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Astron. Astrophys. 354, 610-620 (2000)

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2. Predicted relations

The nonlinear convective pulsating models adopted in this paper are computed with four values of the stellar mass ([FORMULA]5, 7, 9, 11) and three chemical compositions (Y=0.25, Z=0.004; Y=0.25, Z=0.008; Y=0.28, Z=0.02), taken as representative of Cepheids in the Magellanic Clouds and in the Galaxy. The basic assumptions on the input physics, computing procedures and the adopted mass-luminosity relation have been already discussed in Bono et al. (1998), Bono et al. (1999a [Paper I]) and Paper III and will not be repeated. Here we wish only to remark that the whole instability strip moves toward lower effective temperatures as the metallicity increases and, as a consequence, that the bolometric magnitude of the pulsators increases as the metal abundance increases (see Fig. 2 in Paper II).

2.1. Period-luminosity

From the bolometric light curves of our pulsating models and adopting the grid of atmosphere models provided by Castelli et al. (1997a,b), we first derive the predicted intensity-weighted mean magnitude [FORMULA] of pulsators for the BVRIJK passbands  2. The fundamental pulsators plotted in Fig. 1, where the different lines depict the blue and red limits of the pulsation region at the various compositions, show that the metal abundance effects on the predicted location and width of the instability strip are significantly dependent on the photometric passband. In particular, the models confirm early results (e.g., see Laney & Stobie 1994; Tanvir 1999 and references therein) that infrared magnitudes are needed to reduce both the intrinsic scatter and the metal abundance dependence of the PL relation.

[FIGURE] Fig. 1. Location in the [FORMULA] and [FORMULA] plane of fundamental pulsators with different chemical compositions and masses (from left to right: 5, 7, 9 and 11[FORMULA]). The lines depict the predicted edges of the instability strip.

This matter has been already discussed in Paper II on the basis of the predicted [FORMULA] and [FORMULA] relations derived under the assumption of a uniformly populated instability strip. However, since the PL is a "statistical" relation which provides the average of the Cepheid magnitudes [FORMULA] at a given period, we decide to estimate the effect of a different pulsator population. Thus, following the procedure outlined by Kennicutt et al. (1998), for each given chemical composition we populate the instability strip 3 with 1000 pulsators and a mass distribution as given by [FORMULA] over the mass range 5-11[FORMULA]. The corresponding luminosities are obtained from the mass-luminosity relation derived from canonical evolutionary models, i.e. with vanishing efficiency of convective core overshooting (see Bono & Marconi 1997; Paper II; Paper III).

Fig. 2 shows the resulting [FORMULA] distribution of fundamental pulsators with the three selected metallicities. We derive that the pulsator distribution becomes more and more linear by going toward the infrared and that, aiming at reducing the intrinsic scatter of PL, the pulsator distributions in the [FORMULA], [FORMULA] and [FORMULA] planes are much better represented by a quadratic relation. The dashed lines in Fig. 2 show the quadratic least square fit ([FORMULA]), while the solid lines refer to the linear approximation ([FORMULA]). The values for the coefficients a, b and c of each [FORMULA] relation are listed in Table 1 (quadratic solution) and Table 2 (linear solution), together with the rms dispersion ([FORMULA]) of [FORMULA] about the fit. However, we wish to remark that the present solutions refer to a specific pulsator distribution and that different populations may modify the results. As a matter of example, if the longer periods (log[FORMULA]1.5) are rejected in the final fit, then the predicted linear PL relations become steeper and the intrinsic dispersion in the BVR bands is reduced (see Table 3).

[FIGURE] Fig. 2. Period-magnitude distribution of fundamental pulsators at varying metallicity and photometric bandpass. Dashed lines and solid lines refer to the quadratic and linear PL relations given in Table 1 and Table 2, respectively.


Table 1. Theoretical PL relations for fundamental pulsators. Quadratic solutions: [FORMULA].


Table 2. Theoretical PL relations for fundamental pulsators. Linear solutions: [FORMULA].


