Clump-scale Gas Infall in High-mass Star Formation: A Multitransition View with James Clerk Maxwell Telescope HCN (4–3) Mapping

Gas infall motions play a crucial role in high-mass star formation and are characterized by observable signatures of blueshifted asymmetric spectral line profiles (“blue profiles”). However, the connection between blue profiles and infall motions is unclear due to complex gas motions at parsec scales. In this study, we present the results of an HCN (4–3) mapping survey conducted with the James Clerk Maxwell Telescope, toward 38 massive clumps exhibiting blue profiles in HCO+ (3–2). We extract 34 HCN cores from the 38 observed fields. The core-averaged spectra show various line profiles, indicating that blue-profile HCO+ (3–2) does not guarantee the same in HCN (4–3). Through non-LTE radiative-transfer calculations, we attribute the low detection rate of high-J blue profiles to a combination of insufficient HCN (4–3) opacity and the intricate gas motion across different density layers. A comparison between the MALT90 and Bolocam Galactic Plane Survey line surveys highlights the importance of appropriate tracers, high spectral resolution, and column density thresholds when searching for blue profiles. We select 11 reliable infall candidates and adopt the Hill5 model to fit the infall velocity of 0.2–1.6 km s−1, corresponding to 5%–74% of freefall velocity. Assuming a spherically collapsing model, we estimate the median and mean mass infall rates to be 4.5 × 10−3 and 7.6 × 10−3 M ⊙ yr−1, respectively. The consistency of the mass infall rates among different transitions suggests a steady accretion process from the clump gas envelope to the inner region.


Introduction
Massive stars (>8 M e ) play a predominant role in the energy budget of galaxies via their radiation, wind, and supernova events, but mass-assembly processes including gas accretion or infall motions remain unclear.On the other hand, gravitational infall is a basic step in star formation theory (Larson 1969;Shu et al. 1987), and is expected in both "core-fed" (McLaughlin & Pudritz 1996;McKee & Tan 2003) and "clump-fed" massive star formation models (Bonnell et al. 2001;Wang et al. 2010;Vázquez-Semadeni et al. 2019).As such, identifying and studying the accretion flows that collect the material out of which stars form, either directly or indirectly, is an important aspect of understanding the mass assembly of massive stars (Fuller et al. 2005;Sun & Gao 2009;Jackson et al. 2019).However, massive stars form in complex environments and over large distances; thus, the features of individual cores embedded in a massive star-forming clump are averaged together into a single-dish beam, making infall motions harder to observe (e.g., Reiter et al. 2011;Liu et al. 2016;Yuan et al. 2017;Pillai et al. 2019;Huang et al. 2023) and observational evidence of collapse difficult to interpret (Evans 1991;Myers et al. 2000;Wu & Evans 2003;Wu et al. 2007).
Self-absorbed, optically thick line profiles serve as phenomenological evidence of the infall within star-forming regions.When examining the emission arising from the infalling envelope positioned on the far side of a protostar, a proportional blue (Doppler) shift emerges, attributable to the velocity gradient toward the core.This blueshifted emission evades absorption by foreground layers that are warmer or at a substantially different velocity (see Figure 1 in Evans 2002), thereby leading to an excess of emission on the blueward side of the source velocity within the line profile (Walker et al. 1986;Zhou et al. 1993;Walker et al. 1994;Mardones et al. 1997;Evans 2002).Notably, in instances where the source exhibits moderate optical thickness, a distinct blueward skew characterizes the line profile.Conversely, strongly selfabsorbed sources manifest two discernible peaks, with the blue peak outshining the red peak to a moderate or significant degree.The depth of the self-absorption feature is intensified in the presence of substantial temperature gradients within the core, while the asymmetry of the line profile is amplified with pronounced velocity gradients (Reiter et al. 2011).The distinctive line profile, commonly referred to as a "blue asymmetric profile" or simply "blue profile," enables the measurement and quantification of infall motion.
Here, we present a JCMT HARP HCN (4-3) mapping survey of 38 massive clumps with known blue profiles in a pilot single-point HCO + (3-2) line survey, conducted by Schlingman et al. (2011) and Shirley et al. (2013).The paper is organized as follows.Section 2 describes the sample selection, JCMT HARP observations and data reduction, and clump distance estimation.Results are presented in Section 3. Discussions follow in Section 4. Finally, we give a summary and prospectus of the survey in Section 5.

