Digging into the Interior of Hot Cores with the ALMA (DIHCA). III. The Chemical Link between NH₂CHO, HNCO, and H₂CO

We constructed moment 0 maps of molecular lines. At first, we checked the spectra around continuum cores within a single beam size in each high-mass star-forming region and confirmed the detection of each molecular species. Information on lines used in the moment 0 maps that were checked for detection is summarized in Table 1. These lines are not blended with other lines in most of the target regions.

Table 1. Information on Lines Used in Moment 0 Maps

Species	Transition	Frequency	Eup/k	Binding Energy a	Binding Energy b	Binding Energy c
	(${J}_{{K}_{a},{K}_{c}}-J{{\prime} }_{{K}_{a}^{\prime} ,{K}_{c}^{\prime} }$)	(GHz)	(K)	(K)	(K)	(K)
CH3CN	JK = 123 − 113	220.709017	133.16	4680	5906	4745–7652
NH2CHO	113,9 − 103,8	233.897318	94.16	5468	8104	5793–10,960
HNCO	100,10 − 90,9	219.798274	58.02	4684	⋯	⋯
H2CO	32,2 − 22,1	218.475632	68.09	2050	5187	3071–6194

Notes.

^a Values of binding energy were calculated by Gorai et al. (2020). ^b Values of binding energy were calculated for crystalline ice by Ferrero et al. (2020). ^c Values of binding energy were calculated for amorphous nature to mimic the interstellar water ice mantles by Ferrero et al. (2020).

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We present moment 0 maps (contour maps) overlaid on the continuum map (gray scale) in Figure 1 as an example. Moment 0 maps toward all of the regions are presented in Figures 9(a)–(d) in Appendix A. Information on noise levels and contour levels of each map are summarized in Tables 8 and 9 in Appendix A.

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Emission of CH₃CN has been detected from all of the sources except for IRAS 18337-0743. In this source, no molecular lines have been detected. We therefore do not present the data toward this field in this paper. The detection rate of the CH₃CN line is 96.7% ($\tfrac{29}{30}$). We identified hot molecular cores (HMCs) based on the moment 0 maps of CH₃CN. If two or more HMCs have been identified in a high-mass star-forming region, we labeled numbers for HMCs in order of integrated intensity, from the highest to the lowest. We identified 44 HMCs in total over the 29 high-mass star-forming regions.

Emission of the H₂CO and HNCO lines has been detected from all of the high-mass star-forming regions, except for IRAS 18337-0743. Emission of NH₂CHO has been detected from 23 high-mass star-forming regions, and the detection rate is 76.7% ($\tfrac{23}{30}$).

Moment 0 maps of all of the molecular lines show almost similar peaks, and they are usually consistent with the continuum peaks within the beam size. The continuum emission of Band 6 mainly comes from dust emission in most of the sources. The G5.89-0.37 region is more evolved and shows unique features in both continuum and moment 0 maps. The continuum emission in this region is dominated by the free–free emission, and shows an explosive dispersal event (Fernández-López et al. 2021). These results imply that molecules are enhanced in shock regions produced by the expanding motion.

Spatial distributions of NH₂CHO are the most compact feature compared to the other species. This is expected, because the calculated binding energy of NH₂CHO is the highest among the observed species (see Table 1). We will discuss this point further in Section 4.2.

We have conducted line identification and analyzed molecular lines at HMCs with the Markov Chain Monte Carlo (MCMC) method assuming the thermodynamics equilibrium (LTE) assumption in the CASSIS software (Vastel et al. 2015). The positions of HMCs are indicated as red crosses in the moment 0 maps (Figures 9(a)–(d)), and their exact positions are summarized in Table 8. The upper level energies, Einstein coefficients, degeneracies, and partition functions are taken from the CDMS database (for ¹³CH₃CN and CH₃CN; Endres et al. 2016) and the Jet Propulsion Laboratory (JPL) spectral line catalog (for NH₂CHO, HNCO, H₂CO; Pickett et al. 1998). The spectra were obtained within a single beam area.

