CONSTRAINING THE ABUNDANCES OF COMPLEX ORGANICS IN THE INNER REGIONS OF SOLAR-TYPE PROTOSTARS

Figure 1 shows the maps of the continuum emission of IRAS 2A and IRAS 4A at 143 and 165 GHz obtained after natural weighted cleaning. Parameters of the continuum emission (integrated flux and deconvolved FWHM size), obtained from elliptical Gaussian fits in the ($u,v$) plane, are given in Table 1. For the two settings, the FWHM size of the continuum emission is slightly smaller than the size of the synthesized beam; the continuum emission is consequently not resolved. In particular, IRAS 4A is known to be a binary system with a 1 8 separation (Looney et al. 2000), as depicted by the two red crosses in Figure 1 that indicate the positions of IRAS 4A-SE and -NW. Although the continuum emission of IRAS 4A is peaked at the southeast (SE) position rather than at the northwest (NW) position for the two settings, we cannot resolve the two sources.

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We estimated the mass of the envelope and the H₂ column density from the continuum fluxes and the sizes derived from elliptical Gaussian fits listed in Table 1. Assuming an optically thin dust emission, the mass M of the envelope is given by

$$\begin{eqnarray}&&M=\frac{{{S}_{\nu }}{{d}^{2}}}{{{K}_{\nu }}{{B}_{\nu }}({{T}_{d}}){{R}_{d}}},\end{eqnarray} \tag{ 1 }$$

where ${{S}_{\nu }}$ is the continuum flux integrated over the Gaussian ellipse and listed in Table 1, d is the distance to the two low-mass protostars (235 pc; Hirota et al. 2008), ${{B}_{\nu }}({{T}_{d}})$ is the Planck blackbody function for a temperature T_d assumed to be 30 K, and R_d is the dust-to-gas mass ratio equal to 0.01. ${{K}_{\nu }}$ is the opacity per dust mass taken from column 5 in Table 5 of Ossenkopf & Henning (1994) (corresponding to grains showing an MRN size distribution covered by thin ice mantles at ${{n}_{{\rm H}}}={{10}^{6}}$ cm⁻³) and extrapolated to 145 GHz (${{K}_{\nu }}=0.38$ g cm⁻²) and 165 GHz (${{K}_{\nu }}=0.48$ g cm⁻²). The column density of H₂ averaged over the Gaussian ellipse can be deduced from M following this formula:

$$\begin{eqnarray}&&N({{{\rm H}}_{2}})=\frac{M}{\mu {{m}_{{\rm H}}}{\Omega }{{d}^{2}}},\end{eqnarray} \tag{ 2 }$$

where M is the envelope mass, μ = 2.38 is the mean molecular mass in units of hydrogen atom masses, ${{m}_{{\rm H}}}$ is the hydrogen atom mass, and ${\Omega }$ is the solid angle subtended by the Gaussian ellipse. The envelope mass M and H₂ column density N(H₂) derived for the two frequencies are listed in Table 1.

For the two sources and for all the settings, we obtained the spectral cubes by subtracting the continuum visibilities from the whole (line+continuum) datacube. For IRAS 4A, the baseline has been flattened by importing the data cubes into CLASS and subtracting a polynomial function to each individual spectrum. The 3.6 GHz wide spectra obtained at the coordinates of IRAS 2A and IRAS 4A-NW with the two WideX backends are presented in Figure 2. We also present the narrowband spectra of CH₃OH and HCOOCH₃ in Figures 3 and 4. The 1σ rms noise in the line-free channels of all spectra is given in the caption of Figures 2–4. The two sources are chemically rich since the WideX spectra toward IRAS 2A and IRAS 4A-NW display ∼200 and ∼170 lines detected with an S/N higher than 3, resulting in a line density of 28 and 23 detected lines per GHz, respectively. For comparison, Maury et al. (2014) detected 86 lines above the 3σ level between 216.9 and 220.5 GHz toward IRAS 2A with the PdBi, resulting in a similar line density of 24 detected lines per GHz.

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The line identification was carried out using the JPL (Pickett et al. 1998) and the CDMS (Müller et al. 2005) spectroscopic catalogs. Line identifications were performed by eye, by taking into account the upper energy level, the line strength, and the velocity of each transition. The detected molecules are listed in Table 2 with the number of detected transitions, the energy range of their upper energy levels, and the spectroscopic reference. We detected about 35 transitions for the main isotopologue of methanol with upper energy levels up to ∼1020 K and about 13 lines for its isotopologue ¹³CH₃OH. In addition to methanol, we report the first multi-line detection of glycol aldehyde HCOCH₂OH, ethyl cyanide C₂H₅CN, and ethanol C₂H₅OH toward low-mass protostars other than IRAS 16293.⁹ We also detected several transitions originating from methyl formate HCOOCH₃, di-methyl ether CH₃OCH₃, and methyl cyanide CH₃CN toward the two sources. Due to the low spectral resolution, we carefully checked that the detected lines do not suffer from any blending with other transitions from similar or other molecules by using the JPL and CDMS databases but also with the Splatalogue database.¹⁰ In total, about 70% of the lines detected at a 3σ level in the two 3.6 GHz WideX correlators are attributed to the complex organics listed in Table 2. Other lines are attributed to the deuterated methanol isotopologues CH₂DOH, CH₃OD, CHD₂OH, other deuterated molecules such as HDO (studied in a previous work; Taquet et al. 2013), DCN, NH₂D, and DCO⁺ (in absorption), and the sulphur-bearing species SO₂, and C³⁴S. The analysis of the deuterated methanol transitions will be published in a separate article. Approximately 20 lines are unidentified.

