CO are predominantly governed by collisions, ratios of H2 CO line fluxes involving different Ka ladders are good tracers of the kinetic temperature, such as para-H2 CO JKaKc = 322–221/303–202, 423 –322/404–303, and 523 –422/505–404 (Mangum & Wootten 1993). Once the kinetic temperature is known, line ratios involving the same Ka ladders yield estimates of the spatial density of the gas, such as JKaKc = 404–303/303–202, 505 –404/303–202, and 524 –423/322–221 (Mangum & Wootten 1993; Mühle et al. 2007; Immer et al. 2016). Transitions connecting the same rotational levels (e.g. J = 3–2 or 4–3) and belonging to either the para- or ortho-H2CO subspecies, but being part of different Ka ladders (e.g. Ka = 0, 2) are particularly useful. These transitions can be measured simultaneously with the same receiver system and their relative strengths (para-H2CO 322–221/303–202, 321–220/303–202, 423–322/404–303, and 422–321/404–303) provide sensitive thermometry. Para-H2CO is therefore possibly the best of the very few molecular tracers that are available for such an analysis of the dense molecular gas. H2CO line ratios have been used to measure physical parameters in our Galactic centre clouds (Qin et al. 2008; Ao et al. 2013; Johnston et al. 2014; Ginsburg et al. 2016; Immer et al. 2016; Lu et al. 2017), star formation regions (Mangum & Wootten 1993; Hurt et al. 1996; Mangum et al. 1999; Mitchell et al. 2001; Watanabe & Mitchell 2008; Nagy et al. 2012; Lindberg et al. 2015; Tang et al. 2017a, 2018), and in external galaxies (Mühle et al. 2007; Tang et al. 2017b).

In this work, we aim to directly measure the kinetic temperature and spatial density towards massive star-forming clumps selected from the ATLASGAL survey making use of the rotational transitions of H2CO (J = 3–2 and 4–3). Our main goals are (a) comparing kinetic temperatures from the gas to temperature estimates based on the dust; (b) searching for a correlation between kinetic temperature and line width, which is expected in the case of conversion of turbulent energy into heat; (c) seeking links between kinetic temperature and star formation rate (SFR) and the evolutionary stage of the massive star-forming regions; and (d) testing the star formation law by correlating the luminosity of the H2CO lines to infrared luminosity.

In Sects. 2 and 3, we describe the measured samples, our H2CO observations, and the data reduction, and introduce the main results. The discussion is presented in Sect. 4. Our main conclusions are summarized in Sect. 5.

2 Sample, observations, and data reduction

We selected the 110 brightest clumps from the ATLASGAL survey (the TOP100 sample) obeying simple IR criteria to cover a range in evolutionary stages as described in Giannetti et al. (2014) and König et al. (2017). These clumps consist almost entirely of clumps that have the potential to form, or are forming, massive stars. Depending on their IR and radio continuum properties, the sample of potentially high-mass star-forming clumps at various evolutionarystages can be separated into four categories: 70 μm weak sources (70w), infrared weak clumps (IRw), infrared bright objects (IRb), and sources containing compact H II regions (H II) (Giannetti et al. 2014; König et al. 2017). Previous work on this sample addressed SiO emission (for parts of the sample, Csengeri et al. 2016), dust continuum characterization (König et al. 2017), millimetre hydrogen recombination lines (for more evolved (i.e. H II regions) parts of the sample; Kim et al. 2017), and temperature structure (Giannetti et al. 2017). The TOP100 is an ideal sampleto study the physical and chemical parameters of the potentially massive star-forming regions at various evolutionarystages.

Sources observed are listed in Table A.1. Our observations were carriedout on 2013 July and December, 2014 September and November, and 2015 April, June, July, and October with the APEX2 12 m telescope located on Chajnantor (Chile). Specific observational details of the 10 measured transitions of H2 CO are listed in Table 1. Five transitions of H2 CO (J = 3–2) were observed with the new MPIfR 1-mm receiver (PI230) with a beam size from 27.6″ to 29.5″ and integration times of 1–3 min. Five H2 CO (J = 4–3) transitions were observed with the FLASH receiver with a beam size ~21.4″ and integration times of 2–4 min. For the PI230 receiver, we used a fast fourier transform spectrometer (FFTS4G) backend with two sidebands (lower and upper). Each sideband has two spectral windows of 4 GHz bandwidth, providing both orthogonal polarizations and leading to a total bandwidth of 8 GHz. An eXtended bandwidth fast fourier transform spectrometer (XFFTS) backend with two spectral windows of 2.5 GHz bandwidth leading to a total bandwidth of 4 GHz was used for the FLASH receiver. These provided velocity resolutions of ~0.08 km s−1 for H2 CO (J = 3–2) and ~0.04 km s−1 for H2 CO (J = 4–3). The observations were performed in position-switching mode with off positions offset from the on position of the sources by (600″, ±600″). We converted the antenna temperatures of the spectra into main beam brightness temperatures for both H2 CO J = 3–2 and 4–3 lines using a factor of 1/0.69. Observed continuum of Mars, Jupiter, and Saturn were used to calibrate the spectral line flux. The calibration uncertainty is about 20%.

Data reduction of spectral lines was performed using CLASS from the GILDAS package3 . To enhance signal-to-noise ratios (S/N) in individual channels, we smoothed contiguous channels to a velocity resolution of ~0.6 km s−1. The line widths tend to be > few km s−1 , so the smoothing has no impact on our results. The typical noise level is ~0.06 K (Tmb scale) for both H2CO (J = 3–2) and H2 CO (J = 4–3) at a velocity resolution of ~0.6 km s−1.

Table 1

Observed H2CO transition parameters.

3 Results

3.1 Overview

Ninety-four sources with H2CO (J = 3–2) transitions and 98 sources with H2CO (J = 4–3) transitions were observed. Nearly all H2CO lines are detected (detection rate ≳ 97%) for the upper energy above ground state, Eu, ( <35 K) towards the targeted massive clumps (see Table 2). For high Eu (>82 K), the H2CO detection rate ranges from 82% to 85%. Non-detections are associated with 70w and IRw sources (see Sect. 2 for the definitions), which are typically associated with the early cold evolutionary stages of massive clumps. Two para-H2 CO (322–221 and 321 –220) transitions (Eu ~ 68 K) show a lower detection rate (~70%), which is caused by the fact that para-H2 CO is the less abundant of the two H2 CO symmetry species and the source of the weaker K = 2 transitions. High detection rates of H2 CO indicate that this species is commonly formed in massive star-forming clumps and is present during all their evolutionary stages.

Examples of H2 CO line spectra are presented in Fig. 1. Line parameters are listed in Tables A.2–A.5, where velocity-integrated intensity, ∫Tmbdv, local standard of rest velocity, V lsr, full width to half maximum line width (FWHM), and peak main beam brightness temperature, Tmb, were obtained from Gaussian fits. The rest frequencies of the ortho-H2CO 432–331 and 431–330 transitions are nearby (see Table 1 and Fig. 1). These two lines are blended in all of our sources, so that Gaussian fits are of limited value and are not part of our tables.

Table 2

Observed H2CO transitions and detection rates.

thumbnail Fig. 1

Observed H2CO spectra (grey) towards AGAL008.684−00.367. Green lines indicate the Gaussian fit results.

3.2 Source size correction

The para-H2CO J = 3–2 (beam size ~ 28.6″) and 4–3 (beam size ~ 21.5″) lines we observed were obtained by single pointing observations with different receivers, so the area covered by our J = 3–2 and 4–3 transitions is slightly different. We compare the integrated intensities of H2 CO, irrespective of the beam size with 870 μm flux densities in Fig. 2. This comparison shows that the H2 CO integrated intensities follow the 870 μm intensity distribution. Apparently dense gas traced by H2 CO is associated well with the dust traced by 870 μm emission in the massive star-forming clumps. Mapping observations of para-H2 CO (303–202, 322 –221, and 321 –220) towards the Orion molecular cloud 1 (OMC1) with the APEX telescope also show that para-H2 CO integrated intensity distributions agree well with the dust emission observed at 850 μm (Johnstone & Bally 1999; Tang et al. 2018). Previous observations of H2 CO (404–303, 423 –322, 422 –321, 432 –331, and 431 –330) towards massive clumps in the W33 region with the APEX telescope (Immer et al. 2014) also indicate that H2 CO distributions are consistent with the dust emission traced by 870 μm. Hence, we assume that the source sizes of H2 CO are the same as the full width to half power source sizes of the 870 μm dust emission derived from Csengeri et al. (2014). We correct for beam dilution by calculating Tmb′$T_{\rm mb}^{\prime}$ = Tmb/ηbf with beam-filling factor ηbf = θs2$\theta_{\rm s}^2$/( θs2$\theta_{\rm s}^2$+ θbeam2$\theta_{\rm beam}^2$). Here θbeam and θs denote beam and source size, respectively. The results of ηbf and the para-H2 CO 404–303/303–202 integrated intensity ratio (I′ (404–303)/I′(303–202)) corrected with ηbf are listed in Table A.6.

thumbnail Fig. 2

Comparison of integrated intensities of H2 CO and 870 μm continuum flux densities. The solid line corresponds to Y = X in the given units.