Sample Selection
The Bolocam Galactic Plane Survey (BGPS) imaged 170 deg 2 sky at 1.1 mm using Bolocam (for a survey description, see Aguirre et al. 2011) and cataloged 8358 continuum clumps (version 1.0.1 catalog; Rosolowsky et al. 2010).As a follow-up work, Schlingman et al. (2011) and Shirley et al. (2013) successively performed a single-pointed spectroscopy survey toward 1882 and 4705 BGPS clumps using the 10 m Submillimeter Telescope (SMT) in HCO + (3-2) and N 2 H + (3-2) with a spectral resolution of 1.1 km s −1 .Shirley et al. (2013) then integrate and present a complete spectroscopic catalog of HCO + (3-2) and N 2 H + (3-2) observations for 6194 sources in the BGPS v1.0.1 catalog between 7.5° l 194°.Among the sample, 80 sources show self-absorbed line profiles where HCO + (3-2) shows two peaks and an absorption dip over the span of at least three channels (3.3 km s −1 ) with the N 2 H + (3-2) line profile having a single peak.Of these, 48 sources are identified as blue asymmetric profiles, by comparing the optical thick HCO + (3-2) lines to the optically thin N 2 H + (3-2) lines.These sources serve as excellent highmass, large-scale collapse candidates (Shirley et al. 2013), which are the parent sample in our work.Due to the limitation of observing time, a subsample of 38 clumps (including one adopted from the JCMT archive) are chosen as target fields (fields hereafter) in this work.The entire sample-selection procedure is outlined in Figure 1, elucidating the process through which the sample is curated, meticulously avoiding biases in relation to essential physical parameters such as distance, clump mass, or luminosity.It should be noted that the BGPS clumps provide an unbiased representation of the galactic star-forming regions, affirming that the subsample maintains representativeness, and consequently that the outcomes of this study demonstrate a representative character.
All the fields are covered by legacy surveys of Spitzer, Herschel, and ATLASGAL, enabling us to obtain the infrared properties.We first retrieve the clump parameters including size, dust temperature, luminosity, mass, and peak column density from Urquhart et al. (2018), which are then corrected for by the updated distance (see Section 2.3).The corrected clump-scale infrared properties are summarized in columns (8)-( 12) of Table 1.The sample expands a wide range in (i) evolutionary stages from infrared dark clouds (IRDCs) to infrared bright UCHII regions, (ii) dust temperature from 9.7-34.4K, and (iii) mass from 1 × 10 2 -6 × 10 3 M e .
We used the 16 pixel HARP for the front-end, and the Auto-Correlation Spectrometer and Imaging System (ACSIS) for the    back-end (Buckle et al. 2009).HARP is a single side-band receiver comprising a 16-receptor array arranged on a 4 × 4 grid.At the observing frequency, HARP has an angular resolution of 14″, and a main-beam efficiency of η mb = 0.61.The footprint of the full array is ¢ ´¢ 2 2 .The "HARP5 Jiggle-Chop" scanning mode is used to fill in the 30″ spacing between the receptors, thereby resulting in a ¢ ´¢ 2 2 map with the pixel size of 6″, which is slightly over Nyquist sampling.The resultant scanning coverage for each field is highlighted by the yellow frame in Figure 2. Note that two or three receptors are not operational in our observations, so the frames are usually incomplete squares, except for BG012.889+00.490.ACSIS was set for a bandwidth of 250 MHz with 8194 channels, centered at the frequency of HCN (4-3) after Doppler shift.A uniform channel width of ∼0.03 MHz then leads to a velocity resolution of 0.026 km s −1 .The position-switch mode was performed after the whole telescope had moved away from the source and onto a reference position, which is especially chosen for each target based on the absence of CO and dust emission.During observations, the weather condition had a precipitable water vapor (PWV) range of 1.575-2.575mm or a τ 225 GHz of 0.08-0.12(Band 3).9The typical on-source time for each map is 40 minutes, or equivalently 1.6 minutes for each HARP pixel.
The data were first calibrated and reduced by the pipeline introduced by Jenness et al. (2015).The processed HARP-ACSIS data were converted into FITS format and then downloaded from the CADC's data collection. 10The orientations of the maps are determined via K-mirror rotation, which are different between observing fields, depending on the elevation of observation.To ensure consistency, we regrid the maps to align the y-axis to the north.We convert the velocity in the barycentric frame to that in the local standard of rest (LSR).We smooth the velocity resolution to a uniform value of 0.2 km s −1 to enhance the signal-to-noise ratio (S/N) for further spectral line analyses.The achieved rms noise level for each field is listed in column (9) of Table 1, with an average value of 0.10 (± 0.02) K at a channel width of 0.2 km s −1 .

Distance Estimation
Reliable velocity determination is crucial for estimating a set of other physical properties for the clumps.We take advantage of the velocity at the local standard of rest (V LSR ) derived from N 2 H + (3-2) or HCO + (3-2) in Shirley et al. (2013).We observe the following workflow to obtain the distance estimation for each source.First, for each source we check if any distance is already given in the references.If the distance is donated by kinematic distance or not given at all, we update the distance with the parallax-based Bayesian maximum-likelihood distance estimation approach (version 2.4.1;Reid et al. 2016Reid et al. , 2019)).Note that if one source is located outside the solar circle (i.e., the distance from the Galactic center R gc > 8.5 kpc) or is at a tangential point, we will calculate one unique distance.However, if one source is located within the solar circle (i.e., R gc < 8.5 kpc), two possible distances are obtained (one near, one far).This degeneracy is commonly referred to as the kinematic distance ambiguity (KDA).To address KDA, we follow the methods described in Urquhart et al. (2018), which test several criteria one by one to determine the distance.First, we search the SIMBAD database for any previous distance estimation.We then choose the one closest to the value reported in the literature.If no reference is found, then we check whether the source elevation (z) to the Galactic midplane for the farther distance is larger than 120 pc.11If this is the case, then the closer one is adopted.Having completed the above workflow, the distances and their references are listed in columns (5)-( 6) of Table 1.

Detection of HCN (4-3) Emission
We generate the moment zero (M0) maps to show how the emission of HCN (4-3) is distributed.For each field, we first extract the velocity range of (V LSR − dV, V LSR + dV ), where dV = 10 km s −1 , to cover the majority of HCN line emission.Then we integrate the spectra within the velocity range at each pixel and obtain the M0 maps shown in Figure A1.
Most of the HCN emission shows core-like condensed structures, although some show more extended and irregular ones.We then apply an automatic source-extraction algorithm, SExtractor12 (Bertin & Arnouts 1996), to the M0 maps to extract HCN emission sources.The advantages of SExtractor in our case are as follows: (i) to reduce background emission; (ii) to support local rms noise input to serve as pixelwise thresholds; and (iii) to deblend the potentially blended sources in one field.The algorithm procedure and the parameter settings are described in Appendix A in detail.As a result, a total of 34 HCN sources are extracted and fitted by 2D Gaussian profiles, shown by green solid ellipses in Figure A1.Since the HCN sources have physical sizes of 0.08-0.35pc (column (9) of Table 2), which are much smaller than the massive clumps of ∼1 pc, we define them as HCN "cores" hereafter.For further spectral line analyses, we also assigned a circle with diameter of 30″ (which is the beam size of SMT at 270 GHz; Shirley et al. 2013) to the six fields where no HCN was detected, shown as green dashed circles in Figure A1.The basic fitted parameters of the 34 HCN cores including offsets (along the x-and y-axes) from the field center, major and minor axes (θ maj and q min ), position angle (PA), and peak flux (F peak ) are listed in columns (3)-( 8) of Table 2.
Following the method of Rosolowsky et al. (2010) and Contreras et al. (2013), the deconvolved angular radius is written as where σ maj and s min are calculated from q 8 ln 2 maj and q 8 ln 2 min , respectively.The σ bm is the averaged dispersion size of the beam (i.e., q 8 ln 2 bmaj , where θ bmaj ; 14″ is the JCMT beam at the frequency of HCN (4-3)).η is a factor that relates the dispersion size of the emission distribution to the determined angular radius of the object.We have elected to use a value of η = 2.4, which is the median value derived for a range of models consisting of a spherical emissivity distribution (Rosolowsky et al. 2010).Therefore, the physical size of the core is derived from (where D is the distance), which is listed in column (9) of Table 2. Some of the cores have sizes comparable to the beam size, rendering them unresolved.In these cases, column (9) of the corresponding rows is marked with "L" as a notation.

