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I am trying to understand the topic of "Metallicity estimates", either HII regions or planetary nebula. For this latter, the electron temperature (and density) must be known. I am studying the formalism of Osterbrock.

I know, for instance, that if you use the lines [OIII]4363 and [OIII]5007 the electron temperature can be computed. In the same way, the ratio between [OII]3729 and [OII]3726 provides an estimation of electronic density.

Actually, I also know that, for the case of [OIII], its metastable excited states are relatively separated. For the case of [OII], such metastable excited states are very close. Such separation, in the metastable excited states, is crucial to compute the temperature and density.

My point is that I do not understand why this separation is important, from a physical point of view. Why the ratio between these lines can estimate such properties?

You can check examples of what I am talking about in "Electronic Density" and "Electronic Temperature" sections of this link: http://www.astronomie-amateur.fr/feuilles/Spectroscopie/NGC2392.html

I hope to make me understood.

Thank you very much for your help.

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Forbidden lines like these arise in thin gases, where collisional de-excitation is unlikely. Forbidden lines become "quenched" by collisional de-excitation when the densities reach levels that depend on the detailed quantum physics of each transition (i.e. different for every line/transition).

If you study the ratio of a pair of lines at densities well below their "quenching thresholds", the ratio will depend only on the relative occupation of their upper levels, which in turn depends on the temperature, but is insensitive to density.

If on the other hand you choose a pair of lines with very similar upper energy levels, the ratio will be insensitive to temperature. However, if one transition is around or even above its "quenching threshold", then the ratio becomes very sensitive to density.

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