In the line identification, we determined representative V_LSR values at each core using the data of the CH₃CN lines. We applied the representative V_LSR when we identified the lines of the other species, which enables us to pin down molecular lines in line-rich HMCs.

The column density (N), excitation temperature (T_ex), line width (FWHM), and systemic velocity (V_LSR) were treated as free parameters in the MCMC method. The size parameter was fixed, and then the beam filling factor was assumed to be one. We show the observed spectra (black curves) and the best-fitting model (red curves) toward the G10.62-0.38 HMC1 in Figure 2 as an example. We derived the excitation temperatures (T_ex) and column densities of ¹³CH₃CN by fitting its six K-ladder lines (J_K = 13_K − 12_K ; K = 0 − 5, E_up/k = 78.0 − 256.9 K; see Table 2). We analyzed the ¹³CH₃CN spectra, instead of CH₃CN, because the low-K lines of the main isotopologue suffer from self absorption in some cases. The lines of ¹³CH₃CN are found to be optically thin (τたう < 0.8). The J_K = 13₆ − 12₆ line of ¹³CH₃CN has been detected in some sources, but this line could not be fitted well with the other lines simultaneously. This could happen, if emission of this line with a very high upper-state energy (335.5 K) comes from smaller regions than the other lines with lower upper-state energies. In general, the best-fitting models underestimate peak intensity of lines with high E_up. This is likely caused by a significant beam dilution effect, where emission of higher E_up lines arises just in the hotter and inner region and does not fill up the observing beam. If the lines of ¹³CH₃CN have not been detected (seven cores), we fitted the spectra of the K-ladder lines of CH₃CN (J_K = 12_K − 11_K ; K = 0 − 4, E_up/k = 68.9 − 183.1 K; Table 2).

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Table 2. Information on Lines Used in MCMC Analyses

Species	Transition	Frequency	Eup/k	logAij
	${J}_{{K}_{a},{K}_{c}}-J{{\prime} }_{{K}_{a}^{\prime} ,{K}_{c}^{\prime} }$	(GHz)	(K)
13CH3CN	JK = 130 − 120	232.234188	78.0	−2.9667
	JK = 131 − 121	232.229822	85.2	−2.9693
	JK = 132 − 122	232.216726	106.7	−2.9772
	JK = 133 − 123	232.194906	142.4	−2.9907
	JK = 134 − 124	232.164369	192.5	−3.0103
	JK = 135 − 125	232.125130	256.9	−3.0368
CH3CN	JK = 120 − 110	220.747261	68.9	−3.0342
	JK = 121 − 111	220.743011	76.0	−3.0372
	JK = 122 − 112	220.730261	97.4	−3.0465
	JK = 123 − 113	220.709017	133.2	−3.0624
	JK = 124 − 114	220.679287	183.1	−3.0857
NH2CHO	113,8 − 103,7	234.316254	94.2	−3.06215
	113,9 − 103,8	233.897318	94.1	−3.06449
	114,7 − 104,6	233.746504	114.9	−3.09332
	114,8 − 104,7	233.735603	114.9	−3.09334
	115,7 − 105,6	233.595412	141.7	−3.13302
HNCO	100,10 − 90,9	219.798320	58.0	−3.82082
	101,9 − 91,8	220.585200	101.5	−3.82055
H2CO	32,2 − 22,1	218.475632	68.1	−3.80403

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We converted the column densities of ¹³CH₃CN to those of CH₃CN using the following formula (Yan et al. 2019):

$$\begin{eqnarray}&&{}^{12}{\rm{C}}{/}^{13}{\rm{C}}=5.08{D}_{\mathrm{GC}}+11.86,\end{eqnarray} \tag{ 1 }$$

where D_GC (in kpc units) is the distance from the Galactic Center. The D_GC values are calculated from the galactic coordinate and the distance between the Sun and the Galactic center (8 kpc; Eisenhauer et al. 2003). We summarize the distance between the Sun and the target high-mass star-forming regions (D), D_GC, and the calculated ¹²C/¹³C ratio in Tables 3 and 4. Figure 3 shows the relationships between D_GC and the derived column density of CH₃CN. The Spearman's rank correlation coefficient (ρろー) is derived to be +0.05. This means that no artificial effects occurred during the conversion from N(¹³CH₃CN) to N(CH₃CN) and the conversion was successful.