Averaged Spectra from HCN Cores
The HCN (4-3) lines are extracted from the defined regions (including 34 HCN cores and six circles, defined in Section 3.1).We first smooth the velocity resolution to a uniform value of 0.2 km s −1 , to enhance the S/N for further spectral line analyses.Then the baselines of the spectra are subtracted; the baseline-free spectra are shown in Figure 3.The S/Ns are defined as the ratio of T peak to σ.In our analyses, the spectra with low S/N (<2) are classified as nondetections of HCN emission, while others are solid detections.We also visually double-check the spectra to exclude the potential temperature jump at bad channels.We note that although the field BG034.259+00.222has a detection in the north, the HCN (4-3) line has a large velocity deviation (∼40 km s −1 ) from the systematic velocity.In addition, the detected core BG034.259+00.222C1 is near the edge of the field.As such, we assume that BG034.259+00.222C1 is not correlated with the clump and it is excluded in the further discussion.Another notice is that although BG015.123-00.558has no detection in the field, the averaged spectrum from the central circle shows a S/N ∼ 2 detection of emission.The nondetection in the SExtractor algorithm could be due to extended and diluted emission.As a result, 34 of the 40 spectra show solid HCN detection, and six are designated as nondetection spectra.
For the HCN spectra with solid detections, we fit them with a single Gaussian model using the Python package PySpecKit, as shown in the upper-right corner of each panel in Figure 3.The Gaussian parameters, including the amplitude, centroid velocity, and line width, as well as their uncertainties are listed in columns (2)-(4) of Table 3.For the six nondetection spectra, columns (2)-(4) of Table 3 are filled with "L."We also flag the nondetection spectra with "N" in column (9).

Synergy with Previous Line Surveys
Previous line surveys conducted by Schlingman et al. (2011) and Shirley et al. (2013) not only serve as a guide for our follow-up survey (see Section 2.1), but also provide a large legacy value for spectral analyses in our work.The N 2 H + (3-2) lines have been observed to be optically thin (Shirley 2015), which can therefore be used to determine the systematic velocity and velocity dispersion of massive clumps.
Two important caveats warrant consideration in our analysis.First, the N 2 H + (3-2) lines were observed using the SMT, whose beam size is approximately twice that of the JCMT.Consequently, the N 2 H + (3-2) lines may reflect the systematic velocity of the entire clump or the dense inner region within the clump, rather than the velocity of the central dense core as indicated by the HCN (4-3) lines.In essence, a coherence in velocity between parent clumps and HCN cores should underpin our discussions concerning line profiles, as discussed in Section 3.4.
Second, due to the larger physical coverage of the SMT's beam, encompassing approximately 4 times the area of the JCMT's beam, more turbulent motion should be included.Consequently, broader line widths are anticipated.This implies that the N 2 H + (3-2) lines should be narrower than they would   2).Offsets along the x-and y-axes in the equatorial coordinate are listed in columns (3)-( 4).Fitted parameters including FWHM major axis, minor axis, PA, and peak flux are listed in columns (5)-( 8).The core size deconvolved with the beam is listed in column (9). a The name of the core is in format of "Field+CN," where "Field" is the name of fields in Table 1 and "CN" donates the core ID (C1, C2, K).
b Flag for core detection.0 = no detection; 1 = one core; 2 = two cores.c Unresolved cores are shown with "L." be if observed within the JCMT's beam.When comparing with the JCMT results in Section 3.4, we should always keep in mind that the line width of the N 2 H + (3-2) line could be overestimated.
Here, we check the consistency between the velocity derived from the HCN (4-3) lines, V LSR,HCN , and that fitted by optical thin lines N 2 H + (3-2), V LSR,thin , from Shirley et al. (2013).As shown in Figure 4, V LSR,HCN and V LSR,thin always share the same values, within the uncertainty, indicating a good correspondence between the two surveys, which establishes the basis for the blue-profile analyses in this paper.

Variety of Observed Spectral Line Profiles
As shown in Figure 3, the averaged HCN (4-3) line shapes differ from core to core, with some showing asymmetric profiles or double-peak profiles (non-Gaussian).To distinguish line profiles and study the distribution statistically, we adopt the definition of velocity difference by Mardones et al. (1997): where the difference between the peak velocity of HCN (4-3) V HCN,peak and the systematic velocity derived from optically thin line V sys is normalized by the FWHM of the thin line dV thin .The normalization makes it convenient and robust to set a uniform criterion to distinguish different line profiles, especially for samples with a wide range of line widths.We first calculate the velocity at the peak intensity as V HCN,peak , which is listed in column (5) of Table 3.To obtain V sys , we then retrieve the fitting results of N 2 H + (3-2) lines from Shirley et al. (2013) where the N 2 H + (3-2) lines are thought to be optically thin and taken as tracers of systematic velocity.V sys for each core is marked as an orange dashed line in Figure 3.If two cores are in one clump, then they share the same V sys and dV thin , which are listed in columns (6)-( 7) of Table 3.By Equation (2), normalized δV is then calculated and listed in column (8).We designate those with δV < −0.25 as significant blue profile (denoted with BP hereafter), those with δV > 0.25 as significant red profile (RP hereafter), and those with −0.25 < δV < 0.25 as single component (S hereafter) that have insignificant asymmetric profiles.The designation is listed in column (9) of Table 3.
We note that the second caveat in Section 3.3 can cause underestimation of δV due to systematic overestimation of dV thin as described in Equation (2).Consequently, there exists the possibility of bias, where the criteria for defining red or blue profiles (i.e., δV > 0.25 or < −0.25) might be more stringent than intended.This could potentially classify marginally satisfactory line profiles as nonasymmetric, resulting in a bias that reinforces the definition of pronounced line profiles but may also elevate the false-negative rate for weak line profiles.To address this potential bias, a secondary assessment should be conducted through visual inspection.Two instances, BG009.212-00.202C1 and BG023.968-00.110C1,exhibit blueshifted double peaks with δV values of −0.13 and −0.14, respectively.Despite not meeting the δV threshold, they are designated as "BP" due to their distinctive characteristics.Additionally, BG027.317+00.175C1,while satisfying the BP criterion, possesses a low S/N, which is then labeled as "S."Furthermore, BG030.772-00.801C1 and BG049.210-00.342C1,despite having δV < −0.25, each features only a single peak.As a result, they are then classified as S. Consequently, the final identification designates 14 cores as BP (referred to as HCN-BP cores) and four cores as "RP."

