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5 Discussion

In this section are presented the main characteristics of the OH flux variations over the entire sample of stars.

5.1 Comparison of the OH and optical curves

We first note that the general shape of the OH and optical variabilty curves is the same. When secondary maxima exist in the optical variability curve, they are also observed in the OH variability curves. This behaviour is well illustrated by the variability curves of the first data set (i.e., from the epoch [45000 $^{\rm JD}-$45400$^{\rm JD}$]) of R Aql, especially, from the 1667 MHz red peak curves (cf. Figs. 9e and h in comparison with Fig. 9a). These secondary maxima can be seen in the variability curves of some components (cf. Fig. 34 cycle (7): the variability curve of the component located at V=-17.08 km s-1 of the U Her 1665 MHz blue peak).

These secondary maxima are not only observed in the OH maser emission. Indeed, they were present in ${\rm H_{2}O}$ maser emission observed by Gómez Balboa & Lépine (1986) for W Hya.

The periods determined by the fitting program show discrepancies of 1 to 20 days in comparison with those determined from the optical curves (Campbell 1985), which represents less than 6% of the period. The periods determined from the optical curves are in fact mean values determined from numerous optical cycles. Because of discrepancies between the various determinations of the optical periods, the mean optical period is reliable within 10 to 20%. The 6% deviation found between the optical and OH periods is in good agreement between the two estimations. Moreover, the estimated periods of R Aql, RR Aql and RS Vir obtained by Herman & Habing (1985) and van Langevelde et al. (1990), determined from the same rough data, are in agreement with our estimations within 10 days. On the other hand, we notice a large discrepancy between our estimations of the OH asymmetry factor and those determined by van Langevelde et al. (1990). The estimations of Herman & Habing (1985) are in better agreement with our results.

In order to determine the phase delay between the optical and OH curves as well as the OH period and the OH asymmetry factor $\Delta f_{0 \; {\rm OH}}$ we have used the fitting program developed by David et al. (1996). We have chosen, when possible, two consecutive cycles and/or the best sampled cycle for each star. The results of the OH curve fitting are shown Fig. 42 and Table 2.

\par\includegraphics[width=8.8cm,clip]{} \end{figure} Figure 42: Results of the fitting of the best sampled OH curve of each star

In this table are given in Col. (1), the name of the sources of the study and in Cols. (2), (3) and (4) respectively, the line, polarization and peak in which the integrated flux curve fitting was performed. The period, asymmetry factor and date of the OH maximum deduced from the OH curve fitting are given in Cols. (5), (6) and (8). Moreover, for comparison Col. (7) gives the optical asymmetry factors obtained from the Campbell catalogue. Finally, the date of the closest optical maximum and the calculated phase delay between optical and OH maxima are given in Cols. (9) and (10).

The phase delay between the OH and the optical maximum was found to range between 10 to 20% of the period.

Even though the general shape of the OH and optical curves is the same, one still notes one peculiar difference: the asymmetry (due to a rise time from a minimum to a maximum which is different from the decline time from a maximum to a minimum) is less important in OH curves due to a smaller phase delay between optical and OH mimima than between maxima. This fact can be clearly observed from the variability curves of the S CrB main lines in the third set of observations (cf. Figs. 18f and 18g in comparison with the corresponding part of the optical light curve). Thus, if we define the asymmetry factor as:

f_{0} = \frac{\rm ascending \; time \; from \; a \; minimum \; toward \; a \; maximum}
{\rm period}
\end{displaymath} (2)

and we label the difference of asymmetry factor between OH and optical light curve

\Delta f_{0} = f_{\rm0 \; OH} - f_{\rm0 \; optical} ,
\end{displaymath} (3)

then, except for UX Cyg which presents a very weak $\Delta f_0$ ( $\Delta f_{\rm0 \; UX \; Cyg} = 0.03$) the difference of asymmetry factor is of $0.11 \le \Delta f_{0} \le 0.25$ (cf. Table 2). This leads to the question: what kind of physical process can cause this behaviour?

The two main hypotheses which could lead to the lower value of the OH asymmetry factor are: either the number of OH molecules varies with the phase, or OH varies in phase with the infrared emission. This second hypothesis is more likely to be correct.

