Digging into the Interior of Hot Cores with ALMA (DIHCA). IV. Fragmentation in High-mass Star-forming Clumps

Figure 3 show the ALMA 1.33 mm dust continuum images of all targets. The internal structure of all the observed high-mass star-forming regions is well resolved and presents a variety of morphologies. Some of the DIHCA targets have emission highly concentrated near the clump center (e.g., IRAS 16547–4247, NGC 6334I, and G351.77-0.54), while in others the emission is relatively widespread (e.g., IRAS 16562–3959, G11.92-0.61, and G333.46-0.16). Some clumps present linearly aligned compact structures, resembling a fragmenting filament (e.g., G333.23-0.06, NGC 6334I(N), and G35.03 +0.35A).

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Individual figures on each region can be found in Appendix D. The ALMA observations reveal that each clump has several compact structures, known as cores. Below, we describe how we uniformly identify these cores, as well as how we derive their physical properties.

We adopted the astrodendro package (Rosolowsky et al. 2008), which uses a dendrogram algorithm to compute hierarchical structures. We applied astrodendro to the continuum data without primary-beam correction, and identified the cores to measure their properties (integrated flux, peak flux, core size, and position). The astrodendro identifies three structures: leaf, branch, and trunk. We define the minimum structure, “a leaf,” as a core. The input parameters for dendrogram are three: min_value (), min_delta (), and min_npix ().

For min_value, we set to remove nonreliable sources (σ is the rms noise written in Table 2). For min_delta, which helps to distinguish cores from neighboring structures, we adopt . The minimum pixel number, min_npix, was selected to be the corresponding number of pixels contained in a synthesized beam. After core identification, core fluxes were corrected by the primary-beam response. We identified 579 leaves in total and removed six leaves located at the edge of the primary beam with less than 30% response.

Table 3 lists the peak flux, integrated flux, position, and size of cores obtained from the dendrogram. The identified cores are labeled in ascending order from 1 in descending order of the peak flux of the core. The final number of identified cores (hereafter ) used for analysis is 573. On average, each clump contains 19 cores. The number of identified cores varies from region to region ranging from 5 to 38 (see Table 5 for each clump). The wide range of may be attributed to the differences in clump properties, the noise level (or mass sensitivity), and the linear resolutions. A summary of core physical properties per clump is presented in Table 5.

In Figure 4, the top columns show as a function of distance, rms noise level, and mass sensitivity. We also present the number of identified cores, , within the same physical area in the bottom columns. In this case, the total number of cores is calculated inside the circular area centered at the phase center of the observation whose diameter is the minimum FWHM among all observed targets (∼25″∼0.16 pc at 1.3 kpc). These figures indicate that the identified number of cores does not depend on the target distance, but the core number within the same physical area strongly depends on the distance. The dependencies of on rms noise level and mass sensitivity are weak.

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More quantitatively, we applied Spearman’s rank correlation test. The test provides the Spearman’s rank correlation coefficient (ρ_s ) which is a measure to estimate the strength of the correlation between two variables for the classical and nonparametric test. Nonparametric tests are valid for small sample sizes and do not have to assume any distribution of the variables such as population distribution to be Gaussian. The value of ρ_s is between −1 and 1. The closer ρ_s is to +1 or −1, the stronger the positive or negative correlation, respectively. A value of 0 means no correlation. The derived values of ρ_s are shown at the top of each panel of Figure 4. Each correlation with is weak, with ∣ρ_s ∣ ≲ 0.3 (Figure 4, top panels). However, there are no statistically significant correlations because the probability value p > 0.05 for each test, corresponding to the probability that the correlation is a false relationship. On the other hand, the correlation between within 0.16 pc and distance is strong, with ∣ρ_s ∣ ≳ 0.5 and p ≪ 0.05 (Figure 4, bottom panels). This is because the closer the target clumps, the higher the spatial resolution (∝d) and the mass sensitivity (∝d²).

