A&A 489, 589-600 (2008)
DOI: 10.1051/0004-6361:200809354

Detection of vibrationally excited methyl formate in W51 e2

K. Demyk^1, - G. Wlodarczak¹ - M. Carvajal²

1 - Laboratoire de Physique des Lasers, Atomes et Molécules, UMR CNRS 8523 Université Lille 1, 59655 Villeneuve d'Ascq Cedex, France
2 - Departamento de Fisica Aplicada, Facultad de Ciencias Experimentales, Universidad de Huelva, 21071 Huelva, Spain

Received 4 January 2008 / Accepted 30 June 2008

Abstract
Context. Hot cores in molecular clouds, such as Orion KL, Sgr B2, W51 e1/e2, are characterized by the presence of molecules at sufficiently high temperatures to populate their low-frequency vibrationally excited states significantly. Complex organic molecules, such as methyl formate, ethyl cyanide or dimethyl ether, are characterized by a dense spectrum both in the ground state and in the excited states and lines from vibrationally excited states certainly participate to the spectral confusion.
Aims. Following a laboratory study of the first torsional excited mode of methyl formate, we search for methyl formate, HCOOCH₃, in its first torsionally excited state ( $\upsilon _{\rm t} = 1$) in the molecular cloud W51 e2.
Methods. We performed observations of the molecular cloud W51 e2 in different spectral regions at 1.3, 2, and 3 mm with the IRAM 30 m single dish antenna.
Results. Methyl formate in its first torsionally excited state ( $\upsilon _{\rm t} = 1$ at 131 cm^-1) is detected for the first time toward W51 e2. We detect 82 transitions among which 46 are unblended with other species. For a total of 16 A-E pairs in the observed spectrum, 9 are unblended; these 9 pairs are all detected. All transitions from excited methyl formate within the observed spectral range are detected and no strong lines are missing. The column density of the excited state is comparable to that of the ground state. For a source size of 7 $^{\prime \prime }$, we find that ${T_{\rm rot}} = 104 \pm 14$ K and $N = 9.4^{+4.0}_{-2.8} \times 10^{16}$ cm^-2 for the excited state and ${T_{\rm rot}} = 176 \pm 24$ K and $N = 1.7^{+.2}_{-.2} \times 10^{17}$ cm^-2 for the ground state. Lines from ethyl cyanide in its two first excited states ( $\upsilon _{\rm t} = 1$, torsion mode at 212 cm^-1) and ( $\upsilon _{\rm b} = 1$, CCN in-plane bending mode at 206 cm^-1) are also present in the observed spectrum. Blending problems prevent a precise estimate of its abundance, although as for methyl formate, it should be comparable to the value derived for the ground state for which we find ${T_{\rm rot}} = 103 \pm 9$ K and $N = 3.7^{+0.6}_{-0.4} \times 10^{15}$ cm^-2 for a 7 $^{\prime \prime }$ source size.
Conclusions. With regard to the number of lines of excited methyl formate and ethyl cyanide detected in W51 e2, it appears that excited states of large molecules certainly account for a significant number of unidentified lines in spectral survey of molecular clouds.

Key words: ISM: molecules - ISM: abundances - ISM: individual objects: W51 e2 - radio lines: ISM - line: identification - methods: observational

1 Introduction

W51 e2 is a hot core part of the W51 H ${\rm _{II}}$ region located in the Sagittarius spiral arm at a distance of about 7-8 kpc. It is a region of high-mass star formation. W51 e2 and W51 e1 appear to be important star-forming cores. They exhibit a rich chemistry, comparable to that observed either in Orion or Sagittarius. Numerous large organic molecules have been observed in their direction. CH₃CN and CS maps were studied by Zhang et al. (1998). Formic acid (HCOOH) was mapped by Liu et al. (2001). Methyl formate (HCOOCH₃) and ethyl cyanide (CH₃CH₂CN) were observed in several studies (Liu et al. 2001; Remijan et al. 2002; Ikeda et al. 2001). Ikeda et al. (2001) studied ethylene oxide (c-C₂H₄O) and its isomer acetaldehyde (CH₃CHO). Acetic acid (CH₃COOH) was detected by Remijan et al. (2002) with a fractional abundance of (1-6) $\times$ 10^-2 relative to HCOOCH₃. Glycine, whose presence may be inferred from the observations of acetic acid with which it shares common structural elements, was not detected in W51 e2 (Snyder et al. 2005). Trans-ethyl methyl ether was however detected in W51 e2 (Fuchs et al. 2005).

