Cool Bottom Processes on the Thermally Pulsing Asymptotic Giant Branch and the Isotopic Composition of Circumstellar Dust Grains

We examine the effects of cool bottom processing (CBP) on the isotopic ratios ¹⁸O/¹⁶O, ¹⁷O/¹⁶O, ¹⁴N/¹⁵N, ²⁶Al/²⁷Al, C/O, and N/O in the convective envelope during the thermally pulsing asymptotic giant branch (TP-AGB) phase of evolution in a 1.5 M_☉ initial mass star of solar initial composition. We use a parametric model that treats extra mixing by introducing mass flow between the convective envelope and the underlying radiative zone. The parameters of this model are the mass circulation rate () and the maximum temperature (T_P) experienced by the circulating material. The effects of nuclear reactions in the flowing matter were calculated using a set of static structures of the radiative zone selected from particular times in a complete stellar evolution calculation. The compositions of the flowing material were obtained, and the resulting changes in the envelope determined. No major shifts in the star's energy budget occur from the imposed CBP if log T_P < 7.73. Using structures from several times on the TP-AGB, it was found that the results for all species except ²⁶Al were essentially independent of the time chosen if log T_P > 7.6. Abundant ²⁶Al was produced by CBP for log T_P > 7.65. While ²⁶Al/²⁷Al depends on T_P, the other isotopic ratios depend dominantly on the circulation rate. The relationship is shown between models of CBP as parameterized by a diffusion formalism within the stellar evolution model and those using the mass-flow formalism employed here. They are shown to be effectively equivalent. In general, the CBP treatment readily permits calculation of envelope compositions as affected by different degrees of extra mixing, based on stellar structures computed by normal stellar evolution models. Using these results, the isotopic ratios under conditions of C/O < 1 and C/O > 1 are compared with the data on circumstellar dust grains. It is found that the ¹⁸O/¹⁶O, ¹⁷O/¹⁶O, and ²⁶Al/²⁷Al observed for oxide grains formed at C/O < 1 are reasonably well understood. However, the ¹⁵N/¹⁴N, ¹²C/¹³C, and ²⁶Al/²⁷Al in carbide grains (C/O > 1) require that many of their stellar sources must have had ¹⁴N/¹⁵N at least a factor of 4 lower than the solar value. This allows a self-consistent description of all these isotopes in most SiC grains. The rare grains with ¹²C/¹³C < 10 cannot be produced by any red giant or AGB source, nor are they reconcilable with novae sources.