Astron. Astrophys. 321, 311-322 (1997)

4. Data analysis

First, we derived parameters of the sources analytically making several simplifications. Second, we made statistical equilibrium calculations to improve the obtained values and estimate their accuracy. We did not analyze DR 21MetC because of poor signal-to-noise ratio of its 96 GHz spectrum.

4.1. Analytical approach

To derive parameters of the sources analytically we had to make several assumptions. We assumed that the same homogeneous sources were observed in all the lines at 96 and 157 GHz. We also assumed a two-temperature model for methanol excitation (see for example Menten et al., 1986). This implies that the excitation temperatures of the transitions within the same K -ladders (in particular, ) can be fitted by a single rotational temperature , and that the transitions (in particular, ) have the same excitation temperature . The latter assumption is roughly in agreement with the results of statistical equilibrium calculations in the wide range of temperatures and densities that is typical for Galactic star-forming regions for levels with and . Note that in the case of two-temperature excitation, , defined above, may be very different from the rotational temperature obtained using standard rotational diagram technique. We neglected frequency differences between the lines in the same series both for 96 and 157 GHz.

The method used in our study has considerable advantages in comparison with the widely used rotational diagram technique. First, the model with two excitation temperatures is in good agreement with the results of statistical equilibrium calculations for dense gas, and in any case better represents the real excitation temperatures of different transitions than a model with a single rotational temperature. Second, this method allowed us not to use the groundless assumption that all the lines in consideration are optically thin.

First, we determined and the optical depths of the lines at 157 GHz. We did this in a way similar to that used by Menten et al. (1988b) to analyze the lines in Orion.

The optical depth in the line centre is

where µ is the dipole moment, which is equal to 1.412 Debye for b-type methanol transitions, and S and are the line strength and excitation temperature, respectively. is the molecular column density in the lower level divided by the statistical weight of the level; is the full width at half maximum of the line in hertz. Using (1) together with the Boltzmann formula (2) for the levels in the backbone ladder and

where is the energy difference between the lower levels of the 157 GHz transitions under consideration, and , one can easily show that the optical depth of any 157 GHz line can be expressed through the optical depth of the line:

Combining (1)-(3) with the radiation transfer equation

where B is the Planck function, one can show that the ratio of the observed brightness temperature of any 157 GHz line and the line, , can be expressed by the equation:

which shows that in the two-temperature model these ratios depend on two unknown parameters, and . We obtained a set of values using data from Table 3 and derived and from (5) by a least-squares method. The results are presented in Table 4.

Table 4. Parameters of the sources obtained analytically. Columns: 1 - source name; 2 - optical depth of the line; 3 - kinetic temperature; 4 - methanol column density; 5 - calculated brightness temperature of the line; 6 - adopted distance to the source; 7 - calculated source linear size; 8 - hydrogen number density; 9 - methanol abundance (assuming equal abundances of A and E methanol)

At a second step, we determined , the excitation temperature of the transitions. For this purpose, we determined the ratios of populations of the levels and , using our data on the and lines. These ratios are connected with via the expression

where and are methanol column densities in the upper and lower levels, which are and in our case, divided by statistical weights of the levels. Under our basic assumptions, the excitation temperatures of both 96 GHz lines in consideration are equal to . Taking into account that the intensities of the 96 GHz lines are typically quite different, it is reasonable to suggest that the lines are optically thin. Statistical equilibrium calculations confirm this suggestion (see next section). In this case one can obtain the following expression for the ratio of the and level populations using (1) and (4) together with our basic assumptions

Using (6) and (7) we can obtain . For the sources in consideration, proved to be of the order of 6-10 K, i.e. much less than the kinetic temperatures of the same sources.

Note that in the analysis above, only relative line intensities were used. Thus, these results are free from calibration errors.

Knowledge of , , , and the optical depths of the 157 GHz lines allowed us to obtain some other source parameters. We used and to estimate the real, not beam-averaged, Raleigh-Jeans brightness temperature of the line . This can be obtained using the radiation transfer equation (4). Knowledge of allowed us to obtain beam-filling factors, equal to , where are the main-beam brightness temperatures. Assuming the source brightness temperature distribution to be Gaussian, we can estimate the source half-power widths and the source linear sizes D, if the distances are known.