Table 2.  List of Molecules Detected toward IRAS 2A and IRAS 4A

	IRAS 2A		IRAS 4A		Ref.
Molecule	${{N}_{{\rm lines}}}$	${{E}_{{\rm up}}}$	${{N}_{{\rm lines}}}$	${{E}_{{\rm up}}}$
		(K)		(K)
CH3OH	34	14–1022	35	14–1022	1
HCOOCH3	20	43–248	20	43–237	2
CH2DOH	13	33–230	13	33–230	3
13CH3OH	13	14–222	12	14–222	4
CH3OCH3	8	11–314	7	11–314	5
C2H5OH	8	37–216	7	37–216	6
CH3CN	7	40–390	7	40–390	7
CHD2OH	6	20–67	6	20–67	8
HCOCH2OH	4	53–68	7	53–177	9
SO2	4	24–102	4	24–102	10
C2H5CN	4	63–130	3	63–130	11
CH3OD	1	40	1	40	12
H$_{2}^{13}$CO	1	10	1	10	13
H2C18O	1	22	1	22	14
HC3N	1	75	1	75	15
NH2CHO	1	30	1	30	16
CH2CO	1	41	1	41	17
C34S	1	14	1	14	18
DC3N	1	62	⋯	⋯	15
HDO	1	319	1	319	19
DCN	1	10	1	10	20
NH2D	1	183	1	183	21
D2CO	1	21	1	21	14

References. (1) Xu et al. (2008), (2) Ilyushin et al. (2009), (3) Pearson et al. (2012), (4) Xu & Lovas (1997), (5) Lovas et al. (1979), (6) Endres et al. (2009), (7) Cazzoli & Puzzarini (2006), (8) Parise et al. (2002), (9) Carrol et al. (2010), (10) Alekseev et al. (1996), (11) Fukuyama et al. (1996), (12) Anderson et al. (1988), (13) Müller et al. (2000), (14) Dangoisse et al. (1978), (15) Lafferty & Lovas (1978), (16) Johnson et al. (1972), (17) Fabricant et al. (1977), (18) Bogey et al. (1982), (19) Messer et al. (1984), (20) De Lucia & Gordy (1969), (21) Cohen & Pickett (1982).

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We estimated the FWHM of the lines detected with the narrowband correlators giving a spectral resolution of 0.4 MHz, and with the WideX correlator providing a spectral resolution of 1.95 MHz, through a Gaussian fit of the spectra obtained at the coordinates of IRAS 2A and IRAS 4A-NW. Table 3 lists the line widths of the transitions detected with the two correlators. The uncertainties in the linewidths are due to the statistical errors from the Gaussian fit and to the uncertainty from the low spectral resolution. In this work, the low spectral resolution dominates the uncertainty of the linewidths. Tables A1–A9 of the Appendix list the properties of all detected transitions along with the FWHM linewidths. For the two sources, the FWHM of the CH₃OH transitions detected at high spectral resolution is about 3 km s⁻¹, while the widths of the HCOOCH₃ lines vary between 0.9 and 2.8 km s⁻¹ when the lines of the -E and the -A states do not overlap. The linewidths derived with the PdBi are similar to the linewidths of other CH₃OH and HCOOCH₃ transitions derived by Maret et al. (2005) and Bottinelli et al. (2004a). Due to the low spectral resolution, the linewidths derived from the WideX correlators result from the convolution of the intrinsic linewidths with the spectral resolution of 1.95 MHz. The top panel of Figure 5 shows the deconvolved FWHM of CH₃OH, ¹³CH₃OH, HCOOCH₃, and CH₃CN, showing a high number of transitions detected at a high S/N, as a function of the energy of the upper level toward the two sources. For this purpose, we excluded several CH₃OH and HCOOCH₃ transitions that are blended with each other. No clear trend can be deduced for the two sources. The fluctuation of the deconvolved FWHM linewidths between 2 and 8 km s⁻¹ seems to be due to their high uncertainties. The bottom panel of Figure 5 compares the linewidths deduced from the narrowband correlators giving a high spectral resolution of 0.4 MHz with the deconvolved linewidths from the WideX spectra for the lines detected with the narrowband correlators. For all the lines, the FWHM linewidth deconvolved from the WideX spectra is higher than the FWHM linewidth deduced from the narrowband correlator, but the differences remain within the uncertainties.

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For all the transitions, the interferometric maps of the IRAS 2A and IRAS 4A protostars have been obtained by integrating the flux over ${{V}_{{\rm LSR}}}\pm {\Delta }V$, where ${{V}_{{\rm LSR}}}$ is the system velocity of the source and ${\Delta }V=3$ km s⁻¹ following the FWHM linewidths of the CH₃OH transitions listed in Table 3. In practice, due to the low resolution of the WideX backends, the line emission is integrated over three channels. Figures 6 and 7 show a compilation of the integrated line maps toward IRAS 2A and IRAS 4A obtained after natural weighted cleaning. For species where several transitions were detected, two maps showing a low-energy and a high-energy transition are presented. Tables A1–A9 of the Appendix list the properties of all detected transitions along with their FWHM sizes of emission and their position angle (PA) derived from the modeling of the visibilities assuming elliptical Gaussians, circular Gaussians, or point sources if Gaussian fits were not possible.

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For the two sources, the emission of most molecules is limited to the inner regions near the protostars. In IRAS 4A, the compact emission of all COM transitions originates from the -NW source although the SE protostar is brighter in the continuum. In contrast, no molecular lines seem to originate from IRAS 4A-SE, as also observed in previous interferometric observations of H$_{2}^{18}$O and other complex organics by Persson et al. (2012). The low-energy transitions of CH₃OH (the transition at ∼145.094 GHz including several non-resolved transitions toward IRAS 2A not shown in this work and all the transitions with ${{E}_{{\rm up}}}\;\leqslant$ 120 K toward IRAS 4A) show an extended emission consistent with the position of molecular outflows. For IRAS 2A, the interferometric map of the CH₃OH transitions at ∼145.094 GHz displays a slightly redshifted emission located 7'' to the north from IRAS 2A, consistent with the direction of an outflow previously detected at small scales with CO by Jørgensen et al. (2007). For IRAS 4A, the emission of the low-energy CH₃OH lines extends in the bipolar outflow along an N-S direction and seems to peak at $\sim 10^{\prime\prime}$ south from the protostars, consistent with the south lobe detected in SiO by Choi (2005). From Figure 7, it is clear that the inner hot corino does not dominate the flux of the weakly excited transitions of methanol in IRAS 4A (see also Maret et al. 2005). The emission of the HC₃N transition is also spatially resolved and shows an elongation of its emission toward the SW direction for IRAS 2A and toward the N direction for IRAS 4A. The spatial distribution and the kinematics of the outflow driven by IRAS 2A and IRAS 4A will be analyzed in a future work.

