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

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3. Variation of parameters

In this paper we investigate whether the model of Paper I which successfully accounted for the brightness of the strongest observed [FORMULA] methanol masers can also account for the [FORMULA] observations. We begin with the model conditions which gave peak brightness for the 12 GHz [FORMULA] line (model D2 of Paper I), then explore the parameter space by varying certain of the model conditions, singly and in combination. Several hundred runs were done for each species. We present the results in terms of the brightness of the 6 GHz [FORMULA] line, [FORMULA], and the brightness temperature ratio of the 6 GHz to 12 GHz lines, [FORMULA]. Generally speaking the 6 GHz and 12 GHz transitions become inverted under the same conditions in our model, and the 6 GHz maser is brighter, as is usually observed.

Our standard model has the following parameters: molecular hydrogen density [FORMULA], gas kinetic temperature [FORMULA], dust continuum temperature [FORMULA], beaming parameter [FORMULA], dust filling factor [FORMULA], dust optical depth at [FORMULA]   [FORMULA], H II region dilution factor [FORMULA], and methanol specific column density [FORMULA]. The last quantity corresponds to the methanol column density divided by line width, or equivalently to the methanol abundance divided by velocity gradient. Usually this quantity is considered in terms of the methanol fractional abundance X and velocity gradient [FORMULA]. The cited value appears, for example, with [FORMULA] and [FORMULA]. Such velocity gradient corresponds to a line width [FORMULA] over a maser dimension of [FORMULA].

These standard parameters are equivalent to the D2 model described in Paper I. With the collision model of Peng & Whiteoak (1993b) we find the brightness temperature of the 6 GHz maser to be [FORMULA], while the brightness of the 12 GHz maser becomes [FORMULA], and so the ratio is [FORMULA]. This degree of maser brightness is sufficient to account for the observations of the 12 GHz maser in W3(OH) (Menten et al. 1988), but not the 6 GHz maser (Menten et al. 1992). By varying the parameters of the model we seek conditions under which [FORMULA] approaches [FORMULA] while [FORMULA] remains of order [FORMULA] so that [FORMULA] in order to model W3(OH). The data can also be used to deduce conditions in other sources for which observations of both lines are available. Indeed, the standard model represents saturated masing, and the ratio [FORMULA] is close to the median observed value of 3.2 (Caswell et al. 1995b).

Fig. 1 shows the effects on [FORMULA] and [FORMULA] of varying certain of the model parameters one at a time about the standard conditions. [FORMULA] and [FORMULA] are fixed at the standard values in all runs.

[FIGURE] Fig. 1a and b. Effects of varying the model parameters one-at-a-time from the D2 standard conditions of Paper I on the 6 GHz brightness [FORMULA] and the 6 GHz to 12 GHz brightness ratio [FORMULA]. The position of the standard model is marked with an arrow in each case.

It is apparent from Fig. 1a that the brightness ratio [FORMULA] can be substantially increased by reducing the hydrogen density [FORMULA] to approximately [FORMULA]. We would like to note here that reduction of [FORMULA] with a fixed value of X leads to a corresponding reduction of the specific column density which is actually responsible for the increase in [FORMULA]. This happens because the brightness of the 6 GHz line begins to increase at a lower value of column density than for the 12 GHz line. The ratio [FORMULA] is not particularly sensitive to the other model parameters (Fig. 1 b-f), and remains [FORMULA] so long as the beaming parameter [FORMULA]. However, reducing [FORMULA] from its initial value of [FORMULA] to [FORMULA] brings the 6 GHz [FORMULA] brightness temperature [FORMULA] down by two orders of magnitude (Fig. 1a), far below the W3(OH) observations. The graph suggests that only a reduction of [FORMULA] can substantially enhance [FORMULA] in compensation (Fig. 1f). Thus the combined 6 and 12 GHz observations in W3(OH) define a maser regime both further away from the H II region and of more moderate column density than was initially assumed. These considerations are borne out by further calculations, described below, but first the individual model parameters are discussed.

Fig. 1a displays the effect of varying the hydrogen number density [FORMULA], while keeping the methanol fractional abundance fixed at [FORMULA]. Both [FORMULA] and [FORMULA] peak at [FORMULA], which is our standard model. At higher hydrogen densities both transitions switch over to absorption as the effects of collisions become more dominant. At lower densities [FORMULA] increases to a maximum of 89 at [FORMULA], but [FORMULA] is only [FORMULA] at this density. So reducing the density with a fixed value of fractional abundance produces an increased brightness ratio, but greatly diminishes the actual brightness of both masers.

