3 Results and analysis

Appendix A (in the electronic version of this paper only: see appendix for details) contains the molecular spectra observed towards each UC HII region. Figure 1 shows examples from four sources. In Appendix B (available only in electronic form from CDS: see appendix for details) we list the line detections for each source by frequency, with identifications and line parameters. Hot core sources are some of the richest known sources in molecular line emission. In total, we have identified more than 150 molecular lines in some sources, in less than 10 GHz total bandwidth. The variation between sources is great, with the number of molecular lines identified with reasonable signal-to-noise varying by a factor of 20 between chemically-rich and chemically-poor sources. Some sources show many high-excitation lines from complex molecules, whereas others show little emission. In G45.47, G10.30, G43.89 and G13.87, only a few molecular lines were detected: C¹⁷O and C¹⁸O, SO, C³⁴S, and weak emission from the lowest excitation energy lines of CH₃CCH and CH₃OH. G45.45 and G45.12 showed no emission lines in the 239 GHz band, and were not observed at other frequencies. We label these "line-poor" sources. In contrast, the sources G34.26, G10.47 and G31.41 show emission from high excitation energy lines of CH₃OH, CH₃CCH and CH₃CN, and also lines of C₂H₅CN, CH₂CHCN, CH₃CHO, CH₃¹³CN, ¹³CH₃OH, (CH₃)₂O, H₂¹³CO, HCOOCH₃, SiS, and SO₂. Five further sources, G5.89, G9.62, G12.21, G29.96 and G75.78, are also line-rich, showing a number of high excitation lines. The 20'' offset positions observed in G9.62, G10.47, G29.96, G31.41 and G34.26 show only a small number of lines, with spectra similar to the line-poor sources.

Below, we describe the techniques used in the analysis of the molecular line data and then examine the line excitation molecule-by-molecule.

 

Figure 1: Examples of spectra in the six main frequency bands for four sources (G34.26, G10.47, G29.96 and G45.47). Lines from the main band (upper frequency scale) are identified on the G34.26 spectra and those from the image band (lower frequency scale, identifications in parentheses) on the G10.47 spectra. Spectra for all the sources are available electronically (see Appendix A)

 

Figure 1: continued

3.1 Analysis

We have studied the excitation of the hot cores using a number of techniques, and some quantitative methods are reused for several different molecules. To avoid repetition, we briefly outline these techniques here. The results of the excitation studies for individual molecules are given below.

In order to make comparisons with chemical models, we are particularly interested in molecular column densities, as these translate into fractional abundances where the total hydrogen column density is known. For one or two detected transitions of a molecule, it is only possible to put a lower limit on the column density, making assumptions about the excitation conditions. Where multiple transitions involving different energy levels are observed for the same molecule, then it is possible to make a more detailed examination of the excitation. Molecules which have multiple transitions covering a wide range of excitation energies in a small frequency range, such as CH₃OH, CH₃CCH and CH₃CN, are the most useful for this purpose as the relative line strengths are not affected by calibration and beam size differences.

Line strengths $T_{\rm R}^*$ in local thermal equilibrium are a function of kinetic temperature $T_{\rm kin}$, molecular column density $N_{\rm mol}$ and source size $\theta_{\rm S}$ given by the following equations:  

$$T_{\rm R}^*\hbox{(peak)} = {\theta_{\rm S}^2\over\theta_{\rm S}^2 + \theta_{\rm B}^2} T_{\rm kin}(1-{\rm e}^{-\tau}),$$ (1)

where $\theta_{\rm B}$ is the telescope beam FWHM and the optical depth at line peak $\tau$ is a function of $T_{\rm kin}$ and source-averaged column density $N_{\rm mol}$, 

$$\tau = {2\pi^2\over 3\epsilon_0 k} (S\mu^2g_{\rm I}g_{\rm K}... ...{\rm u}/kT_{\rm kin}}\over T_{\rm kin} Q(T_{\rm kin})}\biggr).$$ (2)

Here, $\Delta v$ is the velocity FWHM for the molecule and transition under consideration, k is the Boltzmann constant and $\epsilon_0$the dielectric constant, $\nu$ and $E_{\rm u}$ are the rest frequency and upper level energy of the transition, S is the line strength, $\mu$ the permanent dipole moment and $g_{\rm I}$, $g_{\rm K}$ are the reduced nuclear spin degeneracy and K-level degeneracy respectively. The constants are correct for S.I. units. The partition function $Q(T_{\rm kin})$ used here follows the high-temperature approximations for each molecule type given by Turner (1991). These approximations are valid for $kT_{\rm kin}/h \ll B$ (or the largest rotation constant), which for the molecules seen in our survey is the case for temperatures $\ge$ 20 K.

