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Astron. Astrophys. 334, 953-968 (1998)

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4. The solar model

4.1. The early convective core

We first computed a bunch solar evolutionary tracks with metal abundances [FORMULA] and [FORMULA], bracketing the actual value of [FORMULA] (Grevesse 1984; Grevesse & Noels 1993). We started from homogeneous main sequence and followed evolution up to an age of [FORMULA] Gyr, slightly larger than the presently accepted solar age of [FORMULA] Gyr. Pre-main sequence phases with deuterium burning were not accounted for; so we started the computations with no [FORMULA]. The initial [FORMULA] abundance [FORMULA] was adjusted for each track to fit the observed solar luminosity, and the fit of the observed radius was achieved by fine tuning of the FST parameter [FORMULA].

In all, 20 tracks were computed as detailed in Table 1. In all the [FORMULA] cases, instantaneous mixing and overshooting ([FORMULA]) were present; when [FORMULA] coupled-diffusion plus overshooting (Eq. (4)) were adopted. The [FORMULA] case can be seen as representative of both instantaneous mixing and/or coupled-diffusion, since for the small solar convective core in early main sequence both treatments give almost exactly the same results. For larger mass stars, when CNO burning in convective core dominates the structure, the case is different as we will see later on.


Table 1. Legend of the labels for the evolutionary sequences computed. For the ten choices of convection modeling and diffusion displayed with the last digit, the two choices of metallicities are displayed by the penultimate digit

Some of the main results are shown in Table 2, where also the thickness of the external convective envelope (in fractions of the solar radius) and the present surface helium abundance (when sedimentation is active) are reported. Note en passant that models with Z=0.020 and sedimentation seem to better fit the helioseismologically "observed" thickness of the convective envelope [FORMULA] (Christensen-Dalsgaard et al. 1991). Conclusions about the solar metal abundance from these data would be however premature, since any further small revision of radiative opacities, or even an overshooting from the bottom of convection by a few thousand Km (smaller than the upper limit of [FORMULA] suggested by Basu & Antia, 1997) could still change the picture.


Table 2. Thickness of the convective envelope and [FORMULA] abundance at the surface of the models computed. The second and fifth columns give the initial [FORMULA] abundance adopted to fit the present solar luminosity

For completeness, we also show in Table 3 the neutrino fluxes in SNU predicted for all the solar models computed. Our results are largely consistent with those by Bahcall & Pinsonneault (1992), both with and without helium sedimentation. Overshooting, even if slightly changing the central [FORMULA] abundances as discussed in the following, does not substantially modify the [FORMULA] fluxes, also because the more than 80% of the solar [FORMULA] is consumed via the [FORMULA] channel.


Table 3. Computed neutrino fluxes. Results are expressed in Solar Neutrino Units (SNU)

Let us now discuss the behavior of the small, early solar convective core, as depicted in Fig. 1 for the Z=0.017 case (the Z=0.020 case shows only small quantitative differences). Even if the argument is not a strict novelty, it will help to elucidate the qualitative differences between instantaneous and diffusive overshooting.

[FIGURE] Fig. 1. Evolution of the initial solar convective core versus time, for various choiches of the [FORMULA] (diffusive overshooting) or [FORMULA] (instantaneous overshooting) parameter. The larger is the extra-mixing, the longer the duration of the convective core

At the beginning of ZAMS, when Log [FORMULA], the initial [FORMULA] abundance is quite large ([FORMULA]), and transformation of [FORMULA] into [FORMULA] is responsible for the generation of [FORMULA] of the total luminosity very close to the center [FORMULA], very stiff with T). The [FORMULA] ratio (and the radiative gradient) is sufficiently large to keep alive a convective core. [FORMULA] is then consumed and convection tends to die in [FORMULA] Myr (test computations with initial [FORMULA] do not show a very early convective core).

In the meantime, [FORMULA] begins to be produced; its equilibrium concentration reaches a maximum and, when [FORMULA] is exhausted and [FORMULA] increases to allow the [FORMULA] chain to fully power the star, also central [FORMULA] decreases. Since [FORMULA], again quite stiff), a "large" abundance of [FORMULA] could keep alive a convective core too. Actually, central conditions in the Sun (and also in stars of mass up to [FORMULA]) turn out such that, when the [FORMULA] chain takes over, the central equilibrium concentration of [FORMULA] is just slightly smaller than that required to maintain central convection. At this very point the presence of overshooting can play a role, as discussed in the next section.

4.2. Instantaneous vs. diffusive overshooting

In general solar conditions, the lower is T, the larger the equilibrium abundance of [FORMULA], which is then minimum at the center of the star and increases outwards, reaching a peak abundance around [FORMULA]. Overshooting, then, mixes the core with surrounding matter overabundant in [FORMULA] with respect to the central equilibrium concentration. This leads to overproduction of luminosity, maintaining large the radiative gradient and convection alive. [FORMULA] is ineffective for this purpose since, once burned, is not produced any more. Figure 2 shows the small difference in [FORMULA] abundance profiles with and without diffusive overshooting, sufficient to make the difference between an early death and a prolonged existence of a convective core.

[FIGURE] Fig. 2. The [FORMULA] abundance profile (in units of [FORMULA]) in the central part of the star at [FORMULA] yrs, without overshooting (solid line) and with two different values of the diffusive overshooting parameter [FORMULA] (dotted line) and [FORMULA] (dashed line). The larger the mixed region, the larger is also the central [FORMULA] abundance, with prolongued life of the convective core

Let us finally discuss the different behaviors of the convective core with either instantaneous or diffusive overshooting. In the former case, the mixing boundary is plainly shifted outwards. According to the present results, instantaneous overshooting of [FORMULA] would suffice to give a steady convective core for the Sun until H -exhaustion and, in turn, to a detectable gap at the Turn-Off of old open clusters. Consistently with Maeder & Meynet (1987), we conclude that an overshooting this large must be excluded for solar mass stars.

More tricky is the case of diffusive overshooting, where mixing is a somewhat slow process which is almost complete only close to the Schwarzschild boundary, exponentially vanishing at large distances. For the Sun, diffusion with [FORMULA] is roughly similar to instantaneous mixing with [FORMULA] (Fig. 1). And yet, no strict equivalence can be established, since a detailed analysis of the numerical results shows that, with [FORMULA], partial diffusive mixing reaches farther than [FORMULA] beyond the formal convective boundary.

This test gives an hint about the profound differences between the two parameters [FORMULA] and [FORMULA]: to get almost the same amount of mixed matter, the numerical value of the latter must be lower than that of the former. Moreover, in the solar case [FORMULA]. According to Eq. (4) it is then likely that, for more massive stars with large convective cores ([FORMULA]), instantaneous overshooting with [FORMULA] should be almost equivalent to diffusive overshooting with [FORMULA] as long as the amount of mixed matter is concerned. We then conclude that:

a) with diffusive overshooting, [FORMULA] is a conservative estimate which does not modify the overall framework of solar evolution (also thanks to the "non-local" flavor arising from [FORMULA]);

b) the same value of [FORMULA] is expected to mimick, for large mass stars, an instantaneous overshooting around [FORMULA], which is a "reasonable" choice according to Maeder & Meynet (1991) and Stothers & Chin (1992),

In the next of this paper, we will then compute evolutionary tracks and isochrones not only with [FORMULA], but also with [FORMULA] and, in some cases, with [FORMULA]. Comparisons among theoretical results and to observations will then help us deciding whether or not [FORMULA] is a sensible choice.

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998