Maran et al. (1977), in a study of long-term infrared behaviour of Mira-type variables, note that the infrared maxima occur after the visual maxima but that no phase difference could be observed between infrared and visual minima.

The variations at 10351 ${\rm\AA}$ (i.e., filter "104") as well as the variations of the composite index D=0.18T1+T2+V1 as defined by Lockwood (1972) (i.e., were T1 and T2 are color indexes based on TiO measurements and V1 is an index based on VO measurements) are displayed for R Aql, S CrB, R LMi and U Her in Figs. 43a,b,c and d respectively.

\par\includegraphics[width=8.8cm,clip]{} \end{figure} Figure 43: Variations at 10 351 ${\rm\AA}$ (i.e., filter "104") [circles with solid line] as well as the variations of the composite index D=0.18 T1+T2+V1 [squares with dashed-line] defined by Lockwood (1972) were T1 and T2 are color indexes based on TiO measurements and V1 is an index based on VO measurements. Also displayed is the corresponding part of the light curve (AFOEV, private communication) for R Aql, S CrB, R LMi and U Her [triangles with dotted line]. The scale of the D index and the optical curve have been changed as given in each sub-boxes to fit with the 104 scale allowing a direct comparison of the locations of minima and maxima of the three quantities

Also displayed in these figures is the corresponding part of the light curve (AFOEV, private communication). The scale of the D index and the optical curve has been changed as given in each sub-box to fit with the 104 scale, allowing a direct comparison of the locations of minima and maxima relative to the three displayed quantities. It is clear from these figures that the minima of the three curves occur at the same phase (this is clearly seen Fig. 43d for the minimum of U Her which is particularly well sampled) while the optical curves of the four stars always peak at maximum before the 104 and D curves, roughly about 0.05 phase earlier.

Moreover, from the Fig. 34 of Harvey et al. (1974), which displays the infrared observations of U Her at 1.2, 1.6, 2.2, 3.5, 4.8, 10 $\mu $m from 1968 to 1972, one can see that the optical minima indeed fit with the infrared ones while we observe a phase delay between the optical maxima and the infrared ones. It even seems that the infrared maxima are more delayed at longer wavelength in comparison with the optical maxima. Indeed, from Table A3 of Harvey et al. (1974) one see that the relative phase of the infrared maximum at 10 $\mu $m is systematically greater than the maxima at shorter infrared wavelengths 1.2, 1.6 and 2.2 $\mu $m. As the minima occur simultaneously we conclude that the asymmetry factor increases with wavelength. This leads to the conclusion that the contribution of far infrared radiation in the pumping of standard OH maser is rather strong, although the recent calculations of Collisson & Nedoluha (1993) do not confirm this.

Finally, the OH asymmetry values given Table 2 have a mean value of $f_{\rm0 \; OH}= 0.53$. Thus, there is no systematic trend for the OH asymmetry factor to be inferior or superior to $f_{\rm0 \; OH}=0.50$.

5.2 Polarization behaviour of the OH integrated flux

The polarization behaviour is quite different from one star to another; no evidence of a relation between the strength of the polarization and the IRAS color indexes was found (i.e., with the thickness of the dust shell). Nor was there a tendency for any of the emission to be more left or right-hand polarized. However, in all the thickest shells (i.e., R Aql, RS Vir, S CrB and UX Cyg) the general polarization trend is similar for the two main lines.

The 1612 MHz integrated flux is faintly polarized most of the time ($\leq$ 30%). On the other hand, the main lines show a degree of polarization ranging from 0 to 60%. Surprisingly, it appears that the 1667 MHz emission can be more polarized than the 1665 MHz emission (cf. integrated flux of RS Vir Fig. 12 or integrated flux of U Her in both main lines for cycle (7) in Fig. 32).

Inter-peak signal shows no or very faint polarization. For the two standard peaks, the degree of polarization is not constant. These changes differt for the front and back part of the shell, as clearly observed in R LMi. The observed changes in polarization seem to be tied to the variation of the mean integrated flux value as inferred from the main line variations of R LMi and UX Cyg. These changes seem to be tied also to the change of $\Delta I_{\rm min-max}$ of the integrated flux (cf. the behaviour of U Her). Nevertheless, this link is not simple, since for R LMi the degree of polarization decreases and increases in the same manner as the mean integrated flux while for UX Cyg the behaviour is opposite.