The rotational temperature of most of these large molecular species is high. Liu et al. (2001) estimated the rotational temperature in W51 e2 to be in the range 200-300 K. From high resolution observations of CH₃CN analyzed with statistical equilibrium models, Remijan et al. (2004) derived the kinetic temperature in W51 e2 to be $T_{{\rm kin}} = 153$(21) K. At such temperature, low-energy vibrational excited states can be significantly populated. Transitions from vibrationally excited states have indeed been observed in other sources such as in Sgr B2(N-LMH) for C₂H₃CN, CH₃CH₂OH (Nummelin et al. 1998) and CH₃CH₂CN (Mehringer et al. 2004). Lines from torsionally excited methyl formate have been identified in Orion KL (Kobayashi et al. 2007).

In this study, we present the first detection of excited methyl formate and ethyl cyanide in the molecular cloud W51 e2. Our objective was to detect methyl carbamate (NH₂COOCH₃), an isomer of glycine (NH₂CH₂COOH), which has a larger dipole moment, making its detection more favorable than glycine. It was not detected. However, a few strong unidentified lines in the data attracted our attention and were attributed to methyl formate in its first torsional excited state. Further observations confirmed this assertion and lead to the detection of excited ethyl cyanide in W51 e2. The first vibrationally excited state of methyl formate is the CH₃ torsion mode, $\nu_{18}$, hereafter referred to as $\upsilon _{\rm t} = 1$, at 131 cm^-1 (188 K). The rotational spectrum in this excited state was measured and analyzed by Ogata et al. (2004). Ethyl cyanide has two close vibrationally excited states, the CCN in-plane bending mode, $\nu_{13}$, hereafter called $\upsilon _{\rm b} = 1$, at 206 cm^-1 (296 K) and the CH₃ torsion mode, $\nu_{21}$, hereafter called $\upsilon _{\rm t} = 1$, at 212 cm^-1 (305 K). A preliminary analysis of the rotational spectrum in these two excited states is presented by Mehringer et al. (2004).

The observations and methods used for the data analysis are described in Sects. 2 and 3, respectively. The study of methyl formate and ethyl cyanide in the ground and excited state is presented in Sects. 4 and 5, respectively. The search for methyl carbamate and glycine is presented in Sect. 6. Our discussion is completed in Sect. 7.

   
2 Observations

The observations were performed with the IRAM 30 m antenna at Pico Veleta (Spain) in June 2003 and June 2006. W51 e2 was observed at the position $\alpha$(2000) = 19$^$23$^\prime$43.9 $^{\prime \prime }$ and $\delta$(2000) = 14$^$30$^\prime$34.8 $^{\prime \prime }$ in position switching mode with the OFF position located at $\alpha$ = 300 $^{\prime \prime }$ and $\delta = 0$ $^{\prime \prime }$.

Several spectral windows in the 80-250 GHz range were observed to include as many transitions as possible for the searched molecules (excited methyl formate and ethyl cyanide for the 2006 data and methyl carbamate and glycine for the 2003 data) and as few transitions as possible of other molecules with numerous strong lines (such as methyl formate and ethyl cyanide in the ground state and dimethyl ether).

All lines were observed with an array of 4 receivers (in single-side band mode) set at the appropriate frequencies. The spectrometers used were a low resolution 1 MHz filter bank and an autocorrelator with a spectral resolution in the 40-320 kHz range, split between different receivers. Focus and pointing were checked regularly by observing the nearby ultra compact HII region K 3-50A. The rejection of the image band (USB) was about 26 dB at 3 mm, 12 dB at 2 mm, 15 dB at 1.3 mm, and 10 dB at 1.1 mm. The system temperature was typically 100-200 K, 200-700 K, 200-700 K, and 400-1500 K at 3, 2, 1.3, and 1.1 mm, respectively. The total usable ON + OFF integration time varied from 30 to 50 min depending on the frequency range. The beam size was 22 $^{\prime \prime }$, 17 $^{\prime \prime }$, and 10.5 $^{\prime \prime }$ at 3, 2, and 1.3 mm, respectively. The spectra are presented in main beam temperature unit, which is calculated from the antenna temperature: $T_{\rm mb} = F_{\rm eff}$/ $B_{\rm eff} \times T_{\rm A}^*$. The data were reduced using the GILDAS package.

