# 8.1.5. Non-equilibrium ionisation (NEI) calculations¶

Our current implementation of time-dependent, non-equilibrium ionisation spectra is based upon the work of Kaastra & Jansen (1993), KJ hereafter. The method consists essentially upon calculating the transition matrix (containing ionisation and recombination rates) for a grid of temperatures, calculating the eigenvalues and eigenvectors of these matrices and also the inverse eigenvector matrix, and storing those on disk. The ionisation balance is calculated then by dividing the temperature history in several steps, with the steps defined in such a way that the temperature over each time interval is close to one temperature grid point. Then for each step the time evolution of the plasma is calculated using eigenvalue decomposition and using the pre-calculated coefficients for the relevant temperature. For more details see KJ.

Here we make a few changes with respect to KJ. First, KJ adopted a prescribed history of the Hydrogen density, and calculate the corresponding electron density at each time from the ion concentrations at that time. This was motivated by the argument that for a fixed plasma element the number of Hydrogen atoms remains constant over time, while the number of electrons depends upon the degree of ionisation and may be time-dependent, thus not known before the ionisation balance has beeen solved completely. But the situation is slightly more complicated. If the plasma is not (nearly) completely ionised, then for normal abundances there must be a considerable population of neutral Hydrogen or Helium. But in that case charge exchange reactions become important, and should be taken into account in the transition matrix. This however is impossible to incorporate in the KJ method, since in this case there are 3 free parameters ( , and ) instead of 1 ( ), and storage of matrices etc. becomes physically impossible. Knowing that the method of KJ becomes inaccurate for low ionisation rates we decided to leave the prescribed concept and return to the prescribed concept. This has the advantage that the equations for the ionisation balance of each chemical element can be calculated independently from the other elements; KJ need to calculate the concentrations of all elements for each time step in order to determine the conversion from Hydrogen to electron density.

Another rationale behind the above-mentioned modification is the following. For self-similar supernova remnants, a class of problems where non-equilibrium ionisation applies, the hydrodynamics is inconsistent with a time- or place varying electron/Hydrogen ratio: that would introduce another scale length and destroy the assumptions behind self-similarity. Thus, for SNR models it is necessary to assume that the plasma is (almost) completely ionised in order to assure self-consistency. We will neglect small changes in the electron/Hydrogen ratio of SNRs therefore.

The second modification concerns the temperature grid. Contrary to KJ, we use a keV grid instead of a K grid. Further, since we now include 30 chemical elements instead of 15, we had to reduce the step size of the grid from 0.05 in to 0.10, in order not to get disk storage problems. This change however is compensated by a better interpolation technique used here. KJ just rounded all temperatures to the nearest grid point. Here we interpolate the calculated concentrations between two temperature grid points linearly, which is one order of magnitude ( ) better. A rough estimate learns that the error in the method of KJ per decade is which gives for an error of order 0.05, while the current implementation has typical errors per decade of which gives a typical error of 0.01 per decade.