By using the outputs from astrodendro analysis, we defined the physical quantities of each core. The position of each core is defined as the position of the pixel having the peak flux. The mass () is calculated with Equation (4), assuming that the core temperature is the same as that of the parent clump. The surface density, column density, and volume density are estimated, respectively, using Equations (6), (7), and (8), replacing M_cl and R_cl by and , respectively. The core radius () was defined as half of the geometric mean of the FWHM_major and FWHM_minor provided by astrodendro ¹³ (that is, ). The calculated physical quantities of all the identified cores are listed in Table 4, and Table 5 as the average values for each clump.

Table 4. Calculated Physical Properties of Identified Cores

Clump	Core	Position (ICRS)				nave(H2)	Σave(H2)	Npeak(H2)
Name	ID	R.A. (h:m:s)	Decl. (d:m:s)	(M⊙)	(au)	(×108cm−3)	(g cm−2)	(×1023cm−2)
G11.1-0.12	ALMA1	18:10:28.25	−19:22:30.37	2.36	450	7.71	32.59	32.20
G11.1-0.12	ALMA2	18:10:28.30	−19:22:30.62	0.91	340	6.89	22.04	17.61
G11.1-0.12	ALMA3	18:10:28.09	−19:22:35.17	0.75	770	0.50	3.57	4.55
G11.1-0.12	ALMA4	18:10:28.06	−19:22:35.67	0.26	380	1.39	4.97	4.35
G11.1-0.12	ALMA5	18:10:28.19	−19:22:33.77	0.53	710	0.44	2.94	3.52

Note.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table 5. Core Average Properties Per Clump

Source	1σ Mass								Mean
Clump	Sensitivity		Min	Max	Median	Min	Max	Median	Σave(H2)	Npeak(H2)
	(M⊙)		(M⊙)	(M⊙)	(M⊙)	(au)	(au)	(au)	(g cm−2)	(×1023 cm−2)	(×108 cm−3)
G333.23-0.06	0.118	24	0.72	132.0	2.7	559	1704	807	28.7	35.0	3.3
G335.579-0.272	0.087	6	0.98	127.2	10.8	386	1143	730	90.5	155.8	11.8
IRAS 16547–4247	0.020	38	0.12	68.8	0.4	222	1128	344	20.0	26.6	5.4
IRAS 16562–3959	0.009	22	0.07	9.1	0.4	150	606	247	41.8	55.3	18.9
NGC 6334I	0.025	24	0.18	58.9	0.5	105	486	150	246.1	247.4	144.4
NGC 6334I(N)	0.013	36	0.12	29.3	0.4	82	521	134	104.0	168.2	72.4
G29.96-0.02	0.106	16	0.91	44.6	3.9	596	1628	772	37.5	36.2	5.0
G34.43 + 0.24MM1	0.072	12	0.75	107.3	2.1	346	953	486	60.1	82.8	10.9
G35.03 + 0.35A	0.011	15	0.10	5.2	0.4	262	890	430	9.1	12.7	1.9
G35.20-0.74N	0.017	24	0.13	9.5	0.6	270	658	385	16.1	23.8	4.0
IRAS 18151–1208	0.009	16	0.08	8.3	0.4	322	1142	477	9.1	12.6	1.7
IRAS 18182–1433	0.030	10	0.29	20.5	4.4	430	774	564	45.2	50.2	8.5
IRDC 18223–1243	0.035	15	0.29	27.1	0.8	395	1539	581	10.5	14.6	1.6
G10.62-0.38	0.133	30	1.15	807.3	4.7	574	2452	864	35.5	40.3	3.7
G11.1-0.12	0.025	5	0.26	2.4	0.7	341	768	452	13.2	12.4	3.4
G11.92-0.61	0.030	18	0.22	42.2	0.8	390	1691	617	11.2	28.6	1.5
G5.89-0.37	0.034	22	0.28	74.7	2.3	364	1918	757	64.2	69.4	9.5
IRAS 18089–1732	0.018	13	0.16	64.7	0.4	298	1174	430	21.2	44.4	3.9
IRAS 18162–2048	0.003	11	0.02	7.9	0.0	144	1042	220	25.1	63.6	9.0
W33A	0.016	21	0.17	14.6	1.1	312	1218	666	12.5	22.2	2.0
G14.22-0.50S	0.021	10	0.26	6.3	0.9	261	862	481	20.0	30.6	4.7
G351.77-0.54	0.017	23	0.16	9.1	1.2	130	353	217	104.1	112.7	49.1
G24.60 + 0.08	0.030	10	0.37	11.3	1.8	377	852	496	24.2	32.3	5.0
IRAS 18337–0743	0.030	12	0.19	4.6	0.6	410	1189	540	6.6	8.8	1.2
G333.12-0.56	0.049	28	0.33	28.6	1.7	419	1367	642	19.2	25.3	3.0
G335.78 + 0.17	0.048	19	0.39	44.6	1.5	407	1347	687	19.0	37.0	2.8
G333.46-0.16	0.027	22	0.20	35.0	0.8	322	1468	535	12.4	17.2	2.2
G336.01-0.82	0.037	27	0.22	15.6	1.6	350	1047	558	21.9	23.4	4.4
G34.43 + 0.24MM2	0.029	30	0.23	11.5	1.5	331	943	582	17.9	23.5	3.5
G35.13-0.74	0.016	14	0.11	6.2	0.7	241	693	340	32.7	42.6	9.6

