THE PROPERTIES OF MASSIVE, DENSE CLUMPS: MAPPING SURVEYS OF HCN AND CS

The sample in the survey is a subset of the larger sample of dense clumps associated with H₂O masers that have been surveyed with several CS transitions (Plume et al. 1992, 1997), most of which have been mapped with CS 5–4 and dust emission (Shirley et al. 2003; Mueller et al. 2002). Table 1 lists the information on sources that have been mapped in this survey. The sources mapped in CS 5–4 (Shirley et al. 2003) have virial masses within the nominal core radius (R_CS) ranging from 30 M_☉ to 2750 M_☉, with a mean of 920 M_☉. The sources in this category have infrared luminosities ranging from 10³ to 10⁷ L_☉ and most contain compact or ultracompact H ii (UCHII) regions. These dense clumps are mostly quite massive and have been well studied by several tracers; we will focus on these sources for statistical studies of the properties of massive dense clumps. To extend the sample toward lower luminosities for the study of L'_HCN–L_IR correlation, we selected 14 IRAS sources from outflow surveys (Zhang et al. 2005; Wu et al. 2004) and a few lower mass clumps from other publications. These added sources are listed in part 2 of Table 1, separated by a line from the well-studied massive clumps in Table 1. These clumps have various luminosities; in this paper they are not included in our statistics of the properties of massive dense clumps.

We mapped massive clumps in the HCN 3–2, HCN 1–0, CS 7–6, and CS 2–1 transitions. Because of observing constraints, maps of a particular transition are not available for all sources in our sample, but we have attempted to map as many sources as possible in all the tracers. In addition to mapping the clumps in the main transitions, we also observed the optically thin isotopes of HCN and CS (H¹³CN 3–2 and C³⁴S 5–4) at the center position of most of these massive clumps to constrain models and to study infall in massive clumps. Observations were taken between 1996 and 2007, with information presented in Table 2.

Maps of HCN 1–0 and CS 2–1 were made with the 14 m telescope of the Five College Radio Astronomy Observatory (FCRAO). The 16-element focal plane array (SEQUOIA) was used, with typical system temperatures 100–200 K. A velocity resolution of 0.1 km s⁻¹ was achieved with the 25 MHz bandwidth on the dual channel correlator (DCC). We converted the measured T*_A to T_R via T_R = T*_A/(η_FSSη_c), with η_FSS = 0.7. The value of η_c depends on source size; for the typical map in this study (∼10'), η_C = 0.7. The map size was extended until the edge of the HCN J = 1 → 0 and CS J = 2 → 1 emission was reached, typically at the 2σ level (mean σ ∼ 0.3 K km s⁻¹), so we could obtain the total line luminosity.

All the other observations were made at the 10.4 m telescope of the Caltech Submillimeter Observatory⁶ (CSO). The observing date, line frequency, beam size, and main beam efficiency of each line are listed in Table 2. The rest frequencies of HCN 3–2, H¹³CN 3–2, and C³⁴S 5–4 lines have been updated (Ahrens et al. 2002; Gottlieb et al. 2003) after some of our observations. We have corrected our data to the new frequencies listed in Table 2. The position switching mode was used; reference positions were checked when necessary. This is important for tracers that may show absorption features in the spectra (for example, HCN 3–2). We checked all the reference positions for HCN 3–2 emission and found them to be free from emission (T*_A < 0.5 K). Pointing was checked periodically using planets and CO-bright stars. The pointing accuracy was usually better than 6'' for all the runs. Map sizes of HCN 3–2 and CS 7–6 were extended to at least the point where the emission was one third of the peak strength, but usually maps extend to as low as one tenth of the peak strength (typically the 3σ level, with the mean σ ∼ 1 K km s⁻¹). The beam sizes and efficiencies were fairly constant for all transitions except for the CS 7–6, for which various different receiver optics were in place at different times. All data were scaled by the efficiency for the run on which they were taken to obtain the values of T*_R given in tables and used in the analysis.

