3 Evolution of Li provided by planetary material

3.1 Standard model framework

The standard model assumes that no exchange of material occurs between the convective envelope and the stable radiative interior of the star. Therefore, the surface chemical composition reflects only the composition of the well mixed convective envelope. In this subsection we present the evolution of lithium in the stellar surface due to the nuclear burning taking place in the convective envelope.

Assuming solar chemical composition, a planet of 1 $M_{\rm J}$ supplies $\sim$ $1.4\times 10^{44}$ atoms of ⁶Li and $\sim$ $1.7\times 10^{45}$ atoms of ⁷Li. What is the evolution of these atoms during the main sequence lifetime of the star? The nuclear time scale, $\tau_{\rm N}$, is an estimation of how long these atoms could survive to the nuclear burning in the convective envelope. $\tau_{\rm N}$, is defined as the time needed to decrease the superficial abundance of the the isotope J by a factor e, and it has been obtained from the nuclear reaction rates at each layer in the convective zone, and assuming instantaneous mixing. In Fig. 1 we plot $\tau_{\rm N}$ for $^6{\rm Li}$ and $^7{\rm Li}$ at 100 Myr as a function of effective temperature. We plot the curves corresponding to two different values of metallicity. We see that for $\log \mbox{${T}_{\rm eff}$ }\ge 3.76$ ($\sim$5700 K), a large fraction of the $^6{\rm Li}$ from the planet will survive for a long time in the convective envelope of the star. The value of $\tau_{\rm N}$ given in the plot is a minimum, since during the main sequence lifetime, the temperature at the bottom of the convective zone decreases, and so the nuclear time scale increases. The effect of increasing the metallicity is to reduce the nuclear time scale at a given $T_{\rm eff}$. A factor 2 higher metallicity implies a factor 3 lower time scale at 5500 K, a factor 2 at 5700 K and only a reduction of 10% for $T_{\rm eff}$ higher than 5800 K.

Figure 2 presents the evolution (based on models and hypothesis described in Sect. 2) of $^6{\rm Li}$ and $^7{\rm Li}$ abundance for several stellar masses, and for two values of metallicity: [Fe/H]=0.0 (left panel) and [Fe/H ]=+0.34 (right panel). That temporal dependence is obtained computing the nuclear depletion of $^6{\rm Li}$ and $^7{\rm Li}$ in the stellar convective envelope for each temporal step in the evolution and assuming instantaneous mixing. We plot only values of lithium abundances corresponding to times longer than the time of planet accretion, and lower than the dredge-up of the convective envelope. In the upper panels, we plot the curves corresponding to the $^6{\rm Li}$ that result from the accretion of a planet with $\sim$1, 5, and 10 $M_{\rm J}$ in stars with masses between 0.9 and 1.2 ${M}_{\odot }$ (for 1.25 and 1.3 ${M}_{\odot }$ the enhancement of $^6{\rm Li}$ is slightly larger because the mass of the convective envelope is smaller, and the curves are parallel to those of 1.2 ${M}_{\odot }$). This preliminary and simple modelling provides strong support for the interpretation by Israelian et al. (2001) of the $^6{\rm Li}$ detection in HD 82943. According to our models, assuming a $\mbox{${T}_{\rm eff}$ }\sim 6000$ K, age of 5 Gyr and metallicity [Fe/H $]\sim 0.32$ (Santos et al. 2000) this star has a mass of $\sim$1.2  ${M}_{\odot }$. As can be seen in Fig. 2 right panel, it should preserve any ingested $^6{\rm Li}$ during all the main sequence lifetime.

Figure 2: Evolution of 6Li (upper panels) and 7Li (lower panels) provided by dissolution of planets in the convective envelope of stars with masses from 1.2 to 0.7  ${M}_{\odot }$ (numbers indicate the mass of the star) and metallicities: [Fe/H]=0.0 (left panels) and [Fe/H]=+0.34 (right panels). $^6{\rm Li}$: for each stellar model there are three curves corresponding to (from down to top) $\sim$1, 5 and 10 $M_{\rm J}$ of planetary material dissolved. $^7{\rm Li}$: as for 6Li, but we also plot the curve corresponding to no planet accretion. The $^7{\rm Li}$ curves corresponding to: 1.0, 1.1 and 1.2  ${M}_{\odot }$ (left panel), and 1.1, 1.15 and 1.2  ${M}_{\odot }$(right panel) are concentrated in the top of the respective figures.

The $^6{\rm Li}$ abundances plotted in Fig. 2 were obtained assuming a complete dissolution of the planet in the stellar convective zone. For solar metallicity, the masses of the convective envelope in stars with 1.2, 1.1 and 1.0  ${M}_{\odot }$ are 0.006 ${M}_{\odot }$, 0.017 ${M}_{\odot }$ and 0.034 ${M}_{\odot }$, respectively. Numerical simulations by Sandquist et al. (1998) indicate that as the mass of the convective envelope decreases, the fraction of planet that is dissolved is reduced. If only 0.3 of a Jupiter-like planet is dissolved in the convective zone of a 1.2  ${M}_{\odot }$ star, the corresponding curve of $^6{\rm Li}$ will only decrease in $\sim$0.2 dex.

