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.
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.
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 by sine curves:
For group II, we have reduced the 3 free parameters to 2: and ; is taken equal to 0.25 so the maximum is fixed at . 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) when a lower result is obtained. At last, we exclude the stars for which:
Among the 186 Miras with a number of observations greater than 3,
165 finally remained.
4.2. Period-temperature relationship
A least-square polynomial fit gives:
being the standard deviation.
For 121 Miras of the Galaxy, Glass & Feast (1982) obtained:
is the temperature of the best black body fit they have made of JHKL observations. These authors have argued that is similar to . The scatter () 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.
© European Southern Observatory (ESO) 1997
Online publication: July 8, 1998