The central molecular zone (CMZ) of the Milky Way, located at the Galactic Center with a radius of approximately 300 pc, exhibits distinctive characteristics compared to the star formation environment in the solar neighborhood (J. D. Henshaw et al. 2023). There is molecular gas with a mass exceeding 10 7 M e (M. Morris & E. Serabyn 1996;S. N. Longmore et al. 2013;C. Battersby et al. 2024), an average gas number density of approximately 10 4 cm -3 (T. A. D. Paglione et al. 1998;S. N. Longmore et al. 2013), a high gas temperature of 50-100 K (Y. Ao et al. 2013;A. Ginsburg et al. 2016;N. Krieger et al. 2017), a strong magnetic field of about 1 mG (T. Pillai et al. 2015;X. Lu et al. 2024), and strong turbulence of Mach numbers of around 30 (J. M. Rathborne et al. 2014;J. D. Henshaw et al. 2016). A. T. Barnes et al. (2017) found an incipient star formation rate for the CMZ of about 0.09 ± 0.02 M e yr -1 , which is one order of magnitude lower than expected based on the dense gas-star formation relationship (S. N. Longmore et al. 2013;J. Kauffmann et al. 2017b;X. Lu et al. 2019a). A possible explanation for this could be that the clouds in the CMZ are in an early stage of evolution or that strong turbulence suppresses the collapsing and fragmentation of the clouds (J. M. D. Kruijssen et al. 2014; M. R. Krumholz et al. 2017).
The initial mass function (IMF) is the distribution of stellar masses of zero-age main sequence stars. It is crucial for various fields in astrophysics, from star formation to galaxy evolution. It is believed to be universally applicable (P. Kroupa 2002; N. Bastian et al. 2010), although potential variations with respect to local environments have been found (e.g., A. M. Hopkins 2018; J. Li et al. 2023). The high-mass end of the IMF roughly follows a power law of the following form:
µ a -M dN d M log
. P. Kroupa (2002) fitted a power-law index of α = 1.35. However, there is still controversy over the origin of the IMF and its dependence on the environment.
Stars form in dense cores (E. A. Bergin & M. Tafalla 2007). Therefore, it has been suggested that the IMF is related to the core mass function (CMF), an analog of the IMF that describes the distribution of core masses in a star formation region (P. Hennebelle & G. Chabrier 2008; S. S. R. Offner et al. 2014;E. Ntormousi & P. Hennebelle 2019). Numerical simulations have produced CMFs with a shape consistent with that of the IMF (R. S. Klessen 2000; P. Padoan & Å. Nordlund 2011). Therefore, studying the CMF is crucial for exploring the origin of the IMF and investigating the early stages of star formation. Recent results from the ALMA-IMF large program (A. Ginsburg et al. 2022;F. Motte et al. 2022;Y. Pouteau et al. 2023) suggested that as molecular clouds evolve from quiescent to burst phases, the CMF may change from Salpeterlike to top-heavy, and then it might return to Salpeter-like as the clouds approach the end of their star-forming phase.
The CMZ is a specific target of interest for studying the relation between the CMF and IMF in an extreme star-forming environment. The IMF in young massive star clusters in the CMZ has been measured to be top-heavy, i.e., with an overpopulation of higher mass stars with respect to the "canonical" power-law form of µ - M dN d M log 1.35 (B. Hußmann et al. 2012; M. W. J. Hosek et al. 2019). Meanwhile, previous ALMA observations of dense cores in the CMZ clouds found signatures of top-heavy CMFs (e.g., X. Lu et al. 2020). However, there has been a dearth of constraints on the temperature of dense cores in the CMZ while estimating the core masses. Usually, a uniform core temperature (e.g., 20 K) is assumed, which could be biased and lead to large uncertainties in the CMFs (see Appendix D of X. Lu et al. 2020). Furthermore, the analysis of the virial equilibrium of dense cores in the CMZ has been hindered because of a lack of spectral line observations that are able to measure the turbulent linewidth. Without the knowledge of the virial states of cores, we cannot separate gravitationally bound cores that are deemed to collapse and form stars from unbound cores that do not form stars.
To more robustly characterize the masses and virial states of cores in the CMZ, we employ the H 2 CO and CH 3 CN molecular lines to constrain the temperatures and velocity dispersions of cores in three massive molecular clouds in the CMZ. The rotational transitions of H 2 CO and the CH 3 CN K-ladder are widely used thermometers in observations of dense gas in clouds and cores (e.g., D. Downes et al. 1980;J. H. Bieging et al. 1982;R. Zylka et al. 1992; J. G. Mangum et al. 2008;Y. Ao et al. 2013;X. D. Tang et al. 2013). In particular, we focus on the following H 2 CO transitions:
and 2 2,1 -2 2,0 (E u /k = 68.11 K, f rest = 218.76007 GHz). Their critical densities at the temperature of 100 K is ~3 × 10 5 cm -3 .
For CH 3 CN, the transitions of interest include the K-ladder from J K = 12 0 -11 0 (E u /k = 68.86 K, f rest = 220.7472 GHz) to 12 7 -11 7 (E u /k = 325.899 K, f rest = 220.5944 GHz), with a critical density at 100 K of ~5 × 10 6 cm -3 . The cloud sample includes the relatively quiescent Dust Ridge cloud e (cloud e hereafter), the 20 km s -1 cloud in an intermediate star formation state, and the actively star-forming Sgr C (X. Lu et al. 2019bLu et al. , 2021)). These clouds are selected from a representative sample of massive clouds in the CMZ (J. Kauffmann et al. 2017aKauffmann et al. , 2017b) ) and show clear fragmentation in our ALMA 1.3 mm continuum observations at a resolution of 2000 au (X. Lu et al. 2020). Additionally, we incorporate the impact of external pressure into the virial analysis by using the SMA molecular line data from X. Lu et al. (2019b).
