A numerical tool for the calculation of non-equilibrium ionisation states in the solar corona and other astrophysical plasma environments

¹ NASA Goddard Space Flight Center, Solar Physics Lab., Code 671, 8800 Greenbelt Road, Greenbelt, MD 20771, USA e-mail: stephen.bradshaw@nasa.gov
² Department of Computational and Data Sciences, George Mason University, 4400 University Drive, MSN 6A2, Fairfax, VA 22030, USA

Received: 1 August 2008
Accepted: 19 May 2009

Abstract

Context. The effects of non-equilibrium processes on the ionisation state of strongly emitting elements in the solar corona can be extremely difficult to assess and yet they are critically important. For example, there is much interest in dynamic heating events localised in the solar corona because they are believed to be responsible for its high temperature and yet recent work has shown that the hottest (≥10⁷ K) emission predicted to be associated with these events can be observationally elusive due to the difficulty of creating the highly ionised states from which the expected emission arises. This leads to the possibility of observing instruments missing such heating events entirely.

Aims. The equations describing the evolution of the ionisaton state are a very stiff system of coupled, partial differential equations whose solution can be numerically challenging and time-consuming. Without access to specialised codes and significant computational resources it is extremely difficult to avoid the assumption of an equilibrium ionisation state even when it clearly cannot be justified. The aim of the current work is to develop a computational tool to allow straightforward calculation of the time-dependent ionisation state for a wide variety of physical circumstances.

Methods. A numerical model comprising the system of time-dependent ionisation equations for a particular element and tabulated values of plasma temperature as a function of time is developed. The tabulated values can be the solutions of an analytical model, the output from a numerical code or a set of observational measurements. An efficient numerical method to solve the ionisation equations is implemented.

Results. A suite of tests is designed and run to demonstrate that the code provides reliable and accurate solutions for a number of scenarios including equilibration of the ion population and rapid heating followed by thermal conductive cooling. It is found that the solver can evolve the ionisation state to recover exactly the equilibrium state found by an independent, steady-state solver for all temperatures, resolve the extremely small ionisation/recombination timescales associated with rapid temperature changes at high densities, and provide stable and accurate solutions for both dominant and minor ion population fractions. Rapid heating and cooling of low to moderate density plasma is characterised by significant non-equilibrium ionisation conditions. The effective ionisation temperatures are significantly lower than the electron temperature and the values found are in close agreement with the previous work of others. At the very highest densities included in the present study an assumption of equilibrium ionisation is found to be robust.

Conclusions. The computational tool presented here provides a straightforward and reliable way to calculate ionisation states for a wide variety of physical circumstances. The numerical code gives results that are accurate and consistent with previous studies, has relatively undemanding computational requirements and is freely available from the author.