  If NGC 6397 contains a large fraction of "second generation" stars (>70%
according to recent analysis), the helium abundance of its stars might also be
affected, show some star-to-star variation, and be larger than the standard Big
Bang abundance Y~0.24. Can we derive constraints on this issue from the
analysis of the main sequence width and from its luminosity function? We build
up new models for the turnoff masses and the main sequence down to the hydrogen
burning minimum mass, adopting two versions of an updated equation of state
(EOS) including the OPAL EOS. Models consider different initial helium and CNO
abundances to cover the range of possible variations between the first and
second generation stars. We compare the models with the observational main
sequence. We also make simulations of the theoretical luminosity functions, for
different choices of the mass function and of the mixture of first and second
generation stars, and compare them with the observed luminosity function, by
means of the Kolmogorov Smirnov --KS-- test. The study of the width of the main
sequence at different interval of magnitude is consistent with the hypothesis
that both generations are present in the cluster. If the CNO increase suggested
by spectroscopic observation is taken into account the small helium spread of
the main sequence in NGC 6397 implies a substantial helium uniformity (DY~0.02)
between first and second generation stars. The possible spread in helium
doubles if an even larger increase of CNO is considered. The luminosity
function is in any case well consistent with the observed data.

Solar and Stellar Astrophysics