We present the first quantitative computations explaining the presence in the Large Magellanic Cloud of some AGB stars which share the properties of being Carbon stars ( surface C/O > 1 ) and Lithium rich .
A self–consistent description of time–dependent mixing , overshooting and nuclear burning was required .
The products of nucleosynthesis at the stellar surface turn out to be very sensitive to the temperature at the base of the outer convective envelope ( T _ { bce } ) during the quiescent phase of hydrogen burning .
Lithium production is obtained for T _ { bce } \geq 4 \cdot 10 ^ { 7 } K ( Hot Bottom Burning ) , but T _ { bce } \geq 6.5 \cdot 10 ^ { 7 } K is necessary to cycle into Nitrogen the carbon previously convected to the stellar surface by the third dredge up .
Therefore , Li–rich C stars can occurr for T _ { bce } ’ s in this small range of temperatures .
We then identify a possible -narrow- range of masses and luminosities for this peculiar evolutionary stage .
Comparison of these models with the luminosities of the few Li–rich C stars in the LMC provides an independent distance indicator ( within \sim 0.25 mag ) for the LMC .
Present data and models are consistent with ( m - M ) _ { 0 } \sim 18.7 , but a better determination would be possible by refining the observations and the theoretical models .