The structure of the paper is organized as follows. In Section 2, we describe the observations and data reduction. In Section 3, we present the observational results. We then discuss the virial parameters and the CMF within the molecular clouds in Section 4. Finally, we summarize our findings in Section 5. Throughout the paper, we adopt a distance of 8.277 kpc to the Galactic Center (GRAVITY Collaboration et al. 2022) and assume that all the clouds under consideration are at this distance.
The ALMA observations have been presented in X. Lu et al. (2020Lu et al. ( , 2021)), where details of the observational setups and data reduction can be found. Here, we reiterate the information that is related to our analysis in the current work.
The ALMA 12-m array observations in Band 6 toward the three clouds were conducted in the C40-5 and C40-3 configurations during 2017 April and July under the Project 2016.1.00243.S (PI: Q. Zhang). The correlators were configured to cover frequency ranges of 217-221 GHz and 231-235 GHz, with a uniform frequency resolution of 0.977 MHz (equivalent to a velocity resolution of 1.3 km s -1 ). In this paper, we present the results derived from the 1.3 mm continuum and molecular line data of H 2 CO transitions 3 0,3 -2 0,2 , 3 2,2 -2 2,1 , and 3 2,1 -2 2,0 , as well as the CH 3 CN transitions 12 0 -11 0 to 12 7 -11 7 .
The calibration and imaging of the data were performed using the Common Astronomy Software Application (CASA) version 5.4.0 (CASA Team et al. 2022). The calibrated data were then imaged using the tclean task with a robust Briggs parameter of 0.5. The achieved angular resolution of the images is approximately 0 . 24 × 0 . 18. The sensitivity of the spectral line data is in the range of 1.6-2.0 mJy beam -1 per 1.3 km s -1 channel.
In X. Lu et al. (2020), the identification of dense cores in the three clouds has been carried out using the dendrogram algorithm implemented in the astrodendro package 23 upon the continuum images, where the "leaves" of the dendrograms were considered as dense cores. The three key parameters in astrodendro, i.e., the minimum intensity, the minimum significance, and the minimum area of leaves, were set to 4σ, 1σ, and one synthesized beam size (i.e., 30.125 pixels), respectively, where σ = 40 mJy beam -1 . In cloud e, Sgr C, and the 20 km s -1 cloud, 89, 274, and 471 cores were identified, respectively. The 1.3 mm continuum flux within the boundary of each leaf, defined by the 4σ contour, without removing any background contribution from the larger structures within the dendrogram hierarchy, was adopted as the flux of a core. Here, we directly adopted the core catalogs of X. Lu et al. (2020), and extracted the H 2 CO and/or CH 3 CN spectra toward the cores for fitting.
As discussed in X. Lu et al. (2020), the maximum recoverable scale as determined by the short baselines in the ALMA array during our observations is 7″ (~0.3 pc), whereas we focus on the dense cores at much smaller scales of 2000 au. The missing flux issue of ALMA as an interferometer unlikely affects the measurement of the fluxes of the cores significantly.
Figure 1 illustrates the integrated intensity maps of the H 2 CO 3 0,3 -2 0,2 and CH 3 CN 12 0 -11 0 /12 1 -11 1 transitions in the three molecular clouds. Note that no masking was applied to the moment maps. The velocity ranges adopted for integrating the intensities are marked in the panels. We integrated the 12 0 -11 0 and 12 1 -11 1 transitions of CH 3 CN because the two lines are blended, and therefore cannot be separated.
As shown in Figure 1, we found that the CH 3 CN emission is mostly only detected toward the 1.3 mm emission. However, the H 2 CO 3 0,3 -2 0,2 emission is not always associated with the 1.3 mm continuum, especially toward Sgr C and the 20 km s -1 cloud where a significant number of filamentary structures in the H 2 CO emission are observed. A subset of these filaments have been identified as protostellar outflows in X. Lu et al. (2021). The others are likely related to pc-scale shocks, which will be discussed in other works of the series (K. Yang et al. 2025, in preparation). For the H 2 CO emission spatially coincident with the identified cores, we assumed that it traces dense gas in the cores, and therefore can be used to estimate the gas temperature and linewidth of the cores. In Appendix C, we compared H 2 CO gas associated with the cores and that not associated with any cores, and found that the two samples of H 2 CO emissions present statistically different temperatures, with the H 2 CO associated with the cores showing lower temperatures.
The left-hand panels in Figure 2 provide examples of the observed mean spectra of H 2 CO within dense cores from the three clouds, showing the three transitions 3 0,3 -2 0,2 , 3 2,2 -2 2,1 , and 3 2,1 -2 2,0 . The right-hand panels in Figure 2 present the observed mean spectra of CH 3 CN within dense cores from the three clouds, encompassing lines from 12 0 -11 0 to 12 7 -11 7 . Gaussian fits were employed on the mean H 2 CO and CH 3 CN spectra to estimate temperatures and velocity dispersions of the dense cores, which were later used for estimating the masses and virial states of the cores.
Note that when both H 2 CO and CH 3 CN lines are detected toward a core, the fitting result using CH 3 CN was adopted because in such cases the H 2 CO lines are always optically thick and show self-absorption making the fit impractical (see Figure 3).
For the fitting of the H 2 CO lines, we employed the FFTL codefoot_2 that assumes the local thermodynamic equilibrium (LTE) conditions. The LTE is likely achieved in the cores given the high densities of 10 6 cm -3 (P. F. Goldsmith 2001).
Nonetheless, to validate the results, we also performed fitting using the non-LTE method implemented with RADEX (F. F. S. van der Tak et al. 2007) within the PySpecKit package (A. Ginsburg et al. 2022b), with results presented in Appendix A. We found consistent temperatures and velocity dispersions between the LTE and non-LTE methods, as shown in Figure A4.
