Correlations of Methyl Formate (CH₃OCHO), Dimethyl Ether (CH₃OCH₃) and Ketene (H₂CCO) in High-mass Star-forming Regions

We perform multi-component two-dimensional Gaussian fits to 870 $\mu$m continuum emission using the CASA-$imfit$ function. A total of 145 dense cores were resolved in 11 high-mass star-forming regions (Chen et al., 2024). We have checked the spectra towards the 145 cores one by one, and found that 19 separate cores in 9 regions have multiple line emissions of CH₃OCHO or CH₃OCH₃ or H₂CCO. From Figure 1 and 8, these 19 cores show rich line emission. The ALMA 870 $\mu$m continuum emission from 9 high-mass star-forming regions is shown in Figure 2. The positions of 19 line-rich cores are labeled. For the other two of the 11 high-mass star-forming regions (IRAS 14382-6017 and IRAS 17204-3636), the line transitions of the three molecules were not detected. The absence of HMCs from these two sources was affirmative by cross-matching with the ALMA Band 3 dataset (Qin et al., 2022).

For the estimation of the core masses and the molecular hydrogen column densities, the commonly used method assumes that the dust emission is optically thin and in local thermodynamic equilibrium (LTE). Under these assumptions, the core masses and source-averaged H₂ column densities can be estimated by the expressions (Kauffmann et al., 2008):

where S_ν is integrated flux density, $\eta$ = 100 is gas-to-dust mass ratio (Lis et al., 1991; Hasegawa et al., 1992), D is the distance to the core (Liu et al., 2020) and the uncertainty takes 10% (Baug et al., 2020; Liu et al., 2022) when calculating the uncertainty of the core mass, $\kappa$_ν is the dust mass absorption coefficient, B_ν(T) is the Planck function at dust temperature T, $\mu$ = 2.8 is the mean molecular weight of the gas (Kauffmann et al., 2008), m_H is the mass of the hydrogen atom, and $\Omega$ is the solid angle corresponding to the deconvolved size of the core. In our case, $\kappa$_870μm takes a value of 1.89 cm² g^-1, which is interpolated from the table given in Ossenkopf & Henning (1994), assuming grains with thin ice mantles and a gas density of 10⁶ cm^-3. The dust temperatures of dense cores are assumed to be the rotation temperatures of CH₃OCHO, which is considered to be dust temperature probe (Favre et al., 2011). The parameters of 19 line-rich cores are listed in Table 1. In Table 1, we also list whether these cores are associated with outflows. The presence of outflows is identified by searching for red-blue lobes around continuum sources using the CO (3-2), HCN (4-3), and SiO (2-1) lines (Baug et al., 2020).

The source-averaged optical depths of the continuum can be calculated by the following formula (Frau et al., 2010; Gieser et al., 2021):

The derived $\tau$_870μm ranges from 6.2 × 10^-3 to 1.4 × 10^-1 for the 19 dense cores, so the optically thin assumption is reasonable. I15520, I16060, I16071, I16076, I16351, and I17220 were found to be associated with the ultra-compact (UC) Hii regions traced by the H40$\alpha$ lines (Qin et al., 2022; Liu et al., 2022). Free-free emissions from UC Hii regions in the submillimeter band are weak, which have less contribution to the continuum flux.

We extracted the spectra from the continuum peak positions of these dense cores. Then the spectral line transitions are identified using the eXtended CASA Line Analysis Software Suite (XCLASS¹¹1https://xclass.astro.uni-koeln.de; Möller et al. 2017). XCLASS searches for molecular line parameters in the Jet Propulsion Laboratory (JPL²²2http:///spec.jpl.nasa.gov; Pickett et al. 1998) and the Cologne Database for Molecular Spectroscopy (CDMS³³3http://cdms.de; Müller et al. 2001, 2005). The value of the partition functions in the XCLASS have 110 different temperature intervals between 1.072 and 1000 K. Assuming that the molecular gas satisfies the LTE condition, the XCLASS solves the radiative transfer equation and produces synthetic spectra for specific molecular transitions by taking into account beam dilution, dust attenuation, line opacity, and line blending. In the XCLASS modeling, the input parameters are the size ($\theta$), rotational temperature (T), column density (N), line width ($\Delta$V), and velocity offset (V_off) of each molecule. In our study, we took the deconvolved sizes (see Table 1) of the continuum sources as molecular component sizes. To better fit the rotation temperature, column density, and line width parameters, we further employed Modeling and Analysis Generic Interface for eXternal numerical codes (MAGIX; Möller et al. 2013). The MAGIX optimizes these molecular component parameters within the given range and provides the corresponding error estimates. The parameter ranges are set based on the initial guesses provided by XCLASS fitting. Note that in our case the optical depths of the continuum cores are less than 1.4 × 10^-1 and then the dust attenuation effect can be ignored.

