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Astron. Astrophys. 324, 211-220 (1997)

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2. Description of the model

Our model for pumping the 6 GHz [FORMULA] transition of interstellar methanol is the same as that presented in Paper I for the 12 GHz [FORMULA] transition. The masers are assumed to arise in portions of methanol-rich gas within a spherically expanding cloud and can be beamed. (In the large velocity gradient model beaming is expressed by [FORMULA] in definitions of Castor (1970). We treat LVG as a convenient approximation for studying the vast parameter space and think that actual beaming mainly comes from the maser source geometry). When the masers lie along the line of sight to an H II region it provides a source of background radiation for amplification by the maser lines, in the tail of the free-free continuum spectrum. The actual maser pumping is done by warm dust which surrounds the methanol-rich portions of matter, providing an infrared continuum source to pump the first and second torsionally excited states of methanol. Dust with the necessary properties is known to be present in the vicinity of ultracompact H II regions (see Walmsley 1995 for a recent review, and Wink et al. 1994 for recent W3(OH) observational data). Model equations are given in the Appendix.

Actually, the torsionally excited states play the role of transmitters of population between the levels of the ground state which form the maser transition. Although the details of operation of the pumping mechanism are quite complicated (see Sobolev & Deguchi 1994b) in Paper I it was shown that the most prominent pump cycles have similar characteristics. They are: excitation to a level of the 2nd torsionally excited state followed by spontaneous decay through the levels of the 1st torsionally excited state to the levels of the ground state. Radiative rates for methanol are such that spontaneous decay favours downward transitions to levels of the ground state with K quantum number different from that of the initial state. This is the pattern which causes the appearance of masers.

It is worth mentioning that the Rosseland theorem, which works in the presence of diluted black-body radiation, can be applied to this pumping mechanism. This theorem was proved for the case of dilution factors [FORMULA] by Sobolev et al. (1985) and can be relatively easy proved for the case when the external radiation field is determined by emission of dust with finite optical depth using a thermodynamic approach from the above paper. The theorem states that radiative processes lead to splitting of higher energy photons into sets of photons with lower energy. The efficiency of this process is greatest when the higher energy photon splits into photons with equal energy. This causes a much higher efficiency of pumping through the levels of the 2nd torsionally excited state than can be obtained by pumping through the 1st torsionally excited levels, as demonstrated in Paper I.

Symmetry considerations allow two distinct species of methanol, conventionally labelled A and E. Methanol molecules of one symmetry species cannot be converted at significant rates into the other by the normal interstellar radiative and collisional excitation processes; rather, the partitioning between the two species is determined during the chemical formation of methanol. Excitation modelling therefore treats the two species quite independently, although they must of course coexist under the same conditions, and with approximately equal abundance (depending on the temperature of formation). Here we model both species, since we are interested in comparing the [FORMULA] and [FORMULA] transitions.

Methanol energy levels up to rotational quantum number [FORMULA] and torsional quantum number [FORMULA] were calculated according to De Lucia et al. (1989), and Einstein coefficients for radiative transitions within and between the torsional levels were taken from Cragg et al. (1993). As described in Paper I, the number of ground state levels included is determined mainly by the kinetic temperature, and the corresponding levels in the torsionally excited states were included also. In the majority of calculations described in this paper, there were 279 levels of A -species methanol (93 in each torsional state), and 282 levels of E -species methanol (94 in each torsional state).

The details of excitation of methanol in collisions with hydrogen and helium are unfortunately unknown. Here we adopt the collision model of Peng & Whiteoak (1993b), based on double resonance experiments on E -species methanol by Lees & Haque (1974). There are some differences in the numerical results reported here for the 12 GHz [FORMULA] transition from those reported in Paper I, due to an error in implementing the collision model, which has now been corrected. We use the same propensity rules for the A -species, although there are no corresponding experimental results. As in Paper I, collisional transitions were included only between levels of the torsional ground state. For the A -species the rate of [FORMULA], [FORMULA] asymmetry doublet transitions also must be specified. We set Peng & Whiteoak's [FORMULA] factor to unity for these transitions, but we also report some calculations where these transitions were forbidden altogether. In addition, we examine the effects of nonselective collisions.

An examination of the role of numerous line overlaps has shown that they produce a negligible effect on the [FORMULA] line intensity, and so we do not consider them further in the present paper. We plan to describe the overlap effects in a subsequent paper on the other methanol masering transitions, some of which are enhanced by overlap.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998