MERLIN and VLA data have provided the evidence that stellar H₂O maser emission at 22GHz is located in a shell expanding from LPV stars (e.g. Reid & Menten 1990; Bowers & Johnston 1994; Yates et al. 1994; Colomer et al. 2000). The shell appears clumpy and incomplete at these resolutions. Emission originates from the inner parts of the CE of Mira-type stars, from regions of diameter $4{-}7 \times10^{14}$ cm, which are comparable in extent to those in which dust grains form and grow, and in which the expanding envelope has not yet reached terminal velocity. 22GHz masers are believed to probe circumstellar gas in which acceleration away from the star takes place via radiation pressure on dust and subsequent gas-grain collisions (Chapman & Cohen 1986).

Other H₂O masers are well-known to exist in the evolved circumstellar environment, see Table 1. For example, maser emission at 321, 325 and 183GHz is common (Menten et al. 1990; Menten & Melnick 1991; Yates et al. 1995; González-Alfonso et al. 1998). However, all information about the location of these masers in the CE must be inferred from spectral line profiles at present. Their use as tools for probing circumstellar conditions has therefore been limited. Knowledge of the location of common stellar H₂O masers in the CE could make them especially useful for deducing the precise evolutionary stage of the star, in combination with other maser observations (Lewis 1989). We note that, in the case of H₂O masers, the type of line profiles observed also provide a good indication of the evolutionary status of the LPV star (e.g. Takaba et al. 1994).

In this paper, we combine a hydrodynamic pulsation model of a LPV of relatively low mass loss rate, a Mira-type variable, with an H₂O maser propagation model. The aim of this work is to reproduce the observed features of stellar H₂O masers, to investigate the physical conditions leading to such emission and to predict the spatial structure of the submillimetre masers in the CE before the advent of ALMA.

We note that the performance of these simulations depends on the accuracy of current LPV stellar pulsation and maser models. The pulsation model, which was developed by Prof. G. Bowen and is based on Bowen (1988), loses mass at a rate of $\dot{M} = 1.8\times 10^{-7}~M_{\odot}$ yr^-1, mainly through a combination of stellar pulsation-driven shock waves and radiation pressure on dust. The maser saturation radiation transport model was developed by Dr. J. Yates, and is based on Doel et al. (1995) and Gray et al. (1995). These models are described in Sect. 2. Observational data for stellar H₂O masers are described and compared with the results of our simulations in Sects. 3, 4 and 5.

**Table 1:** Astrophysical H₂O masers detected to date. G = ground vibrational state; $\nu _{2}$ = vibrationally excited bending mode; O, P = ortho or para-H₂O; $E_{\rm u}$ = energy of the upper level of the masing transition above ground state.
Transition	v-state	Ortho/	$\nu$	$E_{\rm u}/k$
		Para	(GHz)	(K)
6 $_{16} \rightarrow 5_{23}$	G	O	22	644
$4_{40} \rightarrow 5_{33}$	$\nu _{2}$	P	96	3060
$3_{13} \rightarrow 2_{20}$	G	P	183	205
$5_{50} \rightarrow 6_{43}$	$\nu _{2}$	O	232	3451
10₂₉ $\rightarrow$ 9₃₆	G	O	321	1863
5 $_{15} \rightarrow 4_{22}$	G	P	325	470
17 $_{4\,13} \rightarrow 16_{7\,10}$	G	O	355	5786
4 $_{14} \rightarrow 3_{21}$	G	O	380^*	324
7 $_{53} \rightarrow 6_{60}$	G	P	437	1526
6 $_{43} \rightarrow 5_{50}$	G	O	439	1089
6 $_{42} \rightarrow 5_{51}$	G	P	471	1091
1₁₀ $\rightarrow$ 1₀₁	$\nu _{2}$	O	658	2361

^* Has never been observed towards a stellar environment.

1 Introduction