The FFTL code performs forward modeling, in which model spectra are constructed based on the input temperature, column density, velocity dispersion, and the centroid velocity. Then, the difference between the model spectra and the observed one is minimized using the lmfit packagefoot_3 to derive the best-fit model spectrum, whose input parameters (most notably, the temperature and the velocity dispersion) are taken as the final fitting result.
When selecting dense cores for fitting H 2 CO, we required the H 2 CO 3 0,3 -2 0,2 spectra to have a signal-to-noise ratio (S/N) ratio higher than 5, where the rms noise of the mean spectra was measured between the frequency range of 218.60-218.65 GHz that is mostly line-free. Note that there are several line-rich hot cores where emission of certain lines still exists in this frequency range. The rms noise for these hot cores could be overestimated. Then, there were two scenarios:
(i) When one or both of the other two H 2 CO transitions, 3 2,2 -2 2,1 and 3 2,1 -2 2,0 , have S/N ratios higher than 3, we fit the three transitions simultaneously using the FFTL code. (ii) When both of the two transitions have S/N ratios lower than 3, we derived an upper limit for the temperature: we used the 3σ value as the upper limit for the peak intensity of the two lines, and got the upper limit of the line ratio between these two transitions and the 3 0,3 -2 0,2 transition. This line ratio was then converted to a temperature upper limit following a best-fit power-law relation between the line ratio and the temperature, which is derived from the best-fit temperatures in scenario (i). There are 24 cases in the three clouds where we take this approach to estimate upper limits of the core temperature. Such upper limits are marked with "<" in front of the temperature values in Table 1. Details of the line ratio-temperature relation can be found in Appendix B.
For the cores where CH 3 CN 12 0 -11 0 /12 1 -11 1 lines have a S/N ratio higher than 5, we also fit the CH 3 CN 12 0 -11 0 to 12 7 -11 7 transitions to derive the temperature and velocity dispersion. The rms noise was measured from the line-free channels in the frequency range of 220.55-220.58 GHz. We utilized the emanon codefoot_4 with the LTE assumption, which adopts the same forward-fitting approach as FFTL.
As mentioned above, for all the cores with CH 3 CN detections, the H 2 CO lines are also detected. We adopted the fitting results of CH 3 CN because the H 2 CO lines are optically thick and exhibit pronounced self-absorption, and therefore cannot be fitted robustly (Figure 3). We note that a systematic bias may exist between the best-fit parameters from the two lines because they trace different gas components within the cores given their different excitation conditions. CH 3 CN likely traces the hotter interiors of the cores, whereas H 2 CO traces more extended dense gas.
To validate the fitting results of CH 3 CN, we conducted a cross-check using the XCLASS package that assumed the LTE conditions as well (T. Möller et al. 2017). As shown in Figure A5 of Appendix A, we found consistent results between the emanon code and the XCLASS package.
In the end, for cloud e, Sgr C, and the 20 km s -1 cloud, 15, 120, and 85 cores were successfully fitted with H 2 CO, and 4, 17, and 12 cores were fitted with CH 3 CN. Figure 4 shows an overview of the core temperature. All the H 2 CO and CH 3 CN spectra detected toward the cores and best-fit results are presented Figures A1 and A2. Histograms of temperatures and velocity dispersions are presented in Figure A3. The fitting results for the first 10 dense cores in each of the three clouds are showcased in Table 1. A complete catalog of fitting results for all the dense cores within the three clouds can be found in a machine-readable format.
In Table 1, physical properties of the dense cores within the three molecular clouds are provided. The catalog includes 19, 137, and 97 dense cores in cloud e, Sgr C, and the 20 km s -1 cloud, respectively, whose H 2 CO and/or CH 3 CN spectra were successfully fitted.
We assumed that gas and dust in the cores are coupled, and therefore their temperatures are equal (T dust = T gas ). This assumption are justified because the gas densities in the cores are expected to be sufficiently high (10 6 cm -3 ; P. F. Goldsmith 2001): the densities were estimated to be ~10 7 cm -3 assuming a T dust of 20 K in X. Lu et al. (2020); if higher temperatures of several hundreds K were adopted (see Section 3.2), the resulting densities would be an order of magnitude lower but still ~10 6 cm -3 .
As discussed in Section 3.2.1, when both CH 3 CN and H 2 CO lines are detected toward a core, we adopted the bestfit LTE temperature of the former. When only H 2 CO is detected, the best-fit LTE temperature using the H 2 CO lines is adopted. If neither CH 3 CN nor H 2 CO is detected, a dust temperature of 20 K is adopted, which is the assumption in X. Lu et al. (2020) and corresponds to the typical value of dust temperatures in cold cores (E. A. Bergin & M. Tafalla 2007).
