Is There a Dynamic Effect in the Screening of Nuclear Reactions in Stellar Plasmas?

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Abstract
We consider a fully ionized hydrogen plasma and investigate the two following questions. Does an ion's kinetic energy affect its electrostatic potential energy when it is in statistical equilibrium with a plasma? In other words, is there a difference between the potential energy of a test particle moving through a plasma and the potential energy of a particle in a statistical equilibrium? Are the plasma effects on the rate of nuclear reactions dependent on the kinetic energy of the reacting particles? Carraro and coworkers assumed that the potentials of a test particle and of a particle in a statistical equilibrium are identical and applied the potential of a test particle to evaluate the effect of the plasma on nuclear reactions. However, criticism raised recently by several authors, including Weneser and Gruzinov, claims that the two potentials are not identical, and hence the potential of a test particle, which depends on its kinetic energy, cannot be applied to evaluate the effect of the plasma on nuclear reactions in cores of stars. We examine the relevant statistical mechanics assumptions and in view of the inexistence of a complete rigorous proof, check them numerically. We find the following. The long-term average potential energy of a particle in a statistical equilibrium with the plasma is independent of the particle's kinetic energy, at least for particles with kinetic energies of up to 10kT (which is at present our numerical limit). The potential energy of a test particle differs from the long-term average potential energy of a particle in a statistical equilibrium. We define the "plasma effect" on the nuclear reaction rate and calculate it directly. We find that it depends on the relative kinetic energy of the interacting particles. Using the same numerical calculation, we find the dependence of a particle's potential energy in a statistical equilibrium on its kinetic energy and the potential of a test particle and the dependence of the plasma effects on the relative kinetic energy of the interacting particles. We conclude that a dynamic effect on the screening does exist. In view of our results, we discuss the possibility of measuring it in the laboratory and clarify the difference between reactions in a plasma in thermal equilibrium and a laboratory scattering experiment. The frequently made assumption that a test particle can be assumed in evaluating the kernel of plasma kinetic equations is found to be unjustified. We show that, under the conditions prevailing in the solar core, the mean field approximations are not sufficient to describe the screening, and fluctuations play a crucial role.