** Up:** Age-luminosity relations for low-mass stars

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:

The Lagrangian spatial grid (in relative
mass
)
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
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.

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
-elements (see [Salaris & Weiss 1998] for details). The tables, which
have a common *T*--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
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.

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
.

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.

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.

We use the [Caughlan & Fowler 1988] reaction rates
and the Salpeter formula for weak screening. The nuclear network
follows the evolution of ,
,
,
,
,
,
,
,
.
All other species in the
*pp*-chain and CNO-cycles are assumed to be in equilibrium (this is
justified because it assumes only that -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.

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).

** Up:** Age-luminosity relations for low-mass stars

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