$s$-Processing in AGB Stars Revisited. II. Enhanced $^{13}$C Production Through MHD-Induced Mixing

Slow neutron captures are responsible for the production of about $50\%$ of elements heavier than iron, mainly, occurring during the asymptotic giant branch phase of low-mass stars ($1$ $\lesssim M$/M$_{\odot}$ $\lesssim$ $3$), where the main neutron source is the $^{13}$C($\alpha$,n)$^{16}$O reaction. This last is activated from locally-produced $^{13}$C, formed by partial mixing of hydrogen into the He-rich layers. We present here the first attempt at describing a physical mechanism for the formation of the $^{13}$C reservoir, studying the mass circulation induced by magnetic buoyancy and without adding new free parameters to those already involved in stellar modelling. Our approach represents the application, to the stellar layers relevant for $s$-processing, of recent exact, analytical 2D and 3D models for magneto-hydrodynamic processes at the base of convective envelopes in evolved stars in order to promote downflows of envelope material for mass conservation, during the occurrence of a dredge-up phenomenon. We find that the proton penetration is characterized by small concentrations, but extended over a large fractional mass of the He-layers, thus producing $^{13}$C reservoirs of several $10^{-3}$ M$_{\odot}$. The ensuing $^{13}$C-enriched zone has an almost flat profile, while only a limited production of $^{14}$N occurs. In order to verify the effects of our new findings we show how the abundances of the main $s$-component nuclei can be accounted for in solar proportions and how our large $^{13}$C-reservoir allows us to solve a few so far unexplained features in the abundance distribution of post-AGB objects.