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The emission originating from all COM transitions except some CH₃OH and HC₃N lines is not spatially resolved by the array since their FWHM emission size deduced from the fit to the visibilities is lower than the synthesized beam of the interferometer. Consequently, the source sizes presented in this work can only be used as upper limits. These observations are qualitatively consistent with previous models and observations suggesting that methanol and COMs mostly come from the inner hot corino. In this region, these molecules show a jump of their abundance when the temperature is higher than the temperature of ice sublimation ($T\sim 100$ K). Maret et al. (2004) estimated a size for the hot corinos of IRAS 2A and IRAS 4A of 0 45 by reproducing the formaldehyde emission observed with single-dish telescopes with an abundance jump of two orders of magnitude at $r\sim 50$ AU from the central protostar. The luminosities assumed for IRAS 2A and IRAS 4A by Maret et al. (2004) are lower by a factor of 2.25 and 1.5, respectively, than the luminosities assumed in this work. Assuming that the temperature profile is governed by the Stefan–Boltzmanns law implies a difference in the temperature of a factor of 1.2 and 1.1, respectively. The assumption of higher luminosities for the two sources would therefore increase the size of the corino by a few AU. Maury et al. (2014) estimated an FWHM size of 0 4–0 9 for the hot corino of IRAS 2A through the use of the more extended A configuration of the PdBi array.

First estimates of the excitation temperatures and the column densities of observed molecules in the hot corinos of IRAS 2A and IRAS 4A have been obtained from the rotational diagram (RD) analysis by assuming optically thin emission and an LTE population of the levels. Since the emission of most transitions is not spatially resolved with the PdBi, we measured the flux of all transitions originating from a circular mask with a diameter equal to the major axis of the synthesized beam size of the telescope (∼2 1–2 6). Tables A1–A9 of the Appendix list the measured flux for all transitions and in the two sources.Astrochemical/dynamical models predict low abundances of COMs in the the dense regions of protostellar envelopes where the dust temperature is lower than the temperature of ice sublimation ${{T}_{{\rm ev}}}$ (∼100 K) due to the efficient depletion, while the abundance profiles show a strong jump once $T={{T}_{{\rm ev}}}$ (see Aikawa et al. 2008; Taquet et al. 2014) induced by thermal evaporation in the hot corino. We therefore assumed that all the flux measured in the $\sim 2^{\prime\prime}$ mask comes from the hot corino region. The emission sizes of the COMs are not necessarily similar, because of their different binding energies (see Jaber et al. 2014 for the example of IRAS 16293). Nevertheless, we assumed a same hot corino size ${{\theta }_{s}}$ of 0 5 for all COMs. The linewidth at FWHM ${\Delta }V$ is fixed to 3 km s⁻¹, which represents an average value of Table 3 and previous observations by Maret et al. (2005) and Bottinelli et al. (2007). For molecules with only a few lines detected within a narrow range of upper energy levels (glycolaldehyde, or ethyl cyanide) or for molecules showing only one detection, we assumed two values for the rotational temperature T_rot: ${{T}_{{\rm rot}}}={{T}_{{\rm rot}}}$(CH₃OH) and ${{T}_{{\rm rot}}}={{T}_{{\rm evap}}}$(ice) = 100 K. As already shown in several published observational works studying the emission of methanol toward high-mass and low-mass hot cores (Parise et al. 2006; Bisschop et al. 2007; Isokoski et al. 2013; Zapata et al. 2013), it is likely that low-energy transitions of CH₃OH are optically thick toward the center of protostellar envelopes, giving rise to an underestimation of their population in the rotational diagrams. Consequently, we excluded the CH₃OH transitions with ${{E}_{{\rm up}}}\leqslant 200$ K from our RD analysis. The rotational temperatures and total column densities of all species derived toward the two sources are summarized in Table 4.

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Table 4.  Results from the Rotational Diagram Analysis