Table 3 presents the peak temperature of the line (T*_R), integrated intensity (∫T*_Rdν), LSR velocity, and FWHM linewidth of CS 2–1, CS 7–6, and C³⁴S 5–4 transitions for massive dense clumps. Central position spectra of CS 2–1 and CS 7–6 have also been observed by Plume et al. (1992, 1997), who obtained CS 2–1 lines with the IRAM 30 m telescope at Pico Veleta, and CS 7–6 lines from the CSO. Comparing to the results in Plume et al. (1992, 1997), the observed parameters of CS 7–6 are consistent for the two CSO observations. The mean ratio of FWHM linewidth in the current observations to those from Plume's earlier data is 1.1 ± 0.4, with median ratio of 1.1. The mean ratio for ∫T*_Rdν is 1.1 ± 0.7, with a median ratio of 1.0. For the results of CS 2–1 observations, the IRAM data give a stronger T*_R and larger ∫T*_Rdν; the ratio of ∫T*_Rdν for the IRAM data to current FCRAO data is 3.3 ± 1.8. This difference comes from the different telescope sizes and beam sizes of the IRAM and the FCRAO. But the linewidth from the two observations are close: the FWHM ratio of IRAM CS 2–1 data to FCRAO data is 1.1 ± 0.2, and the median is 1.1.

Table 4 presents the line information for HCN 3–2 and its isotopologue H¹³CN 3-2. The mean integrated intensity ratio of HCN 3–2 to H¹³CN 3-2 is 9.7 ± 6.8. This ratio is smaller than the usual ¹²C/¹³C isotopic abundance ratio, partly due to the large fraction of self-absorption features presented in HCN 3–2 lines. About half of the sources in the massive core sample show double peaks. The description of line profiles is in Table 4.

Table 5 lists the observational results for HCN 1–0. HCN 1–0 has three hyperfine components (F = 1–0, F = 2–1, F = 1–1), separated by −7.064 km s⁻¹ (F = 1–0 to F = 2–1) and 4.842 km s⁻¹(F = 1–1 to F = 2–1). We tried to fit the spectra with three Gaussian components with the same linewidth. The peak temperatures of every component, the LSR velocity of the main component (F = 2–1), the linewidth, and the integrated intensity are presented in Table 5. Massive clumps usually have large linewidths; sometimes the linewidth is wider than the separation of hyperfine components, so that two or three of these components will overlap. Sometimes one or two hyperfine components are optically thick enough to show absorption. We have noted these features in Table 5. If the HCN 1–0 line is optically thin and the hyperfine levels are populated according to LTE, the line ratios between the three hyperfine components are I(F = 1–0): I(F = 2–1): I(F = 1–1) = 1:5:3. The observed ratios for most of our clumps (Table 5) do not agree with these optically thin ratios, as expected for lines that are almost surely optically thick. This hyperfine anomaly has also been reported in other massive star-forming regions (e.g., Pirogov et al. 1996; Pirogov 1999).

Contour maps of HCN 1–0, HCN 3–2, CS 2–1, and CS 7–6 are presented in Figures 1–23. These maps show a variety of morphologies, with some multiple peaks. HCN 1–0 and CS 2–1 maps cover a much larger area (typically greater than 7' × 7') than the higher J transition maps. A substantial fraction of these low-J transition maps show multiple peaks. For HCN 1–0, 16 out of 53 maps (30%) show multiple peaks; For CS 2–1, 21 out of 53 maps show multiple peaks (40%). This ratio is lower for high-J HCN and CS tracers. Only eight out of 42 HCN 3–2 maps (19%), and one of 51 CS 7–6 (2%) present multiple peaks. For CS 5–4 maps (Shirley et al. 2003), this ratio is 16% (nine out of 55). The majority of HCN 3–2 and CS 7–6 maps show a single, well-peaked distribution of intensity, similar to earlier maps of the CS 5–4 transition or the 350 μm dust continuum emission (Shirley at al. 2003; Mueller et al. 2002).

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Considering only sources with maps of multiple lines of the same species, there are 37 dense clumps mapped in both HCN 1–0 and HCN 3–2; six clumps that are single clumps in HCN 1–0 maps break into multiple clumps in HCN 3–2. Among the 38 clumps mapped by both CS 2–1 and CS 5–4, four single clumps in CS 2–1 maps show multiple components in CS 5–4 maps. This effect may be due to the higher resolution of the HCN 3–2 and CS 5–4 observations. For the four clumps that have been resolved by CS 5–4 but not CS 2–1 maps, they are either not observed by CS 7–6, or the other components are beyond the CS 7–6 mapping area. Consequently, we do not know if they are also separate peaks in the CS 7–6 maps.