   
3 Analysis

For the data analysis, we assume that local thermodynamic equilibrium (LTE) is reached, i.e. we assume that the excitation, rotational, and vibrational temperatures are equal to the kinetic temperature in the emitting region and that the lines are thermalized, i.e. their level population is described by a Boltzmann distribution at that temperature. The validity of this assumption is discussed in Sect. 7.

The data were analyzed using the classical rotational diagram method to estimate the rotational temperature and the column density with their uncertainties for the different identified species. We adopted the formulation from Turner (1991), corrected for beam dilution effects:

$${\ln\left(\frac{3kW}{8{\pi}^3{\nu}S{\mu}^2g_ig_k}\right) = \ln\left(\frac{N}{Q}\right) - \frac{E_{\rm u}}{kT} - \ln~(b)}$$ (1)

where W is the integrated line intensity in K km s^-1, $\nu$ the line frequency, $S{\mu}^2$ the line strength in Debye², g_i the reduced nuclear spin statistical weight, g_k the K-level degeneracy, Q is the partition function, ${E_{\rm u}}$ the upper state energy, N is the total column density, and T is the excitation temperature. Assuming a Gaussian beam, the beam dilution factor b is given by:

$$b = \frac{\theta_{\rm s}^2} {\theta_{\rm s}^2 + \theta_{\rm tel}^2}$$ (2)

where ${\theta_{\rm s}}$ and ${\rm\theta_{tel}}$ are the source and telescope beam size in arcsecond, respectively.

Beam dilution effects were taken into account both in the rotational diagram analysis and in the emission modeling (see below). The emission region in W51 e2 was observed to be smaller than 10 $^{\prime \prime }$ for most organic molecules (Liu et al. 2001; Remijan et al. 2002). Consequently beam dilution effects were important at low frequency at which the IRAM 30 m antenna beam size is significantly larger (29 $^{\prime \prime }$ at 86 GHz).

We compared the observed spectrum with simulated spectra calculated using a simple emission model at local thermodynamic equilibrium (LTE). The expression for the simulated main beam temperature for one molecule was thus:

$$T_{\rm mb} = b \times (J - J_{\rm bg}) \times (1 - {\rm e}^{-\tau})$$ (3)

where J is the source function:

$${J = \frac{h\nu}{k} \times \left({\rm e}^{h\nu/kT} -1\right)^{-1}}$$ (4)

and

$${J_{\rm bg} = \frac{h\nu}{k} \times \left({\rm e}^{h\nu/k\times 2.7} -1\right)^{-1}}$$ (5)

$\tau$ is the optical depth, summed over all the transitions of the molecules:

$$\tau = \sum_i \frac{c^2}{8\pi\nu^2}N_{\rm tot}\frac{g_{\rm ... ...nu) {\rm e}^{-E_l/kT}\left(1-{\rm e}^{-E_{\rm ul}/kT} \right)$$ (6)

and $\Phi(\nu)$ is the line profile:

$${\Phi(\nu) = \frac{1}{\sqrt{\pi\Delta\nu_{\rm D}}} \times {\rm e}^{- \left(\nu-\nu_i\right)^2/\Delta\nu_{\rm D}^2}}$$ (7)

${A_{\rm ul}}$ is the Einstein coefficient, ${E_{\rm ul}}$ the energy of the transition, ${E_{\rm l}}$ the energy of the lower state, ${g_{\rm u}}$ is the upper state degeneracy, Q the partition function, ${\rm\Delta\nu_D}$ is the Doppler width of the line, b is the beam dilution correction factor, and ${N_{\rm tot}}$ is the total column density.