For molecules with multiple observed transitions, our first approach is to make rotation diagrams. This is a graphical method of estimating temperature and beam-averaged column density (e.g. Turner 1991) based on the relationship between column density, temperature and measured line intensity given in Eq. (1). The rotation diagram method further assumes that the line emission is optically thin, i.e. $\tau$ is small and $\rm (1-e^{-\tau}) \simeq \tau$.In this case, Eqs. (1) and (2) reduce to the single equation  

$$N_{\rm mol} = \frac{3k\epsilon_0}{2\pi^{2}}\frac{\int T_{\rm... ...{\rm K}} \,Q(T_{\rm kin}) {\rm e}^{{E_{\rm u}}/{kT_{\rm kin}}}.$$ (3)

Writing $L = 3k_{\rm B}\epsilon_0\int T_{\rm R}^* \,{\rm d}v / 2\pi^{2}\nu S\mu^{2} g_{\rm I}g_{\rm K}$ and $T_{\rm rot} = T_{\rm kin}$, Eq. (3) can be rearranged to give  

$$\log_{10}L = \log_{10}\left(\frac{N_{\rm mol}}{Q(T_{\rm rot}... ...ht) - \frac{E_{\rm u}}{k}\frac{\log_{10}{\rm e}}{T_{\rm rot}},$$ (4)

which on a plot of $\log_{10}L$ vs. $E_{\rm u}/k$ is a straight line with gradient of $-(\log_{10}{\rm e})/T_{\rm rot}$ and y-intercept of $\log_{10}(N_{\rm mol} / Q(T_{\rm rot}))$. $N_{\rm mol}$ and $T_{\rm rot}$ are determined by a least-squares straight line fit to the data. Rotation diagrams are appropriate where emission is optically thin and from a source in thermal equilibrium which is characterised by a single temperature and density.

Secondly, we model line strengths using a full local thermal equilibrium (LTE) model which predicts line strengths as a function of kinetic temperature $T_{\rm kin}$, source-averaged molecular column density $N_{\rm mol}$, and source size $\theta_{\rm S}$ according to Eqs. (1) and (2) without assuming the optical depth $\tau$ is small.

These equations are then valid for transitions with significant optical depths as well as for optically thin lines, but still require that the lines are thermalised. Critical densities for many of our observed transitions of CH₃CN and CH₃OH are over $10^{6} \hbox{ cm}^{-3}$, and these cores are one of the few places in which thermalization is a realistic possibility. For molecules such as CH₃CN, observations of lines from isotopomers imply that the optical depth is significant in the main line, and the rotation diagram method is not sufficient. Our LTE modelling approach is similar to the modified rotation diagram method used by Olmi et al. (1996b) for the hot core sources G10.47 and G31.41. We find that the rotation diagram method is highly unreliable when applied to small numbers of transitions, to transitions with significant optical depth, or to observations which sample more than one set of physical conditions along the line of sight. Throughout, we have obtained more consistent and convincing results using simple LTE models for the excitation. We note that the LTE model is fitting $T_{\rm R}^*$directly rather than a logarithmic function of $T_{\rm R}^*$.

A third method of analysis would be to use a more advanced statistical equilibrium radiation transfer treatment as an alternative to our rotation diagram/LTE model approach. This type of approach was ruled out partly because of the extra time requirements of additional complexity, and partly because for the simple molecules we have observed too few lines, and for the complex species such as methanol and methyl cyanide the required data on collision cross-sections are unavailable for the high energy states. Comments on the validity of the LTE assumption are given for each molecule.

We estimated line parameters by fitting Gaussians or by taking linewidth and peak brightness directly from the spectra and using $\int T_{\rm R}^* \,{\rm d}\nu \simeq 1.06*T_{\rm A}^* \Delta\nu/\eta_{\rm fss}$. Where multiple transitions from the same molecule were found in a single spectrum, we fit the lines simultaneously with Gaussians of the same width, fixed at the known line frequencies.

For molecules with only one or two detected transitions we have employed a modification of the rotation diagram Eq. (3) to evaluate a lower limit to the beam-averaged column density, $N_{\rm min}$.An excitation temperature for this equation is determined by evaluating the turning point of the temperature dependent part (${\rm d}/{\rm d}T(Q(T_{\rm kin}){\rm e}^{E_{\rm u}/kT_{\rm kin}}) = 0$). It may be shown that for linear molecules $T_{\rm kin} = E_{\rm u}/k$ and for symmetric and asymmetric top molecules $T_{\rm kin} = 2E_{\rm u}/3k$. Evaluating the second derivative shows that the turning point is a minimum and hence $N_{\rm min}$ can be found, as shown in Eqs. (5).

$$\begin{eqnarray} N_{\rm min} & = & \frac{3k\epsilon_0}{2\pi^{2}}\frac{\int T_{\r... ... &&\hspace{0.3in} \textnormal{for symmetric and asymmetric tops.}\end{eqnarray}$$ (5)

Where we have not detected a molecule of interest, we have evaluated an upper limit to the beam-averaged column density ($N_{\rm max}$). Our approach is to first determine a 3$\sigma$ detection limit for the line temperature $T_{\rm R}^*$ in order to find an upper limit to the detectable integrated intensity. An assumption of the gas temperature must also be made to evaluate $N_{\rm max}$ from Eq.  (3).

Source geometry is a factor in the excitation models. In Sect. 4.1 we show that some of these sources have a dense, hot, chemically rich compact core and a halo of cooler gas. The hot core components are much smaller than the JCMT beam. Where this is the case, we take the hot gas to be concentrated centrally in one condensation (see discussion). We assume circular symmetry when estimating source sizes and masses from the observed beam filling factors.

3.2 Results by molecule

  We discuss the results molecule by molecule below, starting with the larger molecules with many lines in our frequency bands, and working down to molecules for which single lines only were observed.

