Analytical Model of Time-Dependent Ionization in the Envelopes of Type II Supernovae at the Photospheric Phase

We investigate a simplified kinetic system of the hydrogen atom (two levels plus continuum) under conditions of a type IIP supernova at the plateau phase that realistically describes the basic properties of the complete system. We have found the Lyapunov function for the reduced system using which we have analytically obtained the ionization freeze-out effect on long time scales. Since the system completely recombines in the equilibrium approximation on long time scales, which does not occur in reality, this result confirms the necessity of allowance for the time-dependent effect in the kinetics during the photospheric phase in a supernova explosion.