Screening corrections to the electron capture rates in dense stars by the relativistically degenerate electron liquid

We calculate the screening corrections to the electron capture rates in dense stars by the relativistically degenerate electron liquid. In order to calculate the screening corrections we adopt the linear response theory, which is widely used in the field of solid-state physics and liquid-metal physics. In particular, we use the longitudinal dielectric function for the relativistically degenerate electron liquid derived by Jancovici. We calculate the screening potential at the position of the nucleus. By using this screening potential one can calculate the screening corrections to the electron capture rates. We present accurate analytic fitting formulae which summarize our numerical results. These fitting formulae will facilitate the application of the present results. The screening corrections to the electron capture rates are typically a few percent.