Figure 2 shows that there is a mass range in which $^6{\rm Li}$ could survive. In fact, for stars with masses larger than 1.1  ${M}_{\odot }$ (for $Z_{\odot}$, or 1.15  ${M}_{\odot }$ for $Z=2~Z_{\odot}$) the superficial abundance of $^6{\rm Li}$ does not change with age. The reason is that during their main sequence lifetime, the temperature at the base of the convective zone is not high enough to produce efficient burning of ⁶Li. As the stellar mass decreases, the depth of the convective zone increases, and then, even if a large quantity of planetary material is dissolved in the stellar envelope, $^6{\rm Li}$ will be depleted in a short time scale ($\sim$10⁸ yr).

The lower panels of Fig. 2 presents the predictions for ⁷Li according to standard model. Note that, opposite to ⁶Li, the $^7{\rm Li}$ isotope is depleted on the PMS only by a small factor, and that stars with masses larger than 0.9  ${M}_{\odot }$ will not deplete a significant amount during the main sequence lifetime. If these stars were able to dissolve 10 $M_{\rm J}$ in their convective envelope, the effect on the abundance would be to introduce only a small enhancement, 0.2 dex at most. The reason is shown in Eq. (1), where we see that the increase of abundance depends strongly on the depletion factor of $^7{\rm Li}$ in the convective envelope before the accretion of the planetary material.

3.2 Impact of non-standard transport processes

Observations of lithium in open clusters show that the atmospheric abundance decreases during the main sequence lifetime. In order to explain this main-sequence depletion different mixing processes are invoked (e.g. Michaud & Charbonneau 1991), and we should consider their effect on the evolution of $^6{\rm Li}$ in the stellar mass range where standard models predict complete preservation. Among these processes are: a) microscopic diffusion, that consists roughly in the sink of heavy elements with respect to the light ones due to the gravitational potential; b) turbulent diffusion which causes mixing of material between the convective zone and the deep layers where the temperature is high enough to burn lithium.

Concerning the first one, the diffusion velocities are similar for $^6{\rm Li}$ and ⁷Li, but effects of microscopic diffusion are negligible for the effective temperature and ages considered here (e.g. Richer & Michaud 1993). Turcotte et al. (1998) showed that for 1.3  ${M}_{\odot }$ star, the effect of microscopic diffusion on Li is at most a depletion by a factor 0.55, and only after 2 Gyr. In order to test the effect of turbulent mixing we have considered a diffusion coefficient produced by internal waves that fits the evolution of $^7{\rm Li}$ abundances between the Pleiades and M 67 clusters quite well (Montalbán & Schatzman 2000): $^6{\rm Li}$ would decrease only in 0.2 dex after 5 Gyr in a 1.2  ${M}_{\odot }$ star ($Z_{\odot}$). However, in a 1.1  ${M}_{\odot }$ ($Z_{\odot}$) star, $^6{\rm Li}$ will decrease 0.6 dex in the first Gyr and 1.7 dex after 5 Gyr. Of course, these results depend strongly on the behaviour of the diffusion coefficient close to the base of the convective zone. We recall that the diffusion coefficient produced by internal waves has large amplitude at the CZ boundary, and then it decreases rapidly downwards. Therefore, the results are very sensitive to the distance between the base of the convective zone and the $^6{\rm Li}$ burning layer. We have also tested a parametric turbulent diffusion coefficient $D_{\rm T}=D_0(\rho_{\rm cz}/\rho)^3$ as proposed by Proffitt & Michaud (1991) ( $\rho_{\rm cz}$ is the density at the base of convective zone). A value of $D_0 \sim 1.5\times 10^3~{\rm cm}^{2}~{\rm s}^{-1}$ produces, after 5 Gyr, a $^7{\rm Li}$ depletion of: 1.6 dex for 1  ${M}_{\odot }$ ($\sim$5800 K), 0.2 dex for 1.1  ${M}_{\odot }$ ($\sim$6000 K), and <0.1 dex for 1.2  ${M}_{\odot }$ ($\sim$6100 K) (these values are in good agreement with estimations by Pinsonneault et al. 2001 using mixing induced by rotation), and causes a $^6{\rm Li}$ depletion of 0.9 dex at $\mbox{${T}_{\rm eff}$ }\sim 6000$ K, 0.05 dex at $\mbox{${T}_{\rm eff}$ }\sim 6100$ K, and a complete depletion after 1 Gyr for $\mbox{${T}_{\rm eff}$ }\sim 5800$ K.

From observations it is inferred that the efficiency of non-standard mixing decreases with the age in the main-sequence. So what happens if the accretion of the planetary material takes place in a $^7{\rm Li}$ depleted star at a sufficently old age for extra-mixing to be inefficient? The stellar $^7{\rm Li}$ abundance could be significanly enhanced for a long period of time. For instance, if a 1 $M_{\rm J}$ planet were dissolved in our Sun's convective envelope would produce an enhancement of 0.3 dex, if 10 $M_{\rm J}$ were accreted the corresponding increase would be 0.9 dex.