Table 3. Theoretical PL relations for fundamental pulsators with [FORMULA]. Linear solutions: [FORMULA].

As a whole, each predicted [FORMULA] relation seems to become steeper at the lower metal abundances, with the amplitude of the effect decreasing from B to K filter. Moreover, the slope and the intrinsic dispersion of the predicted PL relation at a given Z decrease as the filter wavelength increases (see Fig. 3 for the linear relations), in close agreement with the observed trend (e.g. see Fig. 6 in Madore & Freedman 1991).

[FIGURE] Fig. 3. Slope and intrinsic dispersion of the predicted linear PL relations with different bandpass.

In closing this section, let us observe that the adopted way to populate the instability strip allows PL relations only for the "static" magnitudes (the value the star would have were it not pulsating), while the observations deal with the mean magnitudes averaged over the pulsation cycle [([FORMULA]) if magnitude-weighted and [FORMULA] if intensity-weighted]. It has been recently shown (Caputo et al. 1999 [Paper IV]) that the differences among static and mean magnitudes and between magnitude-weighted and intensity-weighted averages are always smaller than the intrinsic scatter of the PL relation ([FORMULA] mag for V magnitudes and [FORMULA] mag for K magnitudes). Here, given the marginal agreement of the coefficients in Table 1, Table 2 and Table 3 with the values given in Paper IV, we conclude that the additional effect on the intrinsic scatter of the PL relations, as due to different Cepheid populations, is negligible, provided that a statistically significant sample of variables is taken into account.

2.2. Period-color and color-color

The three methods of deriving the mean color over the pulsation cycle use either ([FORMULA]), the average over the color curve taken in magnitude units, or [FORMULA], the mean intensity over the color curve transformed into magnitude, or [FORMULA], the difference of the mean intensities transformed into magnitude, performed separately over the two bands. In Paper IV it has been shown that there are some significant differences between static and synthetic mean colors, and that the predicted ([FORMULA]) colors are generally redder than [FORMULA] colors, with the difference depending on the shape of the light curves, in close agreement with observed colors for Galactic Cepheids. As a matter of example, the predicted difference [FORMULA] ranges from 0.02 mag to 0.08 mag, whereas [FORMULA] is in the range of 0.014 mag to 0.060 mag.

Fig. 4 shows the pulsator synthetic mean colors as a function of the period for the three different metallicities. Since, given the finite width of the instability strip, also the PC relation is a "statistical" relation between period and the average of the mean color indices CI, we follow the procedure of Sect. 2.1. It is evident from Fig. 4 that the pulsator distribution shows a quadratic behavior, as well as that the intrinsic dispersion of PC relations allows reddening estimates within [FORMULA] mag, on average. However, as far as they could help, we present the predicted relations [FORMULA]logP at the various compositions. The relations are plotted in Fig. 4 and the A and B coefficients are listed in Table 4 together with the rms deviation of the color about the fit ([FORMULA]). Note that also the slope of the predicted PC relations depends on the metal abundance, increasing with increasing Z.

[FIGURE] Fig. 4. Location in the period-color plane of fundamental pulsators with different chemical compositions and masses. The lines depict the linear PC relations given in Table 4.


Table 4. Theoretical Period-Color relations.

As for the color-color (CC) relations, Fig. 5 shows that the spurious effect due to the finite width of the instability strip is almost removed. For this reason, from our pulsating models we derive a set of color-color (CC) relations correlating [FORMULA] with [FORMULA], [FORMULA], [FORMULA] and [FORMULA], but similar relations adopting ([FORMULA]) or [FORMULA] colors can be obtained upon request. The predicted CC linear relations are given in Table 5, while Fig. 5 shows the pulsator distribution in the color-color plane. One may notice that the intrinsic scatter of the theoretical relations is very small, with the rms dispersion of [FORMULA] about the fit smaller than 0.01 mag.

[FIGURE] Fig. 5. The color-color plane of fundamental pulsators. The lines depict the CC relations at varying metallicities (see Table 5).


