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Astron. Astrophys. 363, 675-691 (2000)

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8. Results

For different sets of modeling parameters we have computed evolutionary tracks using the convection theories of Böhm-Vitense (1958, MLTBV) and Canuto & Mazitelli (1991 , 1992, MLTCM). Using the [FORMULA] minimization described in Sect. 6, we have selected the best fits taking into account the accuracy of the observational constraints. Table 1 and Table 5 (bottom panel) present the characteristics of [FORMULA] Cen A & B and Sun models computed with the same input physics. Fig. 1 presents the evolutionary tracks in the [FORMULA] versus [FORMULA] H-R diagram. We also performed [FORMULA] minimization with the constraint [FORMULA], but, for brevity sake, we do not report the results as they do not significantly differ from those obtained without this constraint. For all [FORMULA] Cen A models a convective core, caused by an enhancement of the CNO nuclear energy generation, appears as soon as the central temperature becomes slightly larger than 17 M K and the mass fraction of hydrogen [FORMULA]. Fig. 1, Table 1 and Table 5 clearly show that various sets of modeling parameters allow to verify the observational constraints within the error bars. In absence of seismological observations there is no criterion available to discriminate between models. Table 1 and Table 5 show that, within the confidence domains, the values of the mixing length parameter of the two components are very close.

[FIGURE] Fig. 1. Evolutionary tracks in the H-R diagram of models [FORMULA], [FORMULA] (full), [FORMULA], [FORMULA] (dashed) and [FORMULA], [FORMULA] (dot-dash-dot). Dashed rectangles delimit the uncertainty domains. Top panel: full tracks from PMS. The stellar evolution sequences are initialized on the pre main-sequence soon after the deuteron ignition. The "+" denote 1 Gyr time intervals along the evolutionary tracks. Middle left and right panels: enlargements around the observed [FORMULA] Cen A, & B loci. Bottom left and right panels: loci of [FORMULA] Cen A & B models computed with the MLTBV convection theory and modeling parameters within the confidence domains presented in Table 5 ; full triangle: [FORMULA] changes only, full star: [FORMULA] changes only, empty circle: [FORMULA] changes only and full dot: [FORMULA] changes only.

8.1. MLTBV models

Models [FORMULA] & [FORMULA] have an age [FORMULA] Myr compatible with the estimate of Pourbaix et al. (1999). The values of the convection parameters are almost equal ([FORMULA]) and close to the value derived if they are forced to be identical in the [FORMULA] minimization. They are quite close to the value obtained by Noels et al. (1991) but differ notably from the estimate of Pourbaix et al. (1999) though remaining within their large confidence domains. They are smaller than the solar value [FORMULA]. In [FORMULA] a convective core is formed at time [FORMULA] Gyr a few Myr just before [FORMULA]. Fig. 1, bottom panels, shows the loci in the H-R diagram reached by the [FORMULA] & [FORMULA] models when one modeling parameter changes within its uncertainty domain, the other modeling parameters being fixed. The limited extents of the permitted solutions are consequences of limits in metallicity. Table 6 shows the partial derivatives of the observed parameters with respect to the modeling ones. Our results are similar to those of Brown et al. (1994), except for the derivatives with respect to the age which are very dependent on the exact evolutionary stage on the main sequence. As already noticed by Guenther & Demarque (2000), the luminosities are very sensitive to the helium mass fraction (see the large values of their derivatives with respect to [FORMULA] in Table 6), which in turn gives a narrow confidence level for the initial helium abundance (see Table 1).


[TABLE]

Table 6. Partial derivatives of observables [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] with respect to modeling parameters [FORMULA](Myr), [FORMULA], [FORMULA], [FORMULA] and [FORMULA] of models [FORMULA] & [FORMULA].


For both components the surface lithium depletion is not sufficient enough to fit the observed values.

Models [FORMULA] & [FORMULA] with overshooting of convective cores have an age significantly larger than models [FORMULA] & [FORMULA]. Indeed their outer convective zones penetrate deeper. For model [FORMULA] this depletion is marginally compatible with the observation, while it is not large enough for model [FORMULA]. All these differences with models [FORMULA] & [FORMULA], of course, result from the overshooting of convective cores but in a roundabout way via the [FORMULA] minimization which adjusts the modeling parameters as a whole. The convective core of model [FORMULA] is formed at time [FORMULA] Gyr.

Models [FORMULA] & [FORMULA] are calibrated according to the Guenther & Demarque (2000) observational constraints and masses. The age, initial helium mass fraction and metallicity and the convection parameters are significantly larger than those derived using the mass values of Pourbaix et al. (1999). We have obtained similar modeling parameters from the [FORMULA] minimization using models with Guenther & Demarque (2000) masses and our observational constraints. Therefore, the differences between the modeling parameters of models [FORMULA] & [FORMULA] and [FORMULA] & [FORMULA] mainly result from the mass differences, but also from the disparity of observing targets, although in a less extent. For models [FORMULA] & [FORMULA] the convection parameters are close to the solar value. In the model [FORMULA] the convective core is formed at the time [FORMULA] Gyr, larger than in other MLTBV models and the lithium depletion at the surface is marginally compatible with the observed value. The value of the central hydrogen mass fraction is half that obtained in the other MLTBV models. In model [FORMULA] the lithium depletion is compatible with the observation.

