Theoretical periods and theoretical amplitudes, as obtained for
each given assumption about stellar mass, can be finally collected
to produce a theoretical period-amplitude diagram,
thus allowing to investigate the connection
of these two pulsational parameters with the evolutionary parameters.
Figure 12 (click here) shows the distributions into this theoretical
Bailey diagram when the stellar temperature is varied across the
instability region, for selected values of the stellar luminosity
and for the two choices 
(solid lines) and
(dashed lines), keeping everywhere Y=0.24.
The pulsational amplitudes plotted in this figure are referred to four
different luminosity levels and cover the overall instability region.
Figure 12: Theoretical Bailey diagram, bolometric
amplitude versus the logarithm of the period for a fixed helium
content (Y=0.24) and two different mass values 
(solid lines), 
(dashed lines).
Symbols concerning the luminosity levels are the same as in Fig. 10
Bearing in mind the distribution reported in Figs. 10 (click here) and 11,
the topology of data plotted in Fig. 12 (click here) can be easily understood
in terms of the relation which connects the periods with effective
temperatures, luminosities and masses.
By using these relations, for a given value of the mass and for a
fixed luminosity level, the period scales according to
(or 3.3) 
and
the distributions shown in Fig. 10 (click here) can be therefore
translated into the distributions shown in Fig. 12 (click here). Moreover,
by increasing the luminosity level the periods increase by about
.
Such an occurrence removes the ``degeneracy" of F amplitudes with
stellar luminosity and produces the period shift observed
in Fig. 12 (click here) for a fixed value of the mass.
According to these simple arguments, we finally find
that the amplitude distribution, by increasing the mass, moves toward
shorter periods, thus giving full account of the Bailey diagram
morphology disclosed in Fig. 12 (click here).
Adopting such a pulsational scenario, it is obviously interesting to match pulsational constraints with the predictions of the evolutionary theory. For a quick look on this new scenario let us first discuss the distribution of HB stars in the Bailey diagram by assuming that all stars are just in their ZAHB location. Under this assumption, for each given value of the metallicity we derived the set of stellar masses populating the instability strip by interpolating the evolutionary tracks given by CCP, each mass being characterized by proper values of both effective temperature and luminosity. The regular behavior of F pulsators allows a straightforward linear interpolation of the dependence of the amplitude on stellar mass, temperature and luminosity. On this basis evolutionary data can be easily transformed into the Bailey diagram for F pulsators as given in Fig. 13 (click here) for the three different choices of metal abundance Z=0.0001, Z=0.0004 and Z=0.001.
The procedure previously outlined supplies a relevant result that can be summarized as follows:
the difference of the evolutionary structure of the pulsators predicted on the basis of different assumptions about the stellar metallicity plays a minor role in the distribution of the pulsators in the Bailey diagram.
As suggested by Brocato et al. (1996) on a pure observational basis, this is due to the fact that moving from metal poor OoII to metal rich OoI pulsators the decrease in mass is largely balanced by the decrease in luminosity as can be seen in Fig. 8 (click here).
Unfortunately, the distribution of FO pulsators appears much less predictable. Due to the rather intricate amplitude behaviors disclosed in Figs. 10 (click here) and 12, interpolation over the whole range of theoretical data appears not very useful, until a much finer set of nonlinear luminosity amplitudes becomes available. However, the regular behavior of the decreasing branch allows again a linear interpolation of the dependence of such a feature on stellar parameters. On this basis the distribution of FO pulsators could be predicted.
We eventually underline that the distribution of F pulsators in the Bailey diagram, even when taking into account the off ZAHB evolution, would be affected only marginally, since these variables are characterized by a well defined amplitude-temperature relation which presents a negligible dependence on both stellar mass and luminosity level (see Figs. 9 (click here)-11). The conclusions previously drawn for F pulsators cannot be extended to FO pulsators due to the strong dependency of their pulsational amplitudes on luminosity level. As a consequence the distribution of these variables in the Bailey diagram, with the exception of pulsators located along the decreasing branch, might be partially changed by the off ZAHB evolution.
Figure 13: The distribution in the theoretical Bailey diagram
of ZAHB fundamental and first overtone pulsators as expected
according to evolutionary
prescriptions for the labeled assumptions about the stellar metallicity.
The distribution of the decreasing branch of first overtone pulsators
is also sketched. See text for further explanations