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The excitation temperature derived from ¹³CH₃CN or CH₃CN is known to be a good thermometer (e.g., Hernández-Hernández et al. 2014; Hunter et al. 2014). In the case of the other species, several lines (see Table 2) with similar upper-state energies have been clearly detected, and it is difficult to determine their excitation temperatures and column densities simultaneously by fitting their lines. We then applied the values of excitation temperature derived from the analyses of ¹³CH₃CN or CH₃CN. We checked the spectra at all of the positions and selected lines that can be clearly identified and are not affected by contamination by other lines in most positions. ¹⁰ Table 2 summarizes information on lines used in the MCMC method. If the lines of the target species do not show the single Gaussian feature (e.g., possible contamination with other lines), we excluded them from the fitting to derive their column densities precisely. The derived column densities and excitation temperatures are summarized in Table 3. Assuming isothermal gas and using the excitation temperature derived from ¹³CH₃CN or CH₃CN, we found all other lines to be optically thin. The results of V_LSR and FWHM derived by the MCMC method are summarized in Table 10 in Appendix B.

We derived the column densities of H₂, N(H₂), from the continuum data using the following formula (e.g., Sabatini et al. 2022):

$$\begin{eqnarray}&&N({{\rm{H}}}_{2})=\displaystyle \frac{{F}_{1.3\mathrm{mm}}\gamma }{{B}_{1.3\mathrm{mm}}({T}_{\mathrm{dust}}){\rm{\Omega }}{\kappa }_{1.3\mathrm{mm}}{\mu }_{{{\rm{H}}}_{2}}{m}_{{\rm{H}}}},\end{eqnarray} \tag{ 2 }$$

where F_1.3mm, γがんま, B_1.3mm(T_dust), Ωおめが, κかっぱ_1.3mm, ${\mu }_{{{\rm{H}}}_{2}}$, and m_H are the continuum peak flux at 1.3 mm, the gas-to-dust ratio, the Planck function at 1.3 mm with a dust temperature, the beam solid angle, the H₂ mean molecular weight (2.8), and the mass of the hydrogen atom, respectively. We adopted a value of κかっぱ_1.3mm = 0.9 cm² g⁻¹ (e.g., Ossenkopf & Henning 1994; Sanhueza et al. 2019; Sabatini et al. 2022). We assumed that the dust temperature is equal to the excitation temperature of ¹³CH₃CN (or CH₃CN). The gas-to-dust ratios (γがんま) are calculated using the following formula (Giannetti et al. 2017):

$$\begin{eqnarray}&&\mathrm{log}(\gamma )=0.087{D}_{\mathrm{GC}}+1.44.\end{eqnarray} \tag{ 3 }$$

The continuum peak flux, gas-to-dust ratios, and derived H₂ column densities are summarized in Table 4.

We search for correlations among the observed species to investigate their chemical links. As mentioned in Section 1, NH₂CHO has been considered to be closely chemically linked with HNCO (e.g., López-Sepulcre et al. 2015, 2019). In addition, chemical simulations suggest gas-phase formation of NH₂CHO from H₂CO in hot core regions (Gorai et al. 2020). On the other hand, CH₃CN is not expected to be directly related to the other observed species.

We use abundances derived as X(a) = N(a)/N(H₂), where a represents the molecular species, because comparisons using the column densities may mislead correlation coefficients. If we investigate correlations using column densities, the total gas mass or total gas column density represented by N(H₂) could be the lurking third variable. We made plots to investigate this possibility as shown in Figure 4. The outliers, labeled as "IRAS 16562-3959 HMC2" and "IRAS 18162-2048," show lower molecular column densities; nevertheless they have the highest H₂ column densities. IRAS 16562-3959 HMC2 is an ultracompact H II (UC H II) region (Guzmán et al. 2018; Taniguchi et al. 2020). IRAS 18162-2048 possesses a highly collimated magnetized radio jet (Carrasco-González et al. 2010). This jet may produce UV radiation via the strong shocks (Girart et al. 2017). Thus, the lower molecular column densities in these two sources are likely caused by destruction of molecules by the UV radiation.