8.2. MLTCM models

In agreement with Canuto & Mazitelli (1991 , 1992) convection theory, the convection parameters of models [FORMULA] & [FORMULA] are close to unity and close to the solar value. The age is slightly smaller than the solar one but larger than that of models [FORMULA] & [FORMULA]; the outer convection zone is deeper. According to the observations, the lithium of model [FORMULA] is almost totally depleted but practically no depletion occurred in models [FORMULA] and the solar one. The convective core of model [FORMULA] is formed at time [FORMULA] Gyr.

8.3. Seismological analysis of [FORMULA] Cen A & B

The stars [FORMULA] Cen A & B are solar-like stars. The oscillations of such stars may be stochastically excited by the convection as in the case of the Sun. The amplitudes of solar-like oscillations have been estimated for stars of different masses and ages (Houdek 1996; Houdek et al. 1999). The models we have calibrated are close in the H-R diagram, but have different internal structure and they could be discriminated with the help of seismology. The properties of the stellar oscillations are related to the variation along the radius of the sound speed c and of the Brunt-Väissälä frequency N (Christensen-Dalsgaard & Berthomieu 1991). The p-mode frequencies are mainly related to c, while the g-modes, with lower frequencies, are determined essentially by N. We have computed for all our models the adiabatic frequencies of oscillation for modes of low degrees [FORMULA]=0 to 3, which may be detected by future observations. Table 8 presents the p-mode frequencies for the models [FORMULA] and [FORMULA]. As in the solar case (Provost et al. 2000), the lowest frequency p-mode oscillations of [FORMULA] Cen A have a mixed character between p- and g-mode behavior and are very sensitive to the structure of the central stellar regions.

8.3.1. p-mode oscillations

The frequencies of p-modes of given degree [FORMULA] are almost quasi-equidistant. Fig. 2 represents the variation of [FORMULA] as a function of the frequency for models [FORMULA] and [FORMULA]; n is the radial order of the mode. [FORMULA] represents the large spacing between the mode frequencies. It is roughly constant at large frequencies, larger than [FORMULA] Hz for [FORMULA] Cen A and [FORMULA] Hz for [FORMULA] Cen B. Below these frequencies it varies within 10%. In the lower frequency range, [FORMULA] has an oscillatory behavior which is the signature of the helium ionization zone (e.g. Gough 1991). We note that it is important to take into account such a variation while searching peak equidistancy in the p-mode power spectrum.

[FIGURE] Fig. 2. Variations of the large frequency separations between modes of consecutive radial order [FORMULA] for p-modes of degree [FORMULA] (full point) and [FORMULA] (open point) as a function of the frequency for [FORMULA] Cen A & B (models [FORMULA] & [FORMULA]). The relation between the frequencies and the radial order are taken from Table 8. As asymptotically predicted, [FORMULA] is almost constant at high frequency.

Another characteristic quantity of the oscillation spectrum is the small separation, i.e. the difference between the frequencies of modes with a degree of same parity and with consecutive radial order:

[EQUATION]

This small quantity is very sensitive to the structure of the core mainly to its hydrogen content. Asymptotic analysis predicts that it is proportional to [FORMULA]. Fig. 3 gives [FORMULA] and [FORMULA] as a function of the frequencies for the models [FORMULA] and [FORMULA]. For a given model, these two quantities are very close at high frequencies, as expected from asymptotic approximation and they vary almost linearly with frequency for radial orders n larger than about 16 for [FORMULA] Cen A and 10 for [FORMULA] Cen B.

[FIGURE] Fig. 3. Variations of the small frequency differences [FORMULA] and [FORMULA] as a function of the frequency for [FORMULA] Cen A & B (models [FORMULA] & [FORMULA]). Same symbols as in Fig. 2.

In the high frequency range the large and small frequency spacings which characterize the p-mode spectrum are usually estimated by analytical fits of the frequencies and of the small frequency separations. The numerical frequencies are fitted by the following polynomial expression (Berthomieu et al. 1993b):

[EQUATION]

The quantity [FORMULA] varies almost linearly with the frequency or the radial order. The small spacing is estimated by using the fit:

[EQUATION]

As in the solar case, we consider a set of 9 modes centered at [FORMULA] 21, which corresponds approximately to the middle of the range of the expected excited frequencies (i.e. [FORMULA] mHz) for [FORMULA] Cen A, according to Houdek (1996). The fitted coefficients [FORMULA] are very small. The quantity [FORMULA] does not much depend on the degree, so that [FORMULA] and it characterizes the large frequency spacing. The small spacings are conventionally measured by [FORMULA]. Table 7 shows the quantities [FORMULA], [FORMULA], [FORMULA] and [FORMULA] which have been computed for the models of [FORMULA] Cen A & B given in Table 5.


