A comparative study of three modes of s-process nucleosynthesis in extremely metal-poor AGB stars

Carbon-enhanced metal-poor (CEMP) stars in the Galactic halo have a wide range of neutron-capture element abundance patterns. To identify their origin, we investigated three modes of $s$-process nucleosynthesis that have been proposed to operate in extremely metal-poor (EMP) Asymptotic Giant Branch (AGB) stars: the convective 13C burning, which occurs when hydrogen is engulfed by the helium flash convection in low-mass AGB stars, the convective 22Ne burning, which occurs in the helium flash convection of intermediate-mass AGB stars, and the radiative 13C burning, which occurs in the $^{13}$C pocket that is formed during the inter-pulse periods. We show that the production of $s$-process elements per iron seed ($s$-process efficiency) does not depend on metallicity below $[{\rm Fe}/{\rm H}]=-2$, because 16O in the helium zone dominates the neutron poison. The convective 13C mode can produce a variety of $s$-process efficiencies for Sr, Ba and Pb, including the maxima observed among CEMP stars. The 22Ne mode only produce the lowest end of $s$-process efficiencies among CEMP models. We show that the combination of these two modes can explain the full range of observed enrichment of $s$-process elements in CEMP stars. In contrast, the 13C pocket mode can hardly explain the high level of enrichment observed in some CEMP stars, even if considering star-to-star variations of the mass of the 13C pocket. These results provide a basis for discussing the binary mass transfer origin of CEMP stars and their subgroups.