Molecule	Nhc a	Trot	Xb	$X_{{\rm meth}}^{c}$	Nhc(SD)d	${{X}_{{\rm meth}}}$(SD)d
	(cm−2)	(K)		(%)	(cm−2)	(%)
	IRAS 2A
CH3OHe	(1.2 ± 0.4) × 1018	179 ± 62	(2.5 ± 0.9) × 10−7	⋯	1.4 × 1017	⋯
13CH3OH	(4.8 ± 1.3) × 1016	164 ± 43	(9.6 ± 2.5) × 10−9	⋯	⋯	⋯
HCOOCH3	(6.4 ± 1.9) × 1016	200 ± 61	(1.3 ± 0.4) × 10−8	1.9 ± 0.8	$\lt$ 1.2 × 1017	$\lt$ 85
CH3CN	(1.0 ± 0.2) × 1016	289 ± 63	(2.0 ± 0.4) × 10−9	0.30 ± 0.10	1.5 × 1015	1
CH3OCH3	(4.1 ± 1.6) × 1016	154 ± 62	(8.2 ± 3.3) × 10−9	1.2 ± 0.6	$\lt$ 7.2 × 1016	$\lt$ 53
C2H5OH	(5.1 ± 2.2) × 1016	325 ± 140	(1.0 ± 0.4) × 10−8	1.5 ± 0.8	⋯	⋯
HCOCH2OH	7.8 × 1015	179f	1.6 × 10−9	0.23 ± 0.06	⋯	⋯
	2.5 × 1015	100f	5.0 × 10−10	0.074 ± 0.019	⋯	⋯
C2H5CN	1.2 × 1015	179f	2.4 × 10−10	0.036 ± 0.010	$\lt$ 1.7 × 1016	$\lt$ 13
	6.9 × 1014	100f	1.4 × 10−10	0.021 ± 0.005	⋯	⋯
HC3N	7.0 × 1014	179f	1.4 × 10−10	0.021 ± 0.005	⋯	⋯
	7.1 × 1014	100f	1.4 × 10−10	0.021 ± 0.006	⋯	⋯
H213CO	6.6 × 1015	179f	1.3 × 10−9	0.20 ± 0.05	⋯	⋯
	2.1 × 1015	100f	4.3 × 10−10	0.063 ± 0.017	⋯	⋯
NH2CHO	1.2 × 1016	179f	2.3 × 10−9	0.35 ± 0.09	⋯	⋯
	4.3 × 1015	100f	8.7 × 10−10	0.13 ± 0.03	⋯	⋯
CH2CO	7.0 × 1015	179f	1.4 × 10−9	0.21 ± 0.05	⋯	⋯
	2.6 × 1015	100f	5.2 × 10−10	0.077 ± 0.020	⋯	⋯
	IRAS 4A
CH3OHe	(6.3 ± 3.1) × 1017	300 ± 151	(1.7 ± 0.9) × 10−8	⋯	2.0 × 1017	⋯
13CH3OH	(5.1 ± 1.5) × 1016	197 ± 56	(1.4 ± 0.4) × 10−9	⋯	⋯	⋯
HCOOCH3	(5.2 ± 3.3) × 1016	141 ± 90	(1.4 ± 0.9) × 10−9	1.5 ± 1.0	1.0 × 1017	52
CH3CN	(6.5 ± 2.9) × 1015	360 ± 162	(1.8 ± 0.8) × 10−10	0.18 ± 0.10	2.6 × 1015	1
CH3OCH3	(3.1 ± 1.0) × 1016	86 ± 27	(8.5 ± 2.6) × 10−10	0.87 ± 0.37	$\lt$ 4.5 × 1016	$\lt$ 22
C2H5OH	(4.4 ± 1.4) × 1016	221 ± 69	(1.2 ± 3.7) × 10−9	1.2 ± 0.5	⋯	⋯
HCOCH2OH	(8.9 ± 3.4) × 1015	124 ± 48	(2.4 ± 0.9) × 10−10	0.25 ± 0.12	⋯	⋯
C2H5CN	2.3 × 1015	300f	6.2 × 10−11	0.064 ± 0.018	1.9 × 1015	$\lt$ 0.92
	8.2 × 1014	100f	2.2 × 10−11	0.023 ± 0.007	⋯	⋯
HC3N	6.7 × 1014	300f	1.8 × 10−11	0.019 ± 0.005	⋯	⋯
	6.8 × 1014	100f	1.8 × 10−11	0.019 ± 0.005	⋯	⋯
H213CO	5.3 × 1015	300f	1.4 × 10−10	0.15 ± 0.04	⋯	⋯
	1.1 × 1015	100f	3.1 × 10−11	0.032 ± 0.009	⋯	⋯
NH2CHO	8.5 × 1015	300f	2.3 × 10−10	0.24 ± 0.07	⋯	⋯
	2.1 × 1015	100f	5.7 × 10−11	0.059 ± 0.017	⋯	⋯
CH2CO	1.6 × 1016	300f	4.2 × 10−10	0.43 ± 0.12	⋯	⋯
	4.0 × 1015	100f	1.1 × 10−10	0.11 ± 0.03	⋯	⋯

Notes:

^aColumn densities averaged over a source size of 0 5 (see text). ^bThe abundances relative to H₂ are obtained from N(H₂) derived at 145 GHz in Table 1 assuming a homogeneous H₂ column density within the beam. ^cThe abundance ratios are relative to the ¹³CH₃OH column density multiplied by 70 (see text). ^dColumn densities derived from previous single-dish observations and scaled to a source size of 0 5. CH₃OH: Maret et al. (2005); other COMs in IRAS 2A: Bottinelli et al. (2007); other COMs in IRAS 4A: Bottinelli et al. (2004a). ^eN_hc and T_rot have been derived from the rotation diagram neglecting the transitions with ${{E}_{{\rm up}}}\lt 105$ K (see text). ^fWhen they could not be derived from the rotation diagram, the rotational temperatures have been assumed to be equal to the temperature of evaporation of water-ice and the rotational temperature of CH₃OH.

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For most molecules, observational data can be reasonably well fitted by a straight line with some scattering, likely due to opacity or non-LTE effects. For most of species, the column densities derived toward IRAS 2A and IRAS 4A are very similar. Column densities of CH₃OH are about (6–12) × ${{10}^{17}}$ cm⁻², while the column densities of complex organics range between $6\times {{10}^{14}}$ and 6 × ${{10}^{16}}$ cm⁻². The rotational temperatures derived for most COMs are generally higher than 100 K in IRAS 2A and in IRAS 4A. Although they do not necessarily reflect the kinetic temperatures, the high rotational temperatures found in the inner protostellar envelope are in good agreement with the kinetic temperatures expected in hot corinos ($T\gt 100$ K). The CH₃CN population distribution can be used to estimate the kinetic temperature of the warm inner envelope because the CH₃CN population distribution over the different K ladders observed for J = 9 can only be modified by collisions as radiative transitions are prohibited (see Wang et al. 2010 for a more detailed discussion of the CH₃CN population in hot cores). If the energy states are thermalized and the transitions are optically thin, the kinetic temperatures within the hot corinos of IRAS 2A and IRAS 4A would be close to 290 and 360 K, respectively. As we will see in the next section, the temperatures are probably overestimated since the CH₃CN transitions are likely optically thick.

We derived a ¹²C/¹³C abundance ratio of CH₃OH of 26 and 12 in IRAS 2A and IRAS 4A, respectively. The two values are lower than the ¹²C/¹³C abundance ratio of ∼70 expected in the local ISM (Boogert et al. 2002; Milam et al. 2005; Wirström et al. 2011) by a factor 2.7 and 5.8, respectively. We verified that using the same rotational temperature for CH₃OH and ¹³CH₃OH only modifies the ¹²C/¹³C abundance ratios by a few percent at most. The low ¹²C/¹³C abundance ratio might be due to the different ranges of excitation of the observed transitions used to derive the column densities. We used excited transitions (with ${{E}_{{\rm up}}}\gt 200$ K) to derive N(¹²CH₃OH), whereas only weakly excited ¹³CH₃OH transitions (with ${{E}_{{\rm up}}}\;\lt$ 225 K) have been detected. The low ratio measured in the two protostars suggests that excited transitions of CH₃OH (${{E}_{{\rm up}}}\gt 200$ K) are also optically thick. An overestimation of the ¹³CH₃OH column density by the RD best fit, due to the large uncertainty on the fluxes, is also possible. An analysis taking the opacities into account is therefore required to clarify this issue.