3.3 Opacities of H2CO

To determine the gas kinetic temperatures, Tkin , spatial densities, n(H2), and para-H2COcolumn densities, N(H2CO), we used the RADEX non-LTE model (van der Tak et al. 2007) offline code4 with collision rates from Wiesenfeld & Faure (2013). Uncertainties in the collisional excitation rates directly affect the derived volume densities, while kinetic temperature appears to be less affected by collisional excitation rate uncertainties (see Sect. 3.4). The RADEX code needs five input parameters: background temperature, kinetic temperature, H2 density, H2CO column density, and line width. For the background temperature, we adopted 2.73 K. Model grids for the H2CO lines encompass 40 densities (n(H2) = 104–108 cm−3), 40 H2CO column densities (N(H2CO) = 1012–1016 cm−2), and 40 temperatures ranging from 10 to 400 K. For the line width, we used the observed line width value.

The value of N(para-H2CO) depends on para-H2CO 303–202 and/or 404–303 integrated intensities and the para-H2CO 404–303/303–202 ratio (Mangum & Wootten 1993; Tang et al. 2017a). If the para-H2CO 303–202 and 404–303 lines are optically thick in our dense massive clumps, this would cause high para-H2CO 404–303/303–202, 321–220/303–202, and 422–321/404–303 ratios. Higher ratios imply higher spatial densities and kinetic temperatures, respectively (Mangum & Wootten 1993; Ao et al. 2013; Ginsburg et al. 2016; Immer et al. 2016; Tang et al. 2017a,b, 2018). In order to understand the impact of the line optical depth, we modelled the optical depth of para-H2CO 303–202 and para-H2CO 404–303 integrated intensities, and the para-H2CO 404–303/303–202 ratio at a kinetic temperature of 55 K (see Sect. 3.4) in Fig. 3 (or see Fig. G.2 in Immer et al. 2016). Changing the kinetic temperature, weakly affects the optical depth of the para-H2CO 303–202 and para-H2CO 404–303 lines (less than by a factor of few). The para-H2CO 404–303/303–202 ratio is then also not greatly changed ( ≲30%; not shown here). The figure demonstrates that para-H2CO 303–202 is optically thin (τ < 1) at column density N(para-H2CO) < 1×1014 cm−2 and spatialdensity 104−8 cm−3. At higher column density (N(para-H2CO) > 5×1014 cm−2), the para-H2 CO 303–202 becomes optically thick (τ > 5). The optical depth of para-H2 CO 404–303 shows a similar behaviour (slightly lower values) with respect to that of para-H2 CO 303–202 (not shown here; or see Fig. G.2 in Immer et al. 2016). Considering the observed ranges of integrated intensities of para-H2 CO 404–303 (typical value ~20 K km s−1) and para-H2 CO 404–303/303–202 ratios (typical value ~1.0) accounting for relevant beam-filling factors from Sect. 3.2 (see Tables A.4 and A.6), the optical depths of para-H2 CO 303–202 and 404 –303 range from ~0.012 to ~1 in our sample. Compared to the para-H2 CO 303–202 and 404 –303 lines, para-H2 CO 322–221, 321 –220, 423 –322, and 422 –321 lines have higher upper energies above the ground state (Eu > 68 K, see Table 1), so they have lower optical depths (τ ≪ 1). Therefore, the influence of the para-H2 CO 303–202 and 404 –303 optical depths is weak for our determination of spatial density and kinetic temperature.

In our sample the observed Tmb (312–211/303–202) ratios range from 0.74 to 1.83 with an unweighted average of 1.29 ± 0.02 (see Tables A.2 and A.3; errors given here and elsewhere are standard deviations of the mean). For the Tmb (313–212/303–202) ratio, it ranges from 1.03 to 2.35 with an unweighted average of 1.56 ± 0.03 (see Tables A.2 and A.3). The relation between Tmb (312–211/303–202) and H2 CO optical depth, indicated by Sasselov & Rucinski (1990) in their Fig. 2, suggests that for at least 30% of our sample (Tmb (312–211/303–202) ≲ 1.19) the ortho-H2 CO 312–211 and 313 –212 lines are optically thick (τ ≳ 5).

thumbnail Fig. 3

Example of RADEX non-LTE modelling of the N(H2CO)–n(H2) relation for AGAL008.684−00.367 at a kinetic temperature of 55 K (see Sect. 3.4). Black dashed and solid lines are para-H2 CO 404–303 integrated intensities and para-H2CO 404–303/303–202 integrated intensity ratios, respectively. Tothe measured parameters, para-H2 CO 404–303 integrated intensity (orange solid and dashed lines represent observed value and uncertainty) and para-H2 CO 404–303/303–202 integrated intensity ratio (white solid and dashed lines) are corrected by the relevant beam-filling factors (see Table A.6). The colour map shows the optical depth of the para-H2 CO 303–202 line. The purple line in the upper green area corresponds to optical depth τ(para-H2CO 303–202) = 1.0.

3.4 Kinetic temperature

As discussed in Sect. 1, the intensity ratios of H2 CO lines involving different Ka ladders yield estimates of the kinetic temperature of the gas (Mangum & Wootten 1993). For our observed transitions of H2 CO, para-H2 CO 321–220/303–202, 322 –221/303–202, 422 –321/404–303, and 423 –322/404–303 ratios can be useful thermometers to derive the kinetic temperature. Para-H2 CO 322–221/303–202 and 321 –220/303–202 ratios trace the kinetic temperature with an uncertainty of ≲25% below 50 K (Mangum & Wootten 1993). Para-H2 CO 422–321/404–303 and 423 –322/404–303 ratios trace the kinetic temperature with an uncertainty of ≲25% below 75 K (Mangum & Wootten 1993). The para-H2 CO 322–221/303–202 and 423 –322/404–303 line ratios are slightly affected by the spatial density (not shown here; for para-H2 CO 322–221/303–202 see Lindberg et al. 2015 and Tang et al. 2017a). Therefore in this work we use the para-H2 CO 321–220/303–202 and 422 –321/404–303 integrated intensity ratios to derive the kinetic temperature, which also have been used for the Galactic central molecular zone (CMZ) clouds (Ginsburg et al. 2016; Immer et al. 2016).

We ran RADEX to calculate the observed para-H2 CO 321–220/303–202, 422 –321/404–303, and 404 –303/303–202 integrated intensity ratios corrected by the relevant beam-filling factors assuming these transitions of para-H2 CO are optically thin (see Sect. 3.3). In Fig. 4, an example is presented to show how the parameters are constrained by the line ratio distribution of para-H2 CO, accounting for different beam-filling factors in the Tkin –n(H2)parameter space. We used the column density derived from the para-H2CO 404–303 integrated intensity and para-H2CO 404–303/303–202 ratio accounting for the beam-filling factors derived in Sect. 3.2 to constrain the kinetic temperature. Figure 4 shows that para-H2CO 321–220/303–202 and 422–321/404–303 line ratios are sensitive to the gas kinetic temperature (see the black solid lines in Fig. 4), while being relatively independent of spatial density. The integrated intensity ratio I′ (404–303)/I′(303–202) is sensitive to the gas spatial density at high temperature (Tkin > 40 K), where the I′ (404–303)/I′(303–202) ratio becomes relatively independent of kinetic temperature. At low temperature (Tkin < 40 K), this ratio is influenced almost entirely by the gas kinetic temperature because Tkin becomes lower than the excitation difference of the involved states. Therefore, para-H2 CO 321–220/303–202 and 422 –321/404–303 ratios combined with the para-H2 CO 404 –303/303–202 ratio are good tracers to constrain kinetic temperature and spatial density of dense gas in warm regions (gas temperature >30 K, for lower Tkin the levels of the J = 3–2 and 4–3 Ka > 0 lines are too far above the ground state) of massive star-forming clumps. Because the two J = 4, Ka = 2 levels are located 80–90 K above the ground state (see Table 1), the radiative transfer models start to become insensitive to temperatures in excess of 150 K. Hence, temperatures >150 K have to be considered sceptically and should cautiously be interpreted as ≳150 K (Mangum & Wootten 1993; Ginsburg et al. 2016; Immer et al. 2016). The derived kinetic temperatures are listed in Table A.6.

We note a para-H2CO 321–220/303–202 ratio bump at kinetic temperature >50 K and spatial density 105.5−7.0 cm−3 in Fig. 4 (or see Fig. 13 in Mangum & Wootten (1993) and Fig. F.1 in Lindberg et al. (2015)) because the excitation temperature of the para-H2CO 321–220 line rises much faster than that of the para-H2CO 303–202 line with increasing spatial density and/or kinetic temperature (Mangum & Wootten 1993). Kinetic temperatures obtained for a given para-H2CO 321–220/303–202 ratio vary more than by ≳20% at kinetic temperature >60 K and spatial density 105.0−7.0 cm−3. This large “bump” in the para-H2CO 321–220/303–202 contour (see Fig. 4 upper panel) probably leads to an overestimate of the kinetic temperature from the para-H2CO 321–220/303–202 ratio. Para-H2CO 422–321/404–303 is also influenced by a bump, this time at kinetic temperature >100 K and spatial density 105.5−7.0 cm−3 (see Fig. 4 lower panel or Fig. 13 in Mangum & Wootten 1993). Kinetic temperatures derived from the para-H2CO 422–321/404–303 ratio vary less than ≲20% for Tkin < 150 K and spatial density 105.0−7.0 cm−3. It appears that the para-H2CO 422–321/404–303 ratio is more stable and accurate to trace gas kinetic temperature than the para-H2CO 321–220/303–202 ratio at Tkin < 150 K and spatial density 105.0−7.0 cm−3.