In hot cores, the temperature is such that the low-energy vibrational and/or torsional excited modes are significantly populated. We therefore used the vibrational-rotational partition function, $Q{\rm _{rv}}$, instead of the pure rotational partition function. Assuming non-interacting harmonic vibrational levels and rigid rotor levels, the ro-vibrational partition function was approximated by (see Gordy & Cook 1984):

$${Q_{\rm rv} = \prod_i\left(1-{\rm e}^{-h{\nu}_i/kT}\right)^{-d_i} \times Q_{\rm rot}}$$ (8)

where ${\nu}_i$ is the frequency of the vibrational mode i, d_i its degeneracy, and $Q{\rm _{rot}}$ is the rotational partition function. $Q{\rm _{rot}}$ is approximated by:

$${Q_{\rm rot}} = \sigma \times \sqrt{{\frac{\pi(kT)^3}{h^3ABC}}}$$ (9)

where ${\rm\sigma}$ is the symmetry number (see Gordy & Cook 1984). At LTE, the temperature used to calculate the rotational and vibrational partition function was the same: we therefore assumed implicitly that $T{\rm _{vib}} = T {\rm _{rot}}$, hypothesis that may not be valid (see Sect. 7). For methyl formate that has one internal rotor, rotational transitions are split into A and E components and ${\rm\sigma}$ is equal to 2. To calculate the partition function, we considered the first excited state of methyl formate at 131 cm^-1. For ethyl cyanide, we used the partition function given by Mehringer et al. (2004), which is calculated by summing the rotational states of the ground state and the first two excited states of ethyl cyanide at 206 and 212 cm^-1. For methyl carbamate, we used the partition function in the ground vibrational state given by Groner et al. (2007).

Table 1: Laboratory measurements, calculated frequencies and line strengths for methyformate transitions in the first torsionally excited state used in the present detection.

   
4 Torsionally excited methyl formate

The predictions for the methyl formate lines are based on the work on this species by Carvajal et al. (2007). In this work, all experimental data available on the ground and excited states (3496 and 774 microwave lines, respectively) in the 7-200 GHz frequency range, covering the J values up to 43 in the ground state and up to 18 in the first excited state $\upsilon _{\rm t} = 1$, were collected from previously published studies (Ogata et al. 2004; Plummer et al. 1986; Demaison et al. 1984; Oesterling et al. 1999; Plummer et al. 1984). Carvajal et al. (2007) also added 434 new lines of methyl formate in the ground state, measured in Lille in the 567-669 GHz spectral range and corresponding to transitions with J and K values of up to 62 and 22, respectively. This dataset was fitted with almost experimental accuracy (root-mean-square deviations of 94 kHz and 84 kHz for the 3496 (774) lines of the ground torsional state and of the excited state $\upsilon _{\rm t} = 1$, respectively) using the so-called rho axis method (RAM) described in the literature (Hougen et al. 1994) and a model extended to include perturbation terms through to the eighth order. The spectroscopic parameters and the details of the fitting procedure are given by Carvajal et al. (2007) in which a table presenting all the fitted experimental frequencies, measurement uncertainties, calculated frequencies, observed-calculated values, line strengths, energy levels as well as identifications of the transitions, is available as Supplementary data.

For the detection of lines in the W51 e2 spectrum, we provided a line-list of predicted line-center frequencies and line intensities based on an internal rotation model (RAM or Rho Axis Method). This method and the code developed was used for several molecules detected in the interstellar medium: acetaldehyde, CH₃COH, (Kleiner et al. 1996), acetamide CH₃CONH₂, (Hollis et al. 2006), and acetic acid CH₃COOH, (Ilyushin et al. 2008). The laboratory measurements and predicted line frequencies of transitions in the first excited torsional state $\upsilon _{\rm t} = 1$ of methyl formate in the spectral range used for the present detection are presented in Table 1 with the line assignment, the observed-calculated value, the experimental accuracy, the calculated uncertainty, the line strength, and the energy of the lower level.

More than eighty transitions from torsionally excited methyl formate, 82 precisely, are detected in the source among which 46 are not blended. Taking into account possible blending, we find that all lines from torsionally excited methyl formate, predicted to be sufficiently intense to be detectable, are present in the observed spectra. Furthermore, no lines, such as unobserved strong methyl formate lines, contradicts the identification in the observed spectral range. The internal rotation of the methyl group of methyl formate splits the transitions into A and E components of the same intensity. Among the 16 observed A-E pairs, 9 are not blended (see Fig. 1). The intensity of the lines in a A-E pair, when no lines of the pair is blended, is consistent with the expected ones, strengthening the identification of excited methyl formate in this source. The detected lines (observed frequency, integrated intensity, and line intensity) are listed in Table 2 with the laboratory or calculated frequency, the quantum numbers of the transition, its line strength, and lower state energy. The first column of Table 2 indicates the line number used to label the lines in Fig. 1. The number in parenthesis indicates the A or E line associated with the transition when it is observed. However, about half of the lines are blended. Comments were added in Table 2 to indicate line blenders when they are identified.