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We conducted the Spearman's correlation coefficient test, and the derived p-values and correlation coefficients (ρろー) are indicated in Figure 4. These statistical values mean that the molecular column densities have weak positive correlations with the H₂ column density. We then decided to use the abundances to investigate chemical relationships among molecular species in the following parts of this paper. Table 5 summarizes the derived abundances of each species, calculated from values summarized in Tables 3 and 4.

Figure 5 shows relationships between each pair of molecular species. We conducted the Spearman's rank correlation test for each pair. The Spearman's correlation coefficients (ρろー) are derived to be 0.96 and 0.91 for pairs of NH₂CHO-HNCO and NH₂CHO-H₂CO, respectively. These high correlation coefficients imply that NH₂CHO may be chemically linked with HNCO and H₂CO. These results are consistent with the previous studies and the prediction by chemical simulations (Gorai et al. 2020).

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We fitted the plot of NH₂CHO versus HNCO, as López-Sepulcre et al. (2015) analyzed. The best power-law fit is X(NH₂CHO) = 0.07X(HNCO)^0.92(±0.08). López-Sepulcre et al. (2015) derived the best power-law fit as X(NH₂CHO) = 0.04X(HNCO)^0.93, using data toward YSOs with various stellar masses, from low- through intermediate- to high-mass stars. Our result agrees with that derived by López-Sepulcre et al. (2015) well. Our data set contains a sample of targets with larger abundances of NH₂CHO (≈3 × 10⁻¹⁰ − 10⁻⁷) than those in López-Sepulcre et al. (2015; ≈3 × 10⁻¹¹ − 10⁻⁸), and our sample is the largest one in high-mass star-forming regions, to our best knowledge. The same best power-law fits for different ranges of abundances likely mean that the chemistry (formation and destruction processes) of NH₂CHO is common around YSOs with various stellar masses.

As we found a tight correlation between NH₂CHO and H₂CO, we also fitted the plot of NH₂CHO versus H₂CO with the same method. The best power-law fit is X(NH₂CHO) = 0.35X(H₂CO)^1.07(±0.09).

The second and third rows of Figure 5 show correlations between CH₃CN and NH₂CHO/HNCO/H₂CO. Although there are no expected direct chemical relationships between CH₃CN and the other species, there is a possibility that CH₃CN is positively correlated with the other species. The plots show more scatter compared to the two plots in the first row, suggestive of weaker correlations. The Spearman's correlation coefficients are derived to be 0.83, 0.85, and 0.74 for pairs of CH₃CN–NH₂CHO, CH₃CN–HNCO, and CH₃CN–H₂CO, respectively. These correlation coefficients are lower than those for pairs NH₂CHO–HNCO and NH₂CHO–H₂CO, but they still indicate strong correlations (ρろー ≥ 0.7). These correlations may mean that all of the species react to temperature in the same manner rather than chemical links (Quénard et al. 2018). To explore this point further, we will consider a partial correlation test in the next subsection.

As we derived in Section 4.1, all of the species show positive correlations. However, as Quénard et al. (2018) pointed out, the observed correlations may not indicate chemical links, but they could reflect that all of the species act similarly against temperature. For example, assuming that molecules form on dust surfaces in the cold starless core stage and sublimate into the gas phase in the hot core regions with temperatures above 100 K, their abundances increase as the temperature increases. In this subsection, we will conduct a partial correlation test to investigate pure chemical links excluding the effect of temperature.

Figure 6 shows relationships between excitation temperatures and molecular abundances. We found that all of the molecular abundances show positive correlations with the excitation temperatures; ρろー values are 0.81, 0.78, 0.67, and 0.76 for NH₂CHO, HNCO, H₂CO, and CH₃CN, respectively. Hence, the correlation among each species derived in Section 4.1 may be fake, and the temperature may act as the lurking third variable.

