5 Location of $\mathsf{H_{2}}$O masers in the CE

In our simulated data, the location of the stellar H₂O maser sites which produced significant maser emission is shown in Figs. 4, 6, 7 and 8 for the 22, 321, 325 and 183GHz masers respectively. In order to show the location of the brightest maser components in the plane of the sky, maser sites are marked by black circles centred on their projected positions. These circles have diameters which are linearly proportional to the intensity of a maser component integrated over the component spectral line profile ( $\int\, I\, {\rm d}v$). At each frequency, we define $I_{\max}$ to be the maximum value of the velocity-integrated component intensity, denoted by $I_{\max,22}$, $I_{\max,321}$, $I_{\max,325}$ and $I_{\max,183}$. The stellar disk, of radius 1.1 AU at this epoch, is represented by the grey disk at (0,0). Note that the figures are not plotted to the same scale. The largest diameter in each plot corresponds to the $I_{\max}$ at that particular frequency and diameters are not scaled between plots. The ratio of $I_{\max,22}$: $I_{\max,321}$: $I_{\max,325}$: $I_{\max,183}$ is 85:269:48:1 respectively.

Figure 4: H2O masers in the 6 $_{16} \rightarrow 5_{23}$ transition at 22GHz generated in the CE of the model M-Mira. a) Intensities of the masers are linearly proportional to the diameters of the black circles shown. Circles are centred on the maser projected coordinates. The grey disk represents the stellar disk. b) The same data is redrawn as a contour plot, where the lowest contour is set at 1$\%$ of the maximum in the plot. The levels are clipped in order to represent the brightest maser emission.

   
5.1 22GHz masers

Around half (44$\%$) of the three thousand Monte Carlo sites initially distributed in the CE yielded 22GHz maser emission in the line-of-sight. Bright components ( $I_{22}>0.1 ~I_{\max,22}$) lie in an irregularly-shaped distribution in Figs. 4a,b within 9  R_* (10 AU) of the star, with very low gain emission extending much further. No strong emission originates from regions over the stellar disk, indicating that bright 22GHz masers amplify tangentially in the model CE. A requirement for producing strong maser emission is that long amplification paths are available. The tangential velocity gradients in the 22GHz maser zone are typically significantly less than those along radial paths, see Table 3.

The extent and spatial structure of the calculated 22GHz images agree with those of many Mira observations (see e.g. Marvel1997; Colomer2000), noting that the results should be compared with objects of similar ${\dot{M}}$. In this connection we show observations of 22GHz emission detected towards R Cas using the VLA by Colomer et al. (2000) in Fig. 5. With a period of 430 days, a stellar radius of $1.5\pm0.2$ AU (Haniff et al. 1995) and a $\dot{M} = 3.4\times 10^{-7}~M_{\odot}$ yr^-1 (Truong-Bach et al. 1999), R Cas is a M-Mira fairly similar to our model star of $\dot{M} = 1.8\times 10^{-7}~M_{\odot}$ yr^-1 and R_* = 1.1 AU.

Figure 5: VLA data for R Cas from Table 6 in Colomer et al. (2000). Here spot diameter is plotted proportional to velocity-integrated intensity and the linear offset scale is calculated using a Hipparcos distance to R Cas of 106.7 pc (Whitelock & Feast 2000). The mean photospheric diameter of R Cas is $28.6\pm 3.9$ mas for fundamental mode pulsation (Haniff et al. 1995). The stellar position is unknown in this figure.

In the R Cas observations, the H₂O maser distribution is represented in terms of 12 Gaussian sources occupying a spatial extent of $15 \times 9~ R_{*}$ ( $23 \times 13$ AU). The computed image and the observational image are similar in appearance in that the total maser output is dominated by a few bright components/blends of components in an irregularly shaped distribution. In the observed image, the minimum component $\int I~ {\rm d}v$ which occurs within the $23 \times 13$ AU extent is 0.03 $I_{\rm max,obs}$. In the calculated image, components of 0.03 $I_{\max,22}$ occur out to a similar extent of diameter 25.2 AU. However, in the calculated image, this includes emission from 157 Monte Carlo sites rather than around 12, suggesting that we may have oversampled our model CE. Alternatively, large numbers of components may be blended together in the VLA observations, or the low intensity sites could be obscured by noise.

The projected positions of the 22GHz maser components are related to their location in the CE model in Fig. 9. Figure 9a shows that the peak in the radial distribution of 22GHz emission occurs at 2.7  R_*, with secondary peaks at both 4.5 and 7.3 R_*. The physical conditions in these regions are shown in Figs. 9b, c and d. All components of intensity >0.01 $I_{\max,22}$ occur within 13.6  R_* of the stellar position. However, extremely weak "diffuse'' emission, of intensity as low as $2 \times10^{-6}$ $I_{\max,22}$, extends much further, out to a radius of 25.5 R_* in the case of 22GHz masers.

A wide range of physical conditions evidently gives rise to population inversion in the 6 $_{16} \rightarrow 5_{23}$ transition. However the highest gain masers occur in relatively high temperature and high density regions, see Table 3. Some of these conditions lie outside of the parameter space investigated by both Y97 and Neufeld & Melnick (1991). In the model CE, bright emission commonly occurs from components with higher kinetic temperatures than those considered in previous water maser models. We note that these higher temperatures appear to be in conflict with observations by Reid & Menten (1997). In this connection, very recent oxygen-rich hydrodynamical stellar models by Höfner (private communication) do not show such high temperature spikes. The use of such models in future work should yield results which are more compatible with these observations.