Note. , , , Σ_ave(H₂), N_peak(H₂), and correspond to the number of cores, core mass, core radius, core average values of mean surface density, peak column density, and mean volume density.

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Figure 5 shows the relationship between the radius and mass of all identified cores color-coded by the distance with the plotted lines of constant column densities between 10⁻¹ and 10³ g cm⁻², including the theoretical threshold of 1 g cm⁻² suggested by Krumholz & McKee (2008) as the minimum surface density necessary for the formation of high-mass stars. Almost all cores satisfy this threshold. The mean surface density range is mostly distributed between ∼1 and 100 g cm⁻² and some of the most massive cores have ≥100 g cm⁻². There is a clear trend between radius and distance due to the broad distance range. The cores identified in clumps with the closest distances, ≲2 kpc, have somewhat higher column densities than those of cores in other clumps (see Table 1).

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Note that several clumps contain hot cores or H ii region (e.g., IRAS 18162–2048 and G5.89-0.37) and there is a discrepancy between clump and core average temperatures. This leads to an overestimation of the core mass and density, especially in the hot cores with the highest temperatures. Indeed, Taniguchi et al. (2023) found 44 hot cores using the CH₃CN line. The derived excitation temperatures, which can represent the gas and dust temperature of the hot cores under LTE conditions, are larger than 70 K (see also Table 3 in Taniguchi et al. 2023).

In the current work, we mostly focus on the core separation and leave a more detailed analysis including the core mass once temperatures at high resolution are better constrained for the larger sample. For the moment we note that the distribution of core masses is comparable to the estimated thermal Jeans mass of the clumps, which is a result consistent with the subsequent discussion on thermal Jeans length.

The separation of the identified cores is most likely to reflect the fragmentation process. To measure the separations between cores, we applied the geometric algorithm, Minimum Spanning Tree (MST; Gower & Ross 1969) to the identified cores. This is one of the geometrical algorithms in graph theory. The connection of graph vertices with no closed loops is called a spanning tree so that given n vertices, a spanning tree has n − 1 edges. In the case of fragmentation analysis, vertices and edges correspond to core positions and separations, respectively. This analysis has been validated to detect multiple length scales of hierarchical structures (e.g., Clarke et al. 2019). The separation between the nearest neighboring cores, the length of an edge, is regarded as that produced by the fragmentation from the parental structure, following previous studies (e.g., Beuther et al. 2018; Sanhueza et al. 2019; Lu et al. 2020; Zhang et al. 2021; Morii et al. 2024). Here, we used the peak position of each core.

In Figure 6, we show the results of the MST analysis for W33A with a distance of 2.5 kpc, as an example. In this clump, the edges of MST appear to follow the cloud structure relatively well. For W33A, we identified 21 cores whose masses range from 0.17 to 14.6 M_⊙ with a mean of 2.5 M_⊙. In this area, a significant number of cores appears to be distributed along the elongated structure running from northeast to southwest, in which most massive cores reside. The core separations derived from MST range from 1130 to 19800 au with a mean of 4840 au. The mean value is about 6 times larger than the beam size of 830 au. The results from the MST analysis for other clumps are displayed in Appendix D.