A comparison of kinetic temperatures derived from both para-H2 CO 321–220/303–202 and 422 –321/404–303 ratios suggests that the two ratios trace similar temperatures (see Fig. 5). It might have been expected, for example by analogy to NH3 (e.g. Henkel et al. 1987; Mangum et al. 2013a; Gong et al. 2015a,b), that higher excited H2 CO transitions lead to higher Tkin values. Some of the similar kinetic temperatures derived from the para-H2 CO 321–220/303–202 and 422 –321/404–303 ratios (Fig. 5) might be caused by the para-H2 CO 321–220/303–202 ratio bump (Fig. 4, top panel). This excitation effect in the para-H2 CO 321–220/303–202 ratio may result in an overestimate of the kinetic temperaturederived from this ratio with large uncertainty ( ≳20% at kinetic temperature >60 K) at spatial density 105.5−7.0 cm−3.

The para-H2CO line intensity ratios 322 –221/303–202, 321 –220/303–202, 423 –322/404–303 and 422 –321/404–303 can also provide a measurement of the kinetic temperature of the gas assuming local thermodynamic equilibrium (LTE). The kinetic temperature can be calculated from these para-H2 CO transition ratios if the lines are optically thin (see Sect. 3.3) and originate from a high density region (Mangum & Wootten 1993). Following the method applied by Mangum & Wootten (1993) in their Appendix A, TLTE=47.1ln(0.556I(303-202)I(321-220)) K\begin{equation} T_{\rm LTE}=\frac{47.1}{\ln \left(0.556\frac{I(3_{03}-2_{02})}{I(3_{21}-2_{20})}\right)}\,\,\rm K \end{equation}(1)

and TLTE=47.2ln(0.750I(404-303)I(422-321)) K,\begin{equation} T_{\rm LTE}=\frac{47.2}{\ln \left(0.750\frac{I(4_{04}-3_{03})}{I(4_{22}-3_{21})}\right)}\,\,\rm K, \end{equation}(2) where I(303–202)/I(321–220) and I(404–303)/I(422–321) are the para-H2CO integrated intensity ratios. The results of the kinetic temperature calculations from the para-H2CO 303–202/321–220 and 404–303/422–321 integrated intensity ratios are listed in Table A.6. If the assumption of optically thin emission is correct, the kinetic temperatures derived from this method have an uncertainty of ≲30% (Mangum & Wootten 1993). We also compared the kinetic temperatures derived from LTE and RADEX non-LTE calculations (see Fig. 5). It appears that Tnon-LTE is consistently higher than TLTE by ≲25%. This might be caused by the fact that at densities of 106.5 cm−3 (see Sect. 3.5) thermalization is not yet reached (Mangum & Wootten 1993). Therefore, higher Tkin values are needed to compensate for this effect, leading to lower excitation temperatures, and to reproduce data.

thumbnail Fig. 4

Example of RADEX non-LTE modelling of the para-H2 CO kinetic temperature for AGAL008.684−00.367. Black solid and dashed lines are para-H2 CO integrated intensity ratios. Para-H2 CO 404–303/303–202 (blue solid and dashed lines represent observed value and uncertainty, accounting for different beam-filling factors), 321 –220/303–202 and 422 –321/404–303 integrated intensity ratios (top and bottom, red solid and dashed lines) for a para-H2 CO column density 2.8 × 1013 cm−2 are derived from the para-H2CO 404–303 integrated intensity and para-H2CO 404–303/303–202 ratio (see Sect. 3.3).
thumbnail Fig. 5

Top panel: comparison of kinetic temperatures derived from para-H2 CO 321–220/303–202 and 422 –321/404–303 ratios. The dashed line is the result from an unweighed linear fit, Tkin (4_22-3_21/4_04-3_03) = (0.9±0.1)×Tkin(3_21-2_20/3_03-2_02)+(5.8±7.8), with a correlation coefficient, R, of 0.85. Middle and bottom panels: comparisons of kinetic temperatures derived from LTE and RADEX non-LTE calculations for para-H2 CO 321–220/303–202 and 422 –321/404–303 ratios, respectively. The temperature uncertainties are obtained from observed para-H2 CO line ratio errors. Solid lines indicate equal temperatures.

3.5 Spatial density and column density

As described in Sect. 1, with the kinetic temperatureapproximately known, the relative intensity ratio of H2 CO lines involving the same Ka ladders yields estimates of the spatial density of the gas (Henkel et al. 1980, 1983; Mangum & Wootten 1993). For our observed transitions of H2 CO, para-H2 CO 404–303/303–202, 422 –321/321–220 (or 422 –321/322–221), and 423 –322/322–221 (or 423 –322/321–220) ratios are good densitometers to derive the spatial density. The para-H2 CO 303 –202 and 404 –303 lines are thestrongest of the 218 GHz and 291 GHz transitions, respectively,and they are nearly all detected in our sample (see Table 2). Hence, we use the para-H2 CO 404–303/303–202 integrated intensity ratio to derive the spatial density, which has also been used in molecular clouds of the Galactic CMZ (Immer et al. 2016).

We ran RADEX to obtain para-H2 CO column densities and spatial density, and calculated the observed para-H2 CO 303–202 and 404 –303 integrated intensities in K km s−1 units corrected by the relevant beam-filling factors (I →I′). In Fig. 3, an example is presented to show how the parameters are constrained by the corrected integrated line intensity and integrated line intensity ratio distribution of para-H2 CO in the N(para-H2CO)–n(H2) parameter space. This figure shows that at low column density (N(para-H2CO) < 5 × 1014 cm−2) the I′ (404–303)/I′(303–202) ratio accounting for different beam-filling factors (see the black solid lines) is sensitive to the gas spatial density and becomes relatively independent of the para-H2 CO column density, while the kinetic temperature is kept constant at ~55 K (which is close to the actual temperature; see above). At high column density (N(para-H2CO) > 5 × 1014 cm−2) the I′ (404–303)/I′(303–202) ratio does not appear to be sensitive to the gas spatial density and becomes dependent on the column density because the para-H2CO 303–202 transition starts to become optically thick (Mangum & Wootten 1993). The derived results of N(para-H2CO) and spatial density are listed in Table A.6. We used the same method to obtain ortho-H2CO column densities with the observed ortho-H2CO (312–211 and 313–212) integrated intensities, adopting kinetic temperature and spatial density derived from para-H2CO line ratios (see Sect. 3.4 and above) and assuming ortho- and para-H2CO originate from the same region. The obtained results of N(ortho-H2CO) are listed in Table A.6.

As mentioned in Sect. 3.3, the para-H2CO 322–221, 321–220, 423–322, and 422–321 lines are optically thin, so the para-H2CO 422–321/321–220 (or 423–322/322–221) ratio is weakly affected by optical depths. To further check how optical depths influence the para-H2CO 404–303/303–202 ratio, we used the above method with the para-H2CO 422–321/321–220 ratio as well to constrain spatial density. The spatial densities obtained both from para-H2CO 404–303/303–202 (typical value ~1.0) and 422–321/321–220 (typical value ~1.5) ratios yield similar values (n(H2) ~ 2 × 106 cm−3), which confirms that para-H2CO 303–202 and 404–303 lines are not strongly affected by saturation effects when trying to constrain spatial density and kinetic temperature in our sample. However, the ortho-H2CO 312–211 and 313–212 lines are affected by opacities ≳ 1 in parts of our sample (see Sect. 3.3), so the N(ortho-H2CO) may be underestimated in these sources.

The 870 μm continuum source angular sizes range from 22″ to 42″ with an average of 29″ in our sample. If the sizes of H2 CO are much smaller than those of the 870 μm continuum (and/or our beam size; see Table 1), the beam-filling factor is overestimated. If we assume that the H2 CO to 870 μm emission size ratio (θH_2CO/θ870 μm) is 90%, 80%, and 70%, for AGAL008.684−00.367 as an example (see Fig. 3), n(H2) decreases by 6%, 17%, and 23%. Mapping observations of massive clumps in CS (7-6) (Wu et al. 2010) and 350 μm continuum emission (Mueller et al. 2002) show that the median ratio of CS (7-6) emission size to the 350 μm continuum emission sizeis ~0.87 (Liu et al. 2016). Also considering the slightly different beam sizes for para-H2 CO 303–202 and 404 –303 lines (see Sect. 3.2), we conclude that the beam-filling factor does not strongly influence our results for n(H2) constrained from para-H2 CO 404–303/303–202 line ratios.

The statistical weight ratio of ortho- and para-H2 CO and previous H2CO observations in other star-forming regions suggest that the ortho-to-para H2 CO abundance ratio is ≲3 (Kahane et al. 1984; Mangum & Wootten 1993; Dickens & Irvine 1999; J∅rgensen et al. 2005; Guzmán et al. 2011). In most of our sample ( ~95%) the obtained ortho-to-para H2 CO abundance ratios (N(ortho-H2CO)/N(para-H2CO)) range from 1.0 to 3.0 with an unweighted average of 2.0 ± 0.1. Assuming that H2 CO is formed in and expelled from dust grain mantles, this ratio corresponds to a dust temperature of ≲20 K (Kahane et al. 1984; Dickens & Irvine 1999).