Table 2: Detected transitions of the first torsionally excited state of HCOOCH₃ in W51 e2.

The non-blended lines were used to estimate the rotational temperature and column density of excited HCOOCH₃ using the rotational diagram method (Fig. 2, Table 4). Remijan et al. (2002) mapped W51 e2 with BIMA in two transitions of HCOOCH₃ at 228.629 GHz and 90.146 GHz. The size of the emission region in these lines was of about 7 $^{\prime \prime }$ and 12 $^{\prime \prime }$, respectively. Adopting these values for torsionally excited HCOOCH₃, we estimated a rotational temperature and a column density of ${T_{\rm rot}} = 131 \pm 20$ K and $N = 3.4^{+1.5}_{-1.1} \times 10^{16}$ cm^-2, respectively, for a source size of 12 $^{\prime \prime }$, and ${T_{\rm rot}} = 104 \pm 14$ K and $N = 9.4^{+4.0}_{-2.8} \times 10^{16}$ cm^-2, for a source size of 7 $^{\prime \prime }$ (Table 4). A separate analysis of the A and E lines of excited methyl formate provided compatible values, within the uncertainty, for the rotational temperature and column density. Transitions from the ground state of HCOOCH₃ were also detected. Using the rotational diagram method, we measured for HCOOCH₃ $\upsilon_{\rm t}=0$, ${T_{\rm rot}} = 199 \pm 28$ K and $N = 9.6^{+0.9}_{-0.9} \times 10^{16}$ cm^-2 for a 12 $^{\prime \prime }$ source (Fig. 3). If we consider both lines from the ground state and from $\upsilon _{\rm t} = 1$, we find ${T_{\rm rot}} = 154 \pm 8$ K and $N = 5.6^{+4.0}_{-3.7} \times 10^{16}$ cm^-2 for a 12 $^{\prime \prime }$ source (Fig. 3, Table 4). These temperatures and column densities were used to model the emission of methyl formate in the source. The comparison of the modeled spectra with observations is shown in Fig. 1 for a number of lines from HCOOCH₃, $\upsilon _{\rm t} = 1$.

Figure 1: Detected lines from the first torsionally excited state of methyl formate HCOOCH3. The observations (histogram like curve) are compared with LTE emission models of HCOOCH3 $\upsilon _{\rm t} = 1$ with ${\theta }_{\rm s} = 7$ $^{\prime \prime }$, ${T_{\rm rot}} = 104$ K and $N = 9.4 \times 10^{16}$ cm-2 (red curve) and with ${\theta }_{\rm s} = 12\hbox {$^{\prime \prime }$ }$, ${T_{\rm rot}} = 154$ K and $N = 5.6 \times 10^{16}$ cm-2 (green curve). The numbered lines are transitions from torsionally excited HCOOCH3, see Table 2 for the attribution of each line.

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Table 3: Detected transitions of excited CH₃CH₂CN ( $\upsilon _{\rm b} = 1$ and $\upsilon _{\rm t} = 1$) in W51 e2.

Figure 2: Rotational diagram of the torsionally excited state of methyl formate for different source sizes. For a source size of 7 $^{\prime \prime }$ (black stars), we found a rotational temperature of ${T_{\rm rot}}$ =  $104 \pm 14$ K and a column density of $N = 9.4^{+4.0}_{-2.8} \times 10^{16}$ cm-2; for a source size of 12 $^{\prime \prime }$ (red diamonds), we found ${T_{\rm rot}} = 131 \pm 20$ K and $N = 3.4^{+1.5}_{-1.1} \times 10^{16}$ cm-2.

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Figure 3: Rotational diagram of methyl formate in the ground state (black crosses) and methyl formate in both the ground and excited states (red stars). The source size is 12 $^{\prime \prime }$. The rotational temperature and column density for methyl formate in the ground state are ${T_{\rm rot}} = 199 \pm 28$ K and $N = 9.6^{+0.9}_{-0.9} \times 10^{16}$ cm-2 and ${T_{\rm rot}} = 154 \pm 8$ K and $N = 5.6^{+4.0}_{-3.7} \times 10^{16}$ cm-2 when the ground and the first torsionally excited state are combined. For comparison, the rotational diagram for the excited state only is plotted (blue curve).