The overall impression is that low excitation lines may show secondary peaks that are not dense enough to show up in the lines requiring higher excitation, while maps with higher resolution may find multiple peaks within the low-resolution map. Because lines of higher excitation in our study have better spatial resolution, those multiple peaks may not be seen in the lower excitation tracers, just because of instrumental limitations. To check this possibility, we have smoothed the HCN 3–2 maps for four clumps with multiple peaks to the resolution of the HCN 1–0 data and compare them to the HCN 1–0 data (Figure 24). The similarity of the smoothed HCN 3–2 maps to the HCN 1–0 maps supports the suggestion that HCN 1–0 maps with higher resolution would also resolve multiple peaks.

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These few clumps aside, the general appearance of most maps at the current resolution is not one of many cores within a clump, but of single well-peaked distributions. Maps with higher resolution that still retain sensitivity to extended structure are needed to study substructure in these clumps.

The size of the clump is characterized by the angular diameter or linear radius after beam deconvolution. θ_transition is the angular diameter of a circle that has the same area as the half-peak intensity contour:

$$\begin{equation*} \theta _{\rm transition}=2\left(\frac{A_{1/2}}{\pi }-\frac{\theta ^{2}_{\rm beam}}{4}\right)^{1/2}, \end{equation*}$$

and R_transition = θ_transitionD/2, where A_1/2 is the area within the contour of half-peak intensity, θ_beam is the FWHM beam size, and D is the distance to the source. It is important to note that these sizes do not represent sharp boundaries. To show this fact, we chose G10.6-0.4, W3(OH), and S231 as examples, since these sources are round in the contour maps, thus better approximated by a one-dimensional radial profile. In Figure 25, the emission generally declines smoothly to the noise level, when plotted against the angular source size, in units of the FWHM angular extent.

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The calculated sizes of clumps in each transition are listed in Table 6, with the mean and median values given at the end of the table. The statistics on median and mean value include only the massive sources, but not those sources below the horizontal line in the tables, which are mostly not very massive and were added to study the lower luminosity end of the star formation law. The histogram of the deconvolved FWHM sizes (angular size and linear size) of each transition are presented in Figures 26 and 27, in which again we only include massive dense clumps above the horizontal line in the tables. Figure 26 shows the histogram of the deconvolved angular FWHM size of these dense clumps, with a dashed line indicating the beam size in that transition. In HCN 3–2 maps, 29% of dense clumps have their deconvolved FWHM sizes smaller than the beam size. For CS 7–6 maps, this ratio increases to 55%, including one source (G12.21-0.10) not being resolved at all. The beam size effect can be nearly ignored for lower-J transitions; almost all the FWHM sizes of HCN 1–0 and CS 2–1 are well above their beam sizes.

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As Figure 27 makes particularly clear, the size distribution is skewed to low values before it is cut off by our resolution limits. Note that the plots of the lower excitation lines use a larger scale because they cover a substantially larger range of sizes.

For later comparison to work on other galaxies, we compute the line luminosities and the infrared luminosities in the same way as is done for extragalactic studies. The line luminosity of each core, assuming a Gaussian brightness distribution for the source and a Gaussian beam, is (we use HCN 1–0, for example)

$$\begin{eqnarray} L^{\prime }_{{\rm HCN}\scriptsize\hbox{1--0}}&=&23.5\times 10^{-6}\times D^{2}\times \left(\frac{\pi \times \theta _{s}^{2}}{4ln 2}\right)\times \left(\frac{\theta _{s}^{2}+\theta _{\rm beam}^{2}}{\theta _{s}^{2}}\right)\nonumber\\ && \times \int T_{R}dv {\rm K}\, \mbox{km s$^{-1}$}. \end{eqnarray} \tag{ 1 }$$

Here D is the distance in kpc, and θ_s and θ_beam are the angular size of the half-peak HCN 1–0 contour and the beam in arcsecond. This method is analogous to that of Gao & Solomon (2004b), but adapted to Galactic clumps.