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Figure 7 shows the separation distribution and the cumulative ratio for the whole DIHCA sample. The peaks of angular and physical separation are ∼148 and ∼4500 au, respectively, which is defined by the position at the maximum value of the logarithm probability density function (log-pdf) with a Gaussian Kernel (python module gaussian_kde in SciPy). ¹⁴ In angular scale, the peak of the separation distribution is ∼5 times the average angular resolution, ∼03 (Figure 7 top panel). The cumulative distribution of the core separations normalized to the beam size is presented in the middle panel of Figure 7. A total of 80% of core separations are resolved by more than three beams. If we separate the sample by distance, there are no clear differences in the distribution.

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In physical scale and not yet considering projection effects (Figure 7 bottom panel), the separation distribution seems to have a peak at ∼4500 au with a shoulder at around 10³ au and a tail at ≳10⁴ au.

The core separation distribution appears to depend on the spatial resolution. Once the core distribution is classified into four groups with different distances: d < 2 kpc (green histogram), 2–3 kpc (yellow), 3–4 kpc (orange), and 4–5 kpc (red), we find that the shoulder at 10³ au comes from the targets with the closest distance (<2 kpc) and the tail at larger separation is mostly explained by the more distant clumps (≥2 kpc). We will discuss the effect of spatial resolution in more detail in Section 5. The statistics of measured separations of each region are listed in Table 6.

Table 6. Derived Jeans Parameter and Measured Separations of Each Region

Source			S3D (Original)				S3D (Smoothed)
Clump			Min	Max	Median	Mean	Min	Max	Median	Mean
	(M⊙)	(au)	(au)	(au)	(au)	(au)	(au)	(au)	(au)	(au)
G333.23-0.06	1.1	8100	3300	91,600	7800	13,300	⋯	⋯	⋯	⋯
G335.579-0.272	1.1	7500	6200	47,900	7200	15,500	6200	9600	7200	7400
IRAS 16547–4247	1.2	6500	1300	13,500	3500	4400	4300	20,500	10,600	10,200
IRAS 16562–3959	3.6	13,300	1500	38,700	8900	12,200	5800	20,900	10,900	11,600
NGC 6334I	0.5	2500	500	5500	1500	2100	5600	7300	7000	6600
NGC 6334I(N)	0.5	3400	600	14,000	1500	3000	3200	14,400	5600	6800
G29.96-0.02	2.6	11,300	4300	89,400	12,200	18,500	⋯	⋯	⋯	⋯
G34.43 + 0.24MM1	0.8	5400	3000	17,200	6000	8800	2900	17,300	9600	9700
G35.03 + 0.35A	2.0	9800	2800	8000	5000	5000	3000	10,000	7500	6800
G35.20-0.74N	1.7	8900	2400	33,700	4100	5700	3200	14,300	6100	6200
IRAS 18151–1208	2.7	12,100	3200	41,500	6900	11,600	3200	50,900	6800	12,700
IRAS 18182–1433	1.5	9400	3500	20,700	5800	7500	3500	20,700	5800	7500
IRDC 18223–1243	0.8	10,000	2300	26,200	5200	7800	2800	26,200	5200	8200
G10.62-0.38	1.1	5600	3400	31,300	9800	10,900	⋯	⋯	⋯	⋯
G11.1-0.12	1.4	13,400	2700	13,400	5100	6600	2700	13,400	10,400	9200
G11.92-0.61	2.0	13,800	4000	30,700	7700	9100	4000	30,700	9300	10,400
G5.89-0.37	1.3	6000	3200	12,600	6000	6800	3700	14,200	7100	7700
IRAS 18089–1732	1.0	6700	3300	27,300	7100	9100	3300	27,100	9900	11,500
IRAS 18162–2048	1.9	7200	900	8700	1600	3000	6200	11,200	8700	8700
W33A	1.3	8200	2800	10,200	6400	6300	3500	12,400	7800	7900
G14.22-0.50S	0.8	8200	3900	20,000	9300	10,200	2300	24,700	13,800	13,900
G351.77-0.54	0.5	2400	800	3100	2000	2000	11,300	11,300	11,300	11,300
G24.60 + 0.08	1.3	12,600	2600	36,900	8800	10,400	2600	36,800	8800	10,400
IRAS 18337–0743	1.8	14,100	2900	45,200	7300	12,000	2900	45,200	7300	12,000
G333.12-0.56	0.7	7000	3000	20,200	5700	6700	3000	20,200	5800	6800
G335.78 + 0.17	1.3	8500	2500	35,200	5600	9300	2600	35,100	5600	9300
G333.46-0.16	1.3	8000	2500	26,100	7200	9300	3800	30,200	9000	10,900
G336.01-0.82	0.8	6000	2600	25,700	6400	7400	3000	26,800	6900	8100
G34.43 + 0.24MM2	1.2	9100	2700	14,000	4900	5900	3100	14,000	6200	6800
G35.13-0.74	1.0	8000	1400	25,200	3700	6200	5400	33,500	11,700	15,600