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5 Ethyl cyanide

Severals tens of lines from ethyl cyanide in the ground state are observed in the spectra. A large number of these lines are not blended and their energies cover a wide range, allowing us to plot a rotational diagram (Fig. 4). Adopting a source size of 12 $^{\prime \prime }$ as for the ground state of methyl formate, we found a rotational temperature of ${T_{\rm rot}} = 114 \pm 11$ K and a column density $N = 1.7^{+0.3}_{-0.2} \times 10^{15}$ cm^-2. Adopting a smaller source of 7 $^{\prime \prime }$ did not change the rotational temperature significantly but did alter the column density, such that ${T_{\rm rot}} = 103 \pm 9$ K and a column density $N = 3.7^{+0.6}_{-0.5} \times 10 ^{15}$ cm^-2 (Table 4).

Ethyl cyanide appears to be colder than previously found. Liu et al. (2001) adopted a temperature of 200 K and measured an abundance for this molecule of $4 \times 10^{15}$ cm^-2. However their analysis was based on lines of energies lower than 113 K. If we limit ourselves to low energy transitions, we find ${T_{\rm rot}} = 184$ K and a column density $N = 1.3 \times 10^{16}$ cm^-2. Ikeda et al. (2001) fixed ${T_{\rm rot}}$ to be 150 K and found a column density of $N = 7 \times 10^{14}$ cm^-2.

We searched for transitions from the first excited bending mode (in-plane CCN bending mode) of ethyl cyanide at 206 cm^-1 (designated by $\upsilon _{\rm b} = 1$), and from the first torsional excited state, $\upsilon _{\rm t} = 1$, at 212 cm^-1. These 2 states and the CCN out-of-plane bending mode at 378 cm^-1, were studied by Fukuyama et al. (1999) in the 8-200 GHz range for $J \le 16$ and $K_{\rm a} \le 2$. Lines from the two lowest energy states $\upsilon _{\rm b} = 1$ and $\upsilon _{\rm t} = 1$ were detected towards SgrB2 by Mehringer et al. (2004). This paper presents the molecular theory used for the spectral analysis of new measurements of excited ethyl cyanide in the 85-400 GHz range for $J \le 50$ and $K_{\rm a} \le 15$. However, the complete analysis has not yet been published and the prediction for the line frequencies and intensities was obtained from J. Pearson (private communication).

Most lines belonging to the excited states $\upsilon _{\rm t} = 1$ and $\upsilon _{\rm b} = 1$ are blended with other strong lines, identified or not (Table 3). However, few lines allow the column density of vibrationally excited ethyl cyanide to be constrained (Table 3). We estimated the upper limit to the abundance of excited CH₃CH₂CN by comparing the emission spectrum of excited ethyl cyanide simulated and the LTE model with the observed spectra, for different temperatures and column densities (Fig. 5). Adopting the same temperature as the ground state (100 K) and a higher temperature of 200 K, we found that the column density of vibrationally excited ethyl cyanide is 10¹⁶ cm^-2 and $5 \times 10^{15}$ cm^-2, respectively.

Figure 4: Rotational diagram of the ethyl cyanide in the ground state for different source sizes. For a source size of 7 $^{\prime \prime }$ (black crosses) we find a rotational temperature of ${T_{\rm rot}} = 103 \pm 9$ K and a column density of $N = 3.7^{+0.6}_{-0.5} \times 10 ^{15}$ cm-2; for a source size of 12 $^{\prime \prime }$ (red stars) we find ${T_{\rm rot}} = 114 \pm 11$ K and $N = 1.3 \times 10^{16} \times 10^{15}$ cm-2.

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6 Methyl carbamate and glycine

The initial aim of the observations was to search for methyl carbamate (NH₂COOCH₃), an isomer of glycine (NH₂CH₂COOH). Methyl carbamate has a strong dipole moment and is energetically more stable than glycine, two reasons that make it a good candidate for interstellar detection. The rotational spectrum of methyl carbamate in the A torsional substate was studied in the 8-240 GHz frequency range (Bakri et al. 2002; Ilyushin et al. 2006). This study was extended by Groner et al. (2007), who measured transitions of both A and E torsional substates up to 371 GHz. Our first analysis of the observations from 2003 was performed with a prediction of lines frequencies and intensities produced for the A-transitions from the work of Bakri et al. (2002). We then completed a revised analysis of our data from 2003 and 2006 with the predictions given by Groner et al. (2007).