5.3 Amplitudes of the OH integrated flux variations

Excluding UX Cyg, it appears from the measurements of the integrated flux $\Delta I_{\rm min-max}$, that the 1612 MHz line always shows the smallest amplitude of variations, usually of order of 30 - 35% (i.e., $I_{\rm max}/I_{\rm min} \simeq$ 2.0) and hardly ever exceeding 50% (i.e., $I_{\rm max}/I_{\rm min} \simeq$ 3.00). On the other hand, both main lines reach comparable values which are usually greater than 30% and can be as high as 95% (i.e., $I_{\rm max}/I_{\rm min} \simeq$ 40). This fits with the idea that generally the 1612 MHz satellite line emission is considerably more saturated than the main line emissions. For all sources, changes in the amplitude of variations from one cycle to another are evident. Since the two main parameters which govern the degree of saturation are the column density of OH molecules along the line of sight and the velocity field, these changes are the signature of the dynamics of the shell physical conditions. Thus, the quick, large changes observed in the amplitude of variations in the three maser lines of UX Cyg are probably the signature of turbulence in the velocity field of its envelope. This hypothesis is strengthened by the rapidly changing shape of its spectral profile in the three maser lines.

It is striking that the 1665 and 1667 MHz emissions exhibit a similar variability behaviour different from that of 1612 MHz. Indeed, while the 1612 MHz emission usually shows small changes in amplitude it is evident that both main lines can show strong variation amplitude from one cycle to another. Moreover, close observation shows that the 1665 and 1667 MHz emissions exhibit the same variability behaviour for the same side (i.e., front or back part) of the shell. This fact is well illustrated by Fig. 32, showing the integrated flux of U Her at 1665 and 1667 MHz. Indeed, if we compare the OH variabilities of the 1667 MHz red and blue peaks for example (i.e., Figs. 32b and 32d), we see that the general trend is different. We clearly observe a slow increase of the mean value of the blue 1667 MHz peak integrated flux from one cycle to another along the whole set of the displayed observations while the mean value of the red 1667 MHz peak integrated flux is constant from cycle (1) to cycle (5). Besides, the general OH variability trends, displayed in Figs. 32b and 32c (i.e., blue peaks in both main lines) and in Figs. 32d and 32e (i.e., red peaks in both main lines) are obviously similar. This fact is still valid if we compare the general 1665/67 MHz trend with the polarization as illustrated by the main line variations of R LMi Fig. 24. Indeed, we see that the blue part of the 1665 and 1667 MHz emissions show a similar behaviour, exhibiting a rather strong right-handed polarization, especially in cycles (1) and (2) (cf. Figs. 24i and 24g respectively). This behaviour is completely different from the one observed in the red part of the 1665 and 1667 MHz emissions (cf. Figs. 24j and 24h respectively) where the degree of polarization is equal to zero for the whole set of observations.

From the long term OH curves of U Her and R LMi main line integrated flux (cf. Figs. 24i, 24j and 32), we see that the temporal variation of the integrated flux mean value is not a random phenomenon but slowly varies over several cycles.

Previously, Gómez Balboa & Lépine (1986) found the existence of a super period in the ${\rm H_{2}O}$ maser variability curve of R Aql equal to three times the optical period. They also found a super-period in the ${\rm H_{2}O}$ maser emission of W Hya. As they proposed, these super-period phenomena could be the signature of shock wave effects, more precisely of an accumulation of several successive shock waves.

Recently, Bowers & Johnston (1994) have shown, from an interferometric study, that the structure of ${\rm H_{2}O}$ masers varies over several cycles. Also, Danchi et al. (1994) have shown, from an infrared interferometric study, that dust formation undergoes some variations over several cycles as well. Thus, it seems that two levels of variations exist in the circumstellar shell: the well known one over one period which is cyclic and a longer one which may or may not be cyclic. All these long term variations may be correlated with one another. Especially, it is obvious that variations in the dust formation will have direct consequences on the OH masers.

