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

## TL;DR

This paper presents an analytical model of time-dependent hydrogen ionization in Type IIP supernova envelopes, highlighting the importance of non-equilibrium effects during the photospheric phase.

## Contribution

It introduces a simplified kinetic system and derives an analytical Lyapunov function to describe ionization freeze-out, emphasizing the need for time-dependent modeling.

## Key findings

- Analytical expression for ionization freeze-out effect.
- Recombination does not reach equilibrium during the photospheric phase.
- Time-dependent effects are crucial for accurate supernova envelope modeling.

## Abstract

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.