3.2.1 CH₃CN

We observed the J=19-18 (349 GHz) and 13-12 (239 GHz) transitions of CH₃CN (methyl cyanide). CH₃CN is a symmetric top molecule with rotational levels split by the K quantum number, with the different K transitions for each $J+1\rightarrow J$closely grouped in frequency. We observed K transitions from K=0 to 8 for J=13-12 and K=0 to 6 for J=19-18. Excitation energies of the observed lines range from 60 to 430 K.

We detected CH₃CN with a signal-to-noise ratio of more than two in eight sources: G5.89, G9.62, G10.47, G12.21, G29.96, G31.41, G34.26 and G75.78. We failed to detect any lines above $2\sigma$ ($\sim\!0.1 \hbox{ K}$) in six sources: G10.30, G13.87, G43.89, G45.12, G45.45 and G45.47. In G34.26, G10.47 and G31.41, we also detected CH₃¹³CN lines. These are weak and blended with the higher K components of CH₃CN, but line strengths of up to 0.3 K are visible, and the existence of lines at all the expected CH₃¹³CN frequencies confirms the identification. At the 20'' offset positions, we detected the J=13-12 K = 0-3 lines of CH₃CN towards G9.62. No other source observed at the offset position (G10.47, G29.96, G31.41 and G34.26) exhibited any methyl cyanide.

We first analysed the excitation using rotation diagrams, assuming the gas to be optically thin. Figure 2 shows the CH₃CN data plotted in the rotation diagram format. Straight lines fit the data poorly, with attempts at a fit producing unexpectedly high rotation temperatures and positive rather than negative gradients in some cases. By changing the number of datapoints in the fit (removing lines at random) the rotation temperature varied widely, and no confidence could be placed in the results. We conclude that the rotation diagram method is unsuitable for the analysis of CH₃CN in these objects. The assumptions on which it is based must break down indicating high optical depth, anomalous excitation, or source structure.

The LTE model including optical depth produces a better fit to the data, as shown in Fig. 3. In order to estimate line parameters for the LTE excitation model, we used a Gaussian fit, fitting to all ${\rm CH}_3{\rm CN}$ lines in each band simultaneously. We included in our fit the K=0 and 1 lines, which are blended with each other, as the Gaussian fit produces separate estimates for their strengths. We believe the measured strengths for these lines to be less accurate than for $K\ge 2$, so in the modelling below we artificially increased their uncertainties by a factor of two. We did not use the J=19-18 K=3 and 4 lines, as they are confused with CCH lines, and nor did we fit the J=13-12 $K \geq 7$ lines, which are confused with HCOOCH₃ and CH₃¹³CN. We include J=13-12 K=5 and 6. These are also affected by blending with ${\rm CH}_3^{13}{\rm CN}$ but to a lesser extent than the higher K lines. We also include J=13-12 K=4 although there is some contribution to this line from SO₂ in the lower sideband. We did not analyse the G5.89 J=19-18 lines. These observations of G5.89 were made on a separate occasion with slightly different frequency bands, and the CH₃CN lines, though clearly present at levels of up to 1 K, fall across two spectra in the noisy extremities of the bands.

 

Figure 2: CH3CN data in rotation diagram format. We were unable to fit convincing straight lines to these data

 

Figure 3: CH3CN data overlaid with the predictions of the LTE model. The data are plotted as bars (J=13-12) and crosses (J=19-18) with errorbars, and the model predictions as diamonds and squares

The CH₃CN lines all have excitation energies of $\gt 60 \hbox{ K}$.Emission in these lines must be dominated by hot gas, such as we would expect to find in a compact core, and therefore a model with a single temperature and density is appropriate. We use a least-$\chi^2$ fit to choose the three parameters (T, $N_{{\rm CH}_3{\rm CN}}$ and $\theta_{\rm S}$) that produce the closest match to the observed line intensities. To estimate the uncertainties on the estimates of $T_{\rm kin}$, $N_{{\rm CH}_3{\rm CN}}$ and $\theta_{\rm S}$, we use a standard Monte-Carlo technique, adding random noise to the best-fit line intensities to create multiple synthetic datasets, which then produce a distribution of results for each parameter from which the uncertainties can be estimated.

  

Table 3: CH₃CN results: temperature, angular diameter and distance-corrected source size for LTE model fits to the J=19-18 and J=13-12 transitions; line width, virial mass and implied H₂ column density (averaged over source, from virial mass) from the J=13-12 transitions, and beam-averaged column density lower limits from the $J=13-12\,K=2$ line

Temperature and source size estimates are shown in Table 3. Uncertainties in the estimates from the ${\rm CH}_3{\rm CN}$ J=19-18 data are larger than for the J=13-12, reflecting the better signal-to-noise ratio of the J=13-12 observations. Not all parameters could be determined in all sources.

Column density is poorly determined by the excitation model, as is expected for significant optical depths. The line intensities are relatively insensitive to column density as it appears in the negative exponential for $\tau$ only (Eq. (1)), which for high $\tau$ varies only slowly. In Table 3 we give instead the beam-averaged column density lower limits (from the $E_{\rm u} = 109\hbox{ K } J=13-12$ K=2 line). These considerably underestimate the true CH₃CN column densities in the core, as the core is much smaller than the beam (Table 3). In the strongest sources the excitation model fits require CH₃CN column densities of at least $10^{16} \hbox{ cm}^{-2}$. In the three sources in which CH₃¹³CN is detected (G10.47, G31.41 and G34.26) the CH₃¹³CN/CH₃¹²CN line ratios imply optical depths of order 10 in the $J=13-12\,K=3-5$ lines, or CH₃CN column densities of at least $10^{17}\hbox{ cm}^{-2}$, taking [¹²C]/[¹³C] $\simeq 50$. These column densities are consistent with the observed line intensities in CH₃CN and the results of the LTE fit.

