Fundamental problems and basic tests of stellar evolution theory — The case of carbon stars

Carbon stars are thought to be in the asymptotic giant branch (AGB) phase of evolution, alternately burning hydrogen and helium in shells above an electron-degenerate carbon-oxygen (CO) core. The excess of carbon relative to oxygen at the surfaces of these stars is thought to be due to convective dredge-up which occurs following a thermal pulse. During a thermal pulse, carbon and neutron-rich isotopes are made in a convective helium-burning zone. In model stars of large CO core mass, the source of neutrons for producing the neutron-rich isotopes is the 22Ne(α, n)25Mg reaction and the isotopes are produced in the solar system s-process distribution. In models of small core mass, the 13C(α, n) 16O reaction is thought to be responsible for the release of neutrons, and the resultant distribution of neutron-rich isotopes is expected to vary considerably from one star to the next, with the distribution in isolated instances possibly resembling the solar system distribution of r-process isotopes. After the dredge-up phase following each pulse, the 13C is made by the reactions 12C(p,γ) 13N(β+ v) 13C in a zone of large 12C abundance and small 1H abundance that has been established by semiconvective mixing during the dredge-up phase. There is qualitative accord between the properties of carbon stars in the Magellanic Clouds and properties of model stars, but considerably more theoretical work is required before a quantitative match is achieved.The observed paucity of AGB stars more luminous than MBOL ∼ −6 is interpreted to mean that the AGB lifetime of a star more luminous than this is at least a factor of ten smaller than the AGB lifetime of stars less luminous than this, or, at most 105 yr. Since, with current estimates of the 22Ne(α, n)25Mg reaction rate R22, only AGB model stars more luminous than MBOL ∼ −6 can produce s-process isotopes in the solar system distribution, it is inferred that either (1) the current estimates of R22 are too small by one to two orders of magnitude, allowing less luminous AGB stars to contribute, (2) the solar system distribution is not equivalent to the average Galactic distribution, being rather the consequence of a unique injection into the protosolar nebula of matter from a massive intermediate-mass AGB star, or (3) the estimates of the temperatures in the convective shell that are given by extant models are too low by, sav, 10 or 15 percent.The absence of carbon stars more luminous than MBOL ∼ −6 is suggested to be due primarily to the fact that ∼ 106 yr of AGB evolution is necessary to produce surface C/O > 1, rather than to be due to the burning of dredged-up carbon into nitrogen at the base of the convective envelope during the interpulse quiescent hydrogen-burning phase. Thus, the positive correlation between the nitrogen and helium abundances in planetary nebulae is perhaps primarily a consequence of the second dredge-up episode rather than a consequence of processes occurring during the thermally pulsing phase.