| Abstract
| - Context. A 5-night asteroseismic observation of the F8V star HD 203608 was conducted in August 2006 with harps, followed by an analysis of the data, and a preliminary modeling of the star (Mosser et al. 2008). The stellar parameters were significantly constrained, but the behavior of one of the seismic indexes (the small spacing $\deltau_{01}$) could not be fitted with the observed one, even with the best considered models. Aims. We study the possibility of improving the agreement between models and observations by changing the physical properties of the inner parts of the star (to which $\deltau_{01}$ is sensitive). Methods. We show that, in spite of its low mass, it is possible to produce models of HD 203608 with a convective core. No such model was considered in the preliminary modeling. In practice, we obtain these models here by assuming some extra mixing at the edge of the early convective core. We optimized the model parameters using the Levenberg-Marquardt algorithm. Results. The agreement between the new best model with a convective core and the observations is much better than for the models without. All the observational parameters are fitted within 1- σ observational error bars. This is the first observational evidence of a convective core in an old and low-mass star such as HD 203608. In standard models of low-mass stars, the core withdraws shortly after the ZAMS. The survival of the core until the present age of HD 203608 provides very strong constraints on the size of the mixed zone associated to the convective core. Using overshooting as a proxy to model the processes of transport at the edge of the core, we find that to reproduce both global and seismic observations, we must have $\alpha_{{ov}}$ = 0.17 ± 0.03 Hp for HD 203608. We revisit the process of the extension of the core lifetime due to overshooting in the particular case of HD 203608.
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