Note. Thermal Jeans Mass is derived as .

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5.1.1. Core Separation Normalized to the Clump Jeans Length

We compare the core separations measured with MST to a characteristic length scale, the clump thermal Jeans length (). The clump thermal Jeans lengths are calculated from the clump temperature and density listed in Table 1. The Jeans length ranges from 2000 and 14,000 au. The derived values are listed in Table 6.

We show the core separation distribution normalized to the clump Jeans length in Figure 8 (a). The left panel shows the core distribution corrected for projection effects, while the right panel is that normalized to the clump thermal Jeans length. The projection effect on the actual 3D separation distribution has been discussed in Sanhueza et al. (2019) and Li et al. (2021), for example. Assuming that the observed cores have a spherically uniform distribution, a correction factor of 4/π is applied to the projected distance S_obs. This is based on the probability density of the solid angle between core separation and the line-of-sight direction. Thus, the deprojected separation S_3D is estimated as follows;

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The peak of the separation distribution corrected by projection effects is ∼5800 au. The normalized separation appears to have a single peak at with no clear shoulder due to close star-forming regions. The tail at large values also becomes more subtle. The clumps with the closest distances have larger clump densities, and thus the Jeans lengths tend to be smaller. The impact of varying the core identification parameters on the core separation distribution is discussed in Appendix A.

Beuther et al. (2018) observed 20 high-mass star-forming clumps with similar properties to those in the DIHCA sample at a similar angular resolution with NOEMA. They identified 123 cores using clumpfind, finding a 3D core separation distribution with a peak at around 2500 au. To make a direct comparison with their work, we have reanalyzed their images and identified 129 cores using dendrograms under the same parameters we used in the DIHCA sample. The newly derived separation distribution, corrected by projection effects, has a peak at ∼5600 au, consistent with the value derived in the DIHCA sample. The reason why the difference occurred despite a similar number of cores is mainly due to different core identification methods. Despite similar parameter settings with dendrogram, clumpfind tends to identify compact and sharp sources. Thus, separation distribution by clumpfind is distributed on a smaller scale than that of dendrogram.

5.1.2. The Effects of Different Spatial Resolution

As mentioned in Section 4.4, the separation distribution may be affected by the spatial resolution. Because the spatial resolution is proportional to the distance of the target under a similar beam size (∼03 in this work), the broad distance range of the clumps tends to make the separation distribution more flattened. We minimize the effect of having different spatial resolutions in the core separation analysis by smoothing the continuum images. We used the CASA task imsmooth and smoothed the dust continuum images to the angular resolution equivalent to ∼1100 au. We decided to exclude three clumps that are more distant than 4 kpc (corresponding to a spatial resolution of ≳1600 au) to keep the spatial resolution range as small as possible after smoothing. We identified the cores in the smoothed images and made the separation distribution of cores using the MST analysis with the same procedure applied in the original images (Section 4.2). As a result, the total number of identified cores is reduced from 573 in the 30 clumps to 331 in 27 clumps. Figure 8 (b) shows the separation distribution between cores obtained from the smoothed images. The separation distribution is similar to the original. The peak separation is about 6300 au and 0.84 , (∼8%) larger than the values obtained from the original images.

