Solar models with helium and heavy-element diffusion

Helium and heavy-element diffusion are both included in precise calculations of solar models. In addition, improvements in the input data for solar interior models are described for nuclear reaction rates, the solar luminosity, the solar age, heavy-element abundances, radiative opacities, helium and metal diffusion rates, and neutrino interaction cross sections. The effects on the neutrino fluxes of each change in the input physics are evaluated separately by constructing a series of solar models with one additional improvement added at each stage. The effective $1\sigma$ uncertainties in the individual input quantities are estimated and used to evaluate the uncertainties in the calculated neutrino fluxes and the calculated event rates for solar neutrino experiments. The calculated neutrino event rates, including all of the improvements, are ${9.3}_{-1.4\mathrm{}}^{+1.2\mathrm{}}$ SNU for the ${}_{}{}^{37}{}_{}{}^{}\mathrm{Cl}$ experiment and ${137}_{-7\mathrm{}}^{+8\mathrm{}}$ SNU for the ${}_{}{}^{71}{}_{}{}^{}\mathrm{Ga}$ experiments. The calculated flux of ${}_{}{}^7{}_{}{}^{}\mathrm{Be}$ neutrinos is 5.1 (${1.00}_{-0.07\mathrm{}}^{+0.06\mathrm{}}$)×${10}^9$ ${\mathrm{cm}}^{-2\mathrm{}}$ ${\mathrm{s}}^{-1\mathrm{}}$ and the flux of ${}_{}{}^8{}_{}{}^{}\mathrm{B}$ neutrinos is 6.6(${1.00}_{-0.17\mathrm{}}^{+0.14\mathrm{}}$)×${10}^6$ ${\mathrm{cm}}^{-2\mathrm{}}$ ${\mathrm{s}}^{-1\mathrm{}}$. The primordial helium abundance found for this model is $Y=0.278\mathrm{}$. The present-day surface abundance of the model is $Y_s=0.247\mathrm{}$, in agreement with the helioseismological measurement of $Y_s=0.242\mathrm{}\pm 0.003$ determined by Hernandez and Christensen-Dalsgaard (1994). The computed depth of the convective zone is $R=0.712\mathrm{}{R}_{\odot }$, in agreement with the observed value determined from $p$-mode oscillation data of $R=0.713\mathrm{}\pm 0.003R_{\odot }$ found by Christensen-Dalsgaard et al. (1991). Although the present results increase the predicted event rate in the four operating solar neutrino experiments by almost $1\sigma$ (theoretical uncertainty), they only slightly increase the difficulty of explaining the existing experiments with standard physics (i.e., by assuming that nothing happens to the neutrinos after they are created in the center of the sun). For an extreme model in which all diffusion (helium and heavy-element diffusion) is neglected, the event rates are ${7.0}_{-1.0\mathrm{}}^{+0.9\mathrm{}}$ SNU for the ${}_{}{}^{37}{}_{}{}^{}\mathrm{Cl}$ experiment and ${126}_{-6\mathrm{}}^{+6\mathrm{}}$ SNU for the ${}_{}{}^{71}{}_{}{}^{}\mathrm{Ga}$ experiments, while the ${}_{}{}^7{}_{}{}^{}\mathrm{Be}$ and ${}_{}{}^8{}_{}{}^{}\mathrm{B}$ neutrino fluxes are, respectively, 4.5(${1.00}_{-0.07\mathrm{}}^{+0.06\mathrm{}}$)×${10}^9$ ${\mathrm{cm}}^{-2\mathrm{}}$ ${\mathrm{s}}^{-1\mathrm{}}$ and 4.9(${1.00}_{-0.17\mathrm{}}^{+0.14\mathrm{}}$)×${10}^6$ ${\mathrm{cm}}^{-2\mathrm{}}$ ${\mathrm{s}}^{-1\mathrm{}}$. For the no-diffusion model, the computed value of the depth of the convective zone is $R=0.726\mathrm{}{R}_{\odot }$, which disagrees with the observed helioseismological value. The calculated surface abundance of helium, $Y_s=0.268\mathrm{}$, is also in disagreement with the $p$-mode measurement. The authors conclude that helioseismology provides strong evidence for element diffusion and therefore for the somewhat larger solar neutrino event rates calculated in this paper.