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Astron. Astrophys. 349, L5-L8 (1999)

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2. Numerical method

The stellar models are based on the stellar evolution code described by Blöcker (1995b). However, the treatment of the chemical evolution was entirely replaced by a numerical scheme which solves the time dependence of the considered nuclear species - i.e., the changes due to thermonuclear reactions and due to mixing - in one single step. This enables us, in contrast to earlier investigations of very late thermal pulses, to reliably predict the chemical abundance profiles and the nuclear energy generation rates in situations where the time scales of nuclear burning and mixing are comparable. The abundance change for each isotope at each mesh point due to diffusive mixing and nuclear processing is given by

[EQUATION]

where [FORMULA] contains the abundances of all considered isoptopes at the [FORMULA] mesh point, [FORMULA] is the nuclear rate matrix, D is the diffusion coefficient describing the efficiency of convective mixing, r is the radius, m the mass coordinate and [FORMULA] the density. This leads to a set of non-linear equations with [FORMULA] unknowns, where M is the number of grid points and N is the number of isotopes. In the present calculations, M is of the order of 2000, and [FORMULA] as the main thermonuclear reactions for hydrogen burning through the pp chains and the CNO cycle as well as the main helium burning reactions are included. The solution is obtained fully implicit with a Newton-Raphson iteration scheme by making use of the band-diagonal structure of the problem. The scheme converges to sufficient precision within about 3 iterations. A coupled solution of one nuclear reaction at a time and time-dependent mixing, including also the structure equations, has already been applied by Eggleton (1972).

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© European Southern Observatory (ESO) 1999

Online publication: August 25, 1999
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