4 Discussion

4.1 Variations of spatial density and H2CO abundance

The gas spatial densities, n(H2), derived from para-H2CO 404–303/303–202 ratios, range from 6.3 × 105 to 8.3 × 106 cm−3 with an unweighted average of 1.5 (±0.1) × 106 cm−3 (Table A.6); these values agree with the results determined with para-H2CO (505–404/303–202 and 524–423/322–221) and ortho-H2CO (413–414/312–313) ratios from other star-forming regions (Mangum & Wootten 1993; Hurt et al. 1996; McCauley et al. 2011; Lindberg et al. 2015). Mapping the same para-H2CO transitions towards the Galactic CMZ clouds shows that the spatial density of the widespread warm gas is constrained to 104–106 cm−3 (Immer et al. 2016). The spatial densities derived from para-H2CO line ratios in our massive clumps overlap with the values found for high density regions in the Galactic CMZ clouds (see Table A.6 or Fig. 6, and Fig. 3 in Immer et al. 2016). The spatial density deduced from the dust indicates 103–106 cm−3 in our sample (Giannetti et al. 2017), which is lower than the spatial densities that we obtained. This suggests that H2CO (J = 3–2 and 4–3) traces denser gas than the dust emission.

We derived unweighted averaged spatial densities obtained from para-H2CO ratios in sources representing four evolutionary stages consisting of 70 μm weak (70w), mid-infrared weak (IRw), and mid-infrared bright (IRb) sources as well as star-forming clouds with ultra-compact H II regions. The unweighted averaged spatial densities n(H2) are 1.2 (±0.2) × 106, 1.7 (±0.4) × 106, 1.2 (±0.1) × 106 cm−3, and 1.8 (±0.4) × 106 cm−3 in 70w, IRw, IRb, and H II regions, respectively (see Table 3 or Fig. 7). It seems that the averaged spatial densities traced by the para-H2CO 404–303/303–202 ratios do not vary significantly with the evolutionary stage of clumps. This may indicate that the density structure does not evolve significantly as the star formation proceeds. It also suggests that the para-H2CO 404–303/303–202 ratio may be a good densitometer to trace the dense gas at various stages of massive star formation.

The N(para-H2CO) value derived from the para-H2 CO 404 –303/303–202 ratio ranges from 6.4 × 1012 to 6.1 × 1014 cm−2 with an unweighted average of 8.0 (±1.3) × 1013 cm−2 (Table 3), which agrees with the results from other protostellar cores and star-forming regions (Mangum & Wootten 1993; Hurt et al. 1996; Watanabe & Mitchell 2008; Tang et al. 2017a). We also derive averaged column densities of para-H2 CO for the fourevolutionary stages mentioned above. The unweighted average column densities N(para-H2CO) are 1.2 (±0.6) × 1013, 2.7 (±0.4) × 1013, 7.2 (±2.1) × 1013, and 16.9 (±0.3) × 1013 cm−2 in 70w, IRw, IRb, and H II regions, respectively (see Table 3). The fractional abundance X(para-H2CO) = N(para-H2CO)/N(H2) becomes 1.0 × 10−10–1.2 × 10−9 with an average of 3.9 (±0.2) × 10−10, where N(H2) is derived from the 870 μm continuum emission assuming a dust absorption coefficient κ870 = 1.85 cm2 g−1 at 870 μm and adopting the temperature obtained from the dust (König et al. 2017). Therefore the abundance also agrees with the values found in other star formation regions, Galactic centre clouds, and external galaxies (Güsten & Henkel 1983; Zylka et al. 1992; Ao et al. 2013; Gerner et al. 2014; Tang et al. 2017a,b). The unweighted average fractional abundances X(para-H2CO) are 1.8 (±0.7) × 10−10, 2.7 (±0.2) × 10−10, 4.3 (±0.4) × 10−10, and 4.9 (±0.6) × 10−10 in 70w, IRw, IRb, and H II regions, respectively (see Table 3). Averaged variations of fractional abundances of X(para-H2CO) in various stages of star formation amount to nearly a factor of 3, which agrees with observed results in other massive star formation regions (van der Tak et al. 2000a,b; Gerner et al. 2014; Tang et al. 2017a). Therefore, we confirm that H2 CO can be widely used as a probe to trace the dense gas without drastic changes in abundance during various stages of star formation.

The column densities of para-H2CO and the fractional abundances of X(para-H2CO) with corresponding H2 column density, spatial density n(H2), kinetic temperature Tkin(para-H2CO 422–321/404–303), bolometric luminosity, clump mass, and luminosity-to-mass (Lbol/Mclump) ratio are shown in Fig. 6. It is apparent that the para-H2CO column density increases proportionally to the H2 column density, gas kinetic temperature, bolometric luminosity, and Lbol/Mclump ratio in the massive clumps. The fractional abundance of X(para-H2CO) remains stable with increasing H2 column density, spatial density, and mass of clump (Fig. 6). Nevertheless, the scatter in X(para-H2CO) amounts to 0.1–1.2 × 10−9, i.e. to a factor of ~10. The stable (relative to other molecular species; e.g. Tang et al. 2017b) para-H2CO fractional abundances as a function of N(H2) indicate that H2CO is a reliable tracer of the H2 column density.

The luminosity-to-mass ratio is a good evolutionary tracer for massive and dense cluster-progenitor clumps (Molinari et al. 2008, 2016; Liu et al. 2013; Ma et al. 2013; Giannetti et al. 2017). The fractional abundance of X(para-H2CO) shows a weak increasing trend with kinetic temperature, bolometric luminosity, and Lbol/Mclump ratio (see Fig. 6). The H2CO abundances seem to increase with the evolutionary stage of massive clumps. Similar trends were seen in the massive star formation regions studied by Gerner et al. (2014) and Immer et al. (2014). This indicates that H2CO abundances may be enhanced by high temperature, infrared radiation, and clump evolution, which would support a scenario in which H2CO is increasingly released from dust grains into the gas phase during the evolution of the star-forming region.

thumbnail Fig. 6

Column density N(para-H2CO) and fractional abundance X(para-H2CO) vs. column density N(H2) (a, b), spatial density n(H2) (c, d), kinetic temperature Tkin(para-H2CO 422–321/404–303) (e, f), bolometric luminosity (g, h), mass of clump (i, j), and luminosity-to-mass Lbol/Mclump ratio (k, l). The column density and spatial density uncertainties are obtained from observed para-H2 CO line brightness temperature and line ratio errors. The straight lines are the results from unweighed linear fits yielding the given correlation coefficients, R, in the lower right corner of each panel.
Table 3

Averaged parameters in various stages of the massive clumps.

thumbnail Fig. 7

Spatial density derived from para-H2 CO (404–303/303–202) vs. luminosity-to-mass ratio Lbol /Mclump. The dashed line indicates the average spatial density.

4.2 Comparison of kinetic temperatures derived from gas and dust

The gas kinetic temperatures derived from the para-H2 CO (321 –220/303–202 and 422 –321/404–303) line ratios are rather warm, ranging from 43 to >300 K with an unweighted average of 91 ± 4 K; these valuesagree with the results measured with H2 CO in other massive star-forming regions and Galactic centre clouds (Mangum & Wootten 1993; Hurt et al. 1996; Mangum et al. 1999; Watanabe & Mitchell 2008; Nagy et al. 2012; Ao et al. 2013; Ginsburg et al. 2016; Immer et al. 2016; Lu et al. 2017). Most of our clumps, including the detected 70 μm weak clumps, are very warm, which indicates that there is likely ongoing massive star formation in most of our sample. The average kinetic temperatures Tkin are high in early evolutionary stages of the clumps (70w and IRw) (see Table 3), which is consistent with previous observational results measured with para-H2 CO (3–2) in star-forming regions with outflows (Tang et al. 2017a). Sixteen sources of our sample in early evolutionary stages have been observed in SiO (2–1) and (5–4) (Csengeri et al. 2016) SiO emission is detected in all these sources. This indicates that the dense gas probed by H2 CO may be heated by an outflow or shock. Therefore, in early evolutionary stages of the clumps, para-H2 CO traces higher temperature gas that may be related to gas excited by star formation activities (e.g. outflows, shocks) (Tang et al. 2017a).

Parts of our sample have been measured in NH3 (2, 2)/(1, 1) by Wienen et al. (2012). We compare gas kinetic temperatures derived from para-H2 CO and NH3 (2, 2)/(1, 1) against dust temperatures in Fig. 8. This comparison shows that the gas temperatures determined from NH3 (2, 2)/(1, 1) agree with the dust temperatures (also see Giannetti et al. 2017), but are lower than those derived from para-H2 CO (321–220/303–202 and 422 –321/404–303). Previous observations towards the Galactic CMZ, dense massive clumps, and star formation regions indicate that in many cases para-H2 CO (321–220/303–202 and 422 –321/404–303) traces a higher kinetic temperature than the NH3 (2, 2)/(1, 1) transitions and dust (Ao et al. 2013; Ott et al. 2014; Ginsburg et al. 2016; Immer et al. 2016; Tang et al. 2017a, 2018). The difference is likely because the derived kinetic temperatures from NH3 (2, 2)/(1, 1)may reflect an average temperature of cooler and more diffuse gas (Henkel et al. 1987; Ginsburg et al. 2016), while para-H2CO (J = 3–2 and 4–3) ratios trace denser and hotter regions more directly associated with star formation activity (Tang et al. 2017a, 2018).