Several tens of lines from methyl carbamate were searched in spectral regions chosen carefully to avoid or limit confusion with spectral lines from other molecules. However, we did not detect methyl carbamate in our data. Furthermore, most of the lines were fully blended and only a few lines allowed us to constrain an upper limit to the abundance of methyl carbamate in W51 e2. By comparing the simulated emission spectrum of methyl carbamate at different temperatures and source sizes with the observed spectrum (Fig. 6), we constrained the upper limit of methyl carbamate in this source. For a small emission region (7 $^{\prime \prime }$) and warm gas (200 K), we found that $N \le 5 \times 10^{14}$ cm^-2. This upper limit decreases as the rotational temperature and/or the size of the emission source decreases. We measured $N \le 2 \times 10^{14}$ cm^-2 for $T_{\rm {rot}} = 100$ K, and ${\rm {\theta}_s} = 12$ $^{\prime \prime }$, and $N \le 8 \times 10^{13}$ cm^-2 for $T{\rm _{rot}} = 50$ K and ${\rm {\theta}_s} = 30$ $^{\prime \prime }$.

Numerous transitions from both conformers of glycine are within our observed spectral ranges. Glycine is not detected in the spectra. Almost all lines are blended and only a few can be used to estimate roughly the upper limit of glycine in W51 e2. This upper limit varies by an order of magnitude depending on the source size and rotational temperature. Assuming a rotational temperature of 100 K, the upper limit to the column density of glycine is $3 \times 10^{14}$ cm² for a 7 $^{\prime \prime }$ source size, and $6 \times 10^{13}$ cm² for a 30 $^{\prime \prime }$ source size.

Table 4: Temperature and column density of the detected molecules.

Figure 5: Detected lines from the first two excited states $\upsilon _{\rm t} = 1$ and $\upsilon _{\rm b} = 1$ of ethyl cyanide CH3CH2CN. The observations (histogram like curve) were compared with LTE emission models of CH3CH2CN $\upsilon _{\rm t} = 1$ and $\upsilon _{\rm b} = 1$ with ${\theta }_{\rm s} = 7\hbox {$^{\prime \prime }$ }$, T =100 K and N = 1016 cm-2 (red curve) and T = 200 K and $N = 5 \times 10^{15}$ cm-2 (green curve). The numbered lines are excited methyl formate lines (see Table 2). See Table 3 for the identification of the excited CH3CH2CN lines.

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Figure 6: Observed spectrum of W51 e2 (histogram like curve) compared with the LTE emission spectrum of methyl carbamate at different densities, for a 12 $^{\prime \prime }$ source and rotational temperature of 100 K: red curve: N = 1014 cm-2, green curve: $N = 2 \times 10^{14}$ cm-2, blue curve: $N = 4 \times 10^{14}$ cm-2. The methyl carbamate lines are (from Groner et al. 2007): MC1: 142,13-131,12 (E) at 98515.38 GHz and 151,15-140,14 E and A at 98516.99 GHz; MC2: 231,22-222,21 at 155489.10 and 155492.68 GHz for the E and A symmetry, respectively and 232,22-221,21 at 155490.07 and 155493.65 GHz for the E and A symmetry, respectively; MC3 : 1410,5-139,4 (A) and 1410,4-139,5 (A) (blended) at 237922.60 GHz; MC4: 1610,7-159,6 (A) and 1610,6-159,7 (A) at 253099.30 and 253099.37 GHz, respectively.

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7 Discussion

We have found that the rotational temperatures derived from our rotational diagram analysis differ from the kinetic temperature, $T_{\rm kin}$ = 153(21) K, derived by Remijan et al. (2004). The rotational temperature of ethyl cyanide, of between 103 K and 114 K, is lower than $T_{\rm kin}$, independent of the adopted source size. It is higher than $T_{\rm kin}$ if we consider methyl formate in its ground state (176-199 K), lower if we consider only the excited state (104-131 K), and of the same order if we consider simultaneously the ground state and the first excited state of methyl formate (144-154 K). This appears to indicates that the LTE hypothesis is probably not fully valid. However, despite this possible departure from LTE, the rotational diagram method, which assumes implicitly that LTE has been reached and that all temperatures are equivalent (i.e. $T_{\rm exc} = T_{\rm kin} = T_{\rm rot}$ =  $T_{\rm vib}$), is the only method that can be used to estimate the excitation temperature since a statistical analysis is ruled out by the absence of known collision rates for the studied molecules.