We find gas temperatures between 70 and 200 K. These high temperatures are required to excite lines with excitation energies of several hundred kelvin.

The size of the CH₃CN emitting region is small (D < 0.1 pc) in all sources. In the sources with the best signal-to-noise, G34.26 and 10.47, the source sizes suggest that the lower-excitation J=13-12 lines come from a larger region than the J=19-18. The results for the other sources (for which the uncertainties on source size are greater) are also consistent with this suggestion. This is indirect evidence for a temperature or density gradient, as the J=19-18 lines have higher critical densities and higher excitation energies.

There is no direct evidence from CH₃CN for temperature or column density differences between sources, once uncertainties are taken into account. This is partly because the uncertainties of the temperature and source-averaged column density estimates are large. Also, the optical depth in the CH₃CN transitions is significant, so most of the emission we receive comes from the front of the CH₃CN emitting region. In all sources, this will be the nearest region with sufficient temperature and density to excite the lines, ie. approximately the critical density. The size of the hot, compact core from which the CH₃CN lines are emitted appears to vary between sources, with the strongest emission from the largest sources.

Beam-averaged column density upper limits towards the sources with no detected methyl cyanide (G10.30, G13.87, G43.89, G45.21, G45.45 and G45.47) were evaluated and found to be between $8-9 \ 10^{12}$ cm^-2. Upper limits were calculated assuming a gas temperature of 50 K and an average linewidth of 6 MHz.

Upper limits for the 20'' offset positions were calculated, giving CH₃CN column densities (beam-averaged) of less than 1.1 10¹³ cm^-2. CH₃CN was detected at the offset position in G9.62 with a beam-averaged column density lower limit of 1.0 10¹³ cm^-2.The temperature assumed for the offset position was 50 K, based on the temperature derived from the CH₃CCH rotation diagram at this position.

The estimates for G10.47 and G31.41 are consistent with the interferometer results and observations of vibrationally excited ${\rm CH}_3{\rm CN}$ (Olmi et al. 1996a,b). Temperatures are generally higher than those calculated from lower-J transitions of ${\rm CH}_3{\rm CN}$ and from ammonia (Olmi et al. 1993).

Virial masses calculated from CH₃CN are given in Table 3. These are typically a few hundred solar masses. For the strongest sources, by assuming virial equilibrium we estimate $N_{{\rm H}_2} \simeq 10^{25} \hbox{ cm}^{-2}$ and $n_{{\rm H}_2} \simeq 10^{7} - 10^{8} \hbox{ cm}^{-3}$, which is sufficient for the gas to be thermalised. Working again from the assumption of virial equilibrium, and taking $N_{{\rm CH}_3{\rm CN}} = 10^{17} \hbox{ cm}^{-2}$, we estimate a CH₃CN abundance of $X_{{\rm CH}_3{\rm CN}} \simeq 10^{-8}$.

3.2.2 CH₃CCH

The CH₃CCH (propyne) J = 14 - 13 lines lie in the same passband as the ${\rm CH}_3{\rm CN}$ J = 13- 12 lines at 239 GHz. CH₃CCH is again a symmetric top molecule with several transitions closely spaced in frequency. The K = 0 to 3 components were observed in most sources. The K = 0 and K = 1 lines are blended with each other and the K = 2 line is often blended with a line of CH₃CHO, precluding the use of rotation diagrams.

CH₃CCH was detected towards 10 of our 14 sources; G9.62, G10.30, G10.47, G12.21, G13.87, G29.96, G31.41, G34.26, G45.47 and G75.78. The low dipole moment (0.78 Debye) leads to fairly easy thermalisation of CH₃CCH and the fairly weak line temperatures ($T_{\rm R}^{*}< 1$ K) suggests a low optical depth. The excitation energies of the K = 0 to 4 lines range from 90 to 200 K and they require hot gas for excitation. We note that two of these sources were not detected in CH₃CN transitions with lower $E_{\rm U}$.

Lower limits for the beam-averaged column density have been evaluated for the sources in which CH₃CCH was detected and upper limits for the beam-averaged column density have been evaluated in 3 sources where CH₃CCH has not been detected (G43.89, G45.12 and G45.45), and these are given in Table 4. Upper limits were calculated assuming a gas temperature of 50 K and a linewidth of 6 MHz. Note that the remaining source G5.89 was not observed at the 239 GHz band.

  

Table 4: CH₃CCH beam-averaged column density upper and lower limits

The beam-averaged column density of CH₃CCH is fairly similar towards most of the sources, apart from the two most line-rich sources and two of the line-poor sources. The column density lower limits of CH₃CCH towards G34.26 and G10.47 are a factor of 5-6 times greater than the other sources. The sources exhibiting low column densities of CH₃CCH are G45.47 and G13.87, the two line-poor sources for which we have CH₃CCH detections rather than upper limits.

