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The analytic solution is obtained describing kinetic equilibrium of the$\beta$-processes in the nucleonic plasma with relativistic pairs. The nucleons$(n,p)$ are supposed to be non-relativistic and non-degenerate, while theelectrons and positrons are ultra-relativistic due to high temperature$(T>6\cdot 10^9$K), or high density $(\rho>\mu 10^6$g/cm$^3$), or both, where$\mu$ is a number of nucleons per one electron. The consideration is simplifiedbecause of the analytic connection of the density with the electron chemicalpotential in the ultra-relativistic plasma, and Gauss representation of Fermifunctions. Electron chemical potential and number of nucleons per one initialelectron are calculated as functions of $\rho$ and $T$.

Analytic solution for kinetic equilibrium of beta-processes in nucleonic plasma with relativistic pairs