Transitions are considered as detection if the line intensities exceed 3$\sigma$ noise level. CH₃OCHO, CH₃OCH₃, H₂CCO, and CH₃OH lines are detected in all 19 line-rich cores, and ¹³CH₃OH lines are detected in 16 line-rich cores. Figure 1 shows the sample spectra and optical depths of molecular transitions toward I14498 C1. The spectra and optical depths of molecular transitions toward the other 18 cores are presented in Figure 8 of Appendix A. The overall results of the detected spectral lines are summarized as follows:

• 1.

  Most transitions in each core are optically thin ($\tau$ $<$ 1).

• 2.

  In each core, more than three lines are detected for CH₃OCHO, CH₃OCH₃, and H₂CCO, and CH₃OCHO presents more lines than CH₃OCH₃ and H₂CCO.

• 3.

  In all cores, the CH₃OCHO and CH₃OCH₃ have larger line intensities, while the line intensities of H₂CCO are generally smaller.

The rotational transitions of CH₃OCHO, CH₃OCH₃, and H₂CCO detected in 19 dense cores are listed in Table 4 of Appendix B. The upper level energy ranges of CH₃OCHO, CH₃OCH₃, and H₂CCO are 80 to 590 K, 73 to 167 K and 148 to 474 K, respectively. These three molecules, especially CH₃OCHO and H₂CCO, cover a wide range of upper level energies, which is conducive to the constraints of rotational temperatures and column densities.

The MAGIX optimization results for CH₃OCHO, CH₃OCH₃, and H₂CCO in 19 dense cores are shown in Table 2. The column densities mainly range from 10¹⁵ to 10¹⁷ cm^-2 for CH₃OCHO and CH₃OCH₃, and from 10¹⁴ to 10¹⁶ cm^-2 for H₂CCO. The upper level energies of H₂CCO lines are larger than 148 K (see Section 3.2), and the column densities of H₂CCO are lower, which can explain its weaker line intensities than the other two molecules. The rotation temperature ranges of CH₃OCHO, CH₃OCH₃, and H₂CCO are 70 to 170 K, 50 to 137 K and 85 to 173 K, respectively. The rotation temperatures of CH₃OCH₃ in most cores are generally lower than those of CH₃OCHO and H₂CCO, possibly because our 870 $\mu$m observations of the CH₃OCH₃ lines have lower upper level energies (see Table 4) and then the hot components of the cores are not sampled. CH₃OCHO and H₂CCO should trace hotter components than CH₃OCH₃. Considering the overestimation of line widths due to velocity resolution of 0.98 km s^-1 (see Section 2), we adopt the deconvolved line widths by the following formula:

where $\Delta$V is the fitted line width convolved with the velocity resolution $\Delta$v. We computed the molecular abundances relative to H₂ and CH₃OH by the following formula:

where N is the column density of the specific molecule, N (H₂) is the column density of H₂ (see Table 1), and N (CH₃OH) is the column density of CH₃OH (see Table 2). Due to the blending of CH₃OH lines with other molecular lines in 16 line-rich cores, we then fit ¹³CH₃OH line transitions and derive the column densities of CH₃OH by the column densities of ¹³CH₃OH (see Table 2) multiplied by the ¹²C/¹³C ratios from the following formula (Yan et al., 2019):

where R_GC (in kpc) represents the distance from the Galactic Center (Liu et al., 2020). In the other 3 line-rich cores without ¹³CH₃OH detected, the column densities of CH₃OH are obtained through direct fitting, where the CH₃OH lines are not blended. The abundances of CH₃OCHO, CH₃OCH₃, and H₂CCO relative to H₂ and CH₃OH are listed in Table 3.

The integrated intensity maps of CH₃OCHO at 342572 MHz (E_u = 81 K), CH₃OCH₃ at 344358 MHz (E_u = 167 K), and H₂CCO at 343173 MHz (E_u = 148 K) in 9 high-mass star-forming regions are shown in Figure 2. The emission peaks of CH₃OCHO, CH₃OCH₃, and H₂CCO are consistent, and the spatial distribution of the three molecules is similar. These results suggest that there may be physical or chemical links among the three molecules. In addition, there is no difference in the spatial distributions of different energy level transitions of CH₃OCHO in the 9 high-mass star-forming regions, as shown in Figure 9 of Appendix C. From Figure 2 and Figure 9, the line emissions of CH₃OCHO, CH₃OCH₃, and H₂CCO are primarily distributed around intense continuum emission. I15520, I16060, and I17220 are found to be associated with intense UC Hii regions, while I16071, I16076, and I16351 are associated with weaker UC Hii regions (Qin et al., 2022; Zhang et al., 2023). In I15520, I16060, and I17220, the line emissions are offset from the continuum cores, and CH₃OCHO, CH₃OCH₃, and H₂CCO also show irregular emissions in these regions, likely due to the influence of UC Hii regions.