Table 5. Theoretical Color-Color relations.

2.3. Period-luminosity-color

Since the pioneering paper by Sandage (1958), Sandage & Gratton (1963) and Sandage & Tammann (1968), it is well known that if the Cepheid magnitude is given as a function of the pulsator period and color, i.e. if the Period-Luminosity-Color relation is considered, then the tight correlation among the parameters of individual Cepheids is reproduced (see also Laney & Stobie 1986; Madore & Freedman 1991; Feast 1995 and references therein).

In Paper II it has been shown that the intrinsic scatter of the PLC relations in the visual ([FORMULA], [FORMULA]) and infrared ([FORMULA], [FORMULA]) is [FORMULA] mag. Furthermore, we showed in Paper IV that the systematic effect on the predicted visual PLC relations, as due to the adopted method of averaging magnitudes and colors over the pulsation cycle, is always larger than the intrinsic scatter of the relation. For this reason in this paper we present only the predicted PLC relations [FORMULA], but similar relations for magnitude-weighted values can be obtained upon request.

From the least square solutions through the fundamental models we derive the coefficients [FORMULA], [FORMULA] and [FORMULA] presented in Table 6, together with the residual dispersion ([FORMULA]) of [FORMULA] about the fit. Figs. 6-8 illustrate the remarkably small scatter of the PLC relations (see Fig. 1 for comparison). Moreover, one may notice that adopting [FORMULA] color, the metal-rich Cepheids are brighter than metal-poor ones with the same period and color (see lower panel of Fig. 6), whereas the opposite trend holds with [FORMULA] color (see Fig. 8). As a "natural" consequence, the predicted relationship with [FORMULA] color (lower panel of Fig. 7) turns out to be almost independent of the metal abundance.

[FIGURE] Fig. 6. Projection onto a plane of the predicted PLC relations ([FORMULA] and [FORMULA] colors) for fundamental pulsators with different metallicities (see Table 6).

[FIGURE] Fig. 7. As in Fig. 6, but with [FORMULA] and [FORMULA] colors.

[FIGURE] Fig. 8. As in Fig. 6, but with [FORMULA] colors.


Table 6. Theoretical PLC relations.

In order to complete the theoretical framework for classical Cepheids, we have finally considered the Wesenheit quantities W (Madore 1982) which are often used to get a reddening-free formulation of the PL relation. With [FORMULA] giving the absorption in the [FORMULA]-passband, one has


Table 7 gives the coefficients of the theoretical reddening-free PL [hereinafter WPL] relations derived from our fundamental models. Note that present results adopt [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] from the Cardelli et al. (1989) extinction model, but formulations using different ratios of total to selective absorption can be obtained upon request.


Table 7. Theoretical Wesenheit relations.

Figs. 9-10 show the theoretical Wesenheit quantities as a function of the period, together with the predicted relations. From a comparison of Table 7 with Table 2 one derives that the intrinsic scatter of [FORMULA] and [FORMULA] is significantly lower than the dispersion of PL, while no significant improvement occurs with [FORMULA] and somehow larger dispersions are found for [FORMULA] and [FORMULA].

[FIGURE] Fig. 9. Predicted WPL relations of fundamental pulsators with different chemical compositions (see Table 7).

[FIGURE] Fig. 10. Predicted WPL relations of fundamental pulsators with different chemical compositions (see Table 7).

Moreover, the comparison between Figs. 9-10 and Figs. 6-8 discloses the deep difference between PLC and WPL relations (see also Madore & Freedman 1991). The former ones are able to define accurately the properties of individual Cepheids within the instability strip, whereas the latter ones are thought to cancel the reddening effect. As a consequence, provided that the variables are at the same distance and have the same metal abundance, the scatter in observed PLC relations should depend on errors in the adopted reddening, whereas the scatter in observed WPL relations is a residual effect of the finite width of the strip.

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© European Southern Observatory (ESO) 2000

Online publication: February 9, 2000