Time-dependent ionization in dynamic solar and stellar atmospheres. I. Methods

We propose a new numerical method to compute one-dimensional time-dependent wave propagation in stellar atmospheres that incorporates the time-dependent treatment of hydrogen ionization together with an evaluation of radiation losses under departures from local thermodynamic equilibrium (NLTE). The method permits us to calculate acoustic waves and longitudinal magnetohydrodynamic (MHD) tube waves. We have tested the method for the solar atmosphere by calculating the propagation of three types of waves, namely, a monochromatic acoustic wave, a stochastic acoustic wave, and a stochastic longitudinal tube wave. It was found that with a time-dependent treatment of the hydrogen ionization (as well as the Mg ionizations) the degree of ionization (H+/H) and the Mg II/Mg ratio become insensitive to the temperature fluctuations, even in the presence of weak and moderately strong shocks. Only when strong shocks appear do the transition rates become large enough to cause a high correlation between the degree of ionization and the high postshock temperatures. Our calculations show that a mean degree of ionization gets established that increases with height and is very little perturbed by the local temperature fluctuations of the wave. In stochastic calculations, strong shocks appeared periodically (roughly every 3 minutes), which in their postshock regions carried a zone of high or complete ionization. Tests with different numbers of frequency and height points, as well as of the rate of convergence of the Lambda-iteration, were performed.