[TABLE]

Table 7. Theoretical global asymptotic characteristics of the low degree p-mode and g-mode spectrum of the star [FORMULA] Cen A & B. The quantities [FORMULA], [FORMULA] [FORMULA] and [FORMULA] ([FORMULA]), describing the p-mode oscillations, are given in µHz - see definitions Sect. 8.3.1. The characteristic g-mode period [FORMULA] is given in mn. The lower panel presents the variations [FORMULA],[FORMULA], [FORMULA] and [FORMULA] for the models computed with the [FORMULA] theory and extreme modeling parameters [FORMULA], [FORMULA], [FORMULA] and [FORMULA] within the confidence domains presented in Table 1.



[TABLE]

Table 8. Low degree frequencies for [FORMULA] Cen A and B.


These large and small separations of frequency, which characterize the p-mode oscillation spectrum, depend on the stellar mass and age and slightly decrease with the age (Christensen-Dalsgaard 1984; Audard et al. 1995). [FORMULA] and [FORMULA] are mainly related to the envelope structure of the stellar model and they vary proportionally to [FORMULA], while the values of [FORMULA] and [FORMULA] reflect the structure in the core.

The computed values of [FORMULA] are respectively smaller ([FORMULA] larger) than the solar ones due to larger ([FORMULA] smaller) masses of [FORMULA] Cen A ([FORMULA] [FORMULA] Cen B). These values do not depend much on the description of the convection or on the convective core overshoot, as can be seen from a comparison of the models [FORMULA], [FORMULA], [FORMULA] and [FORMULA], [FORMULA], [FORMULA]. On the contrary, the value of [FORMULA] obtained for the oldest model [FORMULA] of [FORMULA] Cen A, which has also a larger mass, is significantly smaller by about [FORMULA]Hz. The result is opposite for [FORMULA] Cen B. All these differences can be accounted for by the differences in mass and radius.

The small separations [FORMULA] and [FORMULA] decrease with stellar age and mass (Christensen-Dalsgaard 1984). The values of [FORMULA] and [FORMULA] obtained for our calibrated models are comparable to those of Pourbaix et al. (1999). As expected, they decrease with the age for the models without core overshoot, and thus are significantly smaller for the [FORMULA] model of [FORMULA] Cen A. Future asteroseismic observations could help to discriminate between such models.

The lower panel of Table 7 presents the variations [FORMULA], [FORMULA] and [FORMULA] for the models computed with the MLTBV theory and extreme modeling parameters [FORMULA], [FORMULA], [FORMULA] and [FORMULA] within the confidence domains presented in Table 1. The variations of [FORMULA] with [FORMULA], [FORMULA] and [FORMULA] are essentially due to the difference of model radii. The small spacings [FORMULA] and [FORMULA] depend mainly on the age and are less sensitive to the differences in radius.

According to Gough (1991), further information on the stellar structure can be provided by low degree oscillations, from the second order difference of frequencies:

[EQUATION]

When plotted as a function of the frequency (Fig. 4), this quantity has a sinusoidal behavior with two different "periods" of order [FORMULA]Hz and [FORMULA]Hz. The larger period has the largest amplitude and is due to the rapid variation of the adiabatic index [FORMULA] in the HeII ionization zone. Its contribution to the frequency is also clearly visible in Fig. 2. The lower period [FORMULA] is an indication for a discontinuity in the derivative of the sound velocity at the base of the convection zone and is the inverse of twice the travel time of the sound from the surface to the base of the convection zone (e.g. Audard & Provost 1994):

[EQUATION]

[FIGURE] Fig. 4. Variations of the differences of frequencies [FORMULA] for p-modes of degree [FORMULA] as a function of the frequency for [FORMULA] Cen A & B (models [FORMULA] & [FORMULA]). The "period" [FORMULA] of the small oscillation is [FORMULA]Hz for [FORMULA] and [FORMULA]Hz for [FORMULA]

The predicted value of [FORMULA] is larger for [FORMULA] Cen B than for [FORMULA] Cen A, mainly due to a deeper convection zone for [FORMULA] Cen B (see Table 5). The corresponding amplitude is sensitive to different processes, like convective penetration below the convection zone (e.g. Berthomieu et al. 1993a).

8.3.2. g-mode oscillations

In the low frequency range, the frequencies are mainly determined by the Brunt-Väissälä frequency N. The period of low degree gravity modes is proportional to a characteristic period:

[EQUATION]

The integral, which defines [FORMULA], is taken in the inner radiative zone, i.e. from the radius of the convective core [FORMULA] for [FORMULA] Cen A, or from the center for [FORMULA] Cen B, to the base of the external convection zone [FORMULA]. For stars without convective core, like the Sun or [FORMULA] Cen B, [FORMULA] depends on the degree of the oscillation, [FORMULA]; for stars with convective core (like [FORMULA] Cen A) [FORMULA] (Christensen-Dalsgaard 1984). Table 7 gives the values of [FORMULA] for the models of [FORMULA] Cen A & B. When the calibration of [FORMULA],Cen A is made with a core overshoot (which increases the extent of the convective core) [FORMULA] is significantly larger (by 23%). The other differences between the values of [FORMULA] for the models of [FORMULA] Cen A & B are accounted for by the differences in mass and in radius.

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Online publication: December 11, 2000
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