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Astron. Astrophys. 317, 761-768 (1997)

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4. Application to Miras. Period-temperature relationship

4.1. Determination of temperatures

Lockwood observed 292 stars among which 256 are M- or MS-Miras; the others are classified S-Miras or semi-regular variables. These oxygen-rich Miras have been observed at several phases, sometimes in different cycles, to give 1501 sets of five-colour measurements. Figure 3 shows the distribution of periods of the sample, which perfectly corresponds to the distribution of the M-Miras taken in the General Catalogue of Variable Stars (Kholopov 1985, 1987). The stars have been preferably observed near maximum and minimum of light curves.

[FIGURE] Fig. 3. Distribution of periods (256 Miras)

All the data (1501 points) are plotted on the plane 78-88/105-104 plane (Fig. 4). The curve calibrated in temperature as determined in Sect. 3 is also drawn. The data follow the curve relatively closely in general, but they are strongly scattered. This is due to the fact that a Mira atmosphere is much more complex than that of a non-variable star.

[FIGURE] Fig. 4. Location of observed indices for all Miras of the sample at all phases. Fill circles are separated by 100 K

Among the 256 M-Miras, 93 are observed more than 5 times (group I) and 93 between 3 and 5 (group II). Miras with one or two observations are not taken into account subsequently. In order to make a determination of temperatures for a number of stars as large and homogeneous as possible, we have fitted the observed indices for each individual Mira as a function of phase [FORMULA] by sine curves:

[EQUATION]

For group II, we have reduced the 3 free parameters to 2: [FORMULA] and [FORMULA] ; [FORMULA] is taken equal to 0.25 so the maximum is fixed at [FORMULA]. We have only kept the Miras which have observed indices distributed in a phase range larger than 0.3. This allows us to fit properly the indices and to obtain values of 78-88 and 105-104 at estimated minimum and maximum for a large sample. The 78-88 index saturates at about 1.8 so we adopted this value when the fit gives a greater estimate. Similarly, we set min(105-104) [FORMULA] when a lower result is obtained. At last, we exclude the stars for which:

[EQUATION]

Among the 186 Miras with a number of observations greater than 3, 165 finally remained.
The resulting values of 78-88 and 105-104 at minimum and maximum are represented by points which often scatter from the curve of Fig. 4. Indeed, each index does not separately give the same temperature. We nevertheless choose to fix a temperature for each couple of values by taking the nearest point of the curve, i.e. the perpendicular projection. The distance between the representive point at minimum or maximum and its projection on the curve gives an uncertainty estimate on temperature.
Table 2 gives, for each star, the period, the group, the averaged temperature [FORMULA], the uncertainty on [FORMULA] and the amplitude of variation ([FORMULA]). The averaged temperature used is [FORMULA]. Indeed, comparison between static and dynamical atmosphere models (Tuchman et al. 1979) shows that is a good approximation of the static temperature. Using the straight mean temperature does not change the results appreciably. We note that the mean temperature amplitude is 640 K. The Mira models developped by Bessell et al. (1989a) predict an effective temperature variation of 680 to 770 K.

4.2. Period-temperature relationship

Figure 5 shows log [FORMULA] versus log P for the 165 Miras of Table 2. There is a clear though scattered linear relation.

[FIGURE] Fig. 5. Period-temperature relationship

A least-square polynomial fit gives:

[EQUATION]

[FORMULA] being the standard deviation.

For 121 Miras of the Galaxy, Glass & Feast (1982) obtained:

[EQUATION]

[FORMULA] is the temperature of the best black body fit they have made of JHKL observations. These authors have argued that [FORMULA] is similar to [FORMULA]. The scatter ([FORMULA]) around their regression line is comparable to ours (cf their Fig. 2).

It is very satisfactory to see that both relations are similar whereas the two methods to determine temperature are totaly different. This indicates that the temperatures obtained here, despite the uncertainty on the method, are globally reliable. It seems clear that the temperature is correlated with the period, but the large scatter prevents us from giving a unique temperature for a given period.

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© European Southern Observatory (ESO) 1997

Online publication: July 8, 1998
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