Assuming optically thin dust emission, the core mass was estimated following R. H. Hildebrand (1983):
where D is the distance to the core (D = 8.277 kpc), S ν is the dust emission flux, η = 100 is the gas to dust mass ratio, κ ν = 0.899 cm 2 g -1 (V. Ossenkopf & T. Henning 1994) is the dust absorption coefficient, T dust is the core temperature, and B ν (T dust ) is the Planck function at the temperature T dust . Note that for the cores with only upper limits of temperatures (see Section 3.2.1), the derived masses would be lower limits, which we have marked with ">" in front of their mass values in Table 1. Also note that the gas-to-dust ratio in the CMZ
) cloud e 17:46:47.19, -28:32:16.93 0.46 1200 L L L L L L L L 0.69 L L 17:46:46.98, -28:32:14.43 0.48 1320 L L L L L L L L 0.68 L L 17:46:48.39, -28:32:10.31 4.07 2250 L L L L L L L L 6.12 L L 17:46:47.05, -28:32:09.91 0.72 1500 138.3 ± 49.7 83 ± 21 L 138.3 ± 49.7 3.86 ± 0.32 4.32 ± 0.63 L 3.86 ± 0.32 0.12 5.50 ± 0.55 15.01 17:46:47.07, -28:32:08.93 21.53 3950 69.2 ± 14.7 73 ± 0.5 94.4 ± 9.7 94.4 ± 9.7 2.73 ± 0.25 1.36 ± 0.24 3.02 ± 0.19 3.02 ± 0.19 3.93 5.21 ± 0.29 4.76 17:46:47.01, -28:32:08.88 0.25 1150 L L L L L L L L 0.79 L L 17:46:46.93, -28:32:08.57 0.2 1030 95.4 ± 34.6 70 ± 5.5 L 95.4 ± 34.6 1.34 ± 0.19 1.34 ± 0.42 L 1.34 ± 0.19 0.05 2.09 ± 0.38 2.31 17:46:47.07, -28:32:07.26 152.95 8160 L L 237.9 ± 16.8 237.9 ± 16.8 L L 2.53 ± 0.1 2.53 ± 0.1 15.29 2.70 ± 0.12 2.97 17:46:46.95, -28:32:08.02 1.77 1820 L L L L L L L L 2.53 L L 17:46:46.91, -28:32:07.58 1.1 1600 L L L L L L L L 1.72 L L Sgr C 17:44:41.38, -29:28:29.90 19.33 4620 125.1 ± 20.3 140 ± 7 L 125.1 ± 20.3 2.34 ± 0.1 2.18 ± 0.32 L 2.34 ± 0.1 4.79 3.43 ± 0.18 0.54 17:44:41.32, -29:28:30.03 0.34 1360 88 ± 28 101 ± 63 L <89.1 1.56 ± 0.2 1.27 ± 0.37 L 1.56 ± 0.2 <0.09 2.61 ± 0.41 0.26 17:44:40.81, -29:28:27.70 0.37 1290 L L L L L L L L 0.84 L L 17:44:41.21, -29:28:26.84 1.1 2330 113.5 ± 42.8 125 ± 16 L 113.5 ± 42.8 1.56 ± 0.18 1.43 ± 0.53 L 1.56 ± 0.18 0.31 2.30 ± 0.33 0.13 17:44:41.20, -29:28:23.94 0.61 1680 43.8 ± 7.4 63 ± 39 L 43.5 ± 7.3 1.24 ± 0.14 1.02 ± 0.28 L 1.24 ± 0.14 0.50 3.02 ± 0.41 0.31 17:44:40.76, -29:28:19.77 0.36 1370 L L L L L L L L 0.77 L L 17:44:40.79, -29:28:19.50 1.15 1310 26.8 ± 7.1 26 ± 26 L <89.9 1.36 ± 0.21 1.35 ± 0.31 L 1.36 ± 0.21 >0.32 4.04 ± 0.78 0.52 17:44:40.78, -29:28:19.09 1.14 1310 73.7 ± 31.2 130 ± 12 L <99.5 1.7 ± 0.33 1.53 ± 0.63 L 1.7 ± 0.33 >0.28 3.15 ± 0.72 0.35 17:44:40.68, -29:28:19.20 0.43 1440 L L L L L L L L 0.90 L L 17:44:40.73, -29:28:19.01
) 17:45:36.41, -29:06:30.01 2 17:45:36.53, -29:06:29.63 0.27 1040 L L L L L L L L 0.36 L L 3 17:45:36.53, -29:06:29.38 0.33 1240 L L L L L L L L 0.51 L L 4 17:45:36.23, -29:06:29.28 0.34 1330 L L L L L L L L 0.55 L L 5 17:45:36.27, -29:06:28.57 0.25 1360 L L L L L L L L 0.52 L L 6 17:45:36.64, -29:06:26.83 0.86 1720 101.5 ± 49.7 125 ± 61 L 101.5 ± 49.7 2.39 ± 0.37 2.99 ± 0.96 L 2.39 ± 0.37 0.21 3.90 ± 0.70 6.12 7 17:45:36.34, -29:06:26.57 0.26 1200 L L L L L L L L 0.42 L L 8 17:45:36.70, -29:06:24.69 0.5 1790 61.2 ± 20.4 69.1 ± 3.1 L 61.2 ± 20.4 3.43 ± 0.51 3 ± 1 L 3.43 ± 0.51 0.21 7.34 ± 1.18 11.18 9 17:45:36.41, -29:06:24.27 0.33 1590 L L L L L L L L 0.71 L L 10 17:45:36.31, -29:06:23.97 0.35 1290 L L L L L L L L 0.46 L L
Note. This table lists physical parameters of the first 10 dense cores from each of the three clouds; the complete version for all cores in the three clouds is available. Columns (2), (3), and (4) present the coordinates, 1.3 mm continuum flux density, and effective radius of the cores, respectively, as determined by astrodendro (X. Lu et al. 2020). Columns (5)-( 7) show temperatures from the LTE fitting of H 2 CO, the non-LTE fitting of H 2 CO, and the LTE fitting of CH 3 CN using EMANON. Column (8) lists the adopted temperature. Columns (9)-( 11) correspond to the velocity dispersions from different approaches, and column (12) shows the adopted value. The masses, Mach numbers, and virial parameters of the cores are presented in columns (13), ( 14), and (15).
(This table is available in its entirety in machine-readable form in the online article.) could be as low as ~50 A. Giannetti et al. (2017). However, this does not affect our discussion of the CMF slope in Section 3.4 where all core masses would scale down at the same rate, nor does it change the virial analysis in Section 3.3.3 where the virial parameters would increase by a factor of 2 while all the conclusions remain unaffected.
