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Astron. Astrophys. 326, 620-628 (1997)

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7. Luminosity, mass and age

The diagram plotted in Fig. 6 shows the luminosity [FORMULA] mean stellar density for models of M = 9, 10, 11 and 12 [FORMULA] calculated by Dziembowski & Pamyatnykh (1993) using OPAL opacities with Z = 0.02. Stellar models corresponding to the parameters (4.33 [FORMULA]   log [FORMULA]   4.37 and 3.58 [FORMULA]   log [FORMULA] 3.88) derived from spectro-photometric observations of [FORMULA]  Cet (cf. Sect. 5) are indicated by dots. Knowledge of the mode of oscillation ( [FORMULA] ) leads to a mean stellar density equal to [FORMULA]. Because the oscillation period P is known with high precision, the accuracy of [FORMULA] is determined by the error in the nondimensional frequency [FORMULA], which increases slightly during the envelope expansion phase of M-S stellar evolution, cf. Dziembowski & Pamyatnykh (1993). In the range of log [FORMULA] and log g considered, [FORMULA] This restricts the possible solutions to the narrow region around the vertical line shown in Fig. 6. The condition that only unstable models for the first overtone of the radial mode should be considered leads to 4.043 [FORMULA]   log [FORMULA] 4.164 for [FORMULA]  Cet. Both [FORMULA] and log L are almost insensitive to moderate changes in chemical composition of stars, due to the fact that the effect of the increase in mean molecular weight (accompanied by decrease of the stellar radius) is well compensated by the effect of the decrease in M. For instance, the increase in Z from 0.02 to 0.03 is compensated by the decrease in mass by about 6 per cent for M-S stellar models with X = 0.70. Similar behaviour is seen when X changes.

[FIGURE] Fig. 6. The diagram log [FORMULA] vs. [FORMULA]   for stellar models of [FORMULA] calculated by Dziembowski &  Pamyatnykh (1993). Models with log [FORMULA] and log [FORMULA] are shown as dots. The vertical line indicates possible solutions for [FORMULA]  Cet oscillating in ( [FORMULA] ) mode.

Another uncertainty in stellar evolution theory is connected with treatment of the convective overshooting phenomenon. Models plotted in Fig. 6 were calculated with no convective overshooting and no mass loss during evolution. As pointed out by Dziembowski & Pamyatnykh (1993), such models are not in conflict with the positions of [FORMULA]  Cep stars on the H-R diagram. There are however motivations for considering these effects on stellar evolution, cf. e.g., Claret & Gimenez (1992), Schaller et al. (1992) and Bressan et al. (1993), where results for a moderate convective overshooting calculated using the OPAL opacities are given. Mass loss by radiatively-driven stellar wind was included in these calculations. One can estimate from these data that, for stellar parameters corresponding to [FORMULA]  Cet, the convective overshooting effect results in higher luminosity by about 0.05 dex., which again can be compensated by a decrease in mass of about 2 per cent. The effect of mass loss is about one order of magnitude smaller. The mass of [FORMULA]  Cet is estimated as [FORMULA], and the age is probably in the range from log(Age) = 7.225 to 7.38 y, as indicated by Dziembowski & Pamyatnykh's (1993) and Schaller's et al. (1992) models, respectively.

The convective overshoot has little effect on the nondimensional frequencies of the radial modes, cf. Dziembowski & Pamyatnykh (1991, 1993). In this case, a detailed analysis of stars located very near the low-temperature boundary of the M-S region may shed some light on this question. On the other hand, Dziembowski & Pamyatnykh (1991) showed that g -modes which enter the p -mode frequency range during stellar evolution become partially trapped in the region containing the outer part of the convective core and the chemically inhomogeneous zone left behind by the shrinking core. Frequencies of such modes are sensitive to the treatment of the convective core boundary. Thus, the detection of the [FORMULA] -mode (cf. Dziembowski & Pamyatnykh 1993) in [FORMULA]  Cep stars would give an important tool for testing stellar theory.

We add that the effect of changing the chemical composition and stellar opacity data, as well as convective overshooting effect, cannot be exactly compensated for by a change of mass. The nonadiabatic parameters [FORMULA] and [FORMULA] change rapidly with [FORMULA], but they are also sensitive to log g, Z and stellar opacity sources, cf. Cugier et al. (1994). These quantities are closely linked to the driving mechanism of stellar pulsations and together with the nonstability criterion are very useful for testing the stellar models. All these circumstances lead us to the conclusion (cf. Sect. 2) that stellar models calculated with OPAL opacities for the initial chemical composition X = 0.70, Y = 0.28 and Z = 0.02 fit best the observations of [FORMULA]  Cet. Having established chemical composition and opacity data, nonadiabatic observables offer determination of the effective temperature and luminosity with high precision. Fig. 2 shows that [FORMULA] and [FORMULA] transformed to observed parameters lead to a shift of the maximum light by about [FORMULA]  rad for [FORMULA] dex, whereas the observed phase lag between light and radial velocity curves seems to be known with an accuracy of the order of [FORMULA] rad, cf. Jerzykiewicz et al. (1988). In the case of log [FORMULA], one can therefore obtain log [FORMULA] for [FORMULA]  Cet. However, further tests performed for other [FORMULA]  Cep stars are needed to establish the final calibration of the stellar opacity data.

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© European Southern Observatory (ESO) 1997

Online publication: October 15, 1997