Unlike HMCs, which are associated with the formation of massive stars, hot corinos are found around low-mass protostars. Typical hot corinos are small in size ($\lesssim$ 200AU) and show a rich chemistry (Cazaux et al., 2003; Bottinelli et al., 2004; Maret et al., 2004; Bottinelli et al., 2007). Intermediate-mass hot cores (IMHCs) provide the link between HMCs and hot corinos, although it has rarely been reported (Sánchez-Monge et al., 2010; Palau et al., 2011; Fuente et al., 2014). In high-mass star-forming regions, dense cores exist within massive reservoirs, and the mass of the stars formed is uncertain. Therefore, it is difficult to identify hot corinos and IMHCs (sometimes they are simply assumed to be HMCs) in high-mass star-forming regions.

In Table 1, we classify 19 line-rich cores into three groups according to their mass (further material accretion or loss may occur), namely 12 high-mass line-rich cores (H; $>$ 8 M_⨀), 6 intermediate-mass line-rich cores (I; 2 – 8 M_⨀), and 1 low-mass line-rich core (L; $<$ 2 M_⨀). Moreover, the 11 high-mass star-forming regions are subsamples of Qin et al. (2022), who reported 60 HMCs from 146 high-mass star-forming regions at ALMA resolutions of $\sim$ 1.2 – 1.9 arcsec. Our currently higher spatial resolution observations reveal that part of the HMCs detected in previous works actually host multiple line-rich cores with different masses. (see Qin et al. 2022 and our Figure 2).

In Figure 3, we compare the CH₃OCHO, CH₃OCH₃, and H₂CCO abundances relative to H₂ in 19 dense cores. CH₃OCHO and CH₃OCH₃ show a strong abundance correlation (the Pearson correlation coefficient r = 0.82). The abundance correlation between CH₃OCH₃ and H₂CCO is significant (r = 0.80). CH₃OCHO and H₂CCO show the stronger abundance correlation (r = 0.93) than the other two pairs of molecules. Figure 4 shows the relationships of the abundances of the three molecules relative to CH₃OH in 19 dense cores. The abundances of CH₃OCHO and CH₃OCH₃ relative to CH₃OH mainly range from 10^-2 to 10^-1, while the abundance of H₂CCO relative to CH₃OH mainly ranges from 10^-3 to 10^-2. The abundances of the three molecules relative to CH₃OH in our observations are consistent with those observed in high-mass star-forming regions (Bøgelund et al., 2019; Peng et al., 2022; Chen et al., 2023; López-Gallifa et al., 2024), intermediate-mass star-forming regions (Palau et al., 2011; Fuente et al., 2014; Ospina-Zamudio et al., 2018), low-mass star-forming regions (Taquet et al., 2015; Lefloch et al., 2017; Jørgensen et al., 2018), Galactic center molecular clouds (Requena-Torres et al., 2006; Requena-Torres et al., 2008), and comets (Biver & Bockelée-Morvan, 2019). The correlation coefficients of the abundances relative to CH₃OH are 0.69 between CH₃OCHO and CH₃OCH₃, 0.75 between CH₃OCH₃ and H₂CCO, and 0.79 between CH₃OCHO and H₂CCO, indicating positive correlations. The obvious abundance correlation relative to CH₃OH between CH₃OCHO and CH₃OCH₃ was also observed in four different interstellar sources, including 1 HMC, 1 hot corino, and 2 comets (López-Gallifa et al., 2024). From Figure 3 and Figure 4, the relative abundances of CH₃OCHO to CH₃OCH₃ are almost constant equal to 1. Chen et al. (2023) also found a constant relative abundance of $\sim$ 1 between CH₃OCHO and CH₃OCH₃  toward 19 protostars with different luminosities. This ratio is consistent with the observations from low-, intermediate- and high-mass star-forming regions (Jaber et al., 2014; Rivilla et al., 2017; Ospina-Zamudio et al., 2018; Coletta et al., 2020; Peng et al., 2022). In Figure 3 and Figure 4, we also compared our results with the molecular abundances relative to H₂ and CH₃OH in other HMCs and hot corinos. The abundance correlations relative to H₂ and CH₃OH in these sources agree with the results in our observations but more dispersed than our samples, which is most likely due to different spatial resolution and spectral setup of these observations. These results suggest that the formation of the three molecules may be closely related, i.e., they may have similar production conditions, or even share a common chemical reaction network.