We used the so-called PD analysis following the method described by Goldsmith & Langer (1999) to investigate the effect of optical depth on the column densities of each level. We applied the PD analysis where four or more transitions were detected for each species. Briefly, the PD analysis includes the influence of optical depths on the level populations assumed to be at LTE, following the formula

$$\begin{eqnarray}&&{\rm ln} (\frac{{{N}_{{\rm up}}}}{{{g}_{{\rm up}}}})={\rm ln} (\frac{{{N}_{{\rm tot}}}}{{{Q}_{{\rm rot}}}})-\frac{{{E}_{{\rm up}}}}{k{{T}_{{\rm rot}}}}-{\rm ln} (\frac{{{{\Omega }}_{a}}}{{{{\Omega }}_{s}}})-{\rm ln} ({{C}_{\tau }}),\end{eqnarray} \tag{ 3 }$$

where ${{N}_{{\rm tot}}}$ is the total column density of the species in question, ${{N}_{{\rm up}}}$ is the observed column density of the upper state of the species with an upper energy ${{E}_{{\rm up}}}$ including the opacity effect, ${{Q}_{{\rm rot}}}$ is the partition function, ${{{\Omega }}_{a}}$ is the beam solid angle, and ${{{\Omega }}_{s}}$ is the source solid angle. ${{C}_{\tau }}$ is given by

$$\begin{eqnarray}&&{{C}_{\tau }}=\frac{\tau }{1-{\rm exp} (-\tau )},\end{eqnarray} \tag{ 4 }$$

with τ being the optical depth. τ can be expressed as

$$\begin{eqnarray}&&\tau =\frac{{{c}^{3}}}{8\pi \nu _{0}^{3}}\frac{{{A}_{{\rm ul}}}}{{\Delta }V}\frac{{{g}_{{\rm up}}}{{N}_{{\rm tot}}}}{{{Q}_{{\rm rot}}}}{\rm exp} (\frac{-{{E}_{{\rm up}}}}{k{{T}_{{\rm rot}}}})({\rm exp} (\frac{h{{\nu }_{0}}}{k{{T}_{{\rm rot}}}})-1),\end{eqnarray} \tag{ 5 }$$

where c is the speed of light, ${{A}_{{\rm ul}}}$ is the Einstein-A coefficient of spontaneous emission, and ${\Delta }V$ is the FWHM fixed to 3 km s⁻¹. We performed a reduced $\chi _{{\rm red}}^{2}$ minimization by running a grid of 125,000 models covering a large parameter space in rotational temperature ${{T}_{{\rm rot}}}$ (50 values between 10 and 500 K), total column density in the source ${{N}_{{\rm tot}}}$ (50 values between 10¹⁵ and 10²⁰ cm⁻²), and source size ${{\theta }_{s}}$ (50 values between 0''.04 and 2''). At LTE, the column density of every upper state ${{N}_{{\rm up}}}$ can be derived for each set of ${{N}_{{\rm tot}}}$, ${{T}_{{\rm rot}}}$, and source solid angle ${{{\Omega }}_{s}}=\theta _{s}^{2}$ according to Equation (3). The best-fit model populations are plotted together with the observed populations of the levels in Figures A1 and A2 and are marked by red cross symbols. Tables 5 and 6 summarize the parameters of the best-fit models.

Table 5.  Results of the PD Analysis of the Methanol Emission

Source	IRAS 2A	IRAS 4A
$\chi _{{\rm red}}^{2}$	1.4	2.1
N(CH3OH) (cm−2)	$5.0_{-1.8}^{+2.9}\times {{10}^{18}}$	$1.6_{-0.8}^{+0.6}\times {{10}^{19}}$
N(13CH3OH) (cm−2)	$7.1_{-2.6}^{+4.2}\times {{10}^{16}}$	$2.3_{-1.1}^{+1.3}\times {{10}^{17}}$
${{\theta }_{s}}$ ('')	$0.36_{-0.04}^{+0.04}$	$0.20_{-0.04}^{+0.08}$
${{T}_{{\rm rot}}}$ (K)	$140_{-20}^{+20}$	$140_{-30}^{+30}$

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Table 6.  Results from the Population Diagram Analysis for the COM Emission