A theoretical study by Winters et al. (1994) has shown that emission of the circumstellar dust shell may have two components: the first which follows the cyclic periodicity of the optical light curve and a second one, spreading over several periods, which is due to the dynamical structure of the dust shell itself. Thus, all these long term variations observed in ${\rm H_{2}O}$, OH and dust emission could indeed be periodic. Confirmation of this idea is important for the understanding of the pulsation mode of such stars.

Inter-peak components are only present in the profiles of the main lines. Thus, the two Miras showing such emission, i.e., RS Vir and RR Aql, do not show any inter-peak component at 1612 MHz. One can see that the red and blue peaks are well detached, with an inter-peak signal equal to zero. The only inter-peak components observed at 1612 MHz are due to eruptive emission as in U Her during 1984 and 1989 (cf. Etoka & Le Squeren 1997).

The inter-peak components do not show any special behaviour either in their OH variability or with respect to the polarization (cf. Figs. 12, 13 and 27). They all show standard OH variations and a weak degree of polarization. The faintness and the weak polarization suggest that these components may be tangential contributions from the outmost parts of the 1665-1667 MHz regions.

5.4 Component behaviour

The number of components observed in the 1612 MHz spectral profiles is generally fewer than in the main lines. Moreover, the 1612 MHz component width is smaller than that in the main lines. These two facts can be explained if the 1612 MHz emission is produced in a less turbulent zone of the envelope.

The components at 1612 MHz are stable over more than 10 years. The spectral shape at 1612 MHz shows no significant changes, while in the main lines, important changes in the number of components and in their strength occur, leading to quite different profiles from one cycle to another.

An indication of an even longer 1612 MHz velocity stability can be deduced from a comparison between the components previously detected by Herman & Habing (1985) for R Aql, RS Vir and RR Aql during the interval of Julian days [43000 - 45300] (i.e., roughly from August 1976 to November 1982) and ours. The velocity of the components detected by Herman & Habing were calculated from their Tables IIa and III where respectively are given, in Col. 7 of the first table their deduced stellar velocity and in Col. 10 of the second table the relative velocity according to the stellar one. Obviously, due to a better resolution in velocity (by about a factor 4) than the Herman & Habing study, we could detect a greater number of individual components. But, it is also clear that all the components detected by them were also detected in the present study (cf. Tables 3,  5 and  23). This then extends the stability of the 1612 MHz components to about 20 years.

All the previously mentioned facts corroborate the idea of a 1612 MHz satellite emission zone more external than the main line one.

In a general way, it is apparent from the spectral decompositions that the spectral FWHM of a single component is less than 1 km s-1. Thus, a greater FWHM is due to strong blending. On the other hand, the narrowest components whose longevity is at least one stellar period have a FWHM of more than 0.3 km s-1, even in spectral profiles obtained with a high spectral resolution of 0.07 km s-1. According to Alcock & Ross (1986), this value corresponds to the local line width of OH molecules at 100 K. It seems that this value is a limit, independent of the spectral resolution. A very few fitted components were found to have a somehow narrower FWHM: 0.2 km s $^{-1} < {\rm FWHM}<0.3$ km s-1. The narrowest features that Fix (1987) could distinguish from his OH spectral study of OH/IR stars at high-resolution (up to 0.014 km s-1) was also 0.2 km s-1. According to the model developed by Alcock & Ross (1986), Fix (1987) determined that this width corresponds to a gas temperature of about 30 K. We found that these kinds of components always have very short longevity: less than 2 months. They may be the signature of local temperature inhomogeneity. If so, we may conclude that the longevity of such local, cold blobs is no more than a few months.

Standard models, based on a uniform outflowing wind, can only create featureless profiles. However, observations at high velocity resolution clearly show that maser spectral profiles are not smooth but are in fact composed of numerous individual components. Modeling studies of Alcock & Ross (1986) have shown that these spectral profiles cannot be explained by a smooth wind. Thus, Fix (1987), from numerical modeling, has found that the actual maser profiles are consistent with emission arising from more than a thousand individual elements. In a more recent study, Zell & Fix (1990) have synthetized 1612 MHz doubly-peaked spectra of some Miras and OH/IR sources. They needed 25 individual elements to synthesize an R Aql type profile while for the other sources of their studies, they need, as did Fix (1987), about a thousand elements.