The total infrared luminosity (8–1000 μm) was calculated based on the four IRAS bands (Sanders & Mirabel 1996), as was done for the galaxy sample of Gao & Solomon (2004a):

$$\begin{equation} L_{\rm IR} = 0.56\times D^{2}\times (13.48\times f_{12}+5.16\times f_{25}+ 2.58\times f_{60}+f_{100}), \end{equation} \tag{ 2 }$$

where f_x is the flux in band x from the four IRAS bands in units of Jy, D is in kpc, and L_IR is in L_☉. Some authors used the bolometric luminosity (L_bol), which is calculated from the spectral energy distribution (SED) of the source (e.g., Mueller et al. 2002). Our statistics show that L_bol calculated in Mueller et al. (2002) is very close to the L_IR calculated in this paper for the same clumps. The ratio of L_IR to L_bol is 1.28 ± 0.90, and the median is 1.09. The relatively large dispersion for this ratio is strongly affected by two sources (S252A and CepA) whose ratio is much larger than others. Excluding these two sources, the mean ratio becomes 1.10 ± 0.25, with the median 1.08. The mean and median values of L_IR are (4.7 ± 1.2)×10⁵ L_☉ and 1.06×10⁵ L_☉, with values ranging from 708 L_☉ to 5×10⁶ L_☉, considering only the sources in the original sample.

The infrared and line luminosities are tabulated in Table 7, with mean and median values for the massive, dense clumps at the bottom. The histograms of luminosity are shown in Figure 28. While the histograms appear to be well-peaked, we are systematically biased against low-luminosity sources. Together with the fact that the histograms plot the logarithm of the luminosity, the highest luminosity sources are clearly rare. For example, the mean value of luminosity is typically three times the median.

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Although all these lines trace "dense" gas, there are differences. Higher J transitions of a given species trace denser parts of the core. Crudely speaking, for a given J level, HCN transitions have higher critical densities than do the corresponding transitions in CS. The dependence on temperature also differs, with the upper level of the CS 7–6 transition lying 66 K above ground, compared to only 26 K above ground for the HCN 3–2 transition.

In Table 8, we list the properties of the transitions and the statistics of the size ratios and luminosity ratios of maps derived from different tracers. Table 9 shows that maps of low-J transitions have larger deconvolved sizes than those of higher J transitions. For instance, the linear size of the HCN 3–2 map is 0.28 of the size of the HCN 1–0 maps, and the ratio is the same for CS 7–6 versus CS 2–1.

Table 8. Ratios of Sizes and Luminosities for Different Tracers^a

Ratio	$\frac{R_{{\rm HCN} 1-0}}{R_{{\rm HCN} 3-2}}$	$\frac{R_{{\rm CS} 2-1}}{R_{{\rm CS} 7-6}}$	$\frac{R_{{\rm HCN} 1-0}}{R_{{\rm CS} 2-1}}$	$\frac{R_{{\rm CS} 5-4}}{R_{{\rm CS} 7-6}}$	$\frac{R_{{\rm HCN} 3-2}}{R_{{\rm CS} 5-4}}$
Meanb	2.97 ± 0.19	4.12 ± 0.45	1.11 ± 0.03	1.42 ± 0.09	1.11 ± 0.04
Median	2.71	3.36	1.07	1.31	1.05
Sdvc	1.06	2.40	0.23	0.64	0.21
	$\frac{L^{\prime }_{{\rm HCN} 1-0}}{L^{\prime }_{{\rm HCN} 3-2}}$	$\frac{L^{\prime }_{{\rm CS} 2-1}}{L^{\prime }_{{\rm CS} 7-6}}$	$\frac{L^{\prime }_{{\rm CS} 2-1}}{L^{\prime }_{{\rm CS} 5-4}}$	$\frac{L^{\prime }_{{\rm CS} 5-4}}{L^{\prime }_{{\rm CS} 7-6}}$	$\frac{L^{\prime }_{{\rm HCN} 3-2}}{L^{\prime }_{{\rm CS} 5-4}}$
Meanb	4.3 ± 0.4	9.4 ± 2.1	4.0 ± 0.4	2.6 ± 0.4	1.6 ± 0.1
Median	3.8	5.9	3.4	1.7	1.5
Sdvc	2.5	11.0	2.3	2.6	0.7

Notes. ^aSizes of CS 5–4 maps come from Shirley et al. (2003). ^bThe uncertainty is the sigma of the mean. ^cThe standard deviation of the ratios.

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Table 9. Properties of Different Transitions

Transition	ncrita	Eupa	neffa	Mean L$^{^{\prime }}$	Mean L$^{^{\prime }}_{{\rm sub}}$c	Mean R	Mean Rsubc
	(100 K)	(K)	(100 K)	K km s pc−2	K km s pc−2	(pc)	(pc)
CS 7–6	2.0 × 107	66	2.6 × 105	64	42	0.28	0.24
CS 5–4b	6.9 × 106	35	6.0 × 104	77	68	0.37	0.31
HCN 3–2	6.8 × 107	26	3.6 × 104	79	101	0.32	0.35
CS 2–1	3.9 × 105	7.1	4.1 × 103	272		1.03
HCN 1–0	4.5 × 106	4.3	5.1 × 103	359		1.13

Notes. ^aFrom Evans (1999). ^bFrom Shirley et al. (2003). ^cStatistics on a subsample of the sources only, which have been mapped by all the three high-J transitions CS 7–6, CS 5–4, and HCN 3–2.