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In order to derive the pure chemical links excluding the lurking third variable, or the temperature, we conducted a partial correlation test. Assuming the local thermodynamic equilibrium (LTE), we use the excitation temperature derived from CH₃CN analyses to represent the gas kinetic temperature and dust temperature. The partial correlation test removes the mutual dependence of the two first variables on the third one. We utilize the excitation temperature derived from the analyses of the CH₃CN (¹³CH₃CN) lines as a representative temperature. We adopted the first-order partial correlation coefficient (e.g., Wall & Jenkins 2012; Urquhart et al. 2018):

$$\begin{eqnarray}&&{\rho }_{\mathrm{12,3}}=\displaystyle \frac{{\rho }_{12}-{\rho }_{13}{\rho }_{23}}{\sqrt{(1-{\rho }_{13}^{2})(1-{\rho }_{23}^{2})}}.\end{eqnarray} \tag{ 4 }$$

Here, we are interested in molecular abundances (1 and 2), and the temperature is considered as the third variable (3).

Table 6 summarizes all of the correlation coefficients for each pair of molecules. The derived partial correlation coefficients are summarized in the last row.

The partial correlation coefficients for HNCO–NH₂CHO and H₂CO–NH₂CHO are still high (>0.8), whereas those for the other pairs with CH₃CN become lower (<0.65). These results most likely indicate that NH₂CHO is chemically linked with HNCO and H₂CO. Thus, the observed correlation between NH₂CHO and HNCO is not the result of having the same response of the two species to temperature, as pointed out by Quénard et al. (2018). On the other hand, CH₃CN is less chemically linked with the other species studied here. We found that the partial correlation test is a strong tool to statistically investigate chemical relationships.

We further investigated the correlation coefficients between the molecular abundances and excitation temperatures. The highest correlation coefficient is derived for NH₂CHO (0.81), followed by HNCO (0.78), CH₃CN (0.76), and H₂CO (0.67). This order matches with the order of the calculated binding energies of each species (Table 1). This also suggests that thermal desorption is important for these molecules. As the emission of a molecule with a high binding energy should be dominated by the central hot core region, the correlation with the temperature should be strong. Hence, the derived correlation coefficients with the temperature suggest that the NH₂CHO emission comes from the central hot core region, whereas the emission of H₂CO could come not only from the inner hot core but also from the outer part of cores. This is consistent with the spatial distribution of the molecular lines (Figures 9(a)–(d)), as mentioned in Section 3.1.

We compare the observed molecular abundances with the chemical models developed by Gorai et al. (2020). We consider a linear density variation with a slope $\tfrac{{\rho }_{\max }-{\rho }_{\min }}{{t}_{{\rm{f}}}-{t}_{{\rm{i}}}}$, where t_f − t_i corresponds to the collapse timescale. Similarly, the temperature increases with a linear slope during the warm-up stage.

The dual-cyclic hydrogen addition and abstraction reactions (Haupa et al. 2019) that occur in the gas phase are included in the models. This cycle consists of four reactions. From HNCO to NH₂CHO, two hydrogen addition reactions occur:

$$\begin{eqnarray}&&\mathrm{HNCO}+{\rm{H}}\to {{\rm{H}}}_{2}\mathrm{NCO},\end{eqnarray} \tag{ 5 }$$

followed by

$$\begin{eqnarray}&&{{\rm{H}}}_{2}\mathrm{NCO}+{\rm{H}}\to {\mathrm{NH}}_{2}\mathrm{CHO}.\end{eqnarray} \tag{ 6 }$$

In a sequence from NH₂CHO to HNCO, two hydrogen abstraction reactions happen:

$$\begin{eqnarray}&&{\mathrm{NH}}_{2}\mathrm{CHO}+{\rm{H}}\to {{\rm{H}}}_{2}\mathrm{NCO}+{{\rm{H}}}_{2},\end{eqnarray} \tag{ 7 }$$

followed by

$$\begin{eqnarray}&&{{\rm{H}}}_{2}\mathrm{NCO}+{\rm{H}}\to \mathrm{HNCO}+{{\rm{H}}}_{2}.\end{eqnarray} \tag{ 8 }$$

