For column densities greater than cm, corresponding to a Thomson depth of unity, neglect of `Comptonization', spectral modification arising from Compton scattering, makes model results incorrect.

In addition, the upper bound on volume density declines as the total column density increases. The reason is that the number of transitions in detailed balance increases rapidly as the product of volume density and column density rises. Once many transitions are in detailed balance, the thermal balance becomes determined by transitions involving rarer species and excited states which are not included in the calculation. As a rough rule of thumb, the product of volume density and column density should not exceed cm.

Tim Kallman 2016-04-29