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This result would not be found in a "raisin-pudding" model of clumps: a roughly uniform pudding (the clump) with multiple dense cores sprinkled randomly throughout. That model would predict similar sizes for maps of transitions sensitive to different density regimes. Such a pattern was seen in early studies of multiple transitions in a small number of sources (Snell et al. 1984; Mundy et al. 1986, 1987). The current results have higher spatial resolution and more complete maps in the high-excitation lines than the early results and do show that higher excitation lines are more centrally condensed. If the clumps are centrally condensed, with either a smooth increase in density with decreasing radius, or an increased fraction of the volume filled by dense "raisins," the differences in size would be predicted as the lines from higher-J transitions would be strongly enhanced in the denser regions. Detailed models of individual dense clumps do support density gradients with power laws providing good fits (n(r) ∝ r^−p with 〈p〉 = 1.8; Mueller et al. 2002).

The linewidth at the peak of the dense clumps for different tracers shows an interesting trend. As presented in Table 10, the mean and median FWHM linewidth of the dense clumps increase for higher-J transitions, from about 4.8 km s⁻¹ for CS 2–1, to about 5.9 km s⁻¹ for CS 7–6. If we look at a optically thin tracer, like C³⁴S 5–4, the mean FWHM is 4.7 km s⁻¹, which is closer to that of the CS 2–1 than higher-J CS transitions.

Studies of molecular clouds have established a positive correlation of cloud size, as mapped with a single tracer, and linewidth of that tracer (Larson 1981; Myers 1985), which can be interpreted in terms of the virial theorem. The linewidth–size relation within individual cores or clumps is much less well-established (e.g., Evans 1999). Here we refer to the useful distinction introduced by Goodman et al. (1998) between four different kinds of relationships. A Goodman Type-4 relation plots linewidths of a single tracer measured over regions of increasing radius; such plots show little correlation (Goodman et al. 1998; Barranco & Goodman 1998). Goodman Type-1 relations plot sizes measured in multiple tracers for multiple cores or clumps. Such plots generally show positive correlations when using tracers of modest density applied to low-mass and intermediate-mass cores or clumps; for example, Caselli & Myers (1995) found Δv ∝ r^0.53±0.7 with a correlation coefficient of 0.81 using NH₃, CS 2–1, and C¹⁸O 1–0 maps of eight starless cores.

In contrast, studies of the cores in this sample with Goodman Type-2 relations (single tracer, multicore) have shown no correlation between linewidth and size (Plume et al. 1997 for CS 7–6) or a weak (r = 0.43) correlation (Shirley et al. 2003 for C³⁴S 5–4). In both cases, the linewidths were 4–5 times larger than would be predicted from the usual linewidth–size relations developed from studies of clouds and low-mass cores. With maps of multiple tracers toward many cores, we could construct Goodman Type-3 relations (multitracer, single core) for each core. Instead, we try a modified version (Type 5?) in which we use the mean value over many cores for each transition.

Figure 29 shows the mean linewidth versus the mean size of the dense clumps for CS 2–1, CS 5–4, CS 7–6, and HCN 1–0 transitions. HCN 3–2 lines have serious self-absorption, therefore they are excluded from this figure. HCN 1–0 has three hyperfine components; we took the mean linewidth of the strongest component (F = 2–1) for the plot. The mean linewidth of the tracer has an inverse correlation with the mean size of the dense clumps. For example, while the CS 2–1 sizes are on average four times larger than those of CS 7–6, the mean of the linewidth ratio of CS 7–6 to CS 2–1 is 1.18 ± 0.34 for the massive dense clumps; the median ratio is 1.16. The CS 2–1 emission traces a larger, quieter area than higher-J transitions, while CS 5–4 and CS 7–6 trace more compact and apparently more turbulent regions. Alternatively, the larger linewidths toward the center could in part reflect higher inflow motions. A similar trend was seen in observations and models of G10.6 (Keto 1990). It is also possible that the turbulence (here refers to all unresolved gas motions, including inflow, outflow, complex fluid motions, etc.) has radial structure, which is higher close to the forming protostars and therefore is being picked up better by the higher-J, dense gas tracers. The usual size–linewidth relations certainly do not apply to these regions where massive stars are forming.