Temperatures towards our selected massive clumps have been measured with CO, CH3OH, CH3CN, and CH3CCH (Giannetti et al. 2017). We compare gas kinetic temperatures derived from para-H2CO (422–321/404–303), CO, CH3OH, CH3CN, and CH3CCH in Fig. 8. It shows that the gas temperatures determined from para-H2CO are higher than those derived from CO, CH3OH (cold component), CH3CN (cold component), and CH3CCH, but are lower than those obtained from the CH3OH and CH3CN hot components. This indicates that para-H2CO (J = 3–2 and 4–3) ratios may trace dense gas in layers intermediate between those of CH3CCH and CH3CN (hot component) and the latter are likely most closely related to recently formed massive stars.

The dust temperatures of our sample are obtained from spectral energy distribution (SED) fitting to Herschel HiGal data at 70, 160, 250, 350, and 500 μm and ATLASGAL data at 870 μm by König et al. (2017). The results are listed in Table A.1. The derived dust temperature range in our observed sources is 11–41 K with an unweighted average of 25 ± 7 K. Previous observations show that the temperatures derived from gas and dust are often in agreement in the active dense clumps of Galactic disk clouds (Dunham et al. 2010; Giannetti et al. 2013; Battersby et al. 2014), but do not agree in the Galactic CMZ (Güsten et al. 1981; Ao et al. 2013; Ott et al. 2014; Ginsburg et al. 2016; Immer et al. 2016; Lu et al. 2017). As in the CMZ, the gas kinetic temperatures derived from para-H2CO show higher values than the dust temperature with no apparent correlation (correlation coefficient R ~ 0.2) between Tdust and Tgas (see Fig. 8).

It is commonly expected that the gas and dust are thermally coupled in the densest regions (n(H2) > 104.5 cm−3) (Goldsmith 2001) because at such densities interactions between dust and gas become sufficiently frequent. The dust emission at mid-infrared (MIR) emission traces primarily warm dust components (Helou 1986). Dust temperatures derived from MIR multi-filter data agree with gas temperatures derived from multi-inversion transitions of NH3 in external galaxies (Melo et al. 2002; Tomono et al. 2006; Ao et al. 2011; Mauersberger et al. 2003). Combining the MIR data for our sample, the fit of the warm gas emission in the SED shows a cold and a warm component (see König et al. 2017). Our dust temperatures are taken from the cold component of the SED fitted results. Dust emission at far-infrared (FIR) emission originates primarily from colder dust components that may not be directly associated with star formation activity (Schnee et al. 2009; Bendo et al. 2012; Mangum et al. 2013a), therefore the dust temperatures derived from FIR measurements rarely exceed 50 K in star formation regions of our Galaxy and external galaxies (e.g. Henkel et al. 1986; Gao & Solomon 2004a; Bernard et al. 2010; Mangum et al. 2013a; Guzmán et al. 2015; Merello et al. 2015; He et al. 2016; Lin et al. 2016; König et al. 2017; Yu & Xu 2016; Tang et al. 2017a; Elia et al. 2017). This suggests that the HiGal dust emission may trace colder dust components that may not be used as a proxy for dust and gas kinetic temperatures (at least traced by H2 CO) in dense regions with massive star formation activity.

thumbnail Fig. 8

Top panel: comparison of kinetic temperatures derived from para-H2 CO (321–220/303–202 and 422 –321/404–303, cyan and blue points) and NH3 (2, 2)/(1, 1) (black squares) ratios against the dust temperatures. NH3 kinetic temperatures are selected from Wienen et al. (2012). The cyan and blue straight lines are the results from unweighed linear fits for gas temperatures derived from para-H2 CO (321–220/303–202 and 422 –321/404–303, respectively. Bottom panel: comparisons of gas temperatures derived from para-H2 CO (422–321/404–303), CO, NH3 (2, 2)/(1, 1), CH3 OH, CH3 CN, and CH3 CCH. Temperatures of CO, CH3OH, CH3 CN, and CH3 CCH are taken from Giannetti et al. (2017). The black straight lines in both panels indicate equal temperatures.

4.3 Line width-luminosity relation

The observed line widths of para-H2 CO (404–303) range from 1.2 to 12.8 km s−1 with an unweighted average of 5.3 ± 0.2 km s−1. Using a mean unweighted kinetic temperature Tkin ~ 91 K and averaged line widths of H2 CO, the thermal and non-thermal line widths are 0.15 and 2.25 km s−1, respectively; σT = kTkinmH_2CO$\sqrt{\frac{kT_{\rm kin}}{m_{\rm H_2CO}}}$ and σNT = Δv28ln2-σT2$\sqrt{\frac{\mathrm\Delta v^2}{8\ln 2}-\sigma_{\rm T}^2}$ where k is the Boltzmann constant, Tkin is the kinetic temperature of the gas, mH_2CO is the mass of the formaldehyde molecule, and Δv is the measured FWHM line width of H2 CO. The thermal line width is significantly lower than the non-thermal line width. The sound speed (as = kTkinμmH$\sqrt{\frac{kT_{\rm kin}}{\mu m_{\rm H}}}$, where μ = 2.37 is the mean molecular weight for molecular clouds and mH is the mass of the hydrogen atom) is ~0.54 km s−1 at temperature 91 K; hence, the Mach number (given as M = σNT/as) is 4.2, which agrees with the results of the high-mass clumps (mean value ~3.5 derived from NH3; Wienen et al. 2012) and the Bolocam Galactic Plane Survey (BGPS) sources (mean value ~3.2 derived from NH3; Dunham et al. 2011). This indicates that these massive clumps are turbulent and H2 CO line widths are influenced strongly by supersonic non-thermal motions in our samples.

Previous observations of NH3 (Wouterloot et al. 1988; Myers et al. 1991; Harju et al. 1993; Ladd et al. 1994; Molinari et al. 1996; Jijina et al. 1999; Wu et al. 2006; Urquhart et al. 2011, 2015), C18 O (Saito et al. 2001; Ridge et al. 2003; Maud et al. 2015), and 13CO (Wang et al. 2009; Lundquist et al. 2015) suggest that the line width is correlated with luminosity, which indicates the presence of a link between formed stars and velocity dispersion. We investigate the line width-luminosity relation in the case of the dense gas tracer H2CO. We plot the line width-luminosity relation in Fig. 9. For the line width of para-H2CO (404–303) and bolometric luminosity, the least squares linear fit result is (3)\begin{eqnarray} \log\mathrm\Delta v({\rm H_2CO~4_{04}{-}3_{03}})&=&(0.11\pm0.01)\times\log L_{\rm bol} \nonumber \\ &&+(0.23\pm0.06). \end{eqnarray}(3)

The correlation coefficient, R, is 0.64. Other transitions of H2CO show similar Δv(H_2CO)–Lbol correlations (not shown here). The slope (0.11 ± 0.01) of the Δv(H_2CO)–Lbol correlation agrees with previous results found with C18O (Saito et al. 2001) and 13CO (Wang et al. 2009), but is lower than that found with NH3 (Wouterloot et al. 1988; Myers et al. 1991; Jijina et al. 1999; Wu et al. 2006; Urquhart et al. 2011, 2015). The correlation appears consistent with the idea that the internal velocity dispersion of the dense clumps can be used to determine the mass of the formed stars (Saito et al. 2001).

We also derive averaged line widths of para-H2CO (404–303) discriminating between the four evolutionary stages introduced in Sect. 2. The unweighted averaged line widths are 3.9 ± 0.4, 4.8 ± 0.2, 4.9 ± 0.3, and 7.4 ± 0.4 km s−1 in 70w, IRw, IRb, and H II regions, respectively (see Table 3). It seems that the velocity dispersion slightly increases with the first three evolutionary stages, 70w, IRw, and IRb. A significant change appears to occur between the first three and the fourth (H II) evolutionary stage. This suggests that the more evolved and more luminous objects tend to be associated with more turbulent molecular cloud structures (Wang et al. 2009).

thumbnail Fig. 9

Line width of the para-H2CO (404–303) transition vs. bolometric luminosity of the measured sources. The straight line is the result from an unweighed linear fit.

4.4 Non-thermal velocity dispersion-temperature relation

Previous observations of NH3 and H2 CO (e.g. Wouterloot et al. 1988; Molinari et al. 1996; Jijina et al. 1999; Wu et al. 2006; Urquhart et al. 2011, 2015; Wienen et al. 2012; Lu et al. 2014; Immer et al. 2016; Tang et al. 2018)suggest that the line width is correlated with kinetic temperature. It is suggested that the correlation between kinetic temperature and line width is due to a conversion of turbulent energy into heat in the Galactic central clouds (e.g. Güsten et al. 1985; Ginsburg et al. 2016; Immer et al. 2016).

Here we examine whether there is a relationship between turbulence and temperature in our massive clumps. We adopt the non-thermal velocity dispersion (σNT ) of para-H2 CO in good approximation as proxy for the turbulence, and the kinetic temperatures of para-H2 CO (321–220/303–202 and 422 –321/404–303) as the gas kinetic temperature (see Fig. 10). For the non-thermal velocity dispersion of para-H2CO and kinetic temperature, the least squares linear fit results are listed in Table 4. The non-thermal velocity dispersion of para-H2CO is significantly positively correlated with the gas kinetic temperature by a power law of the form Tkin ∝ σNT0.66-1.06$\sigma_{\rm NT}^{0.66-1.06}$, which is consistent with results found with NH3 and H2CO in other star formation regions (Wouterloot et al. 1988; Molinari et al. 1996; Jijina et al. 1999; Wu et al. 2006; Urquhart et al. 2011, 2015; Wienen et al. 2012; Lu et al. 2014; Tang et al. 2018). The gas is heated by turbulent energy according to the approximate relation Tkin ∝ Δv0.8−1.0 (gas kinetic temperature measured with NH3 and H2 CO) in molecular clouds of the Galactic centre (Güsten et al. 1985; Mauersberger et al. 1987; Immer et al. 2016), which is consistent with our result (only in terms of slope, not of intercept and absolute value). All this implies that the gas may be heated by turbulent motions in our massive clumps on scales of ~0.1–1.8 pc.

