So I've been attempting to find ionic abundances using visual emission spectra of planetary nebula, and I ran across an equation to do that in the paper Abundances of planetary nebulae NGC 40 and NGC 6153. I've tried looking up some of the definitions for the variables to get a sense of the information needed to be plugged in to find abundance, and I've found a large number of almost word for word reproductions of the same paragraph I found the equation in (1,2,3,4,5,6,7,8) but little success actually coming to a better understanding of it.
The ionic abundances have been determined using the following equation: $$\frac{N_{ion}}{N_p}=\frac{I_{ion}}{I_{H_β}}N_e\frac{λ_{ul}}{λ_{H_β}}\frac{α_{H_β}}{A_{ul}}\left(\frac{N_{u}}{N_{ion}}\right)^{-1}$$ where $I_{ion}/I_{H_β}$ is the measured intensity of the ionic line compared to Hβ, $N_p$ is the density of ionized hydrogen, $λ_{ul}$ is the wavelength of this line, $λ_{H_β}$ is the wavelength of Hβ, $α_{H_β}$ is the effective recombination coefficient for Hβ, $A_{ul}$ is the Einstein spontaneous transition rate for the line, and $N_u/N_{ion}$ is the ratio of the population of the level from which the line originates to the total population of the ion. This ratio has been determined using a five level atom.
I can understand some of it, but there are some pretty significant gaps too. Specifically:
- Is $N_{ion}$ the density of the ion in the nebula?
- What is $N_e$?
- How is the effective recombination coefficient for Hβ found?
- What is the Einstein spontaneous transition rate for the selected line?
I feel like these might be better served with their own individual questions, and I don't really expect them to be given answers here. I'll ask them separately if the need still ends up existing, but for now I'm concerned with understanding where this equation comes from as a whole. Any help towards that end would be much appreciated.
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