## 3. Variation of parametersIn this paper we investigate whether the model of Paper I which successfully accounted for the brightness of the strongest observed methanol masers can also account for the observations. We begin with the model conditions which gave peak brightness for the 12 GHz 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 line, , and the brightness temperature ratio of the 6 GHz to 12 GHz lines, . 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 , gas kinetic temperature
, dust continuum temperature
, beaming parameter , dust
filling factor , dust optical depth at
,
H II region dilution factor , and
methanol specific column density . 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 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 , while the brightness of the 12 GHz maser becomes , and so the ratio is . 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 approaches while remains of order so that 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 is close to the median observed value of 3.2 (Caswell et al. 1995b). Fig. 1 shows the effects on and of varying certain of the model parameters one at a time about the standard conditions. and are fixed at the standard values in all runs.
It is apparent from Fig. 1a that the brightness ratio
can be substantially increased by reducing the
hydrogen density to approximately
. We would like to note here that reduction of
with a fixed value of Fig. 1a displays the effect of varying the hydrogen number density , while keeping the methanol fractional abundance fixed at . Both and peak at , 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 increases to a maximum of 89 at , but is only 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 ,
calculations were also done over the range of densities
. 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
and ratio on specific
column density for different values of hydrogen number density.
Elsewhere in this paper we use the more traditional set of parameters
including and
Fig. 2a illustrates the fact that when densities are lower
than cm Fig. 2b shows that there is a narrow specific column density
window ( cm The methanol fractional abundance
In the majority of calculations reported here, we assume the
abundance of both symmetry species to be equal,
. The combined statistical weight of the
Fig. 1c shows the effects of varying the beaming parameter from 1 to 100. We conclude that is required to account adequately for the brightest 6 and 12 GHz methanol masers . Greater values of beaming were included to seek simultaneous high values of and . Although greater beaming increases the peak value of , it also shifts the peak 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 when and , suggesting that observations of in some sources are indicative of a different maser geometry.
Fig. 1d shows the effects of varying the dust temperature between 50 and 500 K. There is a steep rise in maser brightness when 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 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 , exceeds K at , and exceeds 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 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
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 In Fig. 5 we show the effects of varying the H II region dilution factor between our standard value of and . 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 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 this threshold is achieved at greater values of negative optical depth, which determines the brightness. Hence, reduction of permits higher values of 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 both the 6 GHz brightness and become large as 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 and , we find K, K and . The ratio increases further as is further reduced.
Finally we have done calculations with reduced and [A]/[E] enhanced, in order to maximise the ratio while maintaining very high levels of maser brightness. Some representative results are summarized in Table 1.
© European Southern Observatory (ESO) 1997 Online publication: May 26, 1998 |