Infall Candidates Identified by Line Mapping
Statistically, infall motion is the most likely interpretation of the observed BPs.However, in individual cases, it is not the only possibility.Rotation and outflows can also produce BPs (e.g., Wu & Evans 2003;Wu et al. 2007).Resolved mapping observations are needed to investigate the nature of BPs.The rotation of a core always exhibits BPs and RPs at different spatial positions, which are mistaken as BPs in single-pointed observations.In a similar manner, outflow lobes are easily ruled out if redshifted emission is predominately from an extended wing.A profile that survives these tests provides a strong indication of infall, and the source can be seen as a candidate for collapse.
To provide a better visualization of mapping observations, we present spectral line grids for each HCN-BP core in Figure B1 to exclude other possibilities of producing BPs.The spectra located in the core mask are first averaged from the 2 × 2 pix 2 box and smoothed to a velocity resolution ∼0.4 km s −1 .Then the spectra are overlaid on the green elliptical footprints of HCN-BP cores.
The mapping of three HCN cores, BG023.968-00.110C1,BG029.397-00.095C1, and BG030.719-00.081C1,all present various but coherent line profiles among the core.In other words, although the averaged spectrum over the core shows a significant BP, the individual spectra at different positions can change from blue to red profiles continuously.This variation can also be seen from the moment one (M1) maps in the color map (Figure B1).The details of calculating the rotation axis can be found Appendix B.
Finally, a total of 11 HCN-BP cores survive the "mapping" tests and provide a strong indication of infall motion.We also check whether there are central heating sources to build the temperature gradient in these massive clumps.Although it has the lowest luminosity-to-mass ratio, of approximately 0.2, among the IRDCs, BG028.565-00.236still exhibits molecular outflows and H 2 O/CH 3 OH masers at higher angular resolution   7).Asymmetric parameter calculated by Equation (2).HCN line profile identification is listed in column (9): "BP" = blue profile, "RP" = red profile, and "S" = single-peaked profile; "N" = nondetection.a Masers are checked and used to correct the velocity (Xi et al. 2015(Xi et al. , 2016).b S = single component; B = blue profile; R = red profile; N = no detection.(Lu et al. 2015).These findings suggest the presence of active star formation and central heating sources, not to mention other sources with bright pointlike or even extended infrared emission.The discussion in Section 4.1 will further strengthen the argument here, since the high-J transition traces the denser (therefore inner) regions where the temperature gradient is guaranteed.Therefore, the BPs in the 11 HCN cores are most likely to be induced by infall motion.The subsample hereafter serve as promising candidates of infall in massive star-forming regions.

What Leads to a Variety of Line Profiles at Multi-J Transitions?
As demonstrated in Section 3.4, only 14 out of 38 clumps have BPs seen in HCN (4-3) lines, contributing to a profile retention rate of 36.8% from low-to high-level transitions (low-/high-J in abbreviated form, where "J" represents the quantum number of the rotation transition).In addition, there are four other clumps with RPs in HCN (4-3), while others have only one peaked or even no detection.Since all the clumps have evident BPs in HCO + (3-2), it is natural to ask what leads to the inconsistency of line profiles at dual-J transitions.
We attribute the main factor for the inconsistency of profiles at multi-J transitions to the difference in critical densities. 13In our case, the critical density of HCO + (3-2) is 1.6 × 10 6 cm −3 at 10 K and 1.4 × 10 6 cm −3 at 20 K.By contrast, the critical density of HCN (4-3) is 3.0 × 10 7 cm −3 at 10 K and 2.3 × 10 7 cm −3 at 20 K, which is approximately 20 times higher.Thus, different infall tracers, such as low-/high-J transitions of HCO + or HCN species, should trace different parts or layers of dense star-forming clumps (Xie et al. 2021).As such, the infall profiles will be presented best when the opacity of the source and the critical density of the tracer are well matched, as argued in Wu & Evans (2003).

Two Possible Scenarios
Given the different critical densities between HCO + (3-2) and HCN (4-3), there are two possible scenarios for our observed variety/inconsistency of line profiles at multi-J transitions: 1.While gas infalls in the outer envelope of massive clumps, the bulk motion can become more complex or even prohibited due to feedback from stars, such as outflows and stellar winds, or other dynamic processes occurring at a certain density layer.In some cases, the motion may even be reversed, resulting in an expanding motion.Consequently, there are multiple possibilities for bulk motion at the layer that HCN (4-3) traces, leading to a low detection rate of BPs at high-J transitions.2. The optical depth of molecular lines is determined primarily by the kinetic temperature, T kin , and the column density of the molecule, , where N H 2 represents the column density of molecular hydrogen and X mol denotes the abundance of the molecule.Due to variations in both N mol and T kin within our sample, the optical depth of the high-J transition τ(HCN (4-3)) can vary significantly.Consequently, in some clumps τ (HCN (4-3)) may not be sufficiently high to produce asymmetric line profiles, even if there is still gas infall motion present.
To distinguish between the two scenarios, we can compare the predicted line profiles with the observed ones.For the first scenario, the fraction of distinct profiles should be determined by the likelihood of different types of bulk motion (infall, outflow/expansion, and static).For the second scenario, the detection rate of line profiles should be lower in clumps with lower column density, while the high-J transition line should maintain the same profile as the low-J transition line in clumps with a higher column density.Figure 5 shows that the fractions of both RPs and nonasymmetric profiles systematically decrease while the fraction of BPs increases.Since all clumps have a BP HCO + (3-2), the rising trend of the fraction of HCN (4-3) with BPs suggests that the high-J transition still conveys the same bulk motion information, but only in highdensity clumps.
Caveats.We acknowledge that the peak column density, N H 2 , is based on an angular resolution of 21″, which is coarser than that of the JCMT.If the source has a centralized density distribution, the column density at the higher angular resolution should be higher than that at the lower angular resolution.Considering a Gaussian distribution of density and assuming that dust emission is optically thin, we can calculate how much column density is underestimated by N  : , ; 1, 3 where Ω 1 and Ω 2 are the JCMT and ATLASGAL beam solid angles, respectively, and is a Gaussian density model with a dispersion of σ.For a typical value σ = 20″ in our , indicating that there can be a moderate systematic underestimation of column density if observed in the JCMT beam.However, conversely, we can smooth the JCMT lines into the same resolution of 21″, which is the same as that of the column density.Since the profiles we discuss are from the averaged lines inside the cores where the profiles should be coherent (see Section 3.4), it is safe to compare the two in the context of a 21″ resolution.