We have also detected CH₃CCH at 20'' offset positions in two sources (G34.26 and G9.62). The offset positions should characterise the conditions in the halo. The K = 0 to 3 components were detected at both these offsets, with narrow enough linewidths to separate the K = 0 and 1 components blended at the central position. From a rotation diagram analysis of the offset positions, the halo component towards G34.26 has a rotation temperature of 38 $\pm$ 3 K and a column density of 4.6 $\pm$ 0.7 10¹⁴ cm^-2. The halo component towards G9.62 has a slightly higher rotation temperature of 46 $\pm$ 10 K and a slightly lower beam-averaged column density of 2.6 $\pm$ 1.0 10¹⁴ cm^-2.

Comparing CH₃CCH column densities of a few $\times$ 10¹⁴ cm^-2 with the H₂ hot core column densities from CH₃CN (above) gives fractional abundances of a few times 10^-11. This abundance estimate is a lower limit on the true abundance as the column densities in G34.26 and G9.62 at the offset positions are similar to the on-source positions, which suggests that the CH₃CCH is excited over a larger region than CH₃CN. CH₃CCH emission traces hot gas and is likely to come from a more compact, denser region than the CO.

  

Table 5: CH₃OH hot core component results: temperature, source-averaged column density, source diameter, distance, FWHM, virial mass, H₂ source-averaged column density (from virial mass) and fractional abundance

3.2.3 CH₃OH

We observed lines of CH₃OH (methanol) with frequencies in the 216, 241, 337, 346 and 349 GHz bands. The excitation energies of these lines range from 30 K to 800 K. The higher excitation energy lines, in the 337 GHz frequency range, are $v_{\rm t} = 2$ vibrationally excited states of CH₃OH. Of ground state lines with excitation energy above 150 K, we observed the 14(1) - 14(0) $\pm\quad E_{\rm u} = 260$ K transition at 349.104 GHz and the 12(1) - 12(0) $\pm\quad E_{\rm u} = 190$ K transition at 336.865 GHz. We observed many lines with excitation energies of less than 150 K.

 

Figure 4: CH3OH data (bars with errorbars) overlaid with the predictions of the LTE hot core plus cold halo model (diamonds)

We detected CH₃OH in all sources. The 12(1) - 12(0) $\pm$and 14(1) - 14(0) $\pm$ transitions were observed in G5.89, G9.62, G12.21, G10.47, G29.96, G31.41, G34.26, and G75.78. Vibrationally excited lines and ¹³CH₃OH lines were only observed in G34.26, G10.47 and G31.41. CH₃OH lines were detected at the 20'' offset positions in all five sources where these were observed. The ¹³CH₃OH lines are confused with CH₃CCH, HCOOCH₃ and other lines but are clearly present in these three sources with line strengths of a few times 0.1 K.

To estimate the line parameters, we fit Gaussians simultaneously to groups of lines with similar frequency. In the 337 GHz band, there are many vibrational lines of CH₃OH and it is difficult to separate and identify individual lines, but we estimate line strengths of up to 1 K in these transitions.

As in the CH₃CN case, rotation diagram analysis failed to produce satisfactory results. The fits to the data were generally poor. When A-type and E-type CH₃OH were analysed separately, derived rotation temperatures differed by a factor of more than four in every source. The observed CH₃OH lines span a wide range of excitation energies (from 30 to more than 500 K) and trace both hot and cold gas, so that the emission cannot be fitted with a single straight line corresponding to a single temperature component, and this plus moderate optical depths is why the rotation diagram method fails.

Evidence from other molecules such as CH₃CN and CO which show a hot, dense compact core and a cooler halo component leads us to try a two-component LTE model for the seven sources which show $E_{\rm u} \gt 150$ K lines. This produces a reasonable fit to the ground state CH₃OH lines. In the two-component model, the line emission is assumed to originate from two regions, a hot core and a cool halo, each characterized by temperature, CH₃OH column density, and source size.

$E_{\rm u} \gt 100$ K states are populated almost entirely by the hot core component and we use these to determine its physical parameters. Using the ratios between the $E_{\rm u} = 115$, 197 and 260 K lines (J(K)= 5(4)-6(3)A, 12(1)-12(0) and 14(1)-14(0)) we estimate $N_{{\rm CH}_3{\rm OH}}$ and $T_{\rm kin}$ using Eqs. (3) and (4). The line strength in the $E_{\rm u} = 197$ K line, which is the strongest, is then used to estimate the core size. Results for the hot core are given in Table 5 and for the cold halo in Table 6. The cold halo component is assumed to have an angular size sufficient to fill the telescope beam and a temperature of 15 K, and we estimate the beam-averaged column density from the $E_{\rm u} = 35$ K line.

For the six sources without $E_{\rm u} \gt 100$ K lines, beam-averaged column density lower limits were estimated from the observed low $E_{\rm u}$ lines, and these are also given in Table 6. Again because of reduced frequency coverage we were unable to make an analysis for G5.89, but the strengths of the 14(1) - 14(0) and 12(1) - 12(0) transitions indicate that it too has a hot CH₃OH core. The beam-averaged column density lower limits in these sources probably do not underestimate the true column densities by much, as the emission is likely to be extended (as in the sources for which the source size was determined) and at a similar temperature to that for which the lower limit was determined, 32 K.