Reactions (6) and (8) are barrierless, whereas Reactions (5) and (7) have activation barriers of 2530–5050 K and 240–3130 K, respectively. In the models of Gorai et al. (2020), the gas-phase reaction between NH₂ and H₂CO to form NH₂CHO is included:

$$\begin{eqnarray}&&{\mathrm{NH}}_{2}+{{\rm{H}}}_{2}\mathrm{CO}\to {\mathrm{NH}}_{2}\mathrm{CHO}+{\rm{H}}.\end{eqnarray} \tag{ 9 }$$

The hydrogenation reaction leads to the formation of CH₃CN on dust surfaces in the cold phase. In contrast, its formation by a dust-surface radical–radical reaction between CN and CH₃ is efficient during the initial warm-up phase. In addition, CH₃CN could form by the following barrierless reaction on dust surfaces:

$$\begin{eqnarray}&&{\rm{H}}+{{\rm{H}}}_{2}\mathrm{CCN}\to {\mathrm{CH}}_{3}\mathrm{CN}.\end{eqnarray} \tag{ 10 }$$

The major gas-phase formation route of CH₃CN is the dissociative recombination reaction of CH₃CNH⁺. Destruction routes of CH₃CN on dust surfaces are cosmic-ray-induced desorption, photodissociation, and thermal and nonthermal desorption. Furthermore, the models of Gorai et al. (2020) include the successive hydrogenation reactions of CH₃CN to lead finally to ethanimine (CH₃CHNH).

There are six physical parameters in the models of Gorai et al. (2020); the maximum density (${\rho }_{\max }$ [cm⁻³]), maximum temperature (${T}_{\max }$ [K]), collapsing timescale (t_coll [yr]), warm-up timescale (t_w [yr]), post-warm-up timescale (t_pw [yr]), and initial dust temperature (T_ice [K]). Gorai et al. (2020) ran two types of models, Model A and Model B, as summarized in Table 9 in their paper. In Model A, the physical parameters are fixed with the following values; ${\rho }_{\max }={10}^{7}$ cm⁻³, ${T}_{\max }=200$ K, t_coll = 10⁶ yr, t_w = 5 × 10⁴ yr, t_pw = (6.2–10) × 10⁴ yr, and T_ice = 20 K. For Model B, ${\rho }_{\max }$, ${T}_{\max }$, and t_coll varied in ranges of 10⁵–10⁷ cm⁻³, 100 − 400 K, and 10⁵–10⁶ yr, respectively. The warm-up timescale, post-warm-up timescale, and initial dust temperatures were fixed at 5 × 10⁴ yr, 10⁵ yr, and 20 K, respectively, in Model B.

Figure 7 shows comparisons between the observed abundances and the modeled abundances obtained by Model A (Gorai et al. 2020). As the observed values, we plot the four representative values from the whole hot core populations; maximum, average, median, and minimum values (Table 7).

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The minimum abundances agree with the model around 1.02 × 10⁶ yr (T ≈ 100 K). The minimum observed CH₃CN abundance agrees with model at the almost same age within a factor of 1.3. The minimum abundance of NH₂CHO is derived at IRAS 16562-3959 HMC2, and those of HNCO and H₂CO are derived in IRAS 18162-2048. The derived excitation temperatures of CH₃CN in these HMCs are 88 K and 94.3 K, which are lower than the other sources and close to the modeled value of 100 K. In addition, in the case of IRAS​ ​18162-2048, as mentioned before, the ionized jet may explain the lower molecular column densities.

