one of the doublet’s transitions changes the emergent intensity in the other line. In  K only the inner core was marginally affected. In  k the core was changed by 1.7% and the emission peaks were changed by less than 8.5%. In each doublet, K and H as well as k and h, both lines share similar formation properties, so in the following we focus on the  K and the  k lines only.

The emergent intensities as well as the formation heights slightly change owing to all these simplifications in our model atoms. These changes are relatively small. Since we are not concerned about detailed properties, they do not change the overall results and conclusions of the paper.

Charge conservation {#sec:linerized_eqs}
-------------------

A correct electron density is important for non-LTE modeling of chromospheric lines because the collisions with electrons affect the coupling of the source function to the local conditions of the gas. The model atmosphere does not explicitly contain the electron density, but MURaM implicitly assumes LTE electron densities through its equation-of-state. LTE is however a bad approximation of the electron density in the chromosphere and transition region, and the electron density should ideally be computed including non-equilibrium ionization of 