3 Discussion

The variability of the H₂O maser emission is most probably due to impact of shock waves on the maser generation region, located at distances of about 10¹⁴ cm from the star's centre (Rudnitskij & Chuprikov 1990). The circumstellar masers in the SiO and OH lines are also affected by the shock. The SiO masers, closest to the stellar surface, display an outburst first, then there is a rise in the H₂O and main OH lines, and after that the OH 1612-MHz satellite line may rise. This sequence was traced by us for the star R Leo: after its strong flare in the H$\alpha$ line in May, 1996 (Esipov et al. 1999) there was a rise in SiO (Cho et al. 1998; Herpin et al. 1998) and H₂O (Esipov et al. 1999). A similar sequence of the SiO and H₂O line events in early 1980s in R Leo was followed by an OH flare in 1985 (Etoka & Le Squeren 1997), when the shock reached the OH maser generation region.

U Ori showed dramatic variations in the OH lines in 1974, when the type of its OH emission changed (Cimerman 1979; Jewell et al. 1979, 1981). The events consisted of the sudden appearance of strong 1612-MHz radiation, coinciding with the disappearance of the 1667-MHz radiation. Then the 1612-MHz radiation declined and the 1667-MHz line reappeared. Unfortunately there are no data on systematic SiO or H₂O line monitoring at nearby previous epochs. However, it is known that at least in September, 1976 the H₂O emission in U Ori was quite weak, below $\sim12$ Jy (Spencer et al. 1979).

The next activity cycle of U Ori started in late 1970s. The exact date of its onset is not known, since there are no H$\alpha$ or SiO data available. As mentioned above, the H₂O maser flared between June and October, 1980. Wallerstein (1985) notes that the H₂O flare we observed in 1980 may be related to the emission-line activity, observed by him in the preceding star's cycles, and hence to shocks emerging from the star. Another, weaker flare of up to $\sim450$ Jy km s^-1 at the end of 1983 is probably connected with the SiO flare of the end of 1982, observed by Nyman & Olofsson (1986).

In the intervals between the sequences of the superactivity events (H$\alpha$ - SiO - H₂O - OH(main) - OH(1612)) the H₂O maser U Ori is weak. In addition to the absence of the shock-wave pumping, the cause for this may be the lack of an appropriate H₂O maser medium.

Rudnitskij & Chuprikov (1990) proposed a model, in which the H₂O maser emission is generated within a "quasi-stationary layer'' (QSL) of gas (the term is due to Hinkle et al. 1982), which is located at $r\sim10^{14}$ cm from the star's centre and has an approximately zero velocity with respect to the star. The gas temperature $T_{\rm g}$ in QSL is on the average about 800 K, the density $n_{\rm H}$ may be 10¹¹ cm^-3. QSL represents a temporary stop of the mass lost by the star, the matter is infalling toward the stellar surface from the QSL inner boundary and outflowing from the outer one. The physical conditions at the QSL radius are favourable for dust formation and H₂O maser effect in the above scheme.

In the model of Rudnitskij & Chuprikov (1990) the H₂O maser is pumped by a moderate-strength shock ( $v_{\rm s}\sim6-10$ km s^-1) reaching QSL and creating nonequilibrium conditions in the postshock matter, where the gas is heated promptly and the dust remains cooler for some time. Thus, a collisional-radiative pumping on rotational H₂O transitions ($C\!Rr$ in the classification of Strel'nitskij 1984) can operate. The effectiveness of the mechanism is proportional to the difference of $T_{\rm g}$ and $T_{{\rm dust}}$. In the absence of a shock the dust temperature is controlled mainly by stellar radiation. After the shock passage the temperature rises, and as $T_{{\rm dust}}\to T_{\rm g}$, the efficiency of the H₂O pumping decreases.

The above model of a thin H₂O masering shell, expanding outward behind a shock front, is supported by the H₂O interferometry data of Lada et al. (1981) and Bowers & Johnston (1994). Their maps of U Ori indicate that the maser emission at the most negative ("bluemost'') velocities is approximately centred on the stellar position, thus being located at the near side of the circumstellar envelope, approaching the observer, whereas the emission at velocities close to V_* is distributed (though nonuniformly, in several clumps) over a wider region, probably outlining the entire H₂O shell. The emission redshifted with respect to V_* is spread in a smaller region, also within the expectations of the same model. Furthermore, the entire velocity set of the H₂O shell is visible in the high-sensitivity H₂O line profiles of Bowers (1992) as a tail of faint emission ( $F_\nu\sim100$ mJy), extending from V_* to as far as $V_{{\rm blue}}\sim -$49.5 km s^-1, marking the bluemost boundary of the profile and yielding an estimate of the shock velocity $v_{\rm s}\sim V_{{\rm blue}}-V_*\sim11.4$ km s^-1, in agreement with the value accepted in our model (10 km s^-1). Stronger shocks may mark some light cycles of the star when occasional higher-intensity emission, accessible to our observations (less sensitive than those of Bowers 1992) appeared at blueshifted velocities. This emission is seen in Fig. 2 as "ripple'' at velocities of -41 to -43 km s^-1, especially in 1981 (soon after the giant flare) and in 1997. Unfortunately, our velocity coverage of this portion of the spectrum is not complete enough.