Although Fig. 1 displays the effects of varying other model parameters only at fixed hydrogen density [FORMULA], calculations were also done over the range of densities [FORMULA]. We found it very helpful to consider the dependence of maser characteristics on hydrogen density when plotted as a function of specific column density. This comes from the fact that in the definition of optical depth the specific column density represents a factor containing all physical parameters of the source apart from the excitational pattern. The brightness of masers has exponential dependence on the value of optical depth and much weaker linear dependence on the value of source function. For the strong masers this difference is very greatly pronounced. Hence, for the current study the scale is almost completely determined by the value of specific column density. This is illustrated in Fig. 2, showing the dependence of the brightness temperature [FORMULA] and ratio [FORMULA] on specific column density for different values of hydrogen number density. Elsewhere in this paper we use the more traditional set of parameters including [FORMULA] and X.

[FIGURE] Fig. 2. Effects of varying the hydrogen density [FORMULA] and methanol specific column density [FORMULA] on a the 6 GHz brightness [FORMULA] b the 6 GHz to 12 GHz brightness ratio [FORMULA]. [FORMULA] takes the values [FORMULA] on successive curves.

Fig. 2a illustrates the fact that when densities are lower than [FORMULA] cm-3 the curve for [FORMULA] remains almost invariant: exponential growth until the negative optical depth reaches approximately 11 (this value is determined mainly by [FORMULA]), and subsequent gradual saturation which prevents [FORMULA] from exceeding some upper limit. For the models shown on the plot saturation begins at specific column densities about [FORMULA]. At this point we merely want to show that [FORMULA] cannot be infinitely increased. The dependence of the upper limit on the remaining parameters will be discussed further. It should be noted that the curve for [FORMULA] cm-3 reaches substantially lower peak brightness than the others. This indicates that the methanol maser sources are not likely to have such high densities.

Fig. 2b shows that there is a narrow specific column density window ([FORMULA] cm-2 s for [FORMULA]) in which the 6 GHz maser is brighter than the 12 GHz maser by 1 to 2 orders of magnitude. This contrasts with the modest ratios ([FORMULA] about 3) found at other column densities, irrespective of other model parameters. As was noted above this comes about because the 6 GHz maser switches on at a lower column density than the 12 GHz maser. Fig. 2b shows that the peak value of [FORMULA] grows with decreasing number density, while its position shifts to slightly higher specific column density. This may be used as a guideline in searching for bright models with high values of [FORMULA]. However, if the maser source is an isolated clump, hydrogen number densities in bright maser sources ([FORMULA]  K) cannot be lower than [FORMULA] cm-3 without violating the upper limits on methanol fractional abundance ([FORMULA]) and source size ([FORMULA] for the observed maser spots) under the assumption of plausible values for line width.

The methanol fractional abundance X was varied over the range [FORMULA] (assuming [FORMULA]). As shown in Fig. 3, increasing the fractional abundance produces the 6 GHz maser brightness peak at progressively lower density, but the peak in the brightness ratio also shifts to a correspondingly lower density. Thus changes in X alone are not sufficient to give simultaneous high values of [FORMULA] and [FORMULA]. Neither maser increased significantly in peak brightness at abundances beyond our standard value of [FORMULA], showing the effects of saturation. This exceptionally high level of methanol abundance is required to explain the observed W3(OH) maser brightness, and is discussed further in the next section.

[FIGURE] Fig. 3. Effects of varying the methanol fractional abundance X and hydrogen density [FORMULA] on a the 6 GHz brightness [FORMULA] b the 6 GHz to 12 GHz brightness ratio [FORMULA]. X takes the values [FORMULA] on successive curves.

In the majority of calculations reported here, we assume the abundance of both symmetry species to be equal, [FORMULA]. The combined statistical weight of the A -species ([FORMULA] in the C [FORMULA] (M) molecular symmetry group) is balanced by the double torsional degeneracy of the E -species ([FORMULA] in the notation of Lees 1973). However, because the ground state energies of the two species differ by [FORMULA], if the chemical equilibrium between the two species was set at low temperature, the A -species will be slightly more abundant. If we assume that the high methanol abundance in the maser regions is a result of evaporation from grain mantles, where the methanol may have been formed at temperatures as low as 10 K, then the abundance ratio between the two symmetry species becomes [FORMULA]. Under our standard model conditions if we increase column density by a factor of 1.44 for the A -species only, we find that [FORMULA] is not significantly altered; however there is more effect at low density, and [FORMULA] increases to 469 at [FORMULA].

Fig. 1c shows the effects of varying the beaming parameter [FORMULA] from 1 to 100. We conclude that [FORMULA] is required to account adequately for the brightest 6 and 12 GHz methanol masers [FORMULA]. Greater values of beaming were included to seek simultaneous high values of [FORMULA] and [FORMULA]. Although greater beaming increases the peak value of [FORMULA], it also shifts the peak [FORMULA] value to lower density, as shown in Fig. 4. Thus greater beaming, like greater methanol fractional abundance, can generate brighter masers at a lower hydrogen density, but not the high values of the ratio also required for W3(OH). In contrast, the 12 GHz maser is brighter than the 6 GHz maser [FORMULA] when [FORMULA] and [FORMULA], suggesting that observations of [FORMULA] in some sources are indicative of a different maser geometry.