Similarly as done in Figure 4, we check for a correlation between the number of cores from the smoothed images with observational quantities. Figure 9 shows the correlations between distance, rms noise, and mass sensitivity. For both the total number of cores and the number of cores within 0.08 pc, there is a weak or no correlation with each quantity.

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5.1.3. The Effects of Completeness

To compute the mass completeness limit, we insert a circular, beam size, flat core in each image per trial. The positions of the artificial cores are determined with a uniform probability in the field of view. Then, we applied a dendrogram to identify cores. We repeated this simulation 2000 times. Sometimes, the inserted core overlaps and merges with an existing core. In such a case, the core is counted as one detection. Figure 10 shows the completeness analysis for a given mass. The core detection rate increases with increasing core mass. For most of the clumps, the detection rate shows a similar dependence on the core mass. In a few regions, the morphology of the continuum emission affects the actual detection rate. For example, the detection rate of NGC 6334I has a bump at around 0.4 M_⊙. The clump looks centrally condensed, and the overlap of the inserted cores may somewhat reduce the detection rate. The G10.62-0.38 region shows the poorest mass completeness since it is one of the most distant clumps in our sample (d = 4.95 kpc). However, the detection rate reaches more than 90% by 3 M_⊙ for all the clumps, except for three targets that have the worst mass sensitivity (G333.23-0.06, G29.96-0.02, and G10.62-0.38, which correspond to the three clumps located at distances larger than 4 kpc).

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To explore the effect of having different mass sensitivities, we set the minimum mass for the smoothed images as 1 M_⊙ and 3 M_⊙ and applied the MST to measure the core separations. For M_⊙ and 3 M_⊙, the core separation peaked at 7200 au and 7100 au, respectively (see also Figure 8 (c), (d)). Figure 11 summarizes the dependence of the core separation on the minimum core mass. The marker and error bar illustrate the peak position and the dispersion of the separation distribution. The higher the mass restriction, the smaller the sample size, and the larger the dispersion of the separation distribution. Except for no restriction case, all of the peak positions are within the margin of error and the peak separation appears not to be sensitive to the minimum core mass. Thus, we conclude that the core separation peaks at around 7000 au, irrespective of the observational bias such as the different angular resolution and mass sensitivity. The peak values of each separation distribution defined by different mass limits, as shown in Figure 11, are listed in Table C1.

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In Section 5.1, we concluded that the peak separation is comparable to the thermal Jeans length of the clumps and there is no need to invoke turbulent Jeans fragmentation. However, numerical simulations on cluster formation suggest that a number of density structures are created by diverse physical processes, e.g., fragmentation, merging, accretion, and turbulence, and such structure formation proceeds hierarchically (e.g., Klessen & Burkert 2000; Bate 2012; Myers et al. 2013; Federrath 2015; Wu et al. 2017; Hennebelle & Inutsuka 2019; Vázquez-Semadeni et al. 2019). Local turbulence tends to compress and form elongated denser substructures, such as filaments and sheets. Within these denser regions, which have weaker turbulence, fragmentation into cores is expected to take place. Filaments and sheets can also be generated as a result of anisotropic gravitational contraction (e.g., Burkert & Hartmann 2004; Heitsch et al. 2009; Gómez & Vázquez-Semadeni 2014). Then, protostars (or sink particles) created by fragmentation can gain additional mass from the surroundings so that their final stellar masses are determined (Bate 2019; Pelkonen et al. 2021). Here, we consider such a case in which the turbulent clumps create hierarchical structures during their global contraction, forming sheets or filaments as substructures from which dense cores are formed by gravitational fragmentation.

Below, we compare the clump thermal Jeans length with the characteristic fragmentation scale of the filaments and sheets. For simplicity, we first consider an isothermal axisymmetric cylindrical cloud in dynamical equilibrium. The equilibrium solution was derived by Stodólkiewicz (1963). According to this work, the cloud radius can be expressed as

where ρ_cyl is the density of the cylinder. The fragmentation wavelength of this cylindrical cloud has been obtained by several authors (e.g., Larson 1985; Inutsuka & Miyama 1992). Stodólkiewicz (1963) obtained the critical wavelength above which the perturbation grows in time. The wavelength of the maximum linear growth rate is about twice the critical wavelength, which is about 8 times longer than the cylinder radius.