We calculated the sound speed (c s ) and the Mach number () following
where k B is the Boltzmann constant, T core the temperature of the core derived in Section 3.2, μ is the mean molecular weight and is taken to be 2.37 (J. Kauffmann et al. 2008), m H is the mass of the hydrogen atom, σ NT is the nonthermal velocity dispersion, and σ tot is the total (thermal and nonthermal) velocity dispersion. The H 2 CO and CH CN 3 lines were fitted to estimate the velocity dispersion σ obs , as described in Section 3.2.1. Then, the channel width of V ch = 1.35 km s -1 was subtracted quadratically to derive the deconvolved velocity dispersion:
The total velocity dispersion, which includes the thermal and nonthermal components, was then derived following
where the molecular weight μ mol is 28 for H 2 CO and 44 for CH 3 CN.
Figure 5 shows the relation between the nonthermal velocity dispersion (σ NT ) and sound speed (c s ) of the cores in the three clouds. The Mach numbers of the cores are listed in Table 1, with their errors estimated from the error propagation. The mean Mach numbers in cloud e, Sgr C, and the 20 km s -1 cloud are 3.2, 4.4, and 4.3, respectively, suggesting supersonic motions in the cores. The result confirms that the cores in the three clouds are subject to supersonic nonthermal motions (e.g., turbulence, infall, or rotation), which is the typical condition for gas in the CMZ (e.g., J. M. Rathborne et al. 2014).
As shown in Section 3.2, 253 dense cores have sufficiently high S/N ratios in H 2 CO or CH 3 CN to estimate temperatures and velocity dispersions. We then performed virial analyses to assess dynamical states of these cores. We considered the contributions of kinetic energy (Ω K ), gravitational potential energy (Ω G ), and external pressure (Ω P ) from surrounding gas in the virial equilibrium analysis.
Assuming a spherically symmetric Gaussian density distribution for the gravitational potential energy, the gravitational term (Ω G ) can be derived following (F. Bertoldi & C. F. McKee 1992):
where R is the effective core radius. We took the core area A core from astrodendro and derived R following /p = R A core . The parameter a is a constant whose value depends on the powerlaw index of the radial density distribution r µ - r k p , a = (1k p /3)/(1 -2k p /5). Here, we took k p = 2 and a = 5/ 3, which is the case for self-gravitating cores (F. Bertoldi & C. F. McKee 1992).
The kinetic energy of a core of mass M core and 1D velocity dispersion σ tot is given by
Finally, following, e.g., H.
Kirk et al. (2017) and S. Scibelli et al. (2023), the external pressure term Ω P is given by ( ) p W = -P R 4 . 8 P out 3 P out is external pressure given by ( ) r s m s = = P mn , 9 out out tot,out 2 H out tot,out 2
where σ tot,out is thermal plus nonthermal velocity dispersion of the gas surrounding the core. We assumed that the cores are enveloped by gas at larger scales, as captured by the SMA observations at a resolution of 4″ in X. Lu et al. (2019b), and adopted the gas density n out estimated using the SMA 1.3 mm continuum (assuming a dust temperature of 20 K) and the velocity dispersion σ tot,out from the SMA N 2 H + or CH 3 OH lines (X. Lu et al. 2019b). The adopted values for the cores are listed in Table 1. Note that the SMA observations also suffered from the missing flux issue and filtered out spatially extended emission. However, such an emission should represent more diffuse gas enclosing the gas component captured by the SMA, and thus not directly exerting pressure on even the small cores in the ALMA observations.
We considered the equilibrium between gravitational and kinetic energies ( )
and derived the virial parameter α vir as
If α vir is lower than the critical value of 2, a core would be gravitationally bound, otherwise it would be unbound (J. Kauffmann et al. 2013). Figure 6 depicts the distribution of the virial parameters of the dense cores within the three clouds, in which the dashed horizontal line marks the critical value of 2. It can be seen that the majority of the dense cores (247/253) are in an unbound state.
If we additionally considered the contribution of external pressure ( )
and then derived the virial parameter as
We list the virial parameters of the cores that take external pressure into account in Table 1, and plot the results in Figure 7 (e.g., K. Pattle et al. 2015;H. Kirk et al. 2017). Histograms of the virial parameters in the three individual clouds are shown in Figure 8.
As shown in Figure 7, taking the external pressure into account in the virial equilibrium leads to a greater number of cores being bound (154/253), including 10/19, 108/137, and 36/97 cores in cloud e, Sgr C, and the 20 km s -1 cloud, respectively, characterized by α vir,p < 2. Moreover, the majority of the cores (239/253) have Ω G /Ω P < 1, indicating that the cores are predominantly confined by external pressure instead of self gravity, including 18/19, 134/137, 88/97 cores in cloud e, Sgr C, and the 20 km s -1 cloud, respectively.
Our finding contrasts with results toward clouds in the Galactic disk, where most of the cores are already bound, even without considering the contribution of external pressure in the virial equilibrium (e.g., J. Kauffmann et al. 2013;S. Li et al. 2023;Y. Cheng et al. 2024), and reveals that nonthermal motions in the form of external pressure and internal support dominate gas dynamics in the spatial scale of individual cores
To fit a power law to the high-mass end of the CMF, we employed the maximum likelihood estimation (MLE) method (A. Clauset et al. 2009) implemented in the plfit package,foot_6 following the form
The power-law index (α) and the lower limit of the power-law relation (M min ) were simultaneously determined during the fitting (see e.g., X. Lu et al. 2020). We fit the CMFs for each cloud as well as for the three clouds combined, as shown in Figure 9. Note that we have excluded those sources with α vir,p > 2 and therefore being unbound. In the end, 80, 245, and 410 cores were considered for cloud e, Sgr C, and the 20 km s -1 cloud, respectively. The best-fit power-law indices α for each individual cloud are between 1.09 and 1.38. For the combined CMF of the three clouds, the best-fit α is 1.33 ± 0.14. These power-law indices are higher than those reported in X. Lu et al. (2020), which were derived from CMFs of the same sample of dense cores, yet with an assumed universal dust temperature of 20 K. In Appendix D, we conduct several tests, e.g., changing the assumed temperatures for the cores without H 2 CO/CH 3 CN line detections from 20 to 30 K or 50 K, using the temperatures derived from the non-LTE method, and only using cores with temperatures estimated from the H 2 CO/CH 3 CN lines, and found that the CMFs consistently show higher power-law indices than those in X. Lu et al. (2020).