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While we speak of a transition as "tracing" gas of a certain density, this formulation is too simplistic; the emission from a transition increases smoothly and strongly over a wide range of densities and then levels out just above the critical density. (This description applies only to lines with frequencies high enough that radiative transitions stimulated by cosmic background radiation or continuum emission from dust are unimportant; Evans 1989.) Nonetheless, it is common to speak of a transition as tracing its critical density. If anything, the opposite is true, as transitions fail to provide useful probes of density above their critical density. A more sensible fiducial density is the effective density (Evans 1999), which is the density needed to produce a line with a radiation temperature of 1 K for typical conditions. For these comparisons, we use the critical and effective densities calculated for a kinetic temperature of 100 K (Evans 1999). Both these measures are given in Table 9.

The critical densities (n_crit) of the HCN 1–0 and CS 2–1 lines differ by an order of magnitude, but their effective densities (n_eff), are very similar. The similarity of their mean sizes indicates that n_eff better reflects the excitation requirements. For HCN 3–2, CS 5–4, and CS 7–6, their sizes are also similar while the critical and effective densities make opposite predictions; CS 7–6 has a much higher n_eff than the other two, but HCN 3–2 has the highest n_crit. The excitation energy of the upper level (E_up) is considerably higher for the CS 7–6 transition. If we consider only the sources that have been mapped by all three high-J transitions CS 7–6, CS 5–4, and HCN 3–2 (Columns 5 and 7 in Table 9), statistics show that the mean size and luminosity of the dense clumps increase with decreasing E_up of the transitions, as well as decreasing n_eff, while the correlation with n_crit is weaker. The tracers that need higher excitation energy or density are confined to smaller regions in the dense clumps, consistent with centrally condensed structures. In addition, n_eff better reflects the excitation requirement than n_crit.

Empirically, we find that the HCN 1–0 and CS 2–1 lines form a matched set, while HCN 3–2 and CS 5–4 lines make another pair; each pair has very similar sizes and fairly similar luminosities in the mean. The CS 7–6 lines are close to the properties of the HCN 3–2 and CS 5–4 pair, but still more compact in size.

The virial mass of the dense clumps with power-law density distributions (n(r) ∝ r^−p) is calculated by

$$\begin{equation} M_{\rm Vir}(R)=\frac{5R\Delta v^{2}}{8a_{1}a_{2}Gln2} \approx 209\frac{(R/1 {\rm pc})(\Delta v/1 \mbox{km s$^{-1}$})^{2}}{a_{1}a_{2}} M_{\odot } \end{equation} \tag{ 3 }$$

$$\begin{equation} a_{1}=\frac{1-p/3}{1-2p/5}, \ p < 2.5. \end{equation} \tag{ 4 }$$

where a₁ is the correction for a power-law density distribution and a₂ is the correction for a nonspherical shape (Bertoldi & McKee 1992). For aspect ratios less than 2, a₂ ∼ 1 and can be ignored for our sample. To calculate a₁ we take the average p value (1.77) from Mueller et al. (2002) for all massive dense clumps. A detailed discussion of calculating the virial mass of dense clumps in this sample can be found in Shirley et al. (2003).

The virial masses calculated from each transition (M_Vir(R_transition)), for HCN 1–0, 3–2, CS 7–6, and 2–1, are tabulated in Table 11. Rather than using the linewidth of the tracer, we used the linewidth of C³⁴S 5–4 for all these tracers because it is more reliably optically thin than is the 3–2 line of H¹³CN. For lower mass sources (those below the horizontal line in Table 11), their C³⁴S 5–4 lines are too weak to detect or to give a Gaussian fit except for IRAS20188+3928. The FWHM linewidth of CS 2–1 is closest to that of C³⁴S 5-4 in our sample, with a ratio of linewidth for CS 2–1 to C³⁴S 5–4 of 1.2 ± 0.1 and a median of 1.1. So, for these lower mass clumps, we used the CS 2–1 linewidth to calculate their virial mass, and assigned an additional 20% uncertainty to their linewidth uncertainties. While the linewidths vary among transitions by about 25%, some of this variation must be caused by optical depth differences, and the differences in any case, are small. The sense of these calculations is then an estimate of the mass within the FWHM contour of each tracer. The mean ratio of M_vir for CS 2–1 to CS 7–6 is 4.0, with a median of 3.7; the mean ratio for HCN 1–0 and 3–2 is 2.9, with a median of 2.8. One would expect these ratios to reflect the size ratios, which are 4.1 and 3.0 (the mean ratio), respectively.

