When helium fusion comes to an end at the core, convection mixes the products of the CNO cycle. The second dredge-up occurs in stars with 4–8 solar masses. As a result of the convective mixing, the outer atmosphere will display the spectral signature of hydrogen fusion: the 12C/ 13C and C/ N ratios are lowered, and the surface abundances of lithium and beryllium may be reduced. The first dredge-up occurs when a main-sequence star enters the red-giant branch. As a result, the fusion products are mixed into the outer layers of the stellar atmosphere where they can appear in the spectrum of the star. We find that dredge-up and the transformation into a carbon star occur at significantly smaller core masses (0.584 and 0.608 M solar, respectively) than in previous calculations for 3 M solar.A dredge-up is a period in the evolution of a star where a surface convection zone extends down to the layers where material has undergone nuclear fusion. During the 14th pulse, sufficient carbon has been brought to the surface to produce a carbon star. The 3-M solar model is evolved through 20 thermal pulse events, and we find third dredge-up after the seventh pulse. Strong hot-bottom burning is observed after the fifth pulse. We obtain a maximum helium luminosity of 1.7 × 10 9 L solar and significantly deep dredge-up after the second pulse. For the 5-M solar star, we evolve through 25 thermal pulses and their associated third dredge-up events. We introduce the concept of a viscous mesh in order to improve the numerical stability of the calculations. We present results of the evolution of asymptotic giant branch (AGB) stars of 3 and 5 M solar with solar metallicity calculated with the Eggleton stellar evolution code (STARS), which has a fully implicit and simultaneous method for solving for the stellar structure, convective mixing and nuclear burning.
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