  

Table 6: CH₃OH cold halo component column densities. Source-averaged column density estimates are from core plus halo model. Beam-averaged column density lower limits are given for the five sources for which we did not detect high excitation lines. Fractional abundances are calculated using H₂ column densities estimated from CO for sources where we have estimates of both $N_{{\rm CH}_3{\rm OH}}$ and $N_{{\rm H}_2}$

We could not fit the $v_{\rm t} = 2$ lines, which have excitation energies of > 500 K, with any LTE model. These were observed in the three richest sources G34.26, G10.47 and G31.41. These lines can be excited by infrared emission from hot dust at a wavelength of $\sim 25 ~\mu{\rm m}$ (Lovas et al. 1982). Vibrational transitions of CH₃CN were observed by Olmi et al. (1996b) in G31.41 and G10.47 and may be excited by similar infrared wavelengths, and high excitation transitions of CS in G34.26 may also be excited in this way (Goldsmith et al. 1983; Hauschildt et al. 1993). To excite these lines requires a dust temperature of several hundred kelvin, and this emission must come from a small region surrounding the exciting source.

To produce the observed ¹³CH₃OH line strengths of $\sim\!0.2$ K in G34.26, G31.41 and G10.47, an optically thick ultracompact core is required in addition to the compact component in the core-halo model. The 0.1 pc compact core component in the LTE model with column densities of a few times $10^{17}\hbox{ cm}^{-2}$ cannot account for the ¹³CH₃OH line brightness temperatures, which require optical depths greater than 10 in the corresponding ¹²CH₃OH lines, whereas the hot component in the core-halo model produces optical depths of less than 5. We can estimate the properties of the optically thick ultracompact core as follows. Assuming an excitation temperature of $\sim\!100$ K producing line brightness temperatures of 0.2 K, we estimate its size to be $\sim\!1\hbox{$^{\prime\prime}$}$ or 0.02 pc at 5 kpc (in the optically thick case, $T_{\rm R}^* \simeq {\theta_{\rm S}^2/(\theta_{\rm S}^2 + \theta_{\rm B}^2)}T_{\rm ex}$). This size is comparable to the source sizes seen in CH₃CN J=19-18. The source-averaged methanol column density must be at least $10^{18} \hbox{ cm}^{-2}$ to give the observed $^{13}\hbox{CH}_3\hbox{OH}$ line strengths. If methanol fractional abundances are $3\ 10^{-8}$ as in the compact core (see below) then hydrogen column densities in the ultracompact core are over $10^{26} \hbox{ cm}^{-2}$ and gas densities are over $10^9 \hbox{ cm}^{-3}$.

The CH₃OH transitions have critical densities up to more than $10^{6} \hbox{ cm}^{-3}$. This condition is probably fulfilled in the compact core, but in the halo region subthermal excitation seems probable. However, if the populations were subthermal we would expect underpopulation in the higher-K transitions, which have higher critical densities, and we see no evidence for this.

The hot core temperature, $\sim\!50$ K, falls in between the values derived from CH₃CN and CO, suggesting that the CH₃OH is tracing an intermediate volume of gas (Table 5). The temperature varies little from source to source, suggesting that the observed lines select for gas at this temperature, probably because of optical depth effects (the transitions have optical depths of a few). CH₃OH hot core column densities and source sizes differ by a factor of up to 3. In most cases the smaller sources have lower column densities (exceptions are G12.21 and G29.96). Absolute diameters for the compact core components are also shown in Table 5. G75.78 has the smallest hot CH₃OH core, and G10.47, G34.26, G31.41 and G12.21 have the largest.

The virial mass, resultant H₂ column density (source-averaged) and CH₃OH fractional abundances for the CH₃OH cores are given in Table 5. We use the linewidth of the $E_{\rm u} = 197$ K J(K)= 12(1)-12(0) line for the hot component. Virial masses range from 100 - 1300 $M_{\hbox{$\odot$}}$ and H₂ column densities from 3 - 12 $10^{24} \hbox{ cm}^{-2}$. The fractional abundances given are the ratio of CH₃OH source-averaged column density from the hot component model to H₂ source-averaged column density from the virial mass.

Column densities in the cold halo are over two orders of magnitude lower than in the hot core. The values given as lower limits for G13.87, G43.89, G45.12, G45.45 and G45.47 are good estimates of the true column densities as these cold halo components are optically thin and extended. All sources in which a hot CH₃OH core was not seen have low halo column densities.

CH₃OH fractional abundances for the hot cores in five out of seven sources are $3\ 10^{-8}$, from comparing the CH₃OH source-averaged column density with the H₂ source-averaged column density calculated from the virial mass. Exceptions are G34.26, which has twice this value, and G29.96, which has an extremely high abundance of $1.4 \ 10^{-7}$ as a result of its low virial mass. The CH₃OH abundance in the halo component ranges from 0.2 - 4 10^-9, at most an order of magnitude less than in the cores.

CH₃OH conditions in the 20'' offset positions are similar to the conditions in the cold halo. Assuming temperatures of 20-30 K, the column densities at the offset positions are reduced from the on-source column densities by a factor of between 2 and 10.

3.2.4 HCOOCH₃

Lines of HCOOCH₃ (methyl formate) were observed in most of the line-rich sources (G9.62, G10.47, G29.96, G31.41 and G34.26). The detected lines range in excitation energies from 30-500 K and were all of A-type ${\rm HCOOCH}_3$, apart from one E-type line seen in G34.26. The observed lines were fairly weak ($T_{\rm R}^{*} <$ 0.5 K) and broad ($\Delta v \sim$7 km s^-1). The lines generally occur in pairs which are close in frequency and usually blended, complicating the analysis somewhat. Rotation diagrams for methyl formate were attempted in three sources (G10.47, G31.41 and G34.26), but the ${\rm HCOOCH}_3$ data did not fit well to the rotation diagrams and no convincing results were obtained. No LTE fit was attempted because the lines are weak and blended. Lower limits to the beam-averaged column density were evaluated for most sources where HCOOCH₃ was detected. No such analysis was possible for G9.62, as there were no unblended lines observed. For G75.78, the line identifications were not secure, and we have calculated an upper limit for this and the remaining sources.