Table 11. The Virial Mass, Surface Density, and Average Volume Density of Dense Clumps^a

Source	HCN 3 → 2			HCN 1 → 0			CS 7 → 6			CS 2 → 1
	log(Mvir)	Σ	log($\overline{n}$)	log(Mvir)	Σ	log($\overline{n}$)	log(Mvir)	Σ	log($\overline{n}$)	log(Mvir)	Σ	log($\overline{n}$)
	(M☉)	(g cm−2)		(M☉)	(g cm−2)		(M☉)	(g cm−2)		(M☉)	(g cm−2)
G121.30+0.66	2.17(12)	0.69	5.54	2.35(10)	0.45	5.19	1.79(16)	1.65	6.31	2.44(11)	0.37	5.01
G123.07−6.31	2.68(16)	2.58	6.15	3.24(16)	0.71	5.03	2.52(17)	3.75	6.48	3.21(15)	0.76	5.09
W3(OH)	2.99(05)	1.40	5.60	3.43(03)	0.51	4.72	2.74(05)	2.54	6.11	3.39(05)	0.56	4.80
S231	2.24(10)	0.33	5.03	2.60(09)	0.14	4.30	2.02(10)	0.55	5.47	2.61(09)	0.14	4.29
S235	2.08(09)	0.38	5.21	2.49(09)	0.15	4.38	1.90(10)	0.58	5.57	2.42(08)	0.18	4.53
S255	2.25(08)	0.71	5.53				2.15(10)	0.91	5.74
RCW142	2.97(06)	1.60	5.70	3.45(06)	0.53	4.74	2.73(03)	2.73	6.16	3.23(05)	0.87	5.17
W28A2(1)				3.28(04)	0.95	5.20	2.70(01)	3.66	6.37	3.20(04)	1.15	5.36
M8E	2.07(05)	0.32	5.10	2.55(18)	0.10	4.13	1.78(08)	0.61	5.67	2.42(10)	0.14	4.40
G9.62+0.19	3.49(04)	1.01	5.14	3.89(06)	0.40	4.34	3.39(10)	1.26	5.33	3.86(12)	0.43	4.40
G8.67−0.36	3.17(06)	0.68	5.04				2.84(06)	1.45	5.70
W(31)	3.89(09)	1.05	4.97				3.96(18)	0.89	4.82
G10.60−0.40	3.41(02)	1.19	5.28	3.85(03)	0.44	4.42	3.46(03)	1.06	5.18	3.80(04)	0.49	4.51
G12.21−0.10				4.63(24)	0.09	2.98			7.42	4.69(16)	0.08	2.85
G12.89+0.49							2.32(07)	1.49	5.98
W33cont							3.36(03)	0.45	4.68
G13.87+0.28				3.22(08)	0.04	3.10	2.41(10)	0.23	4.71	3.09(06)	0.05	3.35
G14.33−0.64	2.18(11)	0.57	5.42				2.25(10)	0.48	5.27
G23.95+0.16				3.12(23)	0.04	3.18	2.51(13)	0.15	4.40	3.09(22)	0.04	3.24
W43S	3.12(07)	0.29	4.50	3.64(06)	0.09	3.46	2.76(07)	0.64	5.21	3.53(05)	0.11	3.68
G31.41+0.31							3.11(05)	1.39	5.54
G31.44−0.26							3.09(11)	0.26	4.45
G32.05+0.06							3.47(21)	0.70	4.91
W44	2.97(09)	0.42	4.82	3.49(13)	0.13	3.78	2.79(08)	0.63	5.18	3.29(08)	0.20	4.18
G35.58−0.03				3.51(21)	0.54	4.72	2.86(14)	2.41	6.02	3.46(11)	0.61	4.82
G35.20−0.74							2.81(10)	1.13	5.55
W49S							3.73(24)	0.93	4.96
G45.07+0.13	3.52(10)	0.71	4.90	3.90(09)	0.30	4.13	3.39(10)	0.95	5.15	3.66(09)	0.52	4.62
G48.61+0.02	2.73(13)	0.08	3.88	3.47(07)	0.02	2.41	2.31(14)	0.22	4.72	3.40(14)	0.02	2.54
W51W							2.96(15)	0.22	4.40
W51M	3.98(10)	1.00	4.89	4.51(17)	0.29	3.82	3.75(03)	1.68	5.34	4.44(03)	0.34	3.96
G59.78+0.06				2.53(10)	0.11	4.14				2.54(11)	0.10	4.13
S87				2.60(06)	0.12	4.18				2.57(06)	0.13	4.24