Large Variations In Optical Depths of HCN (4-3)
The optical depth of HCN (4-3) is calculated by RADEX,14 a computer program that calculates the strengths of the atomic and molecular lines of interstellar clouds, which are assumed to be homogeneous (van der Tak et al. 2007).The presumed excitation conditions are as follows: (i) a background temperature of 2.73 K; (ii) a collision partner (H 2 ) volume density of 10 5 -10 6 cm −3 in the HCN cores (see details in Appendix C); and (iii) a HCN (4-3) line width of 8.8 km s −1 , which is the mean value of the observed spectra.We construct a 100 × 100 grid to calculate the optical depth of HCN (4-3), τ pred , based on two variables, the kinetic temperature T kin and the column density of the HCN molecule N HCN , within the parameter space defined by the observed values.
Calculation grids are presented in Figure 6, where the contour levels of τ = 1 and 10 are represented by solid white lines.We utilize the sample of 38 observed massive clumps to predict the optical depths, τ pred .In our calculations, we assume that the kinetic temperature T kin is equal to the dust temperature, T dust .The column density of HCN, N HCN , is determined using the , where N H 2 is the H 2 column density and X HCN is the abundance of HCN relative to H 2 , which depends on the evolutionary stage (Martinez & Paron 2023).We assign different X HCN values to clumps with distinct evolutionary types based on column (7) of Table 1 ( ) and for type 3, = ´-X 3.0 0.6 10 HCN 9 ( ) .Additionally, considering the observed HCN (4-3) lines exhibit various velocity widths (dV HCN ) ranging from 5 to 15 km s −1 , we calibrate the N HCN values to account for the effect of dV HCN using Equation (3) in Remijan et al. (2004).
Figure 6 illustrates that the optical depths of clumps exhibiting BPs (τ B ) consistently exceed those of nonasymmetric profiles (τ NA ) or RPs (τ RP ).Furthermore, in the regime of = n 10 H 6 2 cm −3 , all clumps demonstrate τ B  1, and even in the regime of = n 10 H 5 2 cm −3 , the top five clumps (indicated with labels) maintain a high level of opacity.Conversely, for a significant proportion of clumps exhibiting red and nonasymmetric profiles, the optical depths do not exceed 1 under any given conditions.The outcome is in alignment with the findings presented in He et al. (2016), where the identified infall candidates exhibit elevated H 2 column densities and H 2 volume densities in contrast to the clumps where infall motions were not detected.We acknowledge that there are still several clumps, in particular for BG013.882-00.143C1, with comparable τ pred but displaying a nonasymmetric profile or a RP.In other words, even with enough optical depth, these clumps do not show BPs any longer.Therefore, the observations are likely to support hybrid scenarios wherein an adequate optical depth is crucial for inducing BPs but the inner motion can also be complicated.A BP in low-J transition should not guarantee BPs in high-J transition.
Caveats.First, T kin is assumed to be the dust temperature averaged through the clump, T dust , which is a rough estimate.But, as shown in Figure 6, T kin has much smaller variations in τ pred than N H 2 does.This suggests that the potential bias arising from the uncertainty in T kin is mitigated.Second, there is no one-to-one correspondence between X HCN and each individual source.Consequently, although these caveats result in a relatively rough estimation of t -HCN 4 3 ( ) , the relative values of t -HCN 4 3 ( ) are reliable, allowing for qualitative analysis and further investigation.

Triple-J Transition Lines in a Subsample
We cross-match the 48 BP clumps (in HCO + (3-2) lines as reported by Shirley et al. 2013) with MALT90 surveys15 (Jackson et al. 2013) of their low-J transition counterpart HCO + (1-0) lines (reported by He et al. 2015He et al. , 2016)).Since the SMT and the Mopra are located in different hemispheres, we only have six sources which overlap with both the J = 1-0 and J = 3-2 transitions as a subsample.
Table 4 provides a compilation of line profiles from a subsample consisting of sources from Shirley et al. (2013), He et al. (2015He et al. ( , 2016)), and our work.Among the six sources with BP HCO + (3-2) lines, four sources consistently exhibit BP HCO + (1-0) lines, while the remaining two sources, BG009.212-00.202 and BG012.889+01.480,display RP HCO + (1-0) lines.For the two sources with RP HCO + (1-0) lines, the classification in column (7) in Table 1 indicates that they are both in a more evolved stage, which aligns with their extended infrared emission shown in Figure 2. BG012.889+01.480(also I18089-1732) was reported to contain a nearly face-on disk (Sanhueza et al. 2021) with a collimated SiO (5-4) bipolar outflow (Beuther et al. 2004).If the outflow direction is perpendicular to the disk plane, then the inclination angle of the outflow axis should be small and the ( )( ) in a beam of 21″.The blue, red, and gray colors stand for blue profiles, red profiles, and nonasymmetric profiles (including single peaked and nondetection), respectively.outflow motion can provide enough expanding effects along the line of sight.The argument can be further verified in the case of BG009.212-00.202 by high-resolution observation.Once verified, it is likely that the RPs are a result of outflows and bulk expansion.Previous studies have demonstrated that HCO + (3-2) lines are capable of tracing infall motion in both the early (Xie et al. 2021) and late stages of massive starforming regions (Fuller et al. 2005;Reiter et al. 2011;Klaassen et al. 2012).However, our work, although based on a limited sample size, suggests that low-J transitions such as HCO + (1-0) may be more susceptible to contamination from other bulk motions present in the outer low-density layers.On the other hand, high-J transitions like HCO + (3-2) appear to be more reliable for tracing infall motion in a more evolved stage of high-mass star-forming regions.
For the four sources with HCN (4-3) observations, we calculate the optical depths of the HCN (4-3) lines based on input parameters including the column density of HCN (N HCN ), the kinetic temperature (T kin ), the line width (dV HCN ), and vthe olume density of the collision partner (n H 2 ).The first three parameters are directly retrieved from Tables 1 and 3, while the last one, n H 2 , is given a range of (10 5 , 10 5.5 , and 10 6 cm −3 ), which is similar to what has been done in Figure 6.Overall, higher n H 2 results in a higher level of thermalization due to the collision process, and then results in a higher excitation temperature T ex and t )  .Therefore, the optical depth is the main reason for the variations of line profiles in HCN (4-3).
However, we still note that the conclusion is not significant because of the limited sample size.Therefore, it is encouraged to survey multi-J transitions in a much larger and less biased sample and to check whether the conclusion holds or not.