Another parameter of interest is the methanol column density. We estimated methanol column densities in the following way. First, we calculated real (not beam-averaged) methanol column densities for the level, , using (1). Knowledge of and allowed us to obtain, within the frame of the two-temperature model, the populations of all the rotational levels, and thus to obtain E -methanol column densities. We multiplied the result by 2 to take into account A-methanol (we made the usual assumption that the A and E methanol abundances are equal). Then we estimated the methanol number densities , by dividing by D.

Our next step was to estimate very roughly the hydrogen number densities. We used the two-level approximation, i.e., we chose a transition with known excitation temperature, determined collisional constants, which depend on temperature and density (see formula 10), from the solution of the corresponding statistical equilibrium equation, and then calculated the density.

The excitation temperature of a line depends both on direct collisional and radiative transitions between upper and lower levels and transitions through intermediate levels. The former are taken into account by the two-level approximation, while the latter require computer modelling. To obtain a density close to the real one, we should choose a line with maximum weight of direct transitions. We believe that the best transitions for our purpose are the transitions. Their Einstein coefficients of radiation decay are largest among all rotational transitions for which we can determine excitation temperature, and direct transitions between upper and lower levels determine their relative population to a large degree (see Pelling, 1975). We chose the line at 351 GHz and had to take into account its optical depth; we used the following procedure. First, we calculated the line excitation temperature using the ratio of populations of the and levels, which were obtained at the previous step. Then we calculated the parameter , the probability per scattering that a photon is lost by collisional de-excitation. This can be obtained from ) using formulae presented by Hummer and Rybicki (1971). For our purpose, we arbitrarily chose the relation between and for the outer layers of an optically thick cloud

Knowledge of allows us to calculate the rate of collisional de-excitation of the transition using the solution of the statistical equilibrium equation in the form

where is the Einstein coefficient for radiative decay from level u to level l, , and , respectively (Hummer and Rybicki, 1971). is connected to density via the expression for collisional constants

which is in agreement with the experimental results of Lees and Haque (1974); the same collisional constant was used for this transition in the statistical equilibrium calculations (see next section).

Note, however, that although the transitions are the best ones determining density using the two-level approximation, even for them it is not clear if we can neglect transitions through intermediate levels. The validity of the two-level aproximation does not follow from our basic assumptions. Only satisfactory agreement between the densities obtained as described above and those obtained by statistical equilibrium calculations (see next section) shows the validity of this approach for dense, warm gas.

Knowing the methanol and hydrogen densities, we can determine the methanol abundance , dividing by . The densities and abundances obtained are presented in Table 4.

4.2. Statistical equilibrium calculations

Statistical equilibrium (SE) calculations are necessary to check and improve the results obtained in the previous section. The approach used in this paper is that of Olmi et al. (1993). We used an LVG code kindly made available by C. M. Walmsley. The free parameters of the model are the kinetic temperature , the molecular hydrogen number density , and the methanol E density . Velocity gradients were chosen such that the methanol E column density divided by the linewidth for the plane (see below) for each source was equal to the same parameter obtained analytically. We neglected any external radiation, except the microwave background, to make the problem tractable. Some arguments showing the absence of a strong radiation field will be given elsewhere.

To obtain SE parameters, one should make statistical equilibrium calculations for a number of parameter sets and choose sets which are in agreement with the observational data. The agreement between our observations and the SE models can be evaluated by comparing the 96 and 157 GHz line intensity ratios of a model with the corresponding observed ratios. For the 96 GHz series we used the ratios of E-methanol lines and for the 157 GHz series we used , . The lines in W 51Met3, W75N, and Cep A are much broader than the other 96 GHz lines. It appears that broad components dominate in these lines, and the ratios for narrow components cannot be derived from our data. Therefore, we excluded the lines from further analysis in these sources.

Following Olmi et al. (1993), we defined to be

where are the rms errors of the observed ratios. The best-fit model can be found by minimizing . To find the minimum value, we calculated for a number of parameter sets for each source. There are 3 free parameters of the model, and to find the minimum one should vary all of them. The resulting distribution is a three-dimensional figure in the space. However, varying all the parameters in three-dimensional space requires an enormous amount of computing time. Therefore we made three cross-sections of the three-dimensional figure in the , and planes around the "initial guesses", i.e., the values obtained analytically.

