To approach the problem of comparing the present theoretical scenario with observational data, in Fig. 14 (click here) we show the observational Bailey diagram given by Sandage (1990, and references therein) for pulsators belonging to the two well studied clusters M 15 (OoII) and M 3 (OoI). A quick look to the distributions plotted in this figure reveals a series of relevant similarities with theoretical results that are worth listing:
i) F pulsators in both OoI and OoII clusters arrange along a rather continuous sequence.
ii) The OoII prototype M 15 lacks F pulsators
in the upper portion of the amplitude distribution, whereas
a large amount of M 3 (OoI prototype)
variables populate this region.
iii) The lack of large amplitude F pulsators in M 15 is accompanied by the clear evidence for a decreasing branch of FO pulsators. On the contrary, M 3 shows a much less clear evidence of such a decreasing branch, if any.
Figure 14: The observational Bailey diagram, blue amplitude versus
the logarithm of the period, for RR Lyrae variables belonging to
galactic globulars M 3 (solid circles) and M 15 (open circles) as given
by Sandage (1990)
According to previous findings, we can conclude that the proposed theoretical scenario offers an interesting clue to an evolutionary interpretation of the observed features shown by the Bailey diagram supporting the occurrence of F (or FO) pulsators in OoI (OoII) clusters. However, a more detailed comparison reveals several quantitative mismatches which will be discussed in detail.
To allow such a quantitative comparison with actual cluster pulsators, theoretical light curves have been transformed through Kurucz's (1992) evaluation of bolometric corrections and color-temperature relations to derive amplitudes in the B photometric band, which is the most adopted observational parameter for RR Lyrae amplitudes. As it is well known, when the luminosity of the pulsator decreases the star becomes cooler and the bolometric correction increases. As a consequence, B amplitudes for RR Lyrae pulsators are expected to be larger than the corresponding bolometric amplitudes, an occurrence which of course justifies the widespread use of such an observational parameter. Panel a and b of Fig. 15 (click here) show the predicted amplitude-temperature and amplitude-period relations transformed in terms of B magnitudes.
Figure 15: The theoretical a) amplitude-temperature and b) amplitude-
period relations for RR Lyrae variables as given in terms of blue
amplitudes A(B)
On the basis of this new pulsational topology the analysis given in Fig. 13 (click here) can now be repeated for providing theoretical predictions to be compared with observations. We report in Table 3 (click here) for the metal content Z=0.0001 the expected pulsational properties of F and FO pulsators along the ZAHB.
Table 3: Predicted ZAHB fundamental and
first overtone pulsators for Z=0.0001 (mass and luminosity are in
solar units)
If we take into account only the decreasing branch of the FO amplitudes displayed in the left panel of Fig. 15 (click here), for not too large luminosities, we derive the following relation:
On the basis of such a relation we can attempt a prediction for variable stars located in the OR region, i.e. for stars already evaluated as F pulsators at the top of the F-branch. Data for these FO pulsators are also reported in Table 3 (click here).
The comparison with observational data for pulsators belonging
to M 15 is shown in Fig. 16 (click here). In the same figure are also
plotted the evolutionary
paths of F pulsators evolving off their ZAHB loci. Comparison
with data for M 15 discloses a surprising agreement.
However, the same figure shows that theory foresees the occurrence
of large amplitudes (
) for FO pulsators
which are not present in the cluster.
Figure 16: Observational data for M 15 pulsators are compared
with the theoretical distribution for Z=0.0001 ZAHB pulsators.
In the same figure are also shown the evolutionary paths for the
labeled values of stellar masses
The procedure has been repeated adopting Z=0.0004, assumed as a suitable choice for OoI clusters like M 3. Data for both F and FO pulsators are reported in Table 4 (click here), whereas in Fig. 17 (click here) theoretical predictions are compared with observational data for the quoted clusters. The agreement appears now somehow less satisfactory, since some F pulsators tend to have lower periods than predicted by the theory. If this occurrence is not an artifact of forcing the results for the lower luminosity levels into a common linear interpolation, we find no obvious origin for such a behavior.
Table 4: Predicted ZAHB fundamental and first overtone
pulsators for Z=0.0004
Figure 17: Observational data for M 3 pulsators are compared
with the theoretical distribution for Z=0.0004 ZAHB pulsators.
In the same figure are also shown the evolutionary paths for the
labeled values of stellar masses
As for F pulsators once again we find that theoretical amplitudes for FO pulsators are larger than observed values. However, we look at the comparison in Fig. 17 (click here) as an evidence of the lack of the decreasing branch, disregarding the small amplitude variations at the larger periods as an evidence for the present location of the decreasing branch phase.
As a final point, in Fig. 18 (click here) are compared theoretical and observational data concerning the dependence of blue amplitudes on stellar effective temperatures. According to Brocato et al. (1996) the pulsator mean colors have been first corrected for reddening and then transformed into the true colors of the static models, using the procedure given by Bono et al. (1995). Temperatures have been finally derived adopting the color-temperature relation as given by Kurucz (1992).
We find a further disagreement between observations and theory. Indeed observational amplitudes appear smaller for each given value of the effective temperature or, conversely, for each given amplitude the temperature appears about 300 K larger than predicted by the theory. Note that data in Fig. 18 (click here) assume a reddening value E(B-V)=0.07 for M 15 (Zinn 1985). By adopting the reddening value suggested by Sandage (1990), i.e. E(B-V)=0.11, the discrepancy for this cluster further increases with 200 K. The reader is referred to Brocato et al. (1996) and to Silbermann & Smith (1995) for a detailed discussion concerning the M 15 reddening evaluations.
As a conclusion, the theoretical pulsational scenario we are dealing with, although it presents relevant similarities with the actual behavior of cluster pulsators, is far from reaching a tight correspondence with the available observational data.
Figure 18: Theoretical expectations concerning the B
amplitude-temperature relations for Z=0.0001 or Z=0.0004 are compared
with observational data for the cluster M 15 and M 3, respectively.
Equilibrium temperatures are from Bono et al. (1995)
One could take the comparison in Fig. 18 (click here) as an evidence that our theoretical approach could be affected by a systematic overestimate of amplitudes. As a matter of fact, the difference in temperature of about 300 K for a given amplitude appears much larger than the usual estimates of errors in the color-temperature relation of pulsating stars. If this is the case, the surprising agreement we found for the Bailey diagram of metal poor clusters should be an artifact of balancing errors.
However, in order to properly set the present observational scenario concerning the systematic errors in the RR Lyrae effective temperature scale, it is worth mentioning that, by adopting the V-R color-temperature relation provided by Vanderberg & Bell (1985) instead of Kurucz's relation, Silbermann & Smith (1995) obtained a systematic shift toward cooler temperatures. Most interesting, they also found that by adopting the temperature calibration derived by Longmore et al. (1990) on the basis of V-K colors, the RR Lyrae temperatures are in average 300 K cooler than the temperature obtained by using the V-R colors. Here we can only present this contradictory scenario as something to be explored before reaching the final goal of reading the location of RR Lyrae pulsators in the Bailey diagram in terms of the evolutionary status of cluster stars.