The observed average and median abundances of HNCO and NH₂CHO are most close to the modeled values around (≈1.04–1.05) × 10⁶ yr (T ≈ 165–200 K). The temperature ranges that can reproduce the observed average and median values agree with the average excitation temperature at the whole hot core populations (≈162 K). Thus, the tendencies of the observed abundances for both HNCO and NH₂CHO agree with the modeled results. The median value of H₂CO is close to the modeled abundance around ≈1.03 × 10⁶ yr, which is almost consistent with the ages constrained by HNCO and NH₂CHO ((≈1.04–1.05)×10⁶ yr), but in general, the modeled H₂CO abundances tend to be lower than the observed ones. The observed maximum abundances cannot be explained by Model A, implying that assumed physical parameters are not suitable for the hot core with the maximum molecular abundances.

Figure 8 shows a comparison between the representative observed abundances and modeled abundances obtained by Model B (Gorai et al. 2020). The left panels show the dependences on different t_coll and ${\rho }_{\max }$, and the right panels show the dependences on different t_coll and ${T}_{\max }$, respectively. The initial dust temperature (T_ice) is fixed at 20 K, and t_pw is 10⁵ yr. The modeled abundances plotted here are the values at the end of the simulations; t = t_collapse + t_warm−up (5×10⁴ yr)+ t_{post warm−up} (10⁵ yr). As seen in Figure 8, the maximum observed abundances could not be reproduced by the models in most cases, because the plotted modeled abundances are the values at the end of the simulation. We show the modeled maximum abundances obtained by Model B in Figure 10 in Appendix C. The modeled maximum abundances with some cases consist with the observed maximum abundances by less than 1 order of magnitude.

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In Figure 8, we found low abundances of H₂CO and high abundances of NH₂CHO in the models. We attribute these results to the uncertainty of the reaction rate coefficient for the gas-phase reaction between NH₂ and H₂CO (Reaction (9)). We used an αあるふぁ value of 5.00 × 10⁻¹², but Skouteris et al. (2017) derived a value of 7.79 × 10⁻¹⁵ by their quantum chemical calculation. However, the γがんま value provided by Skouteris et al. (2017) was ≈5.5 times lower than that of Barone et al. (2015). Here, we used the γがんま value from Skouteris et al. (2017). If the αあるふぁ value for this reaction becomes lower, as suggested by Skouteris et al. (2017), the conversion rate from H₂CO to NH₂CHO should be slow. In addition to the αあるふぁ, βべーた and γがんま values used in Gorai et al. (2020) for this reaction, we have also evaluated the modeling results considering the αあるふぁ, βべーた, and γがんま values available in Skouteris et al. (2017). We did not find significant changes for our best-fit results that are obtained comparing with observation. Only a minor change in collapsing time has been noticed.

The maximum abundances of HNCO and NH₂CHO can be reproduced by the models with ${\rho }_{\max }={10}^{5}\mbox{--}{10}^{6.5}$ cm⁻³, as seen in the left panels. The average and median abundances of these two species agree with the following three models; (1) ${\rho }_{\max }={10}^{6}$ cm⁻³ and t_coll ≈ 8 × 10⁵ yr, (2) ${\rho }_{\max }={10}^{6.5}$ cm⁻³ and t_coll = 4 × 10⁵ yr, and (3) ${\rho }_{\max }={10}^{7}$ cm⁻³ and t_coll = 2 × 10⁵ yr. The models with higher ${\rho }_{\max }$ values need to collapse rapidly to reproduce the observed abundances. Their minimum abundances cannot be reproduced by the models, because the adopted maximum temperature (200 K) is too high for these sources (88–95 K; see in the previous paragraph). In fact, the minimum abundances can be reproduced by the model with ${T}_{\max }=100$ K in the right panels, as discussed later.