Figure 9b shows that our simulated 22GHz masers occur in the region of the CE in which acceleration of circumstellar material yields the steadily-outflowing stellar wind, as observed e.g. by Richards et al. (1996). Model component lifetimes can be roughly estimated by the crossing time required to traverse this zone. At an average outflow velocity of 4 kms^-1, a bright component travelling from around 2 to 12 AU in the CE survives 12 years. Component proper motions should be therefore be of the order 0.85 AU yr^-1 in stars of low ${\dot{M}}$.

5.2 321GHz masers

A small minority of Monte Carlo sites (16$\%$) produced maser emission at 321GHz. Figure 6 shows that, in the projected map, the brightest 321GHz maser components occur within 3.5  R_* (4 AU) of the stellar position. It is also evident from Fig. 6 that no bright 321GHz emission orginates from regions directly in front of the stellar disk, indicative of a tangential amplification process. We note that 321GHz components may achieve greater intensity than those of the other masers discussed here. At this epoch of our calculations, $I_{\rm max,321}$ exceeds $I_{\max,22}$ by a factor of three.

Turning to the location of 321GHz masers in the model CE, Fig. 9a shows that the peak in the radial distribution of 321GHz maser emission occurs between $\sim$2-3.5 R_*. Again it is the combination of suitable pumping conditions, in conjunction with a low line-of-sight velocity gradient, which produces a high gain maser of long amplification path. Bright 321GHz maser components are the rarest of all the four masers discussed here. Only thirty-three sites of $I_{321}> 0.01~I_{\max,321}$ resulted from our calculations, existing out to 4.5 R_* in the CE. The remaining, very weak emission is contained within a radius of 11.3 R_*.

In summary, it is clear that very few sites in the model CE are suitable for producing strong 321GHz emission. These sites probe the very innermost region of the CE, a region also probed by SiO masers (see H96). This result is unsurprising since the $E_{\rm u}$ of the 321GHz H₂O transition is very similar to that of v=1 SiO masers, both lying at around 1800 K above ground state. Further calculations are required in order to determine whether 321GHz maser emission may be originating from the same volumes of gas as those giving rise to bright SiO masers. Given that the 10₂₉ $\rightarrow$ 9₃₆ transition lies at a relatively high energy, it follows that 321GHz maser emission requires conditions of high gas density, temperature and radiation field energy density, and that it is not pumped by the larger range of conditions which leads to 22GHz maser emission in the CE. Y97 also identifies a relatively restricted range of conditions over which the 321GHz maser transition is strongly inverted.

Figure 6: 321GHz emission calculated at this epoch. See caption of Fig. 4 for more details.

   
5.3 325GHz masers

The number of sites contributing to 325GHz emission is similar to that of 22GHz maser emission (49$\%$). At this epoch, Fig. 7 shows that the brightest 325GHz maser components occur within a radius of 11 R_* (12 AU) of the stellar position in the projected map. The stellar disk is partially obscured in Fig. 7, but no bright emission occurs from over the star. Tangential amplification is prevalent, as for the 22 and 321GHz masers.

In the CE model, very weak emission extends to a radius of 30  R_* from the stellar position, with sites of >0.01  $I_{\max,325}$ occurring out to a radius of 18  R_*. In Fig. 9a it is clear that bright 325GHz emission occurs over a similar region of the CE to that occupied by 22GHz masers. Although $I_{\max,22}$ exceeds $I_{\max,325}$ by a factor of around two at this epoch, there is a higher success rate of producing masers at 325GHz both than at 22 and 321GHz. The observed strength of the 325GHz maser is due to a large number of maser spots of moderate intensity, rather than a few very bright maser components. The 325GHz maser can exist out to larger radial distances than both the 22GHz and 321GHz masers, indicating it can be pumped in regions of lower gas density and temperature. The transition which must be inverted to produce this maser is relatively low-lying, with an $E_{\rm u}$ of 470 K above ground state.

Figure 7: As for Fig. 4 but for 325GHz emission. See Sect. 5.3 for details.

   
5.4 183GHz masers

A large majority of sites (87$\%$) produced masers at 183GHz. Figure 8 shows the projected spatial structure of emission predicted by our simulations. Compared with the other masers discussed in the present work, the radial distribution of 183GHz emission is very extended. Sites of >0.01 $I_{\max,183}$ exist out to 35  R_* (39 AU), with weak emission extending out to a radius of 48  R_*. Unlike the other masers discussed here, both radial and tangential amplification is evident for the 183GHz emission. For components lying relatively far from the star, the tangential velocity gradient may exceed that in the radial direction, see Table 3 and Fig. 9b.

An additional feature of the 183GHz maser morphology, unlike that of the other stellar H₂O masers, is that emission is not dominated by a small number of very intense components. Rather, the strength of this maser is provided by a large number of contributing components of similar, weak intensity: $I_{\max,22}>I_{\max,183}$ by a factor of 85. Figure 9a shows how 183GHz emission is more evenly distributed over radius than the 22, 321 and 325GHz maser emission. The upper level of the $3_{13} \rightarrow 2_{20}$ 183GHz transition lies at an energy of 205 K above ground state. The physical conditions leading to the brightest 183GHz emission are shown in Table 3. A regime of low $T_{\rm k}$ and n(H₂) favours 183GHz emission, and the maser may therefore be pumped in regions relatively far from the star. However, the transition can also be inverted in a high kinetic temperature and density regime. These pumping conditions are in agreement with the results of Y97, in which the 183GHz maser transition was found to be strongly inverted over a large range of conditions of low $T_{\rm k}$ with low n(H₂) and of high $T_{\rm k}$ with high n(H₂). The region of the CE occupied by the brightest 183GHz components is shown in Figs. 9b, c and d.

Figure 8: As for Fig. 4 but for 183GHz emission. See Sect. 5.4 for details.