Figure 10 displays the cumulative distribution functions (CDFs) of core masses derived from constant dust temperatures of 20 K (X. Lu et al. 2020) and from the updated core temperatures in the current work. The p-values of two-sample Kolmogorov-Smirnov (K-S) tests between the two CDFs are 0.63, 2.9 × 10 -7 , and 0.01, for the three clouds, respectively, and 2.9 × 10 -9 for the three clouds combined. Usually, when p < 0.05, one considers that the two samples in the K-S test are not drawn from the same distribution. As such, the core masses estimated with the updated temperatures do represent a statistically different distribution than those reported in X. Lu et al. (2020), except for cloud e where the number of cores is lower, and therefore the statistics may not be robust.
We noted that the estimated power-law indices of the highmass ends of the CMFs are consistent with that of the canonical IMF with α = 1.35 (P. Kroupa 2002). That is, the CMFs of the three clouds exhibit a Salpeter-like shape in their high-mass ends. The implications of these findings will be discussed in Section 4.2.
Figure 8 shows histograms of the virial parameters in the three clouds. The orange bars represent cores with virial parameters α vir p < 2, which are expected to be bound. Toward the three clouds, there are 10, 108, and 37 cores that fall within this bound state. These cores will likely collapse and form stars.
Meanwhile, the blue bars indicate that a large population of cores are unbound. As seen in Figures 7 and 8, even after considering the contribution of external pressure, there are still 9 out of the 19 cores in cloud e, 29 out of the 137 cores in Sgr C, and 61 out of the 97 cores in the 20 km s -1 with α vir p > 2. The situation is more pronounced in the 20 km s -1 cloud compared to Sgr C, while in cloud e there are statistically not enough cores with molecular line detection to reach a conclusion.
The different fractions of bound or unbound cores in Sgr C and the 20 km s -1 cloud may reflect their different evolutionary phases regarding star formation, or different cloud-scale dynamics when they orbit around the Galactic Center. To discriminate between these scenarios or explore other alternatives, a large sample of clouds in the CMZ at various evolutionary phases and orbital positions is needed to determine the viral states of dense cores. Our observations tentatively lend stronger support to the latter case:
1. The estimate of the virial parameter is clearly affected by the measurement of velocity dispersions. When a core is associated with outflows, the velocity dispersion is likely to be overestimated because outflows broaden the H 2 CO linewidth (H 2 CO has been found to trace outflows in these clouds; X. Lu et al. 2021). Additionally, the presence of hot molecular cores, which are an advanced evolutionary phase of high-mass star formation, will also broaden the observed linewidth due to high temperatures (up to a few 10 2 K) and strong feedback. In such cases, the higher ratios of unbound to bound cores in the 20 km s -1 cloud might suggest that star formation activities in the 20 km s -1 cloud are more active than in Sgr C. However, this appears inconsistent with our observations. The number of outflows in the 20 km s -1 cloud and Sgr C are 20 versus 18, respectively, while the numbers of hot molecular cores (represented by cores with CH 3 CN emission identified in the current work) are 12 versus 17, respectively. These numbers are not significantly different. We conducted a test and found that after excluding cores associated with outflows and hot cores, the distribution of their virial parameters remained largely unchanged. 2. The different positions and evolutionary stages of the two clouds in the CMZ may lead to different cloud-scale dynamics. The 20 km s -1 cloud is closer to the center of gravitational potential in most of the 3D models of the CMZ (see Section 4.3.2 of J. D. Henshaw et al. 2023) and therefore is expected to experience stronger tidal effects than Sgr C. In certain models (e.g., J. M. D. Kruijssen et al. 2015), the 20 km s -1 cloud may have recently passed or will soon pass through the pericenter of an elliptical orbit, which results in a strong tidal effect. This tidal compression would drive turbulence and deform the cloud, during which transient substructures such as unbound over-densities may emerge. These substructures are not the usually bound dense cores, but are sometimes referred to as "starless cores" and will disperse over a few dynamical timescales (S. S. R. Offner et al. 2022).
In this sense, the upcoming ALMA CMZ Exploration Survey (ACES) results, which have systematically surveyed the CMZ in continuum emission and molecular lines in the 3 mm band at a resolution of ~2″, will be critical to discerning evolutionary phases and star formation states of clouds in the CMZ and exploring the origin of the different fractions of bound versus unbound cores.
In this study, we have updated the temperatures and therefore the masses of 19, 137, and 97 cores in cloud e, Sgr C, and the 20 km s -1 cloud, respectively, by fitting the H 2 CO or CH 3 CN lines. Figure 1 shows that the cores with H 2 CO or CH 3 CN line emission are usually bright in the 1.3 mm continuum emission. In X. Lu et al. (2020), when a universal dust temperature of 20 K was assumed, these cores were The red sticks attached to the bottom horizontal axes denote the core masses obtained with best-fit temperatures from H 2 CO or CH 3 CN, while the blue ones represent the core masses assuming a dust temperature of 20 K. The vertical black-dashed lines represent the 5σ mass sensitivity (0.27 M e ) at a dust temperature of 20 K. The vertical magenta-dashed lines represent the minimum core masses of the power-law fittings as determined by plfit, and the slanted blue-dashed lines represent the bestfit power-law relations with an arbitrary normalization. Note that we chose not to plot histograms of core masses because we do not bin the masses to fit the power law but carry out the power-law fitting using all the actual data above the minimum core masses following the MLE method. estimated to be massive and populate the high-mass ends of the CMFs. With the updated temperatures, their masses become lower, and thus the high-mass ends of the CMFs become steeper and consistent with the shape of the Salpeter-like IMF, as presented in Section 3.4.
