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Astron. Astrophys. 318, 783-790 (1997)

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5. Comparison with the observations

For each of these models we locate the binary in the HRD and determine the mass and age of each component by interpolation. We seek for models complying both with the dynamical mass ratio and the assumption of coeval formation of the binary components. In principle, coeval formation of binary components implies both components have the same age. However, there are uncertainties in the ages derived from the theoretical models, mainly due to the fact that the duration of the mass accretion phase is not taken into account in models evolving at constant mass (Mazzitelli 1989). The observed and theoretical mass accretion rates for the early phases of PMS evolution suggest the mass accretion phase can be relatively long (of the order of some million years) and different from star to star (e.g. Mazzitelli 1989; Palla 1993 and references therein). Hence, it is reasonable to assume coeval formation if the ages derived for each binary component differ by no more than 20 [FORMULA], i.e. if the value of [FORMULA] lays approximately within 0.09. The results obtained are presented in Table 3. We see that all models are compatible with the assumption of coeval formation for a K4 primary and a K5 secondary, since the difference in age is always less than 20 [FORMULA]. However, this spectral type combination fails to reproduce the dynamical mass ratio. All models lead to mass ratios higher than the dynamical one ([FORMULA]). The M5 value is the one that stands closest to the dynamical value (1.21) while model M4 clearly leads to the worst result (1.36). On the other hand, the dynamical mass ratio is reproduced by all models for the K4-K4 and K5-K5 combination, but none of them agrees with the assumption of coeval formation of the components. The K5-K4 combination fails to reproduce the dynamical mass ratio and does not agree with the assumption of coeval formation.


Table 3. Theoretical mass and age ratios for the different models. Values correspond to the four possible locations of the binary in the HRD as listed in Table 1

Anyway these results were obtained using nominal effective temperatures and luminosities. They do not take into account the observational uncertainties. In fact, the conversion from spectral type to effective temperature depends on the scale used. However, the main contribution to the uncertainties in this quantity comes from the fact that this conversion is quantizised. It is well known that stars within a given spectral type can have different effective temperatures. There are also uncertainties in the luminosities specially due to uncertainties in the evaluation of the bolometric correction, extinction, distance and "veiling". Taking this into account and following the analysis of Lee (1992) and Hartigan et al. (1994) we may assume an uncertainty of [FORMULA] in [FORMULA] and of [FORMULA] in [FORMULA]. These error bars define for each component an uncertainty domain in the HRD (see Fig. 2). In the following we shall consider that each domain is well represented by five [FORMULA] - [FORMULA] combinations, corresponding to the four vertexes of the box limiting the domain and its center. This choice allows twenty five possible combinations of primary-secondary locations in the HRD.

[FIGURE] Fig. 2. Uncertainty domains for different primary (filled circle) - secondary (open circle) spectral type combinations: a  K4-K4, b  K4-K5, c  K5-K4, d  K5-K5. Each box, centred on the nominal values of [FORMULA] and [FORMULA], is [FORMULA] wide and [FORMULA] high. Tracks and isochrones correspond to the reference model. Isochrones are presented for [FORMULA]  (A), [FORMULA]  (B) and [FORMULA]  (C) years

Therefore, we repeated the mass-age determinations for each of these combinations and compared the results obtained with the observational data. Only model combinations that fitted the dynamical lower mass limits were retained. For these combinations we checked again for agreement with the dynamical mass ratio interval, as well as the constraint of coeval formation. The results are presented in Fig. 3. Only the points inside the dashed box fulfil the two requirements mentioned above.

[FIGURE] Fig. 3. Theoretical mass ratios versus age differences between the binary components for different primary-secondary locations in the HRD. Each plot corresponds to a different model: a  M1, b  M2, c  M3, d  M4, e  M5. Points inside the dashed boxes correspond to primary-secondary combinations having mass ratios within the dynamical limits and ages compatible with the assumption of coeval formation of the binary components

As expected, the results for models differing only in the opacities (M1, M2 and M3) are rather similar, allowing only K4-K4 and K4-K5 combinations. The same happens with model M5 computed using the CM model with [FORMULA]. All these models agree well with the observational constraints (Figs. 3a,b,c,e). On the contrary, we see that observations place severe constraints on the model computed without Coulomb corrections in the EOS (Fig. 3d). The relatively small number of points in Fig. 3d results from the fact that for this model a significant number of primary-secondary combinations violates the dynamical lower mass limits.

Lee (1992) made a similar analysis for the NTTS 162814-2427 binary using the theoretical tracks computed by VandenBerg with the MLT approximation and the theoretical tracks of Mazzitelli where the CM model is used with [FORMULA]. For [FORMULA] he found good agreement between the Mazzitelli models and the observational data. On the contrary, he found the observations led to stringent constraints on the VandenBerg models.

For the K4-K5 combination models M1, M2 and M3 lead to primary and secondary masses close to 1.10 and [FORMULA], respectively, and an age of the order of 3.7 million years. For the same spectral type combination model M5 leads to slightly higher masses: 1.15 and [FORMULA] for the primary and secondary, respectively. This results from the fact that M5 tracks are cooler than the previous ones, yielding higher masses for a given effective temperature and luminosity (see Fig. 1d). The age derived using M5 rounds up to 4.0 million years, which is very similar to the values obtained with models M1, M2 and M3. Unfortunately, Lee does not give the mass and age values for which the observational data and the theoretical model he tested do agree. However, from his results and using only nominal values of the effective temperature and luminosity for the K4-K5 combination and [FORMULA] may suggest the following values: [FORMULA], [FORMULA] and an age around 3.1 million years for the VandenBerg models, or [FORMULA], [FORMULA] and an age of the order of 4.1 million years for the Mazzitelli models.

The K4-K4 combination leads to a relatively wide range of masses and ages even within a given model. The primary and secondary masses range from 1.10 (M1) to 1.24 (M5) and 0.98 (M1) to 1.13 (M5) solar masses, respectively. The age ranges from 3 (M1) to 5 (M5) million years.

We can estimate the uncertainty in the mass and age determinations due to the observational uncertainties by looking at the spread in the masses and ages obtained for a given set. For M1, masses and ages can differ by up to a factor two and five, respectively. The results for the other sets are rather similar. This considerable mass spread is mainly due to the uncertainties in the effective temperature and the fact that the HRD tracks near the location of the binary are almost vertical. On the other hand, the age spread is caused not only by the uncertainties in the luminosity, but also by the uncertainties in the effective temperature since the isochrones are clearly not horizontal.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998