We obtained the distribution for each source. However, the -test showed that these models should be rejected for all objects except OMC-2. One possible reason for that is wrong Gaussian fitting. Several lines both at 96 and 157 GHz are blended, especially the lines at 157 GHz. We must fix some parameters to fit these lines. Therefore, the intensities may be determined incorrectly, and the actual errors may exceed the formal errors obtained by the fitting procedure. However, the most probable explanation is that the real sources are more complex than the models used in the statistical equilibrium calculations. Taking into account that the structure of the sources cannot be established without further high-resolution observations, it seems reasonable to find, using the least-squares method, sets of parameters providing the best agreement between model and observations. The scatter of the line intensity ratios depends on the unknown deviations of the real source structures from the model source structures rather than on the errors of observation. We do not know how these deviations affect line intensities, and therefore assumed that the relative deviations of line intensities are equal. We used the procedure described by Mulvey (1963) to estimate source parameters and the rms deviations of line intensities.

The best-fit model can be found by minimizing the function , defined as

Here are weight factors inversely proportional to the rms deviations of the ratios . As we assumed that the relative rms deviations of line brightness temperatures , equal to , are the same for all lines, is equal to , where is unknown and should be evaluated. Weight factors should be chosen in the form . To find the minimum value, we made cross-sections in the same planes as previously to find the minimum value for each source except OMC-2.

Knowing the minimum value, one can evaluate and find the function M, which is proportional to and is distributed with degrees of freedom, using the technique described by Mulvey (1963).

Knowledge of the M distribution in the , , and planes allowed us to find contours enclosing "true" parameter sets. We used the technique, described by Lampton et al. (1976).

Contours showing the M distribution ( distribution for OMC-2) are presented in Figs. 2-3. Thick lines represent contours enclosing "true" parameter sets at the confidence level. The best-fit parameters are presented in Table 5; Table 6 shows the model brightness temperatures of the and lines for the best-fit parameter sets. Using them, we obtained source sizes, as described in the previous section. Methanol abundance was obtained dividing the E-methanol density by the hydrogen density and multiplying by two to take into account methanol A. Figs. 2c-3c show that the increase of density together with the decrease of E-methanol density and, reversely, the increase of E-methanol density together with the decrease of density in the range of E-methanol densities do not change ; one can understand this taking into account that the increase of both density and methanol density leads to an increase of the degree of thermalization of a source and therefore affects the ratios of the line intensities in the same manner. For E-methanol densities smaller than approximately all the lines in consideration become optically thin and the dependence of intensity ratios from methanol density disappears. Thus, Figs. 2c-3c show that the accuracy of density and especially methanol density determination, using the ratios of the line intensities, is low. One must know line brightness temperatures rather than their ratios to obtain these parameters. However, using Figs. 2c-3c one can determine accurately either density or E-methanol density provided that the other parameter is known. Fig. 3c shows that for ON1 the product is approximately constant (about ).

Fig. 2. Contours showing the M distribution (or the distribution for OMC-2) obtained using 96 and 157 GHz data only. Column a shows the distribution in the plane, b the distribution in the plane, and c the distribution in the plane. The horizontal axis in column a is , the vertical axis is , K. The horizontal axis in column b is , the vertical axis is , K. The horizontal axis in column c is and the vertical axis is . Thick contours correspond to threshold M (or for OMC-2) values, i.e. they enclose "true" parameter sets with a confidence level. Other contours correspond to the levels 0.8, 1.2, 1.5, 2, and 4, multiplied by the threshold values. Shadowed regions in the planes show areas enclosing "true" parameter sets at the confidence level, obtained using 96, 157, and 133 GHz data. The methanol densities for the cross-sections are 1.5E-2 for W3(OH), 1.7E-2 for OMC-2, 6.6E-4 for S235, 6.7E-4 for NGC 2264, and 1.9E-3 for 34.26+0.15. The densities for the cross-sections are 1.6E+6 for W3(OH), 2.8E+6 for OMC-2, 1.0E+6 for S235, 0.7E+6 for NGC 2264, and 2.0E+6 for 34.26+0.15. The kinetic temperatures for the cross-sections are 55 K for W3(OH), 20 K for OMC-2, 22 K for S235, 20 K for NGC 2264, and 25 K for 34.26+0.15.