Recent para-H2CO mapping observations of molecular clouds in the Galactic CMZ show that the warm dense gas is heated most likely by turbulence (Ao et al. 2013; Ginsburg et al. 2016; Immer et al. 2016). Following the method applied by Tang et al. (2018) in their Eq. (2), (4)\begin{eqnarray} 3.3\times10^{-27}~n~\sigma_{\rm NT}^3~L^{-1} &=& 4\times10^{-33}~n^2~T_{\rm turb}^{1/2}(T_{\rm turb}-T_{\rm dust}) \nonumber \\ &&+ 6\times10^{-29}~n^{1/2}~T_{\rm turb}^3~{\rm d}v/{\rm d}r, \end{eqnarray}(4)where the gas density n is in units of cm−3 , the velocity gradient dv/dr is in units of km s−1 pc−1, the one-dimensional non-thermal velocity dispersion σNT is in units of km s−1 , and the cloud size L is in units of pc; we determined the gas kinetic temperature caused by turbulent energy. We computed the gas kinetic temperature assuming turbulent heating dominates the heating process. We assumed a cloud size of ~1 pc (e.g. Dunham et al. 2010, 2011; Rosolowsky et al. 2010; Urquhart et al. 2014; He et al. 2015; Wienen et al. 2015; König et al. 2017; Yuan et al. 2017), a velocity gradient dv/dr = 1 km s−1pc−1, the above-mentioned (Sect. 4.3) averaged non-thermal velocity dispersion of 2.25 km s−1 measured with H2 CO, and an averaged gas spatial density ~106 cm−3 derived from H2CO line intensity ratios (Sect. 3.5). If the averaged dust temperature (Tdust ~ 25 K; derived from HiGal and ATLASGAL data; see Sect. 4.2) and averaged gas temperature (Tkin ~ 91 K; derived from the para-H2COline ratios; see Sect. 4.2) are adopted as the dust temperatures, the gas kinetic temperatures due to turbulence motions Tturb, are 55 and 88 K, respectively. The obtained Tturb values are slightly lower than the averaged gas kinetic temperature (Tkin ~ 91 K) derived from the para-H2CO line ratios. This indicates that turbulent heating significantly contributes to gas temperature in these massive clumps on scales of ~0.1–1.8 pc, which agrees with previous observational results with H2CO in the Orion molecular cloud 1 (OMC-1; Tang et al. 2018). Apparently, turbulent heating plays an important role in heating the dense gas in massive star-forming clumps (Pan & Padoan 2009).

thumbnail Fig. 10

Non-thermal velocity dispersion (σNT ) vs. gas kinetic temperature for para-H2 CO. Top panel: gas kinetic temperatures were derived from para-H2 CO 321–220/303–202 line ratios. Bottom panel: gas kinetic temperatures were derived from para-H2 CO 422–321/404–303 line ratios. The straight lines are results from unweighed linear fits.
Table 4

Kinetic temperature vs. H2 CO non-thermal velocity dispersion.

4.5 Correlation of gas temperature with luminosity

Previous observations of our selected massive clump temperatures determined from CO, NH3, CH3CN, CH3CCH, and CH3OH (Wienen et al. 2012; Giannetti et al. 2014, 2017) suggest that these clumps are heated by radiation from internal massive stars. The comparison between the kinetic temperature and luminosity further helps us to understand the internal heating of embedded infrared sources upon their surrounding dense gas.

To investigate how the kinetic temperatures traced by para-H2CO correlate with luminosity in these massive clumps, we compared the gas kinetic temperature to the bolometric luminosity obtained from MSX, WISE, Herschel HiGal and ATLASGAL data (König et al. 2017). A comparison between gas kinetic temperatures derived from para-H2CO (321–220/303–202 and 422–321/404–303) and the bolometric luminosity is shown in Fig. 11. The least squares linear fit results are (5)\begin{eqnarray} \log L_{\rm bol} &=& (2.53\pm0.54) \times \log T_{\rm kin}({\rm 3_{21}{-}2_{20}/3_{03}{-}2_{02}}) \nonumber \\ && -(0.39\pm1.06) \end{eqnarray}(5) and (6)\begin{eqnarray} \log L_{\rm bol} &=& (2.46\pm0.52) \times \log T_{\rm kin}({\rm 4_{22}{-}3_{21}/4_{04}{-}3_{03}}) \nonumber \\ && -(0.32\pm0.99), \end{eqnarray}(6)with correlation coefficients, R, of 0.53 and 0.50, respectively. This shows that higher temperatures traced by H2 CO are associated with more luminous sources. This result is expected if dense gas probed by H2 CO is illuminated or heated by massive stars inside or adjacent to the clouds. The correlations between gas temperature and bolometric luminosity are weak. The bolometric luminosity and gas temperature derived from para-H2 CO are related by a power law of the form Lbol ∝ Tkin2.5±0.5$T_{\rm kin}^{2.5\pm0.5}$, where the power-law index is not very far from that of the Stefan–Boltzmann law (L ∝ Tkin4$T_{\rm kin}^4$). This also suggests that the dense gas is heated most likely by activity from associated massive stars.

Mapping NH3 observationsof massive star formation regions (Lu et al. 2014; Urquhart et al. 2015) shows that in some cases the gas is heated by radiation from external sources. Owing to a lack of H2 CO source structure information for our sample, we cannot exclude that external heating is contributing in some sources of our sample. Therefore, a detailed mapping study of the temperaturestructure of our structure using formaldehyde is needed. We intend to map a part of our sample with formaldehyde to reveal details of the gas heating mechanism in the future.

thumbnail Fig. 11

Kinetic temperatures derived from para-H2 CO (321–220/303–202 and 422 –321/404–303, black squares and blue points) vs. bolometric luminosity. The straight lines are the results from unweighed linear fits.

4.6 Gas temperature and clump evolution

To investigate whether the kinetic temperatures traced by para-H2 CO are influenced by massive young stellar objects (YSOs) at different evolutionary stages, we derived averaged kinetic temperatures obtained from the para-H2 CO (422–321/404–303) ratios for the four evolutionary stages outlined in Sect. 2. The unweighted average kinetic temperatures Tkin are 52 ± 6, 73 ± 4, 81 ± 6, and 110 ± 8 K in 70w, IRw, IRb, and H II regions, respectively (see Table 3). From this it is clear that the averaged gas kinetic temperature increases with the evolutionary stage, which confirms the trends measured with CO, NH3 , CH3 CN, CH3 CCH, CH3 OH, and dust emission in our sample and in other massive star-forming clumps (Giannetti et al. 2014, 2017; Guzmán et al. 2015; Molinari et al. 2016; He et al. 2016; Yu & Xu 2016; König et al. 2017; Yuan et al. 2017; Elia et al. 2017). This indicates that the gas temperature probed by para-H2 CO is related to the evolution of the clumps.

As mentioned in Sect. 4.1, the luminosity-to-mass ratio, Lbol/Mclump, is a good evolutionary tracer for massive and dense cluster-progenitor clumps, which defines a continuous evolutionary sequence in time. We plot the relation between kinetic temperature derived from para-H2 CO (321–220/303–202 and 422 –321/404–303) ratios and Lbol/Mclump ratios in Fig. 12. The plot shows that the kinetic temperature traced by para-H2 CO is indeed a rising function of the luminosity-to-mass ratio, which is consistent with results found from CH3 CN, CH3 CCH, and CH3 OHin massive star-forming clumps (Molinari et al. 2016; Giannetti et al. 2017).

It seems that massive stars reach the main sequence above a threshold of Lbol/Mclump ~ 10 L⊙/M⊙ (Giannetti et al. 2017), thus strongly increasing their energy output. The Lbol/Mclump ≳ 10 L⊙/M⊙ clumps are associated with IRb and H II regions in our sample (also see Giannetti et al. 2017), indicating late evolutionary stages (see Table A.1). For Lbol/Mclump ≳ 10 L⊙/M⊙, the gas temperature and the luminosity-to-mass ratio are related by power laws of the form (7)\begin{eqnarray} \log T_{\rm kin}({\rm 3_{21}{-}2_{20}/3_{03}{-}2_{02}})&=&(0.14\pm0.05)\times\log (L_{\rm bol}/M_{\rm clump}) \nonumber \\ && +(1.79\pm0.09) \end{eqnarray}(7)

and (8)\begin{eqnarray} \log T_{\rm kin}({\rm 4_{22}{-}3_{21}/4_{04}{-}3_{03}})&=&(0.16\pm0.04)\times\log (L_{\rm bol}/M_{\rm clump}) \nonumber \\ && +(1.68\pm0.08), \end{eqnarray}(8)

with correlation coefficients, R, of 0.37 and 0.45, respectively. The power-law indices are consistent with those derived from CH3 CN, CH3 CCH,and CH3OH in our and other massive clumps (T ∝ (Lbol∕Mclump)0.12−0.22, Molinari et al. 2016; Giannetti et al. 2017). A correlation between gas temperature and Lbol/Mclump ratio indicates that the dense gas appears to be heated by the newly formed massive stars during the late evolutionary stages of clumps. The Lbol/Mclump < 10 L⊙/M⊙ clumps are well associated with 70w and IRw indicating earlier evolutionary stages. For these sources the relation of gas temperature with the Lbol/Mclump ratio does not follow the above trend. The temperature in these sources may not yet be greatly affected by the gas that is heated by internal power sources (Molinari et al. 2016). Instead it may be related to gas excited by star formation activities, such as outflows and shocks (Tang et al. 2017a).

thumbnail Fig. 12

Kinetic temperatures derived from para-H2 CO (321–220/303–202 and 422 –321/404–303, black squares and blue points) vs. luminosity-to-mass ratio Lbol /Mclump. The straight lines are the results from unweighed linear fits for clumps with Lbol /Mclump ≳ 10 L⊙/M⊙.

