Standard solar models, with and without helium diffusion and the solar neutrino problem

We first show that, with the same input parameters, the standard solar models of Bahcall and Ulrich; of Sienkiewicz, Bahcall, and Paczy\ifmmode \acute{n}\else \'{n}\fi{}ski; of Turck-Chi\`eze, Cahen, Cass\'e, and Doom; and of the current Yale code all predict event rates for the chlorine experiment that are the same within \ifmmode\pm\else\textpm\fi{}0.1 SNU (solar neutrino units), i.e., approximately 1% of the total calculated rate. We then construct new standard solar models using the Yale stellar evolution computer code supplemented with a more accurate (exportable) nuclear energy generation routine, an improved equation of state, recent determinations of element abundances, and the new Livermore (OPAL) opacity calculations. We evaluate the individual effects of different improvements by calculating a series of precise models, changing only one aspect of the solar model at a time. We next add a new subroutine that calculates the diffusion of helium with respect to hydrogen with the aid of the Bahcall-Loeb formalism. Finally, we compare the neutrino fluxes computed from our best solar models constructed with and without helium diffusion. We find that helium diffusion increases the predicted event rates by about 0.8 SNU, or 11% of the total rate, in the chlorine experiment; by about 3.5 SNU, or 3%, in the gallium experiments; and by about 12% in the Kamiokande and SNO experiments. The best standard solar model including helium diffusion and the most accurate nuclear parameters, element abundances, radiative opacity, and equation of state predicts a value of 8.0\ifmmode\pm\else\textpm\fi{}3.0 SNU for the $^{37}\mathrm{Cl}$ experiment and ${132}_{\ensuremath{-}17}^{+21}$ SNU for the $^{71}\mathrm{Ga}$ experiment. The quoted errors represent the total theoretical range and include the effects on the model predictions of $3\ensuremath{\sigma}$ errors in measured input parameters. All 15 calculations since 1968 of the predicted rate in the chlorine experiment given in this series of papers are consistent with both the range estimated in the present work and the 1968 best-estimate value of 7.5\ifmmode\pm\else\textpm\fi{}2.3 SNU. Including the effects of helium diffusion and the other improvements in the description of the solar interior that are implemented in this paper, the inferred primordial solar helium abundance is $Y=0.273$. The calculated depth of the convective zone is $R=0.707{R}_{\ensuremath{\bigodot}}$, in agreement with the value of $0.713{R}_{\ensuremath{\bigodot}}$ inferred by Christensen-Dalsgaard, Gough, and Thompson from a recent analysis of the observed $p$-mode oscillation frequencies. Including helium diffusion increases the calculated present-day hydrogen surface abundance by about 4%, decreases the helium abundance by approximately 11%, and increases the calculated heavy-element abundance by about 4%. In the Appendix, we present detailed numerical tables of our best standard solar models computed both with and without including helium diffusion. In the context of the MSW (Mikheyev-Smirnov-Wolfenstein) or other weak-interaction solutions of the solar neutrino problem, the numerical models can be used to compute the influence of the matter in the sun on the observed neutrino fluxes.