The minimum H₂CO abundance is reproduced by the models with ${\rho }_{\max }={10}^{6.5}\mbox{--}{10}^{7}$ cm⁻³, while the average and median values agree with the model with ${\rho }_{\max }={10}^{5.5}$ cm⁻³. In addition, the models with ${\rho }_{\max }={10}^{5}$ cm⁻³ and 10⁶ cm⁻³ reproduce the average and median abundances around t_coll ≈ (1–2) × 10⁵ yr. The observational results of H₂CO tend to show agreements with lower-${\rho }_{\max }$ models compared to the cases of HNCO and NH₂CHO (${\rho }_{\max }={10}^{6}\mbox{--}{10}^{7}$ cm⁻³). These results would be consistent with the fact that the H₂CO emission shows more spatially extended features than the other molecules (Figures 9(a)–(d) in Appendix A). The maximum value, derived in IRAS 18182-1433 HMC1, has not been reproduced by the models in Figure 8, but the modeled maximum abundances with ${\rho }_{\max }={10}^{5}$ cm⁻³ agree with the observed maximum value within 1 order of magnitude (see Figure 10 in Appendix C).

As seen in the right panels of HNCO and NH₂CHO in Figure 8, the maximum abundances prefer shorter collapsing time with a highest ${T}_{\max }$ of 400 K. The maximum abundances of the both molecules are derived in IRAS 18182-1433 HMC1. The excitation temperature at HMC1 is the highest value (≈278 K) among the observed hot cores. This is consistent with the agreement of the maximum abundance with the highest ${T}_{\max }$ model. On the other hand, HMC2 and HMC3 in the same region do not show high abundances, and NH₂CHO has not been detected at HMC3. Hence, in this high-mass star-forming region, it is likely that an massive young stellar object (MYSO) in HMC1 was first born rapidly, and the other sources were slowly formed later.

The average and median values of HNCO and NH₂CHO are consistent with the models with ${T}_{\max }=150\mbox{--}400$ K. The best agreed conditions are t_coll = (2–3) × 10⁵ yr, and the models with lower ${T}_{\max }$ values need to collapse quickly.

In the case of H₂CO, the minimum observed abundance agrees with the models within 1 order of magnitude except for ${T}_{\max }=400$ K, but the other three representative values disagree with the models. This could be explained by the fact that the observed H₂CO abundances prefer the models with low-${\rho }_{\max }$ values, while the right panels show results with ${\rho }_{\max }={10}^{7}$ cm⁻³.

The observed CH₃CN abundances disagree in most cases, but CH₃CN does not take part in the formation network of NH₂CHO. This disagreement does not undermine our analysis for NH₂CHO. The minimum value agrees with the models of ${T}_{\max }=100$ K and t_coll ≈ (3–6) × 10⁵ yr within 1 order of magnitude. As seen in Figure 10 in Appendix C, the observed CH₃CN abundances agree with the modeled maximum abundances with low-${\rho }_{\max }$ models (10⁵ − 10^5.5 cm⁻³), not at the end of the simulations. This could mean that CH₃CN is deficient at the age of the end of the simulations due to its destruction, and the CH₃CN gas may trace chemically younger gas around MYSOs.

We note that there are still uncertainties in the reaction rate constants for the dual-cyclic hydrogen addition and abstraction reactions. We tested changing the αあるふぁ value by 2 orders of magnitude (from αあるふぁ = 10⁻¹⁰ to αあるふぁ = 10⁻¹²). The modeled HNCO abundance becomes lower by 3 orders of magnitude, and the NH₂CHO abundance increase by 1 order of magnitude. Hence, the modeled HNCO abundance with αあるふぁ = 10⁻¹² cannot explain the observed abundances, and the rate constants used by Gorai et al. (2020), αあるふぁ = 10⁻¹⁰, seem to be more reasonable. However, measurements to estimate the rate constants of the reactions involved are necessary to better understand relationships among these species in various astronomical environments.

In summary, most of the sources observed by the DIHCA project are at the hot core stage with temperatures of 150–400 K and densities of 10⁶–10⁷ cm⁻³, judging from HNCO and NH₂CHO. These values are consistent with typical hot core values. On the other hand, the observed H₂CO and CH₃CN abundances prefer models with lower density conditions of ∼10⁵–10^5.5 cm⁻³. This is consistent with the spatial distribution of the observed emission. Thus, the contribution from the outer parts of cores is larger in the case of H₂CO and CH₃CN, while NH₂CHO and HNCO emission comes from inner central cores. These are consistent with the expectation based on the calculated binding energy as discussed in Section 4.2.