2 The stellar evolution code

We are using the Garching stellar evolution code, which is a derivative of the original Kippenhahn-code ([Kippenhahn et al. 1967]) developed and improved over the years. While new properties of the code have always been documented in the corresponding publications, we summarize them here again for completeness:

Numerical aspects:

The Lagrangian spatial grid (in relative mass $M_{\rm r}/M$) adopts itself to structure changes. Its resolution is controlled by an algorithm ensuring that the partial differential equations are solved with a given accuracy ([Wagenhuber & Weiss 1994]). Since all composition changes are calculated between two models of two successive evolutionary ages, the evolution of temperature and density during this time-step has to be given at each grid-point. We use a predictor-corrector scheme for this ([Schlattl 1996]; [Schlattl et al. 1997]). The assumption of constant T and $\rho$ can be used as an alternative, but requires time-steps smaller by about a factor of 2-5. Nuclear burning and particle transport processes (convection and diffusion) are calculated either simultaneously in a single iterative scheme with a generalized Henyey-solver ([Schlattl 1999]) or separately in a burning-mixing-burning-...sequence. In the latter case, the network solves the linearized particle abundance equations in an implicit way. In both cases a number of time-steps, which are smaller than that between the two models and which are adopting to composition changes, are followed until the whole evolutionary time-step is covered.

Opacities:

We use as the sources for the Rosseland mean opacities the latest OPAL tables ([Iglesias & Rogers 1996]) and the molecular opacities by [Alexander & Ferguson 1994]. Both groups provided us (Rogers 1995, private communication, and Alexander, 1995, private communication) with tables for exactly the same compositions including the enhancement of $\alpha$-elements (see [Salaris & Weiss 1998] for details). The tables, which have a common T-$\rho$-grid, can smoothly be merged and together with electron conduction opacities ([Itoh et al. 1983]) result in consistent tables for all stellar interior conditions encountered. The interpolation within a single table is done by bi-rational two-dimensional splines ([Spaeth 1973]), which contain a free parameter allowing the transition from standard cubic to near-linear interpolation. This avoids unwanted spline oscillations but guarantees that the interpolant is always differentiable twice. We then interpolate in a $3\times 3$ cube in X-Z-space to the grid-point's composition by two independent polynomial interpolations of degree 2. The cube of table compositions is chosen such that the central point is closest to the actual composition under consideration.

Equation of state:

We use, where possible, the OPAL equation of state ([Rogers et al. 1996]) with the interpolation procedures provided by the same authors along with the EOS tables. For low densities and temperatures, we use our traditional Saha-type EOS for a partially ionized plasma or approximations for a degenerate electron gas (see [Kippenhahn et al. 1967]; [Weiss 1987]; [Wagenhuber 1996] for details). However, we did not calculate models not being covered by the OPAL EOS for the larger part. This limits the mass range to $\ge 0.6\,M_\odot$.

Convection:

Convection is treated in the standard mixing-length approach. No overshooting or semi-convection is considered. The mixing-length parameter is calibrated with a solar model calculated without diffusion; the resulting value for the physical input employed here is 1.59 pressure scale heights. Note that it would be slightly different for solar models including diffusion. Convective mixing is either assumed to be instantaneous or treated as a fast diffusive process in the case that all processes affecting the chemical composition are treated simultaneously.

Neutrino emission:

Energy losses due to plasma processes are included according to [Haft et al. 1994] for plasma-neutrinos and [Munakata et al. 1985] for photo- and pair-neutrinos.

Nuclear reactions:

We use the [Caughlan & Fowler 1988] reaction rates and the Salpeter formula for weak screening. The nuclear network follows the evolution of $^1{\rm H}$, $^3{\rm He}$, $^4{\rm He}$, $^{12}{\rm C}$, $^{13}{\rm C}$, $^{14}{\rm N}$, $^{15}{\rm N}$, $^{16}{\rm O}$, $^{17}{\rm O}$. All other species in the pp-chain and CNO-cycles are assumed to be in equilibrium (this is justified because it assumes only that $\beta$-decays are faster than p-captures). The network can also treat later burning phases, but this is of no concern here because calculations were stopped at the onset of the core helium flash.

Diffusion:

We consider the diffusion of hydrogen and helium. While our code also allows for metal diffusion, we ignored this here for reasons of CPU economy. Experience from solar models shows that the effect of metal diffusion on the interior evolution is only a fraction of that of H/He diffusion. This is confirmed by test calculations in which metal diffusion was included (see next section). The various coefficients of the particle diffusion equations are calculated according to [Thoul et al. 1994] with the routine provided kindly by A. Thoul (1997, private communication). If diffusion is considered, it turned out to be both more accurate and numerically stable to treat diffusion and burning in a single numerical algorithm (see above).