4.7 Comparisons of H2CO luminosity, bolometric luminosity, and clump mass

The line luminosities of dense molecular gas tracers (e.g. HCN, CS, and HCO+) are found to be approximately linearly correlated with far-infrared luminosities (LFIR ∝Lmolecule) in both Galactic dense clumps and galaxies (Gao & Solomon 2004a,b; Wu et al. 2005, 2010; Schenck et al. 2011; Ma et al. 2013; Zhang et al. 2014; Liu et al. 2016; Stephens et al. 2016), which indicates a link between the SFR represented by infrared luminosities and dense molecular gas mass indicated by molecular line luminosities. Line luminosities of dense molecular gas (e.g. HCN, CS, HCO+, N2H+, SO) appear to be linearly related to the mass of dense gas most relevant to star formation (Wu et al. 2010; Reiter et al. 2011; Liu et al. 2016).

Observations of H2CO K-doublet transitions (ΔJ = 0, ΔKa = 0, ΔKc = ±1) in our Galaxy and external galaxies show that H2 CO traces a denser, more compact component of molecular clouds than low-excitation transitions of CO or HCN (Mangum et al. 1993, 2008, 2013b). Our selected H2 CO transitions may have similar characteristics. The critical densities of H2 CO 313 –212, 312 –211, 303 –202, and 404 –303 transitions are ~4 × 105, ~6 × 105, ~6 × 105, and ~1 × 106 cm−3 (at kinetic temperature 50 K; Shirley 2015), respectively. Following Wu et al. (2005) and assuming Gaussian brightness distributions for the sources and a Gaussian beam, H2 CO line luminosities LH_2CO′$L^{\prime}_{\rm H_2CO}$ can be derived with LH_2CO′=23.5×10-6×D2×(π×θs24ln2)×(θs2+θbeam2θs2)×∫TmbdV.\begin{equation} L^{\prime}_{\rm H_2CO} = 23.5 \times 10^{-6} \times D^2 \times \left(\frac{\pi \times \theta^2_{\rm s}}{4\ln 2} \right) \times \left(\frac{\theta^2_{\rm s}+\theta^2_{\rm beam}}{\theta^2_{\rm s}} \right) \times \int T_{\rm mb}{\rm d}V. \end{equation}(9)

Here D is the distance in kpc from König et al. (2017), and θs and θbeam are the sizes of the line emission source and of the beam in arcsecond. As described in Sect. 3.5, we assume that the extent of the H2 CO emission is the same as that of the 870 μm dust emission derived from Csengeri et al. (2014). The resulting LH_2CO′$L^{\prime}_{\rm H_2CO}$ values are listed in Table A.7.

We present the correlations between clump masses and H2 CO luminosities in Fig. 13. The power-law fitted results are listed in Table 5. These show that the Mclump– LH_2CO′$L^{\prime}_{\rm H_2CO}$ relations are strongly correlated and that correlation coefficients range from 0.84 to 0.93 for various H2CO transitions. The power-law correlations of Mclump– LH_2CO′$L^{\prime}_{\rm H_2CO}$ are found to be slightly sublinear and are consistent with results of, for example HCN, CS, HCO+, N2H+, and SO found in massive dense clumps (Wu et al. 2010; Reiter et al. 2011; Liu et al. 2016). This indicates that LH_2CO′$L^{\prime}_{\rm H_2CO}$ of our observed eight transitions provides good tracers for the mass of dense gas and confirms that Lmolecule′$L^{\prime}_{\rm molecule}$ of dense molecular tracers does reliably probe the mass of dense molecular gas.

The Lbol– LH_2CO′$L^{\prime}_{\rm H_2CO}$ relations are plotted in Fig. 14. We fit power-law relations of Lbol– LH_2CO′$L^{\prime}_{\rm H_2CO}$ for eight different H2CO transitions. The fitted results are listed in Table 5. The bolometric luminosities and the H2CO luminosities are related with a slope range of 0.98–1.19 and correlation coefficients ranging from 0.79 to 0.82 for different H2CO transitions. Considering the uncertainties, the correlations are nearly linear. The correlations of Lbol– LH_2CO′$L^{\prime}_{\rm H_2CO}$ for different H2CO transitions are consistent with previous observational results of, for example, HCN, CS, HCO+, SiO, HC3N, C2H in massive dense clumps (Wu et al. 2005, 2010; Ma et al. 2013; Liu et al. 2016; Stephens et al. 2016). This indicates that the mass of dense molecular gas traced by the H2CO line luminosity is well correlated with star formation.

Observations of dense clumps show that their evolutionary stage impacts the LIR– Lmolecule′$L^{\prime}_{\rm molecule}$ relation (Liu et al. 2016; Stephens et al. 2016). We distinguish two evolutionary classes of clumps in their early stage (70w and IRb) and late stage (IRb and H II regions), respectively, in Fig. 14, and the power-law fitted results are listed in Table 5. Considering uncertainties of the fitted slopes of Lbol– LH_2CO′$L^{\prime}_{\rm H_2CO}$ correlations, we find approximately similar linear correlations of Lbol– LH_2CO′$L^{\prime}_{\rm H_2CO}$ for various transitions in both evolutionary stages of the clumps. This suggests that the LIR– LH_2CO′$L^{\prime}_{\rm H_2CO}$ relations are only weakly influenced by the evolutionary stage of the clumps in our sample. We also compare the Mclump– LH_2CO′$L^{\prime}_{\rm H_2CO}$ relations in the two evolutionary stages in Fig. 13, and the power-law fitted results are listed in Table 5. Apparently, clumps in an early stage are closer to sublinear (slopes of 0.63–0.89) and clumps in a late stage tend to exhibit more linear slopes (0.80–1.01). For the early stage, the Mclump– LH_2CO′$L^{\prime}_{\rm H_2CO}$ data show alarger scatter (see Fig. 13). This may be due to some clumps with lower luminosity ( <103 L⊙), which are likely in an early evolutionary stage with large derived uncertainties of the mass of the clump. The Mclump– LH_2CO′$L^{\prime}_{\rm H_2CO}$ is found to be strongly correlated, with correlation coefficients ranging from 0.92 to 0.94 in the late stages of clumps. This indicates that LH_2CO′$L^{\prime}_{\rm H_2CO}$ (J = 3–2 and 4–3) traces well the mass of warm dense molecular gas associated with bright infrared emission and H II regions in massive star-forming clumps.

thumbnail Fig. 13

Mclump vs. LH_2CO′$L^{\prime}_{\rm H_2CO}$ for eight transition lines of H2 CO. Black squares indicate clumps classified as early stage (70w and IRw) and blue circles indicate clumps classified as late stage (IRb and H II regions) (see Sect. 2 for an introduction to these sources). The black, blue, and red lines are the results from linear fits for early stage, late stage, and all sources, respectively.
Table 5

Clump mass and bolometric luminosity vs. H2 CO line luminosity.

thumbnail Fig. 14

Lbol vs. LH_2CO′$L^{\prime}_{\rm H_2CO}$ for eight transition lines of H2 CO. Black squares indicate clumps classified as early stage (70w and IRw) and blue circles indicate clumps classified as late stage (IRb and H II regions). The black, blue, and red lines are the results from linear fits for early stage, late stage, and all sources, respectively.

5 Summary

We have measured the kinetic temperature and spatial density with H2 CO (J = 4–3) and (3–2) rotational transitions and compare the derived temperatures with values obtained from NH3 with dust emission, line width, and bolometric luminosity for the ATLASGAL TOP100 massive star-forming clumps at various evolutionary stages using the 12 m APEX telescope. The main results are the following:

Using the RADEX non-LTE model, we derived the gas kinetic temperature and spatial density, modelling the measured para-H2 CO 321–220/303–202, 422 –321/404–303, and 404 –303/303–202 ratios. The gas kinetic temperatures derived from the para-H2 CO 422–321/404–303 and 321 –220/303–202 line ratios are very warm, ranging from 43 to >300 K with an unweighted average of 91 ± 4 K. Spatial densities of molecular gas derived from the para-H2 CO 404–303/303–202 line ratios yield 0.6–8.3 × 106 cm−3 with an unweighted average of 1.5 (±0.1) × 106 cm−3.

The fractional abundance X(para-H2CO) does not vary considerably during the various stages of massive star formation, ranging from 1.0 × 10−10 to 1.2 × 10−9 with an average of 3.9 (±0.2) × 10−10, confirming that H2CO does reliably trace the H2 column density.