Molecule	Nhc	Trot	Source Size	Xa	$X_{{\rm meth}}^{b}$
	(cm−2)	(K)	('')
	IRAS 2A
CH3OH	5.0−1.8+2.9 × 1018	140−20+20	0.36−0.04+0.04	1.0−0.4+0.6 × 10−6	⋯
13CH3OH	7.1−2.6+4.2 × 1016	140−20+20	0.36−0.04+0.04	1.4−0.5+0.8 × 10−8	⋯
HCOOCH3	7.9−1.6+4.6 × 1016	160−30+50	0.36c	1.6−0.3+0.9 × 10−8	1.6−1.0+1.1
CH3CN	2.0−0.4+1.2 × 1016	130−40+230	0.36c	4.0−0.8+2.4 × 10−9	0.40−0.25+0.28
CH3OCH3	5.0−1.0+2.9 × 1016	110−20+60	0.36c	1.0−0.2+0.6 × 10−8	1.0−0.6+0.7
C2H5OH	7.9−4.0+4.6 × 1016	270−80+230	0.36c	1.6−0.8+0.9 × 10−8	1.6−1.2+1.1
HCOCH2OH	6.8 × 1015	140d	0.36d	1.4 × 10−9	0.14−0.08+0.05
C2H5CN	1.5 × 1015	140d	0.36d	3.0 × 10−10	0.030−0.017+0.011
HC3N	9.3 × 1014	140d	0.36d	1.9 × 10−10	0.019−0.011+0.007
H213CO	6.3 × 1015	140d	0.36d	1.3 × 10−9	0.130.07+0.05
NH2CHO	1.2 × 1016	140d	0.36d	2.4 × 10−9	0.24−0.14+0.09
CH2CO	6.8 × 1015	140d	0.36d	1.4 × 10−9	0.14−0.08+0.05
	IRAS 4A
CH3OH	1.6−0.8+0.6 × 1019	140−30+30	0.20−0.04+0.08	4.3−2.1+2.5 × 10−7	⋯
13CH3OH	2.3−1.1+1.3 × 1017	140−30+30	0.20−0.04+0.08	6.2−3.0+3.5 × 10−9	⋯
HCOOCH3	5.0−1.8+5.0 × 1017	60−10+20	0.20c	1.4−0.5+1.4 × 10−8	3.1−2.1+3.5
CH3CN	6.3−1.3+3.6 × 1016	200−40+110	0.20c	1.7−0.4+1.0 × 10−9	0.39−0.24+0.30
CH3OCH3	1.6−0.3+0.9 × 1017	80−20+40	0.20c	4.3−0.9+2.5 × 10−9	1.0−0.6+0.8
C2H5OH	1.6 × 1017	140d	0.20d	4.3 × 10−9	1.0−0.6+0.5
HCOCH2OH	4.8 × 1016	140d	0.20d	1.3 × 10−9	0.30−0.17+0.15
C2H5CN	6.4 × 1015	140d	0.20d	1.7 × 10−10	0.040−0.023+0.020
HC3N	2.9 × 1015	140d	0.20d	7.8 × 10−11	0.018−0.010+0.009
H213CO	1.1 × 1016	140d	0.20d	3.0 × 10−10	0.0690.040+0.034
NH2CHO	1.9 × 1016	140d	0.20d	5.1 × 10−10	0.12−0.07+0.06
CH2CO	3.4 × 1016	140d	0.20d	9.2 × 10−10	0.21−0.12+0.10

Notes:

^aThe abundances relative to H₂ are obtained from N(H₂) derived at 145 GHz in Table 1 assuming a homogeneous H₂ column density within the beam. ^bThe abundances relative to CH₃OH were computed from N(CH₃OH) derived from the PD analysis of the CH₃OH and ¹³CH₃OH emissions and adapted for the same source size. ^cThe source size was assumed to be equal to that of methanol when the size could not be constrained. ^dThe source size and rotational temperatures were assumed to be equal to those of methanol.

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We started the PD analysis by simultaneously modeling the population distribution of ¹²CH₃OH and ¹³CH₃OH. For this purpose, we assumed a ¹²C/¹³C abundance ratio of 70 following Boogert et al. (2002) and the same rotational temperature for the two isotopologues. The population modeling of high-energy ¹²CH₃OH (400 $\lt \;{{E}_{{\rm up}}}\;\lt$ 1100 K) and low-energy ¹³CH₃OH optically thin transitions allowed us to constrain the rotational temperature through the determination of the slope of the level populations, leaving only a degeneracy between ${{N}_{{\rm tot}}}$ and ${{\theta }_{s}}$. Since the optical depth τ of each level depends on the total column density ${{N}_{{\rm tot}}}$, low-energy optically thick transitions from ¹²CH₃OH can be used to constrain ${{N}_{{\rm tot}}}$ and ${{\theta }_{s}}$.

Table 5 presents the results of the PD analysis of the methanol population distribution. The methanol emission is relatively well modeled by the PD model for the two sources. The PD analysis converges toward one single set of input physical parameters (${{N}_{{\rm tot}}}$, ${{\theta }_{s}}$, ${{T}_{{\rm rot}}}$) with $\chi _{{\rm red}}^{2}$ of about 1.5–2 and with uncertainties up to 50% at a 1σ level for the column densities. The PD model was able to reproduce the population of most transitions within their uncertainties except the population of some low-energy transitions of ¹³CH₃OH in IRAS 2A and of ¹²CH₃OH in IRAS 4A, which tend to be underestimated. The rotational temperatures of methanol deduced from the PD analysis are similar in IRAS 2A and IRAS 4A (∼140 K). However, the source size derived for IRAS 2A is larger than for IRAS 4A ($\sim 0\buildrel{\prime\prime}\over{.} 36$ versus $\sim 0\buildrel{\prime\prime}\over{.} 20$). The source size deduced for IRAS 2A is in good agreement with the size estimated by Maret et al. (2004), but we found a smaller source size for IRAS 4A, by a factor of 2.5. The source size of IRAS 2A of 0 36, corresponding to a radius of 42 AU at 235 pc, is also consistent with the FWHM emission size of the CH₃OH transitions deduced by Maury et al. (2014) from elliptical Gaussian fits that range between 25 and 70 AU.

For all other COMs, the low number of observed transitions and the relatively high uncertainty on the derived column density of each level generate a degeneracy between the input parameters and prevent the PD model from converging toward one single set of input parameters: the observations are overfitted and can be reproduced by a large range of parameters giving ${{\chi }^{2}}$ lower than 1. Therefore, we decided to fix the source size for the COM emission to the size of the methanol emission, assuming that all COMs will evaporate with methanol in spite of their slightly different binding energies. Even by fixing the source size, the analysis of the glycolaldehyde and ethyl cyanide populations toward the two sources and of the ethanol population toward IRAS 4A did not allow us to converge toward one set of ${{N}_{{\rm tot}}}$ and ${{T}_{{\rm rot}}}.$ For other molecules, we were able to deduce a unique column density with relatively small uncertainties. As seen in Figures A1 and A2, the observed population distribution is well reproduced by the best-fit model since most of the column densities of upper energy levels predicted by the best-fit model lie within the range of uncertainties of the observed data. The best-fit models of the PDs generally consist of lower rotational temperatures and higher total column densities than the values derived with the RDs in order to reproduce the population of the optically thick transitions. For instance, the spread of the population distribution of the low upper energy levels (${{E}_{{\rm up}}}\leqslant 120$ K) of CH₃OH, ¹³CH₃OH, HCOOCH₃, or CH₃CN are explained by opacity effects. For these species, transitions showing a decrease of their population are optically thick with τ higher than 1.

