Astron. Astrophys. 342, 809-822 (1999) 4. AnalysisThe lines identified in this survey have been analysed using the LTE method of Turner (1991). For optically thin emission from molecules in LTE the column density () can be written as where is the integrated intensity of the line, is the line frequency, S is the line strength, µ is the permanent electric dipole moment, and are the reduced nuclear spin degeneracy and the K-level degeneracy of the molecule respectively. is the energy of the upper level of the line and is the rotational temperature of the molecules. The integrated intensity of each line has been calculated using , where the line is assumed to be gaussian in shape. The partition functions used are interpolated from the values given in the JPL molecular line database (Poynter & Pickett 1985). We have used Eq. 1 in two different forms, to put lower limits on the column densities of molecules with one or two detected lines and where possible to constrain the temperature and column density of the gas with the rotation diagram method. All column densities quoted in this paper are beam averaged. 4.1. Rotation diagram analysisEq. 1 may be rearranged to give where . If Eq. 2 is plotted with as abscissa and as ordinate it is the equation of a straight line with a gradient of and a y-intercept of . These parameters have been determined by least-squares fitting of a straight line to the data. The errors in and originate from the uncertainty in the integrated intensity of the line () which has been plotted as error bars on the rotation diagram (Fig. 3).
Care must be taken with the rotation diagram approach. As noted in the previous section the underlying assumptions are that the gas is in LTE, is optically thin and can be characterised by a single temperature. Optically thick emission will affect the temperature derived from the rotation diagram, in some cases lowering the derived temperature or mimicking a two-component temperature distribution (e.g. Serabyn & Weisstein 1995). The only molecule in our survey with sufficient detected lines for a rotation diagram to be constructed is methanol (CH_{3}OH). The lines of methanol are fairly weak ( K) and are expected to be optically thin. There is a possibility that the methanol lines are picked up in the fringes of the telescope beam and may originate from the core, where they exhibit optical depths of roughly 10 (Hatchell et al. 1998a). The linewidths of the lines seen in the halo survey are much narrower than those seen in the hot core survey of Paper I (an average of 3 MHz in the halo survey as opposed to 9 MHz in the hot core survey), suggesting that they originate from the colder gas of the halo and not the core. We can be reasonably confident that the methanol emission originates from the halo and is likely to be optically thin, i.e. the rotation diagram approach is valid and the results of the analysis should not be affected by pick-up of the hot core gas. Methanol possesses two different forms, the A and E-types, which must be analysed separately. Four lines of CH_{3}OH E-type were detected and a rotation diagram yields a rotation temperature K and a column density = 2.5 cm^{-2} (as shown in Fig. 3). This compares with the values from the central position of Paper I of K and cm^{-2}. The A-type form of methanol was detected in only one line and has been analysed using the column density lower limit method described in the next section. 4.2. Lower limits to column density for molecules with one or two detected linesFor molecules with only one or two detected lines we have employed
a modification of Eq. 1 to determine a lower limit to the column
density, . To evaluate the rotational
temperature we set the derivative of the temperature dependent part of
Eq. 1 to zero, i.e. , where
is the appropriate partition
function for each molecular type. It can be shown that for linear
molecules and for symmetric and
asymmetric top molecules . Forming
the second derivative shows that these turning points are minima and
hence can be found as shown in
Eq. 3. where form (a) is used for linear molecules and form (b) is used for symmetric and asymmetric tops. This analysis was performed for the remaining molecular lines in our survey. Again care must be taken that the lines are optically thin and in LTE. High optical depth will underestimate the integrated intensity of the line, reducing the "correct" estimate of . As is a lower limit to the column density it follows that in cases of high optical depth the lower limit will be reduced, but will however still be a lower limit to the column density of the gas. The column density lower limit () and assumed excitation temperature () are summarised in Table 2 alongside the corresponding values from Paper I. Table 2. Assumed excitation temperature and lower limits to column density for molecules with one or two detected lines. The third column contains the corresponding values of column density observed toward the centre of the cloud from Paper I. The values for SO_{2} and A-type CH_{3}OH had sufficient detected transitions to form rotation diagrams and their rotation temperatures (in brackets) and column densities are given. Other species in the core had only one or two detected transitions and the core column densities are lower limits for these species. The SO_{2} rotation diagram from Paper I was evaluated for lines with E/k 250 K. © European Southern Observatory (ESO) 1999 Online publication: February 23, 1999 |