The spatial densities traced by H2CO do not vary significantly with the evolutionary stage of massive clumps. This may indicate that the density structure does not evolve significantly as the star formation proceeds.

A comparison of kinetic temperatures derived from para-H2CO, NH3 (2, 2)/(1, 1), and the dust emission indicates that para-H2CO traces a distinctly higher temperature than the NH3 (2, 2)/(1, 1) transitions and dust.

The H2CO line widths correlate with the bolometric luminosities and increase with the evolutionary stage of the clumps, which suggests that high luminosities tend to be associated with more turbulent molecular cloud structures.

The non-thermal velocity dispersion of H2CO is positively correlated with the gas kinetic temperature, which indicates that the dense gas may be heated by dissipation of turbulent energy in those massive clumps.

A weak positive correlation between gas temperature and bolometric luminosity suggests that the gas might be heated by the activity of the embedded young massive stars.

The average gas kinetic temperature clearly increases with the evolutionary stage of the massive clumps. For Lbol/Mclump ≳ 10 L⊙/M⊙, we find a rough correlation between gas kinetic temperature and Lbol/Mclump ratio, which traces the evolutionary stage of the massive clumps (Molinari et al. 2016; Giannetti et al. 2017).

The strong correlations between H2CO line luminosities and clump masses are approximately linear during the late evolutionary stages of clumps, which indicates that LH_2CO (J = 3–2) and (4–3) reliably trace the mass of warm dense molecular gas associated with bright infrared emission and H II regions. During the earlier evolutionary stages of clumps, the correlation may be slightly sublinear. The Mclump– LH_2CO′$L^{\prime}_{\rm H_2CO}$ correlation appears to be influenced by the evolutionary stage of the clumps.

The H2CO line luminosities are nearly linearly correlated with bolometric luminosities over about four orders of magnitude in Lbol of our massive clumps, suggesting that the mass of dense molecular gas traced by the H2CO line luminosity is well correlated with star formation. The Lbol– LH_2CO′$L^{\prime}_{\rm H_2CO}$ relation seems to be weakly affected by the evolutionary stage of the clumps.

Acknowledgements

The authors are grateful for the valuable comments of the referee Jeff Mangum. We thank the staff of the APEX telescope for their assistance in observations. We also thank Nina Brinkmann for her help with data calibration. This work acknowledges support by The National Natural Science Foundation of China under grant 11433008, The Program of the Light in China’s Western Region (LCRW) under grant XBBS201424, and The National Natural Science Foundation of China under grant 11373062. This work was partially carried out within the Collaborative Research Council 956, subproject A6, funded by the Deutsche Forschungsgemeinschaft (DFG). C. H. acknowledges support by a Chinese Academy of Sciences President’s International Fellowship Initiative for visiting scientists (2017VMA0005). This research has used NASA’s Astrophysical Data System (ADS).

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2

This publication is based on data acquired with the Atacama Pathfinder EXperiment (APEX). APEX is a collaboration between the Max-Planck-Institut für Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory.

All Tables

Table 1

Observed H2CO transition parameters.

In the text
Table 2

Observed H2CO transitions and detection rates.

In the text
Table 3

Averaged parameters in various stages of the massive clumps.

In the text
Table 4

Kinetic temperature vs. H2 CO non-thermal velocity dispersion.

In the text
Table 5

Clump mass and bolometric luminosity vs. H2 CO line luminosity.

In the text

All Figures

thumbnail Fig. 1

Observed H2CO spectra (grey) towards AGAL008.684−00.367. Green lines indicate the Gaussian fit results.
In the text
thumbnail Fig. 2

Comparison of integrated intensities of H2 CO and 870 μm continuum flux densities. The solid line corresponds to Y = X in the given units.
In the text
thumbnail Fig. 3

Example of RADEX non-LTE modelling of the N(H2CO)–n(H2) relation for AGAL008.684−00.367 at a kinetic temperature of 55 K (see Sect. 3.4). Black dashed and solid lines are para-H2 CO 404–303 integrated intensities and para-H2CO 404–303/303–202 integrated intensity ratios, respectively. Tothe measured parameters, para-H2 CO 404–303 integrated intensity (orange solid and dashed lines represent observed value and uncertainty) and para-H2 CO 404–303/303–202 integrated intensity ratio (white solid and dashed lines) are corrected by the relevant beam-filling factors (see Table A.6). The colour map shows the optical depth of the para-H2 CO 303–202 line. The purple line in the upper green area corresponds to optical depth τ(para-H2CO 303–202) = 1.0.
In the text
thumbnail Fig. 4

Example of RADEX non-LTE modelling of the para-H2 CO kinetic temperature for AGAL008.684−00.367. Black solid and dashed lines are para-H2 CO integrated intensity ratios. Para-H2 CO 404–303/303–202 (blue solid and dashed lines represent observed value and uncertainty, accounting for different beam-filling factors), 321 –220/303–202 and 422 –321/404–303 integrated intensity ratios (top and bottom, red solid and dashed lines) for a para-H2 CO column density 2.8 × 1013 cm−2 are derived from the para-H2CO 404–303 integrated intensity and para-H2CO 404–303/303–202 ratio (see Sect. 3.3).
In the text
thumbnail Fig. 5

Top panel: comparison of kinetic temperatures derived from para-H2 CO 321–220/303–202 and 422 –321/404–303 ratios. The dashed line is the result from an unweighed linear fit, Tkin (4_22-3_21/4_04-3_03) = (0.9±0.1)×Tkin(3_21-2_20/3_03-2_02)+(5.8±7.8), with a correlation coefficient, R, of 0.85. Middle and bottom panels: comparisons of kinetic temperatures derived from LTE and RADEX non-LTE calculations for para-H2 CO 321–220/303–202 and 422 –321/404–303 ratios, respectively. The temperature uncertainties are obtained from observed para-H2 CO line ratio errors. Solid lines indicate equal temperatures.
In the text
thumbnail Fig. 6

Column density N(para-H2CO) and fractional abundance X(para-H2CO) vs. column density N(H2) (a, b), spatial density n(H2) (c, d), kinetic temperature Tkin(para-H2CO 422–321/404–303) (e, f), bolometric luminosity (g, h), mass of clump (i, j), and luminosity-to-mass Lbol/Mclump ratio (k, l). The column density and spatial density uncertainties are obtained from observed para-H2 CO line brightness temperature and line ratio errors. The straight lines are the results from unweighed linear fits yielding the given correlation coefficients, R, in the lower right corner of each panel.
In the text
thumbnail Fig. 7

Spatial density derived from para-H2 CO (404–303/303–202) vs. luminosity-to-mass ratio Lbol /Mclump. The dashed line indicates the average spatial density.
In the text
thumbnail Fig. 8

Top panel: comparison of kinetic temperatures derived from para-H2 CO (321–220/303–202 and 422 –321/404–303, cyan and blue points) and NH3 (2, 2)/(1, 1) (black squares) ratios against the dust temperatures. NH3 kinetic temperatures are selected from Wienen et al. (2012). The cyan and blue straight lines are the results from unweighed linear fits for gas temperatures derived from para-H2 CO (321–220/303–202 and 422 –321/404–303, respectively. Bottom panel: comparisons of gas temperatures derived from para-H2 CO (422–321/404–303), CO, NH3 (2, 2)/(1, 1), CH3 OH, CH3 CN, and CH3 CCH. Temperatures of CO, CH3OH, CH3 CN, and CH3 CCH are taken from Giannetti et al. (2017). The black straight lines in both panels indicate equal temperatures.
In the text
thumbnail Fig. 9

Line width of the para-H2CO (404–303) transition vs. bolometric luminosity of the measured sources. The straight line is the result from an unweighed linear fit.
In the text
thumbnail Fig. 10

Non-thermal velocity dispersion (σNT ) vs. gas kinetic temperature for para-H2 CO. Top panel: gas kinetic temperatures were derived from para-H2 CO 321–220/303–202 line ratios. Bottom panel: gas kinetic temperatures were derived from para-H2 CO 422–321/404–303 line ratios. The straight lines are results from unweighed linear fits.
In the text
thumbnail Fig. 11

Kinetic temperatures derived from para-H2 CO (321–220/303–202 and 422 –321/404–303, black squares and blue points) vs. bolometric luminosity. The straight lines are the results from unweighed linear fits.
In the text
thumbnail Fig. 12

Kinetic temperatures derived from para-H2 CO (321–220/303–202 and 422 –321/404–303, black squares and blue points) vs. luminosity-to-mass ratio Lbol /Mclump. The straight lines are the results from unweighed linear fits for clumps with Lbol /Mclump ≳ 10 L⊙/M⊙.
In the text
thumbnail Fig. 13

Mclump vs. LH_2CO′$L^{\prime}_{\rm H_2CO}$ for eight transition lines of H2 CO. Black squares indicate clumps classified as early stage (70w and IRw) and blue circles indicate clumps classified as late stage (IRb and H II regions) (see Sect. 2 for an introduction to these sources). The black, blue, and red lines are the results from linear fits for early stage, late stage, and all sources, respectively.
In the text
thumbnail Fig. 14

Lbol vs. LH_2CO′$L^{\prime}_{\rm H_2CO}$ for eight transition lines of H2 CO. Black squares indicate clumps classified as early stage (70w and IRw) and blue circles indicate clumps classified as late stage (IRb and H II regions). The black, blue, and red lines are the results from linear fits for early stage, late stage, and all sources, respectively.
In the text

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