If the cylinder density is f times higher than the clump density, ρ_cyl = f ρ_cl, the fragmentation scale of the cylinder can be expressed as

For an isothermal sheet, the thickness of the equilibrium sheet (Spitzer 1978) can be given as

The fragmentation scale for the equilibrium sheet (e.g., Larson 1985) can be estimated as

where ρ_sh = f ρ_cl.

Numerical simulations of cluster formation show that the densities of substructures created by either local turbulent compression or anisotropic gravitational contraction are about ≳10 times higher, that is, f ≳ 10 (e.g., Klessen & Burkert 2000; Bate 2012, 2019; Myers et al. 2013; Wu et al. 2017; Hennebelle & Inutsuka 2019; Pelkonen et al. 2021). We also estimate the average density from our ALMA data to roughly evaluate f. Here, we consider that the structures identified as trunks ¹⁵ are the direct precursors of filaments and sheets. We then derive the average density (ρ_trunk) as the total trunk mass divided by the trunk volume for each target. We calculate the volume of a trunk assuming a spherical symmetry with a radius of , excluding trunks that have no substructures (leaves). For all the cases, the trunks having hierarchical structures reside near the center of the FoV. The ratio between the trunk radius and the Maximum Recoverable scale ranges between 0.09 and 1.08, with about 90% of the values smaller than 1. This means that for some trunks, the determined density is a lower limit. The values of f = ρ_cl/ρ_trunk are estimated to be several tens with a mean value of f ≃ 40 and a median value of f ≃ 27. Therefore, considering that for some trunks the density is a lower limit, we conclude that f ∼ 10 is a reasonable estimation.

In such a case, the fragmentation scale of either a cylinder or a sheet is almost equal to the clump thermal Jeans length (within 13%), . This analysis indicates that our peak separation is consistent with both clump thermal Jeans length and the fragmentation scale of a dense cylinder or sheet. However, it is worth noting that if the fragmentation scale is primarily determined by filament fragmentation, the filament width should be about 7000 au/ 4 = 1750 au (≃8.5 × 10⁻³ pc), which is about 10 times smaller than that of the proposed constant filament width (≈0.1 pc) model (André et al. 2014). This implies that filament fragmentation is not a dominant process to create dense cores, or that the filament width is not universal and is significantly smaller than those of nearby star-forming regions. In reality, recent several observations found filaments with small widths in Orion A (e.g., Hacar et al. 2018; Monsch et al. 2018; Suri et al. 2019; Teng & Hirano 2020) and Perseus (e.g., Schmiedeke et al. 2021; Chen et al. 2022). At the average clump density of 10⁶ cm⁻³, the Jeans length is 5600 au or 0.027 pc, and in the central denser parts of the clumps, it is much smaller. Therefore, the Jeans length in the clumps, thus the core separation, is likely to be much shorter than that predicted by fragmentation of the 0.1 pc universal width filaments (André et al. 2014). These might indicate that the filament width is unlikely to be universal in high-mass star-forming clumps.

We compare our results with those of similar previous studies targeting high-mass star-forming regions. The comparative studies are the following three surveys: CORE/20 clumps (Beuther et al. 2018), SQUALO/13 clumps (Traficante et al. 2023), and ASHES/12 clumps (Sanhueza et al. 2019). Our sample is referred to as DIHCA. ASHES is a mosaic observation covering ∼1 arcmin², while the others are single-pointing observations. For CORE, we show the reanalyzed result by astrodendro (see Section 5.1.1).

Figure 12 shows a box plot of the clump luminosity-to-mass ratio (L_bol/M_clump) for each survey. In the ASHES survey, they selected 70 μm-dark objects, representing the earliest evolutionary stage among the samples to be compared. On the other hand, In the CORE sample, luminous objects with >10⁴ L_⊙ were selected, making it the most evolved sample. In SQUALO and DIHCA samples, no thresholds were considered in the luminosity or L_bol/M_clump parameter, and include objects at various evolutionary stages. Figure 13 plots the mass-size relationship of the clumps, indicating that the density of ASHES targets is ∼10⁴–10⁵ cm⁻³, lower than the targets in the other surveys.

