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Astron. Astrophys. 333, 918-925 (1998)

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4. Comparison with the theory of pulsation

In 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 [FORMULA] -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 ([FORMULA] -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.

[FIGURE] Fig. 5. The Bailey diagram for the three RR Lyrae of this study, in comparison with selected theoretical and observational results. For details see the text and Figs. 16a and 16b in BCCIM

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 [FORMULA] = 0.127 mag. Since AQ Lyr has [FORMULA] 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 [FORMULA] = 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 [FORMULA] = 0.06 mag and for CN Lyr [FORMULA] = 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 [FORMULA] = 0.38 mag and thus [FORMULA] = 0.20 mag, in good agreement with the value given by Burstein & Heiles (1982). The same procedure for AQ Lyr provides [FORMULA] =0.33 mag, this means [FORMULA] = 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 [FORMULA], [FORMULA] =0.33 mag and, for [FORMULA], [FORMULA] =0.32 mag, which in terms of reddening means [FORMULA] mag, well within the error of the reddening as derived from Burstein & Heiles (1982) maps.


[TABLE]

Table 6. Selected fundamental quantities for AW Dra, AQ Lyr and CN Lyr.


After correction for reddening, mean colors have been evaluated in three ways: as intensity-weighted ([FORMULA] or [FORMULA]) and as magnitude-weighted [FORMULA] ; 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 [FORMULA]. 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 [FORMULA] 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 [FORMULA] for both AQ Lyr and CN Lyr, whereas for AW Dra one has [FORMULA]. The estimated error on the temperature is about [FORMULA], 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 [FORMULA] for the two metal rich RR Lyrae, and M=0.65-0.75 [FORMULA] for AW Dra. Making use of the relations given by BCCM (corrected by [FORMULA] 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:


[TABLE]

Table 7. Luminosities coming from period-temperature-mass-luminosity relations (see text) by BCCM, for the three RR Lyrae investigated in this study; F and FO mean fundamental and first overtone pulsation mode respectively.


CN Lyr: If this is an F-pulsator, it would be out of the range explored by theory when Z=0.01. However, comparison with results for Z=0.02 suggests that the theoretical rising time of an F-pulsator should be much shorter than observed. Comparison with theoretical predictions for Z=0.01 and Z=0.02 could suggest that this star should be a FO pulsator crossing the strip at large luminosity well above the ZAHB. Even though the star is beyond the limit of the atlas, one finds, e.g., that Z=0.02 FO pulsators with similar temperatures and large luminosities ([FORMULA] = 1.61 and 1.81) show the asymmetric lightcurve disclosed by our observations. Note that in this case one cannot derive the intrinsic color from the quoted Caputo & De Santis (1992) relation, and the reasonable prediction about the CN Lyr reddening should be regarded as obtained by chance. However, as suggested by our referee, on observational grounds CN Lyr (like, e.g., FW Lup in the Lub (1977) sample) has to be regarded as a typical low-amplitude, type b fundamental pulsator, as found towards the red edge of the instability strip. Appropriate theoretical investigation is needed before possible mismatches with the theory can be discussed.

AQ Lyr: There is, apparently, no way to fit the observed lightcurve to theoretical predictions for FO pulsators. The shape of the curve is in good agreement with predictions for F pulsators in the quoted range of luminosities and temperature. Note that the luminosity is the one predicted by BCCIM for metal rich HB stars. However, bearing in mind that the bolometric amplitude is roughly comparable with the amplitude in the V band (Marconi, private communication) theory predicts a larger amplitude. A similar instance has been already discussed in BCCM (see Fig. 18 in that paper).

AW Dra: This star appears to be a classical ab type pulsator. As a matter of fact, in BCCM one finds that a metal poor FO pulsator at the required large luminosity should have a symmetric lightcurve, contrary to observation. Both amplitude and shape of the lightcurve appear in reasonable agreement with predictions for F pulsators, even if the atlas lacks models in the range [FORMULA]. Thus one finds that AW Dra is a fundamental pulsator crossing the strip at [FORMULA], i.e. above the ZAHB luminosity level.

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© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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