S88B	2.15(10)	0.33	5.07	2.69(09)	0.10	4.00	2.06(09)	0.41	5.27	2.43(06)	0.17	4.52
K3-50				4.12(07)	0.38	4.18				4.09(09)	0.40	4.23
ON1				3.54(06)	0.14	3.80	2.81(12)	0.73	5.26	3.49(07)	0.15	3.91
ON2S	3.07(09)	0.23	4.37	3.34(08)	0.12	3.84	2.97(07)	0.29	4.58	3.34(07)	0.12	3.84
ON2N							2.86(11)	0.40	4.85
W75N	2.85(07)	0.66	5.18	3.22(03)	0.28	4.44	2.72(04)	0.88	5.43	3.18(06)	0.31	4.53
DR21S	2.93(03)	1.03	5.43	3.43(03)	0.33	4.43	2.96(06)	0.97	5.38	3.45(05)	0.31	4.39
W75OH	3.22(10)	0.79	5.12	3.59(05)	0.34	4.37	2.97(06)	1.40	5.61	3.54(04)	0.37	4.46
CepA	2.24(06)	1.56	6.05	2.67(07)	0.58	5.19				2.73(07)	0.50	5.07
S158	3.11(06)	0.40	4.72
S158A	2.87(06)	0.35	4.76
IRAS20050+2720	2.06(29)	1.32	6.03	2.53(26)	0.45	5.08				2.52(51)	0.46	5.11
IRAS20106+3545				2.17(24)	0.17	4.64				2.22(23)	0.15	4.55
IRAS20126+4104				2.83(15)	0.20	4.40				2.84(17)	0.19	4.38
IRAS20188+3928	1.75(17)	5.22	7.08	2.19(17)	1.89	6.20				2.22(17)	1.76	6.13
IRAS20216+4107	1.49(34)	0.96	6.10	2.24(20)	0.17	4.62				2.14(21)	0.22	4.81
IRAS20220+3728				3.43(20)	0.09	3.61				3.44(30)	0.09	3.60
IRAS20333+4102	2.14(25)	0.12	4.43	2.33(14)	0.08	4.06				2.30(15)	0.08	4.11
IRAS21391+5802	1.92(20)	1.42	6.14	2.27(16)	0.63	5.44				2.25(15)	0.66	5.47
IRAS22172+5549				2.56(21)	0.19	4.51				2.60(24)	0.17	4.42
IRAS22198+6336				2.29(21)	0.13	4.42				2.32(20)	0.13	4.37
IRAS23011+6126	1.29(25)	1.40	6.45	2.05(21)	0.24	4.93				2.03(24)	0.25	4.97
IRAS23385+6053				3.41(18)	0.13	3.86				3.38(18)	0.14	3.91
IRAS03282+3035	−0.09(32)	0.09	5.34	0.52(21)	0.02	4.13				0.48(24)	0.02	4.19
L483	−0.06(27)	0.07	5.44
L1251B				1.00(09)	0.15	5.13				1.14(16)	0.11	4.85
Meanb	1555	0.78	2.5 × 105	5300	0.29	3.2 × 104	1300	1.09	1.7 × 106	5000	0.33	4.2 × 104
Sigmab,c	445	0.55	0.7 × 105	1760	0.23	0.8 × 104	310	0.91	1.0 × 106	1870	0.27	1.0 × 104
Medianb	890	0.68	1.3 × 105	2700	0.28	1.6 × 104	650	0.89	3.7 × 105	1950	0.31	2.6 × 104

Notes. ^aThe virial mass, surface density, and volume density are calculated for mass within the FWHM contour of each transitions. Values for virial mass and volume density are in log scale, except the mean and median at bottom. Numbers in the parentheses are uncertainties in the log scale, in the unit of 0.01. ^bStatistics on massive dense clumps only, which does not include the sources below the horizontal line in the table. ^cThe σ of the virial mass is the σ of the mean.

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