3 Emitting components

Of the 3000 Monte Carlo sites distributed in the model CE, 44% yielded maser emission at 22GHz; 49% at 325GHz; 16% at 321GHz and 87% at 183GHz. The ranges of physical conditions which give rise to the five brightest components at each frequency are given in Table 3.

Any given component typically produces maser emission from several transitions. This is illustrated in Fig. 2, in which the propagation of masers through velocity-positional space is shown for a single emitting component. The saturation of all four masing lines is clearly evident in Fig. 2. In the case of the 22 and 183GHz emission, self-absorption of the maser photons also occurs. In the case of the 321GHz maser line, a secondary emission shoulder arises from the effect of maser radiation transport under the CVR regime in the presence of a velocity gradient along the direction of maser propagation (cf. Field et al. 1994).

In our simulated data, the ranges of propagation and saturation distances, and the FWHM resulting for the five brightest components at each frequency are given in Table 4, and show that each of the bright components saturates at each frequency. Recalling that the maser propagation distance in the model is given by 3 $\Delta v_{\rm th}$/$\mid$ $\alpha _{\rm los}$$\mid$, constrained to a maximum value of $3\times 10^{12}$ m, it is clear that at 22GHz, the distance at which saturation sets in is between 1-2  $\Delta v_{\rm th}$/$\mid$ $\alpha _{\rm los}$$\mid$. Bright 22GHz maser components are saturated in our calculations, but weaker components may be unsaturated. For example, the brightest maser components at 183GHz yield rather weak and unsaturated emission at 22GHz.

With respect to observations of H₂O maser components at 22GHz, these show that they have a typical linear size of 0.5 AU ( $7.5 \times 10^{12}$ cm), with emission lines of FWHM typically $\sim$1 kms^-1 (Bains 1995; Marvel 1997). From a brightness temperature analysis of the emission observed towards W Hya, Reid & Menten (1990) find that maser components are unsaturated in this Semi-Regular star. Observational evidence that masers may be close to saturation in Supergiants is given by Richards et al. (1999), who observed the spectral line FWHM of 22GHz maser components narrowing with increasing brightness (the unsaturated regime) and rebroadening (under saturation).

Noting that the thermal linewidth for H₂O lines, say at 2000 K, is 2.1 kms^-1, the data in Table 4 show that our synthetic saturated component lineshapes may remain relatively narrow. Since, under the CVR regime, the rate at which population is redistributed over velocity is assumed to exceed the maser simulated emission rate, saturated rebroadening of the lineshapes does not occur (Goldreich & Kwan 1974). The narrowing of component lineshapes due to the initial unsaturated exponential amplification process is retained. Here, the broadest component FWHM result from the secondary gain effect (which does not require saturation) discussed above, rather than via saturated rebroadening. We are therefore unable to comment on this phenomenon in the present work. The lineshapes of H₂O maser components will be addressed in detail in future work.

In our simulations, it is also evident (see Table 3) that bright 22GHz maser components may form within a few AU of the photosphere. However, observations by Reid & Menten (1997) indicate that electrons of sufficient density form a radio photosphere in Mira variables, which extends out to around 2 R_*. The sources of opacity which could be important at 22GHz are proton-electron and H^- free-free bremsstrahlung. As these processes are not included in our model star, we estimate here the effect of a radio photosphere on our results by assuming that the opacity at 22GHz will be greater than unity out to 2 R_* (2.2 AU in our model star). In fact, only a very small number of 22GHz components (28 from a total of 1344) form within this radius in our model calculations. Neglecting these components from our data set would result in only a 0.02% reduction in total maser output at 22GHz. Since the brightest of the components which lies within 2.2 AU achieves only 0.2% of the output of the brightest of all components at 22GHz, these sites have a negligible effect upon our synthetic single-dish spectra and interferometry images. We therefore conclude that the inclusion of these opacity sources would have no significant influence on the outcome of our calculations.

We also note that the temperatures achieved in our model star are rather higher than those indicated by the observations by Reid & Menten (1997). This discrepancy is likely to be due to the lack of molecular coolants included in our model CE. In order to estimate the significance of the higher temperature regimes on our simulated data, we show in Table 5 a breakdown of our results by temperature. These data indicate that sites of $T_{\rm k} > 3000$ K produce a significant contribution to 22 and 321GHz emission, whereas at 325GHz and 183GHz the higher temperature regions provide a more minor contribution. We note that the H₂O molecule is likely to be largely dissociated at 5000 K. As a general rule, for sites of $T_{\rm k} < 2000$ K, 22, 183 and 325GHz emission tends to be produced from roughly the same set of components. At higher temperatures, emitting components additionally tend to yield emission at 321GHz. Other trends for emitting components are identified in Sect. 4.2.