  

Table 7: ${\rm HCOOCH}_3$ column density upper and lower limits

The large range of excitation energies possessed by the observed lines in each source suggests that it is highly likely that ${\rm HCOOCH}_3$ emission originates from regions of differing temperature and density and is thus not well characterised by the rotation diagram approach. The high excitation lines observed and the column density lower limits indicate that the emission must be from warm, fairly dense gas of beam-averaged column density 10¹⁴ - 10¹⁶ cm^-2. Again it is the most line-rich sources (G10.47, G34.26 and G31.41) that possess denser columns of methyl formate. The line-poor sources contain less than $\sim$10¹⁴ cm^-2. This is in agreement, at least for G34.26, with other work by Macdonald et al. (1996) and Mehringer & Snyder (1996) who have found methyl formate emission from G34.26 with temperature 150 K and column density $\sim$10¹⁶ cm^-2.

3.2.5 C¹⁸O and C¹⁷O

We observed four lines of C¹⁷O and C¹⁸O: $\hbox{C}^{18}\hbox{O } 2-1$ (219.560 GHz), $\hbox{C}^{18}\hbox{O } 3-2$ (329.331 GHz), $\hbox{C}^{17}\hbox{O } 2-1$ (224.714 GHz) and $\hbox{C}^{17}\hbox{O } 3-2$ (337.061 GHz). Spectra were not taken of all these lines for all sources.

For sources where three or four CO lines were observed (G9.62, G29.96, G31.41, G34.26 and G75.78), column density, temperature and source size estimates for the CO-emitting region were made using a direct method of calculation, based on an assumption of LTE. Line parameters were measured by fitting Gaussians. From the ratio of the C¹⁸O to C¹⁷O lines, we estimated optical depth in each line (assuming the same excitation temperature in each species and a relative abundance of [C¹⁸O]/[C¹⁷O] = 3.65, Penzias 1981). Temperature and source size were calculated by comparing the line strengths of two transitions of a single species. For these five sources, the total source-averaged CO column density is calculated from each line strength using the temperature and source size estimates and assuming LTE. The results from each line are averaged to produce the quoted $N_{{\rm H}_2}$. Results from different lines are in agreement with each other to within $\pm 0.2 \ 10^{24} \hbox{ cm}^{-2}$.

For the remaining sources, only one or two lines of CO were observed (G5.89, G10.30, G10.47, G12.21, G13.87, G43.89 and G45.47), and we could not determine optical depth, temperature and source size independently. For these sources, we calculate lower limits on the beam-averaged column density from the C¹⁷O J=3-2 line.

Temperatures, source diameters and H₂ column densities derived from CO, assuming $X_{{\rm C}^{18}{\rm O}} = 10^{-7}$, are given in Table 8.

  

Table 8: CO results: temperature and source size; source-averaged H₂ column density assuming $X_{{\rm C}^{18}{\rm O}} = 10^{-7}$; linewidth, virial mass and mass derived from column density

The low temperatures (20-40 K) and large source sizes ($\sim\!20\hbox{$^{\prime\prime}$}$)confirm that CO is tracing material in the ambient cloud or halo region surrounding the compact cores. These low excitation lines are easily excited in cool gas, so the halo emission dominates over the beam-diluted core emission.

Hydrogen column densities from C¹⁸O and C¹⁷O are given in Table 8. Many of the sources have similar column densities, between $5 \ 10^{23} \hbox{ cm}^{-2}$ and $1 \ 10^{24} \hbox{ cm}^{-2}$. The line-weak sources (G10.30, G13.87, G43.89 and G45.47) have particularly low column densities. For G34.26, we estimate a high column density and high temperature. The hot core in this source has the largest angular size (from CH₃CN and CH₃OH results) and this may be influencing the results.

The mass of the halo can be estimated from CO in two ways: from the source-averaged H₂ column density and using the virial theorem. Both estimates are given in Table 8, and range from a few hundred to a few thousand solar masses. Where we have only column density lower limits, we have estimated a virial mass assuming a source size of 0.5 pc (the virial mass varies linearly with source size), and the masses from the column density are lower limits. The masses agree to within a factor of 8, which is reasonable given the uncertainties in the quantities involved. Both $M_{\rm vir}$ and $M_{\rm CD}$ from CO are remarkably consistent with the C³⁴S results of Cesaroni et al. (1991). Errors in the distance affect the mass estimates, as $M_{\rm vir} \propto d$ and $M_{\rm CD} \propto d^2$. However, in G9.62, G29.96, G31.41 and G34.26, the distances would have to be reduced by up to a factor of 5, which is far in excess of any uncertainty in the distance estimates. Note that $M_{\rm CD} \gt M_{\rm vir}$ in four out of five sources for which both masses were determined. This suggests that the linewidth and therefore the virial mass is not being significantly increased by outflows or other systematic velocity variations.