Additionally, Figure D1 illustrates that when the minimum masses of the power-law fitting are fixed to the same values in X. Lu et al. (2020), the power-law indices remain higher than those reported in X. Lu et al. (2020). Meanwhile, to verify the reliability of the data, we test the CMFs composed only of cores with the best-fit temperatures from the H 2 CO or CH 3 CN lines, as shown in Figure D4. For each of the clouds, the number of cores is too low for a robust statistical result. For the three clouds combined, we again found a Salpeter-like shape in the high-mass end.
Therefore, using the updated core temperatures, the three clouds present a Salpeter-like slope in the high-mass end of their CMFs (α ~1.35). This changes the conclusion of topheavy CMFs in X. Lu et al. (2020), and highlights the importance of obtaining reliable core temperatures when fitting the CMFs (P. Dell'Ova et al. 2024).
The Salpeter-like CMFs observed in the three CMZ clouds align with several recent findings toward clouds in the Galactic disk (e.g., Y. Cao et al. 2021;G. Suárez et al. 2021;Y. Pouteau et al. 2023), where Salpeter-like CMFs are also detected. However, unlike the Galactic disk where the IMF is found to be universally consistent with the Salpeter shape, the young massive star clusters in the CMZ are suggested to present topheavy IMFs (B. Hußmann et al. 2012; M. W. J. Hosek et al. 2019), although they can also be explained by tidal stripping of lower-mass stars (S.-M. Park et al. 2020).
There have been debates on the inconsistency between CMFs and IMFs observed toward clouds in the Galactic disk, where the IMFs have the Salpeter-like slope in the high-mass end yet the CMFs sometimes are top-heavy (e.g., Q. Zhang et al. 2015;F. Motte et al. 2018;P. Sanhueza et al. 2019;Y. Cheng et al. 2024;F. Louvet et al. 2024). These results question the direct mapping between the CMF and the IMF, and may suggest a picture of dynamic evolution of core masses through active gas accretion that gradually converges the shape of the CMF to that of the IMF (Y. Pouteau et al. 2023;Y. Cheng et al. 2024). In this study, the inconsistency possibly persists but the shapes of the two mass functions are reversed: the IMFs may be top-heavy, while the CMFs are found to be Salpeter-like. The implication is similar to the studies toward the Galactic disk regions: gas accretion into the cores as well as fragmentation may continuously change the shape of the CMF, such as more active accretion toward more massive cores (I. A. Bonnell et al. 1998Bonnell et al. , 2001;;S. Dib et al. 2010) or more efficient Jeans fragmentation of denser cores (R. B. Larson 2005; B. G. Elmegreen et al. 2008), which may eventually lead to a top-heavy IMF.
We raise several caveats of our study:
1. Core identification: It has been shown that the parameters used in astrodendro affect the outcome of core identification, and therefore the shape of CMFs (X. Lu et al. 2020).
The algorithm adopted for the core identification (e.g., dendrogram versus getsf) also has an impact on the derived CMFs (Y. Cheng et al. 2024).
2. Decoupled gas and dust temperatures: In the Galactic Center environment, the gas temperature and dust temperature are not necessarily well coupled. For example, in shocked regions in the 20 km s -1 cloud, an enhanced gas temperature has been observed (X. Lu et al. 2017), yet shock heating and pressure-volume work may not efficiently heat dust on >2000 au scales. If a core is not internally heated by protostars, it may be possible that the gas temperature is higher than the dust temperature because of shock heating. In Appendix C, we carry out a simple test to compare the H 2 CO emission that is or is not associated with the cores as traced by the 1.3 mm continuum, and find that the H 2 CO compact sources spatially associated with the cores present lower temperatures that the other sample that are presumably heated by shocks. Therefore, the H 2 CO gas in the cores seems to less affected by shock heating, although the possibility cannot be completely excluded. 3. Overestimation of the velocity dispersions: As discussed in Section 3.2.1, outflows traced by H 2 CO and hot cores traced by CH 3 CN exist in the three clouds, which broaden the observed velocity dispersion and lead to overestimation of the virial parameters. In Figure 8, the few cores with virial parameters greater than 10 may be heavily contaminated by outflows or hot cores. Note that several commonly used tracers of dense gas in cores, such as N 2 H + , may be subject to the high ionization in the CMZ, and are therefore not well correlated with dense cores (M. G. Santa-Maria et al. 2021). A careful selection of appropriate molecular lines that are correlated with dense gas in the cores and are less effected by outflows should be carried out, which will be the subject of the next papers in this series. 4. Incomplete temperature measurements in the cores: In Section 3.2.1, we have updated the temperatures of a subset of the cores using the H 2 CO and CH 3 CN lines, which results in a steeper slope for the high-mass ends of the CMFs. The remaining cores do not present sufficient S/N ratios in the H 2 CO or CH 3 CN lines, yet many of them are found to be bound in Section 3.3.3, and therefore could already be forming stars and be heated internally. Their temperatures, as a result, could be higher than the assumed 20 K, and then the core masses could be overestimated, although the temperatures cannot be too high (otherwise molecular transitions such as H 2 CO and CH 3 CN should have been excited). The impact on the shape of the CMFs is an issue to be explored should we obtain multiple-band continuum data in the future to construct dust spectral energy distributions and more robustly measure dust temperatures in the cores.