Fig. 3. Same as in Fig. 2. The methanol densities for the cross-sections are 2.2E-4 for W 51Met3, 1.3E-3 for ON1, 1.4E-3 for W75N, 4.5E-3 for DR 21(OH), 1.5E-2 for S140, and 1.5E-2 for Cep A. The densities for the cross-sections are 1.0E+6 for W 51Met3, 1.5E+6 for ON1, 1.9E+6 for W75N, 2.8E+6 for DR 21(OH), 2.8E+6 for S140, and 0.6E+6 for Cep A. The kinetic temperatures for the cross-sections are 17 K for W 51Met3, 15 K for ON1, 20 K for W75N, 19 K for DR 21(OH), 18 K for S140, and 20 K for Cep A.

Table 5. SE temperatures, densities, and A+E-methanol abundances. Best-fit values together with upper and lower limits are presented for each parameter. Only the 96 and 157 GHz data were used to derive the best-fit parameters, whereas the 96, 157, and 133 GHz data were used to derive upper and lower limits.

Table 6. Line brightness temperatures and source sizes that correspond to the best-fit models. The second column shows the brightness temperatures of the lines at 157 GHz, and the third column, the brightness temperatures of the lines at 96 GHz. Relative rms deviations of line brightness temperatures are presented in the last column

To reduce the suitable ranges of densities and methanol densities, we used observational data for the line at 133 GHz obtained at Kitt Peak (Slysh et al., 1996). This line, unlike the 96 and 157 GHz lines, is inverted in a majority of parameter samples that minimize . Therefore, an increase of methanol density leads to a much larger increase of line brightness temperature at 133 GHz than at 96 and 157 GHz. We added the term to the -distributed function M. Here is the ratio of the and line intensities and is the rms error of the observed ratio. Note that the line was observed in another observing session at a different frequency. Under these circumstances, the accuracy of determination of depends on calibration errors, pointing errors etc. Some sources were observed at low elevations; weather during the observations was sometimes bad; even small pointing errors may lead to errors as large as , because the accurate source positions are unknown and brightness peaks may be offset from the coordinates in Table 2. Therefore we adopted a relative rms error of the 133 GHz line intensity, which is much larger than typical calibration errors at Kitt Peak. It is also much larger than the relative rms deviation of the line brightness temperature (see Table 6) for any observed source; otherwise we should use instead of this value.

The areas enclosing "true" parameter sets are shaded in Figs. 2c-3c. Their borders, as previously, were determined with the technique described by Lampton et al. (1976).

The ratios of the observed and model line intensities for DR 21(OH) are presented in Table 7. The table shows that the ratio R 64 corresponding to the best-fit model is significantly smaller than observed. The same is true for the other sources, except W3(OH). To reach the observed value of R 64, one must increase either methanol abundance or temperature relative to the best-fit model. We believe that this indicates the inhomogeneity in the sources. Probably, the significant part of the emission of the maser line arises in the regions where the kinetic temperature and/or methanol abundance is enhanced relative to the regions where the bulk of thermal emission at 96 and 157 GHz appears. We suggest that the observed 133 GHz intensity is enhanced relative to the intensity of the 96 and 157 GHz sources, and so we believe that the usage of the 133 GHz data may lead to overestimation of the upper and lower limits of methanol density. Therefore we used only the upper limits of methanol densities, obtained using the 133 GHz data, and neglected the lower limits. For sources where the lower limits of methanol density cannot be derived from the 96 and 157 GHz data, we determined them from the obvious relation . We compared the SE and main-beam brightness temperatures for the line.