For species where the PD analysis was not able to converge toward one set of input parameters (namely, glycolaldehyde, ethanol, and ethyl cyanide) and for molecules showing only one detected transition, we fixed the source size and the rotational temperature to the values found for methanol. Results of this analysis are shown in Table 6.

Most transitions of COMs whose collision rates have been computed (CH₃OH, CH₃CN, and HC₃N) have critical densities that range between 10⁵ and 10⁷ cm⁻³ at 100 K. They are therefore likely lower than the densities found in the hot corinos of IRAS 2A and IRAS 4A (${{n}_{{\rm H}2}}\gt 1.3\times {{10}^{8}}$ cm⁻³ following the density profiles of the two envelopes derived by Maret et al. 2004). Given the good fit to the observational data with our LTE PD analysis, it is likely that the observed species are at LTE. Most of the scattering of the population distribution can therefore be attributed to opacity effects only.

Methanol is likely the most abundant complex organics and is believed to be the precursor molecule of several COMs. It is therefore worth comparing the abundance of COMs with respect to methanol to quantify the efficiency of their formation. Moreover, column densities of COMs and methanol have been derived with similar methods and from the same observational data. The estimates of the abundance ratios are therefore more accurate than the absolute abundances derived with respect to H₂ that are likely underestimated due to an overestimate of the H column density, which may include both warm and any cold gas in a shielded disk-like region (see Persson et al. 2012). Tables 4 and 6 list the abundance ratios of the COMs with respect to methanol for the RD and PD analyses. The two targeted sources seem to have a similar chemical composition since the COM abundance ratios differ by only a factor of two at maximum. Methyl formate is the most abundant COM of our sample, with an abundance of 1.5%–3%, followed by ethanol (1%–1.5%) and di-methyl ether (1%). Other COMs are detected with abundances lower than 1%: glycolaldehyde and methyl cyanide show abundances of 0.15% and 0.40%, respectively, while ethyl cyanide is detected with an abundance of 0.03%–0.04%.

Table 4 compares the column densities and abundance ratios deduced from the RD analysis with previous single-dish studies by Bottinelli et al. (2004a), Maret et al. (2005), Bottinelli et al. (2007) carried out toward IRAS 2A and IRAS 4A. The column densities obtained from these previous observations suffer from several limitations: most detected transitions have low upper energy levels, and the large beam of single-dish telescopes encompasses the cold envelope where weakly excited lines may have contaminated the hot corino emission (see the interferometric maps in Figures 6 and 7). Consequently, the rotational temperatures and column densities derived with the PdBi are higher (T_rot = 80–290 and 300–360 K in this work toward IRAS 2A and IRAS 4A versus 100 and 25 K, respectively, and higher column densities up to one order of magnitude), since the interferometric observations probe material closer to the central protostars. The abundance ratios deduced from previous single-dish studies have also higher relative uncertainties than in this work due to the different telescope calibrations since the observations of methanol and COMs have been carried out separately. The abundances relative to methanol derived from our interferometric observations therefore differ from the abundances obtained with the single-dish observations, the latter being usually overestimated. The methyl formate abundance derived in IRAS 2A of 2% is consistent with the upper limit of 85% by Bottinelli et al. (2007). However, the abundance derived in IRAS 4A of ∼3% is 18 times lower than the value derived by Bottinelli et al. (2004a) and using the column density of CH₃OH derived by Maret et al. (2005). The higher abundance of methyl formate with respect to methanol in IRAS 4A derived from single-dish observations is explained by the lower column density of methanol derived in Maret et al. (2005) assuming optically thin emission. For the same reasons, the abundances of methyl cyanide derived in the two sources by our PD analysis are also lower, by a factor of 3–6, than the abundances obtained by Bottinelli et al. (2007). Di-methyl ether, ethanol, and ethyl cyanide have not been detected with single-dish telescopes toward IRAS 2A and IRAS 4A, but their upper limits agree well with our observations.

We report here the first detection of glycolaldehyde in low-mass protostars other than IRAS 16293. Glycolaldehyde co-exists with its isomer methyl formate with a [HCOOCH₃]/[HCOCH₂OH] abundance ratio of $12_{-2}^{+7}$ toward IRAS 2A and of $10_{-4}^{+10}$ toward IRAS 4A. These abundance ratios are similar to the ratios of ∼13 found in the sources A and B of the IRAS 16293 protostellar binary system by Jørgensen et al. (2012) from high angular resolution ALMA observations. They are also consistent with the ratios derived toward SgrB2(N) ranging from 52 in the hot core (Hollis et al. 2001) to 5 found on more extended scales (Hollis et al. 2000).

The COM abundances derived in this work are compared with other published data of low-mass, intermediate-mass, and high-mass hot cores obtained with single-dish and interferometric sub-mm telescopes in Figure 8 showing the COM abundances as a function of the protostar luminosity. Table A10 lists the abundances of selected COMs toward the hot cores shown in Figure 8 along with the references. For this purpose, we only selected observational studies where several transitions of methanol were detected. For most of the works, COMs and methanol were detected simultaneously and the abundance ratios were derived from the main isotopologue CH₃OH, either by assuming optically thin emission and LTE population (MacDonald et al. 1996; Ikeda et al. 2001; Beuther et al. 2007, 2009; Öberg et al. 2011, 2014; Palau et al. 2011), by neglecting optically thick lines in the RD analysis (Bisschop et al. 2007; Isokoski et al. 2013), or by taking the opacity of the lines into account in their model (Nummelin et al. 2000; Qin et al. 2010; Crockett et al. 2014; Neill et al. 2014). Fuente et al. (2014) derived the methanol abundances from the ¹³CH₃OH isotopologue. For IRAS 16293, we combined the CH₃OH absolute abundance derived by Schöier et al. (2002) with the COM abundances obtained by Jaber et al. (2014), both from a radiative transfer modeling, to obtain the abundance ratios. We also derived the glycolaldehyde abundance from Jørgensen et al. (2012).