In this study, we investigate gas dynamics and CMFs of dense cores within three massive clouds in the CMZ using our ALMA 1.3 mm continuum and spectral line observations at 2000 au resolution. We estimate temperatures and velocity dispersions using the H 2 CO or CH 3 CN lines for a total of 253 cores in the cloud e (19 out of 89), Sgr C (137 out of 274), and the 20 km s -1 cloud (97 out of 471). Then, we evaluate their virial equilibrium taking external pressure into consideration, and revisit their CMFs using the newly derived core temperatures, as well as excluding unbound sources. Our main findings are:
1. The high Mach numbers in the cores ( 3-4) suggest the existence of supersonic nonthermal motions down to small spatial scales of 2000 au. 2. In the extreme environment of the CMZ, the contribution of external pressure is critical for the virial equilibrium of the cores, without which most of the cores would be unbound. This differs from the situation of the cores in the Galactic disk where gas is already bound even without considering external pressure. 3. Compared with previous work where the core temperature is assumed to be 20 K universally (X. Lu et al. 2020), we update temperatures of the cores by fitting the H 2 CO and CH 3 CN lines, and find that the high-mass ends of the CMFs of the three clouds exhibit a Salpeter-like slope. Given that the IMF in young massive star clusters in the CMZ may be top-heavy, this finding may suggest a picture of dynamic evolution of the CMF, including gas accretion and fragmentation, that eventually converges to the IMF.
In Figure A5, we compare temperatures derived using the two LTE approaches with CH 3 CN lines. The results show that the temperatures are consistent within the fitting errors.
Figure A6 shows that the temperatures and velocity dispersions of the cores derived from CH 3 CN are generally higher than those derived from H 2 CO (through forced fitting despite the self-absorption). This may be because CH 3 CN traces inner parts of a core where the gas is hotter and more turbulent due to stronger feedback from embedded protostars. Figure A3. Histograms of temperatures and velocity dispersions. The upper panel shows the temperatures and the lower panels shows the velocity dispersions. The blue bars represent the LTE fitting results, while the gray-striped bars represent the non-LTE fitting results. plus at least one of 3 2,2 -2 2,1 and 3 2,1 -2 2,0 lines are detected (i.e., scenario (i)), and fit their line ratios and temperature to derive a line ratio-temperature relation, as shown in Figure B1. The relation between the line ratio (LR) and the temperature (T) follows:
( ) ( ) = T 67.9 exp 0.8LR , B1
which is then used to convert the line ratios to temperatures for scenario (ii).
As discussed in Section 3.1, the H 2 CO line emission is not necessarily associated with the cores traced by the 1.3 mm continuum. To test how well the H 2 CO emission and the 1.3 mm continuum emission are correlated, we select the C1 clump in the 20 km s -1 cloud as an example, where apparent spatially extended H 2 CO emission is seen. We use astrodendro to identify 96 compact sources in the integrated intensity map of the H 2 CO 3 0,3 -2 0,2 line. Then, we clarify them into those associated with the dense cores (core-related) and those not associated (core-unrelated), which consist of 10 and 86 compact sources, respectively, and fit the H 2 CO lines. However, 67 out of these sources cannot be fitted because their 3 0,3 -2 0,2 peak intensities are below 5σ. Figure C1 shows bestfit temperatures and velocity dispersions of the compact sources where LTE fittings can be carried out. The core-related compact sources clearly present lower temperatures than coreunrelated ones.
We further run K-S tests on the temperatures and velocity dispersions of the two samples. As shown in Figure C2, the pvalue of the K-S test on the temperatures is 0.068, suggesting a marginally significant difference between the core-related and core-unrelated compact sources. However, the difference in velocity dispersions is not statistically significant given the high p-value.
Therefore, the H 2 CO emission spatially not associated with the cores may indeed represent a hotter gas component than the dense gas in the cores. Meanwhile, whether H 2 CO detected toward the cores solely traces dense gas remains uncertain, although it may present statistically different temperatures than the core-unrelated component.
We carry out a series of tests on the CMFs, most of which are about different assumptions on the dust temperature. In all cases, the power-law indices of the high-mass ends of the CMFs remain to be higher than those derived in X. Lu et al. (2020) and close to 1.35.
Figure D1 shows the CMFs for the three individual clouds and the three clouds combined, adopting the core temperatures derived in Section 3.2 and fixing the minimum masses for the power-law fit to the same ones in X. Lu et al. (2020). This demonstrates that the higher power-law indices in this work are not a result of different minimum masses in the fit.
Figure D2 shows the CMFs for the three individual clouds and the three clouds combined, where core temperatures derived in Section 3.2 are used, and for cores without H 2 CO or CH 3 CN detections, core temperatures of 30 K are adopted. Similarly, Figure D3 shows the case where core temperatures of 50 K are adopted when H 2 CO or CH 3 CN lines are not detected. The power-law indices are higher than those reported in X. Lu et al. (2020) and can be up to ~2.
Figure D4 shows the CMFs for the three individual clouds and the three clouds combined, adopting the core temperatures derived in Section 3.2. Here, we only include the core masses with updated temperatures, and perform the power-law fitting to the high-mass ends of the CMFs following the same approach as in Section 3.4. For the three clouds combined, one finds a Salpeter-like shape in the high-mass end of the CMF. We note that this test does not imply a new selection criteria that only considers warmer or hotter cores with H 2 CO and/or CH 3 CN emission, but only illustrates that the CMF shape is not affected by excluding the colder cores in Figure 9.
We also test whether adopting the core temperatures derived from non-LTE fits of the H 2 CO lines has any effect on the slope of the CMFs. Figure D5 demonstrates that the results are consistent with those in Figure 9.
Finally, we carry out a Monte Carlo experiment by varying the core temperatures within their corresponding uncertainty ranges, and regenerating the core masses and the CMFs 80000 times. As shown in Figure D6
http://www.dendrograms.org
The Astrophysical Journal, 980:44 (24pp), 2025 February 10 Zhang et al.
https://github.com/xinglunju/FFTL
https://lmfit.github.io/lmfit-py/
https://github.com/xinglunju/emanon, named after a fiction by Shinji Kajio.
The Astrophysical Journal, 980:44 (24pp), 2025 February 10 Zhang et al.
https://github.com/keflavich/plfit