The electrostatic screening of nuclear reactions in the Sun

We apply the method of Shaviv & Shaviv to evaluate from first principles the plasma screening correction to the rate of nuclear reactions under conditions similar to those prevailing in the present-day Sun. We calculate the screening factor for the p-p chain and the CN cycle nuclear reactions. We find the following : 1. The mean field approximation a la Debye-Huckel is not strictly valid under the conditions prevailing in the core of the Sun. A kinetic approach should be implemented. 2. The mean energy exchange between any two scattering particles and the plasma is positive for low relative kinetic energies and negative for high relative kinetic energies. The turnover in a pure hydrogen plasma occurs at about Ekin-rel approximate to 2kT < E-Gamow 6kT for the p-p reaction, for example, and changes with the particular reaction. The net energy exchange, namely, the sum over all pairs of scattering particles in the system, vanishes in equilibrium. 3. The turnover from energy gain to energy loss of the scattering particles is generally below the Gamow peak. The details depend on the masses of the interacting particles relative to the mass of the particles in the ambient plasma. 4. The fluctuations and nonspherical effects are crucial in affecting the screening. We investigate the source of the fluctuations and their power spectrum. In view of the particular behavior of the fluctuations we derive the net effect of the plasma on the reaction rate for each reaction investigated here. 5. The derived screening corrections, which are a function of the relative kinetic energy and depend on the environment, are averaged over the distribution of relative velocities and cross section to obtain the total screening correction. The preliminary results are obtained in two easy-to-handle limits. The p-p reaction is found to be enhanced relative to reaction in vacuum, while higher Z reactions, like the Be-7 + p reaction, are suppressed relative to the classical Salpeter theory. 6. In the Appendix we show the connection between the screening and the coefficients in the Fokker-Planck equation. We use the same computer program to calculate the coefficient in the Fokker-Planck equation and find a behavior that supports the present findings of the general values and behavior of the screening.