The relevance of the LTE assumption within each rotational level may also be evaluated by comparing the cloud density to the critical density of each vibrational state. Collisional rates are not known for molecules as complex as methyl formate or ethyl cyanide. However, it is possible to estimate roughly the critical density for these molecules by adopting the value of methanol interacting with H₂, of the order of a few 10^-11 cm³ s^-1 (Pottage et al. 2004), and by assuming that it is the same for the ground and vibrational excited states. The Einstein coefficients of the rotational transitions are in the range 10^-5-10^-6 s^-1. The critical density is therefore approximately 10⁵-10⁶ cm^-3, i.e. comparable to the hydrogen density in W51 e2, which is estimated to be $n_{\rm H} = 5(2) \times 10^5$ cm^-3 (Remijan et al. 2004). There is thus a competition between collisional and radiative excitation and it is clear that the levels are probably not all thermalized.

The excitation mechanism populating the observed excited state of methyl formate and ethyl cyanide may be inferred in a similar way. The Einstein coefficients for ro-vibrational transitions from the excited states to the ground state are of the order of a few 10^-1 s^-1, far higher than for rotational transitions within the excited state (10^-5-10^-6 s^-1). Adopting the same value for the collisional rate as before (a few 10^-11 cm³ s^-1), the density needed to thermalize the levels in the excited state by collisions must therefore be higher than the critical density, which is of the order of 10¹⁰ cm^-3. The dust in W51 e2 has a temperature of about 140 K (Sollins et al. 2004) and emits efficiently at the wavelength of the excited states of both molecules. It is thus most probable that the excited states of methyl formate and ethyl cyanide are populated by radiative processes rather than collisions.

We have found that the rotational temperature of the gas decreases as the source size becomes smaller. This is surprising since it is expected that the deepest regions of hot cores are also the warmest. The observed transitions are optically thin for the adopted source sizes of 12 $^{\prime \prime }$ and 7 $^{\prime \prime }$. BIMA observations of methyl formate in W51 e2 (Remijan et al. 2002) showed that the emission region is smaller for high energy transitions than for low energy transitions. The size of the emission region was around 12 $^{\prime \prime }$ for the 7_2,5-6_2,4 transition at 90 146 MHz, corresponding to an energy of 10 cm^-1 (15 K), and around 7 $^{\prime \prime }$ for the 18_5,13-17_5,12 transition at 228 629 MHz, corresponding to an energy of 75 cm^-1 (108 K). More interferometric observations are required to locate precisely the emission region of methyl formate and ethyl cyanide in the ground and excited state and understand their temperature distribution.

The presence of methyl formate in torsional excited state in hot cores such as W51 e2 and Orion KL (Kobayashi et al. 2007) is unsurprising. Methyl formate is abundant and its torsional mode has low energy (188 K). Similarly, it is unsurprising that ethyl cyanide is more difficult to detect: it is less abundant and its excited states are at slightly higher energy, around 300 K. However, it is clear that observations of higher signal-to-noise ratio, spectral, and angular resolution, such as forthcoming observations from Herschel and ALMA, will reveal far more lines from the excited states of these molecules but also of abundant large molecules possessing low-frequency vibrational states. For example, dimethyl ether, CH₃OCH₃, has two torsional modes at about 203 and 242 cm^-1 and transitions from these modes should be present in the spectra of hot cores. More generally, a large number of unidentified lines reported in the spectra of hot molecular clouds could be due to transitions from abundant molecules in excited states.

Acknowledgements

The authors would like to thank Alexandre Faure for fruitful discussions, Jean Demaison and Isabelle Kleiner for invaluable help in the methyl formate study and John Pearson for providing us with the excited ethyl cyanide spectrum. We acknowledge all the Pico Veleta IRAM staff for their help during the observations. We thank the referee for his/her constructive comments. This work was supported by the Programme National ``Physico Chimie du Milieu Interstellaire'' and by the European Research Training Network ``Molecular Universe'' (MRTN-CT-2004-512302). M.C. thanks the CNRS (project CERC3) and Junta de Andalucia (project P07-FQM-03014) for financial support.

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