Figures 6 (click here) and 7 (click here) show the distribution of periods for
all computed models with mass M=0.65 and 
which reach
a stable limit cycle either in the F or in the FO mode. In the same
figures we report the linear predictions given by van Albada &
Baker (1971) according to the relations:
where both mass and luminosity are in solar units, whereas the periods are in days.
Figure 6: Fundamental periods versus effective temperatures
for two different stellar masses 
=0.65 (open circles)
and 
=0.75 (solid circles). Solid lines refer
to the nonlinear results of the present computations, whereas dashed
lines show
the linear periods evaluated by using the relations provided by
van Albada & Baker (1971). The luminosity
level of each sequence is also reported at the right of the curves
Figure 7: Same as Fig. 6, but for first overtone pulsators
As expected, nonlinear periods are generally smaller than
linear periods, since convective motions smooth the density
profile
in coincidence of the hydrogen ionization region where the density
inversion takes place, and in the mean time produce a non negligible
increase of the density in the region where the adiabatic exponent
-
- reaches its minimum value. The aftermaths of this effect
together with the nonlinear effects on 
cause the decrease
of the nonlinear periods when compared to linear ones.
The previous leading-term
considerations also provide an explanation of the
physical reason why
this discrepancy tends to increase moving from the blue to the
red side of the instability strip (see Figs. 11 (click here) and 12 in BS).
The new models show that the differences are larger for FO than for F pulsators, whereas BS found that the discrepancy between linear and nonlinear periods was larger for F pulsators. Since the present nonlinear models differ from the models adopted in BS only in the helium abundance, this finding strongly suggests that the FO periods present a non negligible dependence on this parameter.
Linear interpolation among the data plotted in both figures gives the
new relations:
with a correlation coefficient r=1.00.
The differences between the previous relations and the van Albada and Baker
relations -though not negligible- are not expected to produce a
substantial change in the available pulsational scenario,
supporting once more the
pioneering work by van Albada and Baker. However, previous
results give at least a firm warning against the long-standing
attempt to derive the mass of double-mode pulsators by using the
ratio between F and FO periods (
).
Indeed, this quantity
critically depends on both stellar parameters and on even minor
details of the physical and numerical ingredients adopted in the
pulsational codes.
The occurrence and the properties of RR Lyrae models which show
stable double-mode limit cycles will be discussed in a forthcoming paper.
The previous relations, which provide the nonlinear periods for the first two pulsating modes, can be combined with the instability strip topologies given by BCM to derive the behavior of the minimum period allowed in a cluster, i.e. the minimum (fundamentalized) period of the FO pulsator at the blue edge of its instability region:
which appears in good agreement with the preliminary evaluation obtained by Bono et al. (1995). As discussed in the quoted paper the minimum periods provided by the previous relation reproduce quite satisfactorily the minimum periods of variable stars belonging to GGCs, thus supporting once more the canonical evolutionary scenario against the suggested occurrence of an anomalous period shift.
As a further step in our investigation, let us take now into account the canonical evolutionary constraints given by CCP. Figure 8 (click here) shows the expected location of HB evolutionary tracks for selected values of the cluster metallicity. This parameter was chosen such that it should become representative of well known clusters which populate the galactic halo and present a large number of RR Lyrae variables, namely Z=0.0001 (M 15), Z=0.0004 (M 3) and Z=0.001 (M 5).
In the same figure we plot the instability strip
for the two labeled assumptions about the mass of the star, giving,
for each
assumed metallicity,
the maximum mass value (
) allowed by evolutionary models
for populating the HB of a cluster characterized by an age of 15 Gyr
and the expected minimum period of RR
pulsators (see below).
It appears that the instability strip in M 15 should be populated
by stars with mass values included between 0.75 and 
.
More massive stars would be forbidden for the adopted cluster age,
thus giving a straightforward explanation of the blue HB shown by
this cluster.
On the contrary, the instability regions of M 3 and M 5 should be
populated by stars with mass values which range from 0.65 to 
respectively.
These values imply now an amount of mass loss
of the order of 
or 
for M 3 and
M 5 respectively.
Figure 8: The ZAHB location into the
HR diagram for three different values of the cluster
metallicity. In each plot the solid lines show the evolutionary
paths of selected helium burning models for the labeled values of the
masses. Dashed lines and dotted lines report
the instability strip for two different mass values
and 
respectively. For each
given metallicity we also report the name of the typical cluster,
the observed minimum period of fundamental pulsators in that cluster
(log 
), and the maximum mass allowed for populating
the HB (
). See text for further explanations
As a most relevant point, we find
that this evolutionary picture can be easily connected with the
observed minimum period of 
pulsators, i.e. with the
observational evidence at the basis of the Oosterhoff dichotomy.
The first suggestion about the occurrence of the Oosterhoff
dichotomy was produced by the observational evidence that the mean
periods of F pulsators in GGCs, 
, were grouped
around two distinct values, namely 0.65 d (OoII) and 0.55 d (OoI),
with a lack of clusters with 
.
As it was early recognized, such an occurrence is due to the fact
that when compared to OoII clusters, OoI clusters allow the
occurrence of F pulsators characterized by shorter periods.
Indeed, the minimum F period changes from
in OoII clusters
to 
in OoI clusters.
This interesting observational behavior has stimulated an animated debate during the past few years. According to Sandage (1958, 1981, 1982, 1990) the difference in periods should mainly be ascribed to a difference in luminosity: OoII pulsators are more luminous than OoI ones.
In order to explain the F mean period distribution, van Albada
& Baker (1973) suggested an alternative hypothesis:
a difference in the maximum temperature of F pulsators produced
by a hysteresis mechanism. In this context a variable star located
in the OR region will pulsate in the
F mode if it is evolving from lower to higher effective
temperatures, whereas it will pulsate in the FO if
the star is evolving in the opposite direction.
As a consequence of this mechanism, we expect that the
transition between 
and 
occurs either at the
F blue edge or at the FO red edge.
In the already quoted Fig. 8 (click here), the point where the transition is
expected to take place according to the hysteresis hypothesis
is marked with an open circle.
Accordingly one expects that the OR region is
populated by FO pulsators in OoII clusters,
and by F pulsators in the OoI case. According to such an
hypothesis, the
data listed in Table 2 (click here) disclose that the theoretical results
concerning 
overlap surprisingly well with the
observational data when both evolutionary prescriptions
and pulsational results are taken into account.
If one adds the evidence given in BCM for a HR distribution in agreement with the hysteresis prescriptions, we can conclude that the canonical evolutionary scenario appears able to account for several relevant features of RR Lyrae pulsators, without the need of invoking ``ad hoc'' modifications of the current evolutionary knowledge.
Table 2: Theoretically predicted minima of fundamental periods
as a function of cluster metallicity
As a final point, let us note that the data in Table 2 (click here) give a spontaneous indication for the shift in periods within the OoI type clusters (Castellani & Quarta 1987), with F pulsators in M 5 reaching shorter periods with respect to the same type of pulsators in M 3.