An independent confirmation of a thin-shell model for H₂O masers in long-period variables may come also from Colomer et al. (2000) who mapped several Miras and SRs in the 1.35-cm H₂O line on VLA (among them, U Ori was observed, too, but unfortunately not resolved). A 3D model fit by Colomer et al. showed that at least for some stars they observed (U Her, R LMi, R Cas and others) the H₂O maser structure can be best represented as a thin expanding shell, in agreement with our above reasonings.

Note, however, that the actual shape of the circumstellar envelope of U Ori may be not so simple as the spherically-symmetrical structure we have assumed. The results of the OH interferometry, referring to the regions of the envelope slightly outward from the H₂O shell (Reid et al. 1979; Fix et al. 1980; Chapman & Cohen 1985; Bowers & Johnston 1988; Chapman et al. 1991), indicate that mass loss (and hence the distribution of circumstellar matter in U Ori) may have a form of a bipolar outflow, though with some element of spherical symmetry. The same is testified to by detailed modelling of the OH masers around U Ori (Bowers 1991). This asymmetry should be taken into account in further modelling.

QSL hosting the H₂O maser may temporarily vanish and then reappear as a result of an episode of enhanced mass loss by the star (Hinkle et al. 1984). This explains the observed lack of the H₂O maser emission in U Ori and some other Miras during several cycles of light variations (Esipov et al. 1999 and references therein). The flare of the H₂O maser U Ori in 1980-1982 may be associated with rebuilding of QSL after the period of maser absence and weakness in 1976-1980. Infrared data (Danchi et al. 1994) and an analysis of OH data (Etoka & Le Squeren 1997) show that the circumstellar envelope of U Ori is indeed a peculiar one, with little dust located close to the star, and with mass loss in isolated episodes, separated by a decade or so. Hence we can associate this information with temporary disappearance of QSL and of the H₂O maser.

In our model, the phase delay $\Delta\varphi$ between visual and subsequent H₂O maximum is different for different light cycles of U Ori (Fig. 4). Accordingly, the interval between consecutive H₂O maxima varies (Fig. 5). As said above, the dust is not abundant in the shell of U Ori. Since the $C\!Rr$ pumping we assume in our model is quite sensitive to the amount of dust available, then small variations of the dust content and displacements of the inner boundary of dust formation in separate light cycles affect both the maser intensity and phase delay $\Delta\varphi$.

Of interest is also the long-term drift of the H₂O maser velocity centroid (Fig. 7). Note that the dominant maser emission is mainly redshifted with respect to the stated above stellar centre-of-mass velocity -38.1 km s^-1, though, with an account of the probable error of this figure ($\pm1.3$ km s^-1) it may fall within the velocity range shown in Fig. 6. The general trend is a sinewavelike variation with some jumps "blueward'', superimposed onto it, for instance, in 1989-1990, with more blueshifted ripple, see also Fig. 2. However, the flare of 1980-1981 took place at a redshifted velocity (-36.8 km s^-1). This may be due to longer amplification paths, extending from the far, receding side of the expanding shell just after rebuilding of QSL, whereas the sinewave probably reflects the long-term dynamics of QSL: QSL, while present, is as if "breathing'', sometimes infalling, sometimes expanding outward. A detailed discussion of this phenomenon will be given elsewhere.

The correlation of the H₂O maxima following visual maxima with some delay $\Delta\varphi$ is explained as follows. The shock propagating within QSL at $v_{\rm s}\sim6-10$ km s^-1 pumps the H₂O molecules, serving as the source of energy. At the same time the dust, which is a sink of energy, is heated mainly by stellar radiation. The latter, peaking in the infrared, varies lagging behind the visual light by $\Delta\varphi$ of up to 0.4P (Lockwood & Wing 1971). As the infrared radiation is increasing, the pumping effectiveness is falling, the maser emission is going down, too. Thus, the peak of the maser radiation takes place at $\Delta\varphi\sim0.2$ after the visual maximum.

The variability curve of the H₂O maser emission is a result of interplay between the direct shock-wave pumping and heating of dust by stellar radiation; the latter thus indirectly damps the maser by reducing the efficiency of the sink. The shock may be constantly present within the molecular zone coinciding with QSL, while the sink has variable efficiency depending on the infrared stellar irradiation of dust. This scheme (Berulis et al. 1998) is a modification of the model of Rudnitskij & Chuprikov (1990), which included only direct effects of a shock on QSL. It was applied to the H₂O maser RR Aql (Berulis et al. 1998) in an attempt to reconcile obvious correlation of the H₂O maser emission, mimicking with a delay of 0.2-0.3P the visual brightness, with a long travel time of the shock: for a shock front with $v_{\rm s}\sim10$ km s^-1 it may take about two years to reach QSL and to produce the required pumping; the shock then arrives to QSL at an arbitrary stellar phase. The visual-H₂O correlation may still persist, though $\Delta\varphi$ may be quite arbitrary. For instance, in the case of the H₂O maser associated with the semiregular M-type supergiant VX Sgr $\Delta\varphi$ in the course of our observations (1981-1998) was progressively increasing from zero to $\sim1P$ ( $P=732^{\rm d}$) (Esipov et al. 1999). However, in distinction from the giant U Ori, the circumstellar envelope of VX Sgr is much larger, with the H₂O maser region located at $\sim10^{15}$ cm from the stellar centre. Similar estimates for VX Sgr show that for this star $\Delta\varphi$ may in reality be as long as 10-15P. In this case, only constructing a correlation function of the H₂O maser variations and visual light curve at a time interval of a few decades may help to test the model with more confidence.