The spectral decomposition done here is such that about 10 to 20 components of about 0.3 to less than 1 km s-1 are needed to fit the observed spectra. Thus, considering the Zell & Fix theory, each fitted component obtained here may correspond to a whole maser blob including itself a hundred of narrow components very close in velocity (i.e., $\ll 0.2$ km s-1).

Bowers et al. (1989) made interferometric 1612 MHz observations of R Aql and RR Aql and 1667 MHz observations of U Her in early 1985. Some of the components they detected are the same as some of our fitted components. Especially, their OH map of U Her shows about 70% of components having the same velocity as our fitted components within our velocity resolution (0.15 km s-1, which increase up to 95% within their velocity resolution of 0.30 km s-1). Sivagnanam et al. (1990) and Chapman et al. (1994) also observed U Her but in both main lines. Quite good agreement of our spectral decomposition and their detected components (cf. Sect. 4.6.4) was found. All our fitted components for R Aql and RR Aql are present (i.e., within less than 0.15 km s-1) in the maps of Bowers et al. (1989). However, they detected a greater number of components for the red peak of R Aql and for both peaks of RR Aql. We attribute this discrepancy to the large FWHM of our spectral decomposition for the cited sources and peaks, especially for the blue peak components of RR Aql, which exhibit mean FWHM greater than 1 km s-1, the signature of a strong blending.

In all cases, considering that a great part of the maser blobs with a specific velocity belong to the same cloud, spectral decomposition gives a way to determine the variability of a specific blob and, with maps obtained from various epochs, it could allow us to determine the change in variability according to the location of the maser blobs in the shell at a specific moment.

All the fitted components observed over more than one stellar cycle show variations in phase with the optical light curve, taking into consideration the usual delay observed between OH and optical curves.

We often observe that the most external group of components of both standard peaks (i.e., taking the stellar velocity as a reference) exhibits the greatest values of $\Delta I_{\rm min-max}$ while, surprisingly, the most internal component shows lower amplitudes. The small variations usually observed for the most internal component emission can be explained if they arise from a region lying in the outermost part of the OH shell rather than in the inner part (i.e., if it is a tangential contribution).

Moreover, considering the evolution of the OH emission over several consecutive cycles, it is clear that the mean value of the integrated flux (due to an increase or a decrease of the maxima and minima values from one cycle to another) as well as its $\Delta I_{\rm min-max}$ is not a global phenomenon but can be very different from one component to another, even though these components belong to the same peak. As can be seen in Fig. 36, components belonging to the same group (for instance U Her components centered at V=-16.02 and V=-15.20 km s-1 at 1667 MHz) usually show very similar behaviour. Neverthelees, sometimes large discrepancies are observed even from components of the same group as can be seen in Fig. 34 where the U Her components at 1665 MHz located at V=-18.66 and -20.35 km s-1 exhibit a similar behaviour, clearly different from the one for the component at V=-19.61 km s-1 which belongs to the same group. Now, considering the maps obtained by Bowers et al. (1989) we can see that both the V=-18.66 and V=-20.35 km s-1 components (i.e., V=-18.6 and V=-20.2 km s-1 in their maps accounting for the velocity uncertainty) are located in the same area, different from the location of the V=-19.61 km s-1 component (i.e., their component at V=-19.7 km s-1).

It should be noted that the difference in behaviour from one component to another was previously observed in ${\rm H_{2}O}$ maser emission by Cox & Parker (1979). They could follow the variability behaviour of the 2 close components of W Hya, respectively centered at V=+38.2 and V=+40.2 km s-1 and show that the variation in amplitude observed in their 1975-1976 observations was only due to the V=+40.2 km s-1 component while the V=+38.2 km s-1 component behaved regularly.
These time variations in the mean value of the integrated flux and the amplitude of the flux variation from one cycle to the next are clearly observed in the main line emissions and are not linked with any optical fluctuation of the light curves. Thus, these variations can be interpreted as a fluctuation in the degree of saturation or as local inhomogeneities either in density or velocity of the OH or/and pumping sources themselves. These phenomena are hardly observable in the 1612 MHz satellite line because of its greater saturation.

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