Numerical integral of resistance coefficients in diffusion

TL;DR

This paper presents high-accuracy calculations of resistance coefficients in stellar plasma using advanced numerical methods, addressing singularities and providing data for different coupling regimes, with implications for solar models.

Contribution

The paper introduces a precise numerical scheme for calculating resistance coefficients in Coulomb potentials, improving accuracy and addressing singularities in the integral evaluation.

Findings

Resistance coefficients agree with previous results for repulsive potentials.

Significant differences found for attractive potentials in intermediate and strong coupling.

Resistance coefficients slightly weaken the solar model effects compared to previous models.

Abstract

The resistance coefficients in screen Coulomb potential of stellar plasma are evaluated in high accuracy. I have analyzed the possible singularities in the integral of scattering angle. There are possible singularities in the case of attractive potential. This may result in problem for numerical integral. In order to avoid the problem, I have used a proper scheme, e.g., splitting into many subintervals and the width of each subinterval is determined by the variation of the integrand, to calculate the scattering angle. The collision integrals are calculated by using Romberg's method therefore the accuracy is high (i.e., $ \sim 10^{-12}$). The results of collision integrals and their derivatives in $-12 \leq \psi \leq 5$ are listed. By using Hermite polynomial interpolation from those data, the collision integrals can be obtained with an accuracy of $10^{-10}$. For very weak coupled plasma ($\psi \geq 4.5$), analytical fittings for collision integrals are available with an accuracy of $10^{-11}$. I have compared the final results of resistance coefficients with other works and found that, for repulsive potential, the results are basically same to others, for attractive potential, the results in intermediate and strong coupled case show significant differences. The resulting resistance coefficients are tested in the solar model. Comparing with the widely used Cox et al.(1989) and Thoul et al. (1994) models, the resistance coefficients in screen Coulomb potential leads to a little weaker effect in solar model, which is contrary to the expectation of attempts to solve the solar abundance problem.