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The upper panel of Figure 14 plots the relationship between the 1σ mass sensitivity and spatial resolution for each survey. DIHCA and CORE have spatial resolutions up to ∼2000 au, while SQUALO and ASHES have resolutions of ∼2000–8000 au. The mass sensitivity of SQUALO is ≳0.1 M_⊙, while the others have ≲0.1 M_⊙. The lower panel plots the ratio of the estimated thermal Jeans mass to the 1σ mass sensitivity and the ratio of the thermal Jeans length to the spatial resolution. This also shows that SQUALO can only resolve the thermal Jeans mass by ≲10 and the thermal Jeans length by ≲3, which is nearly an order of magnitude smaller than the other surveys. Our tests in Sections 5.1.2 and 5.1.3 showed that the separation distributions of the nearby clumps (green histograms) in Figure 15 differ significantly between (a) original and (b), (c), (d) smoothed, suggesting that the effects of insufficient resolution and mass sensitivity are reflected. It is highly likely that a similar phenomenon is occurring in the separation distribution of SQUALO. Thus, we do not seek a physical meaning in the separation distribution of SQUALO. For the calculation of thermal Jeans mass and thermal Jeans length in each survey, we used the mass, radius, and temperature from Table 1 of Sanhueza et al. (2019) for ASHES, and from Table 3 of Traficante et al. (2023) for SQUALO. For CORE, we estimated the mass and radius using the same method as ours toward SCUBA 850 μm data of 17 clumps. Although the rotational temperature of H₂CO emission lines has been estimated as gas kinetic temperature by Beuther et al. (2018), the temperature range is high at ∼40–160 K except for one region, which deviates from the clump temperatures of other surveys. Therefore, we assumed a temperature of 23.2 K, which is the median temperature of DIHCA clumps. Indeed, Izumi et al. (2024) find evidence that the H₂CO kinetic temperature is not tracing ambient gas, but it is rather tracing the molecular outflows.

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Figure 15 shows the log-pdf of the separation distribution for each survey (same as Figure 10). Comparing the distribution of physical scales in the left panel, it is clear that the distributions of CORE and DIHCA are distinctly different from those of ASHES and SQUALO. These differences in distribution are thought to be mainly due to differences in spatial resolution and evolutionary stage. Spatial resolution determines the lower limit of resolvable separation and inevitably affects the separation distribution. Regarding the difference in the evolutionary stage, the effect of tighter separation due to gravity as evolution progresses may also be reflected (e.g., Traficante et al. 2023; Xu et al. 2024). Interestingly, the peak position of DIHCA and CORE is ∼6000 au. This value has also been found in previous studies such as Xu et al. (2024; ASSEMBLE survey), Tang et al. (2019; W51 North), Palau et al. (2018; OMC-1S), and Lu et al. (2020; CMZ), suggesting the existence of a universal fragmentation scale.

Looking at the distribution of scales normalized by the thermal Jeans length in the right panel of Figure 15, CORE, DIHCA, and ASHES are around ∼0.8. ASHES clumps are at an earlier evolutionary stage compared to CORE and DIHCA, but it shows a separation distribution that is similarly consistent with the thermal Jeans length. In this way, previous studies also show consistent results, robustly supporting the hypothesis that thermal Jeans fragmentation is dominant on a ≲1 pc scale.

This result is supported by other previous studies up to ∼1 pc scale that focus on aspects other than the fragmentation scale. Gutermuth et al. (2011) proposed a simple picture of the thermal fragmentation of high-density gas in an isothermal self-gravitating layer to explain the correlation between YSO surface density and gas column density they found in eight molecular clouds within 1 kpc. Palau et al. (2015) compared the “fragmentation level” with the Jeans mass and Jeans number of 19 massive dense cores with a size of ∼0.1 pc. They found that it shows a good correlation when thermally supported rather than turbulent supported. Thus, the overall results of this work along with the results of previous works seem to consistently support a scenario where thermal Jeans fragmentation is at work.