Table 7. Ratios of line intensities corresponding to the best-fit model and to models with minimum and maximum parameters in agreement with the observations for DR 21(OH). The values of the corresponding parameters are presented in Table 5. The ratios are: R 14 - ; R 24 - ; R 34 - ; R 54 - ; R 0-1 - ; R 1-1 - ; R 64 -

Liechti & Wilson (1996) mapped several methanol sources in the line at 36 GHz using the Effelsberg 100-m radio telescope. In particular, they mapped thermal or quasi-thermal emission in W3(OH), W75N, and DR 21OH. The source sizes in W75N and DR 21(OH) ( and , respectively) nearly coincide with our best-fit values, suggesting that our best-fit estimates are close to the source parameters. The source size of in W3(OH) measured by Liechti & Wilson proved to be smaller than ours. If the source size measured by Liechti & Wilson is the same as in 96 and 157 GHz, this means that the 96 and 157 GHz line brightness temperatures are 3.2 times larger than those presented in Table 6. The best-fit density and methanol abundance will be about cm^-3 and , respectively. We assume, however, that the source size at 36 GHz may be smaller than at 96 and 157 GHz. The line, like the line, belongs to Class I (Menten, 1991) and unlike the 96 and 157 GHz lines, it should be suppressed if the radiation is strong enough, which is probably the case in the vicinity of the compact HII-region (see Cragg et al., 1992; Kalenskii, 1995 for the description of methanol excitation). The linewidth presented by Liechti & Wilson (2.2 km s^-1) is smaller than the linewidths at both 96 and 157 GHz, in agreement with this assumption.

Comparison of the results obtained by the two methods shows that they are roughly in agreement. Tables 4 and 5 show that the source densities obtained by both methods are close to , kinetic temperatures are close to 20 K for all sources except W3(OH), and methanol abundances are of the order of for all sources except Cep A. For our sample of sources, the mean value of SE kinetic temperatures proved to be larger than the mean value of obtained analytically by a factor of 1.5. Densities and methanol abundances agree within a factor of 3, except for W75N and Cep A, where the SE methanol abundances are approximately 5 times higher than the same parameters obtained analytically. The 96 GHz lines proved to be optically thin for the majority of the appropriate parameter sets, including the best-fit ones; however, optical depths of the lines of the order of unity were obtained for models with maximum allowable methanol densities for some sources. Fig. 5 illustrates the agreement between the results. One can see that correlations between brightness temperatures, logarithms of densities, and methanol abundances indeed exist. The kinetic temperatures of all sources except W3(OH) are close to 20 K. Therefore the correlation formally exists (the correlation coefficient is 0.8), but is not reliable.

Fig. 4. Comparison of the SE and analytical source parameters. The horizontal axis represents SE parameters, the vertical axis represents parameters determined analytically. The vertical axis of the left graph represents instead of . The solid lines show regressions between the two estimates. Correlation coefficients are: 0.8 for kinetic and rotational temperatures, 0.9 for brightness temperatures, 0.7 for logarithms of density, and 0.8 for logarithms of methanol abundance.

The agreement between the results obtained by the two methods makes it likely that our estimates represent parameters which are typical for the sources under consideration. However, it is noteworthy that our parameter determinations, both those obtained analytically and those obtained using statistical equilibrium calculations, strongly depend on the adopted collisional constants. We used collisional constants based on the experimental results of Lees et al. (1973) for both techniques. Their revision may require revision of the densities and probably, temperatures. The revision of densities leads to the revision of methanol abundance.

In addition, the role of radiation should be studied. First, the continuum radiation, which we believe to be weak, may, however, be non-negligible. Second, radiation transfer in optically thick clouds is incorrectly described by the LVG technique if there is no velocity gradient in the real sources. This may also lead to significant errors. Thus, our estimates represent real source parameters provided that the adopted collisional constants are close to the real ones, the continuum radiation is negligible, and LVG model adequately describes the clouds.

Further observations are necessary to test our estimates. The best test is high-resolution observations of thermal methanol, suitable to test both our basic assumptions about the structure of the sources and our estimates of the source brightness temperatures and sizes. One can observe the series of lines near 165 GHz and estimate source parameters using 165 GHz line intensities instead of 157 GHz intensities. The 165 GHz lines probably are more suitable for the determination of source parameters, because they are not blended and their intensities can be obtained more accurately. In addition, one can observe ¹³ CH₃ OH lines to test optical depths and methanol column densities and other molecules that trace high-density gas to make independent estimates of gas parameters.

© European Southern Observatory (ESO) 1997

Online publication: June 30, 1998
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