Low Detection Rate of Blue Profiles and Their Connection to Infall Motions
As summarized in Figure 7, two systematic investigations have been undertaken to identify BPs within massive star-forming clumps, employing the ATLASGAL and BGPS follow-up line surveys, respectively.However, the detection rate of the BPs in BGPS clumps is found to be more than 10 times lower than that observed in ATLASGAL clumps.This substantial discrepancy can be attributed to two primary factors.
First, ATLASGAL clumps are observed using low-J transitions, whereas the BGPS clumps employ high-J transitions.Furthermore, it is important to consider that the ATLASGAL line survey implemented a flux threshold of 0.25 Jy at 870 μm.This threshold was applied to ensure the inclusion of clumps with a mass of 200 M e , assuming a distance of 10 kpc and a temperature of 10 K (Jackson et al. 2013).In contrast, the BGPS line survey did not impose any specific flux threshold.Consequently, the absence of a threshold in the BGPS survey leads to the inclusion of a broader range of clumps, including those with lower flux values.Consequently, the opacity of the high-J transition line, especially in low-flux clumps, may not be sufficiently high to produce the characteristic self-absorption signature, as discussed in Section 4.1.2,therefore diluting the overall detection rate of BPs in the BGPS sample.Second, the MALT90 line survey has a spectral resolution that is 10 times greater than that of the BGPS line survey.It should be noted that a low spectral resolution is inadequate for detecting BPs induced by low infall velocities.These factors contribute significantly to the disparity observed in the detection rates of BPs between the two surveys.Moreover, such a comparison of detection rate also encourages further investigations with enhanced spectral resolution and the use of appropriate transition lines.
Nevertheless, it is essential to recognize that studying BPs is one kind of phenomenology after all, and establishing a direct link between such profiles and infall motions remains a challenging endeavor.Despite advancements in observational capabilities, the comprehension of BPs is impeded by limited insights into the physical conditions prevailing within these regions, such as the distribution of temperature and density.Additionally, the intricate nature of infall motions within these clumps, coupled with potential influences from feedback mechanisms, contributes to a notable false-positive rate when inferring infall motion from BPs.To disentangle the intricate implications of BPs, future high-resolution investigations are imperative, encompassing thorough analyses of gas kinematics (refer to Section 4.4 in Xu et al. 2023).
On another front, systematic examinations of BPs also grapple with a significant false-negative rate.For example, the low detection rate of BPs in BGPS clumps does not necessarily imply a low detection rate of infall motion, underscoring the importance of using suitable tracers for identifying infall motion.A careful balance must be maintained between the detectability of BPs and the ability of a tracer to probe the desired depth.Low-J transitions, for instance, tend to produce BPs more readily due to sufficient opacity, albeit primarily tracing the gas envelope.On the other hand, high-J transitions effectively capture inner gas motion but may be too optically thin to generate BPs in clumps with low column densities.

Mapping Clump-scale Global Collapse
JCMT mapping observations with an angular resolution of 14″ have provided valuable insights into the resolution of infall motions.Among the observed clumps with HCN (4-3) BPs, there are five clumps with the highest predicted optical depths (τ pred ), namely BG081.721,BG133.748,BG030.719,BG012.889, and BG133.949.It is worth noting that these five clumps also exhibit the consistently highest S/Ns, as illustrated in Figure 3, suggesting the robustness of RADEX line simulation.Furthermore, they show the most extended emission patterns, as depicted in Figure B1, further emphasizing their significance in the study of infall motion.However, it should be noted that BG030.719presents a unique case in which two separated HCN cores with opposite profiles are observed, resulting in a limited number of pixels available for mapping the infall motion with sufficient sampling.Therefore, excluding BG030.719, the remaining four clumps serve as a subsample that can be effectively utilized for infall motion mapping analyses.
As shown in all four clumps in Figure 8, the HCN (4-3) lines show strong spatial correlation with submillimeter continuum (dust) emission and BP spectra in most of the clumps.Such line profiles are expected for an optically thick tracer of idealized collapsing clouds in which the excitation temperature is rising toward the center.What is important to note here is the extent (at least nine independent beams) over which this spectral signature is observed, and the absence of any other line asymmetries (see Section 3.5), strongly suggesting that all four clumps are undergoing global collapse (see a typical example, of SDC335, in Peretto et al. 2013 andXu et al. 2023).A radiative-transfer model combined with temperature and density profiles derived from far-infrared data can be used to fit the map of line profiles, inferring the infall velocity and mass infall rate.

Infall Parameters Fitted by the Hill5 Model
According to Section 3.5, a sample of 11 HCN cores are undergoing infall motion.To estimate the infall velocity, we used the "Hill5" model, first introduced by De Vries & Myers (2005).The Hill5 model assumes that the excitation temperature in the front of the cloud increases inward as a   The model has five free parameters to fit: (i) the peak excitation temperature T peak , (ii) the velocity dispersion of the molecular line σ, (iii) the optical depth of the core t core , (iv) the systematic velocity v LSR , and (v) the infall velocity of the gas in the core v infall .Formula derivations for the model are presented in detail in Appendix D.
To determine the global accretion rate toward these cores, we fit the average spectra of the HCN (4-3) emission across the cores.The S/N of the 10 spectra has an average of ∼30, satisfying the criterion for the Hill5 model fitting.Although two spectra of BG028.565-00.236C1 and BG039.267-00.589C1have relatively lower S/N  6, the high spectral resolution of 0.2 km s −1 assure enough effective data points for model fitting.Most cores have extended velocity wings that are assumed to be induced by molecular outflows.To reduce the contamination from the wings, we cut out the velocity channels which are non-Gaussian parts in Gaussian fitting process (see Section 3.2).The preserved channels are fitted channels marked as orange bands in Figure 9, with a bandwidth of 30-60 channels to cover the BP features.The uncertainties in the fitting are given by Python package lmfit that explicitly explore the parameter space and determine confidence levels.As initial guesses of the fit we assume t core ranging from 0.1 to 30, a v LSR between v LSR − 5 km s −1 and v LSR + 5 km s −1 , v infall between 0.1 and 4 km s −1 , σ between 0 and s HCN (from Gaussian fitting), and T peak between 2.73 and 30 K.
The fitted spectra are shown in Figure 9, with five parameters shown in the top left of each panel.The blue band indicates the velocity range of the infall motion, that is, from v LSR − v infall to v LSR + v infall .The fitted parameters as well as the velocity channels used for fitting are sorted in columns (3)-(7) of Table 5.

