Stellar oscillations and stellar convection in the presence of an Urca shell

The problem of damping of stellar oscillations in presence of a Ur ca shell is solved analytically in a plane symmetrical approximation. Low-amplitude oscillations are considered. Oscillatory pressure perturbations induce beta reactions of the electron capture and decay in the thin layer around the Urca shell, leading to damping of oscillations. Owing to the non-linear dependence of beta reaction rates on the pulsation amplitude in degenerate matter, even a low-amplitude oscillation damping follows a power law. It is shown that in the presence of the Urca shell the energy losses owing to neutrino emission and the entropy increase resulting from non-equilibrium beta reactions are much smaller than the rate of decrease of the energy of pulsations by the excitation of short-wavelength acoustic waves. The dissipation of the vibrational energy by the last process is the main source of heating of matter. Convective motion in the presence of an Urca shell is considered, and equations generalizing the mean free path model of the convection are derived.