Proc. Neutrino Astrophysics - MPP Theory Group

14
In the depth range where an abundant element is partly ionized Γ1 is greatly affected by the
energy of ionization. Thus the He abundance, in particular by its effect in the zone of partial
HeII ionization, has some influence on the eigenfrequencies of the Sun’s pressure (p) modes
of oscillation. The inversion of observed frequencies yields a mass fraction YS = 0.242 ±0.003
([22], other authors find similar results). It should be noted that the original He abundance,
Y0, is adjusted so that the luminosity of the present model, at age t⊙, equals L⊙. The result
of this procedure is Y0 = 0.27... 0.28, depending on other input to the model. The difference
to the seismically determined YS is appropriate in view of helium settling in the radiative
solar interior, see below.
b) Nuclear Reactions
Concerning the nuclear reactions I shall concentrate on the pp chains, which provide ≈ 99% of
the energy. The reaction rates, and therefore the branching between the three chains leading
to helium, depend on temperature (see below), and on the “astrophysical S-factor” S(E).
This factor depends weakly on the center-of-mass energy E and measures the cross section
after separation of 1/E times the penetration probability through the Coulomb barrier. For
the most important reactions Parker [21] reviews results of S-factors (at zero energy):
p(p,e + ν)d S11(0) = 3.89 · 10 −25 MeV·b ±1%
3 He( 3 He,2p) 4 He S33(0) = 5.0 MeV·b ±6%
3 He( 4 He,γ) 7 Be S34(0) = 533 eV·b ±4%
7 Be(p,γ) 8 B S17(0) = 22.2 eV·b ±14%
It is difficult to assess the errors. S11 is so small that it can only be calculated. The other
S-factors are measured in the laboratory but must be extrapolated to zero energy, and a
correction must be applied for electron screening at low energy. In particular the 14% error
of S17 has been criticized, e.g. [17], as it is derived from diverse experiments with partially
contradicting results in the range 17.9–27.7 eV·b. Perhaps better results will soon become
available as the measurements are extended to lower energy, cf. the contribution of M. Junker
to these proceedings.
c) Equation of State and Opacity
It is appropriate to discuss these two important ingredients to the solar model together, since
both depend on the knowledge of the number densities of the diverse particles, and especially
on the ionization equilibria and the electron density. To first order the Sun consists of a
perfect gas; but significant corrections, at the percent level, arise e.g. from the electrostatic
interaction of the particles (especially at the depth where abundant species are partially
ionized) and from partial electron degeneracy (in the core), cf. [23]. Modern standard models
are usually calculated with a tabulated equation of state and opacity. The most recent tables
from the Lawrence Livermore Laboratory [18] still exhibit unexplained discrepancies of up to
20% as compared to various other calculations; in the energy-generating region of the Sun
the uncertainty probably is much less, 2.5% according to [4].
Generally, the recent opacities [18] are somewhat increased in comparison to earlier results,
e.g. from Los Alamos, mainly because more elements have been included in the calculations.
The increase renders the radiative transport of energy less effective. The solar model responds
with a slightly raised central temperature (cf. discussion below), and a slightly lowered base of