[FIGURE] Fig. 4. Effects of varying the beaming parameter [FORMULA] and hydrogen density [FORMULA] on a the 6 GHz brightness [FORMULA] b the 6 GHz to 12 GHz brightness ratio [FORMULA]. [FORMULA] takes the values 1,3,10,20,100 on successive curves.

Fig. 1d shows the effects of varying the dust temperature [FORMULA] between 50 and 500 K. There is a steep rise in maser brightness when [FORMULA] approachs 150 K, i.e., when the maximum of the dust emission is shifted to the values of frequency corresponding to transitions between the ground and the second torsionally excited state. At [FORMULA] these transitions prevail and the masers (pumped by the dust continuum radiation) become less sensitive to the dust temperature. At lower gas densities there is even less variation. For [FORMULA], [FORMULA] exceeds [FORMULA]  K at [FORMULA], and [FORMULA] exceeds [FORMULA]  K. This 12 GHz brightness is larger than that observed in W3(OH), suggesting that the dust temperature is probably not this high. Our results with [FORMULA]  K should be treated with some caution, since the number of energy levels included in our calculations may be inadequate here (levels of the third torsionally excited state and those of vibrationally excited states are not included).

The major source of uncertainty in excitation modelling of methanol is the collisional excitation rates, which as discussed earlier are based on propensity rules derived from a few experiments on the E -species alone. We therefore tried as an alternative a model involving nonselective collisions (Goldreich & Kwan 1974). The total collision cross section was chosen to be the same as that given by the model of Peng & Whiteoak. We compared the two models over the density range [FORMULA], with other parameters as in our standard model. The major change was that the peak [FORMULA] value at [FORMULA] fell to [FORMULA]  K, while [FORMULA]  K, so that the ratio [FORMULA] was reduced to 1.1. At lower densities there was less variation in maser brightness, and the peak ratio was [FORMULA] at [FORMULA]. We conclude from this that the pumping mechanism which generates the strongest Class II methanol masers in our model is not strongly dependent on the details of the collision model, but that the ratio of 6 to 12 GHz maser brightness temperatures is more sensitive to the choice of these rates. Thus it may not be possible to properly model the A -species masers until accurately calculated collisional excitation rates become available.

In addition, we looked at the effects of changing the kinetic temperature between 20 and 50 K within Peng & Whiteoak's collision model (Fig. 1e), and of forbidding collisional transitions between asymmetry doublet levels in the A -species, but the changes in both cases were very minor. All these results are consistent with the idea that the Class II methanol masers are pumped by radiative processes, and collisional processes are not responsible for the generation of the strongest Class II masers, influencing only quantitative details. The gas-dust temperature difference may in fact be less than assumed, but it is not possible to adequately model rather dense gas with temperatures [FORMULA] with the current truncated set of energy levels. Moreover, additional calculations with a larger set of energy levels have shown that if masers are formed in low density regions, the gas can be even warmer than the surrounding dust.

In Fig. 5 we show the effects of varying the H II region dilution factor [FORMULA] between our standard value of [FORMULA] and [FORMULA]. We also investigated two limiting cases: one where the H II region is removed altogether from the model, and another where it becomes infinitely diluted. The maser brightness increases progressively as [FORMULA] is reduced. This comes from the effect of saturation of the masering transition, which restricts the population inversion when the angle-averaged intensity in the maser line exceeds some threshold value. With lower values of [FORMULA] this threshold is achieved at greater values of negative optical depth, which determines the brightness. Hence, reduction of [FORMULA] permits higher values of [FORMULA] through diminishing influence of saturation. On the other hand, the H II region provides a source of background radiation for amplification which is independent of the distance between the source and the maser. In our model the H II region is about 6300 times brighter than the 2.7 K microwave background and this makes possible to create the strongest masers. At densities below [FORMULA] both the 6 GHz brightness and [FORMULA] become large as [FORMULA] is reduced. This regime is the only set of model conditions found capable of accounting for the observations in W3(OH), under the assumption that the maser spot is an isolated clump. Thus for example when [FORMULA] and [FORMULA], we find [FORMULA]  K, [FORMULA]  K and [FORMULA]. The ratio increases further as [FORMULA] is further reduced.

[FIGURE] Fig. 5. Effects of varying the H II region dilution factor [FORMULA] and hydrogen density [FORMULA] on a the 6 GHz brightness [FORMULA] b the 6 GHz to 12 GHz brightness ratio [FORMULA]. [FORMULA] takes the values [FORMULA] on successive curves.

Finally we have done calculations with reduced [FORMULA] and [A]/[E] enhanced, in order to maximise the ratio [FORMULA] while maintaining very high levels of maser brightness. Some representative results are summarized in Table 1.


Table 1. Selected model results for 6 and 12 GHz brightness temperatures (see text for definition of parameters). All models have [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA].

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998