Non-equilibrium calcium ionisation in the solar atmosphere

Context. The chromosphere of the Sun is a temporally and spatially very varying medium for which the assumption of ionisation equilibrium is questionable. Aims. Our aim is to determine the dominant processes and timescales for the ionisation equilibrium of calcium under solar chromospheric conditions. Methods. The study is based on numerical simulations with the RADYN code, which combines hydrodynamics with a detailed solution of the radiative transfer equation. The calculations include a detailed non-equilibrium treatment of hydrogen, calcium, and helium. Next to an hour long simulation sequence, additional simulations are produced, for which the stratification is slightly perturbed so that a ionisation relaxation timescale can be determined. The simulations are characterised by upwards propagating shock waves, which cause strong temperature fluctuations and variations of the (non-equilibrium) ionisation degree of calcium. Results. The passage of a hot shock front leads to a strong net ionisation of Ca II, rapidly followed by net recombination. The relaxation timescale of the calcium ionisation state is found to be of the order of a few seconds at the top of the photosphere and 10 to 30 s in the upper chromosphere. At heights around 1 Mm, we find typical values around 60 s and in extreme cases up to ∼150 s. Generally, the timescales are significantly reduced in the wakes of ubiquitous hot shock fronts. The timescales can be reliably determined from a simple analysis of the eigenvalues of the transition rate matrix. The timescales are dominated by the radiative recombination from Ca III into the metastable Ca II energy levels of the 4d 2 D term. These transitions depend strongly on the density of free electrons and therefore on the (non-equilibrium) ionisation degree of hydrogen, which is the main electron donor. Conclusions. The ionisation/recombination timescales derived here are too long for the assumption of an instantaneous ionisation equilibrium to be valid and, on the other hand, are not long enough to warrant an assumption of a constant ionisation fraction. Fortunately, the ionisation degree of Ca ii remains small in the height range, where the cores of the H, K, and the infrared triplet lines are formed. We conclude that the difference due to a detailed treatment of Ca ionisation has only negligible impact on the modelling of spectral lines of Ca ii and the plasma properties under the conditions in the quiet solar chromosphere.