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Astron. Astrophys. 336, 352-358 (1998)

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4. Gas phase CH4 modeling

Models of the ro-vibrational spectrum of the [FORMULA] dyad of gaseous CH4 are described in detail elsewhere (Helmich 1996; Boogert et al. 1997, 1998). An essential parameter in these models is the line broadening due to thermal and turbulent motions. The depth of unresolved, optically thick lines increases at larger b values (the Doppler parameter [FORMULA]). For [FORMULA]3 km s-1, this effect is important at N(CH4)[FORMULA] cm-2 for T=20 K and at N(CH4)[FORMULA] cm-2 for T=120 K. The high resolution infrared absorption line study of Mitchell et al. (1988; 1990) reveals optically thin [FORMULA] ro-vibrational lines with [FORMULA] km s-1 toward W 33A and C 7538 : IRS9. This is probably an upper limit, since it is comparable to the instrumental broadening. Rotational emission lines at
(sub-)millimeter wavelengths have widths around [FORMULA]3 km s-1, with wings ranging up to 18 km s-1 (Goldsmith & Mao 1983; Hasegawa & Mitchell 1995; van Dishoeck, priv. comm.). Here, we will adopt a conservative [FORMULA] km s-1 at any gas temperature for both sources, thus minimizing underestimation of the derived CH4 column densities due to optical depth effects.

The CH4 gas temperature can be constrained by the observed depth, and upper limits, of the ro-vibrational lines in the P and R branches, provided these lines are not too optically thick. For W 33A, the detection of lines up to R(6) indicates that the absorbing gas is warm, since this line originates from an energy level of [FORMULA] K above the ground rotational state. The relatively large depth of the R(3) and P(3) lines indicates an excitation temperature of [FORMULA] K (Fig. 2). If the ro-vibrational energy levels are populated according to thermodynamic equilibrium, and the lines are optically thin, the rotational temperature ([FORMULA]) and total CH4 gas column density ([FORMULA]) can be determined unambiguously from a rotation diagram (see e.g. Mitchell et al. 1990 for CO). For CH4, the construction of such a diagram is complicated by the cluster splitting of each J transition. Each ro-vibrational line consists of a number of lines from the [FORMULA], [FORMULA], and [FORMULA]type multiplets (Helmich 1996), with slightly different lower energy levels. The population [FORMULA] of the lower energy level [FORMULA] of cluster line i with rotational quantum number J is given by the Boltzmann equation:


with [FORMULA] the statistical weight of level J, [FORMULA],3, or 5 the statistical weight for the [FORMULA], [FORMULA], or [FORMULA]type multiplets respectively, and [FORMULA] the partition function at rotational temperature [FORMULA]. The equivalent width [FORMULA] of the cluster line originating from level [FORMULA] is then given by (e.g. Spitzer 1978):


with [FORMULA] the oscillator strength of the transition. Now, at the resolution of our observations [FORMULA], we do not resolve the individual cluster lines, and we effectively observe the equivalent width of the sum of all the cluster lines: [FORMULA]. Then the Boltzmann equation becomes:


where we took [FORMULA], which is a very good approximation. We constructed a rotation diagram of CH4, by plotting Eq. 3 logarithmically (Fig. 3), using the observed equivalent widths given in Table 1. In this plot, a straight line can be fitted to the data, with gradient [FORMULA] and abscissa ln([FORMULA]). Unfortunately, both abscissa and gradient depend rather strongly on [FORMULA], and the relation to be fitted is non-linear. For this reason, we took an expansion formula for [FORMULA] from the HITRAN database selection program (Rothman et al. 1992; accurate to 10% for [FORMULA] K) and fitted the following non-linear equation to the points in Fig. 3, using a gradient expansion algorithm (the `Marquardt' method; Bevington & Robinson 1992):


with x and y being the axes of Fig. 3. With this method we find [FORMULA] K, and [FORMULA] cm-2 for the CH4 gas toward W 33A.

[FIGURE] Fig. 2. CH4 gas phase models (upper three) compared to the continuum subtracted spectrum of W 33A (bottom). The model temperatures are (from top to bottom) 20 K, 100 K, and 180 K, at b=3 km s-1, and [FORMULA]. The adopted nomenclature for the ro-vibrational transitions is indicated at the bottom. Each line is a blend of lines (`cluster splittings') at slightly different energy levels. The main components follow the selection rules for P,Q, and R-branches as in linear rotators.

[FIGURE] Fig. 3. Rotation diagrams of the CH4 deformation mode toward W 33A (top) and C 7538 : IRS9 (bottom). The solid lines indicate least square fits to the data, yielding [FORMULA] K. The dotted lines are two temperature models ([FORMULA] K; [FORMULA]=200 K), that limit [FORMULA], provided the lines are optically thin (see text).

CO absorption observations have shown that, besides warm gas, there is a significant cold gas component toward W 33A ([FORMULA]25 K; [FORMULA]; Mitchell et al. 1988). For CH4 the column density of such a cold component is constrained by the depth of the lower rotational lines. It would be visible in the rotation diagram as a steepening of the curve at low energies, provided the lines are optically thin. In Fig. 3 there is no evidence for a pronounced cold component. If we assume that all observed lines are optically thin (i.e. b is large enough), the ratio [FORMULA] can be limited, using the rotation diagram. We constructed a 2 temperature curve, with a minimum possible [FORMULA] cm-2 at maximum [FORMULA] K to fit the higher rotational lines, and [FORMULA] K. Thus, we find that [FORMULA] cm-2, and [FORMULA] (Fig. 3). At these column densities, and [FORMULA] km s-1, the assumption of low optical depth is still valid. For lower b values, optical depth effects need to be taken into account. Any amount of cold CH4 gas can be hidden in the data, provided b is small enough. For example, at [FORMULA] km s-1 the spectrum is consistent with [FORMULA] cm-2, and at [FORMULA] km s-1 this is a factor 5 higher. We note that no such narrow components have yet been found in emission line studies of W 33A, and thus a high [FORMULA] remains to be proven.

For the CH4 lines detected toward C 7538 : IRS9 we also constructed a rotation diagram (Fig. 3). Although, the uncertainties in the equivalent widths are large for each of the detected lines (Table 1), the combination of all lines constrains the gas temperature and column density reasonably well. In the fits to the rotation diagram we included the detection of the R(0) line by Lacy et al. (1991). With the method described above, we find a best fit of [FORMULA] K, and [FORMULA] cm-2 for C 7538 : IRS9. Again, for this source the rotation diagram does not show evidence for a significant cold gas component. When we assume a minimal [FORMULA] cm-2, and maximal [FORMULA] K (from CO; Mitchell et al. 1990), we derive that [FORMULA] for [FORMULA] K. This anti-correlates with CO, for which [FORMULA] is 2 orders of magnitude lower (Mitchell et al. 1990). Similar to W 33A, a much larger [FORMULA] can be hidden in the data if [FORMULA] km s-1, but at present there is no observational evidence for such small velocity dispersions. A summary of the column densities is given in Table 2.


Table 2. Solid and gas phase abundances

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© European Southern Observatory (ESO) 1998

Online publication: July 7, 1998