The stellar models have been computed with the updated version of the ATON2.0 code (Ventura et al. 1998). We constructed models with masses ranging from 0.6 to 1.3 ${M}_{\odot }$ with a step of $\Delta M=0.05$ ${M}_{\odot }$ , and followed the evolution from the pre-main sequence to 7 Gyr. These models use a mixing length theory (MLT) description of convective transport with a parameter of mixing length $\alpha=1.6$ , as given by the solar calibration (Ventura et al. 1998; Montesinos et al. 2001). Two different chemical compositions were adopted: a) initial solar helium abundance of $Y_{\odot}=0.28$ and heavy-elements mass fraction of $Z_{\odot}=0.02$ ([Fe/H]=0.0) (Montesinos et al. 2001), b) Z=0.04, with $\Delta Y/ \Delta Z = 2$ , that is, [Fe/H ]=+0.34.

The code used in our model computations allows to include gravitational settling of helium and thermal diffusion, but it does not consider the diffusion of metals. Diffusion generally produces a reduction of the convection zone, however, Turcotte et al. (1998) showed that the convective envelopes in models with metal diffusion are deeper than in models including only helium settling. Furthermore, their results indicate (see their Fig. 4) that for the higher masses considered in this paper ( $M\leq 1.3 ~{M}_{\odot}$ ) the mass of the convective envelope of a non-diffusion model is actually closer to the full-diffusion one than to a model incorporating He settling only (in fact, the non-diffusion and full-diffusion curves overlap). Because of that, we decided to use stellar structures obtained without including diffusion. Notice also that, a proper treatment of diffusion (Turcotte et al. 1998) has no significant impact on the evolution of lithium abundance in stars with $M\leq 1.3 ~{M}_{\odot}$ . The nuclear burning rates of $^6{\rm Li}$ and $^7{\rm Li}$ in the star's convective envelope were computed for each temporal step in evolution using NACRE nuclear reaction tables (Angulo et al. 1999).

Masses for the engulfed sub-stellar companions ranged, from 1 $M_{\rm J}$ to 40 $M_{\rm J}$ . This upper mass limit was adopted to ensure the complete preservation of $^6{\rm Li}$ (Pozio 1991; Nelson et al. 1993), but, in practice, the maximum mass of the accreted planet was constrained to be less than the mass in the convective zone of the star under consideration.

The hypothesis and approximations adopted in our study are: i) planet accretion takes place just after the star arrives on the main sequence. The analysis presented here does not change if the planet is engulfed at any other moment during the main sequence lifetime. Such case can be dealt by taking into account that the temperature, mass of the convective envelope and the stellar surface lithium abundance will be different since they decrease as age increases. ii) The planet is completely dissolved in the stellar convective envelope. As shown by numerical simulations (Sandquist et al. 1998), the reliability of this assumption depends on the mass of the planet ( $M_{\rm Pl}$ ) and on the mass of the convective envelope $M_{\rm CZ}$ , and also strongly depends on the details of the internal structure of the planet. The smaller the $M_{\rm CZ}$ , the lower the fraction of planet dissolved in the convective zone. That means that, at least for the higher planet masses under consideration, the quantity of lithium provided by the planet must be reduced in a factor equal to the fraction of the planet consumed in the star's convective envelope. iii) The engulfment of a planet has no significant effect in the structure of the star. iv) The planet is instantly mixed in the convective zone of the star. Given that the diffusion coefficient in the convective zone is of the order of $10^{13}~{\rm cm}^2~{\rm s}^{-1}$ , the mixing time is lower than the temporal step in the evolution computation.

The stellar models were followed from the PMS, but we will not use the surface abundance of $^6{\rm Li}$ and $^7{\rm Li}$ given by the models because it is well known that they overestimate PMS lithium destruction (D'Antona & Mazzitelli 1997). For masses lower than 1.4 ${M}_{\odot }$ it is expected that no $^6{\rm Li}$ exists after the PMS (Proffitt & Michaud 1989). As regards surface $^7{\rm Li}$ abundance we adopted the values observed in stars of the Pleiades Cluster (Soderblom et al. 1993). So the stellar surface abundances of lithium isotopes at the moment of the engulfment of the planet are: $N(^6{\rm Li})/N(H)=0$ and $N(^7{\rm Li})/N(H)=F(T_{\rm eff})_{\rm Pleiades}$ . These abundances in the planets are assumed equal to the meteoritic values. That is: $N(^7{\rm Li})_0/N(H)=2\times 10^{-9}$ , and $N(^7{\rm Li})_0/N(^6{\rm Li})_0=12.$

Following Burrows & Sharp (1999) we assume an Anders & Grevesse (1989) solar chemical composition for both the planet and the star. In this case, the difference between the abundance of the isotope J in the star's convective zone before (s) and after (s+p) the dilution of the planet can be written as:

$\begin{displaymath}\epsilon (J)_{\rm s+p} - \epsilon(J)_{\rm s} = \log \left(\frac {x}{x+1}\cdot(1-f)\right) - \log f. \end{displaymath}$

(1)

$\begin{figure} \par\includegraphics[width=6.7cm,clip]{H3476F1.eps}\end{figure}$

Figure 1: $^6{\rm Li}$ (lower curves) and $^7{\rm Li}$ (upper curves) lifetime against $(p,\alpha )$ nuclear reactions (N(Li)/N(Li $)_0=\exp(-t/\tau_{\rm N})$ ) as a function of effective temperature. The curves correspond to standard models at 10⁸ yr with two different metallicities: [Fe/H]=0.0 (solid lines) and [Fe/H ]=+0.34 (dashed lines). The numbers close to the symbols (empty circles for solar metallicity and full circles for [Fe/H ]=+0.34) indicate the mass of the stellar model.

2 Models