Infall Velocity versus Freefall Velocity
Since the freefall velocity represents the typical timescale of the gravitational collapse of a star-forming clump, comparing observed infall velocity to freefall velocity helps to understand how fast the star formation proceeds in these massive clumps.The freefall velocity, v ff , is calculated by where M enc is the mass enclosed within a radius R. Since the HCN cores are mostly even smaller than the clumps, we need to scale down v ff at the HCN core scale.Substituting Equation (C1) into Equation (4), the freefall velocity should   1 into Equation (4), the freefall velocity has a range of 2.0-6.8km s −1 , with mean and median values of 3.6 and 3.2 km s −1 .Therefore, the infall velocity fraction, infall  , defined as the ratio of the infall velocity to freefall velocity, ranges from 5% to 74%, with both mean and median values of 32%.The minimum value is consistent with what has been found in Wyrowski et al. (2016), but the maximum and mean/ median values are systematically larger.However, the large fraction should be due to the different distances in our sample, because the radii of clumps with smaller distance tend to have lower mass (M clump ∝ D 2 , where D is the distance).If we exclude the three nearest clumps, BG081.721+00.57,BG133.748+01.19, and BG133.949+01.06,then the median fraction is ∼20%.
Since the timescale is directly related to the velocity at a given radius, the ratio of the infall timescale (τ infall ∝ 1/v infall ) to the freefall timescale (τ ff ∝ 1/v ff ) is inversely proportional to infall  .This means that the dense region of the clump, as indicated by the HCN cores, will undergo collapse within a few to several tens of freefall timescales.

Mass Infall Rate
Assuming a simplified spherical model, the mass infall rate is calculated by 8.9 10 2.809 10 cm 0.1 pc 1 km s yr , 5 where m H 2 is the molecular weight per hydrogen molecule (m = 2.809; ) .We note that the HCN core BG039.267-00.589C1has not been resolved, so we use the physical scale of the beam size as an upper limit, and therefore the derived mass infall rate is an upper limit as well.The calculated mass infall rate is then listed in column (8) of Table 5.
The mass infall rate exhibits a wide range, ranging from 0.15 to 32.1 ×10 −3 M e yr −1 , which aligns with typical values observed in high-mass clumps (He et al. 2016;Yu et al. 2022).The mean and median values of the mass infall rate are 7.6 × 10 −3 and 4.5 × 10 −3 M e yr −1 , respectively, which are in good agreement with the values derived from HCO + (1-0) lines in a sample of 11 IRDCs (Xie et al. 2021) and HCO + /HNC (1-0) lines in a sample of 33 IRDCs (Pillai et al. 2023).It should be noted that the mass infall rate obtained from HCN (4-3) lines primarily traces the inner regions of massive clumps, while the mass infall rate derived from HCO + (1-0) lines predominantly represents the outer parts or envelopes.Nevertheless, the remarkable consistency in mass infall rates between these two tracers suggests a continuous accretion process from the clump envelope to the inner region.This finding is supported by previous studies indicating minimal variations in mass infall rates during the evolution of high-mass clumps (He et al. 2016).Utilizing multi-J comparisons allows us to establish a connection between mass infall rates at various scales.Therefore, it is crucial to conduct follow-up high-resolution observations (e.g., by the Atacama Large Millimeter/submillimeter Array, the Northern Extended Millimeter Array, or the Submillimeter Array, SMA) to precisely quantify the amount of mass ultimately transferred to the protostars.For instance, Xu et al. (2023) collected threescale observations and revealed a consistent accretion process from clump-scale global collapse to core-scale gas feeding in the case of SDC335.Furthermore, high-resolution observations can improve our understanding of the concept of global collapse.Although a spherical model featuring a collapsing shell can adequately explain the BPs, most observations indicate that the inflows manifest as gas streams or elongated filamentary structures (Kirk et al. 2013;Peretto et al. 2013;Liu et al. 2016;Lu et al. 2018;Xu et al. 2023;Yang et al. 2023).The high-resolution observations of massive clumps with BPs are promising for deepening our understanding of the inner dense gas distribution and kinematics as well (Contreras et al. 2018;Xu et al. 2023).

Conclusions
Leveraging the efficient-mapping advantages of the JCMT HARP instrument, we perform an HCN (4-3) mapping survey of 38 representative massive star-forming clumps in the BGPS, guided with HCO + (3-2) blue asymmetric line profiles (BPs).The high-J transition with a critical density of >10 7 cm −3 , combined with previous low-J transition data, mapping observational mode, and a wide range of physical properties in such a large sample, help deepen our understanding of BPs and their connection to gas infall motion in massive starforming clumps.Our main findings are summarized as follows: 1. We integrate the line intensity of HCN (4-3) lines and produce 38 HCN (4-3) M0 maps, of which 32 have detection and six have no detection.Thirty M0 maps show isolated emission regions, while two M0 maps show double-emission regions.In total, 34 HCN emission cores (HCN cores) are identified by the SExtractor algorithm.The HCN (4-3) spectra extracted from the HCN cores have velocities consistent with those of N 2 H + (3-2) lines, justifying the usage of N 2 H + (3-2) as the systematic velocity tracer.2. The averaged HCN (4-3) lines show various line profiles, including 14 blue, four red, and 22 nonasymmetric profiles, rather than keeping the same BP as the lower-J transition HCO + (3-2) exhibits.Adopting the HCN (4-3) maps, we found intrinsic variations of the line profile in three HCN cores, suggesting potential rotation motion.The remaining 11 HCN cores serve as promising candidates of infall motion in massive starforming regions.3. We find an increasing rate of BPs along the H 2 column density and the opacity of HCN (4-3) lines calculated from the non-LTE radiative-transfer code RADEX, suggesting insufficient opacity should be the main reason for the low profile retention rate of 36.8% (14 BPs out of 38 massive clumps).However, even with sufficient HCN (4-3) opacity, there are still some detections of red or nonasymmetric profiles, which suggests gas undergoing different motions in different density layers, traced by different transitions.4. A six-source subsample has three transitions, HCO + (1-0), HCO + (3-2) and HCN (4-3), with critical density ranging from 4.5 × 10 4 cm −3 to 2.3 × 10 7 cm −3 .Although limited by sample size, single-peaked line profiles have systematically low opacity τ = 1, while BPs have high enough opacity τ  1.Additionally, we find that two sources, namely BG009.212-00.202 and BG012.889+00.490,which exhibit bipolar outflows at relatively small inclination angles, display red profiles in the lowest-J transition of HCO + (1-0).These profiles can be attributed to the presence of expanding gas envelopes along the line of sight. 5. Comparison between two line surveys guided by ATLASGAL (He et al. 2015(He et al. , 2016) ) and BGPS (Schlingman et al. 2011;Shirley et al. 2013) highlights the importance of an appropriate tracer, high spectral resolution, and column density threshold in searching for BPs in a large sample.We also caution that the BP is a phenomenological problem after all, and the connection between BPs and infall needs in future to be calibrated by a multi-J transition line survey for a large sample.6.If all 11 BPs are produced by infall motions, we adopt the Hill5 model to fit the infall velocity of the HCN cores, ranging from 0.2 to 1.6 km s −1 , with mean and median values of 1.0 and 1.1 km s −1 .The infall velocities account for a fraction of 5%-74% of freefall velocity, indicating the HCN cores will collapse within a few to several tens of freefall timescales.7. Assuming a simplified spherical model, the mass infall rate can be calculated, ranging from 0.15 to 32.1 × 10 −3 M e yr −1 , with mean and median values of 7.6 × 10 −3 and 4.5 × 10 −3 M e yr −1 , which is consistent with what has been found in the low-J transition HCO + (1-0).The consistency of the mass infall rate among different transitions (i.e., different density layers) suggests a steady accretion process from the clump gas envelope to the inner region, as proposed by Xu et al. (2023).

