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Astron. Astrophys. 318, 841-869 (1997)

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6. The convective flux as a function of [FORMULA] and [FORMULA]

Because the solar model with "overshooting" better fits the observations than that without "overshooting", it has been extrapolated that models with "overshooting" must fit also the observations of stars with temperatures, gravities, and metallicities different from the solar ones. However, if opacity, rather than convection, is the cause of the too high solar computed energy distribution, "overshooting" could make the fit worse in other stars with different absorption lines and different convection than in the Sun.

To have an idea of the importance of the convection and of the effect of the "overshooting" as a function of the model parameters, we compared, for different [FORMULA], [FORMULA], and [M/H], the [FORMULA] / [FORMULA] - log  [FORMULA] relations for convective fluxes computed with the "overshooting" option switched both on and off. For a few models we added also the relation for the only mixing-length (ML) obtained by dropping both the Lester et al. (1982)(HAO) and the "approximate overshooting" (AO) modifications (Sects. 2.2 and 2.3).

Fig. 11 shows the [FORMULA] / [FORMULA] - and the T- log  [FORMULA] relations for the Sun. The [FORMULA] / [FORMULA] -log  [FORMULA] relations from the SUNK94 model, the SUNNOVERC125 model, and a solar model without the HAO and AO modifications are compared in the upper plot. The T-log  [FORMULA] relations from the SUNK94 and SUNNOVERC125 models are compared in the lower plot. At log  [FORMULA] =0, the difference of [FORMULA] / [FORMULA] from the SUNK94 and the SUNNOVERC125 model is 0.087, namely it is just the value of [FORMULA] / [FORMULA] in the SUNK94 model, because no convective flux is predicted at log  [FORMULA] =0 when the "overshooting" option is switched off. We already discussed in the previous sections the effects of the different convection options on the emerging radiation.

[FIGURE] Fig. 11. Upper plot: the ratio [FORMULA] / [FORMULA] as a function of log  [FORMULA] in the Sun for: (a) the SUNK94 model (full line), (b) the standard ML theory without any modification (crosses), (c) the SUNNOVERC125 model (no "overshooting")(dashed line). Lower plot: the T-log  [FORMULA] relation for the (a) and (c) cases

Fig. 12 shows, on the left, the changes of the [FORMULA] / [FORMULA] - log  [FORMULA] relations as a function of [FORMULA] for [FORMULA] =4 and solar metallicity. The corresponding T-log  [FORMULA] relations are plotted on the right. The convective zone increases with decreasing [FORMULA], but it is gradually shifted towards larger depths, so that the structure of the superficial layers is less and less affected by the convection when [FORMULA] decreases, until the convection zone rises again toward the upper layers for temperatures lower than 4500K, owing to the dissociation of H2. The "overshooting" pushes a small fraction of the convective flux closer to the surface, but its effect is not linearly correlated with the amount of the convective flux, because the differences in the T-log  [FORMULA] relations for models computed with and without "overshooting" increase up to a maximum for [FORMULA] included between 7500 K and 6500 K and then decreases toward the lower temperatures, up to disappear for models with [FORMULA] [FORMULA] 4500 K.

[FIGURE] Fig. 12. On the left: The ratio [FORMULA] / [FORMULA] as a function of log  [FORMULA] for solar metallicity, [FORMULA] =4, and different [FORMULA] ranging from 8500 K to 3500 K. On the right: the T-log  [FORMULA] relations corresponding to the models on the right. The symbols are the same as in Fig. 11

Analogous plots drawn for different gravities, same [FORMULA], and same [M/H] show that the convective flux decreases with decreasing gravity, owing to the decreasing density. At a given effective temperature and gravity, the convective flux increases with decreasing metallicity owing to the increasing gas pressure and decreasing electron pressure which cause a growth of the hydrogen ionization zone.

Table 3 shows which models are affected by convection for gravities ranging from [FORMULA] =5.0 to [FORMULA] =1.0. Furthermore, for the metallicities [M/H]=0 and [M/H]=-3, it lists the models which show the largest difference, at [FORMULA] =1, between the [FORMULA] / [FORMULA] computed with the "overshooting" option switched on and off respectively. The maximum effective temperature of models affected by the different convection options decreases with decreasing [FORMULA].


[TABLE]

Table 3. The parameters of models affected by convection (columns 1 and 2) and the parameters of models which show the largest difference at log  [FORMULA] =0 between [FORMULA] / [FORMULA] computed for the "overshooting" option switched on and off respectively. Columns 1 and 3 are for [M/H]=0 and columns 1,4 are for [M/H]=-3


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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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