In Figure 5, we compare the rotation temperatures of the three molecules. The temperature correlation coefficient of CH₃OCHO and CH₃OCH₃ is 0.62, CH₃OCH₃ and H₂CCO is 0.47, and CH₃OCHO and H₂CCO is 0.64. The results indicate that the rotation temperatures have no significant correlations among the three molecules. The results observed by Coletta et al. (2020) toward 13 high-mass star-forming regions also showed a poor temperature correlation (r = 0.45) between CH₃OCHO and CH₃OCH₃, but the overall temperature ranges of the two molecules are similar. In fact, most of the temperatures of the three molecules in our case are in the range of tens of Kelvin, which is probably the reason why there is no apparent trend in temperature.

The line width relationships of the three molecules are shown in Figure 6. The CH₃OCHO and H₂CCO show strong line width correlation (r = 0.95), followed by CH₃OCHO and CH₃OCH₃ (r = 0.88), and by CH₃OCH₃ and H₂CCO (r = 0.79). This agrees with the line width correlation of CH₃OCHO and CH₃OCH₃ observed in 13 high-mass star-forming regions (Coletta et al., 2020). This implies that the three molecules may trace the similar kinematics.⁴⁴4In our case, the line widths of the three molecules are almost entirely contributed by non-thermal motions: $\Delta$ V_dec $\sim$ $\Delta$ V_NT = $\sqrt{\Delta\rm V_{\rm dec}^{2}-8ln2\frac{\rm kT_{\rm ex}}{\rm m}}$, where $\Delta$V_NT is the line width contributed by non-thermal motions, k is the Boltzmann constant, T_ex is the excitation temperature of the molecule, and m is the molecular mass.

In Figures 3, 5, and 6, the H cores are shifted to the upper right relative to the I and L cores, indicating a tendency for the abundances relative to H₂, temperatures and line widths of the three molecules to be higher in massive line-rich cores. In Figure 7, we also compare molecular abundances relative to H₂ and CH₃OH, rotation temperatures and line widths of the cores with and without detected outflows (see Table 1). The abundances relative to H₂, temperatures and line widths of the three molecules tend to be higher where outflows are detected. These results confirm that massive cores and cores with outflows may have richer chemistry (Jørgensen et al., 2020; Ceccarelli et al., 2023).

The spatial similarities, abundance correlations and line width correlations of CH₃OCHO, CH₃OCH₃, and H₂CCO are obvious. We briefly summarize the chemical models of these three molecules below. In gas-phase chemistry, protonated methanol (CH₃OH${}_{2}^{+}$) can react with CH₃OH to produce CH₃OCH₃  as follows (Charnley et al., 1995; Taquet et al., 2016; Jørgensen et al., 2020):

Balucani et al. (2015) proposed that CH₃OCH₃ could generate CH₃OCHO in cold gas environments through the following reactions:

where CH₃OCH₃ is a precursor to CH₃OCHO. In the case of grain-surface chemistry, the CH₃OCHO and CH₃OCH₃ follow the reactions (Garrod & Herbst, 2006; Garrod et al., 2008):

where methoxide (CH₃O) is the common precursor of CH₃OCHO and CH₃OCH₃. On the surface of the dust grains, the aldehyde (HCO) can produce H₂CCO as follows (Charnley et al., 2001; Charnley & Rodgers, 2005; Krasnokutski et al., 2017):

where HCO is the common precursor of CH₃OCHO and H₂CCO through Eq. (10) (12). The spatial similarities, abundance correlations, and line width correlations of CH₃OCHO, CH₃OCH₃, and H₂CCO, combined with chemical models, suggest that three molecules are chemically related. Taquet et al. (2016) predicted the abundances of CH₃OCH₃ relative to H₂ to be $\sim$ 10^-7 and relative to CH₃OH to be $\sim$ 10^-2 using reaction (8). Our observed abundances of CH₃OCH₃ relative to H₂ and CH₃OH are consistent with this prediction. Garrod et al. (2022) showed a chemical network that includes reactions (8), (9), (10), (11), and (13) to predict the abundances of CH₃OCHO, CH₃OCH₃, and H₂CCO. The three warm-up timescales of 5 × 10⁴ years (fast), 2 × 10⁵ years (medium), and 1 × 10⁶ years (slow) were employed in their models. At the medium timescale, the predicted abundances relative to H₂ are $\sim$ 10^-7 for CH₃OCHO and CH₃OCH₃, $\sim$ 10^-8 for H₂CCO, and relative to CH₃OH are $\sim$ 10^-2 for CH₃OCHO and CH₃OCH₃, $\sim$ 10^-3 for H₂CCO. Our observed abundances of the three molecules relative to H₂ and CH₃OH are consistent with the medium timescale predictions. In conclusion, our observed abundances of the three molecules support both grain-surface and gas-phase chemical pathways for the production of CH₃OCHO and CH₃OCH₃, and grain-surface chemical pathways for the production of H₂CCO.