3.2.6 Sulphuretted species (SO₂, H₂S, SO and C³⁴S)

Sulphuretted molecules have been postulated to be a major chemical diagnostic of evolution in hot cores (Millar et al. 1997; Charnley 1997). We include results from all observed sulphur-bearing molecules in this section. The column densities derived from each molecule are summarised in Table 9.

  

Table 9: Column densities of sulphuretted species. Note that G45.12 and G45.45 have not been observed at any of the frequencies of the lines below and have been omitted from the table and that a dash denotes that the source has not been observed at the frequency of the line(s) in question

Only one transition of H₂S was observed, the 2₂₀ - 2₁₁ line at 216.71 GHz. H₂S was seen towards all sources observed at this frequency (G9.62, G10.47, G29.96, G31.41, G34.26 and G75.78). Lower limits to the column density have been evaluated and are summarised in Table 9.

SO₂ was detected towards 6 sources (G5.89, G9.62, G10.47, G12.21, G34.26 and G75.78), with excitation energies ranging from 100 - 500 K. Sufficient lines for the rotation diagram analysis to be performed were seen towards G10.47, G34.26 and G75.78. Mainly moderate (100-200 K) excitation lines were observed towards G75.78 whereas a mixture of moderate and high excitation lines were observed toward G34.26 and G10.47. The G10.47 rotation diagram data do not fit well to a straight line and the temperature and beam-averaged column density of SO₂ in G34.26 are poorly constrained. Too few lines were observed in each of these sources to construct separate high and low excitation rotation diagrams.

SO was seen towards most of our sources (G5.89, G9.62, G10.47, G12.21, G29.96, G31.41, G34.26, G45.47 and G75.78). The remaining sources (G10.30, G43.89, G45.12 and G45.45) were not observed at the frequencies of the detected SO lines. Only G10.47 possesses sufficient SO lines for a rotation diagram analysis, the temperature determined from this was 29 $\pm$ 11 K. Lower limits for the beam-averaged SO column density have been evaluated for the remaining sources and are summarised in Table 9.

The C$^{34} \hbox{S }7-6$ line at 337.06 GHz was observed towards G5.89, G9.62, G10.47, G12.21, G29.96, G31.41, G34.26, G43.89 and G75.78. No C³⁴S was detected towards G10.30 and the remaining sources G45.12, G45.45, G45.47 and G13.87 were not observed at this frequency. C$^{34} \hbox{S }7-6$ has an excitation energy of 65 K. Lower and upper limit column density analyses were performed and the results are given in Table 9.

3.2.7 Other molecules

We discuss other important molecules detected in the survey (C₂H₅CN, CH₃CHO, CCH and HNCO) in turn.

C₂H₅CN (ethyl cyanide) lines, with excitation energies of 130-290 K, were seen towards G10.47, G31.41 and G34.26. The observed lines were fairly faint with $T_{\rm R}^*$ < 0.5 K. It was found that the C₂H₅CN data from G34.26 did not fit to a rotation diagram. Single lines of CH₂CHCN were detected towards G10.47 and G31.41, with a beam-averaged column density lower limit of $\sim$ 2 10¹⁴ cm^-2 in both cases. This is in agreement with the column densities of 10¹⁴ - 10$^{15} \hbox{ cm}^{-2}$observed in G34.26 by Mehringer & Snyder (1996).

CH₃CHO (acetaldehyde) was tentatively detected in one or two transitions towards most sources in the survey (G9.62, G10.47, G12.21, G29.96, G31.41, G34.26 and G75.78). One transition was identified as the 7(3,4) - 7(2,6) at 236.52 GHz and a second as 12(1,11) - 11(1,10) CH₃CHO at 235.99 GHz. Both transitions are of the E-type form of CH₃CHO. However, CH₃CHO should have many other transitions in the observed bands which we did not detect, placing these identifications are in some doubt. The transition at 236.52 GHz may be HC₃N. Nevertheless, lower limit analyses were carried out for these sources; column densities are similar across all the sources at a few 10¹⁵ cm^-2 with the exception of G10.47 which has a slightly higher beam-averaged column density lower limit of 1.4 10¹⁶ cm^-2.

Two lines of CCH (the ethynl radical) were detected (the J = 9/2-7/2 and 7/2-5/2 doublets of the N = 4-3 transition at 349 GHz) towards G9.62, G10.47, G12.21, G29.96, G31.41, G34.26, G45.47 and G75.78. The remaining sources were not observed at this frequency. The CCH lines are slightly blended with the CH₃CN J = 19 - 18 $K = 3 \hbox{ \& } 4$ lines in most sources; in G34.26 the linewidths are such that the CCH lines are completely blended. Beam-averaged column density lower limits of $\sim\!10^{14} \hbox{ cm}^{-2}$ were evaluated for all sources.

HNCO (isocyanic acid) was detected towards G9.62, G10.30, G10.47, G29.96, G31.41, G34.26, G43.89 and G75.78. Most observed transitions of HNCO were unfortunately blended with each other or with methanol and could not be analysed. However lower limit analyses were possible for all the above sources except G10.30 and G43.89 and revealed that the beam-averaged column density of HNCO is fairly constant across all the sources at $\sim$10¹⁴ cm^-2. HNCO was also detected at 20'' offset positions in G9.62 and G34.26 with a beam-averaged column density lower limit of $\sim$5 10¹³ cm^-2.