Overall, the abundance ratios of COMs estimated toward IRAS 2A and IRAS 4A in this work tend to be lower than the abundances derived in other low-mass protostars from single-dish observations. For example, methyl formate and methyl cyanide have been detected toward five other low-mass sources and show abundances of 5%–50% and 0.8%–1.7% respectively, representing a factor of 1.5–15 and 1.5–3 higher than in our work. Since these observations suffer from the same limitations as the single-dish observations by Bottinelli et al. (2004a, 2007) presented in the previous section (low number of detected transitions, low upper energy levels of detected transitions, large beam encompassing the external envelope), the discrepancy likely comes from the differences in the observational methods and does not necessarily reflect differences in the chemistry between the sources.

Table 7 summarizes the mean abundance ratios of COMs in low-mass, intermediate-mass, and high-mass protostars. Along with Figure 8, they allow us to investigate any possible correlation of the COM abundances with the protostar luminosity. For each molecule, the data have been fitted by a linear curve depicted by the dashed curve whose slope a is shown at the top left of each panel of Figure 8, by considering detected ratios only. It can be noticed that the abundances of the six COMs tend to slightly increase with the protostar luminosity. However, for all species but C₂H₅CN, the increase remains negligible compared to the dispersion of the abundance ratio values. We can conclude that the abundance ratio of these COMs stays relatively constant with the protostar luminosity within six orders of magnitude. In spite of their lower luminosities, inducing lower temperatures, and smaller sizes, low-mass protostars seem to be as chemically complex as high-mass protostars.

We compare the observed abundance ratios in low-mass, intermediate-mass, and high-mass protostars with the results of two astrochemical models in Table 7. The model of Garrod (2013) is a multilayer gas-grain astrochemical model in which COMs are assumed to be mostly formed at the surface of interstellar ices during the warm-up protostellar phase ($30\;K\lt T\lt 100$ K) through radical recombination induced by the UV photodissociation of the main ice components. In the model of Rodgers & Charnley (2001), COMs are only formed by warm gas phase chemistry for a set of constant physical parameters representative of hot cores (${{n}_{{\rm H}}}={{10}^{7}}$ cm⁻³; T = 100 and 300 K) after the sublimation of interstellar ices with typical ice composition, already containing C₂H₅OH with an abundance of 20% with respect to CH₃OH. Although the rates of some key reactions for the formation of COMs have been lowered meanwhile, such as the methyl cation transfer reaction between H₂CO and CH₃OH₂⁺ or the electronic recombination of protonated COMs ions, this model still provides a good basis to estimate the formation efficiency of methyl formate, di-methyl ether, and methyl cyanide in the gas phase.

Table 7.  Averaged COM Abundances (in %) with Respect to Methanol

Molecule	LMP	IMP	HMP	G13	R01
HCOOCH3	$14_{-7}^{+41}$	1.3 ± 0.2	$14_{-6}^{+18}$	0.082–0.84	0.013–0.67
CH3OCH3	$12_{-10}^{+28}$	1.2	$30_{-18}^{+23}$	0.44–0.74	0.62–18
HCOCH2OH	$0.38_{-0.18}^{+0.31}$	/	3.1	0.54–1.3	⋯
C2H5OH	$1.3_{-0.3}^{+0.3}$	$3.2_{-2.0}^{+5.9}$	$6.4_{-4.1}^{+6.6}$	0.50–2.6	15–23
CH3CN	$1.1_{-0.5}^{+0.4}$	0.52	$5.1_{-3.3}^{+7.8}$	0.034–0.45	3.0–3.7
C2H5CN	$0.090_{-0.055}^{+0.11}$	/	$4.1_{-2.8}^{+16}$	0.052–0.79	⋯

Note. LMP, IMP, and HMP stand for low-mass ($L\lt 100\;{{L}_{\odot }}$), intermediate-mass ($100\;{{L}_{\odot }}\lt L\lt {{10}^{4}}\;{{L}_{\odot }}$), and high-mass (${{10}^{4}}\;{{L}_{\odot }}\lt L$) protostars. The two abundances in the column "G13" are the peak abundances of the "Fast" and "Slow" models of Garrod (2013). The two abundances in the column "R01" are the peak abundances of the models including ammonia at T = 100 and T = 300 K of Rodgers & Charnley (2001).

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Both models tend to underpredict the abundance of methyl formate relative to methanol observed in low-mass and high-mass protostars by at least 1–2 orders of magnitude. Moreover, the model of Garrod (2013) also underpredicts the [HCOOCH₃]/[HCOCH₂OH] abundance ratio by two orders of magnitude since it seems to reproduce well the observed abundance of glycolaldehyde. This comparison suggests that the chemical network forming methyl formate either in the gas phase or on ices is still incomplete. Possible alternative branching ratios for the photodissociation of CH₃OH on ices or other gas phase reactions involving HCOOH could enhance its formation. The observed abundances of di-methyl ether and methyl cyanide with respect to methanol can be reproduced by the gas phase model of Rodgers & Charnley (2001) only. The absolute abundances of these two molecules are similar in the two models (10⁻⁸ to 10⁻⁷ for di-methyl ether and 10⁻⁹ to 10⁻⁸ for methyl cyanide), showing that warm gas phase chemistry tends to be as efficient as surface chemistry to produce these COMs. However, the difference in the abundances comes from the efficient destruction of methanol in the warm gas in the model of Rodgers & Charnley (2001) increasing the abundance ratio of COMs. No efficient formation routes in the gas phase have been proposed for glycolaldehyde and ethanol, but their formation at the surface of interstellar ices seems to be efficient enough to reproduce the observations toward low-mass and high-mass protostars. The abundance of ethyl cyanide shows an increase of almost two orders of magnitude between low-mass and high-mass protostars. Grain surface chemistry is able to reproduce the observations toward low-mass protostars but not toward high-mass hot cores. It is also possible that models also missed gas phase reactions, as it was the case for methyl formate, where a new gas phase reaction has been recently recognized by Balucani et al. (2015).