Figure 2 .
Figure 2. Overview of the 38 clumps.Background shows the Spitzer infrared three-color map (blue: 3.6 μm; green: 4.5 μm; red: 8 μm).White contours are the ATLASGAL 850 μm or the JCMT SCUBA-2 850 μm continuum emission for three sources without ATLASGAL data.The JCMT observing fields are outlined by yellow lines, and the field names are shown in the upper left of each panel.The ATLASGAL/JCMT beam and the scale bar of 1 pc are shown on the lower left and right, respectively.

Figure 3 .
Figure 3. Averaged HCN (4-3) lines from the defined HCN cores, whose names are labeled in the top left of each panel.For solid detections, the HCN (4-3) lines are fitted by Gaussian profiles.The 2σ threshold of the best-fitting model is shown with a green dashed line and a red line, respectively.The results of the Gaussian fitting (amplitude A, centroid velocity Δx, and velocity dispersion σ) are shown on the top right.Nondetection spectra are not fitted, and only the baselines (green horizontal lines) are shown.The systematic velocities in previous surveys are marked with orange dashed lines.

Figure 4 .
Figure 4. Consistency between V LSR,HCN and V LSR,thin .Red: red-profile spectral lines; blue: blue-profile spectral lines; black: no clear line profiles.The error bars in two directions are given by spectral line-fitting errors.

Figure 5 .
Figure 5.The number distribution (histogram) and the fraction (line-connected scatter plot) of different line profiles change with peak column density bins -N log H cm 2 2 magnitude along with n H 2 , BG008.458-00.222 and BG011.083-00.536 with single-peaked profiles have t -

Figure 6 .
Figure 6.Simulated grids of predicted optical depth (τ pred ) based on the kinetic temperature (T kin ) and the column density of HCN (N HCN ), with three different volume densities of collisional partner H 2 (n H2 ).The white lines outline the contour levels of τ = 1 and 10.The 38 clumps are shown as filled circles, with blue, red, and gray colors representing blue profiles, red profiles, and nonasymmetric profiles, respecitvely.The five clumps with the highest τ pred are labeled.The color bars are shown in the lower right.
(2)-(4).The critical density at 20 K is listed below the transition lines(Shirley 2015): "BP" = blue profile, "RP" = red profile, and "S" = single-peaked profile.The opacities of HCN (4-3) lines at three different volume densities of the collision partner H 2 are shown in columns (5)-(6).The value of opacity is shown in the form of a(b), denoting a × 10 b .

Figure 7 .
Figure 7.The low detection rate of blue profiles (BPs) in star-forming clumps guided by ATLASGAL and BGPS.In this, 3246 ATLASGAL clumps with 870 μm flux larger than 0.25 Jy are observed in HCO + /N 2 H + (1-0) at a spectral resolution of 0.11 km s −1 by the Mopra MALT90 line survey; 732 clumps have a solid detection of S/N > 3, among which 231 clumps have BPs by the canonical criterion of δV < −0.25.The detection rate of BPs is then 31.6% (He et al. 2015, 2016).Further, 6194 BGPS clumps without flux threshold are observed in HCO + /N 2 H + (3-2) at a spectral resolution of 1.1 km s −1 by the SMT line survey (Schlingman et al. 2011; Shirley et al. 2013); 1795 clumps have a solid detection of S/N > 3, among which 48 have BPs, giving a detection rate of 2.8% (Shirley et al. 2013).

Figure 9 .
Figure 9.The averaged HCN (4-3) lines are indicated by the black solid line, while the best-fit Hill5 model are indicated by the red solid line, with five parameters shown in the upper-left corner.We highlight the velocity range used for Hill5 model fitting in orange color at the bottom.The red band indicates the systematic velocity v LSR .The blue band indicates the velocity span of the infall motion, that is, from v LSR − v infall to v LSR + v infall .

Figure A1 .
Figure A1.The M0 maps of HCN (4-3) in 38 fields.The names are labeled in the top left of each panel.The green solid ellipses mark the HCN cores by SExtractor, while the green dashed circles outline the regions where the spectra are extracted in the nondetection fields.The JCMT beam and its physical scale in each map are shown in the bottom left.

Table 3
Parameters of HCN Emission Lines

Table 4
Line Profiles at Multi-J Transition Lines Source names are inherited from column (1) of Table1.Profiles in multi-J transition lines are shown in columns The Astrophysical Journal SupplementSeries, 269:38 (25pp), 2023 December  Xu et al.be constant over the self-gravitating clumps.Therefore, we can directly compare the infall velocity and the freefall velocity of the HCN cores.As shown in column (5) of Table5, the infall velocity of the 11 clumps has a range of 0.2-1.6 km s −1 , with mean and median values of 1.0 and 1.1 km s −1 .Adopting the clump radius and mass in columns (8) and (11) of Table Evans et al. 2022), n H 2 and R deconv are the volume density and physical radius of the defined HCN core, and v infall is the infall velocity fitted from the Hill5 model.The n H 2 is estimated by N R 2

Table 5
Hill5 Fitting Result of BP HCN (4-3) Lines Note.Core name is listed in column (1).Velocity range used for model fitting is listed in column (2).Hill5 fitting results including optical depth, velocity at local standard of rest, infall velocity, velocity dispersion, and peak excitation temperature are listed in columns (3)-(7).The mass infall rate is listed in column (8).