## 4. Data analysisFirst, 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 approachTo 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 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 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.
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 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 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 calculationsStatistical 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 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 The best-fit model can be found by minimizing the function , defined as Here are weight factors inversely
proportional to the Knowing the minimum value, one can evaluate
and find the function Knowledge of the Contours showing the
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 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
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 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.
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 © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |