## 4. Comparison with the theory of pulsationIn recent times the use of non-linear, non-local, time-dependent convective models by Bono and coworkers has produced a large amount of theoretical predictions concerning RR Lyrae pulsations. (See the most recent papers by Bono et al. 1997a (BCCIM), 1997b (BCCM) and reference therein). Taking advantage of this theoretical framework, in this section we will discuss the pulsational properties of the three studied variables. As a first step, Fig. 5 shows the location in the Bailey (period-amplitude) diagram for our three variables, as compared with selected samples of field RRab and with theoretical results from BCCIM; the field stars were selected to lie within the limits of metallicity given in the figure. Periods and amplitudes of field RR Lyrae are from Blanco (1992), while metallicities are from Layden (1994, 1995) and Layden et al. (1996). In all three panels our variables are reported with four-pointed stars: going toward lower periods one finds AW Dra, CN Lyr and AQ Lyr. One finds that AW Dra lies in a region where only low metallicity pulsators occur. Accordingly one can predict for this variable a metallicity as low as -1.4. On the contrary, both AQ Lyr and CN Lyr clearly are members of the high metallicity group, as expected in particular for CN Lyr ( -0.26). One may notice that both stars lie on the lower envelope of the observed distributions, well below theoretical expectations even for "young" massive pulsators (see Fig. 16b in BCCIM and the discussion in that paper). We conclude that, if these stars are "bona fide" ab-type pulsators, theory has to be improved to account for such kind of unpredicted variables.
To allow a closer comparison with predicted lightcurves one can estimate temperatures from the observed colors, provided that the reddenings are known. For AQ Lyr Burstein & Heiles (1978) provided = 0.127 mag. Since AQ Lyr has values by Suntzeff et al. (1994), one may use the method by Sturch (1966), as improved by Blanco (1992), to test this reddening with an independent estimate. As a result we find for AQ Lyr = 0.13 mag, in excellent agreement with the previous value. For CN Lyr and AW Dra the literature gives no indications. To get the missing values, we again used the reddening maps by Burstein & Heiles (1982), obtaining for AW Dra = 0.06 mag and for CN Lyr = 0.21 mag. According to the quoted authors, the error on these estimates is of the order of 0.03 mag. An alternative way to derive information about reddenings is that of using statistical relations such as those provided by Caputo & De Santis (1992). These authors used the Lub (1977) sample of field ab type variables to derive relations between periods, B amplitudes, metallicities and mean de-reddened colors. Using their Eq. (10) we obtain for CN Lyr = 0.38 mag and thus = 0.20 mag, in good agreement with the value given by Burstein & Heiles (1982). The same procedure for AQ Lyr provides =0.33 mag, this means = 0.10 mag, slightly lower (about 0.03) than the value previously determined from Blanco's reddening law and from Burstein & Heiles (1978) maps but within the errors. Hence we adopt those estimates in order to compute the temperature of the variable in Table 6. For AW Dra there is no available metallicity evaluation in the literature, but we can estimate lower and upper limits for it from our Fig. 5. We get, for , =0.33 mag and, for , =0.32 mag, which in terms of reddening means mag, well within the error of the reddening as derived from Burstein & Heiles (1982) maps.
After correction for reddening, mean colors have been evaluated in three ways: as intensity-weighted ( or ) and as magnitude-weighted ; all these values are reported in Table 5. However Bono et al. (1995, BCS hereafter) have once again shown that the color of the static model does not match exactly any observed mean color. To all these mean colors we thus applied the corresponding correction as tabulated by BCS for Z=0.001; we estimated that errors due to the different metallicity should not exceed few thousandths of magnitude, thus preserving the general trend . After applying the BCS correction the three substantially different mean colors for each star become very similar, so we proceeded to average them and we took this mean value as our best estimate of the RR Lyrae mean colors, as given by in Table 5. Finally, RR Lyrae mean colors have been translated into temperatures by estimating gravities from the period - gravity relation obtained from the period - temperature - luminosity - mass relation by BCCM and using Kurucz (1992) models. One finds for both AQ Lyr and CN Lyr, whereas for AW Dra one has . The estimated error on the temperature is about , largely dominated by the error in the reddening. Temperatures for the three RR Lyrae studied in this paper are reported in Table 6 together with further pertinent quantities. Given period and temperature, one can obtain an estimate of the star luminosity from the period - temperature - luminosity - mass relation, provided that suitable assumptions of the pulsator masses are made. Evolutionary constraints indicate that one can safely assume M= 0.53-0.58 for the two metal rich RR Lyrae, and M=0.65-0.75 for AW Dra. Making use of the relations given by BCCM (corrected by for fundamental metal rich pulsators, as stated by BCCIM) one finally derives the range of luminosity given in Table 7 under the two alternative assumptions about the mode of pulsation. We are now able to compare observed lightcurves with the atlas presented by BCCIM and BCCM. For the various stars one finds:
© European Southern Observatory (ESO) 1998 Online publication: April 28, 1998 |